Ildefonso Jesús

Prof Danielle Diaz

Prof Hilhorst

Haim Brezis

Jérôme Côville

Luca Rossi

Grégoire Nadin

Phuoc Juliette Bou- Hours

 ; the second one is to investigate the deeper mathematical aspects of this model and deal with the new problems where nonlocal and nonlinear diusion are considered.

. We further study the concentration of the species in the cylindrical domain when the exterior domain is changed to be extremely unfavorable.

In the Chapter 2, we focus on conditioning the a sharp criterion for the existence, nonexistence and uniqueness of positive solution of fully semilinear elliptic equation. When the divergence of the drift term is zero, the existence of positive solution can be characterized by the amplitude of the drift term under some fair assumptions on the growth rate. The large time behavior of associated parabolic equation is considered, where we have to deal with the case of possibly unbounded coecients.

The Chapter 3 extends the existence, nonexistence and uniqueness in the second chapter for a quasilinear equation involving p-laplacian operator. The main diculty is that it seems hard to apply the strong maximum principle and thus we make use a variational approach to attain an important comparison principle.

In Chapter 4, we investigate three notion of generalized principal eigenvalues for nonlocal operators in bounded and unbounded domains (eventually R N ). If the kernel is compactly supported, we can also prove the equivalence of these eigenvalues in unbounded domain. We consider the limits of the eigenvalues of the rescaling operator with respect to the diusion. The results are very dierent depending on the rate of rescaling.

In Chapter 5, by the help of the results in Chapter 4, we consider the nonlocal evolution equation and prove that the solution of evolution equation converges to the unique stationary solution, whose existence is directly conditioned by the sign of the generalized principal eigenvalue. The convergences holds in L ∞ (R N ) and L p (R N ), p > 0. In the second part of this chapter, we further investigate the singular limits of the unique positive solution of the rescaling equations. We show that the unique solution of nonlocal equation either approximates the unique solution of local KPP type equation or approximates a solution of reaction-equation, which may not be unique. ∆ + a(x) > 0, then there ε 0 ∈ (0, ∞) so that for all ε ≤ ε 0 there exists no positive solution to (19).

. As a consequence, even in bounded domain, simple sharp survival criteria are already quite involved to obtain. Another diculty inherent to the study of nonlocal equations (18) in unbounded domain concerns the lack of "reasonable" a priori estimates for the solution thus making standard approximations dicult to use in most cases. Chapitre 1 Persistence versus extinction under a climate change in mixed environments 1.1 Introduction and main results

, one sees that when δ goes to 0, α is arbitrarily close to α * . Since ϕ is bounded, obviously for all 0 < α < α * we can choose C(α) (possibly changed if necessary) such that

This implies that U (x 1 , y) decays exponentially as |x 1 | → ∞, uniformly in y. On the other hand, since U is a solution of (1.6), we can use L p estimate for the Neumann problem with p > N and Harnack inequality up to the boundary (see ) to derive

These inequalities end the proof.

Introduction générale

The thesis is devoted to the studies of the local and nonlocal reaction-diusion equations, which arise from various models of ecology, medicine, neutron, and population dynamics involving KPP-type nonlinearity.

Reaction-diusion equations with KPP nonlinearity have been the subject of intensive research after the pioneering work of Kolmogorov, Petrovsky, Piskounov, (KPP), 1937, [START_REF] Kolmogorov | Etude de l'equation de la diusion avec croissance de la quantite de matiere et son application a un probleme biologique[END_REF], where the authors wanted to study long time dynamics of the biological populations modeled by

∂ t u -u xx = 1u -u 2 = f (u) in R.
They showed that u(t, x) converges to traveling wave solutions u(t, x) = v(x -ct) of speed c (unknown) as t → ∞ satisfying v + cv + f (u) = 0; v(-∞) = 0; v(+∞) = 1

if and only if c ≥ c * = 2 (or 2 f (0)). This convergence is uniform in the domain {x ≤ ct}. Note that traveling wave solution is not unique since v(x + a) is another solution for all a ∈ R.

In 2005, Berestycki, Hamel, Nadirashvili [START_REF] Berestycki | The speed of propagation for KPP type problems. I. Periodic framework[END_REF] showed that c ≥ 2 f (0) is sharp, conversely v = 1 is the unique nontrivial solution. Later, Berestycki, Hamel, Rossi [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF] proved that the equation

a ij (x)u ij + b i (x)u i + f (x, u) = 0, in R N
admits at most a positive bounded solution, which will satisfy also inf

R N u > 0 if lim inf |x|→∞ (4α(x)f s (x, 0) -|b(x)| 2 ) > 0, (1)
where 0 < α * ≤ α(x) = inf ξ∈R N |ξ|=1 a ij (x)ξ i ξ j . In these frameworks, the common hypothesis on f is

lim inf |x|→∞ f s (x, 0) > 0.
In this thesis, we mainly focus on the opposite assumption, where f s (x, 0) may be negative, sign-changing and may decay fast up to Hardy potential.

The thesis is a collection of results from several independent topics related to the local and nonlocal models and new assumptions. Therefore, it will be presented by a chronological scheme, where each chapter is treated as a separate part of the thesis. The thesis consists of ve chapters and is organized as follows :

the two rst chapters study various aspects of the PDE models in cylindrical domain and the whole space. We further extend the rst results for p-Laplacian operator in chapter 3. Finally, chapter 4 and 5, also the most abstract parts of the thesis, involve the studies of nonlocal equations, where we respectively present some developments of generalized principal eigenvalues for nonlocal linear operators and positive solutions for nonlinear equations.

I. Persistence versus extinction under a climate change in mixed environments.

Our rst interest is to understand the dynamics of the population under the eect of a climate change, which is modeled by

u t -∆u = f (t, x 1 -ct, y, u) t > 0, x ∈ Ω, (2)
where Ω is of cylindrical or partially periodic domain, f is of KPP-type and the scalar c > 0 is a given forced speed. This type of equation has rst been introduced from the pioneering works of Berestycki, Diekmann, Nagelkerke and Zegeling [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], also of Potapov and Lewis [START_REF] Potapov | Climate and competition : the eect of moving range boundaries on habitat invasibility[END_REF], where the authors want to characterize the persistence versus extinction of the species who are sensitive to the climatic condition. More precisely, in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], the authors already studied the time-independent version of Eq. ( 2) in one dimensional space

u t -u xx = f (x -ct, u) x ∈ R. (3)
A typical f considered in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] is

f (x, s) = -sm x < 0 and x > L sm 1 - s K 0 ≤ x ≤ L, (4)
for some positive constants m, m , L, K and more general assumption is given by

f s (x, 0) ≤ -m, ∀|x| ≥ R. (5)
Higher dimensional versions was later studied by Berestycki and Rossi [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF], where [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] is also imposed. This assumption involved a completely unfavorable environment at innity, i.e. the favorable zone has compact support. Note that, f s (x, 0) is understood as the initial per capita rate of growth. In Chapter 1, we extend the model of (3) in two frameworks, for which condition [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] is no longer true. The rst is for an innite cylindrical domain with Neumann boundary condition :

u t -∆u = f (x 1 -ct, y, u) t > 0, x ∈ Ω ∂ ν u(t, x 1 , y) = 0 t > 0, x ∈ ∂Ω, (6)
where Ω = R × ω, ω is an open bounded and smooth domain in R N -1 , ν denotes the exterior unit normal vector eld to Ω. In this framework, the environments are assumed to be independent of time. Moreover, we are especially interested in considering the environments of mixed type, which are only assumed to be globally unfavorable at innity. To understand the model, one can think of the environment containing both favorable and unfavorable regions that extend all the way to innity, namely f s (x 1 , y, 0) > 0 and f s (x 1 , y, 0) < 0 respectively as x 1 → ±∞, depending on the location of y. The competitive and mutual inuences between these regions play a major role in characterizing the persistence and extinction of the species in the whole domain. For example of such environments, we refer to the species in alpine ecosystem, their habitats are very complex, heterogeneous or fragmentary and the climate change may give dierent eects on each patch of their environment. Our paper presents a new look on such environments.

In the second framework, we investigate another type of mixed environment with periodic dependence on y and t. More precisely, the equation is now of following type

u t -∆u = f (t, x 1 -ct, y, u) t > 0, x = (x 1 , y) ∈ R N ,
where the nonlinearity reaction is assumed to be periodic in y and t.

We shall use a global condition in terms of spectral property to describe that the environment is globally unfavorable at innity. More precisely, let L 1 and L 2 be as follows

L 1 u = a ij (x)∂ ij u(x) + b i (x)u i (x) + c(x)u, x ∈ Ω L 2 φ = ∂ t φ -a ij (t, y)∂ ij φ(t, y) -b i (t, y)φ i (t, y) -c(t, y)φ(t, y), (t, y) ∈ R × R N -1 ,
with bounded coecients, we dene the following eigenvalues

λ N (-L 1 , O) := sup{λ ∈ R : ∃φ ∈ W 2,N loc (O), φ > 0, (L 1 + λ)φ ≤ 0 a.e in O, ∂ ν φ ≥ 0 on ∂O}. λ1 (L 2 , R × R N -1 ) = sup{λ ∈ R : ∃ φ > 0, φ ∈ C 1,2 t,y (R × R N -1 ), φ is T-periodic in t and periodic in y such that (L 2 -λ)φ ≥ 0 in R × R N -1 }.
Note that if O is bounded, then λ N (-L, O) coincides with λ N , which is the classical Neumann principal eigenvalue. Assumption [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] is respectively replaced in the rst and second framework by : 1) There exists a measurable bounded function µ : ω → R such that µ(y) = lim sup

|x 1 |→∞
f s (x 1 , y, 0) and λ N (-∆ y -µ(y), ω) > 0.

2) There exists a function γ(t, y) ∈ L ∞ (R × R N -1 ), which is periodic in y and T-periodic in t such that γ(t, y) = lim sup

|x 1 |→∞
f s (t, x 1 , y, 0) and λ1 (∂ t -∆ y -γ(t, y), R × R N -1 ) > 0.

We examine the criteria for persistence and extinction phenomena under these new conditions. The similar conclusions are also derived for problem [START_REF] Berestycki | Existence and bifurcation of solutions for an elliptic degenerate problem[END_REF] under Dirichlet boundary condition. Therefore, we are motivated to investigate the concentration of the species in the more favorable region. The problem is described as follows :

We consider equation [START_REF] Berestycki | Existence and bifurcation of solutions for an elliptic degenerate problem[END_REF] in the whole space R N and study the limit of the sequence of traveling fronts with the reaction terms F n (x, s) such that their growth rates are negative outside the cylindrical domain Ω and tend to -∞ as n → ∞. These solutions solve the equations

∆U n + c∂ 1 U n + F n (x, U n ) = 0, x ∈ R N ,
where F n (x, s) = f (x, s) for x ∈ Ω and ∂F n ∂s (x, 0) → -∞ as n → ∞ locally uniformly in R N \ Ω. Set Lφ = ∆φ + c∂ 1 φ + f s (x, 0)φ, x ∈ Ω;

L n φ = ∆φ + c∂ 1 φ + ∂ s F n (x, 0)φ, x ∈ R N . λ D = λ D (-L, Ω) λ n = λ 1 (-L n , R N ).
If λ D < 0, we prove the following limits :

lim n→∞ λ n = λ D . lim n→∞ U n -U ∞ L ∞ (R N ) = 0,
where U ∞ ∈ W 2,N (R N ) vanishes in Ω c and coincides with the unique positive solution of the following equation

∆U + c∂ 1 U + f (x, U ) = 0 x ∈ Ω U (x) = 0 x ∈ ∂Ω.
In the last result of Chapter 1, we prove the monotonicity and the symmetry breaking of the unique traveling front of Eq. [START_REF] Berestycki | Existence and bifurcation of solutions for an elliptic degenerate problem[END_REF].

II. Liouville-type result for semilinear elliptic equations with possibly vanishing and signchanging potentials

In Chapter 2, we study conditions of Liouville-type result in the class of possibly unbounded solutions of the following semilinear elliptic equations

a ij (x)∂ ij u(x) + Kq(x) • ∇u(x) + f (x, u) = 0 in R N , (7)
where f is of KPP-monostable type. The goal is to allow the medium f s (x, 0) to be sign-changing and and it may behave like a slow decay or Hardy potential near innity, namely f s (x, 0) ≤ -C|x| -α in R N \ B R 0 for some C, R 0 > 0 and α ∈ [0, 2]. This considerably extends [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] and partially concerns the case that (1) has not covered. Condition [START_REF] Bates | Traveling waves in a convolution model for phase transitions[END_REF] was used in [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF] to prove the existence of positive bounded solutions of [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF].

The solutions must have positive inma if exist. Coarsely, this condition implies

lim inf |x|→∞ f s (x, 0) > 0, (8)
which is opposite to [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF]. In this chapter, we have the following targets :

• First, we extend the results of [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] to a more general framework and study the new cases in which (1), [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] are not fullled. More precisely, we allow f s (x, 0) to be sign-changing all the way as |x| → ∞ and it may behave like a slow decay or Hardy potential near innity, namely f s (x, 0) ≤ -C|x| -α in R N \ B R 0 for some C, R 0 > 0 and α ∈ [0, 2]. From the biological point of view, these new assumptions refer to the environments that can be unfavorable (α = 0) or unfavorably neutral (α ∈ (0, 2)) or nearly neutral (α = 2) mixing with favorable patches extending to innity. Our results conrm that the unique persistence and extinction of the species can be conditioned by the sign of the generalized principal eigenvalue of the linearized operator of [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF]. This is a delicate conclusion since if (1) holds, we know from [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF] that there exists a unique positive solution of [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF], which has a positive inmum and therefore the species are persistent. To this aim, we will propose a global condition in term of spectral theory. The central role is to develop the notions of generalized principal eigenvalues in unbounded domains, which are inspired from [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF], [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]. These generalizations are originally conjectured by Prof H. Berestycki (personal communication).

• Second, under the setting framework, we further investigate the inuence of the drift term on the existence of nontrivial solutions of [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF]. Indeed, in the works [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], the authors implicitly showed that the existence of positive solution is strongly connected with the amplitude of the drift term. Here, we consider the drifts satisfying divergence-free and some fair growth conditions. By adequate compactness compensations, we partially extend the result of Berestycki-Hamel-Nadirashvili [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF] on the limit of large drift for the rst eigenvalues in bounded domain to the generalized principal eigenvalues in unbounded domain with no periodic assumption. This is in fact an intermediate result that we shall use to derive the existence of sharp threshold of the drift-amplitude, K , such that positive solutions of [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF] exist if and only if K < K . Detailed explanation is postponed until stating the result.

• Third, the long time behaviors of

(|x| + 1) -α ∂ t u = a ij (x)∂ ij u(x) + Kq(x) • ∇u + f (x, u) t ∈ R + , x ∈ R N
are lastly investigated. We prove that, if the initial datum decays enough, solutions of this equation converge as t → ∞ in L p (R N )-weighted norm to the unique positive stationary of [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF]. Here 0 < p ≤ ∞.

We obtain an intriguing counterpart of these results between two cases α ∈ [0, 2) and α = 2. The dierent eects due to the slow decay and Hardy potentials of f s (x, 0) for |x| large are fully understood. Our results also conrm that the Liouville-type result for KPP-type nonlinearity is strongly depended on the behavior near innity of zero-order coecient of the linearized operator about zero.

The central role is the generalized principal eigenvalues. Let us dene it now. The following nice properties of the functions in D α will be frequently used in the proofs :

For α ∈ [0, 2), ϕ ∈ D α , ∀λ ∈ R, λ > 0, lim |x|→∞ ϕ(x)e λ|x| 1-α 2 = ∞ lim |x|→∞ ϕ(x)e -λ|x| 1-α 2 = 0. For α = 2, ϕ ∈ D 2 , ∀λ ∈ R, λ > 0, lim |x|→∞ ϕ(x)|x| λ = ∞ lim |x|→∞ ϕ(x)|x| -λ = 0.
The following new assumptions are considered :

Introduction générale

There exist α ∈ [0, 2] and µ ∈ L ∞ (R N ) satisfying lim sup |x|→∞ {|x| α (f s (x, 0) -µ(x))} ≤ 0 [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF] and one of the followings is true

η α = λ α (-a ij (x)∂ ij -Kq(x).∇ -µ(x), R N ) > 0,
(a ij ) is the identity matrix, I N , divq = 0 and there exists ψ ∈ C 2 loc (R N ) such that 2I N • ∇ψ(x) = -q(x),

η α = λ α (-∆ -µ(x), R N ) > 0. ( 10
)
We examine the Liouville type result in the class of functions φ satisfying lim sup

|x|→∞ ln φ |x| 1-α 2 = 0 if α ∈ [0, 2) and lim sup |x|→∞ ln φ ln |x| = 0 if α = 2.
The existence of threshold value of drift is directly related to the set of rst integrals. Let us dene it now.

Denition 0.0.2. A function z = 0 is called a rst integral of the vector elds q ∈ R N if z ∈ H 1 (R N ) and q • ∇z = 0 almost everywhere in R N . We denote the set of rst integrals of q by I q = {z|z ∈ H 1 (R N ), z = 0, and q • ∇z = 0 a.e in R N }.

We obtain the following result.

Theorem 0.0.3. Let α ∈ [0, 2) assumptions ( 9), [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF] hold and q satisfy some appropriate growth rates. If one has

inf φ∈H 1 (R N ) R N |∇φ| 2 -f s (x, 0)φ 2 R N φ 2 < 0 < inf φ∈Iq R N |∇φ| 2 -f s (x, 0)φ 2 R N φ 2 ,
with the convention that the inmum is +∞ if I q being empty, then there exists a sharp threshold value K such that Eq. ( 7) possesses a positive solution if and only if K < K .

Theorem 0.0.4. Let α = 2 the assumptions of Theorem (0.0.3) hold. Its conclusion remains true if

D 2 ≥ 1 or D 2 < 1 and η 2 > N (1 -D 2 ) -N 2 4 . (11)
We note that [START_REF] Berestycki | Spreading speeds for one-dimensional monostable reaction-diusion equations[END_REF] becomes unconditioned when N ≥ 2.

These results partially extend the result of Berestycki-Hamel-Nadirashvili [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF] on the lower limit of the rst eigenvalue for large drift in bounded domain to generalized principal eigenvalues in unbounded domain with no periodic assumption.

III. Liouville-type result for a quasilinear equation with possibly vanishing and sign-changing potentials

In this chapter, we pursue to understand whether the characterization on the existence, nonexistence and uniqueness for semilinear equations in chapter 2 are still true for the equation involving p-Laplacian and possibly vanishing potentials. In a joint work with Phuoc-Tai Nguyen, Technion Israel Technology Institute, during my visit, we successfully characterize the existence and uniqueness of the following equation

K a [u] + b(x)g(u) = 0 in R N (12)
where K a

where λ R (a) is the rst eigenvalue and rst eigenfunction ϕ R of K a with Dirichlet boundary condition on

Ω satisfying    K a [ϕ R ] = λ R (a)ϕ p-1 R in B R ϕ R > 0 in B R ϕ R = 0 on ∂B R . (14)
we derive the following results :

Theorem 0.0.1. Let α ∈ [0, p), p ≥ 2, and conditions (C1)-(C2) hold, then Eq. [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] cannot possess more than one positive solution such that Theorem 0.0.2. Let α = p, p ≥ 2, and conditions (C1)-(C2) hold. Assume that m > N p (N p + 3p + 2), then Eq. [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] cannot possess more than one positive solution such that

lim sup |x|→∞ u(x) |x| γ 0 < ∞,
where γ 0 is the unique positive solution of algebraic equation

(N p + 1)γ p + (3p + 1)N γ p-1 -m = 0. ( 15
)
Assume in addition that (C3) holds, then the unique positive solution is achieved if and only if λ ∞ (a) < 0.

It has to decay polynomially as |x| → ∞ when exists.

The two last sections are devoted to the studies of the nonlocal models of the previous equations. These are also the most abstract parts of thesis, where will present some developments on the theory of the generalized principal eigenvalue for nonlocal operators in unbounded domains. Furthermore, we also investigate some asymptotic behavior of the rescaling equations and the connection between solutions of local and nonlocal equations, which are just recently emerged in some contemporary researches. For the sake of coherence, we will write a short introduction for each chapters here. More explanations are left for the details of chapter IV and V. These materials are based on a joint work with H. Berestycki and J. Coville.

The main goal is nding survival criteria for a species that has a long range dispersal strategy. As a model species, we can think as trees whose seed and pollens are disseminate on a long range. In ecology a commonly used model that integrate such long range dispersal [START_REF] Grinfeld | Non-local dispersal[END_REF][START_REF] Hutson | The evolution of dispersal[END_REF][START_REF] Lutscher | The eect of dispersal patterns on stream populations[END_REF] is the following :

∂u ∂t (t, x) = J u(t, x) -u + f (x, u(t, x)) in R + × R N , (16)
where u(t, x) is the density of the considered population, J is a dispersal kernel, f (x, s) is a KPP type non-linearity describing the growth rate of the population. The possibility of a long range dispersal is well known in ecology, where numerous data are now available to support this assumptions [START_REF] Cain | Long-distance seed dispersal in plant populations[END_REF][START_REF] Clark | Why Trees Migrate So Fast : Confronting Theory with Dispersal Biology and the Paleorecord[END_REF][START_REF] Fastie | Reid's Paradox of rapid plant migration[END_REF][START_REF] Xhaard | Inferring invasion determinants with mechanistic models and multitype samples[END_REF][START_REF] Deveaux | Estimation de la dispersion de pollen à longue distance à l'echelle d'un paysage agicole : une approche expérimentale[END_REF].

In this setting the tail of the kernel is a measure of the frequency at which long dispersal events occur. A biological motivation for the use of the steady state

J u(x) -u(x) + f (x, u(x)) = 0 in R N . (17)
to describe the evolution of the population comes from the observation that the intrinsic variability in the capacity of the individuals to disperse generates, at the scale of a population, a long range dispersal of the population. The eect of such variability has been investigate in [START_REF] Hapca | Anomalous diusion of heterogeneous populations characterized by normal diusion at the individual level[END_REF][START_REF] Petrovskii | Variation in individual walking behavior creates the impression of a Lévy ight[END_REF] by means of the study of correlated random walks. In such framework, all individuals follow a simple random walks where the diusion coecient follows a probability law, it can be checked that the probability of the density of population will then follow an integro-dierential equation [START_REF] Hapca | Anomalous diusion of heterogeneous populations characterized by normal diusion at the individual level[END_REF][START_REF] Petrovskii | Variation in individual walking behavior creates the impression of a Lévy ight[END_REF][START_REF]Quantitative Analysis of Movement : Measuring and Modeling Population Redistribution in Animals and Plants[END_REF] where a dispersal kernel J describes the probability to jump from one location to another. For compactly supported dispersal kernels J, we derive an optimal survival criteria. We prove that a positive stationary solution of [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF] exists if and only if the principal eigenvalue

J ϕ -ϕ + ∂ s f (x, 0)ϕ + λ p ϕ = 0 x ∈ R N , (18)
is negative. In addition, for any continuous non-negative initial data that is bounded or integrable, we establish the long time behavior of u(t, x). We also analyse the impact of the size of the support of the dispersal kernel on the persistence criteria. We exhibit situations where the dispersal strategy has "no impact" on the survival of the species and situations where the slowest dispersal strategy is not any more an Ecological Stable Strategy. Some generalisations of the survival criteria are also discussed for fat-tailed kernels. The proofs of these results essentially rely on some news properties of the principal eigenvalue λ p in R N that we derive in the rst part of the paper. In particular, we prove the equivalence of three denitions of the principal eigenvalue and identify some asymptotic limits of λ p with respect to some rescaling of the dispersal operator. Furthermore we investigate the asymptotic behavior of the solutions u ε of rescaling equations

1 ε m (J ε u ε (x) -u ε (x)) + a(x)u ε -u 2 ε = 0 in R N , (19)
as ε → 0 and ε → ∞, where m = 0, 0 < m < 2, m = 2 and J ε (x) = 1 ε N J x N . The results are very dierent between theses cases. In particular for the case m = 2, we derive an interesting connection between the local and nonlocal solution, where we prove that the unique positive solution u ε of [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF] approximates in L 2 loc (R N ) norm the unique positive solution of the elliptic equation

K 2,N D 2 (J) 2 ∆v + v(a(x) -v) = 0 in R N , with D 2 (J) := R N J(z)z 2 dz and K 2,N := 1 |S N -1 | S N -1 (σ.e 1 ) 2 dσ = 1 N .
The central role is the spectral theory of the nonlocal operator. Here, in the rst part of this work, we will study carefully three notions of eigenvalues, which may be also useful for other various researches.

IV. The generalized principal eigenvalues for nonlocal operators

We consider the following generalized principal eigenvalues : Denition 0.0.6. Let L Ω [ϕ] = Ω J(x -y)ϕ(y)dy -ϕ(x) and a(x) ∈ L ∞ (Ω). We set

λ p (L Ω + a(x)) := sup{λ ∈ R | ∃ϕ > 0, ϕ ∈ C(Ω), s.t L Ω [ϕ] + a(x)ϕ + λϕ ≤ 0 in Ω} λ p (L Ω + a(x)) := inf{λ ∈ R | ∃ϕ > 0, ϕ ∈ C(Ω) ∩ L ∞ (Ω), s.t L Ω [ϕ] + a(x)ϕ + λϕ ≥ 0 in Ω}.
and the variational characterization Denition 0.0.7. L Ω [ϕ] = Ω J(x -y)ϕ(y)dy -ϕ(x) and a(x) ∈ L ∞ (Ω). Assume further that J is radially symmetric. We set λ v (L Ω + a(x)) := inf ϕ∈L 2 (Ω),ϕ ≡0

Ω Ω J(x -y)(ϕ(x) -ϕ(y)) 2 dxdy -Ω (a(x) -k(x))ϕ 2 (x) dx ϕ 2

L 2 (Ω)
.

where k(x) := Ω J(x -y) dy.

The main achievements on the relation of these eigenvalues are :

i) If Ω is bounded λ p (L Ω + a(x)) = λ p (L Ω + a(x)) = λ v (L Ω + a(x)).
ii) If Ω = R N and J is compactly supported, we also have the same conclusion as i).

For the later, we investigate some asymptotics of the eigenvalue of the following rescaling operators. Set

M ε,m [ϕ] = 1 ε m (J ε ϕ -ϕ),
we obtain the following result Theorem 0.0.8. Let J be compactly supported. Assume a(x) ∈ C(R N ) ∩ L ∞ (R N ) then we have

• When 0 < m ≤ 2 lim ε→+∞ λ p (M ε,m + a(x)) = -sup R N a(x)
• When 0 ≤ m < 2, lim ε→0 λ p (M ε,m + a(x)) = -sup R N a(x)

• When m = 0,

lim ε→+∞ λ p (M ε + a(x)) = 1 -sup R N a(x)
In this case, if a(x) is radially symmetric non increasing and sup R N a(x) < 1 then there is a sharp threshold ε 0 , in the sense that λ p (M ε,m + a(x)) is nonincreasing with respect to ε and it is negative if and only if ε < ε 0 .

• When m = 2 and a(x) is globally Lipschitz, then

lim ε→0 λ p (M ε,2 + a(x)) = λ 1 K 2,N D 2 (J) 2 ∆ + a(x)
where

λ 1 (K 2,N D 2 (J)∆ + a(x)) := inf ϕ∈H 1 0 (R N ),ϕ ≡0 K 2,N D 2 (J) 2 R N |∇ϕ| 2 (x) dx ϕ 2 2 -R N a(x)ϕ 2 (x) dx ϕ 2 2 .
In the second part of this work, we investigate the positive solution of nonlinear equations ( 16)- [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF].

V. The positive solutions of nonlocal equations with KPP nonlinearity

As previously mentioned, we will investigate the existence, nonexistence and uniqueness of Eq. ( 17) and the large time behavior of Eq. ( 16) in L ∞ (R N ) and L 1 (R N ) norms. We further characterize the limits of solutions of [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF] as ε → ∞ and ε → 0 in three case m = 0, 0 < m < 2, m = 2. One of the important results that we achieved up to now is Theorem 0.0.9. Assume that J is compactly supported and f (x, s) = a(x)s -s 2 and let m = 2. Then there exists ε 1 ∈ (0, ∞) so that for all ε ≥ ε * there exists a positive solution u ε to [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF]. Moreover,

lim ε→+∞ u ε -a + ∞ = 0, lim ε→+∞ u ε -a + L 2 (R N ) = 0.
In addition, the following dichotomy holds

• When λ 1 K 2,N D 2 (J) 2
∆ + a(x) < 0, there exists ε 0 ∈ (0, ∞) so that for all ε ≤ ε 0 there exists a positive solution to [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF] and

u ε → v, in L 2 loc (R N ),
where v is the unique bounded non-trivial solution to

K 2,N D 2 (J) 2 ∆v + v(a(x) -v) = 0 in R N . • When λ 1 K 2,N D 2 (J) 2

Introduction and denitions

In a pioneering paper [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], Berestycki et al. studied the inuence of climate change (global warming) on the population dynamics of biological species, who are strongly sensitive to temperature conditions. The authors proposed a mathematical model in R, which is formulated as a reaction-diusion equation with a forced speed c :

u t -u xx = f (x -ct, u) x ∈ R. (1.1)
Here c is the speed of the climate change. A typical f considered in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] is

f (x, s) = -sm for x < 0 and x > L sm 1 - s K for 0 ≤ x ≤ L, (1.2)
for some positive constants m, m , L, K. This nonlinearity expresses that the environment is unfavorable outside a compact set [0, L] and favorable inside. The higher dimensional versions with more general type of f were studied later in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF]. Beside that a similar model was also considered in the context of competing species by Potapov and Lewis [START_REF] Potapov | Climate and competition : the eect of moving range boundaries on habitat invasibility[END_REF], where the authors investigated the co-existence of two species under the eect of climate change and moving range boundaries on habitat invasibility. All these papers assume that the environments are completely unfavorable near innity, i.e the favorable zone has compact support. More precisely, there exist R, m > 0 such that

f s (x, 0) ≤ -m, ∀|x| ≥ R. (1.3)
Note that f s (x, 0) is understood as the initial per capita rate of growth. The main purpose of this paper is to study the criterion for persistence and extinction of species in more general frameworks than the ones considered in these previous works [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF], [START_REF] Potapov | Climate and competition : the eect of moving range boundaries on habitat invasibility[END_REF] and further provide some applications of this theory. Our aim is to deal with the new cases for which condition (1.3) is no longer true. We extend the model of (1.2) in two frameworks. The rst one is for an innite cylindrical 13 domain with Neumann boundary condition:

u t -∆u = f (x 1 -ct, y, u) t > 0, x ∈ Ω ∂ ν u(t, x 1 , y) = 0 t > 0, x ∈ ∂Ω, (1.4)
where Ω = R × ω, ω is an open bounded and smooth domain in R N -1 , ν denotes the exterior unit normal vector eld to Ω. In this framework, the environments are assumed to be independent of time. We are especially interested in considering environments of mixed type, which are only assumed to be globally unfavorable at innity. One can think of the environment containing both favorable and unfavorable regions that extend all the way to innity, namely f s (x 1 , y, 0) > 0 and f s (x 1 , y, 0) < 0 respectively as x 1 → ±∞, depending on the location of y. The competitive and mutual inuence between these regions play a major role in characterizing the persistence and extinction of the species in the whole domain.

Mathematically, we will use a global condition in terms of spectral property to describe that the environment is globally unfavorable at innity. The more detailed explanations of this condition will be given in subsection 1.2.1.

In the second framework, we investigate another type of mixed environment with periodic dependence on y and t. More precisely, the equation is now of following type

u t -∆u = f (t, x 1 -ct, y, u) t > 0, x = (x 1 , y) ∈ R N ,
where the nonlinearity reaction is assumed to be periodic in y and t. The time-periodic dependent reaction has been previously investigated in various frameworks, the interested reader are referred to [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], [START_REF] Du | The periodic logistic equation with spatial and temporal degeneracies[END_REF], [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF], [START_REF] Nadin | Existence and uniqueness of the solutions of a space-time periodic reaction-diusion equation[END_REF], [START_REF] Peng | The periodic-parabolic logistic equation on R N[END_REF]. The main dierence of the present work with respect to these papers is that here f is not assumed to be periodic in x 1 -direction but be shifted with the forced speed c, which can be seen as an eect of climate change. This has been considered in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] for an environment pointwise unfavorable at innity. Our extension here is to consider f s (t, x 1 , y, 0) to be sign-changing depending on the location of y ∈ R N -1 at the time t. We only require f to satisfy a global condition as x 1 → ±∞. The additional diculties are due to the fact that we do not a priori require the solutions to be periodic in y nor in t and also we do not impose any boundary conditions as x 1 → ±∞. The time-periodic dependence of reaction term can be thought of as representation of a seasonal dependence of environment. We further investigate the concentration of the species in regions surrounded by highly hostile environments. More precisely, our aim is to describe the dynamics of the species in the rst framework not only in the cylindrical domain Ω but in the whole space R N under the assumption that the environment outside Ω becomes more and more unfavorable. From the biological point of view one may wonder whether the species still survives if some parts of the environment becomes extremely unfavorable. This question can be addressed by solving the following mathematical problem. We consider equation (1.4) in the whole space R N and study the limit of the sequence of traveling fronts with a reaction term F n (x, s) such that their growth rates are negative outside the cylindrical domain Ω and tend to -∞ as n → ∞. These solutions solve the equations

∆U n + c∂ 1 U n + F n (x, U n ) = 0, x ∈ R N ,
where

F n (x, s) = f (x, s) for x ∈ Ω and ∂F n ∂s (x, 0) → -∞ as n → ∞ locally uniformly in R N \ Ω.
If the species survives, we aim to characterize the limit. This is the object of section 4. Very recently, Guo and Hamel [START_REF] Guo | Propagation and blocking in periodically hostile environments[END_REF] have studied the similar problem on the periodic and not necessarily connected domains without the eect of climate change, namely when c = 0. From a dierent point of view, here we consider the concentration of the species facing a climate change in an innite cylindrical domain Ω when the exterior domain R N \ Ω becomes extremely unfavorable. To this aim, we rst need to ascertain the existence and uniqueness of traveling front for problem (1.4) with Dirichlet boundary condition on ∂Ω. The lack of compactness of Ω as well as the presence of c = 0 and the fact that near innity of Ω, the environment contains both favorable and unfavorable regions are the main diculties to be overcome. Here, we will make use of some recent advances of spectral theory in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF].

Finally, the last result is devoted to the study of symmetry breaking of the fronts in Ω. The main reason leading to the symmetry breaking is the dierence of asymptotic behaviors near ±∞. To be more clear, due to the spectral theory for cylindrical domains developed by Berestycki-Nirenberg in [START_REF] Berestycki | traveling fronts in cylinders[END_REF], under some fair assumptions on the growth rate of f s (x 1 , y, 0) as x 1 → ±∞, we can nd the exact behaviors of the unique solution of (1.4) as x 1 tends to ±∞ depending on c. By conditioning that these behaviors are dierent, we obtain the asymmetry of the solution. In particular, assuming c = 0, we will see that the asymmetry holds when f s (x 1 , y, 0) converges fast enough to the same negative constant as x 1 → ±∞.

In the remainder of this section, we give notations and denitions that are used in the paper. The set Ω denotes an innite straight cylindrical domain Ω = R × ω, where ω is an open bounded and smooth domain in R N -1 . We use the notation x = (x 1 , y) ∈ R × ω for the points in Ω and denote :

Ω + = {x ∈ Ω, x 1 ≥ 0, y ∈ ω} ; Ω -= {x ∈ Ω, x 1 ≤ 0, y ∈ ω}; Ω r = {x ∈ Ω, -r < x 1 < r, y ∈ ω}.
Let O ⊂ R N and L be a uniformly elliptic operator with coecients bounded on O dened by

Lu = a ij (x)∂ ij u(x) + b i (x)u i (x) + c(x)u.
If O is smooth and bounded, it is classical that L admits a unique eigenvalue -λ D (respectively -λ N ) and a unique (up to multiplication) eigenfunction with Dirichlet (respectively Neumann) boundary condition i.e :

Lϕ = -λ D ϕ x ∈ O ϕ = 0 x ∈ ∂O. Lϕ = -λ N ϕ x ∈ O ∂ ν ϕ = 0 x ∈ ∂O.
As is known, the principal eigenpair (eigenvalue and eigenfunction) for an associated elliptic operator plays an important role in deriving persistence results and long time dynamics. In 1994, Berestycki, Nirenberg and Varadhan [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] gave a very simple and general denition of the principal eigenvalue of L for general domains whose boundaries are not necessarily smooth and later Berestycki, Hamel and Rossi [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF] used this approach to dene generalized principal eigenvalues in unbounded domains. More precisely now allowing O to be a smooth and possibly unbounded domain, they dened the generalized Neumann principal eigenvalue as follow

λ N (-L, O) := sup{λ ∈ R : ∃φ ∈ W 2,N loc (O), φ > 0, (L + λ)φ ≤ 0 a.e in O, ∂ ν φ ≥ 0 on ∂O}. (1.5)
When O is bounded, these two notions coincide : λ N (-L, O) = λ N . We adopt this denition in our paper.

Under the assumption a ij , b i , c ∈ L ∞ (O), it is easily seen that λ N (-L, O) is well dened. For related denition and more properties of generalized eigenvalues, the reader is referred to [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF].

Hypotheses and main results

The cylindrical environment without time dependence

The function f (x 1 , y, s) : R × ω × [0, +∞) → R is assumed to be continuous in x 1 , measurable in y, and locally Lipschitz continuous in s. In addition, the map s → f (x, s) is of class C 1 (0, s 0 ) for some positive constant s 0 , uniformly in x. We assume that f (x, 0) = 0, ∀x ∈ Ω. As we will see, the dynamics is controlled by traveling fronts. Thus, we look for the solutions of Eq. (1.4) of the type u(t, x) = U (x 1 -ct, y), which are called traveling front solutions with forced speed c. Such solutions are given by the equation :

       ∆U + c∂ 1 U + f (x, U ) = 0 x ∈ Ω ∂ ν U = 0 x ∈ ∂Ω U > 0 in Ω U is bounded. (1.6)
In the results below, we will require the following hypotheses on f :

∃S > 0 such that f (x, s) ≤ 0 for s ≥ S, ∀x ∈ Ω, (1.7)
s → f (x, s)/s is nonincreasing a.e in Ω and there exist D ⊂ Ω, |D| > 0 such that it is strictly decreasing in D.

(

Both of these conditions are classical in the context of population dynamics. The rst condition means that there is a maximum carrying capacity eect : when the population density is very large, the death rate is higher than the birth rate and the population decreases. The second condition means the intrinsic growth rate decreases when the population density is increasing. This is due to the intraspecic competition for resources.

As has been already mentioned, we are looking for a condition that applies to mixed environments. We assume that there exists a measurable bounded function µ : ω → R such that

µ(y) = lim sup |x 1 |→∞ f s (x 1 , y, 0) and λ µ := λ N (-∆ y -µ(y), ω) > 0.
(1.9)

Condition (1.9) means that in the mixed environment is globally unfavorable at innity in the direction of x 1 . This generalizes the condition

f s (x 1 , y, 0) ≤ -m < 0 for |x 1 | large enough, y ∈ ω, (1.10)
which is used in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF]. Indeed, if µ(y) ≤ -m < 0 one gets λ N (-∆ y -µ(y), ω) ≥ m > 0. Our generalization here aims at allowing f s (x 1 , y, 0) to change sign when |x 1 | is large. An illustration of condition (1.9) will be given in Section 2.1.

We are now ready to state the main results regarding this framework

The existence and uniqueness of traveling front

The existence and uniqueness results are directly conditioned by the amplitude of the speed of climate change and the sign of the principal eigenvalue λ 0 := λ N (-L 0 , Ω), where

L 0 ϕ = ∆ϕ + f s (x, 0)ϕ.
Theorem 1.1.1. Assume that (1.7)-(1.9) hold. Then there exists a unique critical speed c * such that (1.4) admits a traveling front solution, that is a solution of (1.6) if and only if 0 ≤ c < c * . Moreover, when the front exists, it is unique.

This theorem yields an analogue to the results obtained in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF], [START_REF] Potapov | Climate and competition : the eect of moving range boundaries on habitat invasibility[END_REF]. Indeed, under the assumption of type (1.9), one should not expect too many times the same thing. We also point out that the uniqueness of (1.6) is achieved in the class of positive bounded solutions without necessarily prescribing the boundary condition as x 1 → ±∞.

Denition of the critical speed c * By using the Liouville transformation V (x 1 , y)

:= U (x 1 , y)e c 2 x 1 , problem (1.6) is equivalent to            ∆V + f (x 1 , y, V (x 1 , y)e -c 2 x 1 )e c 2 x 1 - c 2 4 V = 0 x ∈ Ω ∂ ν V = 0 x ∈ ∂Ω V > 0 in Ω V (x 1 , y)e -c 2 x 1 is bounded. (1.11)
Linearizing this equation about 0, one gets a self-adjoint operator :

Lw := ∆w + (f s (x, 0) -c 2 /4)w.
We set L 0 ϕ = ∆ϕ + f s (x, 0)ϕ and λ 0 := λ N (-L 0 , Ω) is the generalized Neumann principal eigenvalue of L 0 in Ω. Since f s (x, 0) is bounded, λ 0 is well dened and nite. We are led to Denition 1.1.2. We dene the critical speed by

c * := 2 -λ 0 if λ 0 < 0 (1.12) Proposition 1.1.3. The eigenvalue λ N (-∆ -c∂ 1 -f s (x, 0), Ω) < 0 i 0 ≤ c < c * .
Proof. Let L = ∆ + c∂ 1 + f s (x, 0). Since we do not assume the test-function of (1.5) to be bounded. It immediately follows from the denition (1.5

) that λ N (-L, Ω) = λ N (-L, Ω) = λ 0 + c 2 4 .
The next two results deal with the long time dynamics of the evolution equation (1.4) in L ∞ (Ω) and L 1 (Ω).

Long time dynamics Theorem 1.1.4. Let u(t, x) be the solution of (1.4) with initial condition u(0, x) ∈ L ∞ (Ω), which is nonnegative and not identically equal to zero. Assume that

(1.7) -(1.9) hold. i) If c ≥ c * , then lim t→∞ u(t, x) ∞,Ω = 0; ii) if 0 ≤ c < c * then lim t→∞ (u(t, x 1 , y) -U (x 1 -ct, y)) ∞,Ω = 0,
where U is the unique solution of (1.6) and • ∞,Ω denotes the sup-norm on Ω.

This theorem means that, a species cannot keep pace with a climate change if its speed is too large. This theorem generalizes the results in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF]. Note that, in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF], condition (1.10) was actually used in the proofs, in particular, to derive the exponential behavior at innity. Although our approaches are similar to those in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF], new diculties arise from the non-constant unfavorable characterization at innity, especially to obtain the comparison principle.

The next result is concerned with the L 1 (Ω) convergence of the traveling fronts. This result describes the long time dynamics of the total population. Theorem 1.1.5. Let u(t, x) be the solution of (1.4) with initial condition u(0, x) ∈ L ∞ (Ω) ∩ L 1 (Ω), which is nonnegative and not identically equal to zero. Assume that (1.7) -(1.9) are satised then the same conclusions as in Theorem 1.1.4 hold with the L 1 (Ω) norm besides L ∞ (Ω) norm.

The partially periodic environment with time dependence

We now consider problem (1.4) in partially periodic environments with seasonal dependence. Namely, the reaction term f now depends periodically in the time variable and (1.4) becomes

u t -∆u = f (t, x 1 -ct, y, u) t ∈ R, x = (x 1 , y) ∈ R N , (1.13)
where c > 0 is the given forced speed and f is now assumed to be periodic in y. More precisely, we say that the environment is partially periodic in y and depends seasonally on time if : 1) ∀i ∈ {2, ..., N }, there exist the constants L 2 , ..., L N such that

f (t, x + L i e i , s) = f (t, x, s) ∀t ∈ R, s ∈ R, x ∈ R N
where {e 1 , ...e N } denotes the unit normal orthogonal basis of R N .

2) There exists T > 0, such that

f (t + T, x, s) = f (t, x, s) ∀t ∈ R, s ∈ R, x ∈ R N .
We assume in addition that f (t, x, 0) = 0, f is C 1 with respect to s, f and f s are Holder-continuous with respect to t and x, precisely

∀s > 0, f (•, •, s), f s (•, •, 0) ∈ C α 2 ,α t,x (R × R N ), where C α 2 ,α t,x (I × H), I ⊂ R, H ⊂ R N denotes the space of functions φ(t, x) such that φ(•, x) ∈ C α 2 (I) and φ(t, •) ∈ C α (H)
uniformly with respect to t and x respectively.

We are interested in pulsating fronts of (1.13), namely the solutions of the form u(t, x) = U (t, x 1 -ct, y) > 0. They are obtained from the equation

U t = ∆U + c∂ 1 U + f (t, x, U ) t ∈ R, x ∈ R N U is bounded. (1.14)
Note that U (t, x 1 , y) is not a priori assumed to be periodic in t nor in y.

Here, we require the principal eigenvalue of the linearized operator of Eq. (1.14). More generally, we consider the operators of the form :

Lu = ∂ t u -a ij (t, x)∂ ij u(t, x) -b i (t, x)u i (t, x) -c(t, x)u(t, x), x = (x 1 , y) ∈ R N , (1.15)
where a ij , b i , c i are T -periodic in t and periodic in y with the same period. To dene the generalized principal eigenvalue of L, we assume that the coecients satisfy the regularity condition as mentioned above and the matrix

(a ij (t, x)) is uniformly elliptic, namely a ij , b i , c i ∈ C α 2 ,α t,x (R × R N
) and there exist some positive constants E 1 , E 2 such that for all ξ ∈ R N and (t,

x) ∈ R × R N such that E 1 |ξ| 2 ≤ 1≤i≤j≤N a ij (t, x)ξ i ξ j ≤ E 2 |ξ| 2 . Denition 1.1.6. Let O ⊂ R and Q = {(t, x) = (t, x 1 , y) ∈ R × O × R N -1 }, the generalized principal eigenvalue of L in Q is dened by: λ1 (L, Q) = sup λ ∈ R : ∃ φ > 0, φ ∈ C 1,2 t,x (Q), φ is T-periodic in t and periodic in y such that (L -λ)φ ≥ 0 in Q} . (1.16)
By assuming in addition that

a ij , b i , c i ∈ L ∞ (R N +1
), one can take λ = -sup R×R N f s (t, x, 0) and 1 as a test function to see that λ1 is well-dened and -sup R×R N f s (t, x, 0) ≤ λ1 . We point out that this denition does not make sense if we do not require that the test functions to be periodic in t. Indeed, since (L -λ)(φe αt ) = (L + α -λ)(φe αt ), ∀α ∈ R, if we do not force the periodicity in t, it would yield λ1 = λ1 + α for all α. This kind of eigenvalue seems analogous to the ones introduced by Berestycki and Rossi [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] and Nadin [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF]. However, the dierence is that here we force the test functions to be periodic in t and y but not in x 1 while in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], the test functions are not periodic in any direction of x = (x 1 , y) and in [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF], the test functions must be periodic in both t and x = (x 1 , y).

We further need the two following conditions that are similar to (1.7), (1.8), but take into account the time-periodic dependence of f :

∃S > 0 such that f (t, x, s) ≤ 0 for s ≥ S, ∀t ∈ R, x ∈ R N , (1.17)
s → f (t, x, s)/s is nonincreasing and for all t 0 ∈ [0, T ) there exist D ⊂ (-∞, t 0 ) × R N , |D| > 0 such that it is strictly decreasing in D.

(1.18)

Suppose that a parabolic operator L is dened on R × R N -1 and has the form

Lφ = ∂ t φ -a ij (t, y)∂ ij φ(t, y) -b i (t, y)φ i (t, y) -c(t, y)φ(t, y), y ∈ R N -1 .
Under the assumptions that

a ij (t, y), b i (t, y), c i (t, y) ∈ L ∞ (R × R N -1
) and the matrix a ij (t, x) satises the uniform elliptic condition, we dene the generalized space-time periodic eigenvalue of L :

λ1 ( L, R × R N -1 ) = sup{λ ∈ R : ∃ φ > 0, φ ∈ C 1,2 t,y (R × R N -1 ), φ is T-periodic in t and periodic in y such that ( L -λ)φ ≥ 0 in R × R N -1 }.
It is worth mentioning that, Nadin [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF] has shown the existence and uniqueness of the space-time periodic eigenpair (λ p , ϕ p ) of the eigenvalue problem

       Lϕ p = λ p ϕ p ϕ p > 0 ϕ p (., . + T ) = ϕ p ϕ p (. + L i e i , .) = ϕ p .
For more details, we refer the reader to the Theorems 2.7 in [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF]. Obviously, λ p ≤ λ1 ( L, R × R N -1 ).

Using this notion, we assume that there exists a function γ(t, y) ∈ L ∞ (R × R N -1 ), which is periodic in y and T-periodic in t such that γ(t, y) = lim sup

|x 1 |→∞ f s (t, x 1 , y, 0) and λ1 (∂ t -∆ y -γ(t, y), R × R N -1 ) > 0. (1.19)
Condition (1.19) yields the characterization of the environment expressing that it is globally unfavorable at innity. The new diculties of this problem arise since we deal with the solution in the unbounded domain (whole space) without a-priori assuming that the solutions are periodic in y nor in t. Moreover, the monotonicity in time of solutions of parabolic operators starting by a stationary sub (or super) solution no longer holds.

Let us call Pϕ = ∂ t ϕ -∆ϕ -c∂ 1 ϕ -f s (t, x, 0)ϕ the linearized operator associated with (1.14). In the sequel, we will briey denote by λ1 = λ1 (P, R × R N +1 ). We are now able to state the results of this section : Theorem 1.1.7. Assume that (1.17) -(1.19) hold, then there exists a positive pulsating front (1.14) if and only if λ1 < 0. If it exists, it is unique, T -periodic in t, periodic in y and decays exponentially in |x 1 |, uniformly in y and t.

U ∈ C 1,2 (R × R N ) of
One of the interesting points of this theorem is the loss of compactness since f is not periodic in x 1 and the solution is not a priori assumed to be periodic in y nor in t. We will prove that the uniqueness of (1.14) still holds in this larger class of solution, that is the class of nonnegative bounded solutions. As pointed out in section 1.5 of [START_REF] Nadin | Existence and uniqueness of the solutions of a space-time periodic reaction-diusion equation[END_REF], one cannot expect to show a general uniqueness in the class of nonnegative bounded solutions, even when the coecients of (1.14) are periodic in x = (x 1 , y) and in t under condition λ1 < 0 only. Some extra assumptions are needed. Here the uniqueness holds due to assumption (1.19), which is a key ingredient to derive the exponential behavior of solutions of (1.14) as x 1 → ±∞.

Theorem 1.1.8. Let u(t, x) be the solution of (1.13) with nonnegative initial datum u 0 (x) ∈ L ∞ (R N ) and not identically equal to 0. Assume that (1.17)- (1.19)

hold. i) If λ1 ≥ 0 then lim t→∞ u(t, x) = 0, uniformly in x ∈ R N . ii) if λ1 < 0 then lim t→∞ (u(t, x 1 , y) -U (t, x 1 -ct, y)) = 0,
where U is the unique solution of (1.14), uniformly in x 1 , locally uniformly in y. If, in addition, u 0 is periodic in y or satises

∀r > 0, inf |x 1 |<r,y∈R N -1 u 0 (x 1 , y) > 0, (1.20)
then above convergence is uniform also in y.

The fact that the convergence holds uniformly in x 1 is a consequence of the exponential decay as x 1 → ±∞, which can be derived from (1.19).

Organization of the paper. We divide the rest of the paper into four sections. Section 2 deals with problem (1.4) in the cylindrical domain Ω without time dependence. Section 3 investigates problem (1.13) in partially periodic domain with time dependence. In section 4, we rst study the concentration of the species in Ω of Section 2, when the exterior domain becomes extremely unfavorable and further prove the symmetry breaking of the fronts. Finally, some auxiliary results are contained in the Appendix.

1.2

The cylindrical environment without time dependence

An illustration

Before proving the main results of this section, let us provide an illustration of how theorems (1.1.1)-(1.1.5) apply and why condition (1.9) is useful to describe the heterogeneity of habitat of a species facing a climate change.

