Jong-Shenq Guo

François Hamel

Propagation and blocking in periodically hostile environments *

We study the persistence and propagation (or blocking) phenomena for a species in periodically hostile environments. The problem is described by a reaction-diffusion equation with zero Dirichlet boundary condition. We first derive the existence of a minimal nonnegative nontrivial stationary solution and study the large-time behavior of the solution of the initial boundary value problem. To the main goal, we then study a sequence of approximated problems in the whole space with reaction terms which are with very negative growth rates outside the domain under investigation. Finally, for a given unit vector, by using the information of the minimal speeds of approximated problems, we provide a simple geometric condition for the blocking of propagation and we derive the asymptotic behavior of the approximated pulsating travelling fronts. Moreover, for the case of constant diffusion matrix, we provide two conditions for which the limit of approximated minimal speeds is positive.

Introduction and main results

This paper is concerned with persistence and propagation phenomena for reaction-diffusion equations of the type u t -∇ • (A(x)∇u) = f (x, u) (1.1) * Part of this work was carried out during a visit by the second author at the Department of Mathematics of the National Taiwan Normal University, the hospitality of which is thankfully acknowledged. The first author is partially supported by the National Science Council of the Republic of China under the grant NSC 99-2115-M-032-006-MY3. The second author is supported by the French "Agence Nationale de la Recherche" within the projects ColonSGS and PREFERED. He is also indebted to the Alexander von Humboldt Foundation for its support.

1 in R N or in some unbounded open subsets Ω of R N with zero Dirichlet boundary condition on ∂Ω. Equations of the type (1.1) arise especially in population dynamics and ecological models (see e.g. [25,[START_REF] Shigesada | Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution[END_REF][START_REF] Xin | Analysis and modeling of front propagation in heterogeneous media[END_REF]), where the nonnegative quantity u typically stands for the concentration of a species.

Let us start with the case of the whole space R N . The symmetric matrix field x → A(x) = (A ij (x)) 1≤i,j≤N is assumed to be of class C 1,α (R N ) with α > 0 and uniformly positive definite: that is, there exists a positive constant β > 0 such that

∀ x ∈ R N , ∀ ξ = (ξ 1 , . . . , ξ N ) ∈ R N , Aξ • ξ := 1≤i,j≤N A ij (x)ξ i ξ j ≥ β |ξ| 2 , (1.2)
where | • | denotes the Euclidean norm in R N . We set R + = [0, +∞). The nonlinear reaction term f : R N × R + → R, (x, u) → f (x, u) is assumed to be continuous, of class C 0,α with respect to x locally uniformly in u ∈ R + , of class C 1 with respect to u, and ∂f ∂u (•, 0) is of class C 0,α (R N ). Furthermore, we assume that f (x, 0) = 0 for all x ∈ R N , there exists M > 0 such that f (x, M) ≤ 0 for all x ∈ R N .

(1.

3)

The functions A ij (for all 1 ≤ i, j ≤ N) and f (•, u) (for all u ∈ R + ) are assumed to be periodic in R N . Hereafter a function w is called periodic in R N if it satisfies

w(• + k) = w(•) for all k ∈ L 1 Z × • • • × L N Z,
where L 1 , • • • , L N are some positive real numbers, which are fixed throughout this paper.

If f fulfills the additional Fisher-KPP (for Kolmogorov, Petrovsky and Piskunov) [START_REF] Fisher | The advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]] assumption ∀ x ∈ R N , u → g(x, u) = f (x, u) u is decreasing with respect to u > 0, (1.4) then the large-time behavior of the solutions of the Cauchy problem

u t -∇ • (A(x)∇u) = f (x, u), t > 0, x ∈ R N , u(0, x) = u 0 (x), x ∈ R N (1.5)
is directly related to the sign of the principal periodic eigenvalue λ 1 of the linearized operator at 0 (see [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF]). This eigenvalue λ 1 is characterized by the existence of a (unique up to multiplication) periodic function ϕ ∈ C 2,α (R N ), which satisfies

-∇ • (A(x)∇ϕ) -ζ(x)ϕ = λ 1 ϕ in R N , ϕ > 0 in R N , (1.6)
where ζ(x) = ∂f ∂u (x, 0) for all x ∈ R N . The precise statement of what is known under the additional assumption (1.4) will be recalled just after Proposition 1.1 below.

Our first result, which is a preliminary step before the main purpose of the paper devoted to propagation phenomena in environments with hostile boundaries, is actually concerned with the existence of a minimal positive stationary solution p for problem (1.5) and with the large time behavior of the solutions u of (1.5), when f fulfills the assumption (1.3) alone.

Proposition 1.1 Assume that λ 1 < 0 and (1.3). Then there is a minimal periodic solution p(x) of -∇ • (A(x)∇p) = f (x, p(x)) in R N , p > 0 in R N , (1.7) in the sense that, for any solution q of (1.7), there holds q ≥ p in R N . Furthermore, p ≤ M in R N and, if u 0 : R N → [0, M] is uniformly continuous and not identically 0, then the solution u(t, x) of the Cauchy problem (1.5) is such that lim inf t→+∞ u(t, x) ≥ p(x) locally uniformly with respect to x ∈ R N .

If one further assumes that u 0 ≤ p in R N , then u(t, x) → p(x) as t → +∞ locally uniformly with respect to x ∈ R N .

It is obvious to see that the solution p of (1.7) is not unique in general. Choose for instance A(x) = I N (the identity matrix) and f (x, u) = sin(u) for all (x, u) ∈ R N × R + : the function f satisfies (1.3) with M = π, λ 1 = -1 < 0, but any constant function p(x) = mπ with m ∈ N\{0} solves (1.7). On the other hand, if, in addition to (1.3), the function f satisfies the assumption (1.4), then the solution p of (1.7) is unique, see [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF]. In particular, all solutions of (1.7) are necessarily periodic. Notice that, in the general case of assumption (1.3) alone, Proposition 1.1 still states the existence of a minimal periodic solution p of (1.7) in the class of all positive solutions q, which are not a priori assumed to be periodic. It is also known that, under hypotheses (1.3) and (1.4), the condition λ 1 < 0 of the unstability of 0 is a necessary condition for the existence of the solution p of (1.7) as well: if λ 1 ≥ 0, then all bounded solutions u of (1.5) converge to 0 as t → +∞ uniformly in R N , see [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF]. On the other hand, under the assumptions (1.3), (1.4) and λ 1 < 0, for any non-zero bounded uniformly continuous u 0 : R N → R + , there holds u(t, x) → p(x) as t → +∞ locally uniformly in x ∈ R N (see [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF]). We also refer to [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF][START_REF] Cantrell | The effects of spatial heterogeneity in population dynamics[END_REF][START_REF] Cantrell | On the effects of spatial heterogeneity on the persistence of interacting species[END_REF] for related results in the case of bounded domains with Dirichlet or Neumann boundary conditions, and to [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive media[END_REF][START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF] for results with KPP nonlinearities and periodic or non-periodic coefficients in R N . Lastly, it is worth noticing that Proposition 1.1 and the aforementioned convergence results are different from what happens with other types of nonlinearities f , like combustion, bistable or even monostable nonlinearities which are degenerate at 0: in these cases, the large-time behavior of the solutions u of (1.5) strongly depends on some threshold parameters related to the size and/or the amplitude of u 0 (see e.g. [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF][START_REF] Polacik | Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N[END_REF][START_REF] Roquejoffre | Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders[END_REF][START_REF] Zlatoš | Sharp transition between extinction and propagation of reaction[END_REF]).