We consider Ω = R × (0, 1) and for α ∈ (0, 1), L > 0 the family f α,L (x 1 , y, s) = (ρ L (x 1 ) + µ α (y))s -s 2 , where

ρ L (x 1 ) = 2 on [-L, L] -2 outside [-L, L], µ α (y) = 1 on [0, α] -1 on [α, 1].
These nonlinearities are discontinuous. However, f s is well dened a.e and all our results apply for zero order coecient in L ∞ (see [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF] for further discussion of this extension). For every α ∈ (0, 1), let (λ α , φ α ) be the (unique) eigenpair of the Neumann eigenvalue problem

-φ α -µ α (y)φ α = λ α φ α in (0, 1) ∂ ν φ α = 0 at 0 and 1.
Dividing the equation by φ α and integrating by part, we get

- 1 0 φ 2 α φ 2 α dy - 1 0 µ α (y)dy = λ α .
Hence, λ α ≤ -

1 0 µ α (y)dy = -(2α -1)
. It is also known that λ α is decreasing with respect to α and that α → λ α is continuous. Since λ 0 = 1 we see that there exists a unique α such that λ α = 0 and λ α < 0 i α > α.

For α > α, λ α < 0, it is well-known that there exists a unique positive solution (Berestycki,[START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF]) of -p α -µ α (y)p α + p 2 α = 0 in (0, 1) ∂ ν p α = 0 at 0 and 1.

In this case, the environment is globally favorable at innity and there is always persistence. Indeed, it can be shown in this case that u(t, x 1 , y) → p α (y) as t → ∞ pointwise in Ω. Under the condition α < α, the environment is globally unfavorable at innity. Then, for xed c not too large, for instance c = 1, we claim that there exists a unique L * such that the persistence holds i L > L * .

To prove this, let us call

Q L [φ] = φ + φ + ∂ s f α,L (x 1 , y, 0)φ, where ∂ s f α,L (x 1 , y, 0) = ρ L (x 1 ) + µ α (y) dened on Ω.
Since Ω is unbounded, we cannot dene the classical eigenvalue of Q L on Ω. We make use of the denition (1.5)

. Let us call λ L = λ N (-Q L , Ω) and λ L = λ N (-φ -∂ s f α,L (x 1 , y, 0)φ, Ω). By Propsition 1.1.3, one has λ L = λ L + 1 4 .
Since ρ L is increasing with respect to L, λ L is decreasing with respect to L. Moreover, the map

L → λ L is continuous on [0, ∞]. Indeed, for any L ∈ [0, ∞], let {L n } ∈ [0, ∞] be an arbitrary sequence converging to L, we see that ∂ s f α,Ln (x 1 , y, 0)-∂ s f α,L (x 1 , y, 0) L ∞ (Ω) → 0 as n → ∞.
Arguing as in the proof of Proposition 9.2 part (ii) [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF], we get lim n→∞ λ Ln = λ L . It is worth noting that since Ω is a smooth domain with Neumann boundary condition, we can apply the Harnack inequality up to the boundary (see Berestycki-Caarelli-Nirenberg [START_REF] Berestycki | Inequalities for second-order elliptic equations with applications to unbounded domains. I. A celebration of John F. Nash[END_REF]) as in proof of Proposition 9.2 part (ii) [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]. Moreover, since

∂ s f α,0 (x 1 , y, 0) = -2+µ α (y) ≤ -1 and ∂ s f α,∞ (x 1 , y, 0) = 2 + µ α (y) ≥ 1, by taking 1 as a test-function, we have λ 0 = λ 0 + 1/4 ≥ 5/4 and λ ∞ = λ ∞ + 1/4 ≤ -3/4.
The claim is proved. As we will see in the next section, the equation

q α + cq + f α,L (x 1 , y, q) = 0 in Ω ∂ ν q α = 0 on ∂Ω (1.21)
admits a unique positive solution if and only if λ L < 0. From this result, it follows that for L > L * , as t → ∞ , u(t, x 1 , y) converges to the unique positive solution q(x 1 , y) of (1.21) and u(t, x 1 , y) converges to zero for L ≤ L * . Thus L * is the threshold value. Therefore, the problem is more subtle when α < α. Our theory applies in this case. Even if the persistence is known, the non-persistence and uniqueness are still delicate questions.

The existence and uniqueness of the front

To achieve the existence and uniqueness of Eq. (1.6), the key property is the exponential decay of solution. This estimate is the object of the next section Exponential decay Proposition 1.2.1. Let U be a nonnegative bounded solution of (1.6). Assume that (1.8)-(1.9) hold, then for all

0 < α < α * with α * = -c+ √ c 2 +4λµ 2
, there exists a positive constant C(α) such that

U (x 1 , y) + |∇U (x 1 , y)| ≤ C(α)e -α|x 1 | .
We point out that this proposition generalizes Proposition 3 of [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF]. Moreover, our proof is somewhat simpler than the proof of Berestycki and Rossi in that we do not use the Liouville transformation.

Proof. Consider ϕ ∈ W 2,p (ω) the eigenfunction associated with λ µ ∆ϕ + µ(y)ϕ

+ λ µ ϕ = 0 in ω ∂ ν ϕ(y) = 0 on ∂ω (1.22)
where ν = ν(y) is the outward unit normal on ω. As is well-known by the Hopf lemma, inf ω ϕ > 0.

For any δ ∈ (0, λ µ ), let Lw = ∆w + c∂ 1 w + (µ(y) + δ)w be dened in Ω. Due to (1.8)-(1.9), one has

∆U + c∂ 1 U + f s (x, 0)U ≥ 0 in Ω.
and there exists R = R(δ) > 0 such that

f s (x 1 , y, 0) ≤ µ(y) + δ in Ω \ Ω R , therefore LU ≥ 0 in Ω \ Ω R .
For any p > 0, set

w p (x 1 , y) = e (R+p)(τ -α) e α|x 1 | ϕ(y) + e R(τ +α) e -α|x 1 | ϕ(y) = C 1 e α|x 1 | ϕ(y) + C 2 e -α|x 1 | ϕ(y),
where R, τ, α > 0 will be chosen. Direct computation shows that

Lw p ϕ = C 1 α 2 + ∆ y ϕ ϕ + cα x 1 |x 1 | + µ(y) + δ e α|x 1 | + +C 2 α 2 + ∆ y ϕ ϕ -cα x 1 |x 1 | + µ(y) + δ e -α|x 1 | ≤ (α 2 + cα -λ µ + δ)(C 1 e α|x 1 | + C 2 e -α|x 1 | ) (1.23)
For w p to be a supersolution of L in Ω \ Ω R , it suces to take

α = α(δ) = -c + c 2 + 4λ µ -4δ 2 > 0.
Clearly, α(δ) is decreasing with respect to δ ∈ (0, λ µ ). Choosing τ = α/2 and R large enough , we have the following estimates on ∂(Ω R+p \ Ω R )

w p (x 1 , y) ≥ e Rτ ϕ(y) ≥ e Rτ inf ω ϕ ≥ U (x 1 , y) as |x 1 | = R, y ∈ ω w p (x 1 , y) ≥ e (R+p)τ ϕ(y) ≥ e (R+p)τ inf ω ϕ ≥ U (x 1 , y) as |x 1 | = R + p, y ∈ ω. Fix α, τ and R, we set z(x 1 , y) = wp(x 1 ,y)-U (x 1 ,y) ϕ (y)
. Obviously, L(zϕ) = Lw p -LU ≤ 0. Routine computation yields

L 1 [z] = L[zϕ] ϕ = ∆z + 2 ∇ y ϕ ϕ • ∇ y z + c∂ 1 z + ∆ϕ + (µ(y) + δ)ϕ ϕ z ≤ 0
Observe that z ≥ 0 when |x 1 | ∈ {R, R + p}, y ∈ ω and ∂ ν z = 0 on ∂ω. Moreover, the zero order coecient of L 1 is negative. Hence, by the maximum principle, we get that

U (x 1 , y) ≤ w p (x 1 , y) = e -(R+p)α/2 e α|x 1 | ϕ(y) + e 3Rα/2 e -α|x 1 | ϕ(y) in Ω R+p \ Ω R .
Letting p → ∞, we obtain

U (x 1 , y) ≤ e 3R -c+ √ c 2 +4λµ-4δ 4 e - -c+ √ c 2 +4λµ-4δ 2 |x 1 | ϕ(y) in Ω. Set α * = -c+ √ c 2 +4λµ 2
Proof of Theorem 1.1.1

Let us rst consider the case c < c * . By Proposition 1.1.3, we know that

λ1 := λ N (-∆ -c∂ 1 -f s (x 1 , y, 0), Ω) < 0.
Thanks to Proposition 1, [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF], we have the limit lim R→∞ λ R = λ1 < 0, where λ R is the unique eigenvalue of problem :

       -∆ϕ R -c∂ 1 ϕ R -f s (x, 0)ϕ R = λ R ϕ R x ∈ Ω R ϕ R (x) > 0 x ∈ Ω R ∂ ν ϕ R (x 1 , y) = 0 |x 1 | < R, y ∈ ∂ω ϕ R (±R, y) = 0 y ∈ ω.
Moreover there exists an eigenfunction ϕ ∞ ∈ W 2,N (Ω) associated with λ1 . Fix R > 0 large enough such that λ R < 0, we dene φ(x) as following :

φ(x) = ϕ R (x) x ∈ Ω R 0 otherwise. Since f (x, s) is of class C 1 [0, s 0 ]
with respect to s, for ε > 0 small enough, we see that

∆(εφ) + c∂ 1 (εφ) + f (x 1 , y, εφ) = εφ -λ R + f (x 1 , y, εφ) εφ -f s (x 1 , y, 0) > 0.
Hence, εφ is a subsolution of Eq. (1.6). Since φ is compactly supported, we can choose ε small such that ε sup φ ≤ S, where S is a super solution of Eq. (1.6) given by (1.7). Therefore, by the classical iteration method, there exists a nonnegative solution U satisfying εφ ≤ U ≤ S. Furthermore, thanks to the strong maximum principle, U is strictly positive . The nonexistence and uniqueness are direct consequences of the following comparison principle. Let U and V be respectively super and subsolutions of (1.6). We will now show that V (x) ≤ U (x) in Ω. Indeed, by condition (1.9), there exists an eigenpair (λ µ , ϕ) of (1.22), where ϕ satises inf ω ϕ > 0 due to the Hopf lemma. On the other hand, Proposition 1.2.1 implies that V decays exponentially as |x 1 | → ∞, uniformly in y, therefore, for any ε > 0, there exist R(ε) > 0 such that V (x 1 , y) ≤ εϕ(y) in Ω \ Ω R(ε) . Then, the set

K ε := {k > 0 : kU ≥ V -εϕ in Ω}, is nonempty. Let us call k(ε) := inf K ε . Obviously, the function k(ε) : R + → R is nonincreasing. Assume by a contradiction k * = lim ε→0 + k(ε) > 1. Take 0 < ε < sup Ω V / sup ω ϕ, we have k(ε) > 0, k(ε)U -V + εϕ ≥ 0. By the denition of k(ε), there exists a sequence (x ε 1,n , y ε n ) in Ω such that k(ε) - 1 n U (x ε 1,n , y ε n ) < V (x ε 1,n , y ε n ) -εϕ(y ε n ). Fix ε > 0, we have (x ε 1,n , y ε n ) ∈ Ω R(ε)
for n large enough, therefore (x ε 1,n , y ε n ) converges up to subsequence to some (x 1 (ε), y(ε)) ∈ Ω R(ε) . This limiting point must satisfy :

(k(ε)U -V + εϕ)(x 1 (ε), y(ε)) = 0.
(1.24)

Without loss of generality, we may assume lim ε→0 + y(ε) = y 0 ∈ ω. The case that there exists x 0 such

that |x 0 | = lim inf ε→0 + |x 1 (ε)| < ∞ is ruled out. Indeed, from (2.15), k * < ∞, the function W = k * U -V is
nonnegative and vanishes at (x 0 , y 0 ). Since f is Lipschitz continuous with respect to second variable and k * > 1, we have

-∆W -c∂ 1 W ≥ k * f (x, U ) -f (x, V ) ≥ f (x, k * U ) -f (x, V ) ≥ z(x)W,
for some function z(x) ∈ L ∞ loc (Ω). Thanks to condition (1.9), this inequality holds strictly in D ⊂ Ω, with |D| > 0. The strong maximum principle implies that W cannot achieve a minimum value in the interior of Ω. This means y 0 ∈ ∂ω, but the Hopf lemma yields another contradiction: ∂ ν W (x 0 , y 0 ) < 0.

It remains to consider the case lim

ε→0 + |x 1 (ε)| = ∞. Set W ε = k(ε)U -V + εϕ, we have W ε ≥ 0 and W ε vanishes at (x 1 (ε), y(ε)). Thus there exists r > 0 such that k(ε)U < V in B r (x 1 (ε), y(ε)) ∩ Ω. For ε small enough, k(ε) > 1, we derive from (1.8) for B r (x 1 (ε), y(ε)) ∩ Ω: (∆ + c∂ 1 )W ε ≤ f (x, V ) -k(ε)f (x, U ) -(µ(y) + λ µ )εϕ ≤ f (x, V ) -f (x, k(ε)U ) -(µ(y) + λ µ )εϕ ≤ - f (x, k(ε)U ) k(ε)U (k(ε)U -V + εϕ) - λ µ 2 εϕ - λ µ 2 + µ(y) - f (x 1 , y, k(ε)U ) k(ε)U εϕ.
(1.25)

Take 0 < ε 1, then |x 1 (ε)|γ1, we have f (x 1 , y, k(ε)U ) k(ε)U < µ(y) + λ µ 2 , ∀(x 1 , y) ∈ B r (x 1 (ε), y(ε)) ∩ Ω,
choosing r smaller if necessary. Since λ > 0, we get from (2.46)

-∆W ε -c∂ 1 W ε -(x)W ε > λ µ 2 εϕ > 0 in B r (x 1 (ε), y(ε)) ∩ Ω, where (x) = f (x,k(ε)U ) k(ε)U
is bounded. The strong maximum principle asserts that (x 1 (ε), y(ε)) cannot be an interior point of Ω. Hence (x 1 (ε), y(ε)) ∈ ∂Ω, but then the Hopf lemma yields another contradiction

∂ ν W ε (x 1 (ε), y(ε)) < 0.
We have proved that k * = lim

ε→0 + k(ε) ≤ 1. Letting ε → 0 + , we derive V ≤ lim ε→0 + (k(ε)U + εϕ) ≤ U in Ω.
The uniqueness of Eq. (1.6) is obviously achieved by exchanging the roles of U and V . We end the proof by showing the nonexistence when c ≥ c * . Assume by contradiction that (1.6) possesses a positive solution U when c ≥ c * . One has λ1 = λ N (-∆ -c∂ 1 -f s (x 1 , y, 0), Ω) ≥ 0. Let ϕ ∞ be a generalized principal eigenfunction with Neumann boundary condition associated with λ1 . Without loss of generality, we may assume that 0 < ϕ ∞ (0) < U (0). We derive, from (1.8), that

-∆ϕ ∞ -c∂ 1 ϕ ∞ = (f s (x 1 , y, 0) + λ1 )ϕ ∞ ≥ f (x 1 , y, ϕ ∞ ) in Ω
By Proposition 1.2.1, U decays exponentially as |x 1 | → ∞ uniformly in y. The above comparison principle implies that U (x) ≤ ϕ ∞ (x) for all x ∈ Ω. This contradiction ends the proof.

Remark 1. The globally unfavorable characterization of the environment near innity plays the key role to derive the uniqueness of (1.6). Indeed, if the environment is favorable at innity, that is λ µ < 0 (see (1.9)), from section 2.1, we see that there exists a positive solution p(y) at innity. If f is homogeneous in the traveling direction, i.e independent of x 1 , then for all a ∈ R, U (x 1 + a, y) are also solutions of (1.6) and thus the uniqueness does not hold.

Long time dynamics

In order to study the long time dynamics of Eq. (1.4), we rst prove the following Liouville type theorem for entire solutions (solutions for all t ∈ R). Consider the evolution problem in the cylindrical domain with Neumann boundary condition

∂ t u * = ∆u * + c∂ 1 u * + f (x, u * ) t ∈ R, x ∈ Ω ∂ ν u * = 0 t ∈ R, x ∈ ∂Ω, ( 1
lim n→∞ t n = +∞, lim inf n→∞ u * (-t n , x 0 ) > 0, (1.27)
then u * (t, x) ≡ U (x), where U (x) is the unique solution of (1.6), given by Theorem 1.1.1.

Proof. Set S * = max{S, u * L ∞ (Ω) }, where S is the positive constant given in (1.7), obviously S * is a super solution of stationary equation of Eq. (1.26). Let v(t, x) be the solution of (1.26) starting by v(0, x) = S * , the parabolic maximum principle and standard estimates imply that v is nonincreasing in t and converges locally uniformly in Ω to a stationary solution V (x) of (1.26). That V (x) solves Eq. (1.6). For any h ∈ R, we dene v h (t, x) = v(t -h, x). This function is a solution of (1.26) in (h, +∞) × Ω and satises v h (h, x) = S * ≥ u * (h, x). The parabolic comparison principle thus yields

0 ≤ u * (t, x) ≤ lim h→-∞ v h (t, x) = V (x) ∀t ∈ R, x ∈ Ω. (1.28)
We consider separately two dierent cases: Case 1. c ≥ c * . Theorem 1.1.1 asserts that the stationary equation of Eq. (1.26) only has zero-solution. Namely, V (x) ≡ 0 in Ω. Therefore, the necessary condition for existence of nontrivial entire solution of (1.26) is c < c * .

Case 2. c < c * and (1.27) holds. Theorem 1.1.1 again asserts that the stationary equation of Eq. (1.26) admits a unique positive solution U . We will prove that u * (t, x) ≡ U (x). Assume by contradiction that there exists x 0 ∈ Ω such that u * (t, x 0 ) = U (x 0 ). We will reach a contradiction by proving the following claim.

Claim. There exist ε ∈ (0, 1] and n 0 ∈ N such that for n ≥ n 0 , one has εU (x) ≤ u * (-t n , x).

Assume for a moment that this claim holds true, the concluding argumentation goes as follows. Thanks to (1.8), for any ε ∈ (0, 1], εU is a subsolution of stationary equation of Eq. (1.26). Let w(t, x) be a solution of (1.26) with initial condition w(0, x) = εU (x) and w n (t, x) = w(t + t n , x). We know, by the standard parabolic estimates, that as t → ∞, w(t, x) is nondecreasing, bounded from above by S * and converges locally uniformly in Ω to the unique stationary solution W (x) of Eq. (1.26). The strict positivity of W is derived from the condition c < c * . By the way of setting, one has w n (-t n , x) = εU (x) ≤ u * (-t n , x). The parabolic comparison principle implies that w n (t, x) ≤ u * (t, x) in (-t n , +∞) × Ω. Therefore, by letting n → ∞, one has

u * (t, x) ≥ lim n→∞ w n (t, x) = W (x) locally in R × Ω.
Combining this inequality with (1.28), we obtain

W (x) ≤ u * (t, x) ≤ V (x), ∀t ∈ R, ∀x ∈ Ω. The uniqueness result of Theorem 1.1.1 yields u * ≡ W ≡ V .
It remains to prove the claim. Assume by contradiction that for all ε ∈ (0, 1] and for all n 0 ∈ N there exist n(ε) > n 0 and

x n(ε) ∈ Ω so that εU (x(n ε )) ≥ u * (-t n(ε) , x n(ε) ). Since U is bounded, choosing a sequence ε k → 0 as k → ∞, by a diagonal extraction, one nds sequences (t k ) ∈ R + and (x k ) ∈ Ω such that t k → +∞ and u * (-t k , x k ) → 0 as k → ∞. We set ũk (t, x) = u * (t + t k , x + x k ).
Obviously, ũk (t, x) is bounded from above by S * and satises the equation

∂ t ũk = ∆ũ k + c∂ 1 ũk + f (x + x k , ũk ) t ∈ R, x ∈ Ω, ∂ ν ũk = 0 t ∈ R, x ∈ ∂Ω
By standard parabolic estimates, we get ũk → ũ∞ (up to subsequences) as k → ∞. Thanks to the Lipschitz continuity of f (x, s) with respect to s, there exists a negative constant -M so that ũ∞ satises the equation

∂ t ũ∞ ≥ ∆ũ ∞ + c∂ 1 ũ∞ -M ũ∞ t ∈ R, x ∈ Ω ∂ ν ũ∞ = 0 t ∈ R, x ∈ ∂Ω.
Moreover, ũ∞ (0, 0, 0) = 0. The strong maximum principle implies that ũ∞ (t, x) = 0, ∀t ≤ 0, x ∈ Ω. Choosing t = -2t k , we get lim k→∞ u * (-t k , x) = 0, ∀x ∈ Ω. This contradicts assumption (1.27) and thus completes the proof.

Remark 2. In this proof, we have used dierent arguments from the ones of Berestycki and Rossi, Lemma 3.4 [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF]. More precisely, we choose a solution of Eq. (1.26), w(t, x) starting by a subsolution of stationary equation εU (x), which is not necessarily compactly supported but bounded. On the other hand, we reach the contradiction by showing that lim k→∞ u * (-t k , x) = 0, ∀x ∈ Ω, which diers from the way to show that for all r > 0, lim inf n→∞,x∈Ωr u * (t n , x) > 0 in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF].

We are now ready to prove Theorem 1.1.4 to derive long time behavior of solution of (1.4) in L ∞ (Ω).

Proof of Theorem 1.1.4. Let S := max{S, u 0 L ∞ (Ω) }, where S is the positive constant in (1.7). Then 0 and S are respectively sub and super solution of (1.4). It follows from [START_REF] Lieberman | Second Order Parabolic Dierential Equations[END_REF], by the standard theory of semilinear parabolic equations, that there exists a unique (weak) solution to (1.4) satisfying 0 ≤ u ≤ S with initial condition u 0 (x). We deduce, from the parabolic strong maximum principle, that u(t, x) > 0 ∀t > 0, x ∈ Ω (by extending u(t, x) to larger cylinder to make the "corner" smooth). The locally long time behavior of u follows by applying directly Theorem 1.2.2 and the standard parabolic estimates. Actually, one sets ũ(t, x 1 , y) = u(t, x 1 + ct, y). The solution of this type satises ũ(0, x) = u 0 (x) and satises

∂ t ũ = ∆ũ + c∂ 1 ũ + f (x, ũ) t ∈ R, x ∈ Ω ∂ ν ũ = 0 t ∈ R, x ∈ ∂Ω. (1.29)
To apply Theorem 1.2.2, we only need to verify condition (1.27) when c < c * . Indeed, the rst case c ≥ c * is easily seen. Let (t n ) be a sequence such that t n → +∞ as n → ∞, we infer, by the parabolic estimates and embedding theorems, that the sequence ũ(t + t n , x) converges (up to subsequences) to some nonnegative bounded solution u * (t, x) of Eq. (1.29) as n → ∞ locally in Ω. By Theorem 1.2.2, this limit is identically equal to 0 when c ≥ c * . Consider the case c < c * , we necessarily verify condition (1.27). Let U be the unique solution of stationary solution of (1.29) and (t n ) be such that t n → -∞ as n → ∞. Fix R > 0, the Hopf lemma implies that inf Ω R ũ(1, x) > 0. For ε > 0, the function εU is a subsolution to stationary equation of (1.29) when ε ≤ 1. Take ε small enough such that εU ≤ ũ(1, x) and in Ω R . Hence (t, x) → εU (x) is a subsolution to (1.29) in R × Ω R . The parabolic comparison yields εU (x) ≤ ũ(t + 1, x) for t > 0 and x ∈ Ω R . As a consequence inf t∈R u * (t, 0, y 0 ) ≥ U (0, y 0 ) > 0, for some y 0 ∈ ω.

It remains to show that the convergences hold uniformly in Ω. Assume by contradiction that

lim t→∞ ũ(t, x) = U (x)
is not uniform in x ∈ Ω. This means that there exist ε > 0, (t n ) ∈ R + and (x 1,n , y n ) ∈ Ω such that

lim t→∞ t n = ∞, |ũ(t n , x 1,n , y n ) -U (x 1,n , y n )| ≥ ε ∀n ∈ N.
Since y n ∈ ω, which is bounded, one may assume that y n converges (up to subsequences) to ζ ∈ ω. The locally uniform convergences yields

lim n→∞ |x 1,n | = ∞, therefore lim n→∞ U (x 1,n , y n ) = 0 in both cases c ≥ c * and c < c * . Then we get lim inf n→∞ ũ(t n , x 1,n , y n ) ≥ ε.
The standard parabolic estimates and compact injections again imply that ũ(t + t n , x 1 + x 1,n , y n ) converges (up to subsequences) to ũ∞ (t, x 1 , ζ) uniformly in (-ρ, ρ) × Ω ρ , for any ρ > 0. In particular, ũ∞ satises ũ∞ (0, 0, ζ) ≥ ε and satises the following equation

∂ t ũ∞ ≤ ∆ũ ∞ + c∂ 1 ũ∞ + µ(y)ũ ∞ t ∈ R, x ∈ Ω ∂ ν ũ∞ = 0 t ∈ R, x ∈ ∂Ω.
(1.30)

By condition (1.9), there exists an eigenpair (λ µ , ϕ) of Eq. (1.22) satisfying λ µ > 0. Setting ω(t, y) = S e -λµ(t+h) ϕ(y), we have

∂ t ω -∆ω -c∂ 1 ω -µ(y)ω ≥ 0.
We know, by the Hopf lemma, that inf ω ϕ(y) > 0. Hence, the function

W (t, x) = ũ∞ (t, x) -ω(t, y) satises W (-h, x) ≤ 0 for S large enough. Let us call L 1 = ∂ t -∆ -c∂ 1 -µ(y), then L 1 W ≤ 0 t ∈ R, x ∈ Ω ∂ ν W ≤ 0 t ∈ R, x ∈ ∂Ω. Set W (t, x) = z(t, x)ϕ(y) we have ∂ ν z ≤ 0 on ∂Ω, z(-h, x) ≤ 0 ∀x ∈ Ω and z satises 0 ≥ L 1 W ϕ = z t -∆z - 2 ϕ ∇ y ϕ • ∇ y z -c∂ 1 z + λ µ z.
Since λ µ > 0, the parabolic maximum principle implies that z ≤ 0 in (-∞, -h) × Ω. As a consequence, one gets

0 < ε ≤ ũ∞ (0, 0, ζ) ≤ lim h→-∞ ω(0, ζ) = 0.
This contradiction concludes the proof.

The next result concerns the long time behavior of the solution of Eq. (1.4) in L 1 (Ω). The central diculty is to deal with the case f s (x, 0) being sign-changing as |x| large. To overcome it, we decompose solution of (1.4) into the sum of two integrable functions. The following lemma plays a key role. Lemma 1.2.3. Let w(t, x) be a nonnegative bounded solution of

∂ t w = ∆w + c∂ 1 w + ζ(t, x)w, t > 0, x ∈ Ω ∂ ν w(t, x) = 0, t > 0, x ∈ ∂Ω (1.31)
with initial function w(0,

•) = w 0 ∈ L 1 (Ω) ∩ L ∞ (Ω).
We assume, in addition, that lim t→∞ w(t, x) = 0, pointwise in x ∈ Ω, ζ(x) ∈ L ∞ (Ω) and that there exists µ ∈ L ∞ (ω) satisfying

µ(y) = lim R→∞ sup t>0 |x 1 |≥R ζ(t, x 1 ,
y), and λ N (-∆ y -µ(y), ω) > 0.

(1.32)

Then, there holds

lim t→∞ w(t, x) L 1 (Ω) = 0.
Proof of Lemma 1.2.3. From Eq. (1.31), for any δ > 0, we have

∂ t w -∆w -c∂ w -(µ(y) + δ)w = (ζ(t, x) -µ(y) -δ)w.

Let us call

P := ∂ t -∆ -c∂ 1 -µ(y) -δ, g(t, x) := (ζ(t, x) -µ(y) -δ)w(t, x).
Then, we infer, from the superposition principle, that w = w 1 + w 2 , where (w 1 , w 2 ) is the solution of the system

P w 1 = 0 t > 0, x ∈ Ω ∂ ν w 1 = 0 t > 0, x ∈ ∂Ω, P w 2 = g(t, x) t > 0, x ∈ Ω ∂ ν w 2 = 0 t > 0, x ∈ ∂Ω,
with the initial condition (w 1 , w 2 )(0, x) = (w 0 (x), 0). From condition (1.32), for any δ > 0, there exist R > 0, such that

ζ(t, x 1 , y) ≤ µ(y) + δ, ∀t > 0, ∀(x 1 , y) ∈ Ω \ Ω R .
and there exists an eigenpair (λ µ , ϕ) of Eq. (1.22) satisfying λ µ > 0.

Letting δ < λ µ , we set v 1 (t, x) = e (λµ-δ)t w 1 (t, x)/ϕ(y). Then v 1 satises the equation

   ∂ t v 1 -∆ x v 1 -2∇ y v 1 • ∇ y ϕ ϕ -c∂ 1 v 1 ≤ 0. x ∈ Ω. ∂ ν v 1 ≥ 0 x ∈ ∂Ω.
(1.33)

By the Hopf lemma, we know that inf y∈ω ϕ > 0, then v 1 (0,

•) L ∞ (Ω) ≤ w 0 L ∞ (Ω) / inf ω ϕ. Then, the parabolic maximum principle yields v 1 L ∞ (Ω) ≤ w 0 L ∞ (Ω) / inf ω ϕ. It follows immediately that w 1 (t, •) L ∞ (Ω) → 0 as t → ∞. On the other hand, set v r 1 (t, ρ, y) = r -r v 1 (t, x 1 + ρ, y)dx 1 , we obtain ∂ t v r 1 -∆v r 1 -2 ∇ y ϕ ϕ • ∇ y v r 1 (t, ρ, y) -c∂ ρ v r 1 ≤ 0 (ρ, y) ∈ Ω. (1.34) Since w 0 (0, •) ∈ L 1 (Ω) ∩ L ∞ (Ω), v r 1 (0, ρ, y
) is well-dened a.e on Ω, ∀r > 0. Moreover, there exists a constant M such that v r 1 (0, ρ, y) ≤ M , a.e in Ω, ∀r > 0, M is a supersolution of Eq. (1.34). Then, we infer from the parabolic comparison principle [START_REF] Lieberman | Second Order Parabolic Dierential Equations[END_REF], that

v r 1 (t, •) ≤ M, a.e in Ω, ∀t > 0, ∀r > 0. Therefore, v 1 (t, •) L 1 (Ω) = lim r→∞ ω r -r v 1 (t, x 1 + ρ, y)dx 1 dy ≤ M |ω|, ∀t > 0. As a consequence, we get lim t→∞ w 1 (t, •) L ∞ (Ω) = lim t→∞ w 1 (t, •) L 1 (Ω) = 0.
On the other hand, by assumption, w is bounded, then w 2 is bounded and so it is integrable on any compact set. The same argumentation as of the Proposition 1.2.1 enables us to nd supersolution of the problem satised by w 2 of the form ξ(x 1 , y) = Ce -τ |x 1 | ϕ(y) such that P (ξ) ≥ 0 ≥ g(t, x) in Ω \ Ω R . We have w 2 (0, x) = 0 < ξ(x), ∂ ν w 2 ≤ ∂ ν ξ for y ∈ ∂ω and the fact that inf ω ϕ > 0 allows us to nd a constant C such that w 2 (±R, y) ≤ Ce -τ R ϕ(y), ∀y ∈ ω. Therefore, the parabolic maximum principle implies that

w 2 (t, x) ≤ ξ(x), ∀x ∈ Ω \ Ω R . Since w 2 is bounded, one can choose C large enough so that w 2 (t, x) ≤ ξ(x), ∀x ∈ Ω. Moreover, ∀x ∈ Ω lim t→∞ w + 2 (t, x) = lim t→∞ (w -w 1 ) + (t, x) = 0.
Hence 0 ≤ w = w 1 + w 2 ≤ w 1 + w + 2 , which is integrable on Ω. It follows from Lebesgue's dominated convergence theorem that lim t→∞ w(t, x) L 1 (Ω) = 0 because lim t→∞ w(t, x) = 0 for x ∈ Ω. We thus conclude the proof.

We are now in a position to prove Theorem 1.1.5

Proof of Theorem 1.1.5. The proof is a direct consequence of Lemma 1.2.3. Let u be the solution of (1.4) with u(0, x) = u 0 (x) ∈ L ∞ (Ω) ∩ L 1 (Ω). The function ũ(t, x 1 , y) := u(t, x 1 + ct, y) satises Eq. (1.31) with the same initial condition u 0 . Let W be dened as following :

W (x) = 0 if c ≥ c * U (x) if c < c * , (1.35)
where U (x) is the unique positive solution of Eq. (1.6) when c < c * . Let u, u be respectively the solutions of (1.31) with initial conditions u(0, x) = max{u 0 (x), W (x)} and u(0, x) = min{u 0 (x), W (x)}. We know, from Theorem 1.1.4 that the functions u(t, x 1 -ct, y) and u(t, x 1 -ct, y) converge to W (x) as t → ∞, uniformly with respect to x ∈ Ω. Moreover, the parabolic maximum principle yields

∀t > 0, x ∈ Ω u(t, x) ≥ max{ũ(t, x), W (x)} u(t, x) ≤ min{ũ(t, x), W (x)}.
Therefore, the functions w(t, x) := u(t, x) -W (x) and w(t, x) := W (x) -u(t, x) is nonnegative bounded solution of Eq. (1.31) with

ζ(t, x) = f (x, u) -f (x, W ) u -W ; ζ(t, x) = f (x, W ) -f (x, u) W -u .
Thanks to condition (1.8), one easily sees that ζ and ζ are less than f s (x, 0). Thanks to condition (1.9), ζ and ζ satisfy (1.32). The initial conditions w(0, x), w(0, x) ∈ L 1 (Ω) allow one to apply Lemma 1.2.3 to derive lim

t→∞ u -W L 1 (Ω) = 0 ; lim t→∞ W -u L 1 (Ω) = 0.
This completes the proof because u ≤ ũ ≤ u.

The next section is of independent interest. We are concerned with the existence, uniqueness, long time behavior of pulsating fronts, which are T-periodic in t and periodic in y.

1.3

The partially periodic environment with time dependence Before proving the main results, let us introduce some new denitions and preliminary results that are needed in this section.

Proposition

1.3.1. Let O r = R × (-r, r) × R N -1
, then for any r > 0, there exists a unique real number λ p such that the eigenvalue problem

   Lχ r = λ p (r)χ r a.e in O r χ r = 0
on ∂O r χ r is periodic both in y and t admits a positive solution χ r (t,

x) ∈ C 1,2 t,x (R × (-r, r) × R N -1
), where L is the parabolic operator of the form dened in (1.15). The function χ r (unique up to a multiplication) is called the principal eigenfunction associated with eigenvalue λ p (r) of L in O r .

Proof. The existence and uniqueness of principal eigenvalue for time periodic operator with Dirichlet boundary condition have been derived in [START_REF] Hess | Periodic-parabolic boundary value problems and positivity[END_REF]. For the framework of space-time periodic operator, one can refer to the work of Nadin [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF] and the proof of this theorem is essentially similar to the proof of Theorem 2.7, in [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF]. We omit the proof here. Proposition 1.3.2. There holds λ p (r) → λ p as r → ∞, strictly decreasing in r, where λ p = λ1 (L, R × R N ) dened in (1.15). Moreover, there exists an eigenfunction χ ∈ C 1,2 t,x (R × R N ) associated with λ1 such that Lχ = λ1 χ a.e in R N +1 .

Proof. Assume by contradiction that there exist 0 < r 1 < r 2 but λ p (r 1 ) ≥ λ p (r 2 ). Thanks to the periodicity in y and t and the boundedness of domain in x 1 , there exist α > 0 and x 0 ∈ O r 1 such that αχ r 1 ≥ χ r 2 and αχ r 1 (x 0 ) = χ r 2 (x 0 ). It follows immediately that

L(αχ r 1 -χ r 2 ) ≥ λ p (r 1 )(αχ r 1 -χ r 2 )
a.e x ∈ O r 1 .

The parabolic strong maximum principle implies αχ r 1 ≡ χ r 2 , contradiction. Moreover, it is easily seen, by strong maximum principle, that λ p (r) > -sup Or f s (t, x, 0) ≥ -sup R×R N f s (t, x, 0), ∀r > 0. This implies that the limit λ p = lim r→∞ λ p (r) does exist and bounded from below by λ p . Let us argue that there exists an eigenfunction χ ∈ C 1,2 t,x (R × R N ) which is periodic in y and t such that Lχ = λ p χ. Indeed, thanks to the periodicity in y and t, one can apply the Harnack inequality for the family (χ(r)) r > 0 with normalization χ r (0, 0) = 1 to derive that it is uniformly bounded on any the compact set of R × R N . Then, the standard parabolic estimates imply that χ r → χ, as r → ∞ locally uniformly in R × R N and χ satises Lχ = λ p χ. Moreover, χ is strictly positive, periodic in y and t and satises χ(0, 0) = 1 by the strong maximum principle. Lastly, taking χ as a test super solution for λ p , one nds λ p = λ p .

We are now able to prove Theorem 1.1.7.

Proof of Theorem 1.1.7. Recall that λ1 = λ1 (P, R × R N ). We rst consider the case that λ1 < 0. It then follows from Proposition 1.3.2 that for r > 0 large enough, λ p (r) < 0, where λ p (r) is the space-time periodic principal eigenvalue on O r , with Dirichlet boundary condition, which is dened in Proposition 1.3.1. Let χ r be an eigenfunction associated to λ p (r) in O r , we dene the function :

φ(t, x) = ηχ r (t, x) x ∈ O r 0 otherwise.
For η ∈ R small enough, one obtains immediately that

∂ t φ -∆φ -c∂ 1 φ -f (t, x, φ) = (f s (t, x, 0) + λ p (r))φ R -f (t, x, φ) < 0.
That is, φ is a subsolution of Eq. (1.14) while the constant S given in (1.17) is a super solution of Eq. (1.14). Let us consider the solution u of Eq. (1.14) with the initial condition u(0, x) = φ(0, x). The standard parabolic theory and maximum principle imply that there exists such a solution for any t > 0 and satises φ(t, x) ≤ u(t, x) ≤ S, ∀(t, x) ∈ R × R N +1 . In particular, φ(T, x) ≤ u(T, x), where T is the period of φ and f with respect to t. Consider the function u(t + T, x); it is also a solution of Eq. (1.14) with initial condition u(T, x) ≥ u(0, x), then u(t + T, x) ≥ u(t, x). In particular, u(2T, x) ≥ u(T, x). By induction, one sees that the sequence u n (t, x) = u(t + nT, x) is nondereasing in n and uniformly bounded by S. Therefore, u n (t, x) converges pointwise to a bounded function [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF]. The partial periodicity in y of solution follows from the construction.

U (t, x) such that U is T-periodic in t, φ ≤ U ≤ S, U solves Eq. (1.
Let us postpone for a moment the proof of the necessary condition to prove the uniqueness of the solution. We emphasize that the uniqueness of (1.14) is proved to hold in the class of positive bounded solutions without a-priori assuming to be periodic in y nor in t.

Assume by contradiction that U and U are two positive bounded solutions of (1.14), Theorem A.2, Appendix, yields lim

|x 1 |→∞ U (t, x 1 , y) = lim |x 1 |→∞ U (t, x 1 , y) = 0,
uniformly in y and t. By condition (1.19), there exists a pair (λ, φ) such that λ > 0, φ > 0, φ is periodic in both y and t and ∂ t φ-∆φ-γ(t, y)φ ≥ λφ in R×R N -1 . Thanks to the periodicity, we have inf R N ϕ(t, y) > 0.

For any ε > 0, there exists R(ε) > 0 such that

U (t, x 1 , y) ≤ εϕ(t, y), ∀ |x 1 | ≥ R(ε), y ∈ R N -1 , t ∈ R, (1.36)
and therefore, the set

K ε := {k > 0 : kU ≥ U -εϕ in R × R N } is nonempty. Set k(ε) := inf K ε . Obviously, the function k(ε) : R + → R is nonincreasing. Assume by way of contradiction that k * = lim ε→0 + k(ε) > 1 Take 0 < ε < sup R N +1 U /ϕ, we see that k(ε) > 0, k(ε)U -U + εϕ ≥ 0. The denition of k(ε) yields that there exists a sequence (t ε n , x ε 1,n , y ε n ) in Ω such that k(ε) - 1 n U (t ε n , x ε 1,n , y ε n ) < U (t ε n , x ε 1,n , y ε n ) -εϕ(t ε n , y ε n ). (1.37) From (2.27), we have (t ε n , x ε 1,n , y ε n ) ∈ O R(ε) for n large enough. Taking the sequences (τ ε n ) and (z ε n ) such that t ε n -τ ε n ∈ [0, T ) and y ε n -z ε n ∈ [0, L 2 ) × ... × [0, L N ). For any ε > 0, one sees that U ε n (t, x 1 , y) = U (t + τ ε n , x 1 , y + z ε n ) and U ε n (t, x 1 , y) = U (t + τ ε n , x 1 , y + z ε n ) are solutions of the following equation ∂ t U ε n -∆U ε n -c∂ 1 U ε n = f (t + τ ε n , x 1 , y + z ε n , U ε n ).
Using the priori estimates of solutions (Theorem A.2, Appendix), we deduce that as n → ∞, up to extractions, U

ε n → U ε ∞ and U ε n → U ε ∞ locally uniformly in R N +1 . Moreover, since f is periodic in y and T-periodic in t, there exist τ ε ∞ , z ε ∞ such that by the standard parabolic estimates, we know that U ε ∞ and U ε ∞ are the solutions of : ∂ t U ε ∞ -∆U ε ∞ -c∂ 1 U ε ∞ = f (t + τ ε ∞ , x 1 , y + z ε ∞ , U ε ∞ ). By passing to the limit in (1.37), W ε ∞ = k(ε)U ε ∞ -U ε ∞ +εϕ ε ∞ ≥ 0, where ϕ ε ∞ (t, y) = lim n→∞ ϕ(t+τ ε ∞ , y+z ε n ) satisfying ∂ t φ ε ∞ -∆φ ε ∞ -γ ε ∞ (t, y)φ ε ∞ ≥ λφ ε ∞ , with γ ε ∞ (t, y) = γ(t + τ ε ∞ , y + z ε ∞ ), φ ε
∞ is periodic in y and t, and λ > 0. Moreover, there exist (t(ε),

x 1 (ε), y(ε)) such that (k(ε)U ε ∞ -U ε ∞ + εϕ ε ∞ )(t(ε), x 1 (ε), y(ε)) = 0. (1.38) Note that t(ε) ∈ [0, T ) and y(ε) ∈ [0, L 2 ) × ... × [0, L N ) are bounded with respect to ε. The case that lim inf ε→0 + |x 1 (ε)| < ∞ is ruled out. Indeed, if lim inf ε→0 + |x 1 (ε)| < ∞, there exists a sequence (ε n ) → 0 as n → ∞ such that (t(ε n ), x 1 (ε n ), y(ε n )) → (t 0 , x 0 , y 0 )
as n → ∞, up to subsequences. Moreover, by the partial periodicity in y and the periodicity in t of f , the standard parabolic estimates yield that

U εn ∞ → U 0 ∞ and U εn ∞ → U 0 ∞ locally uniformly in R N +1 satisfying the following equation ∂ t U 0 ∞ -∆U 0 ∞ -c∂ 1 U 0 ∞ = f (t + t 0 ∞ , x 1 , y + z 0 ∞ , U 0 ∞ ). for some t 0 ∞ ∈ [0, T ) and z 0 ∞ ∈ [0, L 2 )×...×[0, L N ). From (2.28) k * < ∞, then the function W = k * U 0 ∞ -U 0 ∞
is nonnegative and vanishes at (t 0 , x 0 , y 0 ). Since k * > 1, The Lipschitz continuity of f with respect to s and condition (1.18) yield

∂ t W -∆W -c∂ 1 W ≥ k * f (t + t 0 ∞ , x 1 , y + z 0 ∞ , U 0 ∞ ) -f (t + t 0 ∞ , x 1 , y + z 0 ∞ , U 0 ∞ ) ≥ f (t + t 0 ∞ , x 1 , y + z 0 ∞ , k * U 0 ∞ ) -f (t + t 0 ∞ , x 1 , y + z 0 ∞ , U 0 ∞ ) ≥ z(t + t 0 ∞ , x 1 , y + z 0 ∞ )W, (1.39)
where

z(t, x) ∈ L ∞ loc (R × R N ).
Hence, the parabolic strong maximum principle implies W = 0 in (-∞, t 0 ) × R N . This is a contradiction because from condition (1.18), the inequality (1.39) hold strictly in some

D ⊂ (-∞, t 0 ) × R N with |D| > 0. It remains to consider the case lim ε→0 + |x 1 (ε)| = ∞. We have shown that W ε ∞ ≥ 0 and W ε ∞ vanishes at (t(ε), x 1 (ε), y(ε)), we infer that there exists a neighborhood O of (t(ε), x 1 (ε), y(ε)) such that k(ε)U < U in O, shrinking O if necessary. Since k * > 1, for ε small enough, k(ε) > 1, we derive from (1.18) for x ∈ O ∂ t W ε ∞ -∆W ε ∞ -c∂ 1 W ε ∞ ≥ k(ε)f (t + τ ε ∞ , x 1 , y + z ε ∞ , U ε ∞ ) -f (t + τ ε ∞ , x 1 , y + z ε ∞ , U ε ∞ ) + (γ(t + τ ε ∞ , y + z ε ∞ ) + λ)εϕ ε ∞ ≥ f (t + τ ε ∞ , x 1 , y + z ε ∞ , k(ε)U ε ∞ ) -f (t + τ ε ∞ , x 1 , y + z ε ∞ , U ε ∞ ) + (γ(t + τ ε ∞ , y + z ε ∞ ) + λ)εϕ ε ∞ ≥ f (t + τ ε ∞ , x 1 , y + z ε ∞ , k(ε)U ε ∞ ) k(ε)U ε ∞ (k(ε)U ε ∞ -U ε ∞ + εϕ ε ∞ ) + λ 2 εϕ ε ∞ + λ 2 + γ(t + τ ε ∞ , y + z ε ∞ ) - f (t + τ ε ∞ , x 1 , y + z ε ∞ , k(ε)U ε ∞ ) k(ε)U ε ∞ εϕ ε ∞ . (1.40)
Condition (1.19) yields, for ε small enough, that

f (t + τ ε ∞ , x 1 , y + z ε ∞ , k(ε)U ε ∞ ) k(ε)U ε ∞ < γ ε ∞ (t, y) + λ 2 , ∀(x 1 , y) ∈ O.
Then, it follows from (1.40) that

∂ t W ε ∞ -∆W ε ∞ -c∂ 1 W ε ∞ -(t + τ ε ∞ , x 1 , y + z ε ∞ )W ε ∞ > λ 2 εϕ ε ∞ > 0 in O, where (t, x) = f (t,x,k(ε)U ) k(ε)U
is bounded. This is a contradiction because the strong maximum principle

asserts that W ε (t, x 1 , y) = 0 in O. As a consequence, we have proved that k * = lim ε→0 + k(ε) ≤ 1. Therefore U ≤ lim ε→0 + (k(ε)U + εϕ) ≤ U in R N +1 .
We derive the uniqueness due to the equivalence of U and U . Note that we do not use the periodicity of y and t of solution in the proof of uniqueness. This thus infers that any positive bounded solutions of Eq.

(1.14) must be periodic in y and T-periodic in t.

To conclude the proof of Theorem 1.1.7, it only remains to prove the necessary condition. Assume by contradiction that λ1 ≥ 0 and Eq. (1.14) admits a solution U , which is T-periodic in t but not necessarily periodic in y. Let χ be a principal eigenfunction associated with λ1 (Proposition 1.3.2) with normalization χ(0, 0) < U (0, 0). Then

∂ t χ -∆χ -c∂ 1 χ -f (t, x, χ) = λ1 χ + f s (t, x, 0)χ -f (t, x, χ) ≥ 0.
Arguing similarly as the proof of uniqueness, we achieve the contradiction :

U ≤ χ in R N +1 .
Before investigating the long time behavior, we point out that the monotonicity in time of solutions starting by a stationary sub (or super) solution of parabolic operator with time-dependent coecients no longer holds. In addition, the boundedness of initial datum does not suce to guarantee that the solutions of Eq. (1.13) converge uniformly to the unique solution of Eq. (1.14) as t → ∞. However, thanks to the periodicity in t of solutions, which obtained by the uniqueness, we will have the locally uniform convergence and under some extra conditions (part (ii), Theorem 1.1.8) we can actually derive the uniform convergence as t → ∞.

Proof of Theorem 1.1.8. Set S := max{S, u 0 L ∞ (Ω) }, S is the positive constant given in (1.17). Then, the function ũ(t, x) = u(t, x 1 + ct, y) satises 0 < ũ ≤ S in R + × R N and solves ∂ t ũ = ∆ũ + c∂ 1 ũ + f (t, x, ũ) t > 0, x ∈ R N , (1.41)
with initial condition ũ(0, u) = u 0 (x). Let w be the solution to (1.41) with initial condition w 0 (x) = S . Clearly, the constant S is T-periodic in t and periodic in y. Arguing as the proof of Theorem 1.1.7, we deduce that the sequence w n (t, x) = w(t + nT, x) is nonincreasing and converges locally uniformly to W (t, x), which is a solution of

∂ t W -∆W -c∂ 1 W -f (t, x, W ) = 0 ∀t > 0, x ∈ R N . (1.42)
Moreover, W (t, x) is T periodic in t and periodic in y. Then

∀r > 0, lim t→∞ sup x∈Or (ũ(t, x) -W (t, x)) ≤ lim t→∞ sup x∈Or (w(t, x) -W (t, x)) = 0. If λ ≥ 0, then W ≡ 0 in R × R N . Therefore, ũ(t, x) → 0 as t → ∞ locally uniformly with respect to x ∈ R N .
This convergence is uniform in x ∈ R N due to the following claim. Claim. lim

min(t,|x 1 |)→∞ ũ(t, x 1 , y) = 0 uniformly in y ∈ R N -1 . (1.43)
Let us postpone for a moment the proof of claim to consider the case λ1 < 0. From Propositions 1.3.1 and 1.3.2, there exists ρ > 0 such that λ p (ρ) < 0. Let χ ρ (t, x) is an associated principal eigenfunction of λ p (ρ), for γ > 0 small enough, one sees that the function

V (t, x) = γχ ρ (t, x) x ∈ O ρ 0 otherwise
is a subsolution of Eq. (1.42). Then if (1.20) holds, there exist γ small enough such that V (0, x) ≤ u 0 (x).

Alternatively, u 0 (x) is periodic in y, then ũ(t, x) is strictly positive, periodic in y and T-periodic in t, then the parabolic strong maximum principle yields V (T, x) ≤ ũ(T, x). In both case, we always can dene ṽ(t, x) is such that ṽ(0, x) = V (0, x) or ṽ(T, x) = V (T, x). It follows immediately by parabolic maximum principle that ṽ(t, x) ≤ u(t, x), ∀t > T, x ∈ R N . Arguing similarly as the proof of Theorem 1.1.7, we deduce that the sequence v n (t, x) = ṽ(t + nT, x) is nondecreasing and converges locally uniformly to P (t, x), which is a solution of

∂ t P -∆P -c∂ 1 P -f (t, x, P ) = 0 ∀t > 0, x ∈ R N .
Moreover, P is strictly positive, periodic in y and T-periodic in t. The uniqueness of Theorem 1.1.7 implies that W = P = U in R N +1 , which is a solution of Eq. (1.14). Assume by contradiction that this convergence is not uniform in x, this means that there exist ε > 0 and a sequence

(t n , x 1,n , y n ) ∈ R + × R N such that lim n→∞ t n = ∞, ∀n ∈ N, |ũ(t n , x 1,n , y n ) -U (x 1,n , y n )| ≥ ε. (1.44)
Due to the locally uniform convergence of ũ, necessarily, the sequence (x 1,n ) is unbounded. We get, by the a priori estimates of U , that U (x 1,n , y n ) → 0 as n → ∞, uniformly in y. But from (1.44), this inference contradicts the claim (1.43). Thus, to conclude the proof, it remains to prove the Claim 1.43.