Remark 1.1

The assumption that u 0 ranges in the interval [0, M] is made to guarantee the global existence and boundedness (from below by 0 and from above by M) of the solutions u of the Cauchy problem (1.5). If f fulfills the KPP assumption (1.4) together with (1.3), or if f (x, s) ≤ 0 for all (x, s) ∈ R N × [M, +∞), then it follows that the solution u exists for all t ≥ 0 and is globally bounded from below by 0 and from above by max M, u 0 L ∞ (R N ) , as long as u 0 is nonnegative and bounded. The same comment also holds for the Cauchy problem (1.14) below with zero Dirichlet boundary condition on ∂Ω.

As a matter of fact, in Proposition 1.1, the negativity of λ 1 immediately implies that the positive periodic functions ε ϕ are subsolutions of (1.5) for ε > 0 small enough, where ϕ is a solution of (1.6). It then follows from the above proposition and the results of Weinberger [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] that, for each unit vector e of R N , there is a positive real number c * (e) > 0 (minimal speed) such that the following holds: for each c ≥ c * (e), there is a pulsating travelling front

u(t, x) = φ(x • e -ct, x)
solving (1.5) and connecting 0 to p, that is, the function φ : R×R N → [0, M], (s, x) → φ(s, x) is periodic in x, decreasing in s, and it satisfies φ(-∞, x) = p(x) and φ(+∞, x) = 0 for all x ∈ R N . Furthermore, such pulsating travelling fronts do not exist for any c < c * (e). We also refer to [START_REF] Nadin | Travelling fronts in space-time periodic media[END_REF][START_REF] Nolen | Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds[END_REF][START_REF] Nolen | Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle[END_REF][START_REF] Xin | Existence of planar flame fronts in convective-diffusive periodic media[END_REF][START_REF] Xin | Analysis and modeling of front propagation in heterogeneous media[END_REF] for other results about pulsating travelling fronts in the whole space R N , including other types of nonlinearities and the case of time-periodic media. Now, based on the previous results in R N , we turn our attention to the main concern of this paper, namely the case when there are hostile periodic patches in the domain under consideration. We deal with persistence and propagation phenomena for reaction-diffusion equations of the type

u t -∇ • (A(x)∇u) = F (x, u), x ∈ Ω, u(t, x) = 0, x ∈ ∂Ω, (1.8)
in an unbounded open set Ω ⊂ R N which is assumed to be of class C 2,α (with α > 0) and periodic. The periodicity means that

Ω = Ω + k for all k ∈ L 1 Z × • • • × L N Z.
Furthermore, the fields A(x) and F (x, u) are assumed to be periodic with respect to x in Ω, to have the same smoothness as before and to fulfill (1.2) and (1.3) above, where x ∈ R N is now replaced with x ∈ Ω. In particular, assumption (1.3) is now replaced with F (x, 0) = 0 for all x ∈ Ω, there exists M > 0 such that F (x, M) ≤ 0 for all x ∈ Ω.

(1.9)

Throughout the paper, we denote

C = Ω ∩ [0, L 1 ] × • • • × [0, L N ]
the cell of periodicity of Ω. The zero Dirichlet boundary condition imposed on ∂Ω means that the boundary is lethal for the species. Note that the unbounded periodic open set Ω is not a priori assumed to be connected. The reason for that will become clear later, once the approximation procedure (1.16) below has been introduced. However, due to the global smoothness of ∂Ω, the set Ω has only a finite number of connected components relatively to the lattice

L 1 Z × • • • × L N Z. That is, there is a finite number of connected compo- nents ω 1 , . . . , ω m of Ω such that ω i ∩ (ω j + k) = ∅ for all 1 ≤ i = j ≤ m and for all k ∈ L 1 Z × • • • × L N Z, and Ω = 1≤i≤m Ω i , where Ω i = k∈L 1 Zו••×L N Z ω i + k. (1.10)
The sets ω i are not uniquely defined, but the sets Ω i are unique (up to permutation), periodic, and Ω i ∩ Ω j = ∅ for all 1 ≤ i = j ≤ m.

In the case of no-flux boundary conditions ν(x) • (A(x)∇u(t, x)) = 0 on ∂Ω when Ω is connected, much work have been devoted in the recent years to the study of propagation of pulsating fronts u(t, x) = φ(x • ect, x), where φ(s, •) is periodic for all s ∈ R and e is any unit vector, for various types of nonlinearities F , in straight infinite cylinders [START_REF] Berestycki | Travelling fronts in cylinders[END_REF][START_REF] Roquejoffre | Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders[END_REF] or in periodic domains [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Hamel | Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity[END_REF][START_REF] Hamel | Uniqueness and stability properties of monostable pulsating fronts[END_REF][START_REF] Liang | Asymptotic speeds of spread and traveling waves for monostable semiflows with applications[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF]. In the case of KPP nonlinearities F , further properties of the minimal propagation speeds can be found in [START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF][START_REF] Smaily | Pulsating travelling fronts: Asymptotics and homogenization regimes[END_REF][START_REF] Heinze | Large convection limits for KPP fronts[END_REF][START_REF] Kinezaki | Spatial dynamics of invasion in sinusoidally varying environments[END_REF][START_REF] Liang | A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations[END_REF][START_REF] Nadin | Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation[END_REF][START_REF] Ryzhik | KPP pulsating front speed-up by flows[END_REF][START_REF] Zlatoš | Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows[END_REF].

In this paper, we consider a larger class of reaction terms F , together with zero Dirichlet boundary condition. Let us first mention that, under the assumption that the equation (1.8) is invariant in the direction x 1 and under appropriate conditions on F , classical travelling fronts

u(t, x) = φ(x 1 -ct, x 2 , . . . , x n )
in straight infinite cylinders (in the x 1 -direction) with zero Dirichlet boundary condition are known to exist (see [START_REF] Muratov | Front propagation in infinite cylinders. I. A variational approach[END_REF][START_REF] Vega | Travelling waves fronts of reaction-diffusion equations in cylindrical domains[END_REF], including the case of some systems of equations). In this case, the profiles φ of these travelling fronts solve elliptic equations or systems. For problem (1.8) in periodic domains, the reduction to elliptic equations does not hold anymore since the equation is not assumed to be invariant in any direction. Recently, existence results for problems of the type (1.8) in connected two-dimensional periodically oscillating infinite cylinders with homogeneous isotropic diffusion (A(x) = I 2 ) and KPP nonlinearities satisfying (1.4) have been established, see [START_REF] Liang | Spreading speeds and traveling waves for abstract monostable evolution systems[END_REF]. In the present paper, the set Ω is periodic in all variables x 1 , . . . , x N and the direction of propagation may be any unit vector e of R N . Actually, one of the novelties of this paper with respect to the previous literature is that the nature of propagation vs. blocking strongly depends on the direction e and on geometrical properties of the set Ω itself. Let λ 1,D denote the principal periodic eigenvalue of the linearized equation at 0 in Ω with zero Dirichlet boundary condition. That is, there exists a function ϕ ∈ C 2,α (Ω), which is periodic in Ω and satisfies (1.11) where ζ(x) = ∂F ∂u (x, 0) for all x ∈ Ω. If Ω is connected, then ϕ > 0 in Ω and ϕ is unique up to multiplication. Otherwise, in the general case, the function ϕ is unique up to multiplication in each set Ω i on which it is positive. More precisely, ϕ can be chosen to be positive on the (largest possible) set Ω = i∈I min Ω i , where I min denotes the set of indices i ∈ {1, . . . , m} for which the principal periodic eigenvalue λ 1,Ω i ,D of the operator -∇ • (A(x)∇)ζ(x) in Ω i with zero Dirichlet boundary condition on ∂Ω i is equal to λ 1,D . That is,

           -∇ • (A(x)∇ϕ) -ζ(x)ϕ = λ 1,D ϕ in Ω, ϕ = 0 on ∂Ω, ϕ ≥ 0 in Ω, max Ω ϕ > 0,
λ 1,D = min 1≤j≤m λ 1,Ω j ,D = λ 1,Ω i ,D for all i ∈ I min .
The following theorem, which is analogue to Proposition 1.1, is concerned with the existence of a minimal nonnegative and non-trivial stationary solution of (1.8) in Ω and the large-time behavior of the solutions of the associated initial boundary value problem, under the assumption that the steady state 0 of (1.8) is linearly strictly unstable. To do so, we introduce the set I -= i ∈ {1, . . . , m}, λ 1,Ω i ,D < 0 .

(1.12)

Theorem 1.2 Assume that λ 1,D < 0, that is I -= ∅. Then there exists a minimal stationary periodic solution p(x) of

     -∇ • (A(x)∇p) = F (x, p(x)) in Ω, p = 0 in ∂Ω ∪ i ∈I -Ω i , p > 0 in i∈I -Ω i , (1.13)
in the sense that any bounded solution q of (1.13) satisfies q ≥ p in Ω. Moreover, for any uniformly continuous function u 0 : Ω → [0, M] which is not identically 0, the solution u(t, x) of the initial boundary value problem

     u t -∇ • (A(x)∇u) = F (x, u), t > 0, x ∈ Ω, u(t, x) = 0, t > 0, x ∈ ∂Ω, u(0, x) = u 0 (x), x ∈ Ω (1.14) is such that lim inf t→+∞ u(t, x) ≥ p(x) (1.15)
locally uniformly with respect to the points x ∈ Ω whose connected components intersect the support of u 0 . If one further assumes that u 0 ≤ p in Ω, then u(t, x) → p(x) as t → +∞ in the same sense as above.

As already emphasized, the periodic open set Ω is not assumed to be connected, this is why the lower bound (1.15) or the convergence of u(t, x) to p(x) at large time can only hold in the (open) connected components C of the intersection of Ω with the support of u 0 (outside these components, the solution u(t, x) stays 0 for all times t ≥ 0). If such a connected component C is included in a set Ω i with i ∈ I -, then Theorem 1.2 implies that u(t, x) is separated away from 0 at large time, locally uniformly in C. However, (1.15) does not say anything about the behavior of u(t, x) when x ∈ i ∈I -Ω i (p(x) = 0 there). Actually, for each Ω i with i ∈ I -, one has λ 1,Ω i ,D ≥ 0 and if F satisfies the additional assumption (1.4) in Ω i , then u(t, x) → 0 as t → +∞ uniformly in x ∈ Ω i , as follows from the same ideas as in [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF].

The remaining part of this paper is concerned with the existence of pulsating fronts and the possibility of blocking phenomena for problem (1.8) with zero Dirichlet boundary condition. The strategy, which is one of the main interests of the paper, consists in approximating the Dirichlet condition on ∂Ω (and even in R N \Ω) by reaction terms with very negative growth rates in R N \Ω, using the previous results and then passing to the singular limit in the stationary solutions and in the pulsating travelling fronts as the growth rates converge to -∞ in R N \Ω. This means that the quantity u lives in the whole space R N , but the space contains very bad regions. We will see that the location of the good vs. bad regions plays a crucial role in the dynamical behavior of the solutions.

For this, let (f n ) n∈N be a sequence of real-valued functions defined in R N × R + such that each function f n : (x, u) → f n (x, u) is continuous, periodic with respect to x ∈ R N , of class C 0,α with respect to x ∈ R N locally uniformly in u ∈ R + , of class C 1 with respect to u with ∂fn ∂u (•, 0) ∈ C 0,α (R N ), and it satisfies (1.3). Here we define N to be the set of all nonnegative integers. Furthermore, we assume that

     f n (x, u) = F (x, u) for all (x, u) ∈ Ω × R + and n ∈ N, (f n (x, u)) n∈N is nonincreasing for all (x, u) ∈ Ω × R + , g n (x, u) → -∞ as n → +∞ locally uniformly in (x, u) ∈ (R N \Ω) × R + , (1.16)
where

g n (x, u) =      f n (x, u) u if u > 0, ∂f n ∂u (x, 0) =: ζ n (x) if u = 0.
The last condition means that the death rate in the region R N \Ω is very high, namely this region becomes more and more unfavorable for the species as n becomes larger and larger. Typical examples of such functions f n satisfying (1.9) and (1.16) are

f n (x, u) = ρ n (x) u + f (u),
where the function f : R + → R is of class C 1 and satisfies f (0) = 0, f (M) ≤ 0, and the functions ρ n : R N → R are periodic, nonpositive, of class C 0,α (R N ), nonincreasing with respect to n, independent of n in Ω, and ρ n → -∞ as n → +∞ locally uniformly in R N \Ω.

For every n ∈ N, let λ 1,n denote the principal periodic eigenvalue of the linearized operator at 0 in R N . That is, there exists a (unique up to multiplication) periodic function ϕ n of class C 2,α (R N ), which satisfies

-∇ • (A(x)∇ϕ n ) -ζ n (x)ϕ n = λ 1,n ϕ n in R N , ϕ n > 0 in R N . (1.17)
We first establish the relationship between the principal eigenvalues λ 1,n of (1.17) and the principal eigenvalue λ 1,D of (1.11), as well as the convergence of the minimal solutions p n of (1.7) with nonlinearities f n to the minimal solution p of (1.13), when λ 1,D < 0.

Theorem 1.3 Under the above notation, the sequence (λ 1,n ) n∈N is nondecreasing and there holds λ 1,n → λ 1,D as n → +∞. Furthermore, if λ 1,D < 0, then the sequence (p n ) n∈N of minimal solutions of (1.7) with nonlinearities f n is nonincreasing and

p n (x) → p ∞ (x) as n → +∞ for all x ∈ R N ,
where, up to a negligible set,

p ∞ is nonnegative, periodic in R N , p ∞ = 0 in R N \Ω, the restriction of p ∞ on Ω is of class C 2,α (Ω) and solves -∇ • (A(x)∇p ∞ ) = F (x, p ∞ ) in Ω, p ∞ = 0 on ∂Ω. (1.18)
Lastly, p ∞ ≥ p in Ω, where p is given in Theorem 1.2.