Let us call (t n , x 1,n , y n ) ∈ R + × R N be a sequence such that the claim (1.43) is not true :

lim n→∞ t n = lim n→∞ x 1,n = ∞, ∀n ∈ N, lim inf n→∞ ũ(t n , x 1,n , y n ) ≥ ε for some ε > 0.
For any n ∈ N, we dene the functions ũn (t, x) = ũ(t + t n , x 1 + x 1,n , y + y n ). It holds that 0 ≤ ũn ≤ S and

∂ t ũn = ∆ũ n + c∂ 1 ũn + f (t + t n , x 1 + x 1,n , y + y n , ũn ) t > -t n , x ∈ R N .
Since f is periodic in y and T-periodic in t, we can assume without loss of generality that t n → t 0 and y n → y 0 (up to a subsequence) as n → ∞. Thanks to (1.18)-(1.19), we deduce, by the parabolic estimates and embedding theorems that ũn converges (up to a subsequence) to ũ∞ locally uniformly in

R × R N satisfying ∂ t ũ∞ ≤ ∆ũ ∞ + c∂ 1 ũ∞ + γ(t + t 0 , y + y 0 )ũ ∞ ∀t ∈ R, x ∈ R N
and ũ∞ (0, 0) ≥ ε. Moreover, condition (1.19) implies that there exists a pair (λ, φ) such that ∂ t φ-∆φ-γ(t+ t 0 , y + y 0 )φ ≥ λφ, φ is periodic in y and t, and λ > 0. Let us dene the function ω(t, x) = S φ(t, y)e -λ(t-h) we have

∂ t ω ≥ ∆ω + c∂ 1 ω + γ(t + t 0 , y + y 0 )ω (t, x) ∈ R N +1 . Then, set L = ∂ t -∆ -c∂ 1 -γ(t + t 0 , y + y 0 ), W (t, x) = ω -ũ∞ and W (t, x) = z(t, x)φ(t, y), we obtain 0 ≤ LW φ = z t -∆z -∇ y z • ∇ y φ φ + λz.
Since the periodicity yields inf R N φ(t, y) > 0 and ũ ≤ S, we can enlarge S so that z(h, x)

≥ 0 in R N . The parabolic comparison principle yields z(t, x) ≥ 0, ∀t ≤ h, x ∈ R N . As a consequence ε ≤ ũ(0, 0) = S φ(0, 0)e λh .
Letting h to -∞, we get a contradiction. This concludes the proof. Note that, if Ω is bounded, λ D coincides with the classical Dirichlet eigenvalue ( [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]). Similar to assumption (1.9), we assume that there exists a measurable bounded function µ : ω → R such that

µ(y) = lim sup |x 1 |→∞ f s (x 1 , y, 0) and λ D (-∆ y -µ(y), ω) > 0. (1.46)
Then, we obtain an analogue of Theorem 1.1.1 as following Theorem 1.4.1. Assume that conditions (1.7)-(1.8) and (1.46) are satised. Then there exists a unique critical speed c * such that the equation

       ∆U + c∂ 1 U + f (x, U ) = 0 in Ω U = 0 on ∂Ω U > 0 in Ω U is bounded. (1.47)
admits a solution if and only if 0 ≤ c < c * . Moreover, the solution is unique when it exists.

Proof. The proof of this theorem is essentially similar to the proof of Theorem 1.1.1. However, there are signicant dierences to be outlined here: i) Existence.

Since the problem is set up with the Dirichlet boundary condition, the Proposition 1, in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF], cannot be applied. However, the Dirichlet boundary condition allows us to use Theorem 1.9, [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] to prove the existence of Eq. (1.47). Indeed, let us call λ D = λ D (-∆ -c∂ 1 -f s (x, 0), Ω) and let c * be dened as Denition 1.2, then the Proposition 1.3 yields that λ D < 0 i 0 ≤ c < c * . Let (λ R , ϕ R ) be the Dirichlet principal eigenvalue and eigenfunction of the problem

   -∆ϕ R -c∂ 1 ϕ R -f s (x, 0)ϕ R = λ R ϕ R x ∈ Ω R ϕ R (x 1 , y) = 0 |x 1 | < R, y ∈ ∂ω ϕ R (±R, y) = 0 y ∈ ω,
we deduce, by Theorem 1.9 in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF], that λ D = lim R→∞ λ R . Note that Theorem 1.9 in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] also deals with the case of nonsmooth domain as the set Ω R of ours. Then the existence of Eq. (1.47) can be obtained in the same way with Theorem 1.

           ∆ Ũ + f (x 1 , y, Ũ (x 1 , y)e -c 2 x 1 )e c 2 x 1 - c 2 4 Ũ = 0 x ∈ Ω Ũ = 0 x ∈ ∂Ω Ũ > 0 in Ω Ũ (x 1 , y)e -c 2 x 1 is bounded.
(

The argument of Theorem 1.1.1 for the nonexistence result can be applied if one can prove that Ũ decays exponentially as |x 1 | → ∞. Moreover, if Ũ decays exponentially, the uniqueness can be obtained by using variational argument as Theorem 2.3 in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF]. Note that the sliding argument as of Theorem 1.1.1 does not work in this case due to the lack of the Hopf lemma under Dirichlet condition. We end the proof by showing that Ũ really decays exponentially.

From assumption (1.46), we know that for any δ > 0 there exists R = R(δ) > 0 such that f s (x 1 , y, 0) ≤ µ(y) + δ when |x 1 | ≥ R. Since ω is bounded, λ D (-∆ y -µ(y), ω) coincides with the classical Dirichlet principal eigenvalue in ω, says λ. There exists an eigenfunction φ ∈ L ∞ (ω), associated with λ, positive in ω, such that ∆ y φ + µ(y)φ + λφ = 0 in ω ; φ = 0 on ∂ω.

One can actually nd a function φ > 0 in ω such that ∆ y φ + (μ(y) + λ) φ = 0 in ω, with μ(y) suciently close to µ(y) and sup ω |μ -µ| < δ, which is chosen later. Set L + δ = ∆ + µ(y) -c 2 /4 + δ, conditions (1.8) and (1.46) yield that for any δ > 0 there exists

R = R(δ) > 0 such that (L + δ) Ũ ≥ 0 in Ω \ Ω R . On the other hand, we set w(x) = Cθ a (x 1 ) φ(y)
, where θ a is the solution of equation

   θ a = (κ + δ)θ a in (R, R + a) θ a (R) = Ce √ κR θ a (R + a) = Ce √ κ(R+a) , with κ = λ/2 + c 2 /4 -2δ and C = sup Ω Ũ (x)e -c 2 x 1 / inf ω φ. Choosing δ < λ/2, direct computation yields (L + δ)w ≤ 0 in Ω \ Ω R . The same argumentation of Proposition 1.2.1 enables us to conclude that U (x 1 , y) ≤ Ce -( √ κ-c/2)|x 1 | φ(y) in Ω.
This concludes the proof of theorem.

By this preliminary, we are ready to present the main result of this section.

Concentration of species in the more favorable region

Based on the characterizations of the persistence and extinction of the species in the cylindrical domain Ω under Dirichlet boundary condition, we study the behavior of the species when a part of their habitat changes to be extremely unfavorable. More precisely, we consider Eq. (1.47) in the whole space R N with two disjointed regions : the cylindrical domain Ω as in Subsection 4.1 and its complement Ω c = R N \ Ω. We shall prove that the species concentrate in the more favorable zone, Ω, and the annihilation occurs in the dead zone Ω c if the death rate in Ω c becomes extremely high. Our goal is to characterize the limit of the sequence U n (x), which are solutions of the equations

∆U n + c∂ 1 U n + F n (x, U n ) = 0 x ∈ R N , U n > 0 and bounded in R N .
(1.49)

For any n, the nonlinearities F n (x, s) are assumed to be continuous with respect to x and of class C 1 with respect to s, F n (x, 0) = 0, ∀x ∈ R N . Moreover, F n (x, s) are assumed to satisfy

∃S > 0 such that F n (x, s) ≤ 0 for s ≥ S, ∀x ∈ R N , (1.50) s → F n (x, s)/s is nonincreasing a.e in R N and there exist D ⊂ R N , |D| > 0 such that it is strictly decreasing in D. (1.51) Let f (x, s) : Ω × [0, +∞) → R satisfy (1.7), (1.8), (1.46)
, ρ n (x) = ∂Fn ∂s (x, 0), we assume further that:

   F n (x, s) = f (x, s) x ∈ Ω, s ∈ R + for all n ∈ N F n (x, s) and ρ n (x) are nonincreasing in n ∀(x, s) ∈ Ω × R + ρ n (x) → -∞ as n → ∞
locally uniformly in Ω c .

(1.52)

Before stating the result, let us briey explain the meaning of this condition and of our achievement. This condition means that the environment of the species outside Ω is unfavorable and it becomes extremely unfavorable as n → ∞. Our result conrms that no species can persist outside Ω under such condition as n → ∞. As is proved in Subsection 4.1 that the species is persistent in Ω if and only if 0 ≤ c < c * . In the following result, we will see that as n → ∞ the species can only persist in Ω and it is immediately mortal outside Ω. Moreover, we will prove that the limit as n → ∞ coincides with the unique solution of Eq. (1.47) in Ω and zero in Ω c . We derive the following result :

Theorem 1.4.2. Let U n (x) be the sequence of of traveling front solution of Eq. (1.49) with

F n (x, s) satises (1.50)-(1.52). If c < c * with c * is given in Theorem 1.
4.1, then the following limit holds

U n (x) → U ∞ (x) as n → ∞, uniformly for x ∈ R N , where U ∞ ∈ W 2,N (R N ) is nonnegative, vanishing in Ω c
and coincides with the unique positive solution of the following equation

∆U + c∂ 1 U + f (x, U ) = 0 x ∈ Ω U (x) = 0 x ∈ ∂Ω. (1.53)
Proof. Some arguments in the proof are inspired from [START_REF] Guo | Propagation and blocking in periodically hostile environments[END_REF].

Let us call L n = ∆ + c∂ 1 + ρ n (x), dened in R N , and the generalized principal eigenvalue of L n as following

λ n = sup{λ ∈ R : ∃φ ∈ W 2,N loc (R N ), φ > 0, (L n + λ)φ ≤ 0 a.e in R N }.
We also denote by λ D the generalized Dirichlet principal eigenvalue of the operator ∆ + c∂ 1 + f s (x, 0) in Ω dened by (1.45). We have the following lemma:

Lemma 1.4.3. There holds that λ n converges increasingly to λ D .

Let us postpone the proof of this lemma for a moment to continue the proof of theorem. As is known, c < c * if and only if λ D < 0. By Lemma 1.4.3 and assumption, we have λ n < λ D < 0. Then Theorem 1.1 [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] yields that the equation

∆U n + c∂ 1 U n + F n (x, U n ) = 0 x ∈ R N 0 ≤ U n ≤ S x ∈ R N . (1.54) admits a strictly positive solution U n . Let V n (x 1 , y) = U n (x 1 , y)e c 2 x 1 , then U n is a solution of Eq. (1.54) if and only if V n is a solution of    ∆V n + F n (x 1 , y, V n (x 1 , y)e -c 2 x 1 )e c 2 x 1 - c 2 4 V n = 0 x ∈ R N V n (x 1 , y)e -c 2 x 1 is bounded.
(

For any n ∈ N, one has lim sup

|x|∈Ω c ,|x|→∞
ρ n (x) < 0, The Proposition 4, [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] yields that V n (x) decays exponentially for x ∈ R N \ Ω and since F n (x, s) = f (x, s) satises condition (1.46) in Ω, Theorem 1.4.1 yields that V n (x) also decays exponentially for x ∈ Ω. Thanks to condition (1.51), we derive, by Theorem 1.1 of [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], that

U n (x) is unique. Moreover, since F n is nonincreasing, for m, k ∈ N, k ≥ m, one has ∆U m + c∂ 1 U m + F k (x, U m ) = -F m (x, U m ) + F k (x, U m ) ≤ 0, x ∈ R N .
Thus U m is a supersolution of equation satised by U k . One can apply the comparison principle, Theorem 2.3 [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], to imply that

U k ≤ U m in R N for k ≥ m.
Then U n is nonincreasing with respect to n and converges pointwise to a nonnegative function U ∞ ≤ S. We will prove now that U ∞ = 0 in Ω c . From above arguments, V n (x) decays exponentially as |x| → ∞. Multiplying V n (x) to Eq. (1.55), we derive, by applying the Stokes formula, that

R N ∇V n • ∇V n = R N F n (x 1 , y, V n e -c 2 x 1 )e c 2 x 1 V n - c 2 4 V 2 n ≤ R N F 0 (x 1 , y, V n e -c 2 x 1 )e c 2 x 1 V n ≤ ≤ max x∈R N ∂ s F 0 (x, 0) R N V 2 0 (x) ≤ M < ∞.
This implies, by Lesbesgue monotone convergence theorem, that the sequence V n converges monotonically to some

V ∞ ∈ H 1 (R N ) as n → ∞, weakly in H 1 (R N )
and strongly in L 2 (R N ). Moreover, taking an arbitrary compact set K ⊂ Ω c , one gets

-(max K ρ n ) K V 2 n ≤ - K ρ n V 2 n ≤ - K F n (x, V n e -c 2 x 1 )e c 2 x 1 V n = - R N |∇V n | 2 - R N c 2 4 V n + R N \K F n (x, V n e -c 2 x 1 )e c 2 x 1 V n ≤ R N \K F n (x, V n e -c 2 x 1 )e c 2 x 1 V n ≤ M
Then, from assumption (1.52), we have max

K ρ n → -∞, ∀K ⊂ Ω c , whence V ∞ = 0 for all compact set in Ω c . This implies V ∞ = 0 a.e in Ω c or in the other words U ∞ = 0 a.e in Ω c . As a consequence, the restriction of U ∞ in Ω belongs to H 1 0 (Ω). Moreover, since F n (x, s) = f (x, s)
in Ω, we have

∆U n + c∂ 1 U n + f (x, U n ) = 0 x ∈ Ω.
The standard elliptic estimates yield that U n → U ∞ as n → ∞ locally uniformly in Ω and moreover U ∞ is a solution of the same equation in Ω in the weak H 1 0 (Ω) sense. Thanks to Theorem 1.4.1, we know that U ∞ is unique and

U ∞ (x) → 0 as |x 1 | → ∞, uniformly in y ∈ ω. Assume by contradiction that the convergence U n → U ∞ as n → ∞ is not uniform, one nds a positive constant ε > 0 and a sequence (x k ) ∈ R N such that for some n 0 ∈ N, |U n (x k ) -U ∞ (x k )| ≥ ε, ∀n ≥ n 0 .
Since the convergence is already locally uniform in R N , we deduce that |x n | → ∞ as n → ∞. Since U n is nonincreasing with respect to n, we deduce that for some 0 < ε 1 < ε, one has

U n 0 (x k ) ≥ ε 1 + U ∞ (x k ).
This is a contradiction since we know that U ∞ (x), U n 0 (x) → 0 as |x| → ∞.

Lastly, to conclude the proof, it remains to prove Lemma 1.4.3, λ n λ D . To this end, we rst show that λ n < λ D , ∀n ∈ N. Since ρ n (x) is nonincreasing in n, one sees that λ n is nondecreasing in n. Assume by contradiction that λ n ≥ λ D for some n. Let us denote by ϕ n and ϕ respectively the principal eigenfunctions associated with λ n and λ D , it holds that :

∆ϕ n + c∂ 1 ϕ n + f s (x, 0)ϕ n = -λ n ϕ n ≤ -λ D ϕ n , x ∈ Ω,
and ϕ n > 0 in Ω. Note that the existence of a positive eigenfunction associated with the generalized principal eigenvalue λ n in unbounded domain is given in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]. Because ϕ n is a supersolution of equation satised by ϕ, if there exists 0 < κ < +∞ such that κϕ ≤ ϕ n in Ω, one can enlarge κ until κϕ touches ϕ n from below at some point. The strong maximum principle implies κϕ ≡ ϕ n in Ω, which is impossible because ϕ = 0 on ∂Ω. In other words, sup{κ ∈ (0, +∞], κϕ ≤ ϕ n in Ω} = +∞. This yields another contradiction since ϕ > 0 in Ω. As a result, λ n < λ D , ∀n ∈ N. Next, we aim to show the limit lim n→∞ λ n = λ D . Since λ n is nondecreasing and bounded from above, there exists

λ ∞ = lim n→∞ λ n ≤ λ D . We shall prove that λ ∞ = λ D .
Observe that, by the transformation φn = ϕ n e c 2 x 1 , we see that φn satises the equation

∆ φn + ρ n (x) φn - c 2 4 φn + λ n φn = 0, x ∈ R N . (1.56)
Let us show that φn decays exponentially. Indeed, as in the proof of Theorem 1.4.1, let φ be the function such that inf ω φ > 0 and ∆ y φ+ μ(y) φ = 0 in ω, with μ(y) suciently close to µ(y)-λ, where λ = λ D (-∆ y -µ(y), ω) > 0. Then, one sees that, (λ n , φn ) is the principal eigenpair of the operator Ln = ∆ + ρ n (x) -c 2 /4 if and only if (λ n , φ n ), with φ n = φn / φ, is the principal eigenpair of the following operator

∆ + 2∇ φ • ∇ φ + c∂ 1 + ρ n (x) φ -µ(y) φ -λ φ φ .
Beside that, by assumptions (1.46) and (1.52), one has

lim sup |x|→∞ ρ n (x) φ -µ(y) φ -λ φ φ < 0 < -λ n . (1.57)
Hence, applying the Proposition 1.11 [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF], we know that λ n is simple, moreover φ n is unique (up to multiplications) and decays exponentially as |x| → ∞. It follows immediately that φn also decays exponentially.

On the other hand, since λ n < λ D < 0 and ρ n (x) is nonincreasing in n, one has

∆ φn + ρ 0 (x) φn ≥ 0.
From above, φn is bounded and φn solves linear equation (1.56), we can normalize ϕ n in such the way sup R N ϕ n = 1. Thanks to conditions (1.46) and lim sup x∈Ω c ,|x|→∞ ρ 0 (x) < 0, the same argumentation of Proposition 8.6 in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] may be applied to derive that there exists an exponential decay function ϕ depending only on ρ 0 (x) such that φn ≤ ϕ in R N . From the equation (1.56), one has

R N |∇ φn | 2 ≤ R N ρ n (x) φ2 n (x) + λ n R N φ2 n (x) ≤ R N ρ 0 (x)ϕ 2 (x) < ∞ (1.58)
This implies that there exists ϕ ∞ ∈ H 1 (R N ) such that φn converges up to subsequence to ϕ ∞ weakly in H 1 (R N ) and strongly in L 2 (K) for all compact set K ⊂ R N . For any compact set K ⊂ Ω c , we derive from (1.58)

-max K ρ n K φ2 n dx ≤ - K ρ n φ2 n dx ≤ - R N |∇ φn | 2 + R N \K ρ n φ2 n dx ≤ sup R N ρ 0 R N ϕ 2 dx.
Since, from (1.52) for all K ⊂ Ω c , -max K ρ n → ∞ as n → ∞, we have ϕ ∞ = 0 a.e in K and then a.e in Ω c . Lastly, again from (1.56), one has

R N |∇ φn | 2 ≤ R N ρ n (x) φ2 n (x) + λ n - c 2 4 ≤ R N ρ 0 (x) φ2 n (x)dx + λ ∞ - c 2 4 .
Since φn ≤ φ, we derive, by Lebesgue dominated convergence theorem that

R N ρ 0 φ2 n (x)dx → R N ρ 0 ϕ 2 ∞ (x)dx = Ω f s (x, 0)ϕ 2 ∞ (x)dx.
Whence, the lower semicontinuity property yields

Ω |∇ϕ ∞ | 2 = R N |∇ϕ ∞ | 2 ≤ lim inf n→∞ R N |∇ φn | 2 ≤ Ω f s (x, 0)ϕ 2 ∞ (x)dx + λ ∞ - c 2 4 (1.59)
By the Liouville transformation, λ D is the principal eigenvalue of a self-adjoint operator. We know from [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] that it has a variational structure.

λ D = inf w∈C 1 c (Ω), w L 2 (Ω) =1 Ω |∇w| 2 -f s (x, 0)w 2 dx + c 2 4 . Since C 1 c (Ω) is dense in H 1 0 (Ω) and ϕ ∞ ∈ H 1 0 (Ω), there exists a sequence w n ∈ C 1 c (Ω) of L 2 (Ω)-norm equal to 1 converges to ϕ ∞ in H 1 (Ω). Combining with (1.59), we derive λ D ≤ Ω |∇w n | 2 -f s (x, 0)w 2 n dx + c 2 4 → Ω |∇ϕ ∞ | 2 -f s (x, 0)ϕ 2 ∞ dx + c 2 4 ≤ λ ∞ . ( 1
λ D = inf w∈H 1 0 (Ω), w L 2 (Ω) =1 Ω |∇w| 2 -f s (x, 0)w 2 dx + c 2 4 .
By assumption (1.46) and the same arguments as (1.56)-(1.57), we can actually prove that λ D is simple. This proof is referred to Proposition 1.11 [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]. In this case, due to the density of C 1 c (Ω) in H 1 0 (Ω), the inma of variational characterization of λ D taken over H 1 0 (Ω) and C 1 c (Ω) are equivalent and equal to the unique λ D . This fact is not true in general, especially when (1.46) does not hold. Here, thanks to linear structure of Eq. (1.56), we can normalize φn of the sup-norm equal 1. Therefore, we can nd a uniform exponential decay, ϕ ≥ φn , depending only on ρ 0 (x), which helps us to compensate the lack of compactness of Ω and R N .

The last result is concerned with a further qualitative property of the fronts of Eq. (1.6), namely the symmetry breaking in x 1 axis. To this aim, the monotonicity and exact asymptotic behavior of the fronts play the crucial role. From Proposition 1.2.1, we know that the fronts U (x 1 , y) decay exponentially as x → ±∞. Therefore, natural questions may arise, which are the right conditions such that the fronts are monotone when |x 1 | large enough and whether they are symmetric in x 1 axis. These questions are addressed in the following by studying the asymptotic behavior of solutions as x 1 → ±∞.

Symmetry breaking of the fronts

Theorem 1.4.4. Let U be a traveling front solution of Eq. (1.6) with f is such that

|f s (x 1 , y, 0) -α(y)| = O(e px 1 ) as x 1 → -∞, and λ α = λ N (-∆ y -α(y), ω) > 0 |f s (x 1 , y, 0) -β(y)| = O(e -qx 1 ) as x 1 → +∞, and λ β = λ N (-∆ y -β(y), ω) > 0 (1.61)
uniformly in y ∈ ω, for some α, β ∈ L ∞ (ω), p, q > 0. We assume further that s → f (x, s) ∈ C 1,r (0, δ) for some r, δ > 0. Then, U is asymmetric if

λ β = λ α + c 2 -2c λ α + c 2 4 ,
where c is the given forced speed of traveling front.

Proof. We investigate at rst the precise asymptotic behavior of solution of Eq.(1.6) on the branch Ω -, By analogy, we derive also the asymptotic behavior on the branch Ω + . According to Proposition 1.2.1 and Theorem A.1, Appendix, for any δ > 0, we have shown that

C 2,δ e κ δ x 1 ≤ U (x 1 , y) ≤ C 1,δ e τ δ x 1 ∀(x 1 , y) ∈ Ω -, (1.62)
where

κ δ = λ α + δ + c 2 4 - c 2 and τ δ = λ α -δ + c 2 4 - c 2 .
Since ω is bounded, we refer to [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF], that there exists a unique (up to a multiplication) eigenfunction ϕ with Neumann boundary condition associated to

λ α : -(∆ y + α(y))ϕ = λ α ϕ in ω, ∂ ν ϕ(y) = 0 on ∂ω.
We rewrite Eq. (1.6) as in the following form

MU = -∆U -c∂ 1 U -α(y)U = H(x 1 , y) H f (x 1 , y) = f (x 1 , y, U ) -α(y)U
By the regularity condition s → f (x, s) ∈ C 1,r (0, δ) and condition (1.61), we have

|H f (x 1 , y)| ≤ |f (x 1 , y, U ) -f s (x 1 , y, 0)U | + |f s (x 1 , y, 0)U -α(y)U | ≤ C 1 e (r+1)τ δ x 1 + C 2 e (p+τ δ )x 1 ≤ C 3 e m(τ δ )x 1 as x → -∞, (1.63)
where m(τ δ ) = min{(r + 1)τ δ , p + τ δ }. By Theorem 4.3 of [START_REF] Berestycki | traveling fronts in cylinders[END_REF], we can write U as follow

U = u 0 (x 1 , y) + u * (x 1 , y),
where (u 0 , u * ) is a solution of system

Mu 0 = 0 x ∈ Ω - ∂ ν u 0 = 0 ∂Ω - Mu * = H f (x 1 , y) x ∈ Ω - ∂ ν u * = 0 ∂Ω -. (1.64)
Moreover, u 0 has a precisely exponential asymptotic behavior as x 1 → -∞, namely there exist λ > 0 and ψ(x 1 , y) = (-x 1 ) k ψ k (y) + ... + ψ 0 (y) ≡ 0 such that

u 0 (x 1 , y) = e λx 1 ψ(x 1 , y) + O(e λx 1 ) ∇u 0 (x 1 , y) = ∇(e λx 1 ψ(x 1 , y)) + O(e λx 1 ), (1.65)
and for any ε > 0, u * satises the inequality

|u * (x 1 , y)| + |∇u * (x 1 , y)| ≤ C ε,δ e (m(τ δ )-ε)x 1 for some C ε > 0. (1.66) Let us dene τ 0 = sup{τ : ∃C τ such that u(x 1 , y) ≤ C τ e τ x 1 in Ω -}.
Inequalities (1.62) yields κ δ ≤ τ 0 ≤ τ δ , for any δ > 0, thus τ 0 is indeed a real number. We want to prove

that τ 0 = λ α + c 2 4 - c 2 . Taking τ < τ 0 , then 0 ≤ u(x 1 , y) ≤ C τ e τ x 1 and moreover |∇u(x 1 , y)| ≤ C τ e τ x 1
by the Harnack inequality. Proceeding as (1.63), we get :

|H f (x 1 , y)| ≤ C 4 e m(τ )x 1
, where m(τ ) = min{(r + 1)τ, p + τ } > τ . As a result of (1.66), we have

|u * (x 1 , y)| + |∇u * (x 1 , y)| ≤ D τ e (τ +m(τ ))x 1 2
.

One sees that as τ τ 0 , τ + m(τ ) 2

τ 0 + m(τ 0 ) 2 > τ 0 . Therefore, there exist ε > 0 and C ε > 0 such that |u * (x 1 , y)| + |∇u * (x 1 , y)| ≤ C ε e (τ 0 +ε)x 1 in Ω -. (1.67)
It follows immediately that ∀τ < τ 0

|u 0 (x 1 , y)| ≤ |u(x 1 , y)| + |u * (x 1 , y)| ≤ C τ e τ x 1 + C ε e (τ 0 +ε)x 1 ≤ (C τ + C ε )e τ x 1 in Ω -.
On the other hand, for δ small enough

u 0 (x 1 , y) = u(x 1 , y) -u * (x 1 , y) ≥ C 2,δ e ( λα+δ+ c 2 4 -c 2 )x 1 -C ε e (τ 0 +ε)x 1 ≥ C 3,δ e ( λα+δ+ c 2 4 -c 2 )x 1 .
Applying Theorem 4.2 of [START_REF] Berestycki | traveling fronts in cylinders[END_REF] to u 0 , we deduce that there is exactly one positive constant λ such that

τ ≤ λ ≤ λ α + δ + c 2 4 -c 2 ,
∀τ < τ 0 and (1.65) holds for a suitable exponential solution w(x 1 , y) = e λx 1 ψ(y). From (1.67), λ cannot be strictly bigger than τ 0 , therefore we must have λ = τ 0 > 0. Since u > 0, we deduce ψ k > 0 and thus Theorem 2.4 of [START_REF] Berestycki | traveling fronts in cylinders[END_REF] yields that ψ(y) is a solution of

-(∆ y + α(y))ψ = (λ 2 + cλ)ψ in ω ψ ν = 0 on ∂ω. (1.68)
Since λ α > 0, Theorem 2.1 of [START_REF] Berestycki | traveling fronts in cylinders[END_REF] implies that (1.68) possesses exactly one positive principal eigenvalue,

that is λ = -c + √ c 2 + 4λ α 2 = τ 0 .
We obtain the precisely asymptotic behavior of U (x 1 , y) as x 1 → -∞. By analogy, we obtain the precisely exponential behavior of U (x 1 , y) as x 1 → +∞. It is precisely exponentially asymptotic as x 1 → +∞ with the exponent λ = -c -c 2 + 4λ β 2 . As a consequence, we have proved that

U (x 1 , y) ∼ C 1 e -( c+ √ c 2 +4λ β 2 )x 1 as x → +∞; U (x 1 , y) ∼ C 2 e ( -c+ √ c 2 +4λα 2 )x 1 as x → -∞,
uniformly in y. This result, in particular, implies that U (x 1 , y) is increasing in (-∞, -R)×ω and decreasing in (R 1 , ∞) × ω for R, R 1 large enough. To achieve the symmetry in x 1 , necessarily, we have

c + c 2 + 4λ β 2 = -c + √ c 2 + 4λ α 2 ⇐⇒ λ β = λ α + c 2 -2c λ α + c 2 4 .
In other words U is asymmetric if

λ β = λ α + c 2 -2c λ α + c 2 4 .
Remark 5. we see that if c = 0, the asymmetry holds even when λ β = λ α . The drift term is therefore the main inducement that makes the front asymmetric. However, we do not know that whether the front is symmetric when

λ β = λ α + c 2 -2c λ α + c 2 4 .
The answer of this question requires more involved analysis. We state this as an open question. Our result applies, in particular, to show that the asymmetry holds when condition (1.61) becomes as (1.2), namely

λ β = λ α = m > 0 and |f s (x 1 , y, 0) + m| = O(e -p|x 1 | ) as |x 1 | → ∞, for some m, p > 0.
Remark 6. We point out that the assumption of the exponential convergences f s (x 1 , y, 0) → α(y) and (1.61) does not hold, the precise exponential behavior of u 0 satisfying Eq. (1.64) may not be true as (1.65) in general. For instance, in one dimensional space, β(y) ≡ -1/2, if f s (x, 0) converges slowly to -1/2, we can take w(x) = xe -x , which is a solution of

f s (x 1 , y, 0) → β(y) in (1.61) is important. Indeed, if
w + 1 2 w + g(x)w = 0 in R \ (2, -∞), g(x) = - 1 2 + 1 2x → - 1 2 as x → +∞.

Appendix

Theorem 1.5.1. Let U be a traveling front solution of (1.6). Assume that (1.9) holds and f is such that lim inf

x 1 →±∞ f s (x 1 , y, 0) ≥ α ± (y) and λ α ± = λ N (-∆ y -α ± (y), ω) > 0, (1.69)
for some functions α ± ∈ L ∞ (ω). Then, for any δ > 0, there exist

A ± > 0 and τ α ± ≥ λ α ± + δ + c 2 4 such that U (x 1 , y) ≥ A -e (τα --c 2 )x 1 ∀(x 1 , y) ∈ Ω -and U (x 1 , y) ≥ A + e -(τα + + c 2 )x 1 ∀(x 1 , y) ∈ Ω + .
Proof. Since (1.9) holds, from Proposition (1.2.1), we know that U (x 1 , y) decays exponentially as |x 1 | → ∞.

Let us denote I = {α -, α + }. We know from [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] that there exist the principal eigenfunctions ϕ i associated with λ i such that for i ∈ I

-∆ϕ i -i(y)ϕ i = λ i ϕ i in ω ∂ ν ϕ i = 0 on ∂ω. For i ∈ I, δ > 0, we set L i = ∆ x + c∂ 1 + i(y) -δ. By assumption (1.69), there exists R = R(δ) > 0 such that L α -U ≤ 0 in Ω -\ Ω R and L α + U ≤ 0 in Ω + \ Ω R . Dene the functions ω α -(x) = e τα -x 1 ϕ α -(y) and ω α + (x) = e -τα + x 1 ϕ α + (y), direct computation yields L α -ω α -= τ 2 α -+ cτ α --λ α --δ ω α -≥ 0 in Ω -\ Ω R if τ α -≥ c 2 + 4(λ α -+ δ) -c 2 ; L α -ω α + = τ 2 α + -cτ α + -λ α + -δ ω α + ≥ 0 in Ω + \ Ω R if τ α + ≥ c 2 + 4(λ α + + δ) + c 2 .
The strong maximum principle and the Hopf lemma yield inf y∈ω ϕ i (y) > 0, inf y∈ω U (-R, y) > 0 and inf y∈ω U (R, y) > 0. Therefore, we can choose the positive constants C δ,i small enough such that the functions satisfy

W α -(x 1 , y) = U (x 1 , y)-C δ,α -ω α -(x 1 , y) ≥ 0 and W α + (x 1 , y) = U (x 1 , y)-C δ,α + ω α + (x 1 , y) ≥ 0 for x 1 = -R, y ∈ ω.
They have the Neumann boundary conditions ∂ ν W i = 0 on ∂Ω and satisfy the inequalities

L i W i ≤ 0 in Ω ± \ Ω R . Set z i (x 1 , y) = W i (x 1 , y)/ϕ i (y), we get ∂ ν z i = 0 on ∂Ω and 0 ≥ L α -W α - ϕ α - = ∆z α -+ c∂ 1 z α -+ 2 ∇ y ϕ α - ϕ α - .∇ y z α --λ α -+ δ z α -x 1 < -R, ∀y ∈ ω, 0 ≥ L α + W α + ϕ α + = ∆z α + + c∂z α + + 2 ∇ y ϕ α + ϕ α + .∇ y z α + -λ α + + δ z α + , x 1 > R, ∀y ∈ ω. Since z α -(x 1 , y) → 0 as x 1 → -∞, z α + (x 1 , y) → 0 as x 1 →
+∞, and zero-order coecients of ellipticoperators with respect to z i are negative, the weak maximum principle is applied to derive

z i ≥ 0 in Ω ± \Ω R .
As a consequence, there exist

τ i ≥ λ i + δ + c 2 4 such that C δ,α + e -(τα + + c 2 )x 1 ϕ α + (y) ≤ U (x 1 , y) in Ω + \ Ω R . C δ,α -e (τα --c 2 )x 1 ϕ α -(y) ≤ U (x 1 , y) in Ω -\ Ω R . (1.70)
By the Harnack inequality, one has inf |x 1 |≤R U (x 1 , y) > 0, we deduce that C δ,i indeed can be chosen such that the inequality (1.70) holds respectively in Ω ± . The proof is complete.

Theorem

1.5.2. Let U ∈ W 1,2 N +1,loc (R × R N )
be a solution of (1.14), where f is such that condition (1.19) holds true. Then there exist two positive constants k and ε such that

∀(t, x) ∈ R × R N U (t, x 1 , y) ≤ ke -ε|x 1 | .
Proof. We only need to prove that the statement holds for x 1 ≥ 0 and by analogy we also derive the result for x 1 ≤ 0. Using the transformation V (t, x 1 , y) = U (t, x 1 , y)e c 2 x 1 , we see that V (t, x) is periodic in y, T-periodic in t and satises the following equation

   V t = ∆V + f (t, x, V e -c 2 x 1 )e c 2 x 1 - c 2 4 V t ∈ R, x ∈ R N V e -c 2 x 1 is bounded. For any R > 0, δ > 0, we denote Q R = R × [0, R] × R N -1 and set L = ∂ t -∆ -γ(t, y) -δ + c 2 4 . From condition (1.19), there exist R = R(δ) > 0 such that LV ≤ 0, ∀(t, x) ∈ Q R . Moreover, since λ γ = λ1 (∂ t -∆ y -γ(t, y), R × R N -1
) > 0, we refer to Theorems 2.7, [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF] that there exists a unique space-time periodic eigenpair (λ γ , ϕ) satisfying

       Lϕ = λ γ ϕ ϕ > 0 ϕ(., . + T ) = ϕ ϕ(. + L i e i , .) = ϕ for i ∈ {1, N -1}.
Fix τ ∈ R and dene the function ω(t, x) = θ a (x 1 )ϕ(t, y)e (τ -t)δ , where

θ a : [R, R + a] → R is the solution of    θ a = (κ + δ)θ a in (R, R + a) θ a (R) = Ce √ κR θ a (R + a) = Ce √ κ(R+a) , 1.5. Appendix
where C = sup R×R N V (t, x)e -c 2 x 1 / inf R×R N -1 ϕ(t, y), κ > 0 would be chosen later. Note that inf R×R N -1 ϕ(t, y) > 0 thanks to the periodicity of ϕ in y and t. Direct calculation yields

θ a (ρ) = C(e ( √ κ+ √ κ+δ)R ) 1 - e √ κa -e - √ κ+δa e √ κ+δa -e - √ κ+δa e - √ κ+δρ + +C(e ( √ κ- √ κ+δ)R ) -e - √ κ+δa + e √ κa e √ κ+δa -e - √ κ+δa e √ κ+δρ .
On the other hand, we have

Lω = -κ -3δ + λ γ + c 2 /4 ω ≥ 0 if and only if κ ≤ λ γ + c 2 /4 -3δ. Moreover, it is easily seen that ω(t, x 1 , y) ≥ V (t, x 1 , y) for t ≤ τ, x 1 ∈ {R, R + a}. Taking 0 < δ < λ γ /3 we can choose κ ∈ c 2 /4, λ γ + c 2 /4 -3δ . Let W (t, x) = ω(t, x) -V (t, x) = z(t, x)ϕ(t, y), one sees that z(t, x) ≥ 0 for t ≤ τ , x 1 ∈ {R, R + a}, y ∈ R N -1 . Since sup t∈R,x∈(R,R+a)×R N -1 V (t, x) < +∞ and inf t≤τ,x∈(R,R+a)×R N -1 ω(t, x) > 0,
there exists t 0 (a) τ , which may depend on a and suciently close to -∞ such that z(t 0 (a), x) ≥ 0 for x ∈ (R, R + a) × R N -1 . In addition that, we have

0 ≤ LW ϕ = ∂ t z -∆z -2∇z. ∇ϕ ϕ + λ γ -δ + c 2 4 z.
Since the zero order coecient of parabolic operator with respect to z is positive, we deduce from the parabolic weak maximum principle that z

(t, x) ≥ 0 in (t 0 (a), τ ) × Q R+a \ Q R for every a > 0. Finally, the classical parabolic regularity implies that V (τ, x) ≤ ω(τ, x) for x ∈ Q R+a \ Q R . Therefore, U (τ, x) ≤ lim a→+∞ θ a (x 1 )ϕ(τ, y)e -c 2 x 1 = C(e ( √ κ+ √ κ+δ)R )e -( √ κ+δ+ c 2 )x 1 .
The arbitrariness of τ enables us to conclude the proof.

Remark 7. In the proof of this theorem, we need not assume that the solution U is periodic in y and T-periodic in t, but the local regularity of solutions plays an important role. On the other hand, as seen from above, it is possible to choose κ = λ γ + c 2 /4 -3δ to obtain

U (τ, x) ≤ C 1 e -( λγ + c 2 4 -2δ+ c 2 )x 1 for τ ∈ R, x 1 ≥ R, y ∈ R N -1
Using the same arguments, we derive that there exist C 2 > 0 :

U (τ, x) ≤ C 2 e ( λγ + c 2 4 -2δ-c 2 )x 1 for τ ∈ R, x 1 ≤ -R, y ∈ R N -1 .
Since U is bounded, one can choose C 1 , C 2 large enough such that these inequalities hold in R N +1 .

Chapitre 2

Liouville-type result for semilinear elliptic equations with possibly vanishing and sign-changing potentials

Introduction and Main Results

Introduction

The paper is devoted to characterizing the set of positive solutions of

a ij (x)∂ ij u(x) + Kq(x) • ∇u(x) + f (x, u) = 0, x ∈ R N , (2.1)
and large time behavior of

(|x| + 1) -α ∂ t u = a ij (x)∂ ij u(x) + Kq(x) • ∇u + f (x, u) t ∈ R + , x ∈ R N , (2.2)
where α ∈ [0, 2] and K ≥ 0.

In a pioneering work, Berestycki, Dickmann, Nagelkerke and Zegeling [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] have proposed a model in R to study the impact of climate change on the population dynamics of species, who are strongly sensitive to temperature conditions. More precisely, they considered the equation

u t -u xx = f (x -ct, u) t ∈ R, x ∈ R, (2.3)
where c > 0 is a given forced speed. A typical f is

f (x, s) = -sm x < 0 and x > L sm 1 - s K 0 ≤ x ≤ L, (2.4)
for some positive constants m, m , K, L. This nonlinearity has the property

f s (x, 0) = -m < 0 x < 0 and x > L m > 0 0 ≤ x ≤ L. (2.5)
Condition (2.5) means that the environment of the species is unfavorable outside a prescribed compact set. Ecologically speaking, the environments satisfying (2.5) are surrounded by unprosperous vicinities, where the source for persistence is so scarce or the climatic condition is unpropitious. Here we say that the environment is favorable (respectively unfavorable and neutral) at the point x 0 if f s (x 0 , 0) > 0 (respectively f s (x 0 , 0) < 0 and f s (x 0 , 0) = 0). The authors in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF] showed that u(t, x) converges to U (x), which is the unique positive solution of

U + cU + f (x, U ) = 0 in R
uniformly as t → ∞. Thus, to understand this dynamics, the characterization of the set of nontrivial stationary solutions plays the central role. Higher dimensional version of this model was later studied in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], where condition (2.5) is relaxed to

lim sup |x|→∞ f s (x, 0) < 0. (2.6)
These equations are the particular cases of (2.1)-(2.2) where α = 0 and q is a constant vector in R N . The use of condition (2.6) is to derive the exponential decay of solutions. The interested readers are also referred to the works of Potapov, Lewis [START_REF] Potapov | Climate and competition : the eect of moving range boundaries on habitat invasibility[END_REF] and Roques et al. [START_REF] Roques | A population facing climate change : joint inuences of Allee eects and environmental boundary geometry[END_REF], where similar models were investigated. On a dierent framework, Berestycki-Hamel-Rossi also proved a Liouville-type result for positive bounded solutions if the following condition holds

lim inf |x|→∞ (4α(x)f s (x, 0) -K|q(x)| 2 ) > 0, (2.7)
where

0 < α * ≤ α(x) = inf ξ∈R N |ξ|=1 a ij (x)ξ i ξ j .
Actually, it is proved in [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF] that the solution is unique and has positive inmum if exists. To this paper, we pursue the following goals :

• First, we extend the results of [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] to a more general framework and study the new cases in which (2.6),(2.7) are not fullled. More precisely, we allow f s (x, 0) to be sign-changing all the way as |x| → ∞ and it may behave like a slow decay or Hardy potential near innity, namely

f s (x, 0) ≤ -C|x| -α in R N \ B R 0 for some C, R 0 > 0 and α ∈ [0, 2].
From the biological point of view, these new assumptions refer to the environments that can be unfavorable (α = 0) or unfavorably neutral (α ∈ (0, 2)) or nearly neutral (α = 2) mixing with favorable patches extending to innity. Our results conrm that the unique persistence and extinction of the species can be conditioned by the sign of the generalized principal eigenvalue of the linearized operator of (2.1). This is a delicate conclusion since if (2.7) holds, we know from [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF] that there exists a unique positive solution of (2.1), which has a positive inmum and therefore the species are persistent. To this aim, we will propose a global condition in term of spectral theory. The central role is to develop the notions of generalized principal eigenvalues in unbounded domains, which are inspired from [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF], [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]. These generalizations are originally conjectured by Prof H. Berestycki (personal communication). Precise description is given in Section 1.2.

• Second, under the setting framework, we further investigate the inuence of the drift term on the existence of nontrivial solutions of Eq. (2.1). Indeed, in the works [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], the authors implicitly showed that the existence of positive solution is strongly connected with the amplitude of the drift term. Here, we consider the drifts satisfying divergence-free and some fair growth conditions. By adequate compactness compensations, we partially extend the result of Berestycki-Hamel-Nadirashvili [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF] on the limit of large drift for the rst eigenvalues in bounded domain to the generalized principal eigenvalues in unbounded domain with no periodic assumption. This is in fact the intermediate result that we shall use to derive the existence of sharp threshold of the drift-amplitude, K , such that positive solutions of (2.1) exist if and only if K < K . Detailed explanation is postponed until stating the result.

• Third, the long time behavior of Eq. (2.2) are lastly investigated. We prove that, if the initial datum decays enough, solutions of Eq. (2.2) converge in L p (R N )-weighted norm to the unique positive stationary of Eq. (2.1). Here 0 < p ≤ ∞.

We obtain an intriguing counterpart of these results between two cases α ∈ [0, 2) and α = 2. The dierent eects due to the slow decay and Hardy potentials of f s (x, 0) for |x| large are fully understood. Our results also conrm that the Liouville-type result for KPP-type nonlinearity is strongly depended on the behavior near innity of zero-order coecient of the linearized operator about zero.

The central role : generalized principal eigenvalues

In bounded domains, one of the classical approaches ([3], [START_REF] Berestycki | Existence and bifurcation of solutions for an elliptic degenerate problem[END_REF]) to investigate the existence and uniqueness for equations like Eq. (2.1), is to look at the rst eigenvalue of the linearized operator of the nonlinear equation. Thanks to the compactness of domain and prescribed boundary condition (Dirichlet or Neumann), the existence and simplicity of the rst eigenvalue is well-established by the Krein-Rutman theory (see [START_REF] Brezis | Functional analysis, Sobolev spaces and partial dierential equations[END_REF]). However, if the domain is not smooth and possibly unbounded, the Krein-Rutman theory cannot be applied directly and in fact the rst eigenvalue may not exist in general. Therefore, appropriate eigenvalues should be looked for to overcome the existing diculties. The notion of generalized principal eigenvalue was rst introduced in the celebrated work of Berestycki-Nirenberg-Varadhan [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] and has been widely used after in many researches of reaction-diusion equations ( [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF], [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF], [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], [START_REF] Nadin | The principal eigenvalue of a space-time periodic parabolic operator[END_REF], [START_REF] Nadin | Existence and uniqueness of the solutions of a space-time periodic reaction-diusion equation[END_REF], [START_REF] Rossi | Liouville type results for periodic and almost periodic linear operators[END_REF]). We also refer to the early works of Pinchover [START_REF]Pinchover Criticality and ground states for second-order elliptic equations[END_REF], [START_REF] Pinchover | On criticality and ground states of second order elliptic equations[END_REF] and López-Gómez [START_REF] López-Gómez | The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems[END_REF] on the eigenvalues for indenite-weight elliptic operators. More recently, Berestycki et al. [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF], [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF], [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] have deeply generalized this notion for unbounded domains in various aspects. In this paper, we use similar approach of these papers to dene a new generalized principal eigenvalue. To this aim, let us provide rst some basic assumptions.

In Eq. (2.1), we denote by (a ij ) with i, j ∈ {1, ..., N } and q(x) = (q 1 (x), ..., q N (x)) respectively the matrix N × N and the N -dimensional drift term. The matrix (a ij ) is assumed to be symmetric but need not be uniformly elliptic. Throughout the paper, we assume that the functions a ij , q i are continuous and bounded in R N , the reaction term f (x, s) : R N × [0, +∞) → R belongs to C 0,γ loc (R N ) with respect to x, locally in s for some γ ∈ (0, 1) and C 1,γ (R N ) uniformly with respect to s ∈ [0, δ] for some δ > 0. We further assume that f (x, 0) = 0 for all x ∈ R N .

Throughout this paper, we denote the linearized operator of Eq. ( 2.1) about 0 by

L[φ] = a ij ∂ ij φ + Kq(x) • ∇φ + f s (x, 0)φ, (2.8)
and sometimes we use L K [φ] to emphasize the dependence on K.

Let us recall the generalized principal eigenvalue in [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF], [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF], [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] that are used in this paper

λ 1 (-L, Ω) := sup{λ ∈ R : ∃φ ∈ W 2,N loc (Ω), φ > 0, (L + λ)φ ≤ 0 a.e in Ω}.
(2.9)

λ 1 (-L, Ω) := inf{λ ∈ R : ∃φ ∈ W 2,N loc (Ω), φ > 0, φ ∈ L ∞ (Ω), (L + λ)φ ≥ 0 a.e in Ω}.
(2.10)

λ 1 (-L, Ω) := sup{λ ∈ R : ∃φ ∈ W 2,N loc (Ω), inf Ω φ > 0, (L + λ)φ ≤ 0 a.e in Ω}.
(2.11)

and we dene the new generalized principal eigenvalues as follows.

Denition 2.1.1. Let α ∈ [0, 2], the generalized principal eigenvalue associated with L in Ω is

λ α (-L, Ω) := sup λ ∈ R : ∃h α ∈ C(Ω), lim x∈Ω,|x|→∞ h α (x) |x| -α = 1, ∃φ ∈ D α , L + λ|x| -α φ ≤ 0 in Ω h + α , (2.12
) where Ω h + = {x ∈ Ω, h(x) ≥ 0} and D α denotes the class of admissible test-functions

D α := φ ∈ W 2,N loc (Ω), φ > 0 in Ω h + , lim sup x∈Ω,|x|→∞ |∇φ(x)||x| α 2 φ(x) ≤ D α and lim x∈Ω,|x|→∞ ln φ(x) |x| 1-α 2 = 0 ,
for α ∈ [0, 2) and for α = 2, D 2 is given by

D 2 := φ ∈ W 2,N loc (Ω), φ > 0 in Ω h + , lim sup x∈Ω,|x|→∞ |∇φ(x)||x| φ(x) ≤ D 2 and lim x∈Ω,|x|→∞ ln φ(x) ln |x| = 0 .
In principle, we dene two dierent notions of generalized principal eigenvalues for α ∈ [0, 2) and α = 2, where the test-functions are assigned dierent growth conditions near innity. Intrinsically, the dierence of growth conditions are due to the dierent decays of fundamental solutions for the linear equations with slow decay and Hardy potentials. Further explanation of these eigenvalues are given in section 2. Note that, if Ω is bounded, all of growth conditions are superuous and obviously λ α (-L, Ω) = λ 1 (-L, Ω). More generally, we refer the interested readers to the work of López-Gómez [START_REF] López-Gómez | The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems[END_REF] for a careful investigation on the existence of weighted-eigenvalue in bounded domain. We are now ready to state main results

Hypothesis and Main Results

(H1) Hypothesis 1 : There exist α ∈ [0, 2] and µ ∈ L ∞ (R N ) satisfying

(|x| + 1) α f s (x, 0) , (|x| + 1) α µ(x) ∈ L ∞ (R N ) lim sup |x|→∞ {|x| α (f s (x, 0) -µ(x))} ≤ 0 (H1a)
and one of the followings is true

η α = λ α (-a ij (x)∂ ij -Kq(x).∇ -µ(x), R N ) > 0, (H1b)
(a ij ) is the identity matrix, divq = 0 and there exists ψ

∈ C 2 loc (R N ) such that 2∇ψ(x) = -q(x), η α = λ α (-∆ -µ(x), R N ) > 0. (H1c)
Remark 8. The particular case f s (x, 0) ∼ -m(|x| + 1) α for some m > 0, α ∈ [0, 2] as |x| → ∞ is also covered by (H1b), where we can take µ(x) = -m(|x| + 1) α and 1 to be a test-function for η α . Note that, in this case, q is not necessarily involved in (H1b).

(H2) Hypothesis 2 :

s → f (x, s) s is nonincreasing for x ∈ R N
and it is strictly decreasing in some set D ⊂ R N , with |D| > 0.

(H3) Hypothesis 3 : For xed K, there exists a function 0

< S K (x) ∈ W 2,N loc (R N ) satisfying a ij (x)∂ ij S K + Kq(x) • ∇S K (x) + f (x, S K ) ≤ 0 a.e in R N , such that lim |x|→∞ ln S K (x) |x| 1-α 2 = 0 if α ∈ [0, 2) and lim |x|→∞ ln S K (x) ln |x| = 0 if α = 2.
Existence and Uniqueness of Eq. (2.1)

(Liouville-type result) Theorem 2.1.2. Let α ∈ [0, 2), θ ∈ R + , Λ = max ij a ij ∞ < ∞
and Hypotheses 2 hold. Eq. (2.1) cannot possess more than one positive solution satisfying

lim sup |x|→∞ [ln u(x)]|x| α 2 -1 ≤ 0. (2.13)
If Hypothesis 3 is besides assumed then Eq. (2.1) possesses a unique positive solution i λ 1 (-L, R N ) < 0.