We point out that, in general, the function p ∞ is not identically equal to the solution p of (1.13) in Ω. However, it is well equal to p in Ω if F fulfills (1.4) in Ω. We refer to Remark 3.1 for more details.

The last result is concerned with the asymptotic behavior as n → +∞ of the pulsating travelling fronts of the type φ n (x • ect, x) connecting 0 to p n (for problem (1.1) in R N with nonlinearities f n ) and of their minimal speeds c * n (e) > 0 in any direction e (when λ 1,n < 0). The limit shall depend strongly on the direction e and blocking phenomena may occur in general.

Theorem 1.4 Assume that λ 1,D < 0 and let e be any given unit vector of R N .

a) The sequence (c * n (e)) n∈N is nonincreasing with limit c * (e) ≥ 0. If all connected components C of Ω are bounded in the direction e in the sense that

sup x∈C |x • e| < +∞, (1.19)
n (t, x) = φ n (x • e -c n t, x) for (1.1) in R N with nonlinearity f n satisfy u n (t, x) → u(t, x) in C 1 t and C 2 x locally in R × Ω, 0 in L 1 loc (R × (R N \Ω))
up to extraction of a subsequence, where u(t, x) = φ(x•e-ct, x) is a classical solution of (1.8) with u t ≥ 0 in R×Ω and φ(s, •) is periodic in Ω for all s ∈ R. Moreover, for any given i ∈ I -, one can shift in time the functions u n so that u(-∞, •) = 0 and u(+∞, •) > 0 in Ω i . c) Assume here that A is constant. If there exist a unit vector e ′ = ±e and two real numbers a < b such that

Ω ⊃ x ∈ R N , a < x • e ′ < b , (1.20)
then c * (e) > 0. If there exist a unit vector e ′ , a point x 0 ∈ R N and a real number r > 0 such that e ′ is an eigenvector of A with e ′ • e = 0, and

Ω ⊃ x ∈ R N , d(x, x 0 + Re ′ ) < r , (1.21)
where d denotes the Euclidean distance, then c * (e) > 0.

Theorem 1.4 provides a simple geometrical condition for the blocking of propagation, in a given direction e, in the presence of hostile periodic patches (by blocking, we mean that c * n (e) → 0 as n → +∞). Consequently, some quantitative estimates of the spreading speeds of the solutions u of the Cauchy problems (1.5) with nonlinearities f n can be derived. Indeed, for any compactly supported function u 0 ≡ 0, the solution u of (1.5) with nonlinearity f n spreads in the direction e with the spreading speed

w * n (e) = min ξ∈S N-1 , ξ•e>0 c * n (e) ξ • e ,
in the sense that lim inf t→+∞ u(t, c t e + x) ≥ p n (x) locally uniformly in x if 0 ≤ c < w * n (e), whereas lim t→+∞ u(t, c t e + x) = 0 locally uniformly in x if c > w * n (e) (see [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive media[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF]). In particular, 0 < w * n (e) ≤ c * n (e). Hence, under condition (1.19), c * (e) = c * (-e) = 0 and the solution u of (1.5) with nonlinearity f n spreads as slowly as wanted in the directions ±e when n is large enough. In this case, since all connected components of Ω are bounded in the direction e, pulsating fronts in the directions ±e for problem (1.8) in Ω make no sense even if, under the notation of part b), the solutions u n can be shifted to converge to a non-trivial solution u of (1.8) in R × Ω: what happens is that, in each connected component of Ω i , u is just a time connection between 0 and a non-trivial steady state.

On the other hand, Theorem 1.4 also gives some simple geometrical conditions, of the types (1.20) or (1.21), for non-blocking in the directions ±e. These conditions mean that Ω contains a slab which is not orthogonal to e, or contains a cylinder in a direction which is not orthogonal to e. We do not know however if these conditions are optimal, even when A is constant. Lastly, Theorem 1.4 shows the existence of pulsating fronts for problem (1.8) in Ω. Assume for instance that Ω is connected, that is m = 1 under notation (1.10). Then, there are pulsating traveling fronts, in the usual sense, in the direction e, connecting 0 to a non-trivial periodic stationary solution of (1.8). Furthermore, if F is of the KPP type (1.4) in Ω, the limiting state is unique and is then equal to the function p = p ∞ given in Theorems 1.2 and 1.3 (see Remark 3.1 below and the end of the proof of Theorem 1.4). However, Theorem 1.4 holds for general monostable functions F which may not be of the KPP type and it gives the first result about the existence of pulsating fronts with zero Dirichlet boundary condition in periodic domains (which may not be cylinders).

Outline of the paper. Section 2 is devoted to the proof of Proposition 1.1 and Theorem 1.2 about the existence of minimal non-trivial stationary solutions p of problems (1.7) and (1.13) respectively, and about the large-time behavior of the solutions u of the Cauchy problems (1.5) and (1.14). Section 3 is concerned with the proof of Theorem 1.3 and the relationship between the minimal solutions p n of problems (1.7) with nonlinearities f n and the minimal solution p of problem (1.13). Lastly, in Section 4, we do the proof of Theorem 1.4 and make clear the role of the geometrical condition (1.19) in the blocking process as n → +∞.

2 Minimal stationary solutions and large-time behavior for the Cauchy problems (1.5) and (1.14) In the first part of this section, we first deal with the elliptic and parabolic problems (1.7) and (1.5) set in the whole space R N with the assumption (1.3) on the nonlinearity f . Namely, we do the proof of Proposition 1.1. It is based on the elliptic and parabolic maximum principles and on the construction of suitable subsolutions. Since some parts of the proof are quite similar to some arguments used in [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF] and [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF], they will only be sketched. In the second part of this section, we will be concerned with the stationary and Cauchy problems (1.13) and (1.14) posed in the set Ω with zero Dirichlet boundary condition. That is, we will do the proof of Theorem 1.2, which will itself be inspired by that of Proposition 1.1, but additional difficulties arise.