Furthermore, for all γ ∈ (0, γ ), one nds

ϕ ∈ D α , C(γ), R(γ) > 0 such that u(x) ≤ C(γ)e γ|x| 1-α 2 ϕ(x) for |x| ≥ R(γ),
where γ is explicitly given as follows (i) if (H1a), (H1b) hold and lim sup |x|→∞ |q(x)||x| α 2 ≤ θ, then γ = γ, which is the unique positive solution of

γ 2 1 - α 2 2 N 2 + 2γ 1 - α 2 N D α + γ 1 - α 2 K Λ θ - η α Λ = 0; (2.14)
(ii) if (H1a), (H1c) hold, in addition that

lim inf |x|→∞ |∇ψ(x)||x| α 2 = θ, lim sup |x|→∞ ψ(x)|x| α 2 -1 = 2θ 2 -α and θ < η α 2KD α , (2.15)
then

γ = γ = γ + - 2Kθ 2 -α with γ + = -2D α + 2 D 2 α + η α + K 2 θ 2 2 -α . (2.16)
In particular, u decays exponentially when it exists.

Observe that, when α = 2, the characterizations (2.14), (2.15), (2.16) are not well-determined, for instance, from (2.15), ψ blows up. In fact, another phenomenon takes place, we are thus led to Theorem 2.1. (2.17)

3. Let α = 2, θ ∈ R + , Λ = max ij a ij ∞ < ∞ and
If Hypothesis 3 is besides assumed then Eq. (2.1) possesses a unique positive solution i λ 1 (-L, R N ) < 0.

Furthermore, for all β ∈ (0, σ ), one nds

ϕ ∈ D 2 , C(β), R(β) > 0 such that u(x) ≤ C(β)|x| β for |x| ≥ R(β),
where σ is explicitly given as follows (i) if (H1a), (H1b) hold and lim sup |x|→∞ |q(x)||x| ≤ θ, then

σ = σ = -b + √ b 2 -4ac 2a , a = ΛN, b = (3N + 2D 2 )Λ + Kθ, c = -η 2 .
(2.18)

(ii) if (H1a), (H1c) hold, in addition that

lim inf |x|→∞ |∇ψ(x)||x| = θ, lim sup |x|→∞ ψ(x) (ln |x|) 2 = θ and θ + θ 2D 2 (N -2) < η 2 2KD 2 , (2.19)
then

σ = σ = β + 2 -Kθ > 0 with β + 1 = (N -2 + 2D 2 ) + (N -2 + 2D 2 ) 2 + 4η 2 + 4K 2 θ 2 2 .
(2.20)

In particular, u decays polynomially when it exists.

Remark 9. In the next result, we will apply the particular case θ = 0 of part ii) to prove Theorem (2.1.5) and (2.1.6). Let us provide some examples of q that fulll (H1a), (H1c) and one of two conditions (2.15) and (2.19) in this case.

Let N ≥ 2, θ = 0 and ψ be the fundamental solutions of Laplace equations ∆ψ 1 (x) = 0 in R N . It is known that ψ 1 has the explicit forms

ψ 1 (x) = ln |x| N = 2 1 |x| N -2 N ≥ 3.
Let R > 0 to be chosen as small as we want and ψ 2 , ψ 3 be as follows

ψ 2 (x) =    ln(|x|) |x| ≥ 1 -ln(|x|) |x| ∈ (R, 1) -ln(R) |x| ≤ R. for N = 2 ; ψ 3 (x) = 1/|x| N -2 |x| ≥ 1 1/R N -2 |x| ≤ R. for N ≥ 3
Obviously, ψ 2 and ψ 2 + 1 (respectively) ψ 3 and ψ 3 + 1 are continuous (not smooth) solutions of ∆φ = 0 in R N (resp. N = 2 and N ≥ 3). Hence, by the classical iteration method and the regularity of Laplace equation, there exists a solution

ψ ∈ C 2 loc (R N ) of ∆ψ = 0 in R N satisfying ψ 2 ≤ ψ ≤ ψ 2 + 1 (reps.) ψ 3 ≤ ψ ≤ ψ 3 + 1 in R N .
Note that ψ need not be unique up to multiplicative and additive constants.

We choose q(x) = ∇ψ(x). Obviously, all component

q i ∈ C 1 loc (R N ), i = {1, .., N }, divq = 0 and lim sup |x|→∞ (ln |x|)|x| α 2 -1 = 0, lim sup |x|→∞ |x| α 2 -1 /|x| N -2 = 0 0 ≤ α < 2 lim sup |x|→∞ ln |x| (ln |x|) 2 = 0, lim sup |x|→∞ 1 (ln |x|) 2 |x| N -2 = 0 α = 2.
These verify (H1c), (2.15) and (2.19). Moreover, if θ = 0, another important class for example of q is the class of constant vectors in R N , for instance, q = -→ c when ψ(x) = -→ c .x, x ∈ R N . Very recently, the constant vector elds are used in the study of the impact of climate change (see [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF]). Theorem (2.1.2) and (2.1.3) extend the existence/nonexistence and uniqueness results in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] even in the case α = 0 and q is a constant vector.

Sharp threshold of drift

Next, we prove the existence of sharp threshold of drift term for which Eq. (2.1) possesses at least one positive solution. This result is in fact a consequence of the necessary and sucient conditions for the existence of Eq. (2.1), which are directly conditioned by the sign of the eigenvalue λ 1 (-L K , R N ) and the continuous monotonicity of λ 1 (-L K , R N ) with respect to K. The goal is to understand the sign-dependence of λ 1 (-L K , R N ) on the large and small K. Here, the interesting diculties are due to the presence of zero's order coecient of linearized operator of Eq. (2.1), f s (x, 0), which allows to change sign and possibly decays fast up to Hardy-potential as |x| → ∞. In [START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF], Berestycki-Hamel-Nadirashvili have already studied the limit of the rst eigenvalue for the large drift in bounded domains with Dirichlet (Neuman) boundary condition or periodic domains. However, the problem changes drastically when the domain becomes unbounded without periodic assumption. The new diculties have been successfully overcome by imposing additional assumptions (H1a) and (H1c). This bears adequate (but dierent between α ∈ [0, 2) and α = 2) decays for the principal eigenfunction associated with the generalized principal eigenvalue λ 1 (-L K , R N ) to compensate for the lack of compactness of domain. In the sequel, we will prove that the limit of the eigenvalues as K → ∞ depends on the existence of rst integrals of the drift term. Let us now introduce this notion Denition 2.1.4. A function z = 0 is called a rst integral of the vector elds q ∈ R N if z ∈ H 1 (R N ) and q • ∇z = 0 almost everywhere in R N . We denote the set of rst integrals of q by

I q = {z|z ∈ H 1 (R N ), z = 0 and q • ∇z = 0 a.e in R N }. (2.21)
We obtain the following results:

Theorem 2.1.5. Let α ∈ [0, 2) and assumptions (H1a), (H1c) (2.15) with θ = 0 be fullled. If the followings hold

inf φ∈H 1 (R N ) R N |∇φ| 2 -f s (x, 0)φ 2 R N φ 2 < 0 < inf φ∈Iq R N |∇φ| 2 -f s (x, 0)φ 2 R N φ 2 , ( 2.22)
with the convention that the inmum is +∞ if I q being empty. Then there exists a sharp threshold K such that Eq. (2.1) possesses a positive solution if and only if K < K .

Theorem 2.1.6. Let α = 2 and (H1a), (H1c), (2.19) with θ = 0, (2.22) hold. The conclusion of theorem (2.1.5) remains true if

D 2 ≥ 1 or D 2 < 1 and η 2 > N (1 -D 2 ) -N 2 4 .
(2.23)

Remark 10. We note that (2.23) becomes unconditioned when N ≥ 2. This condition is indeed used to guarantee that the eigenfunction associated with λ K belongs to L 2 (R N ). This is unnecessary for the case α ∈ [0, 2) in Theorem (2.1.5) since, in that case, we will prove that the eigenfunctions decay exponentially .

Long time behavior in L ∞ and L p norms Theorem 2.1.7. Let α ∈ [0, 2), u(t, x) be the solution of (2.2) starting with nonnegative initial datum u 0 ≡ 0 and a ij , q satisfy the growth conditions as in Theorem (2.1.2). Assume that Hypotheses 2, 3 and one of the followings are true i) (H1a), (H1b) and lim sup |x|→∞

ln u 0 (x) |x| 1-α 2
< -γ 0 , for some γ 0 ∈ (0, γ ), where γ = γ is the positive solution of (2.14).

ii) (H1a), (H1c), (2.19) and lim sup |x|→∞

ln u 0 (x) |x| 1-α 2 < -γ 0 -4Kθ
2-α , for some γ 0 ∈ (0, γ ), where γ = γ is given in (2.16).

If one has

lim inf |x|→∞ sup t>0 u(t, x) e γ 0 |x| 1-α 2 < ∞, (2.24)
then for all 0 < p ≤ ∞

lim t→∞ |u(t, x) -W (x)|e γ 0 |x| 1-α 2 L ∞ (R N ) = 0, (2.25)
where W (x) is either identical to zero or the unique positive solution of Eq. (2.1) corresponding to

λ 1 (-L, R N ) ≥ 0 or λ 1 (-L, R N ) < 0. If u 0 /S K ∈ L ∞ (R N )
, above convergences hold locally on any compact set of R N .

Theorem 2.1.8. Let α = 2, u(t, x) be the solution of (2.2) starting with nonnegative initial datum u 0 ≡ 0 and a ij , q satisfy the growth conditions as in Theorem (2.1.3). Assume that Hypotheses 2, 3 and one of the followings are true i) (H1a), (H1b) and lim sup |x|→∞ ln u 0 (x) ln |x| < -σ 0 , for some σ 0 ∈ (0, σ ), where σ = σ is the positive solution of (2.18).

ii) (H1a), (H1c), (2.15) and lim sup |x|→∞ ln u 0 (x) ln |x| < -σ 0 -2Kθ, for some σ 0 ∈ (0, σ ), where σ = σ is given in (2.20).

If one has

lim inf |x|→∞ sup t>0 u(t, x) |x| σ 0 < ∞, (2.26)
then,

lim t→∞ |u(t, x) -W (x)|(|x| + 1) σ 0 L ∞ (R N ) = 0, (2.27) moreover, if p ∈ (0, ∞) is such that σ 0 ∈ (N/p, σ ), then lim t→∞ |u(t, x) -W (x)|(|x| + 1) σ 0 L p (R N ) = 0, (2.28)
where W (x) is either identical to zero or the unique positive solution of Eq. (2.1) corresponding to

λ 1 (-L, R N ) ≥ 0 or λ 1 (-L, R N ) < 0. If u 0 /S K ∈ L ∞ (R N )
, above convergences hold locally on any compact set of R N .

2.2

Further discussion on the hypotheses and results

Meaning of the hypotheses and results

In this section, we discuss about the meaning of the generalized principal eigenvalues in denition (2.1.1) and of the main results presented above. Let us recall rst the similar generalized eigenvalue used in [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF], that is

λ 1 (L, (R, ∞)) = inf{λ|∃φ ∈ A R such that (L -λ)φ ≤ 0 in (R, ∞)}, (2.29)
where if |R| < ∞, A R is the set of admissible test-functions over (R, ∞):

A R = φ ∈ C 1 ([R, ∞)) ∩ C 2 ((R, ∞)), φ > 0 in [R, ∞), φ /φ ∈ L ∞ ((R, ∞)), lim x→+∞ ln φ(x) x = 0
and A -∞ is the set of admissible test-functions over R:

A -∞ = φ ∈ C 2 (R), φ > 0 in R, φ /φ ∈ L ∞ (R), lim |x|→+∞ ln φ(x) |x| = 0 }.
Obviously, -λ 1 (L, (-∞, ∞)) ≤ λ 0 (-L, R). The authors of [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF] dene the eigenvalues in either half or whole R. In denition (2.1.1), we distinguish dierent notions by the rate of decay of the zero's order coecient. We realize that, for all α ∈ [0, 2), similar notions can be extended in the same manner as (2.29) but not for α = 2. The reason is that if α = 2, the growth rate lim |x|→∞ ln φ(x) |x| 1-α 2 = 0 may not be proper for any supersolution of the operators with Hardy potentials. Necessarily, a reasonable growth rate should be looked for to deal with the case. We impose lim |x|→∞ ln φ(x) ln |x| = 0. The eigenvalue λ α also diers from λ 1 , λ 1 , and λ 1 of Berestycki, Rossi [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] in the sense that we impose the milder constraint on the test-functions lim |x|→∞

ln φ(x) |x| 1-α 2 = 0 for α ∈ [0, 2) and lim |x|→∞ ln φ(x)
ln |x| = 0 for α = 2 instead of asking some constant bounds from above or below on the test-functions. For α = 0, we refer to [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF] for the construction of an eigenfunctions in class of random stationary ergodic coecients. The eigenfunction is not bounded but satises the prescribed growth condition. Further properties of this eigenvalue can also be found in [START_REF] Berestycki | Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, Analysis[END_REF]. If one restricts the set of admissible test-functions, for instance, to be

A -∞ = {φ ∈ C 2 (R N ) ∩ L ∞ (R N ), inf R N φ > 0}
then the corresponding generalized principal eigenvalues are not equal in general. Therefore, the class of random stationary ergodic coecients emphasizes that it is very important to use the milder growth assumption for admissible test-functions.

The set of admissible test-functions D α is quite large. Obviously, all positive constants belong to D α . For α ∈ [0, 2), D α contains all of the functions that grow at most exponentially sublinear and polynomially near innity, for instance, any function having exact asymptotic behavior as |x| → ∞ like Ce λ|x| κ or C|x| λ , where C > 0, λ ∈ R and κ ∈ [0, 1 -α 2 ) belongs to D α . The class D 2 is smaller, it contains all function growing at most as e λ(ln |x|) κ as |x| → ∞ with λ ∈ R and κ < 1. One can take any function having exact asymptotic behavior as e λ(ln |x|) κ and also ln |x| for examples. The following nice properties of the functions in D α will be frequently used in the proofs :

For α ∈ [0, 2), ϕ ∈ D α , ∀λ ∈ R, λ > 0, lim |x|→∞ ϕ(x)e λ|x| 1-α 2 = ∞ lim |x|→∞ ϕ(x)e -λ|x| 1-α 2 = 0. (2.30) For α = 2, ϕ ∈ D 2 , ∀λ ∈ R, λ > 0, lim |x|→∞ ϕ(x)|x| λ = ∞ lim |x|→∞ ϕ(x)|x| -λ = 0. (2.31)
Now, under the sense of λ α , Hypothesis 1 extends (2.5) and (2.6) in two aspects. First, for general q, (H1a) and (H1b) suggest to look at the eigenvalue λ α dened on R N of the upper operator of -L near innity, where the upper operator of -L near innity means

-L[φ] = -a ij (x)∂ ij φ -Kq(x).∇φ -µ(x)φ and lim sup |x|→∞ {|x| α (f s (x, 0) -µ(x))} ≤ 0.
Second, if a ij is the identity matrix and q is divergence-free, (H1c) suggests a more concise condition, that is looking at the sign of λ α of -∆ -µ(x) dened on R N . As a typical example for f , one can think of

f (x, u) = a(x)u -b(x)u p , b(x) > 0, p > 1,
and a(x) has exact asymptotic behavior as O(-|x| -α ) near innity. There, one can take a positive constant to be a test-function for λ α such that (H1b) and (H1c) are satised. Hypothesis 3 requires the existence of supersolution having the same growth as the functions in D α but not necessarily require the growth of the gradient. This allows us to treat more general functions than the functions f (x, u) satisfying the following assumption

∃S ∈ R, S > 0 such that f (x, s) ≤ 0, ∀x ∈ R N , s ≥ S .
For the existence of sharp threshold, although the conclusions of Theorem (2.1.5) and (2.1.6) are similar but the intrinsic phenomena are dierent. Especially, in the compactness argument, for α ∈ [0, 2) the eigenfunction ϕ K associated with λ 1 (-L K , R N ) < 0 decays exponentially while for α = 2 it decays polynomially. Here, thanks to (H1a) and (H1c), we can use the new criteria in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] to achieve the minimal growth at innity of the eigenfunctions (see (2.2.4)-(2.2.5)) to prove the decays of the eigenfunctions.

The interesting point of the long time behavior in Theorem (2.1.7) and Theorem (2.1.8) is that we provide upper critical values of the growth rates for initial data so that u(t, x) converges as t → ∞ to the unique stationary solution (maybe zero) in L p (R N ), for all p > 0. These results are actually obtained thanks to the decays of stationary solutions, where the negativity of λ α plays an important role. Since the stationary solutions may decay very weakly depending on α, they may not be able to keep up with the evolution of ∂ t u. The weight (|x| + 1) -α in front of ∂ t u aims to pull down the growth of ∂ t u so that u(t, x) will converge if started by a suciently decaying initial data. Although we impose some decays on initial data, but these theorems are strong enough since the class of test-functions D α for λ α is considerable and in addition that, by hypothesis (H3), we allow Eq. (2.1) to possess quite large supersolutions, which may be unbounded with suitable growth rates. Moreover, by (2.25), (2.27) and (2.28), we obtain the exact convergences in L p (R N )-weighted norms, which are stronger than L p (R N ) norms. One of the main tools is the extension Phragmén-Lindelöf principle (Theorem 1, [START_REF] Cosner | Asymptotic behavior of solutions of second order parabolic partial dierential equations with unbounded coecients[END_REF]) for parabolic operators with unbounded coecients, which does not seem to be very well known.

Earlier results

Theorem 2.2.1. Let Ω be smooth and coecients of L be bounded. Then, one has i. [Theorem 1.7, (iii) [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]]

λ 1 (-L, Ω) ≤ λ 1 (-L, Ω) ≤ λ 1 (-L, Ω).
ii. [Theorem 1.8 [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]] λ 1 (-L, Ω) = λ 1 (-L, Ω)(= λ 1 (-L, Ω)) if one of the followings holds true: a) L is a self-adjoint, a ij are bounded and either

N = 1 or Ω = R N and L is radially symmetric. b) L = L + γ(x), where L is such that λ 1 (-L, Ω) = λ 1 (-L, Ω) and γ ∈ L ∞ (Ω) is nonnegative and lim x∈Ω,|x|→∞ γ(x) = 0.
c) a ij are bounded, L is a self-adjoint operator and or in non-divergence form satisfying

lim x∈Ω,|x|→∞ |q(x)| = 0; ∀r > 0, ∀β < lim sup x∈Ω,|x|→∞ f s (x, 0), ∃B r (x 0 ) ⊂ Ω s.t. inf Br(x 0 ) f (x, 0) > β. d) λ 1 (-L, Ω) ≤ -lim sup |x|→∞ f (x, 0).
iii. [Proposition 2.2, (vi), [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]] If L is self-adjoint, one has

λ 1 (-L, Ω) = inf φ∈H 1 0 (Ω),φ =0 Ω a ij (x)∂ i φ∂ j φ -f s (x, 0)φ 2 Ω φ 2 .
Theorem 2.2.2 (Proposition 9.2, (ii), [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]). Assume that

a n ij , b n i , c n ∈ L ∞ (R N ). If a ij are uniformly Hölder continuous, b i c are uniformly continuous and a n ij → a, b n i → b i c n → c in L ∞ (R N ), then λ 1 (-L n , R N ) → λ 1 (-L, R N ) as n → ∞, where L n u = a n ij (x)u ij + b n i (x)u i + c n (x)u and Lu = a ij (x)u ij + b i (x)u i + c(x)u. Theorem 2.2.3 (Proposition 4.2 [9]). Let λ R : R + → R be the Dirichlet principal eigenvalue of -L in (2.8) in B R . Then λ R is decreasing and satises lim R→∞ λ R = λ 1 (-L, R N ).
Furthermore, there exists a positive eigenfunction ϕ ∈ W 2,p loc (R N ) for any 1 ≤ p < ∞ associated with

λ 1 (-L, R N ), namely -Lϕ = λ 1 (-L, R N )ϕ a.e in R N . (2.32) Denition 2.2.4. Let Ω be unbounded. A positive function u ∈ W 2,p loc (Ω) satisfying L[u] = 0 a.e in Ω (2.33)
is said to be a solution of minimal growth at innity if for any ρ > 0 and any positive function

v ∈ W 2,p loc (Ω \ B ρ ) satisfying L[v] ≤ 0 a.e in Ω \ B ρ , one nds R ≥ ρ and k > 0 such that ku ≤ v in Ω \ B R .
Theorem 2.2.5 (Theorem 8.4 [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]). If Ω is unbounded, smooth and λ 1 (-L, Ω) satises

λ 1 (-L, Ω) < lim r→∞ λ 1 (-L, Ω \ B r ),
then the eigenfunction ϕ associated with λ 1 (-L, Ω) of (2.32) is a solution of minimal growth at innity and, therefore, λ 1 (-L, Ω) is simple in the class of positive functions.

A-priori estimates

Here, we establish a-priori estimates for solutions (subsolutions) satisfying suitable growth conditions of the linear equation

L[u] = a ij (x)∂ ij u + Kq(x) • ∇u + f s (x, 0)u ≥ 0 (2.
ϕ ∈ D α , C(γ) > 0, R(γ) > 0 u(x) ≤ C(γ)e -γ|x| 1-α 2 ϕ(x)
for |x| ≥ R(γ).

ii) If (H1a), (H1c) and (2.15) are fullled, then for all γ ∈ (0, γ), where γ are given by (2.16), one nds

ϕ ∈ D α , C(γ) > 0, R(γ) > 0 e -ψ K (x) u(x) ≤ C(γ)e -(γ+ 2Kθ 2-α )|x| 1-α 2 ϕ(x) and u(x) ≤ C(γ)e -γ|x| 1-α 2 ϕ(x) for |x| ≥ R(γ).
In particular, u(x) decays exponentially.

Proof. Since Hypothesis (H1) is twofold, the proof is distinguished into two cases :

(i) Case 1. (H1a)-(H1b) and lim sup |x|→∞ |q(x)||x| α 2 ≤ θ hold. There exist α ∈ [0, 2) and µ ∈ L ∞ (R N ) such that for any δ > 0, one nds R = R(δ) > 0 satisfying f s (x, 0) ≤ µ(x) + δ|x| -α and q(x)|x| α 2 ≤ (θ + δ) for |x| ≥ R.
By assumption (H1b), η > 0, for any λ ∈ (0, η), one nds a test function ϕ ∈ D α and

h α ∈ C(R N ) satisfying lim |x|→∞ h α (x)|x| α = 1 such that a ij (x)∂ ij ϕ + Kq(x) • ∇ϕ + µ(x)ϕ ≤ -λh α (x)ϕ for |x| ≥ R. (2.35) Set L µ [φ] = a ij (x)∂ ij φ + Kq(x) • ∇φ + (µ(x) + δ)|x| -α φ,
we see that, for any p > 0, one has

L µ [u] ≥ 0 in B R+p \ B R .
We look for a supersolution of Eq. (2.34) in B R+p \ B R of the form w p (x) = w 1 p (x) + w 2 p (x) = e (R+p) β (τ -γ) e γ|x| β ϕ(x) + e R β (τ +γ) e -γ|x| β ϕ(x), γ, β > 0.

Let us denote for brief that C 1 = e (R+p) β (τ -γ) and C 2 = e R β (τ +γ) and I is the dirac function, we treat rst w 1 p , careful computation yields that

L µ 1 C 1 w 1 p = a ij (x)∂ j γβ(|x| β-2 x i e γ|x| β ϕ(x)) + e γ|x| β ∂ i ϕ(x) + γβKq(x) • x|x| β-2 e γ|x| β ϕ(x) + Kq(x) • ∇ϕ(x)e γ|x| β + µ(x)ϕ(x)e γ|x| β . = a ij (x)γβ (β -2)x i x j |x| β-4 + I(i -j)|x| β-2 + γβx i x j |x| 2β-4 + 2x i |x| β-2 ∂ j ϕ ϕ × × e γ|x| β ϕ + γβKq(x) • x|x| β-2 e γ|x| β ϕ + (a ij (x)∂ ij ϕ + Kq(x) • ∇ϕ + (µ(x) + δ)ϕ) e γ|x| β (2.36)
We choose β = 1 -α 2 > 0. Thanks to (2.35), using the Cauchy-Schwarz inequality, we estimate the right-hand side (RHS) of (2.36) as follow (RHS) of (2.36)

ϕe γ|x| β ≤ a ij (x) γ 1 + α 2 N |x| -α 2 -1 + γI(i -j)|x| -α 2 -1 + γ 2 1 - α 2 N 2 |x| -α + 2γN |x| -α 2 |∇ϕ| ϕ 1 - α 2 + γ 1 - α 2 K|q(x)||x| -α 2 -λh α (x) + δ|x| -α . (2.37) Since ϕ ∈ D α , one has |∇ϕ(x)| ≤ D α |x| -α 2 ϕ(x)
when |x| ≥ R with R large enough, the (RHS) of (2.37) is bounded from above by (RHS) of (2.37)

ϕe γ|x| β ≤ Λ|x| -α γ 1 - α 2 4 N |x| α 2 -1 + γ 1 - α 2 N |x| α 2 -1 + γ 2 1 - α 2 2 N 2 + + 2γ 1 - α 2 N D α + γ 1 - α 2 K Λ (θ + δ) - λ -δ Λ h α (x) |x| -α for |x| ≥ R. (2.38)
Similarly, one has the same computations for w 2 p , in which γ is replaced by -γ. Collecting the all of computations and noting that |x| α 2 -1 → 0 as |x| → ∞, for any xed K, α, in order to have the (RHS) of (2.38) to be negative in R N \ B R , it is sucient to nd γ such that

γ 2 1 - α 2 2 N 2 + 2γ 1 - α 2 N D α + γ 1 - α 2 K Λ (θ + δ) - λ -δ Λ < 0. (2.39)
Observe that as δ → 0, λ is arbitrarily close to η α . Let γ be the positive solution of (2.14), one sees that for any for γ ∈ (0, γ), there exist λ, δ > 0 and R large enough such that (2.39) is satised. This implies that L µ w p ≤ 0 in B R+p \ B R , ∀p > 0. Now, we want to estimate on the boundary of

B R+p \ B R . Since ϕ ∈ D α , there exists a function h(x) → 0 as |x| → ∞ such that ϕ(x) ≥ e h(x)|x| 1-α 2 , for |x| ≥ R.
On the other hand, since u satises (2.13), for any κ > 0, there exists

C = C(κ) > 0 such that u(x) ≤ Ce κ|x| 1-α 2 , for |x| ≥ R.
Choosing arbitrarily γ ∈ (0, γ), τ ∈ (0, γ) and κ = τ /2, we derive, for R large,

Ce τ |x| 1-α 2 ϕ(x)/u(x) ≥ e (τ +h(x)-κ)|x| 1-α 2 ≥ 1, for |x| ≥ R.
In particular, for any p > 0, one has

Cw p (x) ≥ Cw 1 p (x) = Ce τ (R+p) 1-α 2 ϕ(x) ≥ u(x) for x ∈ ∂B R+p , Cw p (x) ≥ Cw 2 p (x) = Ce τ R 1-α 2 ϕ(x) ≥ u(x) for x ∈ ∂B R .
To make use the maximum principle, we set

z = u -Cw p ϕ . Evidently, L µ (zϕ) ≥ 0 in B R+p \ B R for all p > 0 and z ≤ 0 on ∂(B R+p \ B R ). Direct calculation yields 0 ≤ L µ (zϕ) ϕ = ∂ ij z + 2 ∂ j ϕ ϕ ∂ i z + Kq(x) • ∇z + 1 ϕ (∂ ij ϕ + Kq(x) • ∇ϕ + µ(x)ϕ) z ≤ ∂ ij z + 2 ∂ j ϕ ϕ ∂ i z + Kq(x) • ∇z -λh α (x)z.
The zero-order's coecient of operator satised by z is nonpositive, for such R xed, the maximum principle applies implies that z ≤ 0 in B R+p \ B R for all p > 0. As a result,

∀x ∈ B R+p \ B R , u(x) ≤ Cw p (x) = Ce (τ -γ)(R+p) 1-α 2 e γ|x| 1-α 2 ϕ(x) + Ce (τ +γ)R 1-α 2 e -γ|x| 1-α 2 ϕ(x).
Lastly, sending p → ∞, we achieve

x ∈ R N \ B R , u(x) ≤ Ce (τ +γ)R 1-α 2 e -γ|x| 1-α 2 ϕ(x).
In particular, since ϕ ∈ D α , u decays exponentially with the rate O(e -γ|x| 1-α 2 ).

(ii) Case 2. (H1a), (H1c) and (2.15) hold.

Let ψ be given in assumption (H1c), using the transformation

u(x) = e ψ K (x) u(x),
where ψ K = Kψ satisfying 2I N • ∇ψ K (x) = -Kq(x), we convert Eq. (2.34) into self-adjoint form. Plugging u to Eq. (2.34), since divq = 0, we derive after simplifying

(L[u])e -ψ K (x) = ∆ u + -|∇ψ K (x)| 2 + f s (x, 0) u ≥ 0
We would like to use the same type of supersolution as of Case 1, namely

w p (x) = w 1 p (x) + w 2 p (x) = e (R+p) 1-α 2 (τ -κ) e κ|x| 1-α 2 ϕ(x) + e R 1-α 2 (τ +κ) e -κ|x| 1-α 2 ϕ(x),
with suitable positive constants R, τ, γ. Observe that, by assumption (H1a), there exists α ∈ [0, 2) such that for any δ > 0, there exists R = R(δ) > 0 satisfying

f s (x, 0) ≤ µ(x) + δ|x| -α for |x| ≥ R. (2.40)
From (H1c), for any λ ∈ (0, η), where η = λ α (-∆ -µ(x), R N ) > 0, we nd a test function ϕ ∈ D α and

h α ∈ C(R N ) satisfying lim |x|→∞ h α (x)|x| α = 1 such that ∆ϕ + µ(x)ϕ ≤ -λh α (x)ϕ for |x| ≥ R. (2.41)
From (2.15), one has

|∇ψ K (x)| 2 ≥ K 2 θ 2 -δ |x| α for |x| ≥ R = R(δ) large enough.
and thus

∆ u + - K 2 θ 2 -δ |x| α + f s (x, 0) u ≥ 0.
For δ > 0, we set

L µ [φ] = ∆φ + µ(x)φ - K 2 θ 2 -2δ |x| α φ x ∈ R N \ {0}.
Obviously, L µ [ u] ≥ 0 for |x| ≥ R. The constants λ and δ will be chosen later. As in Case 1, we want to nd γ > 0 small enough and R > 0 large enough such that

L µ w p ≤ 0 in B R+p \ B R , ∀p > 0. Since |x| α 2 -1 → 0 as |x| → ∞,
similar computations as (2.36)-(2.38) lead us to determine γ in such the way that

κ 2 1 - α 2 2 + 2κ 1 - α 2 D α -(λ + K 2 θ 2 -2δ) < 0.
As δ → 0, λ is arbitrarily close to η. Let γ + be the positive solution of

κ 2 1 - α 2 2 + 2κ 1 - α 2 D α -(η α + K 2 θ 2 ) = 0, that is γ + = -2D α + 2 D 2 α + η α + K 2 θ 2 2 -α ,
we see that for any κ ∈ (0, γ + ), there exist λ, δ > 0 and R large enough such that ∀p > 0,

L µ w p ≤ 0 in B R+p \ B R . Also from (2.15) lim sup |x|→∞ |ψ(x)| |x| 1-α 2 = 2θ 2 -α , θ < η α 2KD α ,
hence for any κ ∈ ( 2Kθ 2-α , γ + ), we readily nd λ ∈ (0, η α ) and δ > 0 such that lim sup

|x|→∞ |ψ K (x)| |x| 1-α 2 ≤ 2Kθ 2 -α < κ = -2D α + 2 D 2 α + λ + K 2 θ 2 -2δ 2 -α < γ + . (2.42)
The estimates on the boundary B R+p \ B R are similar in Case 1. Since u satises (2.13), for any 0 < ς, there exists

C = C(ς) > 0 such that u(x) ≤ Ce ς|x| 1-α 2 , in R N .
We choose τ and κ such that 0 < -2Kθ 2-α + τ < -2Kθ 2-α + κ < γ and ς = 1 2 (-2Kθ 2-α + τ ). Hence, for R large, one has

Ce τ |x| 1-α 2 ϕ(x) u(x) = Ce τ |x| 1-α 2 ϕ(x)e ψ K (x) u(x) = e (τ -2Kθ 2-α -ς+h(x))|x| 1-α 2 ≥ 1 for |x| ≥ R.
In particular, for any p > 0, C w p ≥ u in ∂(B R+p \ B R ). The remainder of the proof goes exactly as of Case 1. As a consequence, by letting p → ∞, one has

e -ψ K (x) u(x) = u(x) ≤ Ce R 1-α 2 (τ +κ) e -κ|x| 1-α 2 ϕ(x) in R N \ B R .
Due to (2.42), for R large,

u(x) ≤ Ce R 1-α 2 (τ +κ) e -κ|x| 1-α 2 e ψ K (x) ϕ(x) ≤ Ce R 1-α 2 (τ +κ) e -(κ-2Kθ 2-α )|x| 1-α 2 ϕ(x) in R N \ B R . (2.43) Set γ = κ - 2Kθ 2 -α ∈ (0, γ) γ = -2D α + 2 D 2 α + η α + K 2 θ 2 2 -α - 2Kθ 2 -α ,
we achieve statement ii). Since ϕ ∈ D α , u(x) decays exponentially and the proof is thereby complete.

ϕ ∈ D 2 , C(β) > 0, R(β) > 0 u(x) ≤ C(β)|x| -β ϕ(x) in |x| > R(β).
ii) If (H1a), (H1c), (2.19), (2.20) are fullled, then for all β ∈ (0, β), where β is given by

β = (N -2 + 2D 2 ) + (N -2 + 2D 2 ) 2 + 4η 2 + 4K 2 θ 2 2 -Kθ, (2.44) one nds ϕ ∈ D 2 , C(β) > 0, R(β) > 0 e -ψ K (x) u(x) ≤ C(β)|x| -(β+Kθ) ϕ(x) and u(x) ≤ C(β)|x| -β ϕ(x) in |x| > R(β).
In particular, u(x) decays polynomially.

Proof. The scheme of the proof of Theorem 2.3.1 can be applied, thus we only outline the points, which are mainly dierent from the former. (i) Case 1: (H1a)-(H1b) and lim sup |x|→∞ |q(x)||x| ≤ θ By Hypothesis (H1), there exists µ ∈ L ∞ (R N ) such that for any δ > 0, one nds R = R(δ) > 0 satisfying

f s (x, 0) ≤ µ(x) + δ|x| -2 and q(x)|x| ≤ (θ + δ) for |x| ≥ R.
Assumption (H1b) yields that for any λ ∈ (0, η 2 ), one nds a test function ϕ ∈ D 2 and h 2 ∈ C(R N ) satisfying lim |x|→∞ h 2 (x)|x| 2 = 1 such that

a ij (x)∂ ij ϕ + Kq(x) • ∇ϕ + µ(x)ϕ ≤ -λh 2 (x)ϕ for |x| ≥ R. Set L µ [φ] = a ij (x)∂ ij φ + Kq(x) • ∇φ + (µ(x) + δ)|x| -2 φ,
obviously, for any p > 0, one has

L µ [u] ≥ 0 in B R+p \ B R . It is expected to nd a supersolution of Eq. (2.34) in B R+p \ B R of the form w p (x) = w 1 p (x) + w 2 p (x) = C ln(R + p) |x| β ϕ(x) + C|x| -β ϕ(x), β > 0.
Direct computations similarly to Case 1 of Theorem 2.3.1, we see that w p is a supersolution of Eq. (2.34) if β satises the following algebraic inequality

N Λβ 2 + (3N Λ + 2ΛD 2 + Kθ)β -λ + δ ≤ 0
where the constant D 2 appears due to ϕ ∈ D 2 . As δ → 0, λ is arbitrarily close to η 2 , thus β necessarily satises

0 < β < σ = -(3N Λ + 2ΛD 2 + Kθ) + (3N Λ + 2ΛD 2 + Kθ) 2 + 4ΛN η 2 2N Λ .
Thanks to (2.17), for all σ > 0, there exists C = C(σ) such that

u(x) ≤ C(σ)|x| σ for |x| ≥ R
and the growth condition of ϕ yields that there exist g 1 (x), g 2 (x) → 0 as |x| → ∞ and large R

|x| g 1 (x) ≤ ϕ(x) ≤ |x| g 2 (x) |x| ≥ R.
Fix such R, choosing arbitrarily β ∈ (0, σ) and σ ∈ (0, β), then

lim inf |x|=R+p,p→∞ w p u ≥ lim inf |x|=R+p,p→∞ w 1 p u ≥ lim inf |x|=R+p,p→∞ C|x| σ u(x) |x| β-σ |x| g 1 (x) ln(R + p) = ∞.
By using the same argumentation as of Case 1 of Theorem 2.3.1 and the weak comparison principle, we imply that u(x) ≤ w p (x)

x ∈ B R+p \ B R ,
where C = C(β) is a large positive constant independent of u and p. By letting p → ∞, thanks to the growth assumption of ϕ, we obtain the polynomial decay

u(x) ≤ C|x| -β ϕ(x) |x| > R(β).
(ii) Case 2 : (H1a), (H1c) and (2. [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF] We omit the computations because we use the same technique as of Case 2 of Theorem 2.3.1. However, we will precisely provide the type of the supersolution on B R+p \ B R , which is important to derive the decay of solution.

Thanks to divq = 0, we can convert Eq. (2.34) into self-adjoint form by the transformation

u(x) = e -ψ K (x) u(x),
where ψ K = Kψ satisfying 2I N • ∇ψ K (x) = -Kq(x), namely u satises

∆ u + -|∇ψ K (x)| 2 + f s (x, 0) u ≥ 0 x ∈ R N .
For Cβ 1 , β 2 > 0, ϕ ∈ D 2 , by plugging the function

w p = w 1 p + w 2 p = C (ln(R + p)) 2 |x| β 1 ϕ(x) + C|x| -β 2 ϕ(x),
to Eq. (2.34), we see that w p is a supersolution of (2.34) if β 1 and β 2 satisfy the following algebraic equations for some δ > 0 Taking arbitrary constants β 1 ∈ (Kθ, β + 1 ) and β 2 ∈ (Kθ, β + 2 ), there exist

β 2 1 + (N -2 + 2D 2 )β 1 -(η 2 + K 2 θ 2 ) + δ ≤ 0 β 2 2 -(N -2 + 2D 2 )β 2 -(η 2 + K 2 θ 2 ) + δ ≤ 0. Necessarily, 0 < β 1 < β + 1 = -(N -2 + 2D 2 ) + (N -2 + 2D 2 ) 2 + 4η 2 + 4K 2 θ 2 2 0 < β 2 < β + 2 = (N -2 + 2D 2 ) + (N -2 + 2D 2 ) 2 + 4η 2 + 4K
C = C(β), R = R(β) > 0, ϕ ∈ D 2 large enough such that for all p > 0 u(x) ≤ C (ln(R + p)) 2 |x| -β 1 ϕ(x) + C|x| -β 2 ϕ(x) x ∈ B R+p \ B R ,
where we applied the comparison principle on B R+p \ B R . Note that Kθ < β + 2 and β 2 -Kθ ∈ (0, β). By sending p → ∞, thanks to the growth rate of ϕ, we obtain the polynomial decay:

u(x) ≤ C|x| -(β 2 -Kθ) ϕ(x) |x| > R(β).
(2.45)

Existence/Nonexistence and Uniqueness

In this section, we investigate the existence/nonexistence and uniqueness of Eq. (2.1) for every xed constant K. Since Eq. (2.1) is not self-adjoint in general, the variational approach, which was used in Theorem 2.3 [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], does not succeed to apply here. However, replied on the a-priori estimates in Section 2, we are able to prove Theorem 2.1.2 by exploiting the strong maximum principle. To this aim, we will prove a comparison principle, which covers both two cases of Hypothesis 1. First, we observe that, for any xed K, from condition (H1c), u is a solution (resp. sub or supersolution) of Eq. (2.1) if and only if u(x) = e -ψ K (x) u(x) with ψ K = Kψ is a solution (resp. sub or supersolution) of following equation

∆ u(x) -|∇ψ K (x)| 2 u + f (x, ue ψ K (x) )e -ψ K (x) = 0 (2.46)
Therefore, the existence/nonexistence and uniqueness of Eq. (2.46) is equivalent to Eq. (2.1). It will be convenient to denote the following operators and functions

M 1 [u] = a ij (x)∂ ij u(x) + Kq(x) • ∇u and M 2 [ u] = ∆ u -|∇ψ K (x)| 2 u; f 1 (x, s) = f (x, s) and f 2 (x, s) = f (x, se ψ K (x) )e -ψ K (x) .
Our comparison principle is based on the ideas of the analogous result, Theorem 3.3 in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF]. However, the interesting novelties are due to the new hypothesis (H1), which allows f s (x, 0) to change sign and it possibly decays fast up to Hardy potential beside the presence of the nonconstant drift term and nonconstant behavior near innity of f s (x, 0).

Theorem 2.4.1. Let α ∈ [0, 2], u i and v i , i ∈ {1, 2} respectively satisfy the following inequalities

M i [u i ] + f i (x, u i ) ≥ 0 and M i [v i ] + f i (x, v i ) ≤ 0 a.e in R N .
Assume that Hypothesis (H2) holds and inf Br v i (x) > 0 for all r and i ∈ {1, 2}. Then, if (H1a) and (H1b) hold, one has u 1 (x) ≤ v 1 (x) in R N . Alternatively, if (H1a), (H1c) and (2.15) hold, then

u 2 (x) ≤ v 2 (x) in R N .
Proof. Thanks to Hypothesis (H2), one has ∂ s f i (x, 0)s ≥ f i (x, s) for i ∈ {1, 2}. Therefore,

M i [u] + ∂ s f i (x, 0)u i ≥ 0 in R N .
Since ∂ s f i (x, 0) satises Hypothesis (H1), there exists α ∈ [0, 2] such that for any δ > 0, there exists

R = R(δ) > 0 satisfying ∂ s f i (x, 0) ≤ µ(x) + δ(|x| + 1) -α for |x| ≥ R. One nds λ i > 0, ϕ i ∈ D α and h α ∈ C(R N ) such that lim sup |x|→∞ h α (x)|x| α = 1 and M i [ϕ i ] + µ(x)ϕ i ≤ -λ i h α (x)ϕ i for |x| ≥ R. (2.47)
From Theorem (2.3.1)-(2.3.2), there exists a function g α i (x) > 0 such that for R large enough

u i (x) ≤ g α i (x)ϕ i (x), in R N \ B R ,
where

g α i (x) = C i e -γ i |x| 1-α 2 if α ∈ [0, 2) and g α i (x) = D i |x| -β i if α = 2 for some C i , γ i , D i , β i > 0.
Let ε be an arbitrary positive constant, we dene the set

K i ε := {k > 0 : kv i ≥ u i -εϕ i in R N }. Evidently, K i ε is nonempty because g α i (x) → 0 as |x| → ∞, u i > 0 in B r , ∀r > 0 and u i (x) -εϕ i (x) ≤ (g α i (x) -ε)ϕ i ≤ 0, for |x| ≥ R(ε) large enough. (2.48)
Hence there exists the inmum k i (ε) := inf K i ε . The function k i : R + → R is nonincreasing. Assume by contradiction that

k * i = lim ε→0 + k i (ε) > 1.
Taking 0 < ε < sup R N {u i /ϕ}, we get k i (ε) > 0 and

w ε i = k i (ε)v i -u i + εϕ i ≥ 0. By the denition of k i (ε) one nds a sequence (x ε n ) ∈ R N such that k i (ε) - 1 n v i (x ε n ) < u i (x ε n ) -εϕ i (x ε n ). Fix ε > 0, (2.48) implies that (x ε n ) ∈ B R(ε)
for n large enough. Thus, it converges up to a subsequence to some x(ε) ∈ B R(ε) . Then, one nds

w ε i (x(ε)) = 0. (2.49)
The case that lim inf

ε→0 + |x(ε)| < ∞ is ruled out. In fact, if there is x 0 ∈ R such that |x 0 | = lim inf ε→0 + |x(ε)|, by (2.49), k * < ∞, the function w ∞ i = k * v i -u i is nonnegative, vanishes at x 0 and satises -M i [w ∞ i ] ≥ k * i f i (x, v i ) -f i (x, u i ) ≥ [f i (x, k * i v i ) -f i (x, u i )] ≥ σ i (x)w ∞ i ,
where σ i (x) ∈ L ∞ loc (R N ). But, by Hypothesis 2, the inequality is strictly in D with |D| > 0. This contradicts to the strong maximum principle, which asserts that w ∞ i ≡ 0 in R N . It remains to consider the case lim

ε→0 + |x(ε)| = ∞.
As above, we know that w ε i ≥ 0 and it vanishes at x(ε) ∈ B R(ε) . Hence, there

exists a neighborhood O(x(ε)) of (x(ε)) such that k i (ε)v i < u i in O(x(ε)). For ε small enough, k i (ε) > 1,
we derive from Hypothesis 2, for x ∈ O(x(ε)):

-M i [w ∞ i ] ≥ k i (ε)f i (x, v i ) -f i (x, u i ) + µ(x)εϕ i + λ i h α (x)εϕ i ≥ f i (x, k i (ε)v i ) -f i (x, u i ) + µ(x)εϕ i + λ i h α (x)εϕ i ≥ f i (x, k(ε)v i ) k i (ε)v i (k(ε)v i -u i + εϕ i ) + λ i 2 h α (x) + µ(x) - f i (x, k(ε)v i ) k(ε)v i εϕ i + λ i 2 h α (x)εϕ i . (2.50)
Taking 0 < ε 1 then |x(ε)|γ1, we derive from (H1a) and Hypothesis (H2) that

f i (x, k(ε)v i ) k(ε)v i ≤ ∂ s f i (x, 0) < µ(x) + λ i h α (x) 2 , ∀x ∈ O(x(ε)),
shrinking O if necessary. We get from (2.50) that

-M i [w ε i ] -i (x)w ε i > 0 in O(x(ε))
,

where i (x) = f i (x,k(ε)v i ) k(ε)v i ∈ L ∞ (O(x(ε))
) . This contradicts again to the strong maximum principle, which asserts that

w ε i ≡ 0 in O(x(ε)). Therefore, k * i > 1 is ruled out, we actually have k * i ≤ 1. Letting ε → 0 + , u i ≤ lim ε→0 + (k i (ε)v i + εϕ i ) ≤ v i in R N .
This completes the proof of theorem.

Proof of Theorem 2.1.2. Theorem (2.4.1) implies in particular that Eq. (2.1) cannot possess more than one positive solution. It remains to prove the existence and nonexistence of Eq. (2.1).

• λ 1 (-L, R N ) ≥ 0 : nonexistence. Observe that, if (H1c) holds, the transformation u(x) = e -ψ K (x) u(x) converts Eq. (2.1) to be self-adjoint, namely

∆ u -|ψ K (x)| 2 u + f (x, ue ψ K (x) )e -ψ K (x) = 0. (2.51)
Using the same notations M i and f i as before, we see that

λ 1 (-M i -∂ s f i (x, 0), R N ) = λ 1 (-L, R N ) ≥ 0.
We argue by contradiction that there exists at least a positive solution u of Eq. (2.1) or Eq. (2.51).

Let us call φ i , i ∈ {1, 2}, with normalization φ i (0) < u(0), the eigenfunctions associated with λ 1 (-M i -∂ s f i (x, 0), R N ), which are obtained by Theorem (2.2.3). Hypothesis 2 yields

M i [φ i ] + f i (x, φ i ) ≤ M i [φ i ] + ∂ s f i (x, 0)φ i = -λ 1 (-M i -∂ s f i (x, 0), R N )φ i ≤ 0.
Therefore, φ i is a supersolution of the equation

M i [u] + f i (x, u) = 0.
Theorem (2.4.1) applies, we get u(x) ≤ φ i (x) in R N . This contradiction conrms the nonexistence.

• λ 1 (-L, R N ) < 0 : existence.

Thanks to the property (i) of Theorem (2.2.1), we have λ 1 (-L, R N ) ≥ λ 1 (-L, R N ). Whence, λ 1 (-L, R N ) < 0. This implies that there exists a bounded test function φ > 0 and a negative constant λ close enough to

λ 1 (-L, R N ) such that a ij (x)∂ ij φ(x) + Kq i φ i (x) + f s (x, 0)φ(x) + λφ(x) ≥ 0.
By assumption f (x, s) is C 1,γ (R N ) uniformly with respect to s ∈ [0, δ] for some δ > 0, one sees that the function φ = κφ with κ small enough is a subsolution of Eq. (2.1), namely

a ij (x)∂ ij φ(x) + Kq i φ i (x) + f (x, φ i ) ≥ 0 in R N .
On the other hand, by Theorem (2.3.1), we know that there exists positive constants C, γ and ϕ ∈ D α such that for R large enough

φ(x) ≤ Ce -γ|x| 1-α 2 ϕ(x) for |x| ≥ R.
Thanks to Hypothesis 3, there exists a positive supersolution, S K (x) of Eq. (2.1) satisfying

lim |x|→∞ ln |S K (x)||x| -1+ α 2 = 0. Since lim |x|→∞ ln ϕ(x)|x| -1+ α 2 = 0, one has lim inf |x|→∞ S K (x) φ(x) ≥ lim inf |x|→∞ S K (x)e γ 2 |x| 1-α 2 Ce -γ 2 |x| 1-α 2 ϕ(x) = ∞.
Therefore, there exists R > 0 large enough such that S K (x) ≥ φ(x) in R N \ B R . Obviously, for κ suciently small, S K (x) ≥ φ(x) in R N . The classical iteration method allows one to nd a positive solution u for Eq.

(2.1) satisfying φ ≤ u ≤ S K . Theorem (2.3.1) implies that u(x) decays exponentially as |x| → ∞ and this completes the proof.

Remark 11. The existence/nonexistence of positive bounded solution are still true when the supersolution

S K (x) is allowed to satisfy lim sup |x|→∞ ln(S K (x))|x| 1-α 2 < γ ,
where γ is given in (i) of Theorem (2.1.2). In fact, due to Theorem (2.3.1), this growth condition is sucient for proving exponential decay once a solution exists. Here, we use the lower bound of S K (x) to construct a positive solution between φ and S K , but there is another way to construct a compactly supported subsolution and then using the strong maximum principle to derive the positivity on R N for solutions. We refer to [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] for this construction.

As previously mentioned, Theorem (2.1.3) only diers from Theorem 2.1.2 at the growth condition of supersolution S K (x) given by Hypothesis (H3). For α = 2, Hypothesis (H3) provides the existence of supersolution, which grows slower than any polynomials and it is enough due to Theorem (2.3.2) to derive the polynomial decay for solutions of Eq. (2.1). This result presents the fact that if f s (x, 0) decays fast enough, for instance, like a Hardy potential, existence of positive bounded solutions for Eq. (2.1) may not be derived from the existence of fast growing supersolution. Theorem (2.1.3) is actually a direct consequence of Theorem (2.3.2) and Theorem (2.1.2). Its proof is omitted.

Sharp threshold of drift

In this section, we prove the existence of sharp threshold of drift term K for which Eq. (2.1) (when (a ij ) is the identity matrix) admits positive solutions. From Theorem (2.1.2), we know that the existence of positive solutions of Eq. (2.1) is straightly conditioned by the sign of the generalized principal eigenvalue of the linearized operator of Eq. (2.1) about 0. For the simplicity of presentation, we denote by

L K [φ] = ∆φ + Kq i φ i + f s (x, 0)φ, λ K = λ 1 (-L K , R N
) and ϕ K is an associated eigenfunction of λ K given by Theorem (2.2.3), namely

L K [ϕ K ] = -λ K ϕ K in R N . (2.52)
In order to prove the existence of K , we aim at showing that • There exist K 1 and K 2 such that λ K 1 < 0 and λ K 2 ≥ 0.

• λ K is continuous and nondecreasing with respect to K.

These facts are indeed the conclusions of the following lemmas :

Lemma 2.5.1. λ K is continuous and nondecreasing with respect to K. Moreover,

lim K→0 λ K = inf φ∈H 1 (R N ), φ L 2 (R N )=1 R N |∇φ| 2 -f s (x, 0)φ 2 .
Lemma 2.5.2. Let α ∈ [0, 2), K > 0 be such that λ K < 0. If (H1a), (H1c), (2.15) with θ = 0 hold, then one can normalize ϕ K in such the way sup R N ϕ K = 1 and there exist positive constants γ, C(γ) independent of K such that

ϕ K (x) ≤ C(γ)e -γ|x| 1-α 2 , in R N .
Thus, for such K, ϕ K decays exponentially.

Lemma 2.5.3. Let α ∈ [0, 2), I q be the set of all rst integrals dened in (2.21). If (H1a), (H1c), (2.15) with θ = 0 hold and

inf φ∈Iq R N |∇φ| 2 -f s (x, 0)φ 2 R N φ 2 > 0,
with the convention that the inmum is +∞ if I q being empty. Then there exists K > 0 such that λ K ≥ 0 for K ≥ K .

Lemma 2.5.4. Let α = 2, K > 0 be such that λ K < 0. If (H1a), (H1c), (2.19) with θ = 0 hold. For any β ∈ (0, β), where β given in (2.44), one can normalize ϕ K in such the way sup R N ϕ K = 1 and there exists C(β) > 0 independent of K such that

ϕ K (x) ≤ C(β)|x| -β φ(x), in R N .
Thus, for such K, ϕ K decays polynomially .

Lemma 2.5.5. Let α = 2, I q be the set of all rst integrals dened in (2.21). If (H1a), (H1c), (2. [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF]) with θ = 0 hold and

inf φ∈Iq R N |∇φ| 2 -f s (x, 0)φ 2 R N φ 2 > 0,
with the convention that the inmum is +∞ if I q being empty. Then the conclusion of Lemma (2.5.3) remains true.

Assume for the moment that these lemmas hold true, we go through to the

Now, we respectively prove above-mentioned lemmas

Proof of Lemma 2.5.1. Thanks to (H1c), let ψ K = Kψ, the transformation ϕ K (x) = e -ψ K (x) ϕ K (x) converts Eq. (2.52) into the self-adjoint form

L[ ϕ K ] = ∆ ϕ K + -|∇ψ K | 2 + f s (x, 0) ϕ K = ∆ ϕ K + - K 2 4 |q(x)| 2 + f s (x, 0) ϕ K = -λ K ϕ K (2.53)
Clearly, λ K is increasing with respect to K. Since |q(x)| and f s (x, 0) are bounded, one has the uniform convergence

- K 2 4 |q(x)| 2 + f s (x, 0) → f s (x, 0) as K → 0.
Theorem (2.2.2) implies that λ K is continuous with respect to K and λ K → λ 1 (-∆ -f s (x, 0), R N ) as K → ∞. Finally, the variational characterization of λ 1 (-∆ -f s (x, 0), R N ) is derived from the part (iii) of Theorem (2.2.1).

Proof of Lemma 2.5.2. From (H1a), there exists α ∈ [0, 2) such that for any δ > 0, there exist R(δ) such that f s (x, 0) ≤ µ(x) + δ|x| -α for |x| ≥ R(δ).

By (H1c), there exist φ ∈ D α , ξ ∈ (0, η α ) and R 1 > 0 such that

∆φ + µ(x)φ + ξh α (x)φ ≤ 0 in R N \ B R 1 .
Set z K = ϕ K /φ, one obtains, by direct computation from (2.53) that

(L K + λ K )[z K ] = ∆z K + 2∇z K • ∇φ φ + z K ∆φ φ + -|∇ψ K | 2 + f s (x, 0) z K + λ K z K = 0. (2.54)
Therefore, z K is an eigenfunction of L K associated with the principal eigenvalue λ K . Since λ K < 0, choosing δ < ξ, the following inequalities hold

lim r→∞ λ 1 (-L K , R N \ B r ) ≥ lim r→∞ λ 1 (-L K , R N \ B r ) ≥ lim r→∞ (-sup R N \Br c(x)) ≥ 0 > λ K , where c(x) = ∆φ + f s (x, 0)φ(x) -|∇ψ K | 2 φ φ ≤ (-ξ + δ)h α (x) ≤ 0 as |x| → ∞.
Theorem 2.2.5 implies that z K has minimal growth at innity. Moreover, for |x| ≥ R(δ)

(L K + λ K )[1] ≤ ∆φ + (µ(x) + δ(|x| + 1) -α )φ φ + λ K ≤ (-ξ + δ)(|x| + 1) -α + λ K < 0.
Denition (2.2.4) yields that z K is bounded. Since θ = 0 and φ ∈ D α , one has

lim sup |x|→∞ ln ϕ K (x) |x| 1-α 2 = lim sup |x|→∞ ln z K (x)φ(x)e ψ K (x) |x| 1-α 2 = 0.
Since Lϕ K = -λ K ϕ K ≥ 0, thanks to (H1a), (H1c), (2.15), Theorem 2.3.1 part ii) implies that for any

κ ∈ 0, -2Dα+2 √ D 2 α +η α 2-α
, there exist C(κ), R(κ) > 0 and τ ∈ (0, κ) independent of K such that

ϕ K (x) ≤ C(κ)e R 1-α 2 (τ +κ) e -κ|x| 1-α 2 for |x| > R(κ).
Indeed, this is a direct observation from (2.43) by substituting θ = 0. Therefore, ϕ K are bounded solutions of the linear equation Lϕ K = -λ K ϕ K ≥ 0. We can normalize ϕ K in such the way sup R N ϕ K = 1 and nally obtain the uniform bound for some

C 1 (κ) ≥ C(κ) ϕ K (x) ≤ C 1 (κ)e R 1-α 2 (τ +κ) e -κ|x| 1-α 2 in R N .
The proof is thereby complete.

Proof of Lemma 2.5.3. Observe that if there exists K > 0 such that λ K ≥ 0, we complete the proof since λ K is nondecreasing with respect to K. Recall that the eigenpair (λ K , ϕ K ) satises

∆ϕ K (x) + Kq i (x)∂ i ϕ K (x) + f s (x, 0)ϕ K + λ K ϕ K = 0 in R N . (2.55)
Assume by contradiction that λ K < 0 for all K > 0, then

lim sup K→∞ λ K ≤ 0. (2.56)
By Lemma (2.5.2), we can normalize ϕ K in such the way ϕ K L ∞ (R N ) = 1. First, we consider the case I q is not empty. Let z ∈ I q and ε > 0, multiplying Eq. (2.55) by

z 2 ϕ K +ε ∈ W 1,1 (R N ) and integrating over R N , one gets - R N ∆ϕ K z 2 ϕ K + ε -K R N q • ∇(z 2 ln(ϕ K + ε)) + 2K R N z ln(ϕ K + ε)q • ∇z + - R N f s (x, 0) z 2 ϕ K ϕ K + ε = λ K R N z 2 ϕ K ϕ K + ε . (2.57)
By Lemma 2.5.2, the interior elliptic estimates and Harnack inequality yields that for

1 ≤ p < ∞ ϕ K W 2,p (B 1 (x)) ≤ C 2 (K, γ) ϕ K L ∞ (B 2 (x)) ≤ C 3 (K, γ)ϕ K (x), in R N .
Hence the Sobolev injection of W 2,p in C 1 for p > N implies that ∇ϕ

K (x) L ∞ (B 1 (x)) ≤ C 3 (K, γ)ϕ K (x) in R N . We have |∇ϕ K (x)| → 0 as |x| → ∞, then lim R→∞ ∂B R z 2 ϕ K + ε ∇ϕ K • ndS ≤ lim R→∞ z 2 ϕ K + ε L 1 (∂B R ) ∇ϕ K • n L ∞ (∂B R ) = 0.
(2.58) Therefore, by Stokes theorem, the rst term of (2.57) can be estimated by

- R N ∆ϕ K z 2 ϕ K + ε = R N ∇ϕ K • ∇ z 2 ϕ K + ε = R N 2z(ϕ K + ε)∇ϕ K • ∇z -z 2 ∇ϕ K • ∇ϕ K (ϕ K + ε) 2 ≤ R N |∇z| 2 . (2.59) Since ϕ K > 0 is bounded, z 2 ln(ϕ K + ε) ∈ W 1,1 (R N ), it can be approximated in W 1,1 norm by a sequence in (u n ) ∈ C ∞ c (R N ).
Moreover, q is divergence-free, the integration by part yields

R N q • ∇u n = sptun q(x) • ∇u n = 0,
and thus the second term of (2.57) is zero. The third term of (2.57) is obviously equal to zero since z ∈ I q .

As a consequence, we obtain

R N f s (x, 0) z 2 ϕ K ϕ K + ε + λ K R N z 2 ϕ K ϕ K + ε ≤ R N |∇z| 2 .
Since f s (x, 0) is bounded and z ∈ H 1 (R N ), by the Lebesgue dominated convergence theorem, we derive the following limit in L 1 (R N )

g ε (x) = f s (x, 0) z 2 ϕ K ϕ K + ε + λ K z 2 ϕ K ϕ K + ε → f s (x, 0)z 2 + λ K z 2 as ε → 0 It follows that λ K ≤ M = R N |∇z| 2 -f s (x, 0)z 2 R N z 2 .
(2.60)

Multiplying ϕ K to (2.55) and integrating over R N , we derive by Stokes theorem that

- R N |∇ϕ K | 2 + K 2 R N q(x) • ∇(ϕ K ) 2 + R N f s (x, 0)ϕ 2 K = -λ K R N ϕ 2 K . (2.61)
Thanks to the exponential decay of ϕ 2 K and Stokes theorem, the integration by part again yields

R N q(x).∇(ϕ K ) 2 = - R N ϕ 2 K divq = 0. (2.62)
Consequently,

λ K R N ϕ 2 K = R N |∇ϕ K | 2 - R N f s (x, 0)ϕ 2 K . (2.63)
From Lemma 2.5.2, there exists an exponential decay ϕ

(x) = C(γ)e -γ|x| 1-α 2 independent of K such that ϕ K ≤ ϕ in R N . Since λ K < 0 R N |∇ϕ K | 2 dx ≤ R N f s (x, 0)ϕ 2 K ≤ sup R N f s (x, 0) ϕ L 2 (R N ) .
By the normalization of ϕ K , we deduce that there exists a function ϕ ∞ ∈ H 1 (R N ), which is not identically equal to zero, such that ϕ Kn converges (up to subsequence) to ϕ ∞ weakly in H 1 (R N ) and strongly in

L 2 loc (R N ) due to the compact injection H 1 loc (R N ) → L 2 loc (R N ).
The Lebesgue dominated convergence theorem and the lower semicontinuity property imply that

lim n→∞ ϕ Kn L 2 (R N ) = ϕ ∞ L 2 (R N ) and R N |∇ϕ ∞ | 2 ≤ lim inf n→∞ R N |∇ϕ Kn | 2 .
Moreover, we see that ∆ϕ K ∆ϕ ∞ in the sense of distributions. Dividing (2.55) by K n and passing to the limit n → ∞, one gets q • ∇ϕ ∞ = 0 a.e in R N . Therefore,

ϕ ∞ ∈ I q . Since f s (x, 0) ∈ L ∞ (R N ), ϕ K → ϕ ∞ strongly in L 2 (R N )
, moreover λ Kn is nondecreasing and λ Kn ≤ M by (2.60), we can pass to the limit from (2.63)

lim inf n→∞ λ Kn R N ϕ 2 ∞ ≥ R N |∇ϕ ∞ | 2 - R N f s (x, 0)ϕ 2 ∞ (2.64)
By (2.22), we achieve the contradiction of (2.56)

lim inf K→∞ λ K ≥ inf φ∈Iq R N |∇φ| 2 -f s (x, 0)φ 2 R N φ 2 > 0.
This implies that there exists K such that λ K ≥ 0 and the monotonicity of λ K Lemma (2.5.1) implies that λ K ≥ 0 for K ≥ K . Actually (2.60) and (2.64) also implies that λ Kn → +∞ as n → ∞ if I q is empty. The proof is therefore complete.

The arguments of the proofs of lemmas (2.5.2) and (2.5.3) remain valid for lemmas (2.5.4) and (2.5.5), therefore we shall just point out the main dierences without necessarily providing details.

Proof of Lemma 2.5.4. To prove Lemma (2.5.4), we need to apply Theorem (2.3.2) part ii) instead of Theorem (2.3.1) part ii) to derive the polynomial decay for the principal eigenfunctions.

Proof of Lemma 2.5.5. The proof of Lemma 2.5.5 can be followed by the scheme of Lemma (2.5.3). However, we need to verify by line that all the arguments remain valid. We see that (2.58),(2.59) are true for any |∇ϕ K (x)| → 0 as |x| → ∞, without concerning the rate of decay. But in order to apply the Stokes theorem to derive the formulations at (2.61), (2.62), ϕ K and |∇ϕ K | have to decay fast enough. More precisely, ϕ K ∈ L 2 (R N ) and lim R→∞ ∂B R |ϕ K ||∇ϕ K |ds = 0. By Lemma 2.5.4, to guarantee that there exists β ∈ (0, β) such that both the requirements are true, it suces to have

2β > N ⇐⇒ N -2 + 2D 2 + (N -2 + 2D 2 ) 2 + 4η 2 > N,
where we use assumption (2.19) with θ = 0. By a simple calculation, we see that the last inequality is equivalent to condition (2.23). The proof is therefore complete.

Remark 12. If q is a constant vector eld, the results of theorems 2.1.5 and 2.1.6 are the consequence of theorems 2.1.2 and 2.1.3. In fact, by the Liouville transformation, we get

λ 1 (-∆ -q • ∇ -f s (x, 0), R N ) = q 2 4 + λ 1 (-∆ -f s (x, 0), R N ). (2.65)
Under the hypotheses (H1), (H2), and (H3), theorems 2.1.2 and 2.1.3 assert that the existence and nonexistence of (2.1) are directly conditioned by the sign of λ 1 (-∆-q •∇-f s (x, 0), R N ). More precisely, by (2.65), we see that (2.1) admits a positive solution i |q| < -2λ 1 (-∆ -f s (x, 0), R N ) and this is the threshold value.

2.6

Long time behavior in L ∞ and L p norms Proof of Theorem (2.1.7). Thanks to Hypothesis 3, for xed K and α ∈ [0, 2), there exists a supersolution of Eq. (2.1), S K (x) satisfying lim |x|→∞ ln S K (x)|x|

α 2 -1 = 0. Since u 0 (x)/S K (x) ∈ L ∞ (R N ), there exists a constant M > 1 such that |u 0 (x)| ≤ M S K (x). Obviously M S K (x)
is also a supersolution of stationary equation of Eq. (2.2). Let w be the solution of Eq. (2.2) starting by initial condition w(0, x) = M S K (x), w satises

∂ t w ≤ (|x| + 1) α [a ij (x)∂ ij w(x) + Kq(x) • ∇w + f s (x, 0)w] t ∈ R + , x ∈ R N w(0, x) = M S K (x).
(2.66)

Since α ∈ [0, 2], thanks to the growth conditions of a ij (x), q(x), f s (x, 0) and (2.24), we can apply the extension Phragmén-Lindelöf principle (Theorem 1, [START_REF] Cosner | Asymptotic behavior of solutions of second order parabolic partial dierential equations with unbounded coecients[END_REF]) for parabolic operators with unbounded coecients in the following particular way

a ij (x) = (|x| + 1) α a ij (x) ≤ C a |x| α , | q(x)| = |(|x| + 1) α q(x)| ≤ C q |x| α 2 and (|x| + 1) α f s (x, 0) ∈ L ∞ (R N ) and for k > γ 0 lim inf R→∞ sup |x|=R,t>0 w(t, x) e k(1+R 2 ) 1/2 ≤ 0.
We infer that t → w(t, x) is nonincreasing and w(t, x) converges as t → ∞ to a function W (x) pointwise in

R N such that lim sup t→∞ u(t, x) ≤ lim sup t→∞ w(t, x) = W (x) ≤ M S K (x).
Since the coecients are smooth and locally bounded in R N , the L p regularity for second order parabolic equation in [START_REF] Lieberman | Second Order Parabolic Dierential Equations[END_REF] yields that for any R > 0, there exists C R > 0 such that

w (N +1),(t 0 ,t 0 +R)×B R (x 0 ) ≤ C R .
Therefore, as t → ∞, w(t, x) converges to W (x) locally in R N , ∂ i w(t, x) and ∂ ij w(t, x) also converge respectively weakly locally in

L N +1 (R + , R N ) to ∂ i W (x) and ∂ ij W (x) satisfying a ij (x)W ij (x) + Kq i (x)W i (x) + f (x, W ) = 0 in R N 0 ≤ W (x) ≤ M S K (x).
(2.67) Furthermore, we have the following lemma Lemma 2.6.1.

Let (t n ) ∈ R + and (x n ) ∈ R N satisfying lim n→∞ t n = lim n→∞ |x n | = +∞. There exists γ ∈ (γ 0 , γ ) such that lim n→∞ u(t n , x n )e γ |xn| 1-α 2 = 0 (2.68)
Let us postpone the proof of this lemma for a moment to rst show that how (2.68) leads to (2.25) for p = ∞, i.e the uniform convergence.

• λ 1 (-L, R N ) ≥ 0.

Owing to Theorem 2.1.2, we must have W ≡ 0. This shows that u(t, x) converges locally in R N to zero as t → ∞. To prove the uniform convergence, namely (2.25) with p = ∞, we assume by contradiction that it is not true, then there exists ε > 0, (t n ) and (x n ), such that t n , |x n | → +∞ and u(t n , x n )e κ 0 |xn| 1-α 2 ≥ ε for every n ∈ N. This contradicts (2.68) and therefore the statement (i) is conrmed.

• λ 1 (-L, R N ) < 0.

Let λ R and ϕ R be respectively the Dirichlet principal eigenvalue and eigenfunction of L on B R , which are given by Theorem (2.2.3). We have shown that for κ > 0 small enough, there exists R > 0 such that λ R < 0 and the function

U (x) := κϕ R (x) x ∈ B R (x) 0 otherwise
is a weak subsolution of Eq. (2.67). Without loss of generality, we may assume U (x) ≤ u(1, x) and u(1, x) > 0. The Phragmén-Lindelöf principle (Theorem 1, [START_REF] Cosner | Asymptotic behavior of solutions of second order parabolic partial dierential equations with unbounded coecients[END_REF]) implies that the solution v of Eq. (2.2) starting by v(0, x) = U (x) is nondecreasing in t and satises

U (x) ≤ v(t, x) ≤ u(t + 1, x) ≤ w(t + 1, x), (t, x) ∈ R + × R N .
The routine argument implies that v(t, x) converges locally uniformly to a function V satisfying U ≤ V ≤ W ≤ M S K (x) and a ij (x)W ij (x) + Kq i (x)W i (x) + f (x, W ) = 0 a.e in R N . The strong maximum principle implies that V > 0 in R N and Theorem 2.1.2 yields U ≡ V ≡ W , where U is the unique solution of (2.67).

The local convergence in R N , u(t, x) → U (x) as t → ∞ is conrmed. If this convergence is not uniform in R N , there exist the sequences (t n ), (x n ) and ε > 0 such that

lim n→∞ t n = lim n→∞ |x n | = +∞, |u(t n , x n ) -U (x n )|e γ 0 |xn| 1-α 2 ≥ ε for any n ∈ N.
By Theorem (2.3.1), for all γ 1 ∈ (γ 0 , γ ), one has

lim |x|→∞ U (x n )e γ 1 |xn| 1-α 2 = 0. It follows immediately ε 2 < u(t n , x n )e γ 0 |xn| 1-α 2
for n large enough.

This contradicts Lemma (2.6.1).

It remains to prove Lemma (2.6.1). Assume by contradiction that the claim is not true, one nds ε > 0 and the sequences

(t n ) → ∞ and |x n | → ∞ as n → ∞ such that u(t n , x n )e γ 0 |xn| 1-α 2 ≥ ε for any n ∈ N. (2.69)
Observe that, by Hypothesis 2, we have

∂ t u -(|x| + 1) α (a ij (x)∂ ij u + Kq(x) • ∇u + f s (x, 0)u) ≤ 0 t ∈ R, x ∈ R N .
To achieve a contradiction, we shall look for a supersolution of Eq. (2.2), which decays as both t and |x| tend to innity. The proof is distinguished by two cases :

Case 1. (H1a) and (H1b) hold.

Let

P δ = ∂ t -(|x| + 1) α a ij (x)∂ ij -K(|x| + 1) α q(x) • ∇ -(|x| + 1) α µ(x) -δ,
we see that for all δ > 0, there exists R(δ) such that u(t, x) is a subsolution of

P δ [φ] = 0 in R N \ B R .
Let γ be the positive solution of Eq. (2.14). According to Case 1 of Theorem (2.3.1), for any γ ∈ (0, γ), there exist τ, δ ∈ (0, γ), R = R(δ) large enough and a function ϕ ∈ D α associated with λ ∈ (0, η) such that for all p > 0 the function

w γ p (x) = [e (R+p) 1-α 2 (τ -γ) e γ|x| 1-α 2 + e R 1-α 2 (τ +γ) e -γ|x| 1-α 2 ]ϕ(x)
satises

(|x| + 1) α (a ij (x)∂ ij w γ p + Kq(x) • ∇w γ p + f s (x, 0)w γ p ) -δw γ ρ ≤ 0 in B R+p \ B R , ∀p > 0.
(2.70) By (2.24), there exist a sequence (r n ) → ∞ as n → ∞ and a large constant M 1 such that

u(t, x) ≤ M 1 e γ 0 |x| 1-α 2 for |x| = r n , ∀n ∈ N, t ∈ (0, r n ).
There also exist γ 1 ∈ (γ 0 , γ), ϕ ∈ D α and a constant M 2 ≥ M 1 such that for all p > 0

u 0 (x) ≤ M 2 w γ 1 p (x) in B R+p \ B R .
Without loss of generality, we can extract a subsequence of (r n ), which is increasing and still denoted by (r n ). Let n 0 be such that r n ≥ R(δ) for n > n 0 and p n = r n -r n 0 . We choose τ ∈ (γ 0 , γ 1 ). The functions M 2 w γ 1 pn (x) satisfy

M 2 w γ 1 pn (x) ≥ M 2 e τ (rn 0 ) 1-α 2 ϕ(x) ≥ M 2 e γ 0 |x| 1-α 2 ≥ u(t, x) as |x| = r n 0 M 2 w γ 1 pn (x) ≥ M 2 e τ (rn 0 +pn) 1-α 2 ϕ(x) ≥ M 2 e γ 0 |x| 1-α 2 ≥ u(t, x) as |x| = r n ∀t ∈ (0, r n ).
Since w γ 1 pn (x) satises stationary equation (2.72), the parabolic comparison principle implies that u(t, x) ≤

M 2 w γ 1 pn (x) for (t, x) ∈ (0, r n ) × B rn \ B rn 0 , ∀n > n 0 . Letting n → ∞, we obtain u(t, x) ≤ M 2 e (rn 0 ) 1-α 2 (τ +γ 1 ) e -γ 1 |x| 1-α 2 ϕ(x) in [0, ∞) × R N \ B rn 0 Consequently, since γ 0 < γ 1 and ϕ ∈ D α , taking γ ∈ (γ 0 , γ 1 ) one has u(t, x)e γ |x| 1-α 2 ≤ M 2 e (rn 0 ) 1-α 2 (τ +γ 1 ) e -(γ 1 -γ )|x| 1-α 2 ϕ(x) → 0 as |x| → ∞. (2.71)
This contradicts to (2.69) and therefore conrms the rst case when p = ∞.

Case 2. (H1a), (H1c) and (2.15). Observe that the function u(t, x) = e -ψ K (x) u(t, x), where

ψ K = Kψ satisfying 2I N • ∇ψ K (x) = -Kq(x) satises ∂ t u -(|x| + 1) α ∆ u -|∇ψ K (x)| 2 u + f s (x, 0) w(x) ≤ 0 in R N u(0, x) = e -ψ K (x) u 0 (x). Let Q δ = ∂ t -(|x| + 1) α (∆ + µ(x) -|∇ψ K (x)| 2 ) -δ,
we see that for any δ ∈ (0, η), there exists

R = R(δ) such that u(t, x) is a subsolution of Q δ [φ] = 0 in R N \ B R(δ) .
According to Case 2 of Theorem (2.3.1), we know that for any γ ∈ ( 2Kθ 2-α , γ + ) where γ + is given in (2.16), τ ∈ ( 2Kθ 2-α , γ), there exists ϕ ∈ D α associated with λ ∈ (0, η) such that the function

v γ p (x) = [e (R+p) 1-α 2 (τ -γ) e γ|x| 1-α 2 + e R 1-α 2 (τ +γ) e -γ|x| 1-α 2 ]ϕ(x) satises (|x| + 1) α (∆ + µ(x) -|∇ψ K (x)| 2 )v p + δv p ≤ 0 in B R+p \ B R , ∀p > 0. (2.72)
By (2.24), there exist large constant M 3 > 0 and an increasing sequence (r n ) ∈ R such that

lim n→∞ r n = +∞, u(t, x) ≤ M 3 e ( 2Kθ 2-α +δ)|x| 1-α 2 e γ 0 |x| 1-α 2 = M 3 e (γ0+ 2Kθ 2-α +δ)|x| 1-α 2 for |x| = r n , t ∈ (0, r n ),
moreover for δ small enough, we nd M 4 ≥ M 3 such that

u 0 (x) ≤ M 4 e (-γ 0 -4Kθ 2-α -3δ)|x| 1-α 2 ϕ(x).
It follows, for all p > 0 and

x ∈ B R(δ)+p \ B R(δ) , that u(0, x) ≤ M 4 e ( 2Kθ 2-α +δ)|x| 1-α 2 u 0 (x) ≤ M 4 e -(γ 0 + 2Kθ 2-α +2δ)|x| 1-α 2 ϕ(x) ≤ v γ 1 p (x).
where γ 1 = γ 0 + 2Kθ 2-α + 2δ. Choosing δ such that γ 1 ∈ ( 2Kθ 2-α , γ + ) and then taking arbitrary constant τ ∈ γ 0 + 2Kθ 2-α + δ, γ 1 , we see that the functions M 4 v γ 1 pn (0, x), where p n = r n -r n 0 satises

M 4 v γ 1 pn (x) ≥ M 4 e τ (rn 0 ) 1-α 2 ϕ(x) ≥ M 4 e (γ 0 + 2Kθ 2-α +δ)|x| 1-α 2 ≥ u(t, x) as |x| = r n 0 M 4 v γ 1 pn (x) ≥ M 4 e τ (rn 0 +pn) 1-α 2 ϕ(x) ≥ M 4 e (γ 0 + 2Kθ 2-α +δ)|x| 1-α 2 ≥ u(t, x) as |x| = r n , ∀t ∈ (0, r n )
where n 0 is xed in such the way r n 0 ≥ R(δ). Since v γ 1 pn satises (2.72), the parabolic comparison principle implies that for all n > n 0

u(t, x) ≤ M 4 v γ 1 pn (x) for (0, r n ) × B rn 0 +pn \ B rn 0 .
Letting n → ∞, we obtain

u(t, x) ≤ M 4 e R 1-α 2 (τ +γ 1 ) e -γ 1 |x| 1-α 2 ϕ(x) in (0, ∞) × R N \ B rn 0 . Converting back u to u, since ϕ ∈ D α and γ 1 -2Kθ 2-α -γ 0 > 0, taking γ ∈ (γ 0 , γ 1 -2Kθ 2-α ), we get e γ |x| 1-α 2 u(t, x) ≤ M 4 e R 1-α 2 (τ +γ 1 ) e -(γ 1 -2Kθ 2-α -γ )|x| 1-α 2 ϕ(x) → 0 as |x| → ∞. (2.73)
This contradiction conrms Lemma (2.6.1). Therefore the uniform convergence is achieved.

Finally, the convergences in L p (R N )-norms for p ∈ (0, ∞) can be also resulted by Lemma (2.6.1). Indeed, since u(t, x) ≥ 0 and ϕ ∈ D α , we see that the functions in the right-hand side of inequalities (2.71) and (2.73) are independent of t and decay exponentially as |x| → ∞. Those functions belong to L p (R N ) for all p > 0. Either one of the followings holds for all p > 0:

• λ 1 (-L, R N ) ≥ 0. As shown above, u(t, x) → 0 locally uniformly (pointwise) in R N . The Lebesgue dominated convergence theorem rapidly implies e γ 0 |x| 1-α 2 u(t, x) L p (R N ) → 0 as t → ∞ .

• λ 1 (-L, R N ) < 0. Let us consider the functions u(t, x) and u(t, x) with the initial condition respectively u(0, x) = max{u 0 (x), U (x)} and u(0, x) = min{u 0 (x), U (x)}, where U (x) is the unique positive solution of Eq. (2.67). As above, u(t, x) (resp.) u(t, x) converge nonincreasingly (resp. nondecreasingly) to U (x) as t → ∞, locally in x ∈ R N . Hence, u(t, x) ≤ U (x) ≤ u(t, x), ∀t ≥ 0. Observe that the function w(t, x) = u(t, x) -U (x) ≥ 0 satises Eq. (2.2) with the nonlinearity

f (x, u) -f (x, U ) ≤ f (x, U )u/U -f (x, U ) ≤ f s (x,
e γ 0 |x| 1-α 2 w(t, x) L p (R N ) → 0 as t → ∞. Similarly, the function w(t, x) = U (x) -u(t, x) ≥ 0 satises e γ 0 |x| 1-α 2 w(t, x) L p (R N ) → 0 as t → ∞. The proof is complete.
The proof of Theorem (2.1.8) can be obtained by the same technique of Theorem (2.1.7). Therefore, we shall only specify and prove the main points, which yield signicant dierences with the former.

Proof of Theorem 2.1.8. First, we emphasize that the extension Phragmén-Lindelöf principle (Theorem 1, [START_REF] Cosner | Asymptotic behavior of solutions of second order parabolic partial dierential equations with unbounded coecients[END_REF]) can still be applied in the case α = 2. Arguing as in Theorem 2.1.8, we see that the locally large time behavior is thus followed from the L p regularity for second order parabolic equation in [START_REF] Lieberman | Second Order Parabolic Dierential Equations[END_REF]. For the L p (R N ), 0 < p ≤ ∞ large time behavior, it is in fact a consequence of the following lemma. Lemma 2.6.2.

Let (t n ) ∈ R + and (x n ) ∈ R N satisfying lim n→∞ t n = lim n→∞ |x n | = +∞. There exists β ∈ (σ 0 , σ ) such that lim n→∞ u(t n , x n )|x n | β = 0 (2.74)
The proof of this Lemma is transparently analogous to Lemma (2.6.1). Just note that, the decaying condition of initial datum and (2.26) allow us to invoke the parabolic comparison principle under Hypothesis 1. We omit the details.

For proving uniform convergence in (2.27), this is the direct consequence of Lemma (2.6.2) and the fact that for all β ∈ (σ 0 , σ ), lim sup |x|→∞ U (x)|x| β = 0, which derived from Theorem (2.3.2).

The L p convergences are immediately obtained thanks to Lebesgue dominated theorem and the observation that for some β

∈ (σ 0 , σ ), ϕ ∈ D 2 , C > 0 u(t, x) ≤ C(|x| + 1) -σ 0 (|x| + 1) -(β-σ 0 ) ϕ(x) U (x) ≤ C(|x| + 1) -σ 0 (|x| + 1) -(β-σ 0 ) ϕ(x),
where ϕ(x)|x| -κ → 0 as |x| → ∞ for all κ > 0. Obviously, it is sucient to achieve (2.28) is pσ 0 > N . The proof is thereby complete.

Background and the main results

We consider quasilinear elliptic equations of the form

K a [u] + b(x)g(u) = 0 in R N , (3.1)
where

K a [u] = -div(|∇u| p-2 ∇u) -a(x)u p-1 , p ≥ 2, a, b ∈ L ∞ (R N ), b ≡ 0 and g : R + → R + is a continuous function. A typical model of (3.1) is -∆ p u -a(x)u p-1 + u r = 0 in R N , with r > p -1.
By a solution of (3.1) we mean a function u ∈ W 1,p loc (R N ) such that g(u) ∈ L 1 loc (R N ) and u satises (3.1) in the sense of distribution, namely

R N (|∇u| p-2 ∇u.∇φ -a(x)|u| p-2 uφ + b(x)g(u)φ)dx = 0 ∀φ ∈ C ∞ c (R N ). (3.2) A function u ∈ W 1,p loc (R N
) is called a supersolution (respectively subsolution) of (3.1) if g(u) ∈ L 1 loc (R N ) and (3.2) holds with "=" replaced by "≥" (resp. "≤") and with nonnegative test function φ ∈ C ∞ c (R N ). It is well-known that, the type of equation (3.1) has wide applications in various aspects of mathematical biology, physics and especially in population dynamics. In a pioneering work, Berestycki [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF] has studied the existence and uniqueness of the semilinear equation

-∆u = f (u) in Ω u = 0 on ∂Ω, (3.3)
where Ω is bounded, smooth and f is of KPP type (f (0) = 0 and f (u)/u is stricly descreasing). Afterward, this is a subject of extensive research and much more general results have been obtained for the equations in bounded or periodic domains. Amongst those, we cite [START_REF] Cañada | Existence of positive solutions for some problems with nonlinear diusion[END_REF] and [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF], where interesting biological interpretation of (3.1) is carefully discussed. However, works involving unbounded domains (for instance R N ) are much less. In their groundbreaking papers [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF], Berestycki et al. studied more general elliptic equations of the form

- i,j a ij (x)∂ ij u - i q i (x)∂ i u = f (x, u) in R N (3.4)
where the condition imposed is either (see [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF])

lim inf |x|→∞ (4α(x)f s (x, 0) -|q(x)| 2 ) > 0 with α(x) = inf ξ∈R N ,|ξ|=1 a ij (x)ξ i ξ j > α * > 0 (3.5)
or (see [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF])

lim sup |x|→∞ f s (x, 0) < 0. (3.6)
They established a necessary and sucient condition for existence of positive solutions of (3.4) in terms of the sign of the generalized principle eigenvalue of the operator

L[φ] = - i,j a ij (x)∂ ij φ - i q i (x)∂ i φ -f s (x, 0)φ in R N .
Furthermore, they also dealt with the question of uniqueness of positive bounded solutions. Such type of result is called nonlinear Liouville-type result.

Motivated by the above works, the present paper aims at providing a sharp criterion that characterizes existence/nonexistence and uniqueness of positive solutions of (3.1). It is important to understand that which kinds of result in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF][START_REF] Cañada | Existence of positive solutions for some problems with nonlinear diusion[END_REF] remain true under framework of p-Laplacian. Before stating the main results, let us describe the hypotheses on a, b and g. The novelty of our study is that the potential a is allowed to be sign-changing and satises the following property lim sup |x|→∞ a(x)|x| α < -m for some α ∈ [0, p] and m > 0.

(C1)

Condition (C1) involves the case that (3.5) is not covered and may be regarded as an extension of (3.6) in the sense that a possibly decays at innity. As will be shown in the sequel, the decay rate of the potential a has a signicant eect on that of solutions of (3.1): solutions decay exponentially when a is a slow decay potential (i.e. α ∈ [0, p)), while they decay polynomially when a is a Hardy potential (i.e. α = p).

The nonlinearity g is required to satisfy lim s→0 g(s) s p-1 = 0 and g(s) s p-1 is increasing on (0, ∞).

(C2)

Next, we assume that b veries the following relation for some s 0 > 0,

-a(x)s p-1 0 + b(x)g(s 0 ) ≥ 0 a.e. in R N . (C3)
Owing to (C2), it is easy to see that (C3) also holds for every s ≥ s 0 . This reveals that any constant s greater than s 0 is a supersolution of (3.1). Clearly, (C3) is fullled if inf R N b > 0 and lim s→∞ g(s)

s p-1 = ∞. The above assumptions have biological meanings. Condition (C1) refers to the environment being unfavorable, unfavorably neutral or nearly neutral near innity, which are corresponding to the cases of α = 0, α ∈ (0, p) or α = p. This kind of assumption is recently used to describe the eect of global warming (see [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF], [START_REF] Shigesada | Biological Invasions ; Theory and Practice[END_REF]). Condition (C2) means that the intrinsic growth rate decreases when the population density is increasing. This is due to the intraspecic competition for resources and lastly, condition (C3) expresses a saturation eect: when the population density is very large, the death rate is higher than the birth rate and the population decreases.

Under theses assumptions, we will prove that the existence and nonexistence result for (3.1) is directly conditioned by the sign of "an eigenvalue" of K a in the whole domain R N . For the sake of convenience, let us recall briey some basic facts concerning the rst eigenvalue of K a in bounded domains. It is wellknown that for every C 1,β bounded domain Ω ⊂ R N , the operator K a , under Dirichlet boundary conditions, admits a unique rst eigenvalue λ Ω (a) associated with a unique (up to multiplication) positive eigenfunction ϕ Ω ∈ W 1,p 0 (Ω) (see, e.g., [START_REF] García-Melián | Maximum and Comparison principles for operators involving the p-Laplacian[END_REF]Lemma 3]). More precisely, ϕ Ω is a positve solution to

K a [ϕ Ω ] = λ Ω (a)ϕ p-1 Ω in Ω ϕ Ω = 0 on ∂Ω.
Furthermore, ϕ Ω ∈ C 1,β (Ω) for some β ∈ (0, 1) and λ Ω (a) is variationally characterized by λ Ω (a) = inf It is easily seen that λ ∞ (a) > -sup R N a(x) and thus denes a real number. As we will see in Section 4, λ ∞ (a) can be regarded as an "eigenvalue" of K a in the whole domain R N . In fact, when p = 2, it is known [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] that λ ∞ (a) coincides with the generalized principle eigenvalue of K a in R N dened by

λ 1 (a) = sup{λ ∈ R, ∃φ ∈ W 2,N loc (R N ), φ > 0, (K a + λ)[φ] ≤ 0 a.e in R N }.
When p = 2, such a result still remains open. We will show that λ ∞ (a) plays a key role in proving existence/nonexistence result. For α ∈ [0, p) and m > 0, we dene a class of positive solutions of (3.1) as follows

S α = {u > 0 | u is

The strategy of proofs and organization of the paper

The rest of the paper is organized as follows.

In Section 2, due to a delicate construction of supersolutions, we show that any nonnegative solution of (3.1) in S α decays at the rate depending on α, with α ∈ [0, p]. Interestingly, the rates of decay are remarkable between two cases α ∈ [0, p) and α = p. To be more precise, the solutions of (3.1) decay exponentially in the rst case, while they decay polynomially in the second case. The construction relies highly on the weak comparison principle of [START_REF] Damascelli | Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results[END_REF] and a-prior growth conditions of solutions. It is noteworthy that throughout the construction, boundedness of solutions is not required.

In Section 3, by using a variational approach, we establish a comparison principle for (3.1) in R N under an additional assumption of integrability on subsolution of (3.1). Although the technique is inspired from [START_REF] Du | Boundary blow-up solutions and their applications in quasilinear elliptic equations[END_REF], we need to overcome the new diculties arising from the lack of compactness of the domain. It is worth emphasizing that the sliding arguments and variational arguments that are used in the case of linear operators [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] fail to apply in this framework. Our comparison principle handles to compare a sub and a supersolution of (3.1) without further assumption on the gradient of supersolution as in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF]Theorem 2.3], which only enables us to compare a solution and a supersolution of (3.1).

To conclude the paper, in Section 4, we prove Theorem 3.1.1 and Theorem 3.1.2. To this aim, we construct an eigenfunction associated to λ ∞ (a) dened in (3.8). This task can be done due to the help of Harnack inequality (see [START_REF] Serrin | Local behavior of solutions of quais-linear equations[END_REF], [START_REF] Trudinger | On Harnack type inequalities and their application to quasilinear elliptic equations[END_REF]). The existence of positive solution of (3.1) is derived from the negativity of λ ∞ (a) and (C3), while the nonexistence and uniqueness will be consequences of the comparison principle.

A-priori estimates

In this section, we establish a priori estimates on positive solutions of (3.1). We start with the case α ∈ [0, p). Proof. By assumption, for any ε > 0 there exists R = R(ε) such that

∆ p u ≥ m + ε |x| α u p-1 for |x| ≥ R. Set L ε [φ] = -∆ p φ + (m + ε)|x| -α φ p-1 , we see that L ε u ≤ 0 as |x| ≥ R.
For any ρ > 0, consider the function

w ρ (x) = e (R+ρ) 1-α p (τ -ω) e ω|x| 1-α p + e R 1-α p (τ +ω) e -ω|x| 1-α p := w 1 ρ (x) + w 2 ρ (x)
, where ω, τ and R will be chosen later. Observe that, for any φ ∈ C 2 loc (Ω), we have the expansion formula of p-Laplacian

∆ p φ = (p -2)|∇φ| p-4 D 2 φ∇φ, ∇φ + |∇φ| p-2 ∆φ x ∈ Ω. (3.12) Set w ρ (x) = max ij |∂ ij w ρ (x)|
, by Cauchy-Schwarz inequality, one has

∆ p w ρ ≤ (|p -2|N w ρ + |∆w ρ |)|∇w ρ | p-2 . (3.13)
Direct computation yields

∇w ρ (x) = ω|x| -α p 1 - α p x |x| w 1 ρ -ω|x| -α p 1 - α p x |x| w 2 ρ , |∇w ρ (x)| ≤ ω|x| -α p 1 - α p w ρ , ∂ ij w ρ (x) = ω 2 1 - α p 2 x i x j |x| -2-2α p + ω 1 - α 2 p 2 x i x j |x| -3-α p w 1 ρ + ω 2 1 - α p 2 x i x j |x| -2-2α p + ω 1 - α 2 p 2 x i x j |x| -3-α p w 2 ρ + δ i,j ω 1 - α p |x| -1-α p w 1 ρ -δ i,j ω 1 - α p |x| -1-α p w 2 ρ ≤ ω 2 2 1 - α p 2 |x| 1-α p + ω 2 1 - α 2 p 2 + ω 1 - α p |x| -1-α p w ρ w ρ (x) ≤ ω 2 2 1 - α p 2 |x| 1-α p + ω 2 1 - α 2 p 2 + ω 1 - α p |x| -1-α p w ρ (x) ∆w ρ (x) ≤ N ω 2 2 1 - α p 2 |x| 1-α p + ω 2 1 - α 2 p 2 + ω 1 - α p |x| -1-α p w ρ (x).
Applying these computations to (3.13), we derive

∆ p w ρ (x) ≤ N (p -1)ω p-1 1 - α p p-1 ω 2 1 - α p |x| 1-α p + 3p + α 2p |x| -α-1+ α p w p-1 ρ (x). Therefore, set A = N (p -1)ω p-1 1 -α p p-1 3p+α
2p , one has

L ε w ρ = |x| -α w p-1 ρ - N (p -1) 2 ω p 1 - α p p -A|x| α p -1 + m + ε . Since α < p, |x| α p -1 → 0 as |x| → ∞.
For any ρ > 0, to assign w ρ to be a supersolution of

L ε φ = 0 in B R+ρ \ B R , we choose R = R(ε) and ω such that A|x| α p -1 ≤ ε 2 as |x| ≥ R and ω = 2m + ε N p -N 1/p p p -α > ω.
On the other hand, by the growth condition of u, one nds C > 0 such that u(x) ≤ Ce ω|x| 1-α p in R N . Therefore, we can take τ arbitrarily in (ω, ω) and R suciently large such that for any ρ > 0, one has

w ρ (x) ≥ e R 1-α p τ ≥ Ce R 1-α p ω ≥ u(x), as |x| = R w ρ (x) ≥ e (R+ρ) 1-α p τ ≥ Ce (R+ρ) 1-α p ω ≥ u(x), as |x| = R + ρ.
Fix such ω, τ and R. Applying the weak comparison principle on the bounded domain [START_REF] García-Melián | Maximum and Comparison principles for operators involving the p-Laplacian[END_REF] B R+ρ \ B R , we obtain for all ρ > 0 that

u(x) ≤ w ρ (x) = e (R+ρ) 1-α p (τ -ω) e ω|x| 1-α p + e R 1-α p (τ +ω) e -ω|x| 1-α p in B R+ρ \ B R .
Sending ρ → ∞, we obtain

u(x) ≤ e R 1-α p (τ +ω) e -ω|x| 1-α p in R N \ B R .
The fact ω > ω conrms the proof.

When α = p, solutions under consideration decay polynomially as stated in the following proposition.

Proposition 3.2.2. Let α = p, m > 0 and u be a positive function satisfying

lim inf |x|→∞ |x| p ∆ p u u p-1 - m |x| p > 0 and lim sup |x|→∞ u(x) |x| γ 0 < ∞, (3.14)
where γ 0 is the unique positive solution of (3.9). Then,

lim |x|→∞ u(x)|x| γ 0 = 0. (3.15)
Proof. By assumption, for any ε > 0, there exists R = R(ε) such that

∆ p u ≥ m + ε |x| p u p for |x| ≥ R We set L ε [φ] = -∆ p φ + (m + ε)|x| -p φ p-1 , obviously L ε [u] ≤ 0 in R N \ B R .
Similar to Proposition 3.2.1, we will look for a suitable decaying supersolution of L ε [φ] = 0 on B R+ρ \ B R for all ρ > 0. However, the supersolution is no longer of exponential decay. Let C 1 , C 2 , γ > 0, we consider

w ρ (x) = C 1 ln(R + ρ) |x| γ + C 2 |x| -γ
From (3.13), for any φ ∈ C 2 loc (Ω), we have

∆ p φ ≤ (|p -2|N φ + |∆φ|)|∇φ| p-2 ≤ (pN φ + |∆φ|)|∇φ| p-2 x ∈ Ω, where φ(x) = max ij |∂ ij φ(x)|. Direct computations yield |∇w ρ (x)| p-2 ≤ γ p-2 |x| -(p-2) w p-2 ρ (x), |∆w ρ (x)| ≤ (γ 2 + |N -2|γ)|x| -2 w ρ (x) ≤ (γ 2 + N γ)|x| -2 w ρ (x) w ρ (x) ≤ (γ 2 + 3γ)|x| -2 w ρ (x). Therefore, L ε [w ρ ](x) ≥ [-(N p + 1)γ p -(3p + 1)N γ p-1 + m + ε]|x| -p w ρ (x) p-1 .
Let γ ε be the unique positive solution of

-(N p + 1)γ p -(3p + 1)N γ p-1 + m + ε = 0.
Obviously γ ε > γ 0 , which is the unique positive solution of (3.9). By taking γ = γ ε , we see that L ε [w ρ ] ≥ 0 in B R+ρ \ B R for all ρ > 0. To assign w ρ to be a supersolution of L ε [φ] = 0 on B R+ρ \ B R for all ρ > 0, we verify the boundary condition. By the growth assumption of u, we have

lim sup |x|→∞ u(x) |x| γ 0 < ∞.
This yields that there exist C > 0 such that u(x) ≤ C|x| γ 0 for |x| ≥ R. Choosing C 1 = C, for ρ large, one has

u(x) ≤ C|x| γ 0 ≤ C |x| γε-γ 0 ln(R + ρ) |x| γ 0 ≤ w ρ (x) x ∈ ∂B R+ρ .
It is easily chosen C 2 large enough such that u(x) ≤ w(x) on ∂B R . The weak comparison principle implies, for ρ > 0, that

u(x) ≤ w ρ (x) = C 1 ln(R + ρ) |x| γε + C 2 |x| -γε x ∈ B R+ρ \ B R .
Since C 1 , C 2 are independent of ρ, by letting ρ → ∞, we nally derive

u(x) ≤ C 2 |x| -γε x ∈ R N \ B R .
The proof is complete due to γ ε > γ 0 . Now, we are able to prove the comparison principle.

Comparison principle

In this section, we demonstarte the comparison principle for (3.1) in R N which serve to obtain nonexistence and uniqueness result in the next section.

Theorem 3.3.1. Let u 1 , u 2 ∈ C 1 loc (R N ) be respectively positive supersolution and subsolution of (3.1) such that

u 2 ∈ L 1 (R N ) and lim |x|→∞ u 2 (x) = 0. (3.16)
Then

u 1 ≥ u 2 in R N . Proof. For any ε ≥ 0, R ∈ (0, ∞], set D R (ε) = {x ∈ B R : u 2 (x) + ε > u 1 (x) + 2ε}. We will show that D ∞ (0) = ∅. It is sucient to prove that D R 0 (0) = ∅ for any R 0 > 0 such that b ≡ 0 in B R 0 .
Fix a such R 0 and let ε ∈ (0, 1). By (3.16), there exists

R = R(ε) > R 0 such that u 2 (x) < ε/2 for every x ∈ (B R ) c . Denote ε 1 = 2ε, ε 2 = ε and v i = [(u 2 + ε 2 ) p -(u 1 + ε 1 ) p ] + (u i + ε i ) p-1 i = 1, 2 then v i ∈ W 1,p 0 (K R ) with some K R ⊂⊂ B R and v i = 0 out side K R .
Therefore, v i can be approximated by a sequence of functions in C ∞ 0 (K R ). We can use v i , i = 1, 2, as test functions to get

B R (|∇u 1 | p-2 ∇u 1 .∇v 1 -a(x)u p-1 1 v 1 + b(x)g(u 1 )v 1 )dx ≥ 0 (3.17) B R (|∇u 2 | p-2 ∇u 2 .∇v 2 -a(x)u p-1 2 v 2 + b(x)g(u 2 )v 2 )dx ≤ 0 (3.18)
Subtracting (3.18) from (3.17) yields

B R (|∇u 1 | p-2 ∇u 1 .∇v 1 -|∇u 2 | p-2 ∇u 2 .∇v 2 )dx ≥ B R a(x)(u p-1 1 v 1 -u p-1 2 v 2 )dx - B R b(x)(g(u 1 )v 1 -g(u 2 )v 2 )dx. (3.19)
Set

w i = u i + ε i , V i = ∇(ln(u i + ε i )) = ∇u i u i + ε i i = 1, 2 (3.20)
and

I := |∇u 2 | p-2 ∇u 2 • ∇v 2 -|∇u 1 | p-2 ∇u 1 • ∇v 1 .
By a computation, we obtain

I = w p 2 (|V 2 | p -|V 1 | p -p|V 1 | p-2 V 1 • (V 1 -V 2 )) + w p 1 (|V 1 | p -|V 2 | p -p|V 2 | p-2 V 2 • (V 1 -V 2 )). (3.21)
By the Clarkson's inequality [START_REF] Lindqvist | On the equation div(∇u| p?2 ∇u) + λ|u| p-2 u = 0[END_REF], for all vectors X, Y ∈ R N , we have

|X| p -|Y | p -p|Y | p-2 Y (X -Y ) ≥ c p |X -Y | p+(2-p) + (|X| + |Y |) (2-p) + (3.22)
where

c p =      1 2 p -1 if 1 < p < 2 3p(p -1) 16 if p ≥ 2.
It follows from (3.21) and (3.22) that

I ≥ c p (w p 1 + w p 2 ) |V 1 -V 2 | p+(2-p) + (|V 1 | + |V 2 |) (2-p) + ∀p > 1. (3.23)
Combining (3.19) and (3.23) and using (3.20), we obtain

-c p D R (ε) (w p 1 + w p 2 ) |V 1 -V 2 | p+(2-p) + (|V 1 | + |V 2 |) (2-p) + dx ≥ - D R (ε) a(x) u 2 w 2 p-1 - u 1 w 1 p-1 (w p 2 -w p 1 )dx + D R (ε) b(x) g(u 2 ) w p-1 2 - g(u 1
)

w p-1 1 (w p 2 -w p 1 )dx. (3.24)
Consequently,

c p D R (ε) |V 2 -V 1 | p+(2-p) + (|V 1 | + |V 2 |) (2-p) + [w p 1 + w p 2 ]dx + D R (ε) b g(u 2 ) w p-1 2 - g(u 1 ) w p-1 1 (w p 2 -w p 1 )dx ≤ a L ∞ (R N ) D R (ε) u 2 w 2 p-1 - u 1 w 1 p-1 (w p 2 -w p 1 )dx. (3.25)
Suppose by contradiction that D R 0 (0) = ∅. Thanks to the continuity of u i , i = 1, 2, D R 0 (0) is an open set. Let B be a small ball such that B ⊂ D R 0 (0). On the one hand, as B ⊂ D R (ε) for all ε small and g is nonnegative, it follows that

c p B |∇ ln u 2 -∇ ln u 1 | p+(2-p) + (|∇ ln u 1 | + |∇ ln u 2 |) (2-p) + [u p 1 + u p 2 ]dx + B b g(u 2 ) u p-1 2 - g(u 1 ) u p-1 1 (u p 2 -u p 1 )dx ≤ J ε (3.26)
where J ε is the third term in (3.25). Observe that, for ε < 1 and x ∈ D(ε), one has

u 2 w 2 p-1 - u 1 w 1 p-1 (w p 2 -w p 1 ) ≤ 2p(w 2 -w 1 )w p-1 2 ≤ 2p( u 2 L ∞ (R N ) + 1) p-1 u 2 and lim ε→0 u 2 w 2 p-1 - u 1 w 1 p-1 = 0 pointwise in R N ,
Thanks to (3.16), u 2 ∈ L 1 (R N ), we deduce from Lebesgue dominated convergence theorem that

lim ε→0 J ε = 0. (3.27)
Hence by letting ε → 0 in (3.26) we obtain

B |∇ ln u 2 -∇ ln u 1 | p+(2-p) + (|∇ ln u 1 | + |∇ ln u 2 |) (2-p) + [u p 1 + u p 2 ]dx = B b g(u 2 ) u p-1 2 - g(u 1 ) u p-1 1 (u p 2 -u p 1 )dx = 0,
which, together with condition (C2) on g, implies

|∇(ln u 2 -ln u 1 )| ≡ b ≡ 0 in D R 0 (0). (3.28) Since b ≡ 0 in B R 0 , it follows that D R 0 (0) B R 0 and ∂D R 0 (0) ∩ B R 0 = ∅. Hence there is a connected component O of D R 0 (0) such that ∂D R 0 (0) ∩ ∂O ∩ B R 0 = ∅. From (3.28), we deduce that ln u 1 -ln u 2 ≡ constant in O, which in turn implies u 1 = u 2 in O for some > 0. As u 1 = u 2 on ∂D R 0 (0) ∩ ∂O ∩ B R 0 , it follows that = 1, which contradicts O ⊂ D R 0 (0). Therefore, we must have D R 0 (0) = ∅ and thus u 1 ≥ u 2 in B R 0 . Since R 0 > 0 is arbitrarily large, we conclude that u 1 ≥ u 2 in R N .