Proof of Proposition 1.1. Let ϕ be the unique periodic solution of (1.6) such that max R N ϕ = 1. Since the principal periodic eigenvalue λ 1 of (1.6) is assumed to be negative, one can fix

ε 0 ∈ (0, M] so that f (x, s) ≥ ζ(x) s + (λ 1 /2)s for all (x, s) ∈ R N × [0, ε 0 ]. Now, for any ε ∈ (0, ε 0 ], there holds -∇ • (A(x)∇(εϕ)) -f (x, εϕ) ≤ -ε∇ • (A(x)∇ϕ) -ζ(x)εϕ - λ 1 2 εϕ = λ 1 2 εϕ < 0 (2.1)
for all x ∈ R N . In other words, the functions ε ϕ are strict subsolutions of (1.7) for all ε ∈ (0, ε 0 ]. Let now U be the solution of the Cauchy problem (1.5) with initial datum

U 0 = ε 0 ϕ. Since 0 < U 0 ≤ M and f (•, M) ≤ 0 in R N and since U 0 is a subsolution of (1.7), it follows that ε 0 ϕ(x) ≤ U(t, x) ≤ M for all (t, x) ∈ R + × R N
and that U is nondecreasing with respect to t. Furthermore, by uniqueness for the Cauchy problem (1.5), U(t, •) is periodic in R N for each t ≥ 0. From standard parabolic estimates, it follows then that

U(t, x) → p(x) as t → +∞ uniformly with respect to x ∈ R N ,
where p is a C 2,α (R N ) periodic solution of (1.7) such that 0

< ε 0 ϕ = U 0 ≤ p ≤ M.
Let us then show that p is the minimal positive solution of (1.7) (in the class of all positive solutions of (1.7), which are not a priori assumed to be periodic). Let q be any positive solution of (1.7). Let λ 1,B(y,R),D denote the principal eigenvalue of the operator

-∇ • (A(x)∇) -ζ(x)
in the open Euclidean ball B(y, R) of center y ∈ R N and radius R > 0, with zero Dirichlet boundary condition on ∂B(y, R). For each point y ∈ R N and R > 0, the principal eigenvalue λ 1,B(y,R),D is characterized by the existence of a function ϕ y,R of class

C 2,α (B(y, R)), solving      -∇ • (A(x)∇ϕ y,R ) -ζ(x)ϕ y,R = λ 1,B(y,R),D ϕ y,R in B(y, R), ϕ y,R > 0 in B(y, R), ϕ y,R = 0 on ∂B(y, R).
Up to normalization, one can assume that max B(y,R) ϕ y,R = 1, and the functions ϕ y,R are then unique. As done in [START_REF] Berestycki | Liouville type results for semilinear elliptic equations in unbounded domains[END_REF], there holds

λ 1,B(y,R),D → λ 1 as R → +∞,
uniformly with respect to y ∈ R N . Since λ 1 < 0, one can then fix R > 0 large enough so that λ 1,B(y,R),D < λ 1 /2 for all y ∈ R N . Thus, for each y ∈ R N and ε ∈ (0, ε 0 ], the function εϕ y,R satisfies

-∇ • (A(x)∇(εϕ y,R )) -f (x, εϕ y,R ) ≤ -ε∇ • (A(x)∇ϕ y,R ) -ζ(x)εϕ y,R - λ 1 2 εϕ y,R = λ 1,B(y,R),D - λ 1 2 εϕ y,R < 0 (2.2)
in B(y, R). In other words, the functions εϕ y,R are strict subsolutions of (1.7) in the balls B(y, R) for all ε ∈ (0, ε 0 ]. Now, fix y ∈ R N and observe that min B(y,R) q > 0 by continuity of q. It follows then that

ε * y := sup ε ∈ (0, ε 0 ], εϕ y,R ≤ q in B(y, R)
is positive. We shall prove that ε * y = ε 0 . Assume not. Then 0 < ε * y < ε 0 and ε * y ϕ y,R ≤ q in B(y, R) with equality somewhere in B(y, R). Since q > 0 and ϕ y,R = 0 on ∂B(y, R), the functions ε * y ϕ y,R and q are equal somewhere at an interior point, in B(y, R). But ε * y ϕ y,R is a subsolution of (1.7), from (2.2). Since f is (at least) Lipschitz-continuous locally with respect to the second variable, uniformly in x, it follows from the strong elliptic maximum principle that ε * y ϕ y,R = q in B(y, R), which is impossible since the inequality (2.2) is strict. Therefore, ε * y = ε 0 for all y ∈ R N and, in particular, q(y) ≥ ε 0 ϕ y,R (y) for all y ∈ R N .

But, by uniqueness of the principal eigenfunctions ϕ y,R and by periodicity of A and ζ, the function y → ϕ y,R (y) is continuous and periodic in R N . Since it is positive, one gets that min y∈R N ϕ y,R (y) > 0. Therefore, inf R N q > 0. Define now

ε * = sup ε ∈ (0, ε 0 ], εϕ ≤ q in R N ,
where we recall that ϕ is the unique periodic solution of (1.6) such that max R N ϕ = 1. Since q is bounded from below in the whole space R N by a positive constant and since ϕ is bounded, one has ε * > 0. Assume that ε * < ε 0 . Then ε * ϕ ≤ q in R N and there exists a sequence (x k ) k∈N in R N such that

ε * ϕ(x k ) -q(x k ) → 0 as k → +∞.
By writing

x k = x ′ k + x ′′ k with x ′ k ∈ L 1 Z × • • • L N Z and x ′′ k ∈ [0, L 1 ] × • • • × [0, L N ]
, it follows that the functions q k (x) = q(x + x ′ k ) converge, up to extraction of a subsequence, to a solution q ∞ of (1.7) such that ε * ϕ ≤ q ∞ in R N with equality somewhere in R N . As above, one concludes that ε * ϕ = q ∞ in R N , which is impossible since ε * ϕ is a strict subsolution of (1.7), from (2.1). Therefore, ε * = ε 0 , whence ε 0 ϕ ≤ q in R N . The parabolic maximum principle implies that

U(t, x) ≤ q(x) for all (t, x) ∈ R + × R N ,
where we recall that U denotes the solution of (1.5) with initial datum ε 0 ϕ. By passing to the limit as t → +∞, one gets that p(x) ≤ q(x) for all x ∈ R N .

Finally, let u 0 : R N → [0, M] be a uniformly continuous function which is not identically equal to 0, and let u denote the solution of (1.5) with initial datum u 0 . The maximum principle implies that 0 ≤ u(t, x) ≤ M for all (t, x) ∈ R + × R N , and u(t, x) > 0 for all t > 0 and x ∈ R N . With the same notation as above, there exists then ε ∈ (0, ε 0 ] such that

ε ϕ 0,R ≤ u(1, •) in B(0, R),
where we recall that R > 0 was chosen so that λ 1,B(y,R),D < λ 1 /2 for all y ∈ R N . Let v be the solution of (1.5) with initial datum

v 0 (x) = ε ϕ 0,R (x) if x ∈ B(0, R), 0 if x ∈ R N \B(0, R). Since 0 ≤ v 0 ≤ u(1, •) ≤ M in R N , there holds 0 ≤ v(t, x) ≤ u(t + 1, x) ≤ M for all (t, x) ∈ R + × R N .
Furthermore, since v 0 is a subsolution of (1.7) because of (2.2) and v 0 = 0 in R N \B(0, R), it follows from the maximum principle that v is nondecreasing with respect to t. Hence, from standard parabolic estimates, one gets that

v(t, x) → v ∞ (x) as t → +∞ locally uniformly in x ∈ R N , where v ∞ is a solution of (1.7) satisfying v 0 ≤ v ∞ ≤ M in R N .
Notice in particular that v ∞ is positive in R N from the strong maximum principle, since v 0 is nonnegative and not identically equal to 0. But the previous paragraphs yield then v ∞ ≥ p. Therefore,

lim inf t→+∞ u(t, x) ≥ p(x) locally uniformly in x ∈ R N . Lastly, if u 0 ≤ p, then u(t, x) ≤ p(x) for all (t, x) ∈ R + × R N , whence u(t, x) → p(x) as t → +∞ locally uniformly in x ∈ R N .
The proof of Proposition 1.1 is thereby complete.