3.4

Existence/Nonexistence and Uniqueness

We start with an auxilary result which is obtained thanks to Harnack inequality.

Proposition 3.4.1. There exists a positive function

ϕ ∞ ∈ C 1 loc (R N ) satisfying K a [ϕ ∞ ] = λ ∞ (a)ϕ p-1 ∞ weakly in R N . (3.29)
Furthermore, λ ∞ (a) can be variationally characterized by

λ ∞ (a) = inf φ∈W 1,p (R N ) R N |∇φ| p -a(x)|φ| p dx R N |φ| p dx . (3.30)
Proof. For any k > 0, we see that the sequence {λ R (a)} R>k is bounded. Let ϕ R be the associated eigenfunction of λ R (a), normalized by ϕ R (0) = 1. By Harnack's inequality (see, e.g., [92, Theorem 5], [98, Theorem 1]), there exists a constant

C(k) > 0 independent of R such that sup B k ϕ R (a) < C(k) inf B k ϕ R (a) ≤ C(k).
Hence, in light of local regularity for elliptic equations and standard argument, up to a subsequence, {ϕ R } converges in C 1 loc (R N ) to a function ϕ ∞ , which solves (3.29). Moreover, ϕ ∞ (0) = 1 and therefore ϕ ∞ > 0 in R N by Harnack's inequality.

From above we have λ R (a

) ≥ λ R+1 (a) ≥ λ ∞ (a) and lim R→∞ λ R (a) = λ ∞ (a). Obviously, λ ∞ (a) > -sup R N a(x).
Let us call λ v the right-hand side of (3.30), we shall prove that

lim R→∞ λ R (a) = λ v .
Indeed, by the denition of λ v , for all δ > 0, there exists φ δ ∈ W 1,p (R N ) such that

λ v + δ ≥ R N |∇φ δ | p -a(x)|φ δ | p dx R N |φ δ | p dx . Let χ R ∈ C 1 c (R N ) be compactly supported in B R and χ R = 1 in B R-1 , we set φ δ,R = χ R φ δ . The variational characterization λ R (a) (3.7) yields R N |∇φ δ,R | p -a(x)|φ δ,R | p dx R N |φ δ,R | p dx ≥ λ R (a). We nd a constant k > 1 depending only on sup R>1 χ R W 1,∞ (R N ) such that |∇φ δ,R | = |∇φ δ χ R + φ δ ∇χ R | ≤ |∇φ δ | + k|φ δ |. Since φ δ ∈ W 1,p (R N ), the Lebesgue dominated convergence theorem implies that λ v + δ ≥ R N |∇φ δ | p -a(x)|φ δ | p dx R N |φ δ | p dx ≥ lim R→∞ λ R (a) = λ ∞ (a).
The arbitrariness of δ conrms (3.30).

We are ready to present the Proof of Theorem 3.1.1. i. Existence.

By the denition of λ ∞ (a), there exists R 0 > 0 such that λ R (a) < 0. Let ϕ R 0 be the associated eigenfunction of λ R 0 (a) with normalization ϕ R 0 (0) = 1. Set u R 0 (x) = δϕ R 0 where δ > 0 to be chosen later and

M R 0 := max B R 0 ϕ R 0 . In B R 0 , we get -∆ p u R 0 -a(x)u p-1 R 0 + b(x)g(u R 0 ) = δ p-1 ϕ p-1 R 0 λ R 0 (a) + b L ∞ (R N ) g(δϕ R 0 ) (δϕ R 0 ) p-1 ≤ δ p-1 ϕ p-1 R 0 λ R 0 (a) + b L ∞ (R N ) g(δM R 0 ) (δM R 0 ) p-1 (3.31)
By (C2), one can choose δ small, depending only on R 0 , in such a way that δM R 0 < s 0 , where s 0 is the constant in (C3), and the RHS of (3.31) is negative. Put u = χ B R 0 u R 0 . For any R > R 0 , we see that u and s 0 are respectively sub and supersolution of (3.1) in B R and u < s 0 on ∂B R . By applying the sub-supersolution theorem (see for instance [73, Theorem 3.1]), we derive that there exists a weak solution u R of (3.1) in B R such that u ≤ u R ≤ s 0 in B R . By regularity result for quasilinear elliptic equations (see [START_REF] Dibenedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF]) and standard argument, up to a subsequence, {u R } converges in C 1 loc (R N ) to a function u * , which is a weak solution of (3.1) in R N and satises u ≤ u * ≤ s 0 . Since u * (0) ≥ u(0) = δ > 0, by Harnack inequality Introduction In this article, we are interested in nding survival criteria for a species that has a long range dispersal strategy. As a model species, we can think as trees whose seed and pollens are disseminate on a long range. In ecology a commonly used model that integrate such long range dispersal [START_REF] Fife | An integrodierential analog of semilinear parabolic PDEs, Partial dierential equations and applications[END_REF][START_REF] Grinfeld | Non-local dispersal[END_REF][START_REF] Hutson | The evolution of dispersal[END_REF][START_REF] Lutscher | The eect of dispersal patterns on stream populations[END_REF][START_REF] Mollison | Long-distance dispersal of wind-borne organisms[END_REF][START_REF]Quantitative Analysis of Movement : Measuring and Modeling Population Redistribution in Animals and Plants[END_REF] is the following:

∂u ∂t (t, x) = J u(t, x) -u + f (x, u(t, x)) in R + × R N , (4.1)
where u(t, x) is the density of the considered population, J is a dispersal kernel, f (x, s) is a KPP type non-linearity describing the growth rate of the population. The possibility of a long range dispersal is well known in ecology, where numerous data are now available to support this assumptions [START_REF] Cain | Long-distance seed dispersal in plant populations[END_REF][START_REF] Clark | Why Trees Migrate So Fast : Confronting Theory with Dispersal Biology and the Paleorecord[END_REF][START_REF] Fastie | Reid's Paradox of rapid plant migration[END_REF][START_REF] Xhaard | Inferring invasion determinants with mechanistic models and multitype samples[END_REF][START_REF] Deveaux | Estimation de la dispersion de pollen à longue distance à l'echelle d'un paysage agicole : une approche expérimentale[END_REF].

In this setting the tail of the kernel is a measure of the frequency at which long dispersal events occur. A biological explanation that support the used of (4.2) to describe the evolution of the population comes from the observation that the intrinsic variability in the capacity of the individuals to disperse generates, at the scale of a population, a long range dispersal of the population. The eect of such variability has been investigate in [START_REF] Hapca | Anomalous diusion of heterogeneous populations characterized by normal diusion at the individual level[END_REF][START_REF] Petrovskii | Variation in individual walking behavior creates the impression of a Lévy ight[END_REF] by means of the study of correlated random walks. In such framework, all individuals follow a simple random walks where the diusion coecient follows a probability law, it can be checked that the probability of the density of population will then follow an integrodierential equation [START_REF] Hapca | Anomalous diusion of heterogeneous populations characterized by normal diusion at the individual level[END_REF][START_REF] Petrovskii | Variation in individual walking behavior creates the impression of a Lévy ight[END_REF][START_REF]Quantitative Analysis of Movement : Measuring and Modeling Population Redistribution in Animals and Plants[END_REF] where a dispersal kernel J describe the probability to jump from one location to another.

Along this paper we will always make the following assumptions on the dispersal kernel J:

(H1) J ∈ C(R N ) ∩ L 1 (R N ) is nonnegative, radially symmetric .
(H2) J(0) > 0 and has unit mass (i.e.

R N J(z)dz = 1). In the present paper, we focus our analysis on species that have a bounded ecological niche. A simple way to model such spatial repartition consists by considering that the environment is hostile to the species outside a bounded set. This fact translates in our modelling by assuming that f satises:

(H3) f ∈ C 1,α (R N +1 ) is of KPP type, that is :      f (•, 0) ≡ 0, For all x ∈ R N , f (x,
s)/s is decreasing with respect to s on (0, +∞).

There exists S(x) ∈ C(R N ) ∩ L ∞ (R N ) such that f (x, S(x)) ≤ 0 for ∀x ∈ R N . (H4) lim sup |x|→∞ f (x,s) s < 0, uniformly in s ≥ 0.
A typical example of such nonlinearity is given by f (x, s) := s(a(x) -b(x)s) with b(x) > 0 and a(x) so that lim sup |x|→∞ a(x) < 0.

Our main purpose is to seek conditions on J and f that characterize the persistence of the species modelled by (4.1). In this task, we focus our analysis on the description of the set of positive stationary solution of (4.1). That is the set of positive solution of the equation below

J u(x) -u(x) + f (x, u(x)) = 0 in R N . (4.2)
This description is expected to provide useful persistence criteria.

In the literature, persistence criteria have been well studied for the reaction diusion version of (4.1)

∂u ∂t (t, x) = ∆u(t, x) + f (x, t, u(t, x)) in R + × Ω, (4.3)
where Ω is the domain of R N possibly R N itself. Survival criteria have been obtained for various media, ranging from periodic media to ergodic media [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF][START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF][START_REF] Cantrell | Diusive logistic equations with indenite weights : population models in disrupted environments[END_REF][START_REF] Cantrell | Diusive logistic equations with indenite weights : population models in disrupted environments[END_REF][START_REF] Cantrell | On the eects of spatial heterogeneity on the persistence of interacting species[END_REF][START_REF] Fraile | Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation[END_REF][START_REF] Nadin | Existence and uniqueness of the solutions of a space-time periodic reaction-diusion equation[END_REF][START_REF] Nadin | Propagation phenomena for time heterogeneous KPP reaction-diusion equations[END_REF][START_REF] Pinchover | On positive solutions of second-order elliptic equations, stability results, and classication[END_REF][START_REF] Shen | Traveling waves in diusive random media[END_REF]. In the context of global warming, survival criteria have been investigate in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF][START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, II -Cylindrical type domains[END_REF]. For such reaction diusion equations the survival criteria are often obtained by looking at the sign of the rst eigenvalue of the linear problem obtained by linearising (4.3) around the 0 solution. That is the sign of the rst eigenvalue

λ 1 (∆ + ∂ s f (x, 0), Ω) of the spectral problem ∆ϕ(x) + ∂ s f (x, 0)φ(x) + λ 1 ϕ(x) = 0 Ω, (4.4)
associated with the proper boundary conditions (if Ω = R N ). In most situations, for KP P like nonlinearities, the existence of a positive stationary solution to (4.3) is uniquely conditioned by the sign of λ 1 . More precisely, there exists a unique positive stationary solution if and only if λ 1 < 0. If such type of criteria seems reasonable in problems dened on bounded set, it is less obvious for problems in unbounded domains. In particular, in unbounded domains, one of the main diculty concerns the denition of λ 1 .

As shown in [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF][START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF], the notion of rst eigenvalue in unbounded domain can be ambiguous and several denition of λ 1 exists rendering the establishment of a sharp survival criteria quite involved.

For the non-local equation (4.2) less is known and to our knowledge survival criteria have been essentially investigated in some specic situations, periodic media [START_REF] Coville | Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity[END_REF][START_REF] Coville | Pulsating fronts for nonlocal dispersion and KPP nonlinearity Ann[END_REF][START_REF] Shen | Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats[END_REF] or for a version of the problem (4.2) dened in a bounded domain Ω,

∂u ∂t (t, x) = Ω J(x -y)u(t, y) dy -u(t, x) + f (x, u(t, x)) in R + × Ω, (4.5)
[2, [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] Coville | Nonlocal refuge model with a partial control[END_REF][START_REF] García-Melián | On the principal eigenvalue of some nonlocal diusion problems[END_REF][START_REF] Kao | Random Dispersal VS. Nonlocal Dispersal Discrete and Continuous Dynamical Systems[END_REF][START_REF] Shen | Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats[END_REF]. We also quote [START_REF] Berestycki | The periodic patch model for population dynamics with fractional diusion[END_REF] for an analysis of a persistence criteria in periodic media for a non-local version of (4.3) involving a fractional diusion and [START_REF] Rawal | Wenxian Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications[END_REF] for survival criteria in time periodic versions of (4.5). Similarly to the local diusion case, for KP P like non-linearities, the existence of a positive solution to the non-local equation (4.5) can be characterised by the sign of a spectral quantity λ p , called the generalised principal eigenvalue of

Ω J(x -y)φ(y) dy -φ(x) + ∂ s f (x, 0)φ(x) + λ p φ(x) = 0 in Ω. (4.6)
In fact, λ p is dened by

λ p := sup λ ∈ R | ∃ϕ ∈ C(Ω), ϕ > 0, Ω J(x -y)ϕ(y) dy + (∂ s f (x, 0) -1 + λ)ϕ(x) ≤ 0 in Ω .
Unlike the elliptic PDE case, due to the lack of a regularising eect of the diusion operator, the above spectral problem may not have a solution in any reasonable space of functions i.e (L p (Ω), C(Ω)) [START_REF] Coville | Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity[END_REF][START_REF] Coville | Singular measure as principal eigenfunctions of some nonlocal operators[END_REF][START_REF] Kao | Random Dispersal VS. Nonlocal Dispersal Discrete and Continuous Dynamical Systems[END_REF]. As a consequence, even in bounded domain, simple sharp survival criteria are already quite involved to obtain. Another diculty inherent to the study of nonlocal equations (4.6) in unbounded domain concerns the lack of "reasonable" a priori estimates for the solution thus making standard approximations dicult to use in most cases.

Notations

To simplify the presentation, we introduce some notations and various linear operator that we use along this paper:

• B R (x 0 ) will denote the standard ball of radius R centred at the point x 0

• χ R will always refer to the characteristic function of B R (0).

• S(R N ) denotes the Schwartz space.

• For a positive integrable function J ∈ S(R N ), the constant R N J(z)|z| 2 dz will refer to

R N J(z)|z| 2 dz := R N J(z) N i=0 z 2 i dz
• We denote L Ω the continuous linear operator

L Ω : C( Ω) → C( Ω) u → Ω K(x, y)u(y) dy, (4.7)
where

Ω ⊂ R N and K ∈ C( Ω × Ω) is a non negative function satisfying K(x, •) ∈ L 1 (Ω) for all x ∈ Ω.
• L R correspond to the continuous operator L Ω -Id where K(x, y) = J(x -y) and Ω = B R (0)

• We will use M to denote the operators L Ω -Id where K(x, y) = J(x -y) and Ω = R N .

• Finally, M ε will denote the operator M with a rescaled kernel 1 ε N J z ε and M ε,m := 1 ε m M ε Sometimes, we also use the notation β(x) := ∂ s f (x, 0) to avoid the repeatability.

Preliminaries

In this section, we recall some results on the principal eigenvalue of a linear non-local operator L Ω + a(x) where a(x) ∈ C( Ω) ∩ L ∞ (Ω) and some known results about the KPP equation below

∂u ∂t (t, x) = L Ω [u] + f (x, u(t, x)) in R + × Ω, (4.8)
where Ω ⊂ R N is a bounded domain. For simplicity, we divide this section into two subsections, one devoted to the principal eigenvalue and the other dedicated to known survival criteria of (4.8).

The principal eigenvalue problem in bounded domains

In this subsection, we focus on the properties of the solution of the spectral problem

L Ω [ϕ] + a(x)ϕ + λϕ = 0 in Ω. (4.9)
In contrast with elliptic operators, when a(x) ≡ Cste, neither L Ω + a(x) + λ nor its inverse are compact operators and the description of the spectrum of L Ω + a using the Krein-Rutman Theory fails. However as shown in [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF], some variational formula introduced in [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] to characterise the rst eigenvalue of elliptic operators

E := a ij ∂ ij + b i (x)∂ i + c(x), λ 1 (E) := sup{λ ∈ R | ∃ ϕ ∈ W 2,n (Ω), ϕ > 0 so that E[ϕ] + λϕ ≤ 0}, (4.10)
can be transposed to the operator L Ω + a(x). Namely, the following quantity is well dened

λ p (L Ω + a(x)) := sup{λ ∈ R | ∃ ϕ ∈ C(Ω), ϕ > 0 so that L Ω [ϕ] + a(x)ϕ + λϕ ≤ 0}. (4.11)
Also noted in [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF], the generalised principal eigenvalue given by the denition (4.11) is not always achieved. This means that there is not always a positive continuous eigenfunction associated with λ p . However, as proved in [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] Kao | Random Dispersal VS. Nonlocal Dispersal Discrete and Continuous Dynamical Systems[END_REF][START_REF] Shen | Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats[END_REF], when Ω is an open bounded set we can nd some conditions on the coecients that guarantee the existence of a positive continuous eigenfunction. For example there exists a positive continuous principal eigenfunction, if we assume that the function a(x) satises

1 sup Ω a -a(x) ∈ L 1 (Ω ) for some open bounded domain Ω ⊂ Ω.
Another useful criteria that guarantee the existence of a continuous principal eigenfunction was remark in [START_REF] Coville | Pulsating fronts for nonlocal dispersion and KPP nonlinearity Ann[END_REF][START_REF] Coville | Nonlocal refuge model with a partial control[END_REF]: Proposition 4.2.1. Let Ω be a bounded open set and let L Ω be as in (4.7) then there exists a positive continuous eigenfunction associated to λ p if and only if λ p (L Ω + a) < -sup Ω a.

We point the interested reader to [START_REF] Coville | Singular measure as principal eigenfunctions of some nonlocal operators[END_REF] for a more complete description of the positive solution associated to λ p when the domain Ω is bounded.

Next we recall some properties of the principal eigenvalue λ p that we will constantly use along this paper: Proposition 4.2.2. (i) Assume Ω 1 ⊂ Ω 2 , then for the two operators

L Ω 1 [u] + a(x)u := Ω 1 K(x, y)u(y) dy + a(x)u L Ω 2 [u] + a(x)u := Ω 2 K(x, y)u(y) dy + a(x)u
respectively dened on C(Ω 1 ) and C(Ω 2 ) we have

λ p (L Ω 1 + a(x)) ≥ λ p (L Ω 2 + a(x)).
(ii) Fix Ω and assume that a 1 (x) ≥ a 2 (x), then

λ p (L Ω + a 2 (x)) ≥ λ p (L Ω + a 1 (x)).
(iii) λ p (L Ω + a(x)) is Lipschitz continuous in a(x). More precisely,

|λ p (L Ω + a(x)) -λ p (L Ω + b(x))| ≤ a(x) -b(x) ∞ (iv)
We always have the following estimate

-sup Ω a(x) + Ω K(x, y) dy ≤ λ p (L Ω + a) ≤ -sup Ω a.
We refer to [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] Coville | Nonlocal refuge model with a partial control[END_REF] for the proofs of (i) -(iv).

Existence criteria for the kpp-equation in bounded domains

Equipped with the above notion of principal eigenvalue, it has been shown [START_REF] Bates | Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal[END_REF][START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF] that on bounded domains the existence of a positive stationary solution of (4.8) is conditioned by the sign of the principal eigenvalue λ p of the linear operator L Ω + ∂ s f (x, 0). That is to say Theorem 4.2.3 ([2, 39]). Let Ω be a bounded open set and let L Ω be as in (4.7). Assume that f satisfy (H3). Then there exists a unique positive continuous stationary solution ū to (4.8) if and only if λ p (L Ω + ∂ s f (x, 0)) < 0. Moreover, if λ p ≥ 0 then any non negative uniformly bounded stationary solution of (4.8) is identically zero. In addition, we have the following dynamics for any positive bounded continuous solution of (4.8):

(i) When

λ p ≥ 0, lim t→∞ u(t, x) → 0 uniformly in Ω, (ii) When λ p < 0, lim t→∞ u(t, x) → ū uniformly in Ω.
Remark 13. This existence criteria is similar to those known for the reaction diusion version of (4.8) [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF][START_REF] Cantrell | Diusive logistic equations with indenite weights : population models in disrupted environments[END_REF][START_REF] Cantrell | Diusive logistic equations with indenite weights : population models in disrupted environments[END_REF][START_REF] Cantrell | On the eects of spatial heterogeneity on the persistence of interacting species[END_REF][START_REF] Englander | On the construction and support properties of measure-valued diusions on D ⊂ R d with spatially dependent branching[END_REF][START_REF] Fraile | Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation[END_REF].

Generalized principal eigenvalues in unbounded domain

For the rest of the article it is important to understand the eigenvalue problem

L Ω [ϕ] + a(x)ϕ + λϕ = 0 in Ω (4.12)
when the domain Ω ⊂ R N is unbounded and in particular the eigenvalue problem

M[ϕ] + a(x)ϕ = -λϕ in R N . (4.13)
The purpose of this section is to study their properties. To this end, motivated by the works [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF][START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF] for elliptic operators, we start by introducing two denitions of the principal eigenvalue.

Denition 4.3.1. Let L Ω + a(x) be as in (4.7). We set

λ p (L Ω + a(x)) := sup{λ ∈ R | ∃ϕ > 0, ϕ ∈ C(Ω), s.t L Ω [ϕ] + a(x)ϕ + λϕ ≤ 0 in Ω} (4.14) λ p (L Ω + a(x)) := inf{λ ∈ R | ∃ϕ > 0, ϕ ∈ C(Ω) ∩ L ∞ (Ω), s.t L Ω [ϕ] + a(x)ϕ + λϕ ≥ 0 in Ω}. (4.15)
Another useful denition that holds only for self-adjoint operators is Denition 4.3.2. Let L Ω + a(x) be as in (4.7). Assume further that K(•, •) is symmetric (i.e. K(x, y) = K(y, x) for all x, y ∈ Ω). We set

λ v (L Ω + a(x)) := inf ϕ∈L 2 (Ω),ϕ ≡0 Ω Ω K(x, y)(ϕ(x) -ϕ(y)) 2 dxdy -Ω (a(x) -k(x))ϕ 2 (x) dx ϕ 2 L 2 (Ω) . (4.16)
where k(x) := Ω K(y, x) dy.

In the context of the study of nonlocal operators, these denitions are natural extension to the nonlocal case of the denitions known for an elliptic operator. It is worth to mention that those denitions have already been used in the context of the study of (4.2) in several papers [START_REF] Coville | Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity[END_REF][START_REF] Coville | Pulsating fronts for nonlocal dispersion and KPP nonlinearity Ann[END_REF][START_REF] Coville | Nonlocal refuge model with a partial control[END_REF][START_REF] García-Melián | On the principal eigenvalue of some nonlocal diusion problems[END_REF][START_REF] Ignat | San Antolin Lower and upper bounds for the rst eigenvalue of nonlocal diusion problems in the whole space[END_REF], but the relation between all the formulations has never been claried.

For elliptic operators, it is known that the analogous of the three formulations above are equivalent on bounded domain [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF]. This is not necessarily the case for unbounded domains, where examples can be constructed [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF][START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] showing that λ 1 > λ 1 . Since L Ω + a(x) shares many properties with elliptic operators, it is then expected that the three denitions are not necessarily equivalent. It is then important do understand the relation between the dierent quantities λ p , λ p and λ v .

From the denition of λ p and λ v , we can check that the monotone behaviours and the Lipschitz continuity properties satised by λ p ((i) -(iii) of Proposition (4.2.2)) are still true for λ p and λ v . Also, as already been identied for λ p (L Ω + a(x)) in a bounded domain, an intrinsic diculty related to this type of nonlocal operator comes from the possible non-existence of a positive continuous eigenfunction associated to the denition of λ p , λ p or to λ v . In particular in unbounded domain, the existence of an associated positive principal eigenfunction is not known in general. To our knowledge, the only general result known concerns the case of dispersal kernel K that satisfy the following condition:

(H5) There exist positive constants r 0 ≥ r 1 ; C 0 ≥ c 0 so that ∀x, y ∈ Ω,

C 0 χ Br 0 (x) ≥ K(x, y) ≥ c 0 χ Br 1 (x)
For such dispersal kernel, we can show that Theorem 4.3.3 (Sucient condition [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF]). Let Ω ⊂ R N , a ∈ C(Ω) ∩ L ∞ (Ω) and K ∈ C( Ω × Ω) non negative, satisfying condition (H7). Let us denote σ := supΩ a(x) and assume further that the function a(x) satises 1 σ-a(x) ∈ L 1 (Ω 0 ) for some bounded domain Ω 0 ⊂ Ω. Then there exists a principal eigen-pair (λ p , ϕ p ) solution of

L Ω [ϕ] + (a(x) + λ)ϕ = 0.
Moreover, ϕ p ∈ C(Ω), ϕ p > 0 and we have the following estimate

-σ < λ p < -σ,
where σ := sup x∈Ω a(x) + Ω K(x, y) dy .

Main results

We show that three notions of eigenvalue λ p , λ p , and λ v are equivalent when J is a symmetric kernel satisfying (H1 -H2) with compact support. Namely, we show Theorem 4.3.4. Let J be compactly supported and satisfy (H1 -H2). Assume a(x) ∈ C(R N ) ∩ L ∞ (R N ) then we have

λ p (M + a(x)) = λ p (M + a(x)) = λ v (M + a(x)).
It is worth noticing that the condition for such equivalence remains an open problem for general kernels. As a side result, we show that on bounded domain and for any symmetric kernel J satisfying (H1 -H2) we have λ p (L R + a(x)) = λ p (L R + a(x)) = λ v (L R + a(x)), where

λ v (L R + a(x)) := inf ϕ∈L 2 (B R ),ϕ ≡0 1 2 B R ×B R J(x -y)[ϕ(x) -ϕ(y)] 2 dxdy + B R R N \B R J(x -y) dy ϕ 2 (x) dx ϕ 2 L 2 (B R )
.

- B R a(x)ϕ 2 (x)dx ϕ 2 L 2 (B R )
We also obtain a full description of the asymptotic behaviour with respect to ε of the rescaled dispersal operator M ε,m := 1 ε m M ε . More precisely, we obtain Theorem 4.3.5. Let J be compactly supported and satisfy (H1 -H2). Assume a(x

) ∈ C(R N ) ∩ L ∞ (R N )
then we have

• When 0 < m ≤ 2 lim ε→+∞ λ p (M ε,m + a(x)) = -sup R N a(x) • When 0 ≤ m < 2, lim ε→0 λ p (M ε,m + a(x)) = -sup R N a(x) • When m = 0, lim ε→+∞ λ p (M ε + a(x)) = 1 -sup R N a(x)
In this case, if a(x) is radially symmetric non increasing and sup R N a(x) < 1 then there is a sharp threshold ε 0 , in the sense that λ p (M ε,m + a(x)) is nonincreasing with respect to ε and it is negative if and only if ε < ε 0 .

• When m = 2 and a(x) is globally Lipschitz, then

lim ε→0 λ p (M ε,2 + a(x)) = λ 1 K 2,N D 2 (J) 2 ∆ + a(x)
where

D 2 (J) := R N J(z)z 2 dz, K 2,N := 1 |S N -1 | S N -1 (σ.e 1 ) 2 dσ = 1 N
and

λ 1 (K 2,N D 2 (J)∆ + a(x)) := inf ϕ∈H 1 0 (R N ),ϕ ≡0 K 2,N D 2 (J) 2 R N |∇ϕ| 2 (x) dx ϕ 2 2 -R N a(x)ϕ 2 (x) dx ϕ 2 2 .

First properties of these three denitions

Equipped with this criteria, we start by exploring the behaviour of λ p (L Ω +a(x)) with respect to the domain Ω. Namely, we show Lemma 4.3.6. Assume that a achieves its maximum in Ω and let L Ω + a(x) be dened as in (4.7) with K satisfying (H7). Let (Ω n ) n∈R be a sequence of subset of Ω so that lim n→∞ Ω n = Ω, Ω n ⊂ Ω n+1 . Then we have

lim n→∞ λ p (L Ωn + a(x)) = λ p (L Ω + a(x))
Proof. By a straightforward application of the monotone properties of λ p with respect to the domain ((i) of Proposition 4.2.2) we get the inequality

λ p (L Ω + a(x)) ≤ lim n→∞ λ p (L Ωn + a(x)). (4.17)
To prove the equality, we argue by contradiction. So let us assume

λ p (L Ω + a(x)) < lim n→∞ λ p (L Ωn + a(x)), (4.18)
and let λ ∈ R so that

λ p (L Ω + a(x)) < λ < lim n→∞ λ p (L Ωn + a(x)). (4.19)
We claim Claim 4.3.7. There exists ϕ > 0, ϕ ∈ C(Ω) so that (λ, ϕ) is an adequate test function. That is, ϕ satises

L Ω [ϕ] + (a(x) + λ)ϕ ≤ 0 in Ω.
Assume for the moment that the above claim holds then by denition of λ p (L Ω + a(x)) we get a straightforward contraction λ p (L Ω + a(x)) < λ ≤ λ p (L Ω + a(x)).

Hence,

lim n→∞ λ p (L Ωn + a(x)) = λ p (L Ω + a(x))
Let us now prove the Claim Proof of the Claim. First let us denote σ := sup Ω a(x). Since by assumption the function a(•) achieves its maximum in Ω, there exists x 0 ∈ Ω so that σ = a(x 0 ). Moreover there exists n 0 so that for all n ≥ n 0 , σ = sup Ω a(x) = sup Ωn a(x).

Now let χ ε be the following cut-o" functions : χ ε (x) := χ x 0 -x ε where ε > 0 is to be chosen later on and χ is a smooth function such that 0 ≤ χ ≤ 1, χ(z) = 0 for |z| ≥ 2 and χ(z) = 1 for |z| ≤ 1. Finally, let us consider the continuous functions a ε (•), dened by a ε (x) := sup{a(x), σχ ε (x)}. Observe that by construction we have a -a ε ∞ → 0 as ε → 0.

Since Ω n → Ω, by the Lipschitz continuity of λ p (L Ω + a(x)) with respect to a(x) ((iii) Proposition 4.2.2), for ε small enough the inequality (4.19) yields to

λ p (L Ω + a ε (x)) < λ < lim n→∞ λ p (L Ωn + a ε (x)). (4.20)
By construction, by taking n 0 larger if necessary we also have B ε (x 0 ) ⊂ Ω n for all n ≥ n 0 . Moreover, sup Ωn a ε (x) = sup Ω a ε (x) = σ.

Observe now that since a ε ≡ σ on B ε 2 (x 0 ), for all n ≥ n 0 the function

1 σ-aε(x) ∈ L 1 loc (B ε (x 0 )
) and therefore by Theorem 4.3.3 for all n ≥ n 0 there exists ϕ n ∈ C( Ωn ), ϕ n > 0 associated with λ p (L Ωn + a ε (x)). Since x 0 ∈ n≥n 0 Ω n , without loss of generality we can assume that for all n ≥ n 0 the ϕ n can be normalised by ϕ n (x 0 ) = 1. Recall that for all n, ϕ n satises

L Ωn [ϕ n ] + (a ε (x) + λ p (L Ωn + a ε (x)))ϕ n = 0 in Ω n ,
so from (4.20), it follows that (ϕ n , λ) satises

L Ωn [ϕ n ] + (a ε (x) + λ)ϕ n (x) < L Ωn [ϕ n ] + (a ε (x) + λ p (L Ωn + a ε (x)))ϕ n = 0 in Ω n . (4.21)
Let us now dene b n (x) := -λ p (L Ωn + a ε (x)) -a ε (x). Then ϕ n satises

L Ωn [ϕ n ] = b n (x)ϕ n in Ω n . (4.22)
By construction for all n ≥ n 0 we have b n (x) ≤ -λ p (L Ωn 0 +a ε (x))-σ > 0, therefore the Harnack inequality (Theorem 1.4 in [START_REF] Coville | Harnack type inequality for positive solution of some integral equation[END_REF]) applies to ϕ n . Thus for n ≥ n 0 xed and for all compact set ω ⊂⊂ Ω n there exists a constant C n (ω) such that

ϕ n (x) ≤ C n (ω)ϕ n (y) ∀ x, y ∈ ω.
Moreover the constant C n (ω) only depends on x∈ω B r 0 (x) and is monotone decreasing with respect to inf x∈Ωn b n (x). For all n ≥ n 0 , the function b n (x) being uniformly bounded from below by a constant independent of n, the constant C n is bounded from above independently of n by a constant C(ω). Thus we have

ϕ n (x) ≤ C(ω)ϕ n (y) ∀ x, y ∈ ω.
From a standard argumentation, using the normalization ϕ n (x 0 ) = 1, we deduce that the sequence (ϕ n ) n≥n 0 is bounded in C loc (Ω) topology. Moreover, from a standard diagonal extraction argument, there exists a subsequence still denoted (ϕ n ) n≥n 0 such that (ϕ n ) n≥n 0 converges locally uniformly to a continuous function ϕ. Furthermore, ϕ is a nonnegative non trivial function and ϕ(x 0 ) = 1.

Since K satises condition (H7), we can pass to the limit in equation (4.21) using the Lebesgue monotone convergence theorem and get

L Ω [ϕ] + (a ε (x) + λ)ϕ(x) ≤ 0 in Ω.
Hence, we have

L Ω [ϕ] + (a(x) + λ)ϕ(x) ≤ 0 in Ω, since a(•) ≤ a ε (•).
Next we investigate the relation between λ p and λ p . We rst show that Lemma 4.3.8. Let L Ω + a be as in Lemma 4.3.6, then we always have

λ p (L Ω + a(x)) ≤ λ p (L Ω + a(x)).
Proof. First observe that to get the inequality λ p (L Ω + a(x)) ≤ λ p (L Ω + a(x)), it is enough to show that for any δ > 0:

λ p (L Ω + a(x)) ≤ λ p (L Ω + a(x)) + δ.
Let δ > 0 be xed and let us consider the operator L Ω + b(x) where b(x) denotes the function b(x) := a(x) + λ p (L Ω + a(x)) + δ.

We claim that Claim 4.3.9. There exists ϕ ≥ 0 so that ϕ ∈ C c (Ω) ∩ L ∞ (Ω) that satises

L Ω [ϕ] + b(x)ϕ ≥ 0 in Ω.
Note that by proving the claim, we prove the Lemma. Indeed, assume for the moment that the claim holds. Then by construction (ϕ, λ p (L Ω + a(x)) + δ) satises

L Ω [ϕ] + (a(x) + λ p (L Ω + a(x) + δ)ϕ ≥ 0 in Ω.
Thus by denition of λ p (L Ω + a(x)), we have λ p (L Ω + a(x)) ≤ λ p (L Ω + a(x)) + δ.

The constant δ being arbitrary, we get for all δ > 0:

λ p (L Ω + a(x)) ≤ λ p (L Ω + a(x)) + δ.
Proof of the Claim. By construction λ p (L Ω + b(x)) < 0, so by Lemma 4.

b -b ε ∞,ω = b -b ε ∞,Ω ≤ ε, λ p (L ω + b ε (x)) + ε < 0 and there is ϕ p ∈ C(ω), ϕ p > 0 associated to λ p (L ω + b ε (x)). That is ϕ p satises L ω [ϕ p ] + b ε (x)ϕ p = -λ p (L ω + b ε (x))ϕ p in ω. (4.23)
Without loss of generality we can assume that ϕ p ≤ 1.

Let σ denotes the maximum of b ε in ω, then by Proposition 4.2.2 there exists τ > 0 so that -λ p (L ω + b ε (x)) -ε -σ ≥ τ > 0. Moreover, since ϕ p satises (4.23), there exists c 0 > 0 so that inf ω ϕ p ≥ c 0 .

Let us choose ω ⊂⊂ ω such that

|ω \ ω | ≤ c 0 inf{τ, -λ p (L ω + b ε ) -ε} 2 K ∞ ,
where for a set A, |A| denotes the Lebesgue measure of A.

Since ω ⊂⊂ ω and ∂ω are two disjoint closed sets, by the Urysohn's Lemma there exists a continuous function η such that 0 ≤ η ≤ 1, η(x) = 1 in ω , η(x) = 0 in ∂ω. Consider now ϕ p η and let us compute

L ω [ϕ p η] + b ε (x)ϕ p η. Then we have L ω [ϕ p η] + b(x)ϕ p η ≥ -λ p (L ω + b ε (x))ϕ p -K |ω \ ω | -b ε (x)ϕ p (1 -η) -(b ε (x) -b(x))ϕ p (x) ≥ -(λ p (L ω + b ε (x)) + b -b ε ∞,ω )ϕ p - c 0 inf{τ, -λ p (L ω + b ε ) -ε} 2 -b ε (x)ϕ p (1 -η) ≥ -(λ p (L ω + b ε (x)) + ε)ϕ p - c 0 inf{τ, -λ p (L ω + b ε ) -ε} 2 -max{σ, 0}ϕ p ≥ -(λ p (L ω + b ε (x)) + ε + max{σ, 0})ϕ p - c 0 inf{τ, -λ p (L ω + b ε ) -ε} 2 .
Since by assumption

-λ p (L ω + b ε (x)) -ε > 0 and -λ p (L ω + b ε (x)) -ε -σ ≥ τ > 0 it follows from the above inequality that L ω [ϕ p η] + b ε (x)ϕ p η ≥ -(λ p (L ω + b ε (x)) + ε + max{σ, 0})c 0 - c 0 inf{τ, -λ p (L ω + b ε ) -ε} 2 ≥ c 0 inf{τ, -λ p (L ω + b ε (x)) -ε} 2 ≥ 0.
By construction we have ϕ p η ∈ C(ω) that satises

L ω [ϕ p η] + b(x)ϕ p η ≥ 0 in ω, ϕ p η = 0 on ∂ω.
Finally by extending ϕ p η by 0 outside ω and denoting ϕ this extension, we get our desired test function. Indeed, by construction ϕ ∈ C(R N ) ∩ L ∞ (R N ) is a non negative function which satises

L Ω [ϕ] + b(x)ϕ = L ω [ϕ] + b(x)ϕ ≥ 0 in ω, L Ω [ϕ] + b(x)ϕ = L ω [ϕ] ≥ 0 in Ω \ ω.
Hence, ϕ ≥ 0, ϕ ∈ C(Ω) ∩ L ∞ is our desired test function.

Remark 14. Note that assumption (H7) on the kernel K is only needed to reduce the problem dened on a unbounded domain to a problem dened on a bounded problem. Therefore the proof holds as well for Ω bounded and Lemma 4.3.8 holds true without condition (H7). Namely, let Ω be a bounded domain and let L Ω + a(x) be as in (4.7) then

λ p (L Ω + a(x)) ≤ λ p (L Ω + a(x)).
In particular, for any δ > 0, there exists ϕ ∈ C c (Ω) so that

L Ω [ϕ](x) + (a(x) + δ + λ p (L Ω + a(x)))ϕ(x) ≥ 0 in Ω.
Let us prove that in bounded domains, then the three denitions λ p , λ p and λ v are equivalent.

Theorem 4.3.10. Let Ω be a bounded domain of R N and let L Ω + a(x) be dened as in (4.7). Then we have

λ p (L Ω + a(x)) = λ p (L Ω + a(x))
In addition, when L Ω + a(x) is self adjoined, we have

λ p (L Ω + a(x)) = λ p (L Ω + a(x)) = λ v (L Ω + a(x)).
Proof. By Lemma 4.3.8 we already have

λ p (L Ω + a(x)) ≤ λ p (L Ω + a(x)).
It remains to prove the converse inequality. We argue by contradiction and assume that λ p (L Ω + a(x)) < λ p (L Ω + a(x)).

Choose λ ∈ (λ p (L Ω + a(x)), λ p (L Ω + a(x))). Then by denition of λ p and λ p there exists ϕ and ψ non negative continuous function so that

L Ω [ϕ](x) + (a(x) + λ)ϕ(x) ≤ 0 in Ω, (4.24)
L

Ω [ψ](x) + (a(x) + λ)ψ(x) ≥ 0 in Ω. (4.25)
Moreover, ϕ > 0 in Ω. By taking λ smaller if necessary, we can assume that ϕ satises

L Ω [ϕ](x) + (a(x) + λ)ϕ(x) < 0 in Ω.
By a direct computation, we have

Ω K(x, y)ϕ(y) ψ(y) ϕ(y) - ψ(x) ϕ(x) dy > 0.
Since ψ ϕ ∈ C( Ω), the function ψ ϕ achieve a maximum at some point x 0 ∈ Ω. At this point we get the contradiction,

0 < Ω K(x 0 , y)ϕ(y) ψ(y) ϕ(y) - ψ(x 0 ) ϕ(x 0 ) dy ≤ 0.
Thus, λ p (L Ω + a(x)) = λ p (L Ω + a(x)).

In the self-adjoined case to prove the equality of all the three denition, it is enough to prove that λ p (L Ω + a(x)) = λ v (L Ω + a(x)). Let λ > λ p (L Ω + a(x)), then by denition there exists ψ ≥ 0 so that ψ ∈ C(Ω) ∩ L ∞ (Ω) and

L Ω [ψ](x) + (a(x) + λ)ψ(x) ≥ 0 in Ω.

1 2 Ω Ω K(x, y) (ψ(x) -ψ(y)) 2 dxdy - Ω (a(x) + k(x))ψ(x) 2 dx ≤ λ Ω ψ 2 (x) dx (4.28) λ v (L Ω + a(x)) Ω ψ 2 (x) dx ≤ λ Ω ψ 2 (x) dx. (4.29)
Therefore, λ v (L Ω + a(x)) ≤ λ p (L Ω + a(x)). Let us prove now the converse inequality. We argue by contradiction and assume that

λ v (L Ω + a(x)) < λ p (L Ω + a(x)). (4.30)
Let us rst observe that by density of C( Ω) in L 2 (Ω), we can easily check that

-λ v (L Ω + a(x)) = - inf ϕ∈L 2 (Ω),ϕ ≡0 1 2 Ω Ω K(x, y)(ϕ(x) -ϕ(y)) 2 dydx -Ω (a(x) + k(x))ϕ(x) 2 dx ϕ 2 L 2 (Ω) , = - inf ϕ∈L 2 (Ω),ϕ ≡0 -Ω Ω K(x, y)ϕ(x)ϕ(y) dydx -Ω a(x)ϕ(x) 2 dx ϕ 2 L 2 (Ω) , = sup ϕ∈L 2 (Ω),ϕ ≡0 Ω [L Ω [ϕ](x) + a(x)ϕ(x)] ϕ(x) dx ϕ 2 L 2 (Ω) , = sup ϕ∈C( Ω),ϕ ≡0 Ω [L Ω [ϕ](x) + a(x)ϕ(x)] ϕ(x) dx ϕ 2 L 2 (Ω)
.

By (iv) of Proposition 4.2.2 since λ p (L Ω + a(x)) = λ p (L Ω + a(x)) we deduce from (4.30) that σ + dened by

σ + = sup ϕ∈C(Ω) Ω (L Ω [ϕ] + a(x)ϕ)ϕ Ω ϕ 2 (4.31) satises σ + > max Ω a. (4.32)
Now, using the same argument as in [START_REF] Coville | Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity[END_REF][START_REF] Hutson | The evolution of dispersal[END_REF] we deduce that the supremum in (4.31) is achieved. Indeed, it is standard [START_REF] Brezis | Functional analysis, Sobolev spaces and partial dierential equations[END_REF] that the spectrum of L Ω + a(x) is to the left of σ + and that there exists a sequence ϕ n ∈ C(Ω) such that ϕ n L 2 (Ω) = 1 and (L Ω + a(x) -σ + )ϕ n L 2 (Ω) → 0 as n → +∞. By compactness of L Ω : L 2 (Ω) → C(Ω) for a subsequence, lim n→+∞ L Ω [ϕ n ] exists in C(Ω). Then, using (4.32), we see that ϕ n → ϕ in L 2 (Ω) for some ϕ and (L Ω + a(x))ϕ = σ + ϕ. This equation implies ϕ ∈ C(Ω), and σ + is an eigenvalue for the operator L Ω + a(x). Moreover, ϕ ≥ 0, since ϕ + is also minimizer. Indeed, we have

σ + = Ω [L Ω [ϕ] + a(x)ϕ(x)]ϕ + (x) dx ϕ + 2 L 2 (Ω) , = Ω [L Ω [ϕ + ] + a(x)ϕ + (x)]ϕ + (x) dx ϕ + 2 L 2 (Ω) + Ω Ω K(x, y)ϕ -(x)ϕ + (y) dydx ϕ + 2 L 2 (Ω) , ≤ Ω [L Ω [ϕ + ] + a(x)ϕ + (x)]ϕ + (x) dx ϕ + 2 L 2 (Ω) ≤ σ + .
Thus, there exists a non-negative continuous ϕ so that

L Ω [ϕ] + (a(x) + λ v )ϕ = 0 in Ω.
Since λ v < λ p , we can argue as above and get the desired contradiction. Hence,

λ v = σ + = λ p = λ p .
Finally, we investigate the relation between λ p , λ p and λ v when M is a self-adjoined operator. precisely we establish Lemma 4.3.11. Assume that a(x) ∈ C(R N ) ∩ L ∞ (R N ), J is symmetric and J is compactly supported. Then one has

λ p (M + a(x)) ≤ lim inf n→+∞ λ v (L Bn + a(x)) ≤ λ p (M + a(x)),
where (B n ) n∈N be the increasing sequence of ball of radius n centred at 0 Before proving this Lemma, we start by showing some scaling invariance of λ p (M + a(x)). Namely, we have Lemma 4.3.12. Assume that a(x) ∈ C(R N ) ∩ L ∞ (R N ). Then for all ε > 0 one has

λ p (M + a(x)) = λ p (M ε + a ε (x)).
where a ε (x) := a x ε and

M ε [ϕ] := 1 ε N R N J x-y ε ϕ(y) dy -ϕ(x).
Proof. The proof of this invariance is a consequence of the following observation. By denition of λ p (M + a(x)), we have for all λ < λ p (M + a(x)),

M[ϕ](x) + (a(x) + λ)ϕ(x) ≤ 0 in R N ,
for some positive ϕ ∈ C(R N ). Let X = x ε and ψ(x) := ϕ( x ε )then we can rewrite the above inequality as follows

R N J (X -y) ϕ(y) dy -ϕ(X) + (a(X) + λ)ϕ(X) ≤ 0 in R N , R N J x ε -y ψ(εy) dy -ψ(x) + (a ε (x) + λ)ψ(x) ≤ 0 in R N R N J ε (x -z) ψ(z) dy -ψ(x) + (a ε (x) + λ)ψ(x) ≤ 0 in R N ,
with J ε (z) := 1 ε N J z ε and a ε (x) := a x ε . Thus ψ is a positive continuous function that satises

M ε ψ(x) + (a ε (x) + λ)ψ(x) ≤ 0 in R N .
Therefore, λ ≤ λ p (M ε +a ε (x)) and as a consequence λ p (M+a(x)) ≤ λ p (M ε +a ε (x)). The above argument holds true by replacing the role of λ p (M + a(x)) and λ p (M ε + a ε (x)). Hence, we get

λ p (M + a(x)) = λ p (M ε + a ε (x)).
Let us now turn our attention to the proof of Lemma 4.3.11.

Proof of Lemma 4.3.11. The proof follows some ideas developed in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF][START_REF] Berestycki | The periodic patch model for population dynamics with fractional diusion[END_REF][START_REF] Coville | Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity[END_REF][START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF]. Let us call λ p = λ p (L Ω + a(x)) and λ p = λ p (L Ω + a(x)). By Lemma 4.3.12, without any loss of generality we can assume that the support of J is contained in the unit ball. First recall that that for a domain Ω bounded, we have

λ p = λ p = λ v .
Let (B n ) n∈N be the increasing sequence of ball of radius n centred at 0. By monotonicity of λ p with respect to the domain, we have

λ p (M + a(x)) ≤ λ p (L Bn + a(x)) = λ v (L Bn + a(x)) Therefore λ p (M + a(x)) ≤ lim inf n→∞ λ v (L Bn + a(x)).
Thanks to the last inequality, to show λ p (M + a(x)) ≤ λ p (M + a(x)) it is enough to prove that

lim inf n→∞ λ v (L Ωn + a(x)) ≤ λ p (M + a(x)). (4.33)
Observe that to prove (4.33), it is enough to show that for any δ > 0

lim inf n→∞ λ v (L Bn + a(x)) ≤ λ p (M + a(x)) + δ. (4.34)
Let us now x δ > 0 and let us denote µ := λ p (M + a(x)) + δ. By denition of λ p (M + a(x)) there exist

a function ϕ ∈ C(R N ) ∩ L ∞ (R N ), ϕ ≥ 0 satisfying M[ϕ] + a(x)ϕ(x) + µϕ(x) ≥ 0 in R N . (4.35)
Without loss of generality we can also assume that ϕ L ∞ (R N ) = 1. Let χ n be the characteristic function of B n and let w n = ϕχ n . By denition of λ v (L Bn + a(x)), since

w n ∈ L 2 (B n ) we have λ v (L Bn + a(x)) w n 2 L 2 (Bn) ≤ Bn -L Bn [w n ](x) -a(x)w n (x) w n (x) dx. ( 4
I n ϕ 2 L 2 (Bn) ≤ C ϕ L 2 (B n+1 \Bn) ϕ L 2 (Bn) .
Thanks to Lemma 2.6 in [START_REF] Berestycki | The periodic patch model for population dynamics with fractional diusion[END_REF] the right hand side of the above inequality tends to 0. Hence, we get

lim inf n→∞ λ v (L Bn + a(x)) ≤ µ + lim inf n→∞ I n ϕ n 2 L 2 (Bn) = λ p (M + a(x)) + δ (4.48)
Since in the above argumentation δ can be chosen arbitrary the inequality (4.48) holds true for any δ > 0 and the Lemma is proved.