Let us now turn to the proof of Theorem 1.2. Some of the ideas of the proof of Proposition 1.1 can be adapted. However, the case of problems (1.13) and (1.14) in Ω is substantially more involved than the case of the whole space R N , mainly due to the fact that zero Dirichlet boundary condition is imposed on ∂Ω and the connected components of Ω may be bounded or unbounded.

Proof of Theorem 1.2. Remember that the sets Ω i given in (1.10) are all periodic and pairwise disjoint. We first work in each set Ω i for which λ 1,Ω i ,D < 0, that is i ∈ I -. We claim that, for each such index i ∈ I -, there exists a periodic solution

p i ∈ C 2,α (Ω i ) of the stationary problem      -∇ • (A(x)∇ p i ) = F (x, p i (x)) in Ω i , p i = 0 on ∂Ω i , p i > 0 in Ω i .
(2.3) Indeed, let ϕ i be the principal periodic eigenfunction of the operator

-∇ • (A(x)∇) -ζ(x)
in Ω i with zero Dirichlet boundary condition on ∂Ω i . That is, the function ϕ i is periodic, of class C 2,α (Ω i ), and it solves

     -∇ • (A(x)∇ ϕ i ) -ζ(x) ϕ i = λ 1,Ω i ,D ϕ i in Ω i , ϕ i = 0 on ∂Ω i , ϕ i > 0 in Ω i .
(2.4)

Up to normalization, one can assume that max Ω i ϕ i = 1. Now, as in the proof of Proposition 1.1, since λ 1,Ω i ,D < 0, there exists ε 0 ∈ (0, M] such that, for any ε ∈ (0, ε 0 ], the function ε ϕ i is a strict subsolution of (2.3), namely

-∇ • (A(x)∇(ε ϕ i )) -F (x, ε ϕ i (x)) < 0 in Ω i , (2.5)
together with ε ϕ i = 0 on ∂Ω i and ε ϕ i > 0 in Ω i . But since the constant M is a supersolution of this problem, the solution u i of the Cauchy problem

     (u i ) t -∇ • (A(x)∇u i ) = F (x, u i ), t > 0, x ∈ Ω i , u i (t, x) = 0, t > 0, x ∈ ∂Ω i , u i (0, x) = ε 0 ϕ i (x), x ∈ Ω i , (2.6)
is such that ε 0 ϕ i (x) ≤ u i (t, x) ≤ M for all (t, x) ∈ (0, +∞) × Ω i and u i is nondecreasing in t and periodic in x in Ω i . Therefore, there exists a periodic C 2,α (Ω i ) solution p i of (2.3) such that u i (t, x) → p i (x) as t → +∞, uniformly in x ∈ Ω i .

Let now q i be any classical bounded solution of (2.3) and let us prove that q i ≥ p i in Ω i . By definition of Ω i , the set ω i is one of its connected components, and any of its connected components is of the type

ω i + k for some k ∈ L 1 Z × • • • × L N Z.
Two cases may then occur: either ω i is bounded, or ω i is unbounded.

Case 1. Consider first the case when ω i is bounded. Since q i > 0 in ω i (⊂ Ω i ), q i = 0 on ∂ω i (⊂ ∂Ω i ) and F (•, 0) ≡ 0, it follows from Hopf lemma and the compactness of ∂ω i that max

x∈∂ω i ∂ q i ∂ν (x) < 0,
where ν denotes the outward unit normal on ∂Ω. On the other hand, the principal eigenfunction ϕ i of (2.4) is (at least) of class C 1 (ω i ) and ϕ i = 0 on ∂ω i . Hence, the quantity

ε * := sup ε ∈ (0, ε 0 ], ε ϕ i ≤ q i in ω i
is a positive real number, belonging to the interval (0, ε 0 ]. Furthermore, ε * ϕ i ≤ q i in ω i . Since ε * ϕ i is a strict subsolution in ω i ⊂ Ω i , in the sense of (2.5), the strong maximum principle and the Hopf lemma imply that ε * ϕ i < q i in ω i and

∂ q i ∂ν < ε * ∂ ϕ i ∂ν on ∂ω i .
Therefore, there exists η 0 > 0 such that (ε

* + η) ϕ i ≤ q i in ω i for all η ∈ [0, η 0 ]. The definition of ε * then yields ε * = ε 0 , whence ε 0 ϕ i ≤ q i in ω i . The same argument can be repeated in ω i +k for all k ∈ L 1 Z × • • • × L N Z. Therefore, ε 0 ϕ i ≤ q i in Ω i .
By comparing q i with the solution u i of the Cauchy problem (2.6), it follows then as in the proof of Proposition 1.1 that

p i ≤ q i in Ω i . (2.7)
Case 2. Consider now the case when ω i is unbounded. For all y ∈ ω i and R > 0, define

ω i,y,R = z ∈ ω i , d Ω (y, z) < R ,
where d Ω denotes the geodesic distance inside Ω, and set λ 1,ω i,y,R ,D = min

φ∈H 1 0 (ω i,y,R )\{0} ω i,y,R A∇φ • ∇φ -ζφ 2 ω i,y,R φ 2 .
(2.8)

Actually, λ 1,ω i,y,R ,D is the smallest eigenvalue of the operator -∇ • (A∇)ζ in ω i,y,R with zero Dirichlet boundary condition (that is, in the H 1 0 (ω i,y,R ) sense), but, since ∂ω i,y,R may not be smooth in general, the eigenvalue λ 1,ω i,y,R ,D may not be associated with C 1 (ω i,y,R ) eigenfunctions. We first claim that lim sup

R→+∞ sup y∈ω i λ 1,ω i,y,R ,D < 0.
To do so, let ρ : R → [0, 1] be a C ∞ (R) function such that ρ = 1 on (-∞, -1] and ρ = 0 on [0, +∞) and, for all y ∈ ω i and R > 0, denote

ρ y,R (x) = ρ d Ω (x, y) -R for all x ∈ ω i .
These functions ρ y,R are then is W 1,∞ (ω i ). For every y ∈ ω i and R > 1, the restriction of the function ϕ i ρ y,R to ω i,y,R belongs to H 1 0 (ω i,y,R )\{0}, whence

λ 1,ω i,y,R ,D ≤ ω i,y,R A∇( ϕ i ρ y,R ) • ∇( ϕ i ρ y,R ) -ζ( ϕ i ρ y,R ) 2 ω i,y,R ( ϕ i ρ y,R ) 2 ≤ ω i,y,R ρ y,R A∇ ϕ i • ∇( ϕ i ρ y,R ) -ζ( ϕ i ρ y,R ) 2 ω i,y,R ( ϕ i ρ y,R ) 2 + M |ω i,y,R \ω i,y,R-1 | ω i,y,R ( ϕ i ρ y,R ) 2
,

where M = (1 + ∇ ϕ i L ∞ (Ω i ) ) × max x∈Ω, |ξ|=1, |ξ ′ |=1 (A(x)ξ • ξ ′ )
is a positive constant which does not depend on y or R, and |ω i,y,R \ω i,y,R-1 | denotes the Lebesgue measure of ω i,y,R \ω i,y,R-1 . By integrating by parts, it follows then from (2.4) that

λ 1,ω i,y,R ,D ≤ λ 1,Ω i ,D + 2 M |ω i,y,R \ω i,y,R-1 | ω i,y,R ( ϕ i ρ y,R ) 2 .
Since ϕ i is periodic and positive in Ω i (and then uniformly away from 0 in each non-empty set of the type ω δ i := x ∈ ω i , d(x, ∂ω i ) > δ with δ > 0) and since Ω (and hence ω i ) has a smooth boundary, it follows that lim inf

R→+∞ inf y∈ω i |ω i,y,R-1 | -1 ω i,y,R ( ϕ i ρ y,R ) 2 > 0, while lim sup R→+∞ sup y∈ω i |ω i,y,R-1 | -1 |ω i,y,R \ω i,y,R-1 | = 0. Remember that λ 1,Ω i ,D < 0. Therefore, there exists R 0 > 1 such that ∀ R ≥ R 0 , ∀ y ∈ ω i , λ 1,ω i,y,R ,D < λ 1,Ω i ,D 2 .
(2.9)

Let now δ > 0 be any positive constant such that ω δ i = ∅ and let us show that inf ω δ i q i > 0. Assume not and let

ε i > 0 be such that F (x, s) ≥ ζ(x)s+(λ 1,Ω i ,D /2)s for all (x, s) ∈ Ω×[0, ε i ].
There is then a sequence (x n ) n∈N in ω δ i such that q i (x n ) → 0 as n → +∞. Since q i ≥ 0 in ω i and q i = 0 on ∂ω i , it follows from Harnack inequality that max ω i,xn 0 ,R 0 q i ≤ ε i for some n 0 ∈ N large enough.

In particular,

-∇ • (A(x)∇ q i ) -ζ(x) q i ≥ λ 1,Ω i ,D 2 q i in ω i,xn 0 ,R 0 . (2.10)
On the other hand, from (2.8) and (2.9), and owing to the definition of

H 1 0 (ω i,xn 0 ,R 0 ), there is φ ∈ C 1 c (ω i,xn 0 ,R 0 )\{0} (with a compact support which is included in ω i,xn 0 ,R 0 ) such that R[φ] := ω i,xn 0 ,R 0 A∇φ • ∇φ -ζφ 2 ω i,xn 0 ,R 0 φ 2 < λ 1,Ω i ,D 2 .
Now, let ω ′ be any bounded open set of class C 2,α , containing the support of φ, and such that ω

′ ⊂ ω i,xn 0 ,R 0 . It follows that λ 1,ω ′ ,D ≤ R[φ] < λ 1,Ω i ,D /2.
There is then a nonnegative and nontrivial function ϕ ′ ∈ C 2,α (ω ′ ) solving

-∇ • (A(x)∇ϕ ′ ) -ζ(x)ϕ ′ = λ 1,ω ′ ,D ϕ ′ ≤ λ 1,Ω i ,D 2 ϕ ′ in ω ′ (2.11)
with ϕ ′ = 0 on ∂ω ′ . Notice that ϕ ′ may not be positive in ω ′ since ω ′ may not be connected. But ϕ ′ is positive at least in one connected component ω ′′ of ω ′ . Since min ω ′′ q i > 0 and ϕ ′ = 0 on ∂ω ′′ , it follows from (2.10), (2.11) and the strong maximum principle that εϕ ′ ≤ q i in ω ′′ for all ε > 0, which is clearly impossible. One has then reached a contradiction. Hence there holds inf

ω δ i q i > 0 for all δ > 0 such that ω δ i = ∅. (2.12)
It follows then from (2.12), together with the Hopf lemma and the global smoothness of ∂ω i , that sup ∂ω i ∂ q i ∂ν < 0. Therefore, the quantity ε * := sup ε ∈ (0, ε 0 ], ε ϕ i ≤ q i in ω i is a positive real number. From (2.5) and the strong maximum principle, there holds ε * ϕ i < q i in ω i . Furthermore, we claim that inf

ω δ i ( q i -ε * ϕ i ) > 0 for all δ > 0 such that ω δ i = ∅. (2.13)
Assume not. Then there exist δ > 0 such that ω δ i = ∅ and a sequence (

y n ) n∈N in ω δ i such that q i (y n ) -ε * ϕ i (y n ) → 0 as n → +∞. Write y n = y ′ n + y ′′ n where y ′ n ∈ L 1 Z × • • • × L N Z and y ′′ n ∈ C. Notice in particular that d(y ′′ n , ∂Ω) = d(y ′′ n , ∂Ω i ) > δ. Up to extraction of a subsequence, one can assume that y ′′ n → y ∞ ∈ Ω i as n → +∞ with d(y ∞ , ∂Ω) = d(y ∞ , ∂Ω i ) ≥ δ,
and that the functions

x → q i (x + y ′ n ) defined in Ω i converge in C 2 loc (Ω i ) to a solution q i of -∇ • (A(x)∇q i ) = F (x, q i (x)) in Ω i such that q i ≥ ε * ϕ i in B(y ∞ , δ)
⊂ Ω i with equality at y ∞ . The strong maximum principle and (2.5) lead to a contradiction. Thus, the claim (2.13) holds. As above, it follows then from Hopf lemma and the global smoothness of ∂ω i that sup ∂ω i ∂( q i -ε * ϕ i ) ∂ν < 0 and that there exists η 0 > 0 such that (ε * + η) ϕ i ≤ q i in ω i for all η ∈ [0, η 0 ]. Therefore, ε * = ε 0 , whence ε 0 ϕ i ≤ q i in ω i and then in Ω i by repeating the argument in ω i + k for all k in L 1 Zו • •×L N Z. Finally, by comparing q i with the solution u i of the Cauchy problem (2.6), the conclusion (2.7) follows.

Conclusion of the proof. Define the function p in Ω by

p = p i in all the sets Ω i with i ∈ I -, 0 in all the sets Ω i with i ∈ I -.

The function p is periodic, of class C 2,α (Ω), and it solves (1.13). Furthermore, it follows from the previous steps that any bounded solution q of (1.13) is such that q ≥ p in Ω. Lastly, let u 0 : Ω → [0, M] be any uniformly continuous function such that u 0 ≡ 0 in Ω, let u be the solution of the Cauchy problem (1.14) and let ω be a connected component of Ω intersecting the support of u 0 . We shall prove that lim inf

t→+∞ min x∈K (u(t, x) -p(x)) ≥ 0 (2.14)
for any compact set K ⊂ ω. Since 0 ≤ u(t, •) (≤ M) in Ω for all t > 0 and p = 0 in Ω i for all i ∈ I i , it is sufficient to consider the case when ω = ω i + k for some i ∈ I -and some

k ∈ L 1 Z × • • • × L N Z. If ω is bounded, then u(1, •) > 0 in ω and max ∂ω ∂u(1,•) ∂ν
< 0 from the strong parabolic maximum principle. Therefore, u(1, •) ≥ ε ϕ i in ω for some ε ∈ (0, ε 0 ] and

u(t + 1, x) ≥ v(t, x) for all (t, x) ∈ (0, +∞) × ω,
where v is the solution of the Cauchy problem (2.6) in ω with initial datum ε ϕ i in ω and zero Dirichlet boundary condition on ∂ω. Owing to (2.5), v(t, x) is increasing with respect to t (and bounded from above by the constant M), and it converges as t → +∞ uniformly in ω to a solution w of (2.3) in ω such that w ≥ ε ϕ i in ω (whence w > 0 in ω) and w = 0 on ∂ω. It follows as in the study of case 1 above that w ≥ p in ω, which yields (2.14).