When N = 1, the restriction on the kernel can be weakened and we can prove that Lemma 4.3.13. Assume that a(x) ∈ L ∞ (R), K is symmetric and K satises 0 ≤ K(x, y) ≤ C(1 + |x -y|) α for some α > 3 2 . Then one has

λ p (M + a(x)) ≤ lim inf n→+∞ λ v (L (-n,n) + a(x)) ≤ λ p (M + a(x)).
Proof. By arguing as in the above proof, for any δ > 0 there exists ϕ ∈ C(R) ∩ L ∞ (R N ) so that

M[ϕ] + (a(x) + λ p (M + a(x)) + δ)ϕ(x) ≥ 0 in R. and λ v (L (-n,n) + a(x)) w n 2 L 2 ((-n,n)) ≤ µ w n 2 L 2 ((-n,n)) + I n
, where µ := λ p (M + a(x)) + δ), w n := ϕχ (-n,n) and I n denotes

I n := (-n,n) R\(-n,n) K(x, y)ϕ(y) dy ϕ(x) dx (4.49)
As above we achieve our proof by showing

lim inf n→∞ I n ϕ 2 L 2 ((-n,n)) = 0. (4.50)
Let us now treat two cases independently:

Case 1: ϕ ∈ L 2 (R)
Assume that ϕ ∈ L 2 (R), then by the Cauchy-Schwarz inequality in (4.49), we obtain (4.51)

I n ≤ ϕ L 2 (R\(-n,n)) R\(-n,n) n -n K(x, y)ϕ(x) dx
I n ≤ C ϕ L 2 (R\(-n,n)) ϕ L 2 ((-n,n)) . (4.52)
Thus,

lim inf n→∞ I n ϕ 2 L 2 ((-n,n)) ≤ lim inf n→∞ ϕ L 2 (R\(-n,n)) ϕ L 2 ((-n,n)) = 0. Case 2: ϕ ∈ L 2 (R)
Assume now that ϕ ∈ L 2 , then we argue as follows. By using the Fubini Theorem and the Cauchy-Schwarz inequality in (4.49), we obtain

I n ≤ ϕ L 2 ((-n,n)) -n -∞ n -n K(x, y) 2 dx 1 2 ϕ(y) dy + +∞ n n -n K(x, y) 2 dx 1 2
ϕ(y) dy

(4.53) ≤ ϕ L 2 ((-n,n)) Ĩ- n + Ĩ+ n (4.54)
Recall that by assumption there exists C > 0 such that K(x, y) ≤ C(1 + |x -y|) -α with α > 3 2 . So, we have

Ĩ- n ≤ C -n -∞ n -n (1 + |x -y|) -2α dx 1 2 ϕ(y) dy Ĩ+ n ≤ C +∞ n n -n (1 + |x -y|) -2α dx 1 2

ϕ(y) dy

To complete our proof, we have to show that

Ĩ± n ϕ L 2 ((-n,n))
→ 0. The proof being similar in both cases, we only prove it for Ĩ+ n . We claim that Claim 4.3.14. There exists C > 0 so that for all n ∈ N,

+∞ n n -n (1 + |x -y|) -2α dx 1 2 ϕ(y) dy ≤ C.
Assume for the moment that the claim hold true, then (4.50) holds true. Indeed, thanks to (4.54) we have

I n ϕ 2 L 2 ((-n,n)) ≤ C ϕ L 2 ((-n,n)) → 0 when n → ∞.
Hence, in both situation we get

lim inf n→∞ λ v (L (-n,n) + a(x)) ≤ µ + lim inf n→∞ I n ϕ n 2 L 2 ((-n,n))
= λ p (M + a(x)) + δ Since in the above argumentation δ can be chosen arbitrary the above inequality holds true for any δ > 0 and the Lemma is proved Proof of the Claim. Since ϕ ∈ L ∞ and y ≥ n then x ≤ y and we have

Ĩ+ n ≤ ϕ ∞ +∞ n n -n (1 + y -x) -2α dx 1 2 dy ≤ ϕ ∞ √ 2α -1 +∞ n (1 + y -n) α-1 2 dy, ≤ C +∞ 0 (1 + z) α-1 2 dz.
Open problem 1. Is the result in 1D still valid in for N > 1?

We can now prove Theorem 4.

lim R→∞ λ v (L R + a(x)) ≤ λ p (M + a(x)) ≤ λ p (M + a(x)) ≤ lim R→∞ λ v (L R + a(x)) ≤ λ p (M + a(x)).
Therefore, lim

R→∞ λ v (L R + a(x)) = λ p (M + a(x)) = λ p (M + a(x)).
It remains to prove that λ v (M + a(x)) = λ p (M + a(x)) = λ p (M + a(x)). By denition of λ p (M + a(x)) and using the Claim 4.3.9, we can check that

λ v (M + a(x)) ≤ λ p (M + a(x)).
On another hand by denition of λ v (M + a(x)) for any δ > 0 there exists ϕ δ ∈ L 2 (R N ) so that

1 2 R N ×R N J(x -y)(ϕ δ (x) -ϕ δ (y)) 2 dydx - R N a(x)ϕ 2 δ (x) dx ϕ δ 2 L 2 (R N ) ≤ λ v (M + a(x)) + δ. Let us compute I R (ϕ δ ) = 1 2 B R ×B R J(x -y)(ϕ δ (x) -ϕ δ (y)) 2 dydx + B R k R (x)ϕ 2 δ (x) dx -B R a(x)ϕ 2 δ (x) dx ϕ δ 2 L 2 (B R )
, with k(x) := R N \B R J(x -y) dy. Since for all x ∈ R N , lim R→∞ k R (x) = 0, a(x) ∈ L ∞ and ϕ δ ∈ L 2 (R N ) by Lebesgue's monotone convergence Theorem, we get

B R k R (x)ϕ 2 δ (x) dx - B R a(x)ϕ 2 δ (x) dx ≤ δ ϕ δ 2 L 2 (B R ) - R N a(x)ϕ 2 δ (x) dx for R large enough.
Thus we have for R large enough

I R (ϕ δ ) ≤ 1 2 R N ×R N J(x -y)(ϕ δ (x) -ϕ δ (y)) 2 dydx - R N a(x)ϕ 2 δ (x) dx + δ ϕ δ 2 L 2 (B R ) ϕ δ 2 L 2 (B R ) , ≤ ϕ δ 2 L 2 (R N ) ϕ δ 2 L 2 (B R ) (λ v (M + a(x)) + δ) + δ, ≤ λ v (M + a(x)) + Cδ,
for some universal constant C > 0.

By using the denition of λ v (L R + a(x)) we then achieve

λ v (L R + a(x)) ≤ I R (ϕ δ ) ≤ λ v (M + a(x)) + Cδ for R large enough. Therefore lim R→∞ λ v (L R + a(x)) ≤ λ v (M + a(x)) + Cδ. (4.55)
Since (4.55) holds true for any δ, we get

lim R→∞ λ v (L R + a(x)) ≤ λ v (M + a(x)).
As a consequence, we obtain

λ p (M + a(x)) ≤ λ p (M + a(x)) ≤ lim R→∞ λ v (L R + a(x)) ≤ λ v (M + a(x)) ≤ λ p (M + a(x)),
which enforces λ v (M + a(x)) = λ p (M + a(x)).

Further properties of the principal eigenvalue

In this section we investigate further the properties of the principal eigenvalue λ p (M+a(x)) and in particular its behaviour with respect to some scaling limit. We focus on the behaviour of the principal eigenvalue of the spectral problem

M ε,m [ϕ] + (a(x) + λ)ϕ = 0 in R N , where M ε,m [ϕ] := 1 ε m R N J(z)[ϕ(x + εz) -ϕ(x)] dz .
Depending on m, we establish some limits for λ p (M ε,m + a(x)) when ε → 0 and ε → ∞. We analyse three situations: m = 0, 0 < m < 2 and m = 2. m = 0 and m = 2 are two critical situations, so we postpone for the moment their study and start by the simplest case to analyse 0 < m < 2. In this situation, we have Theorem 4.4.1. Let J ∈ C c (R N ) be a radially symmetric non negative function of unit mass and assume that 0 ≤ m < 2. Then we have the following:

• When 0 ≤ m < 2, lim ε→0 λ p (M ε,m + a(x)) = -sup R N a(x) • When 0 < m < 2 lim ε→+∞ λ p (M ε,m + a(x)) = -sup R N a(x) • When m = 0, lim ε→+∞ λ p (M ε + a(x)) = 1 -sup R N a(x).
In this case, if a(x) is radially symmetric non increasing and sup R N a(x) < 1 then there is a sharp threshold ε 0 , in the sense that λ p (M ε,m + a(x)) is nonincreasing with respect to ε and it is negative if and only if ε < ε 0 . Remark 15. The proof of the sharp threshold is linked to the study of the limits of solutions of the rescaling nonlinear equations. Therefore, we postpone it to Theorem 5.1.2, Chapter 5.

When m = 2, the limit as ε → 0 is more delicate and is related to some characterisation of Sobolev Spaces [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Brézis | Ham How to recognize constant functions. Connections with Sobolev spaces[END_REF][START_REF] Ac | A new approach to Sobolev spaces and connections to Γ-convergence[END_REF]. In this situation, we have Theorem 4.4.2. Let J ∈ C c (R N ) be a radially symmetric non negative function of unit mass and assume a(x) is globally Lipschitz . Then

lim ε→0 λ p (M ε,2 + a(x)) = λ 1 K 2,N D 2 (J) 2 ∆ + a(x)
We will now use the notation

I ε,m (ϕ) := 1 ε m 1 2 R N R N J ε (x -y)(ϕ(x) -ϕ(y)) 2 dxdy - R N a(x)ϕ 2 (x) dx ϕ 2 2 , J (ϕ) := D 2 (J) 2 R N |∇ϕ| 2 (x) dx ϕ 2 2 , A(ϕ) := R N a(x)ϕ 2 (x) dx ϕ 2 2 ,
where D 2 (J) := R N J(z)|z| 2 dz. By Lemma 4.4.3, for any ϕ ∈ H 1 we get

I ε,m (ϕ) ≤ ε 2-m J (ϕ) -A(ϕ). ( 4
(M ε,m + a(x)) ≤ λ 1 (D 2 (J)∆ + a(x)) for all ε > 0. Proof of Theorem 4.4.1. Let ϕ ∈ C ∞ c (R N ) so that supp(ϕ) ⊂ supp(a + ).
Then by Lemma 4.3.11 and (4.58) we get for ε small enough

λ p (M ε,m + a(x)) ≤ I ε,m (ϕ) ≤ ε 2-m J (ϕ) -A(ϕ).
To obtain the limit, we just have to remark that by Theorem 4.3.3 we have

-sup R N 1 ε m R N J ε (x -y) dy -1 + a(x) = -sup R N a(x) ≤ λ p (M ε,m + a(x)). Therefore -sup R N a(x) ≤ lim inf ε→0 λ p (M ε,m + a(x)).
On another hand, since a achieves its maximum, says at x 0 , for any ρ > 0 there exists ϕ ρ ∈ H 1 so that supp(ϕ ρ ) ⊂ B ρ (x 0 ) and from the above inequality, we deduce that

lim sup ε→0 λ p (M ε,m + a(x)) ≤ A(ϕ ρ ) = - Bρ(x 0 ) a(x)ϕ 2 ρ (x) dx ϕ ρ 2 2 ≤ -min Bρ(x 0 ) a + (x). Hence, -sup R N a(x) = -a(x 0 ) ≤ lim inf ε→0 λ p (M ε,m + a(x)) ≤ lim sup ε→0 λ p (M ε,m + a(x)) ≤ lim ρ→0 -min Bρ(x 0 ) a + (x) = -a(x 0 ).
Let us now look at the limits of λ p (M ε,m +a(x)) when ε → +∞. The limits when m > 0 is straightforward consequence of the denition of λ p (M ε,m +a(x)). From the denition and by using constant as test function, we always have this estimate

-sup R N 1 ε m R N J ε (x -y) dy -1 + a(x) ≤ λ p (M ε,m + a(x)) ≤ -sup R N - 1 ε m + a(x) . Therefore, -sup R N a(x) ≤ λ p (M ε,m + a(x)) ≤ -sup R N - 1 ε m + a(x)
and for m > 0 we have

-sup R N a(x) ≤ lim ε→+∞ λ p (M ε,m + a(x)) ≤ -sup R N a(x).
Finally, we obtain the last limit as ε → ∞ in the case m = 0. Recall that by Theorem 4.3.3 we have for all ε > 0

-sup R N (a(x)) ≤ λ p (M ε + a(x)) ≤ 1 -sup R N (a(x)). So we have lim sup ε→∞ λ p (M ε + a(x)) ≤ 1 -sup R N (a(x)).
On another hand, for any ϕ ∈ C c (R N ) we have for all ε,

I ε (ϕ) = 1 2 R N R N J ε (x -y)(ϕ(x) -ϕ(y)) 2 dxdy - R N a(x)ϕ 2 (x) dx, = - R N R N J ε (x -y)ϕ(x)ϕ(y)dxdy + R N ϕ 2 (x) dx - R N a(x)ϕ 2 (x) dx, ≥ -ϕ L 2 (R N ) R N R N J ε (x -y)ϕ(x) dx 2 dy 1/2 + R N ϕ 2 (x) dx -sup R N a(x) R N ϕ 2 (x) dx, ≥ -J ε ∞ R N ϕ 2 (x) dx + R N ϕ 2 (x) dx -sup R N a(x) R N ϕ 2 (x) dx, ≥ - J ∞ ε N R N ϕ 2 (x) dx + R N ϕ 2 (x) dx -sup R N a(x) R N ϕ 2 (x) dx.
Therefore by density of C c (R N ) in L 2 (R N ), we have for any ϕ ∈ L 2 (R N )

I ε (ϕ) ϕ 2 L 2 (R N ) ≥ - J ∞ ε N + 1 -sup R N a(x). Thus λ p (M ε + a(x)) ≥ - J ∞ ε N + 1 -sup R N a(x), and
lim inf ε→+∞ λ p (M ε + a(x)) ≥ 1 -sup R N a(x).
Hence,

1 -sup R N a(x) ≤ lim inf ε→+∞ λ p (M ε + a(x)) ≤ lim sup ε→+∞ λ p (M ε + a(x)) ≤ 1 -sup R N a(x).
Let us now deal with the case m = 2 and prove Theorem 4.4.2.

Proof. Let us rewrite I ε,2 (ϕ) in a more convenient way. For ϕ ∈ H 1 (R N )

I ε,2 (ϕ) = 1 ϕ 2 2 1 2ε 2 R N R N J ε (x -y)(ϕ(x) -ϕ(y)) 2 dxdy -A(ϕ) (4.59) = 1 ϕ 2 2 1 2ε 2 R N R N J ε (x -y)|x -y| 2 (ϕ(x) -ϕ(y)) 2 |x -y| 2 dxdy -A(ϕ) (4.60) = 1 ϕ 2 2 D 2 (J) 2 R N R N ρ ε (x -y) (ϕ(x) -ϕ(y)) 2 |x -y| 2 dxdy -A(ϕ), (4.61)
with ρ ε (z) :=

1 ε 2 D 2 (J) J ε (z)|z| 2 . By construction ρ ε is a continuous mollier so that      ρ ε ≥ 0 in R N , R N ρ ε (z)dz = 1 ∀ ε > 0, lim ε→0 |z|≥δ ρ ε (z)dz = 0 ∀ δ > 0.
From the known characterisation of Sobolev spaces in [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Brézis | Ham How to recognize constant functions. Connections with Sobolev spaces[END_REF][START_REF] Ac | A new approach to Sobolev spaces and connections to Γ-convergence[END_REF], we have

lim ε→0 R N R N ρ ε (x -y) (ϕ(x) -ϕ(y)) 2 |x -y| 2 dxdy = K 2,N ∇ϕ 2 2 , with K 2,N := 1 |S N -1 | S N -1 (σ.e 1 ) 2 dσ. Therefore lim ε→0 I ε,2 (ϕ) = K 2,N D 2 (J) 2 ∇ϕ 2 2 ϕ 2 2 -A(ϕ).
From the above inequality, we deduce that

lim sup ε→0 λ p (M ε,2 + a(x)) ≤ λ 1 K 2,N D 2 (J) 2 ∆ + a(x) .
Conversely, let δ > 0, we claim that for all ε > 0 there exists

ϕ ε ∈ C 2 c (R N ) ∩ L ∞ (R N ) so that M ε,2 [ϕ ε ] + (a(x) + λ p (M ε,2 + a(x)) + 2δ)ϕ ε ≥ 0.
Indeed, from the Claim 4.3.9, for all ε, there exists

ψ ε ∈ C c (R N ) ∩ L ∞ (R N ) so that M ε,2 [ψ ε ] + (a(x) + λ p (M ε,2 + a(x)) + δ)ψ ε ≥ 0.
Let η be a smooth mollier of unit mass and with support in the unit ball. Consider

η τ := 1 τ N η z τ . By taking ϕ ε := η τ ψ ε and observing that M ε,2 [ϕ ε ] = η τ (M ε,2 [ψ ε ]) we achieve η τ (M ε,2 [ψ ε ] + (a(x) + λ p (M ε,2 + a(x)) + δ)ψ ε ) ≥ 0, M ε,2 [ϕ ε ] + (λ p (M ε,2 + δ)ϕ ε + η τ (aψ ε )(x) ≥ 0, M ε,2 [ϕ ε ] + (a(x) + λ p (M ε,2 + δ)ϕ ε + R N η τ (x -y)ψ ε (y)(a(y) -a(x)) dy ≥ 0.
Since a(x) is uniformly Lipschitz, we can estimate the integral in the last inequality by

R N η τ (x -y)ψ ε (y)(a(y) -a(x)) dy ≤ R N η τ (x -y)ψ ε (y) a(y) -a(x) y -x |x -y| dy ≤ κτ R N η τ (x -y)ψ ε (y) dy,
where κ is the Lipschitz constant of a. Thus for τ small, says τ ≤ δ 2κ , we achieve

M ε,2 [ϕ ε ] + (a(x) + λ p (M ε,2 + a(x)) + 2δ)ϕ ε ≥ 0. (4.62)
Let us normalize now ϕ ε so that

R N ψ 2 ε (x) dx R N ϕ 2 ε (x) dx = 1. (4.63)
Let us now multiply M ε,ε [ϕ ε ] by -ϕ ε and integrate it over R N , then we have

- R N M ε,2 [ϕ ε ]ϕ ε (x) dx = - R N R N 1 ε 2 J ε (x -y)(ϕ ε (y) -ϕ ε (x))ϕ ε (x) dydx. (4.64) = 1 2ε 2 R N R N J ε (x -y)(ϕ ε (y) -ϕ ε (x)) 2 dxdy (4.65) = D 2 (J) 2 R N R N ρ ε (z) (ϕ ε (x + z) -ϕ ε (x)) 2 |z| 2 dzdx (4.66)
Combining (4.62) and (4.66) it follows that

D 2 (J) 2 R N R N ρ ε (z) (ϕ ε (x + z) -ϕ ε (x)) 2 |z| 2 dzdx- R N a(x)ϕ ε (x) 2 dx ≤ (λ p (M ε,2 +a(x))+2δ) R N ϕ 2 ε (x) dx (
4.67) On another hand, inspired by the proof of Theorem 2 in [START_REF] Brézis | Ham How to recognize constant functions. Connections with Sobolev spaces[END_REF], since ϕ ε ∈ C 2 c (R N ), for all x, z ∈ R N we have by a Taylor expansion

|ϕ ε (x + z) -ϕ ε (x) -z • ∇ϕ ε (x)| ≤ i,j |z i z j | 1 0 t 1 0 |∂ ij ϕ ε (x + tsz)| ds dt. Therefore |z • ∇ϕ ε (x)| ≤ i,j |z i z j | 1 0 t 1 0 |∂ ij ϕ ε (x + tsz)| ds dt + |ϕ ε (x + z) -ϕ ε (x)|,
and for every θ > 0 we have

|z • ∇ϕ ε (x)| 2 ≤ C θ   i,j |z i z j | 1 0 t 1 0 |∂ ij ϕ ε (x + tsz)| ds dt   2 + (1 + θ)|ϕ ε (x + z) -ϕ ε (x)| 2 , ≤ C θ i,j |z i z j | 2 1 0 t 2 1 0 |∂ ij ϕ ε (x + tsz)| 2 ds dt + (1 + θ)|ϕ ε (x + z) -ϕ ε (x)| 2 . Thus R N |z|≤1 ρ ε (|z|) |z| 2 |z • ∇ϕ ε (x)| 2 dzdx ≤ ≤ C θ R N |z|≤1 ρ ε (|z|) i,j |z i z j | 2 |z| 2 1 0 t 2 1 0 |∂ ij ϕ ε (x + tsz)| 2 ds dt dzdx + (1 + θ) R N |z|≤1 ρ ε (|z|) |ϕ ε (x + z) -ϕ ε (x)| 2 |z| 2 dzdx. For ε small supp(ρ ε ) ⊂ B 1 (0), therefore |z|≤1 ρ ε (|z|) |z| 2 |z • ∇ϕ ε (x)| 2 dz = K 2,N |∇ϕ ε (x)| 2 ,
and we have

K 2,N R N |∇ϕ ε (x)| 2 dx ≤ C θ R N |z|≤1 ρ ε (|z|) i,j |z i z j | 2 |z| 2 1 0 t 2 1 0 |∂ ij ϕ ε (x + tsz)| 2 ds dt dzdx +(1 + θ) R N |z|≤1 ρ ε (|z|) |ϕ ε (x + z) -ϕ ε (x)| 2 |z| 2 dzdx. (4.68)
Dividing by ϕ ε 2 2 the above inequality and subtracting A(ϕ ε ) on both side, it follows

K 2,N J (ϕ ε ) -A(ϕ ε ) ≤ C θ ϕ ε 2 2 R N |z|≤1 ρ ε (|z|) i,j |z i z j | 2 |z| 2 1 0 t 2 1 0 |∂ ij ϕ ε (x + tsz)| 2 ds dt dzdx +(1 + θ)I ε,2 (ϕ ε ). (4.69)
By combining now (4.69) with (4.67) and using the denition of λ 1

K 2,N D 2 (J) 2 ∆ + a(x) we get λ 1 K 2,N D 2 (J) 2 ∆ + a(x) ≤ C θ ϕ ε 2 2 R N |z|≤1 ρ ε (|z|) i,j |z i z j | 2 |z| 2 1 0 t 2 1 0 |∂ ij ϕ ε (x + tsz)| 2 ds dt dzdx +(1 + θ)[λ p (M ε,2 + a(x)) + 2δ] (4.70)
Let us now estimate the last integral in the above inequality. By construction we have ∂ ij ϕ ε = ∂ ij η τ ψ ε , so by Fubini's Theorem and standard convolution estimates we get

i,j R N |z|≤1 [0,1] 2 ρ ε (|z|) |z i z j | 2 |z| 2 t 2 |∂ ij ϕ ε (x + tsz)| 2 ≤ i,j |z|≤1 [0,1] 2 ρ ε (|z|) |z i z j | 2 |z| 2 t 2 R N |∂ ij η τ ψ ε (x + tsz)| 2 dx , ≤ i,j |z|≤1 [0,1] 2 ρ ε (|z|) |z i z j | 2 |z| 2 t 2 ∂ ij η τ L 1 (R N ) ψ ε 2 2 , ≤ C N ∂ ij η τ L 1 (R N ) 6 ψ ε 2 2 |z|≤1 ρ ε (|z|)|z| 2 dz,
with C N a universal constant depending only on N . Combining this inequality with (4.70), we get

λ 1 K 2,N D 2 (J) 2 ∆ + a(x) ≤ C θ,N ∂ ij η τ L 1 (R N ) ψ ε 2 2 ϕ ε 2 2 |z|≤1 ρ ε (|z|)|z| 2 dz + (1 + θ)[λ p (M ε,2 + a(x)) + 2δ] (4.71) Thanks to the normalisation (4.63), since |z|≤1 ρ ε (|z|)|z| 2 dz ≤ ε 2 letting ε → 0 in (4.71) yields λ 1 K 2,N D 2 (J) 2 ∆ + a(x) ≤ (1 + θ)[2δ + lim inf ε→0 λ p (M ε,2 + a(x))]. (4.72)
Since (4.72) holds for every θ, δ > 0, we obtain

λ 1 K 2,N D 2 (J) 2 ∆ + a(x) ≤ lim inf ε→0 λ p (M ε,2 + a(x)).
Hence,

λ 1 K 2,N D 2 (J) 2 ∆ + a(x) ≤ lim inf ε→0 λ p (M ε,2 +a(x)) ≤ lim sup ε→0 λ p (M ε,2 +a(x)) ≤ λ 1 K 2,N D 2 (J) 2 ∆ + a(x) .
Remark 17. The characterisation of the limits also holds for a operator 1 ε 2 L R,ε + a(x). Indeed, from Remark 14 for δ > 0, then for all ε > 0 there exists

ψ ε ∈ C c (B R ) so that 1 ε 2 L ε,R [ψ ε ](x) + (a(x) + δ + λ p ( 1 ε 2 L R,ε + a(x)))ψ ε (x) ≥ 0.
Thus by reproducing the above argumentation, we get

λ 1 K 2,N D 2 (J) 2 ∆ + a(x), B R ≤ lim inf ε→0 λ p ( 1 ε 2 L R,ε + a(x)).
To obtain the upper bound, we just observe that for any ϕ ∈ C 2 c (B R ) then we have This is a continuation of Chapter 4, where the fundamental results of generalized principal eigenvalues are applied to derive some deeper results on existence, nonexistence, uniqueness of the positive solutions of (4.3), the long time behavior of (4.1) and the asymptotic of u ε , which are solutions of the rescaling equations

λ p ( 1 ε 2 L R,ε +a(x)) ≤ 1 2ε 2 B R ×B R J ε (x -y)[ϕ(x) -ϕ(y)] 2 dxdy + 1 ε 2+N B R k ε (x)ϕ 2 (x) dx -B R a(x)ϕ 2 (x) dx ϕ 2 L 2 (B R ) , where k ε (x) := 1 ε 2 R N \B R J ε (x -y). Since ϕ is compactly supported, we have 1 ε 2+N B R k ε (x)ϕ 2 (x) dx = 0 for ε small and we deduce that lim sup ε→0 λ p ( 1 ε 2 L R,ε + a(x)) ≤ λ 1 K 2,N D 2 (J) 2 ∆ + a(x), B R .
M ε,m [u ε ] + u ε (a(x) -u ε ) = 1 ε m (J ε u ε -u ε ) + u ε (a(x) -u ε ) = 0 x ∈ R N , (5.1)
for m ∈ [0, 2], as ε → 0 and ε → ∞ .

In the last section, we further provide some extensions for non compactly supported kernels, which possess an adequate decay near innity.

Main Results

Let us now state our main results for this chapter. We rst prove a simple sharp survival criteria assuming that the dispersal kernel J satisfy an extra assumption.

Theorem 5.1.1. Assume that J, f satisfy (H1-H4) and assume further that J is compactly supported. Then, there exists a unique positive solution to (4.2) i λ p (M + ∂ s f (x, 0)) < 0, where

λ p (M + ∂ u f (x, 0)) := sup{λ ∈ R | ∃ϕ ∈ C(R N ), φ > 0 so that M[ϕ] + ∂ s f (x, 0)ϕ + λϕ ≤ 0}, 121
where M denotes the continuous operator M[ϕ] = J ϕ(x) -ϕ(x). Moreover, for any non-negative initial data u 0 ∈ C(R N ) ∩ L ∞ (R N ) we have the following asymptotic behaviour:

• If λ p (M + ∂ s f (x, 0)) ≥ 0, then the solution u ∞ (t) → 0 as t → ∞, • If λ p (M + ∂ s f (x, 0)) < 0, then the solution u -ũ ∞ (t) → 0 as t → ∞
, where ũ denotes the unique positive solution to (4.2)

In addition, if the initial data

u 0 ∈ C(R N ) ∩ L 1 (R N ), then the convergence u(t, x) → ũ holds in L 1 (R N ).
Next, we aim at understanding the eect of the dispersal kernel on the survival of the species. To this end, we analyse the behaviour of the survival criteria under some scaling of the dispersal operator. More precisely, let J ε := 1 ε N J z ε and let M ε denotes the operator M with the rescaled kernel, then we look at the behavior of the solution to (4.2) as ε → 0 or +∞ where the dispersal operator M is replace by α(ε)M ε , with α(ε) ∼ α 0 ε m . These asymptotics represent two possible strategies that are observed in nature. The terms α(ε) refers to a dispersal budget of the species as dened in [START_REF] Hutson | The evolution of dispersal[END_REF]. Roughly speaking, for a xed cost, this budget is a way to measure the dierences between dierent strategies. For a given dispersal cost function of the order of ∼ |y| m , the term α(ε) behaves like α 0 ε m and in the analysis, the dispersal operator is then given by α(ε)M ε . As explained in [START_REF] Hutson | The evolution of dispersal[END_REF], the limit as ε → 0 can be associated to a strategy of producing a lot of ospring but with little capacity of movement. Whereas the limit ε → +∞ corresponds to a strategy that aims at maximizing the possibility to explore the environment at the expense of the number of ospring.

Here, we analysis cases 0 ≤ m ≤ 2 and α 0 = 1, the case m = 0 corresponding to understand the impact of the mean distance on the survival criteria. To simplify the presentation of these asymptotics, we restrict to nonlinearities f (x, s) of the form f (x, s) = s(a(x) -s). However, the proofs apply more generally.

In this situation, we rst obtain Theorem 5.1.2. Assume that J and f satisfy (H1-H4), J is compactly supported and let m = 0. Then there exists ε 0 ∈ (0, +∞] so that for all ε ≤ ε 0 there exists a positive solution u ε to (4.2). Moreover, we have

lim ε→ε 0 u ε (x) = (a(x) -1) + ,
where a(x) + denotes the positive part of a (i.e. a + (x) = sup{0, a(x)}). Assuming further that a is smooth, at least C 0,1 (R N ), we have

lim ε→0 u ε (x) = v(x).
where v is a non-negative bounded solution of

v(x)(a(x) -v(x)) = 0 in R N .
In addition, when a(x) is radially symmetric non increasing and ε 0 < +∞ then ε 0 is sharp, in the sense that for all ε ≥ ε 0 there is no positive solution to (4.2).

The ecological interpretation of this result is that one way of persistence for a species is to match the resource and not move too much. In some situation, ε 0 = +∞ and there is no eect of the dispersal on the survival criteria of the species. A natural condition that ensures that ε 0 = +∞ is (a(x) -1) + = 0.

Main Results

123

In this context, the birth rates exceed all death rates and guarantee the persistence of the population no matter is the dispersal strategy. In particular, the exists a bounded positive solution to (4.2) for any positive kernel J. The uniqueness and the behaviour at innity of the solution are still open questions for general kernels.

When m > 0 then the characterisation of the existence of a positive solution changes and a new picture emerges. In particular, for large ε there is always solution to (4.2) whereas for small ε it may happen that no positive solution exists. Thus, the situation is, in a sense, opposite to the case when m = 0. Non existence for small value of ε appears only when m ≥ 2. More precisely we prove Theorem 5.1.3. Assume that J and f satisfy (H1-H4), J is compactly supported and let 0 ≤ m < 2. Then there exists ε 0 ≤ ε 1 ∈ (0, +∞) so that for all ε ≤ ε 0 and for all ε ≥ ε 1 there exists a positive solution u ε to (4.2). Moreover, we have

lim ε→+∞ u ε -a + ∞ = 0, lim ε→+∞ u ε -a + L 2 (R N ) = 0.
In addition, assuming further that a is smooth, at least C 2 (R N ), we have

lim ε→0 u ε (x) = v(x).
where v is a non-negative bounded solution of

v(x)(a(x) -v(x)) = 0 in R N .
Remark 18. Due to the condition on a(x), we must have v = 0 in {x ∈ R N , a(x) ≤ 0}. The characterization of v in {x ∈ R N , a(x) < 0} are still under investigating.

Theorem 5.1.4. Assume that J and f satisfy (H1-H4), J is compactly supported and let m = 2. Then there exists ε 1 ∈ (0, ∞) so that for all ε ≥ ε * there exists a positive solution u ε to (4.2). Moreover,

lim ε→+∞ u ε -a + ∞ = 0, lim ε→+∞ u ε -a + L 2 (R N ) = 0.
In addition, we have the following dichotomy

• When λ 1 K 2,N D 2 (J) 2
∆ + a(x) < 0, there exists ε 0 ∈ (0, ∞) so that for all ε ≤ ε 0 there exists a positive solution to (4.2) and

u ε → v, in L 2 loc (R N ),
where v is the unique bounded non-trivial solution to

K 2,N D 2 (J) 2 ∆v + v(a(x) -v) = 0 in R N . • When λ 1 K 2,N D 2 (J) 2
∆ + a(x) > 0, then there ε 0 ∈ (0, ∞) so that for all ε ≤ ε 0 there exists no positive solution to (4.2).

The last result clearly accentuates the dependence of the strategy on the cost functions and the structure of ecological niche. Especially when m = 2 the slower dispersal strategy may not be an Ecological Stable Strategy (ESS) in the sense of [START_REF] Smith | The Logic of Animal Conict[END_REF]. This stands in contrast with the results in [START_REF] Hutson | The evolution of dispersal[END_REF] obtained in bounded domain neglecting the structure of the ecological niche. We also point the interested reader to [START_REF] Rs | Evolutionary stability of ideal free dispersal strategies in patchy environments[END_REF] for recent development on this subject.

Finally, we obtained existence/ non-existence criteria when we relax the compactly supported constrain on the dispersal kernel J. In this direction, we investigate a class of kernel J that can have a fat tail but still have some decay at innity. More precisely, assume that (H6)

R N J(z)|z| N +1 < +∞.
Then for such kernel we prove Theorem 5.1.5. Assume that J, f satisfy (H1-H4) and assume further that J satises (H5) then we have

(i) if λ p (M + ∂ s f (x, 0)) > 0 there is no bounded positive solution to (4.2). (ii) if lim R→∞ λ p (L R + ∂ s f (x, 0)) < 0 where L R [ϕ] := B R (0) J(x -y)ϕ(y) dy -ϕ(x),
then there exists a unique positive bounded solution to (4.2).

Although we do not have a sharp criteria for fat tailed kernels, the above existence and uniqueness criteria is almost sharp, in the sense that we can show that

λ p (M + ∂ s f (x, 0)) ≤ lim R→∞ λ p (L B(0,R) + ∂ s f (x, 0)).
We strongly believe that there exists a unique bounded positive solutions to (4.2) when λ p (M+∂ s f (x, 0)) < 0 and no positive solution when λ p (M + ∂ s f (x, 0)) ≥ 0. Possibly, more delicate analyses are required. For the moment, we leave it as an open question.

Open problem 2. Can one obtain a sharp characterization for existence and uniqueness of (4.2) when J only satises (H6) ?

Finally, we also want to stress that although we have a clear description of the existence/non-existence of a positive solution for small ε, the study of the convergence of u ε as ε → 0 is quite delicate. Indeed, in

L ∞ (R N ), the problem v(x)(a(x) -v(x)) = 0 in R N ,
has innitely many bounded non negative solution ( e.g. for any set Q ⊂ R N , the function a + (x)χ Q is a solution) and owing to the lack of regularising eect of the dispersal operator, we cannot rely on standard compactness result in the usual manner to obtain a smooth limit. If for the case m = 2 we could rely on the elliptic regularity and the new description of Sobolev Spaces developed in [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Brézis | Ham How to recognize constant functions. Connections with Sobolev spaces[END_REF][START_REF] Ac | An estimate in the spirit of Poincaré ?s inequality[END_REF][START_REF] Ac | A new approach to Sobolev spaces and connections to Γ-convergence[END_REF] to get some compactness, this characterisation does not allow us to treat the case m < 2. We believe that a new characterisation of Fractional Sobolev space in the spirit of the work of Bourgain, Brezis and Mironescu [START_REF] Bourgain | Another look at Sobolev spaces[END_REF][START_REF] Brézis | Ham How to recognize constant functions. Connections with Sobolev spaces[END_REF] may be helpful to resolve this issue.

Existence/non existence and uniqueness of a non-trivial solution

In this section we construct a non-trivial solution to (4.2) and prove necessary and sucient condition stated in Theorem 5.1.1. For convenience the section is split up into three subsections, each of them respectively devoted to the proofs of existence of a solution, the uniqueness and non-existence.

Existence of a non-trivial positive solution

The construction follows a basic approximation scheme previously used for example in [START_REF] Berestycki | Can a Species Keep Pace with a Shifting Climate ? Bull[END_REF]. To this aim, we introduce the following approximated problem :

L R [u] + f (x, u) = 0 in B(0, R) (5.2)
where B(0, R) denotes the ball of radius R centred at the origin. By Theorem 4.2.3, for any R > 0 the existence of a unique positive solution to (5.2) is conditioned to the sign of λ p (L R + β(x)) where β(x) := ∂ u f (x, 0). By Lemma 4.3.6 since

lim R→+∞ λ p (L R + β(x)) = λ p (M + β(x)) < 0 there exists R 0 > 0 so that ∀R ≥ R 0 , λ p (L R + β(x)) < 0.
As a consequence by Theorem 4.2.3, for all R > R 0 there exists a unique positive solution to (5.2) that we denote u R . Moreover, since for all R > 0, sup B R (0) S(x) is a super-solution to (5.2), by a standard sweeping argument since the solution to (5.2) is unique, we get

∀R > 0, u R ≤ sup B R (0)
S(x) in B(0, R).

On another hand, for any R 1 > R 2 , the solution u R 1 is a super-solution to the problem

L R 2 [u] + f (x, u) = 0 in B(0, R 2 ) (5.3)
So as above by a standard sweeping argument we get

u R 2 ≤ u R 1 (x) in B(0, R 2 ).
Thus the map R → u R is monotone increasing. The idea is to obtain a positive solution to (4.2) as a limit of positive solution of (5.2). To this end we construct a uniform super-solution to (4.2).

Lemma 5.2.1. There exists

ū ∈ C 0 (R N ) ∩ L 1 (R N ), ū > 0 so that ū is a super-solution to (4.2).
Proof. Fix ν > 0 so that ν < -lim sup |x|→∞ β(x) and let R 0 > 1 be such that β(x) ≤ -ν 2 for all |x| ≥ R 0 . Let w be the following function w

(x) = Ce -α|x| ,
where C, α are to be chosen. By direct computations we get for all x ∈ R N \ B R 0 (0):

Since by Lemma 4.3.6 λ p (L R + γ(x)) → λ p (M + γ(x)) = 0, we can take R 1 larger if necessary to achieve

γ(x) + λ p (L R + γ(x)) ≤ - ν 4 for |x| ≥ R 1 .
Now let us consider ψ(x) := Ce -α(|x|-R 1 ) where C and α will be chosen later on. By a straightforward computation, we can see that for all R > R 1

L R [ψ](x) + (γ(x) + λ p (L R + γ(x)))ψ(x) ≤ ψ(x) R N J(z)e α|z| dz -1 - ν 4 for |x| ≥ R 1 ≤ h(α)ψ(x) for |x| ≥ R 1 with h(α) := R N J(z)e α|z| dz -1 - ν 4 .
Since J is compactly supported, by the Lebesgue Theorem, the function h is continuous and h(0) = -ν 4 . By assumption ν > 0, so by continuity of h there exists α 0 > 0 so that h(α 0 ) < 0. Thus we achieve for

α = α 0 L R [ψ](x) + (γ(x) + λ p (L R + γ(x)))ψ(x) ≤ 0 for |x| ≥ R 1 . (5.16)
Recall that by construction the function ϕ n satises

L Rn [ϕ n ](x) + (γ(x) + λ p (L Rn + γ(x)))ϕ n (x) = 0 in B Rn (0). (5.17)
Since J is compactly supported and J(0) > 0 there exists positive constants r 0 ≥ r 1 ; M ≥ m so that M χ Br 0 (x) ≥ J(x -y) ≥ mχ Br 1 (x) for all x, y ∈ R N .

Therefore for n large enough say n ≥ n 0 , we have R n > R 1 + r 0 and by the Harnack inequality, for all n ≥ n 0 we have ϕ n (x) ≤ C(B R 1 , λ p (L Rn + γ(x)))ϕ n (y) for all x, y ∈ B R 1 (0).

The constant C(B R 1 , λ p (L Rn + γ(x))) only depends on x∈B R 1 B r 0 (x) and is monotone decreasing with respect to inf x∈B Rn (γ(x) + λ p (L Rn + γ(x))). For all n ≥ n 0 , the function γ(x) + λ p (L Rn + γ(x)) being uniformly bounded from below by a constant independent of n, the constant C(B R 1 , λ p (L Rn + γ(x))) is bounded from above independently of n by a constant C(B R 1 ). Thus we have for all n ≥ n 0

ϕ n (x) ≤ C(B R 1 )ϕ n (y) ∀ x, y ∈ B R 1 .
In particular, we have for all n ≥ n 0 ,

ϕ n (x) ≤ C(B R 1 )ϕ n (0) = C(B R 1 ) ∀ x ∈ B R 1 . By choosing C > C(B R 1 ), we achieve ψ(x) ≥ C > C(B R 1 ) ≥ ϕ n (x) ∀ x ∈ B R 1 .
Set now w n := ψ -ϕ n , from (5.16) and (5.17) we get

L Rn [w n ](x) + (γ(x) + λ p (L Rn + γ(x)))w n (x) ≤ 0 for R 1 ≤ |x| < R n , (5.18)
w n > 0 for |x| < R 1 .

(5.19)

By a straightforward application of the Maximum principle, it follows that for all n ≥ n 0 we have ϕ n (x) ≤ ψ. Indeed, since w n is continuous, then it achieves a minimum at some point x 0 ∈ B Rn . Assume by contradiction that w n (x 0 ) < 0, then thanks to (5.19) x 0 ∈∈ B Rn \ B R 1 and at this point by (5.18) we have the following contradiction

0 ≥ L Rn [w n ](x 0 ) + (γ(x 0 ) + λ p (L Rn + γ(x)))w n (x 0 ) ≥ B Rn J(x 0 -y)w n (y) dy -w n (x 0 ) + ν 4 |w n (x 0 )|, ≥ B Rn J(x 0 -y)[w n (y) -w n (x 0 )] dy + ν 4 |w n (x 0 )| > 0.
Hence, for all n ≥ n 0 ϕ n ≤ ψ in B Rn which by sending n → ∞ leads to ϕ ≤ ψ in R N which concludes the proof of the Claim.

Long time Behaviour

In this section, we investigate the long-time behaviour of the positive solution u(t, x) of

∂u ∂t (t, x) = J u(t, x) -u(t, x) + f (x, u(t, x)) in R + × R N , (5.20)
u(0, x) = u 0 (x). (5.21)
For any

u 0 ∈ C k (R N ) ∩ L ∞ or in C k (R N ) ∩ L 1 (R N )
, we know that the existence of a solution u(t, x) ∈ C 1 ((0, +∞), C min{1,k} (R N )) respectively u(t, x) ∈ C 1 ((0, +∞), C min{1,k} (R N ) ∩ L 1 (R N )) is a straightforward consequence of the Cauchy-Lipschitz Theorem and of the KP P structure of the nonlinearity f . Before going to the proof of the asymptotic behaviour, let us recall some useful results Lemma 5.3.1. Assume that u 0 (x) is a sub-solution to (5.20), then the solution u(t, x) is increasing in time. Conversely, if u 0 (x) is a super-solution to (5.20) then u(t, x) is decreasing in time.

The proof of this Lemma follows from a straightforward used of the parabolic maximum principle and is let to reader. Let us now prove the asymptotic behaviour of the solution of (5.20) and end the proof of Theorem 5.1.1.

Proof. Let z(t, x) be the solution to Now, to complete the proof we are left to show that u -ũ ∞ → 0 as t → ∞. To this end we follow the argument in [START_REF] Berestycki | Reaction-diusion equations for population dynamics with forced speed, I -The case of the whole space[END_REF]. We argue by contradiction and assume there exists ε > 0 and the sequences

∂z ∂t = J z -z + f (x, z(t, x)) in R + × R N (5.22) z(0, x) = C u 0 ∞ (5.
(t n ) ∈ R + , (x n ) ∈ R N such that lim n→∞ t n = ∞, |u(t n , x n ) -ũ(x n )| > ε, ∀n ∈ N.
(5.32) By (5.31), we already know that u → ũ locally uniformly in R N , so without loss of generality, we can assume that |x n | → ∞. From the construction of ũ, Subsection 5.2.1, we have lim |x|→∞ ũ(x) = 0. Therefore for some R 0 > 0, we have ũ(x) ≤ ε 2 for all |x| ≥ R 0 . The latter combined with (5.31) and (5.32) enforces z(t n , x n ) -ũ(x n ) ≥ u(t n , x n ) -ũ(x n ) > ε, ∀n ∈ N.

(5.33)

We claim that Finally we establish the long time behaviour of the solution u(t, x) starting from an integrable initial datum u 0 ,i.e u 0 ∈ L 1 (R N ) ∩ C(R N ). To do so, we dene two auxiliary functions h(t, x) and v(t, x) that are respectively solution to ∂h ∂t (t, x) = J h(t, x) -h(t, x) + f (x, h(t, x)) in R + × R N , h(0, x) = sup{ũ(x), u 0 (x)}, (5.36) ∂v ∂t (t, x) = J v(t, x) -v(t, x) + f (x, v(t, x)) in R + × R N , v(0, x) = inf{ũ(x), u 0 (x)}.

(5.37)

By construction, from the comparison principle we deduce that v(t, x) ≤ u(t, x) ≤ h(t, x) for all (t, x) ∈

R + × R N . Therefore u -ũ L 1 (R N ) ≤ sup{ h -ũ L 1 (R N ) , v -ũ L 1 (R N ) }.
Thus to prove that u -ũ L 1 (R N ) → 0 it is enough to show that h and v converge to ũ in L 1 (R N ).

Let us show that v converges to ũ in L 1 (R N ). Since ũ(x) is a super solution to (5.37) we deduce v(t, x) ≤ ũ(x) for all x ∈ R N . Let ε > 0 be xed and choose R so that R N \B(0,R) ũ(x) dx ≤ ε 4 then we have

ũ -v L 1 (R N ) = R N \B R (0) (ũ(x) -v(t, x)) dx + B R (0) (ũ(x) -v(t, x)) dx, ≤ 2 R N \B R (0) ũ(x) dx + B R (0) (ũ(x) -v(t, x)) dx, ≤ ε 2 + B R (0)
(ũ(x) -v(t, x)) dx.

Recall that v converges pointwise to ũ as t tends to innity. Therefore, by Lebesgue Theorem for some t(ε) we get for all t ≥ t(ε) , B R (0) (ũ(x) -v(t, x)) dx ≤ ε 2 which enforces

ũ -v L 1 (R N ) ≤ ε.
ε being chosen arbitrary, the latter inequality shows that lim t→∞ ũ -v L 1 (R N ) = 0 which proves that v converges to ũ in L 1 (R N ).

To obtain that h -ũ L 1 (R N ) → 0 we argue as follow. By construction ũ is a sub solution to (5.36), thus ũ(x) ≤ h(t, x) for all (t, x) ∈ R + × R N . Let us denote w(t, x) := h(t, x) -ũ(x). Then w satises for all (t, x) ∈ R + × R N : ∂w ∂t (t, x) = J w(t, x) -w(t, x) + f (x, h(t, x)) h(t, x) -f (x, ũ(x)) ũ h(t, x) + f (x, ũ(x)) ũ w(t, x), ≤ J w(t, x) -w(t, x) + f (x, ũ(x)) ũ w(t, x).

Now thanks to lim |x|→∞

f (x,s) s < 0, there exists κ > 0 and R 0 so that w satises ∂w ∂t (t, x) ≤ J w(t, x) -w(t, x) -κw(t, x) in R + × R N \ B R 0 (0).

(5.38)

Fix now ε > 0. Recall that h(t, x) converges pointwise to ũ, then by Lebesgue Theorem there exists t 0 so that for all t ≥ t 0 , B R 0 (0) w(t, x) dx ≤ κε.

Now let us estimate

R N \B R 0 (0) w(x) dx for t ≥ t 0 . By integrating (5.38) over R N \ B R 0 (0) it yields

∂ R N \B R 0 (0) w(t, x) dx ∂t ≤ R N \B R 0 (0)
J w(t, x) dx -

R N \B R 0 (0) w(t, x) dx -κ R N \B R 0 (0)
w(t, x) dx.

By using Fubini's Theorem, the uniform estimate on w ∞ and the unit mass of the kernel, we can check that for t ≥ t 0 From the later dierential inequality, there exists t(ε) ≥ t 0 so that for all t ≥ t(ε) we have

R N \B R
R N \B R 0 (0)
w(t, x) dx ≤ 2ε.

Hence, we have for all t ≥ t(ε)

w L 1 (R N ) = R N \B R 0 (0) w(t, x) dx + B R 0 (0) w(t, x) dx ≤ 2 + κ |B R 0 (0)| ε,
As above, ε being chosen arbitrary, the latter inequality shows that lim t→∞ w L 1 (R N ) = 0 which proves that h converges to ũ in L 1 (R N ).

Some asymptotics

In this section we analyse the qualitative behaviour of the solution of (4.2) with respect to the size of the support of J. For convenience we investigate the particular situation

1 ε m (J ε u -u) + u(a(x) -u) = 0 in R N (P ε )
where J ε (z) = 1 ε N J z ε with supp(J) = B(0, 1) and a ∈ C 1 (R N ) so that a + ≡ 0.

The later condition on a is necessary to observe the possible existence of a solution. Indeed, if a + ≡ 0 then for any positive constant c 0 we have M[c 0 ] + a(x)c 0 ≤ 0 therefore λ p (M[c 0 ] + a(x)c 0 ) ≥ 0 and for all ε there is no solution to (P ε ) besides 0.

We look for the behaviour of u ε when ε → 0 and → +∞ and try to understand the inuence of m on the resulting limits.

We start by showing some a priori estimate for the solution u ε . Lemma 5.4.1. There exist positive constants C 1 , C 2 , C 3 so that we have for any positive bounded solution u ε of (P ε )

(i) u ε L 2 (R N ) ≤ a + (x) L 2 (R N ) ≤ C 1 , u ε ∞ < C 3 , (ii) R N R N J ε (x -y)(u ε (x) -u ε (y)) 2 dxdy ≤ C 2 ε m (iii) sup supp(a + ) u ε ≥ -λp(Mε,m+a(x)) 2 .
(iv) u ε ≥ (a(x) -1 ε m ) + , Proof. Since by construction the solution is unique and u ε ∈ L 1 (R N ) ∩ L ∞ . Moreover by (P ε ) u ε ≤ M = a ∞ . We obtain (i) by integrating (P ε ) over R N . We get, by Cauchy-Schwart inequalities that

R N u 2 ε (x) dx = R N a(x)u ε (x) dx ≤ R N a + (x)u ε (x) dx ≤ R N [a + (x)] 2 dx R N [u ε (x)] 2 dx.
Therefore,

u ε L 2 (R N ) ≤ a + (x) L 2 (R N ) ≤ C 1 .
To obtain (ii), let us multiply (P ε ) by u ε and integrate over R N , then we get

1 2ε m R N R N J(x -y)(u ε (x) -u ε (y)) 2 dxdy = R N u 2 ε (x)(a(x) -u ε (x)) dx
Since u ε and a(x) are uniformly bounded independently, (ii) holds true with C 2 := 4C 1 M . Observe that (a(x) -1 ε m ) + is always a sub-solution to (P ε ), so by a standard sweeping principle u ε ≥ (a(x) -1 ε m ) + and (iv) holds true. Finally to obtain (iii) we argue as follows. Since u ε is a positive bounded solution of (P ε ) by Theorem 5.1.1 we have λ p (M ε,m + a(x)) < 0. Now since λ p (M ε,m + a(x)) < 0 and J is compactly supported, by regularising a if necessary, we can nd (see the proof of Lemma 4.3.8)

ϕ ε ∈ C c (R N ) so that M ε,m [ϕ ε ] + a(x) + λ p 2 ϕ ε ≥ 0 in R N .
Moreover, we can normalised ϕ ε so that ϕ ε ∞ = 1. Plugging θϕ ε with in (P ε ) it follows that M ε,m [θϕ ε ] + θϕ ε (a(x) -θϕ ε ) ≥ θϕ ε (-λ p 2 -θϕ ε ).

Lemma 5.4.3. Let R, ε be xed and positive then for all η > 0 there exists δ > 0 so that

|λ p (L R + a ε (x)) -λ p (L R + a ε+δ (x))| ≤ η,
where a ε (x) := a(εx).

Let us prove now the continuity of λ p (L R + a ε (x)).

Proof of the Claim. Let ε > 0 and R > 0 be xed. We observe that for all |δ| < ε we have for all

x ∈ R N , a ε+δ (x) = a ε ε+δ ε x therefore

a ε -a ε+δ ∞,R = sup B(0,R) a ε (x) -a ε ε + δ ε x .
Since a ε is a Lipschitz in R N , we have

a ε (x) -a ε ε + δ ε x ≤ K(ε)εδ x ,
where K(ε) is the Lipschitz constant of a ε . Thus

a ε -a ε+δ ∞,R ≤ K(ε)Rεδ.
Hence, by (ii) of Proposition 4.2.2 we get

|λ p (L R + a ε (x)) -λ p (L R + a ε+δ (x))| ≤ K(ε)Rεδ.
Finally, we establish a useful identity, which can be seen as a Green-formula for nonlocal operators. Proof. Thanks to the symmetry of ρ, using standard changes variables we have

R N ×R N ρ(z)[u(x + z) -u(x)]ϕ(x) = = 1 2 R N ×R N ρ(z)[u(x + z) -u(x)]ϕ(x) + 1 2 R N ×R N ρ(-z)[u(x -z) -u(x)]ϕ(x), = 1 2 R N ×R N ρ(z)[u(x + z) -u(x)]ϕ(x) + 1 2 R N ×R N ρ(z)[u(x) -u(x + z)]ϕ(x + z), = - 1 2 R N ×R N ρ(z)[u(x + z) -u(x)][ϕ(x + z) -ϕ(x)], = - 1 2 R N ×R N ρ(z)u(x)[ϕ(x) -ϕ(x -z)] + 1 2 R N ×R N ρ(z)u(x)[ϕ(x + z) -ϕ(x)], = 1 2 R N ×R N ρ(z)u(x)[ϕ(x + z) -2ϕ(x) + ϕ(x -z)].
From the Proposition, for all u ∈ L 2 (R N ), ϕ ∈ C ∞ c (R N ) we straightforwardly get the following identity

R N M ε,m [u](x)ϕ(x) dx = ε 2-m D 2 (J) 2 R N ×R N ρ ε (z)
|z| 2 u ε (x)[ϕ(x + z) -2ϕ(x) + ϕ(x -z)] dxdz (5.39) where ρ ε (z) =

1 ε N D 2 (J) J z ε |z| 2 ε 2 .
Equipped with all these apriori estimates, we can now analyse the asymptotic behaviour of u ε . < 0 and by Theorem (5.1.1) there exists a solution to (P ε ). Moreover the following quantity is well dened ε * := sup{ε > 0 | for all ε < ε, there exists a positive solution to (P ε )}.

In view of (5.41) ε * ∈ (0, +∞] and ε * < +∞ if and only if (a(x) -1) + ≡ 0.

Let us now deal with the limit of u ε as ε → 0 and ε → +∞ and let us start by proving that lim ε→0 u ε (x) = v(x) a.e. (5.42) where v is a non negative bounded solution of v(x)(a(x) -v(x)) = 0 in R N .

(5.43)

Let w ε := a(x) -u ε , then from (P ε ), w ε satisfy -J ε w ε + w ε + u ε (x)w ε (x) = a(x) -J ε a(x).

(5.44)

Multiplying the above equation by w + ε and integrating over R N , it follows that

R N ×R N J ε (x -y)((w + ε ) 2 (x) -w ε (y)w + ε (x)) dxdy + R N u ε (x)(w + ε ) 2 (x) = R N w + ε (x)g ε (x) dx
with g ε (x) := a(x) -J ε a(x).

Let us now estimate the above integrals. First we observe that the double integral is positive. Indeed, since w(y) = w + (y) -w -(x) we get

(u ε ) ε∈[ 1 2 ε * ,ε * ) is uniformly bounded in C 0, 1 2 loc (Q) ∩ C 0, 1 2 (Q c ).
(5.49)

In both case, since a(x) < 0 for |x| >> 1, Q c is a compact set and | Q ∩ Q c | = 0. From (5.49), for all sequence ε n → ε * by a diagonal extraction procedure there exists a subsequence still denoted (u εn ) n∈N that converges locally uniformly in R N \ ( Q ∩ Q c ) to some non-negative function v. By passing to the limit in (P ε ) we can see that v is a bounded non negative solution of

J ε * v(x) -v(x) + v(x)(a(x) -v(x)) = 0 in R N \ ( Q ∩ Q c ). Since Q ∩ Q c is of zero measure v is a solution to L ε * ,R N \( Q∩Q c ) [v] + v(x)(a(x) -v(x)) = 0 in R N \ ( Q ∩ Q c
). Since 0 ≤ λ p (M ε * + a(x)) ≤ λ p (L ε * ,R N \( Q∩Q c ) + a(x)), we deduce that v ≡ 0 which proves the limit.

Remark 21. When a(x) is a radially symmetric non-increasing function we remark that ε * is a sharp threshold. That is for all ε ≥ ε * then (P ε ) does not have any positive solutions. Indeed in this situation the function a ε (x) is monotone non increasing with respect to ε. Thus by (i) of Proposition 4.2.2, for all ε ≥ ε * we have 0 = λ p (M + a ε * (x)) ≤ λ p (M + a ε (x)).

Hence, by Theorem 5.1.1, 0 is the unique non negative solution to (P ε ) for ε ≥ ε * . < 0. Therefore by Theorem (5.1.1) there exists a solution to (P ε ) for small and large ε. The limit of u ε when ε → ∞ is easy to obtain. Indeed it is a straightforward consequence of (iv) of Lemma 5. .

To obtain the limits in L 2 , we just observe that since by Lemma u ε is uniformly bounded in L 2 and converges pointwise to a + , we get u ε a + in L 2 . Moreover by Fatou's Lemma, we have

R N (a + ) 2 (x) dx ≤ lim inf ε→∞ R N u 2 ε (x) dx.
On another hand, by (i) Lemma (5.4.1), we have

u ε L 2 (R N ) ≤ a + (x) L 2 (R N ) .
It follows that

R N (a + ) 2 (x) dx ≤ lim inf ε→∞ R N u 2 ε (x) dx ≤ lim sup ε→+∞ R N u 2 ε (x) dx ≤ R N (a + ) 2 (x) dx
Hence, u ε 2 → a + 2 and by the parallelogram identity u ε → a + in L 2 (R N ) since u ε converges weakly to a + in L 2 .

The case m = 2

In this situation, from Theorem 4. ∆ + a(x)) > 0 then for ε small there exists no positive solution to (P ε ). The limit of u ε when ε → +∞ can be obtain as in the case 2 > m > 0 so we focus only on the limit when ε → 0.

Assume for the moment that λ 1 ( K 2,N D 2 (J) 2

∆ + a(x)) < 0, we will show that u ε → v where v is the positive solution to K 2,N D 2 (J) 2 ∆v + v(a(x) -v) = 0 in R N .

Let (ε n ) n∈N be a sequence of positive number converging to 0 and let u n denote u εn . By Lemma 5.4.1, u n 2 is bounded uniformly and after simple algebraic computation

R N ×R N ρ n (z) (u n (x + z) -u n (x)) 2 |z| 2 dxdz < C,
with C independent of ε. Therefore for any R > 0, we have

B R ×B R ρ ε (z) (u n (x + z) -u n (x)) 2 |z| 2 dxdz < C.
For R > 0 xed, since u n 2 is uniformly there exists a subsequence u n v in L 2 (B R ) and from the characterisation of Sobolev Space [START_REF] Ac | A new approach to Sobolev spaces and connections to Γ-convergence[END_REF][START_REF] Ac | An estimate in the spirit of Poincaré ?s inequality[END_REF], we have u n → v in L 2 (B R ).

By a standard diagonal extraction argument, from the sequence (u n ) n∈N we can then extract a subsequence still denoted (u n ) n∈N which converges to some v in L 2 loc (R N ). Moreover by Lemma 5.4.1 u n is uniformly bounded and there exists δ(λ 1 ) > 0 independent of ε so that max supp(a + ) (u n ) > δ.

Multiplying (P ε ) by ϕ ∈ C ∞ c (R N ) and integrating we get Since v is bounded, by elliptic regularity v is smooth. To conclude we need to prove that v is non trivial. To do so we claim that Claim 5.4.5. There exists R 0 , τ and ε 0 positive constants so that for all ε ≤ ε 0 we have u ε ≥ τ almost everywhere in B R 0 (0).

From the above claim, we deduce that v ≥ τ > 0 a.e. and therefore v ≡ u, the unique smooth non-trivial solution of K 2,N D 2 (J) 2 ∆u + u(a(x) -u) = 0 in R N .

The sequence (ε n ) n being arbitrary, it follows that u ε → u in L 2 loc (R N ). Similarly, if we assume now that λ 1 ( K 2,N D 2 (J) 2

∆ + a(x)) = 0 and there exists a sequence (ε n ) n∈N , ε n → 0 of non trivial solution of (P ε ). The above argumentation then holds true and we get u n → v in L 2 loc (R N ) with v a smooth solution to

K 2,N D 2 (J) 2 ∆v + v(a(x) -v) = 0 in R N . Since λ 1 ( K 2,N D 2 (J) 2
∆ + a(x)) = 0, v ≡ 0 is the only solution and we get u n → 0 in L 2 loc (R N ). Let us complete our proof and establish the claim.

Proof. Let us denote L R,ε the operator

L R,ε [ϕ] := 1 ε 2 B R (0)
J ε (x -y)ϕ(y) dy -ϕ(x) .

Since sup R N a(x) is achieve in R N we regularise a by a σ independently of ε, so that for all ε and R ≥ R 1 the principal eigenvalue λ p (L R,ε + a τ (x)) is associated to a continuous principal eigenfunction ϕ p,ε and |λ p (L R,ε + a τ (x)) -λ p (L R,ε + a(x))| ≤ a σ (x) -a(x) ∞ ≤ κσ, with κ the Lipschitz constant of a.

By the Lipschitz continuity of λ 1 K 2,N D 2 (J) 2

∆ + a(x) with respect to a, we can choose σ small enough so that

λ 1 K 2,N D 2 (J) 2 ∆ + a σ (x) ≤ 1 2 λ 1 K 2,N D 2 (J) 2 ∆ + a(x) < 0.
Recall that

lim R→∞ λ 1 K 2,N D 2 (J) 2 ∆ + a σ (x), B R = λ 1 K 2,N D 2 (J) 2 ∆ + a(x) ,
So we can choose R 0 large so that

λ 1 K 2,N D 2 (J) 2 ∆ + a σ (x), B R 0 ≤ 1 4 λ 1 K 2,N D 2 (J) 2 ∆ + a(x) .
Thanks to Remark 17, we have lim ε→0 λ p (L R,ε +a σ (x)) = λ 1 K 2,N D 2 (J) 2

∆ + a σ (x), B R so for ε small,say ε ≤ ε 0 by choosing σ smaller if necessary, we achieve λ p (L R,ε + a σ (x)) ≤ 1 8 λ 1 K 2,N D 2 (J) 2 ∆ + a(x) for all ε ≤ ε 0 .

Let ϕ p,ε be the principal eigenfunction associated with L R 0 ,ε + a σ (x), then we have To conclude our proof, it is then enough to show that for some well chosen normalisation of ϕ p,ε we have ∆ + a σ (x), B R 0 . Indeed, assume for the moment that (5.59) holds true. Then there exists α > 0 so that αϕ p,ε (x) → αϕ 1 (x) < 1 2 a.e. in B R 0 . Now thanks to (5.58), we can now adapt the proof the proof of (iii) of Lemma 5.4.1 to get for ε small, says ε ≤ ε 1 , for some γ, ε 2 > 0.

L R 0 ,
u ε (x) ≥ - α 32 λ 1 K 2,N D 2 (J) 2 ∆ + a(x) ϕ p,ε (
Since ϕ 1 > 0 in B R 0 , the claim holds true in any smaller ball B R .

To prove (5.59), let us normalise ϕ p,ε by ϕ p,ε L 2 (B R 0 ) = 1. Let k ε be the function

k ε (x) := 1 ε 2 R N \B R 0 J ε (x -y) dy,
Observe that the construction of the super-solution covers a larger class of nonlinearity f (x, u) than those that satisfy H4. As a immediate consequence, the survival criteria obtained in Theorem 5.1.1 is still true for nonlinearity that satises: (H7) There exists µ(x) ∈ C p (R N ) so that :

λ p (M + µ(x)) > 0, lim sup |x|→∞ f (x,s) s -µ(x) ≤ 0 uniformly in s.

From the ecological point of view, such nonlinearity allows to consider a more complex niche structure for the species. In particular, we can consider ecological niches that are the superposition of a compact niche structure with a periodic structure. The condition being that on the periodic structure alone, the species could not survive. The perspective oers by this approach are quite promising and we believe that it may be applied to investigate a climate change version of (4.2).

Proof. The construction of the super-solution in this situation follows the same scheme as for a compactly supported kernel. By assumption since lim sup |x|→∞ (β(x) -µ(x)) ≤ 0, for any δ > 0 there exists R δ > 1 such that β(x) ≤ µ(x) + δ |x| ≥ R δ .

Fix δ < λ p (M + µ(x)) and observe that by denition of λ p (M + µ(x)) there exists a constant δ < λ < λ p (M + µ(x)) and a positive periodic function ϕ so that M[ϕ](x) + (µ(x) + λ)ϕ(x) ≤ 0 for all x ∈ R N .

(5.62)

Let w = C ϕ(x) 1+τ |x| N +1 with C, τ to be chosen.

C 0 τ {|x|>2|z|} J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) dz ≤ ≤ C 0 N +1 i=1 N + 1 i {|x|>2|z|} J(z)(-1) i |z| i τ |x| N +1-i (1 + τ |x + z| N +1 ) dz, ≤ C 0 N +1 i=1 N + 1 i {|x|>2|z|} J(z)|z| i τ |x| N +1-i (1 + τ |x + z| N +1 )
dz.

Since |x| > 2|z|, we have

1 1 + τ |x + z| N +1 ≤ 2 N +1 2 N +1 + τ |x| N +1
and for |x| ≥ R 0 > 1

C 0 τ {|x|>2|z|} J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) dz ≤ ≤ C 0 2 N +1 N +1 i=1 N + 1 i {|x|>2|z|} J(z) |z| i |x| i τ |x| N +1 (2 N +1 + τ |x| N +1 ) dz ≤ C 0 2 N +1 R 0 N +1 i=1 N + 1 i R N J(z)|z| i τ |x| N +1 2 N +1 + τ |x| N +1 dz.

λ

  D (-L, O) := sup{λ ∈ R, ∃φ ∈ W 2,N loc (O), φ > 0, (L + λ)φ ≤ 0 a.e in O, φ = 0 on ∂O}. λ 1 (-L, O) := sup{λ ∈ R, ∃φ ∈ W 2,N loc (O), φ > 0, (L + λ)φ ≤ 0 a.e in O}.

Denition 0.0. 1 . 2 φ-α 2 = 0 ,

 1220 Let α ∈ [0, 2], the generalized principal eigenvalue associated with L in Ω isλ α (-L, Ω) := sup λ ∈ R : ∃h α ∈ C(Ω), lim x∈Ω,|x|→∞ h α (x) |x| -α = 1, ∃φ ∈ D α , (L + λh α (x)) φ ≤ 0 in Ω h + α ,where Ω h + = {x ∈ Ω, h(x) ≥ 0} and D α denotes the class of admissible test-functionsD α := φ ∈ W 2,N loc (Ω), φ > 0 in Ω h + , lim sup x∈Ω,|x|→∞ |∇φ(x)||x| α (x) ≤ D α and lim x∈Ω,|x|→∞ ln φ(x) |x| 1for α ∈ [0, 2) and for α = 2, D 2 is given byD 2 := φ ∈ W 2,N loc (Ω), φ > 0 in Ω h + , lim sup

  [u] = -div(|∇u| p-2 ∇u) -a(x)u p-1 , p ≥ 2, a, b ∈ L ∞ (R N ), b ≡ 0 in R N and g : R + → R + is a continuous function.The following assumptions are made :∃α ∈ [0, p], m > 0 lim sup |x|→∞ a(x)|x| α < -m (C1) lim s→0 g(s) s p-1 = 0 and g(s) s p-1 is increasing on (0, ∞). (C2)∃s 0 > 0 such that -a(x)s p-1 0 + b(x)g(s 0 ) ≥ 0(C3) Using the "global eigenvalue" Denition 0.0.5. Let λ R = λ B R (a), we dene λ ∞ (a) = lim R→∞ λ R (a),

  in addition that (C3) holds, then the unique positive solution is achieved if and only if λ ∞ (a) < 0. It has to decay exponentially as |x| → ∞ when exists.

1 . 4 Further results and applications 1 . 4 . 1

 14141 Similarity of the problem with Dirichlet boundary condition In this subsection, we aim at proving an analogous result of Theorem 1.1.1, where Dirichlet instead of Neumann condition is imposed on the boundary of Ω. This is to prepare for the main goal in the next subsection. To this end, let us dene the generalized Dirichlet principal eigenvalue λ D (-L, Ω) := sup{λ ∈ R : ∃φ ∈ W 2,N loc (Ω), φ > 0, (L + λ)φ ≤ 0 a.e in Ω and φ = 0 on ∂Ω}. (1.45)

  Hypotheses 2 hold. Eq. (2.1) cannot possess more than one positive solution satisfying lim sup |x|→∞ ln u(x) ln |x| ≤ 0.

Chapitre 3 Liouville 1 .

 31 -type result for a quasilinear equations with possibly decaying potentials in R N The contents of this chapter are based on the joint work : Liouville-type result for quasilinear equations with possibly decaying potentials in R N Phuoc-Tai Nguyen(1), Hoang-Hung Vo(2). Department of Mathematics, Technion -Israel Institute of Technology, Haifa 32000, Israel, 2. CAMS -École des Hautes Études en Sciences Sociales, 190-198 avenue de France, 75013, Paris, France, 3.1 Introduction

  every R > 0, denote λ R (a) := λ B R (a). If a ∈ L ∞ (Ω), itis easy to check that the sequence {λ R (a)} is nonincreasing and bounded from below by a -sup Ω a(x) , which leads to Denition 3.1.1. We set λ ∞ (a) := lim R→∞ λ R (a).(3.8)

1 .

 1 a weak solution of (3.1) such that lim sup |x|→∞ Let p ≥ 2. Assume α ∈ [0, p) and conditions (C1)-(C3) hold. Then (3.1) possesses a positive solution in S α if and only if λ ∞ (a) < 0. Moreover, the positive solution is unique in S α and decays exponentially when exists. When α = p, denote by γ 0 the unique positive solution of algebraic equation (N p + 1)γ p + (3p + 1)N γ p-1 -m = 0. (3.9) Dene S p = {u > 0 | u is a weak solution of (3.1) such that lim sup |x|→∞ |x| -γ 0 u(x) < ∞}. Theorem 3.1.2. Let p ≥ 2. Assume α = p, m > N p (N p + 3p + 2) and conditions (C1)-(C3) hold. Then (3.1) possesses a positive solution in S p if and only if λ ∞ (a) < 0. Moreover, the positive solution is unique in S p and decays polynomially when exists.

Proposition 3 . 2 . 1 .

 321 Let α ∈ [0, p), m > 0 and u be a positive function satisfying

3 . 6

 36 there exists a bounded open set ω so that λ p (L ω + b(x)) < 0. By taking ω larger if necessary, we can assume that sup Ω b(x) = sup ω b(x) and b(x) achieves its maximum in ω. By arguing as in the proof of Claim 4.3.7, we can nd b ε so that

(4. 26 )

 26 By construction ψ ∈ L 2 (Ω) and let us multiply (4.26) by -ψ and integrate over Ω. Then we get-Ω Ω K(x, y)ψ(x)ψ(y) dxdy -Ω a(x)ψ(x) 2 dx ≤ λ Ω ψ 2 (x) dx (4.27)

Chapitre 5 On

 5 positive solutions of nonlocal heterogeneous KPP equations The contents of this chapter are based on the joint work : Nonlocal heterogeneous KPP equations in R N Henri Berestycki(1), Jérôme Coville (2), Hoang-Hung Vo (1).

23 )

 23 Since S(x) ∈ L ∞ by choosing C large enough, the constant C u 0 ∞ is a super-solution to (5.20) therefore z(t, x) is a decreasing function and by the parabolic maximum principle we have u(t, x) ≤ z(t, x) for all(t, x) ∈ [0, +∞) × R N , leading to lim sup t→∞ u(t, x) ≤ lim sup t→∞ z(t, x) for all x ∈ R N .(5.24)Now let us consider the approximated parabolic problem∂v R ∂t (t, x) = B R (0) J(x -y)v R (t, y) dy -v R (t, x) + f (x, v R (t, x)) in R + × B R (0), (5.25) v R (0, x) = η R u 0 (x),(5.26)whereη R := η |x| R with η ∈ C(R + ) a smooth cut-o function so that η ≥ 0, η ≡ 1 in [0, 1] and η ≡ 0 in R + \ [0, 2]. By Theorem 4.2.3, for R large enough the solution v R (t, x) converges to u R (x) the unique positive stationary solution of(5.25). By construction since u(t, x) is a super-solution of the problem (5.25), by the parabolic comparison principle we have for all R large enough v R (t, x) ≤ u(t, x) for all (t, x) ∈ [0, +∞) × B R (0). Therefore we have for all R large enoughlim inf t→∞ u(t, x) ≥ u R (x) for all x ∈ B R (0) (5.27)By taking the limit as R → ∞, in the above inequality we getlim inf t→∞ u(t, x) ≥ lim R→∞ u R (x) = ũ(x) for all x ∈ R N (5.28)Note that we can reproduce the above arguments with z(t, x), thus we also getv R (t, x) ≤ z(t, x) for all (t, x) ∈ [0, +∞) × B R (0)(5.29)lim inf t→∞ z(t, x) ≥ lim R→∞ u R (x) = ũ(x) for all x ∈ R N(5.30)By (5.29) z(t, x) is locally uniformly bounded from below and since z(t, x) is a decreasing function of t we get lim t→∞ z(t, x) = z(x) > 0 for all x ∈ R N . Moreover z is a bounded stationary solution to(5.20). By uniqueness of the positive stationary solution, we conclude that z = ũ. Thus we have lim t→∞ z(t, x) = ũ(x) for all x ∈ R N (5.31) Hence by collecting (5.24),(5.28), (5.31) we get for all x ∈ R N ũ(x) ≤ lim inf t→∞ u(t, x) ≤ lim sup t→∞ u(t, x) ≤ lim sup t→∞ z(t, x) = lim t→∞ z(t, x) = ũ(x).

  0 (0) J w(t, x) dx = R N \B R 0 (0) w(t, y) R N \B R 0 (0) J(x -y) dx dy + B R 0 (0) w(t, y) R N \B R 0 (0) J(x -y) d ≤ R N \B R 0 (0) w(t, y) dy + B R 0 (0) w(t, y) dy, ≤ R N \B R 0 (0) w(t, y) dy + κε. Therefore for t ≥ t 0 , w satises ∂ R N \B R 0 (0) w(t, x) dx ∂t ≤ κε -κ R N \B R 0 (0)w(t, x) dx.

Proposition 5 . 4 . 4 .

 544 Let ρ ∈ C ∞ c (R N ) be a radial function, then for all u ∈ L 2 (R N ), ϕ ∈ C ∞ c (R N ) we have R N ×R N ρ(z)[u(x + z) -u(x)]ϕ(x) dzdx = 1 2 R N ×R N ρ(z)u(x)[ϕ(x + z) -2ϕ(x) + ϕ(x -z)] dzdx.

5. 4 . 1

 41 The case m = 0In this situation, from Theorem 4.4.1 we know thatlim ε→0 λ p (M ε + a(x)) = -sup R N a(x)(5.40)lim ε→+∞ λ p (M ε + a(x)) = 1 -sup R N a(x)(5.41)As a consequence for ε small enough we have λp (M ε + a(x)) ≤ -sup R N a(x)2

R 1 2 1 2

 11 N ×R N J ε (x -y)((w + ε ) 2 (x) -w ε (y)w + ε (x)) dxdy = R N ×R N J ε (x -y)((w + ε ) 2 (x) -w + ε (y)w + ε (x)) dxdy + R N ×R N J ε (x -y)w - ε (y)w + ε (x) dxdyTherefore, we getC(ε * )|x -z| ≥ |[1 -a(x) + u ε (x) + u ε (z)][u ε (x) -u ε (z)] + [a(z) -a(x)]u ε (x)|, ≥ |[1 -a(x) + u ε (x) + u ε (z)]||u ε (x) -u ε (z)| -|x -y| |a(z) -a(x)| |x -z| M,and for any x ∈ Q := {y ∈ R N | a(y) < 1} u ε is uniformly Lipschitz in x with a constant independent of ε.Thus (u ε ) ε∈[ 1 2 ε * ,ε * ) is uniformly bounded in C 0, loc (Q). If Q c = ∅, then (u ε ) ε∈[ 1 2 ε * ,ε * ) is uniformly bounded in C 0, loc (R N ). Otherwise, Q c = ∅ and on Q c a(x) ≡ 1. Therefore u 2 ε (x) = J ε u ε and the C 0, 1 2 (Q c ) norm of u ε is bounded independently of ε. Hence,

5. 4 . 2

 42 The case 0 < m < 2In this situation, from Theorem 4.4.1 we know that lim ε→0 λ p (M ε,m + a(x)) = -sup for ε small enough and for large ε we have λp (M ε + a(x)) ≤ -sup R N a(x)2

  4.1 and Lemma 5.4.2 since we have for ε large(a(x) -1 ε m ) + ≤ u ε ≤ a + (x) + 1 ε N 4

2 < 0 2 ∆

 202 4.1 we havelim ε→0 λ p (M ε,m + a(x)) = λ 1 K 2,N D 2 (J) 2 ∆ + a(x)(5.52)lim ε→+∞ λ p (M ε + a(x)) = -sup for large ε we have λ p (M ε + a(x)) ≤ -sup R N a(x)and by Theorem (5.1.1) there exists a solution to (P ε ) for large ε. Whereas the existence of a positive solution for ε small is conditioned to the sign of λ 1 ( K 2,N D 2 (J) + a(x)). When λ 1 ( K 2,N D 2 (J) 2

D 2 (J) 2 R 2 R 2 R 2 R

 2222 N ×R N ρ n (z) |z| 2 u n (x)[ϕ(x + z) -2ϕ(x) + ϕ(x -z)] dxdz + R N ϕ(x)u n (x)(a(x) -u n (x)) dx = 0,where we use(5.39) to computeR N M ε,2 [u n ](x)ϕ(x) dx. Thus we get D 2 (J) N ×R N ρ n (z) |z| 2 u n (x) t z∇ 2 ϕ(x)z dxdz + R N ϕ(x)u n (x)(a(x) -u n (x)) dx = -D 2 (J) N ×R N ρ n (z) |z| 2 u n (x)[ϕ(x + z) -2ϕ(x) + ϕ(x -z) -t z∇ 2 ϕ(x)z] dxdz,where∇ 2 ϕ(x) := (∂ ij ϕ(x)) i,j . Since ρ n (z) is radial, we can see that D 2 (J) N ×R N ρ n (z) |z| 2 u n (x) t z∇ 2 ϕ(x)z dxdz = D 2 (J)K 2,N 2 R N u n (x)∆ϕ(x) dxand we getD 2 (J)K 2,N 2 R N u n (x)∆ϕ(x) dx + R N ϕ(x)u n (x)(a(x) -u n (x)) dx = -D 2 (J) 2 R N ρ n (z) |z| 2 u n (x)[ϕ(x + z) -2ϕ(x) + ϕ(x -z) -t z∇ 2 ϕ(x)z] dxdz,(5.54)Note that since u n converges to v in L 2 loc (R N ) we haveR N ϕ(x)u n (x)(a(x) -u n (x)) dx → R N ϕ(x)v(x)(a(x) -v(x)) dx (5.55)Recall now that ϕ ∈ C ∞ c (R N ), so there exists C(ϕ) and R(ϕ) so that|ϕ(x + z) -2ϕ(x) + ϕ(x -z) -t z∇ 2 ϕ(x)z| < C(ϕ)|z| 3 χ B R(ϕ) (x).Therefore since u n is bounded uniformly,D 2 (J) 2 R N ρ ε (z) |z| 2 u n (x)[ϕ(x + z) -2ϕ(x) + ϕ(x -z) -t z∇ 2 ϕ(x)z] dxdz ≤ CC(ϕ) R N ρ n (z)|z| → 0. (5.56)Passing to the limit ε → 0 in (5.54), thanks to (5.55) and (5.56) we getD 2 (J)K 2,N 2 R N v(x)∆ϕ(x) dx + R N ϕ(x)v(x)(a(x) -v(x)) dx = 0. (5.57) (5.57) being true for any ϕ ∈ C ∞ c this implies that v satises K 2,N D 2 (J) 2 ∆v + v(a(x) -v) = 0 a.e. in R N .

ϕ 1 K 2 ,

 12 p,ε (x) → ϕ 1 (x), a.e. in B R 0 (5.59)ϕ 1 is a positive principal eigenfunction associated with λ N D 2 (J) 2

  x) a.e. in B R 0 , (5.60) which combined with (5.59) enforcesu ε (x) ≥ γϕ 1 (x) a.e. in B R 0 , for all ε ≤ ε 2 ,

M

  [w] + (µ(x) + δ)w(x) = C(1 + τ |x| N +1 ) -1 R N J(x -y) (1 + τ |x| N +1 ) (1 + τ |y| N +1 ) ϕ(y) dy -ϕ(x) + (µ(x) + δ)ϕ(x) , ≤ C(1 + τ |x| N +1 ) -1 R N J(z) (1 + τ |x| N +1 ) (1 + τ |x + z| N +1 ) -1 ϕ(x + z) dz + (δ -λ)ϕ(x) , ≤ C(1 + τ |x| N +1 ) -1 τ R N J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) ϕ(x + z) dz + (δ -λ)ϕ(x) , ≤ w(x) τ R N J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) ϕ(x + z) ϕ(x) dz + δ -λ ,where we use (5.62) and infR N ϕ > 0. Set h(τ, x) = τ R N J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) ϕ(x + z) ϕ(x) dz + δ -λ. Thanks to ϕ ∈ L ∞ (R N ), inf R N ϕ > 0 there exists a positive constant C 0 so that ϕ(x + z) ϕ(x) ≤ C 0 for all x, z ∈ R N .Thus for all x ∈ R N , we haveh(τ, x) ≤ C 0 τ R N J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) dz + δ -λ.(5.63)LetI := C 0 τ R N J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) dz,then we haveI = C 0 τ {|x|≤2|z|} J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) dz + C 0 τ {|x|>2|z|} J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) dz.Let us estimate the rst integral. Since |x| ≤ 2|z| we haveC 0 τ {|x|≤2|z|} J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) dz ≤ C 0 τ 2 N +1 R N J(z)|z| N +1 dz.(5.64)Let us now estimate second term. Since |x + z| N +1 ≥ (|x| -|z|) N +1 , we have

Since for all |x|, τ |x| N +1 2 N

 2 +1 + τ |x| N +1 < 1, we achieve for |x| ≥ R 0 C 0 τ {|x|>2|z|} J(z) |x| N +1 -|x + z| N +1 (1 + τ |x + z| N +1 ) dz ≤ C 0 2 N +1 R 0 N +1 i=1 N + 1 i R N J(z)|z| i dz.(5.65)

  , where c * is dened in Proposition 1.1.3. Conversely, if c < c * and there exist a sequence (t n ) ∈ R as n → ∞ and a point x 0 ∈ Ω such that

	.26)
	we have the following auxiliary result
	Theorem 1.2.2. Assume that conditions (1.7) -(1.9) are satised, then Eq. (1.26) admits a positive
	bounded entire solution only if c < c

*

  .60) Eventually, we obtain λ D = λ ∞ . This completes the proof. Remark 3. In the proof, we have proved a result, which is stronger than what we really need. In fact, to obtain the conclusion of Theorem 1.4.2, one only needs to prove λ n < λ D . However, by proving Lemma 1.4.3, we obtain a more interesting result on the convergence of the eigenvalues and eigenfunctions. This indeed makes Theorem 1.4.2 more transparent and more interesting.

Remark 4. Inequality (1.60) in fact implies that

  Theorem 2.3.2. Let α = 2 and Hypothesis (H1) hold. Let u be a nonnegative subsolution of Eq. (2.34) satisfying (2.17). One of the followings holds true i) If (H1a)-(H1b) and lim sup |x|→∞ |q(x)||x| ≤ θ are fullled, then for all β ∈ (0, σ), where σ is given in (2.18), one nds

  Proof of Theorem 2.1.5 and 2.1.6. The proof is a transparent consequence of above lemmas. By Theorem (2.1.2) and (2.1.3), Eq. (2.1) possesses positive solutions if and only if λ K < 0. Hence, if λ K is continuous, monotone in K and there exists K 1 , K 2 such that λ K 1 < 0 and λ K 2 ≥ 0, one nds immediately K such that λ K = 0. Therefore, we imply that Eq. (2.1) possesses positive solutions if and only if K < K . The lemmas (2.5.1)-(2.5.5) conrm our statement.

  0)w and grows at most as S K (x). The local convergence implies that e γ 0 |x| 1-α

2 w(t, x) → 0 as t → ∞. Theorem (2.3.1), (2.71) and (2.73) assert that e γ 0 |x| 1-α 2 w(t, x) is bounded by the functions of L p (R N ), the Lebesgue dominated convergence theorem again implies that

  .36) Since M[ϕ]w n ∈ L 1 (B n ), from(4.36) and by using (4.35) we achieve Recall that by assumption there exists C > 0 such that J(x -y) ≤ Cχ 1 (|x -y|)). So we get

	λ v (L Bn + a(x)) w n	2 L 2 (Bn) ≤	Bn	-L Bn [w n ](x) -a(x)w n (x) -µw n + µw n w n (x) dx	(4.37)
		≤ µ w n	2 L 2 (Bn) +	Bn	-L Bn [w n ](x) + M[ϕ](x) w n (x) dx	(4.38)
		≤ µ w n	2 L 2 (Bn) +	Bn	R N \Bn	J(x -y)ϕ(y) dy w n (x) dx	(4.39)
		≤ µ w n	2 L 2 (Bn) + I n ,	(4.40)
	where I n denotes
		I n :=				J(x -y)ϕ(y) dy ϕ(x) dx	(4.41)
		Bn		R N \Bn
	Observe that we achieve (4.34) by proving
				lim inf n→∞	I n L 2 (Bn) ϕ 2	= 0.	(4.42)
		I n ≤				J(x -y)ϕ(y) dy w n (x) dx	(4.43)
		Bn	B n+1 \Bn
	By using the Fubini Theorem, Jensen inequality and the Cauchy-Schwarz inequality in (4.43), we obtain
					1/2		2	1/2
	I n ≤	ϕ 2 (y) dy				J(x -y)ϕ(x) dx	dy	,	(4.44)
	B n+1 \Bn			B n+1 \Bn	Bn
							1/2
	≤ ϕ L 2 (B n+1 \Bn)	B n+1 \Bn	Bn	J 2 (x -y)ϕ 2 (x) dx dy	,	(4.45)
							1/2
	≤ ϕ L 2 (B n+1 \Bn)	Bn		B n+1 \Bn	J 2 (x -y) dy ϕ 2 (x) dx	,	(4.46)
	≤ J ∞ ϕ L 2 (B n+1 \Bn) ϕ L 2 (Bn) .	(4.47)
	Dividing (4.47) by ϕ 2 L 2 (Bn) we achieve

  3.4. Proof of Theorem 4.3.4. As in the proof of Lemma 4.3.11, without loss of generality we can assume that the support of J is contained in the unit ball. Now from Lemma 4.3.8 and 4.3.11, we deduce that

  .58) As a straightforward consequence of this inequality, we get the limits in Theorem 4.4.1 Remark 16. When m = 2, another consequence of (4.58) is λ p

  ε [ϕ p,ε ](x) + a(x)ϕ p (x) ≥ -1 8 λ 1 K 2,N D 2 (J) 2 ∆ + a(x) -κσ ϕ p,ε (x) for all ε ≤ ε 0 .

	By choosing σ smaller if necessary,
	-	1 8	λ 1	K 2,N D 2 (J) 2	∆ + a(x) -κσ ≥ -	1 16	λ 1	K 2,N D 2 (J) 2	∆ + a(x)
	and we achieve

L R 0 ,ε [ϕ p,ε ](x) + a(x)ϕ p (x) ≥ -1 16 λ 1 K 2,N D 2 (J) 2 ∆ + a(x) ϕ p,ε (x) for all ε ≤ ε 0 .

(5.58)

Je dédicace cette thèse à.

Chapitre 1. Persistence versus extinction under a climate change

1.2. The cylindrical environment without time dependence

1.3. The partially periodic environment with time dependence

Chapitre 2. Liouville-type result with possibly vanishing and sign-changing potentials

2.4. Existence/Nonexistence and Uniqueness

Chapitre 3. Liouville-type result for a quasilinear equation with possibly decaying potentials

Remerciements

Chapitre 5. Positive solutions of nonlocal heterogeneous KPP equations Bibliographie

-Tai Nguyen with my hearty esteem for their helps, their fruitful discussions and the their collaborations, which play an important role to complete the thesis.

My PhD research is supported by FIRST program of Marie Curie ITN, 7th framework (FP7/2007-2013) of European Commission, grant agreement 238702 in the period 01/2011-12/2013 and partially supported by ERC Grant Agreement 321186 : "Reaction-Diusion Equations, Propagation and Modelling" of European Research Council (FP7/2007-2013) held by Henri Berestycki in the period 01/2014-08/2014. I am also indebted to Centre d'Analyse et de Mathématique Sociales, CNRS-EHESS, France, departments of Mathematics,

 [START_REF] Serrin | Local behavior of solutions of quais-linear equations[END_REF], [START_REF] Trudinger | On Harnack type inequalities and their application to quasilinear elliptic equations[END_REF], we obtain u * > 0 in R N . Finally, rom Proposition 3.2.1, we deduce that u * decays exponentially and thus u * ∈ S α .

ii. Nonexistence and uniqueness. Suppose by contradiction that there exists a function u ∈ S α . Thanks to (C1), Proposition (3.2.1) implies that u satises (3.11). In particular, u satises (3.16). Let ϕ ∞ be a positive solution of (3.29) normalized by ϕ ∞ (0) < u(0). Since λ ∞ (a) ≥ 0, it follows that ϕ ∞ is a positive supersolution of (3.1). By Theorem 3.3.1, we have u ≤ ϕ ∞ in R N . This contradiction completes the proof.

The uniqueness is a direct consequence of the exponential decay property and comparison principle Theorem 3.3.1.

Proof of Theorem 3.1.2. Existence part can be obtained by the same argument as in the proof of Theorem 3.1.1. In order to prove the nonexistence and uniqueness we also follow the above scheme, employing Proposition (3.2.2) instead of Proposition (3.2.1). The proof would be completed if we could show that if u ∈ S p then u satises (3.16) and u ∈ L 1 (R N ) . Indeed, by Proposition (3.2.2), we know that if u ∈ S p then u satises (3.15). Therefore u ∈ L 1 (R N ) if γ 0 > N . Since the map γ → (N p + 1)γ p + (3p + 1)N γ p-1 is strictly increasing, it follows that m > N p (N p + 3p + 2) if and only if γ 0 > N . Thus u ∈ L 1 (R N ). This completes the proof. In this chapter, we collect some new results about the relation of λ p , λ p and λ v in bounded and unbounded domains (eventually R N ). For the sake of coherence, we will present here a common introduction for the two next chapters and do not repeat again in chapter 5. In section 4.1, we will present the derivation of the model, the common hypotheses that are used also for chapter 5 and explain why the investigation of the generalized principal eigenvalues plays the central role.

4.1

where

Moreover we have

To obtain these limits, we start by recalling a known inequality.

Lemma 4.4.3. Let J ∈ S(R N ), J ≥ 0, J symmetric. Then for all ϕ ∈ H 1 (R N ) we have

Proof. Let ϕ ∈ C ∞ c , then by applying the standard Taylor expansion we have

where use the Einstein summation convention

By plugging the Taylor expansion of ϕ (4.56) in the above equality we see that

where we use in the last inequality the standard inequality

). So by using Fubini's Theorem and rearranging the terms in the above inequality, it follows that

, the above inequality holds true for ϕ ∈ H 1 , since obviously the functional

where h(α) is dened by

Since J is compactly supported, thanks to the Lebesgue's Theorems, we can check that h(•) is a smooth (C 2 ) convex increasing function of α. Moreover, we have

Therefore, by continuity of h, we can choose α small says α ≤ α 0 for some α 0 so that h(α) < 0. For such α, we get

Let M := sup B 2R 0 (0) S(x) and choose now C so that C = 2M e 2αR 0 . Let us consider now the continuous function

By direct computation we can check that ū is a super-solution to (4.2). Indeed, for any x ∈ B 2R 0 (0), we have ū = 2M > sup B 2R 0 (0) S(x) which implies that f (x, ū) = f (x, 2M ) ≤ 0 and

We are now in position to construct a solution of (4.2). By Lemma 5.2.1, there exists ū a positive super-solution to (4.2). Therefore for any R > 0, ū is also a positive super-solution to (5.2). Therefore by using a standard sweeping argument, we can check that for all R ≥ R 0 then the unique solution to (5.2)

5.2. Existence/non existence and uniqueness of a non-trivial solution 127

Uniqueness

Having construction of a L 1 (R N ) positive solution to (4.2), the uniqueness of the solutions of (4.2) is then obtained by the following argument. Assume by contradiction that v is another positive solution and assume that v ∈ L ∞ . Then v is a supersolution to (5.2) for any R > 0. Therefore v ≥ u R for all R ≥ R 0 . Since u R is monotone with respect to R, it follows that v ≥ ũ := lim R u R (x). Assume by contradiction that v ≡ ũ and recall that v and ũ satises:

By multiplying (5.6) by v and (5.7) by u we get after integration over R N

(5.9)

Therefore by subtracting the two above equality, we get the contradiction

since ũ ≤ v and f (x, s)/s is decreasing.

Non-existence of a solution

In this section, we deal with the non-existence of positive solution when λ p (M + β(x)) ≥ 0. To simplify the presentation of the proofs we treat the two cases: λ p (M + β(x)) > 0 and λ p (M + β(x)) = 0 separately. The proof in the second case being more involved, we start by showing the non existence results when λ p (M + β(x)) > 0.

Case λ p (M + β(x)) > 0:

In this situation we argue as follows. Assume by contradiction that a positive bounded solution u exists.

By assumption, u satises Let us denote γ(x) := f (x,u(x)) u(x) , then we obviously have

Therefore by denition of λ p we have λ p (M + γ(x)) ≤ 0. By construction γ(x) ≤ β(x), so by combining (5.12) with Proposition 4.2.2, Lemma 4.3.11 and the denition of λ p (M + γ(x)) we can infer that

Therefore

By (i) of Proposition 4.2.2, we then have

Assume for a moment that the claim holds then by arguing as in the uniqueness subsection, since ψ ∈ L 1 we get the following contradiction

Proof of the Claim. For convenience we denote γ(x) := γ + εη. By (5.11), since γ < β we also have

(5.13)

From the later inequality, by using Proposition 4.2.1 and Lemma 4.3.6 we see that there exists R 0 so that for all R ≥ R 0 there exists a positive ϕ R ∈ C( B(0, R)) associated to the principal eigenvalue λ p (L R + γ(x)) of the approximated problem

Take now the increasing sequence (R n ) n∈N := (R 0 + n) n∈N and let (ϕ n ) n∈N be the sequence of function where ϕ n is the positive principal eigenfunction associated to λ p (L Rn + γ(x)). Without loss of generality, we can assume that for all n, ϕ n (0) = 1. Since J(0) > 0 and is compactly supported we can reproduce the arguments developed in Claim 4.3.7 and extract from the sequence (ϕ n ) n∈N a subsequence still denoted (ϕ n ) n∈N which converges locally uniformly to a non negative continuous function ϕ. Moreover, ϕ satises

To conclude the proof of this claim, we characterise the behaviour of ϕ(x) for |x| >> 1. Let us denote 0 < ν < -lim sup |x|→∞ β(x) and x R 1 so that β(x) ≤ -ν 2 for |x| ≥ R 1 .

Claim 5.3.2. For all sequences (t n ) n∈N , (x n ) n∈N so that

Assume for the moment the claim holds true. Then we obtain a straightforward contradiction

Let us prove the Claim. Again we argue by contradiction and assume there exists ε > 0 and sequences

and 0 < z n (t, x) < C u 0 ∞ . Since for all n, z n (0, x) ∈ C ∞ by the Cauchy Lipschitz Theorem we see that

. Thus, there exists C 0 > 0 independent of n so that z n C 1,1 (R + ,C(R N )) < C 0 . From these estimates, the sequence (z n ) n∈N is uniformly bounded in C 1,1 ((0, T ), C 0,1 (R N )) for any T > 0. By a diagonal extraction, there exists a subsequence of (z n ) n∈N that converges locally uniformly to z(t, x). Moreover, thanks to lim |x|→∞ f (x,s) s < 0, there exists κ > 0 so that z(x, t) satises

In addition, for all t > 0, z(t, 0) = lim n→∞ z n (t, 0) ≥ ε. Since z(0, x) is a super-solution of (5.34), by Lemma 5.3.1 the function z(t, x) is monotone decreasing in time. By sending t → ∞, since z ≥ 0, z converges locally uniformly to a non-negative function z that satises

Now let consider the function w(x) := ε 2 e α|x| -z with α to be chosen, then w satises

The left hand side of the inequality is well dened and continuous with respect to α since J is compactly supported. Thanks to R N J(z)dz = 1, by choosing α small enough, we achieve

By construction, since z is bounded lim |x|→∞ w(x) = +∞ and w achieves a minimum in R N says at x 0 . Since w(0) = ε 2 -z(0) ≤ -ε 2 , we have w(x 0 ) < 0. Now at this point, we get the following contradiction

Therefore for 0 < θ ≤ -λp 2 , the function θϕ ε is a sub-solution to (P ε ). By a standard sweeping argument, we get

Since u ε ∈ L 1 (R N ), u ε achieves its maximum at some point, says x 0 . At this point from (P ε ) we have

Thus x 0 ∈ supp(a + ) and u ε ∞ = sup supp(a + ) u ε which proves (iii).

Next we obtain derive some useful super-solution for large ε.

Lemma 5.4.2. There exists ε 0 > 0 so that for all m ≥ 0 and ε ≥ ε 0 any positive bounded solution u ε of (P ε ) satises

Proof. Let δ ∈ (0, N

2 ) and consider the function

We will show that ζ ε is a supersolution to (P ε ) when ε >> 1.

Indeed, we have

where we use in the last inequality that

Thus for ε >> 1, we achieve

Therefore for ε >> 1, by a sweeping argument we get u ε ≤ ζ ε . We end the proof by taking δ = N 4 .

Remark 19. When m = 0 and (a(x) -1) + ≡ 0, the above computation holds as well with ζ ε (x) :=

Thus in this case we have for large ε

Next, we prove some continuity of λ p (L R,ε + a(x)) with respect to ε.

Thus

Let us denote Q := supp(a + ). Since u ε is positive and uniformly bounded, we have supp(w + ) ⊂ Q and

Since a is Lipschitz, by using a Taylor expansion, we can see that

Collecting (5.45),(5.46), we get

then from above estimates we conclude that

Thus u ε converges pointwise almost everywhere to a bounded non-neqative solution of (5.43).

Remark 20. Note that the above proof can be easily adapted to M ε,m for m < 2 as soon as the function a is smooth enough. Indeed, for a ∈ C 2 (R N ), following the above arguments, we get by using the Taylor expansion up to order 2 of a

with C a constant depending on ∇ 2 u ∞ . When a is only Lipschitz, the above argument holds only for M ε,m with m < 1.

Finally, to complete our analysis, we need to check that

We treat separately the following two cases : (i) ε * < +∞, (ii) ε * = ∞. The latter arise when sup R N (a(x) -1) > 0. In this situation, there exists R 0 > 0 so that the continuous function ϕ = (a(x) -1) + ≡ 0 in B R (0) for R ≥ R 0 and we can check that ϕ is a sub-solution to the approximated problem:

(5.48)

Since large constants are super-solutions of (5.48) for any ε ≥ 0, R > R 0 there exists a unique solution ϕ ≤ u ε,R ≤ M . By sending R → ∞ and by the uniqueness of the solution to (P ε ) we have ϕ

Case ε * = +∞:

Thanks to Lemma 5.4.1 and Remark 19, for all x ∈ R N for large ε we have

.

Hence, u ε converge uniformly to (a(x) -1) + .

Case ε * < +∞:

In this situation, the function (a(x) -1) + ≡ 0 in R N and we are reduce to prove that

Note that by denition of ε * we must have λ p (M ε * + a(x)) ≥ 0. Indeed, if not then λ p (M ε * + a(x)) < 0 and by Lemma 4.3.12 λ p (M + a ε * (x)) < 0. Therefore for some R we have λ p (L R + a ε * (x)) < 0. By continuity of λ p (L R + a ε * (x)) with respect to ε, (Lemma 5.4.3) we get for some δ 0 > 0, λ p (L R + a ε * +δ (x)) < 0 for any δ ≤ δ 0 . Hence, λ p (M ε * +δ + a(x)) = λ p (M + a ε * +δ (x)) < 0 for any δ ≤ δ 0 and by Theorem 5.1.1 there exists a positive solution to (P ε ) for all ε ≤ ε * + δ 0 contradicting the denition of ε * . Note also that since ε * < +∞, the construction of the supersolution in Section 5.2 holds for any ε ∈

Let g(x, s) := s(a(x) -1 -s) then for all ε we have

then by multiplying by ϕ p,ε the equation satised by ϕ p,ε and integrating it over B R 0 , we get

Therefore by the characterisation of Sobolev space [START_REF] Ac | A new approach to Sobolev spaces and connections to Γ-convergence[END_REF][START_REF] Ac | An estimate in the spirit of Poincaré ?s inequality[END_REF], along a sequence we have ϕ

Moreover by extending ϕ p,ε and ϕ by 0 outside B R 0 and by arguing as above for any

we get for ε small enough supp(k ε ) ∩ supp(ϕ) = ∅. Thus passing to the limit along a sequence in the above equation yields

(5.61) being true for any ϕ, it follows that ψ is the smooth positive eigenfunction associated to λ 1 normalised by ψ L 2 (B R 0 ) = 1. ψ being uniquely dened, we get ϕ p,ε → ψ in L 2 (B R 0 ) when ε → 0. Thus along any sequence ϕ p,ε (x) → ϕ 1 (x) almost everywhere in B R 0 .

Extension to non-compactly supported kernels

In this section, we discuss the extension of our survival criteria to more general dispersal kernel J and prove Theorem 5.1.5. Observe that the construction of positive solution only required that λ p (L R + β(x)) < 0 for some R, no matter the dispersal kernel J is. Therefore as soon as lim R→∞ λ p (L R + β(x)) < 0 there exists a positive solution to (4.2) with no restriction on the decay of the kernel. Similarly, when λ p (M + β(x)) > 0 the proof of the non-existence of positive bounded solution essentially relies on the inequality between λ p (M + β(x)) and λ p (M + β(x)) obtained in Lemma 4.3.11 which holds for quite general kernels including those satisfying assumption (H5). Concerning the proof of the uniqueness of the positive solution, it relies on the construction of a integrable uniform super-solution of (4.2) which guarantes the existence of a positive L 1 solution to (4.2). Such super-solution still exists for kernels J that satises the decay assumption H5. Indeed, we can show Then there exists ū ∈ C 0 (R N ) ∩ L 1 (R N ), ū > 0 so that ū is a super-solution to (4.2).

Combining (5.64), (5.65) and (5.63), we get for |x| > R 0 h(x, τ

Thanks to (H5), for τ small enough, says τ ≤ τ 1 and R 0 large enough we achieve h(x, τ ) ≤ δ-λ 2 < 0, Hence, for all τ ≤ τ 1 , we have M[w] + (µ(x) + δ)w(x) ≤ w(x)h(x, τ ) ≤ w(x) δ -λ 2 < 0 for all x ∈ R N \ B R 0 .

(5.66)

Fix now τ ≤ τ 1 and x R 0 > R δ so that h(x, τ ) < 0 in R N \ B R 0 (0). Let κ 0 := sup R N \B R 0 (0)

1+τ |x| N +1 . Let 0 < κ < κ 0 and consider the set

By construction since ϕ > 0 in R N we can choose κ small so that