Consider now the case when ω is unbounded. Without loss of generality, up to a translation of the origin, one can assume that k = 0 and ω = ω i . Choose any point y 0 in ω and, from (2.9), let R 0 > 0 be such that λ 1,ω i,y 0 ,R 0 ,D < 0. As above, there is then a function

φ ∈ C 1 c (ω i,y 0 ,R 0 )\{0} such that R ′ [φ] := ω i,y 0 ,R 0 A∇φ • ∇φ -ζφ 2 ω i,y 0 ,R 0 φ 2 < 0 and, if ω ′ is any bounded open set of class C 2,α containing the support of φ and such that ω ′ ⊂ ω i,y 0 ,R 0 ⊂ ω, there holds λ 1,ω ′ ,D ≤ R ′ [φ] < 0.
There is then a nonnegative and nontrivial function

ϕ ′ ∈ C 2,α (ω ′ ) such that -∇ • (A(x)∇ϕ ′ ) -ζ(x)ϕ ′ = λ 1,ω ′ ,D ϕ ′ in ω ′
with ϕ ′ = 0 on ∂ω ′ . Therefore, the function ε ′ ϕ ′ is a subsolution of (2.3) in ω ′ for ε ′ > 0 small enough and one can also assume without loss of generality that ε ′ ϕ ′ ≤ u(1, •) in the compact set ω ′ ⊂ ω. Thus, there holds u(t + 1, x) ≥ v(t, x) for all (t, x) ∈ (0, +∞) × ω, where v is the solution of the Cauchy problem (2.6) in ω with initial datum v 0 = ε ′ ϕ ′ in ω ′ and v 0 = 0 in ω\ω ′ , and zero Dirichlet boundary condition on ∂ω. But v(t, x) is increasing with respect to t and bounded from above by M. It converges locally uniformly in ω to a solution w of (2.3) in ω, such that w ≥ v 0 in ω (whence w > 0 in ω from the strong maximum principle). One concludes as in case 2 above that w ≥ p in ω, which leads to (2.14). Lastly, observe that, if u 0 ≤ p in Ω, then u(t, •) ≤ p in Ω for all t > 0. Hence, (2.14) implies that u(t, x) → p(x) as t → +∞ uniformly in any compact subset K ⊂ ω, where ω is any connected component of Ω intersecting the support of u 0 . The proof of Theorem 1.2 is thereby complete.

3 Relationship between the problems (1.7) with nonlinearities f n and the problem (1.18) This section is devoted to the proof of Theorem 1.3. By using variational arguments, H 1 a priori estimates and Rellich's theorem, we prove the monotonicity and the convergence of the principal periodic eigenvalues of the linearized operators in R N associated with the functions f n , to that of problem (1.11) with zero Dirichlet boundary condition on ∂Ω. Then, we show the monotonicity and the convergence of the functions p n to a solution p ∞ ≥ p of (1.18). The minimality of each solution p n and of p will also be used.

Proof of Theorem 1.3. Let, for each n ∈ N, λ 1,n and ϕ n be the principal eigenvalue and periodic eigenfunction solving (1.17). Let λ 1,D and ϕ solve (1.11), where one can always assume that ϕ > 0 in each Ω i with λ 1,Ω i ,D = λ 1,D , that is i ∈ I min . Call

H 1 per (R N ) = φ ∈ H 1 loc (R N ), φ is periodic , L 2 per (R N ) = φ ∈ L 2 loc (R N ), φ is periodic and C 0 = [0, L 1 ] × • • • × [0, L N ].
For each n ∈ N, there holds

λ 1,n = min φ∈H 1 per (R N )\{0} R n [φ] = R n [ϕ n ],
where

R n [φ] = C 0 A∇φ • ∇φ -ζ n φ 2 C 0 φ 2 .
Since the sequence (ζ n (x)) n∈N is nonincreasing for each x ∈ R N , it follows that the sequence (λ 1,n ) n∈N is nondecreasing. We now claim that λ 1,n < λ 1,D for each n ∈ N. The proof is based on some standard comparison arguments, used 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]. We just sketch it here for the sake of completeness.

Assume that λ 1,n ≥ λ 1,D for some n ∈ N. Pick any index i ∈ I min . Since ζ = ζ n in Ω i ⊂ Ω, there holds -∇ • (A(x)∇ϕ n ) -ζ(x)ϕ n = λ 1,n ϕ n ≥ λ 1,D ϕ n in Ω i
and min Ω i ϕ n > 0. In other words, the periodic function ϕ n is a supersolution of the linear equation satisfied by the periodic function ϕ in Ω i . Since ϕ = 0 on ∂Ω i and ϕ is (at least) of class C 2 (Ω i ), it follows from the strong elliptic maximum principle that the quantity

ε * = sup ε ∈ (0, +∞), εϕ ≤ ϕ n in Ω i
is actually equal to +∞. This is a contradiction since ϕ is positive in Ω i . Therefore, λ 1,n < λ 1,D for all n ∈ N.

As a consequence, the sequence (λ 1,n ) n∈N converges monotonically to a real number λ 1,∞ such that λ 1,∞ ≤ λ 1,D . Let us now show that λ 1,∞ = λ 1,D . Normalize here the eigenfunctions ϕ n so that ϕ n L 2 (C 0 ) = 1. It follows that

C 0 A∇ϕ n • ∇ϕ n = λ 1,n + C 0 ζ n ϕ 2 n ≤ λ 1,∞ + C 0 ζ 0 ϕ 2 n ≤ λ 1,∞ + max R N ζ 0 .
Thus, the sequence (ϕ n ) n∈N is bounded in H 1 (C 0 ). There exists then a function

ϕ ∞ in H 1 per (R N ) such that, up to extraction of a subsequence, ϕ n → ϕ ∞ weakly in H 1 per (R N ) and strongly in L 2 per (R N ). In particular, ϕ ∞ ≥ 0 a.e. in R N and ϕ ∞ L 2 (C 0 ) = 1. Let K be any compact set such that K ⊂ (R N \Ω) ∩ C 0 . For all n ∈ N, one has -max K ζ n K ϕ 2 n ≤ - K ζ n ϕ 2 n = λ 1,n - C 0 A∇ϕ n • ∇ϕ n + C 0 \K ζ n ϕ 2 n ≤ λ 1,∞ + C 0 \K ζ 0 ϕ 2 n ≤ λ 1,∞ + sup C 0 \K |ζ 0 |,
whence ϕ n L 2 (K) → 0 as n → +∞ from (1.16). Thus, ϕ ∞ = 0 a.e. in K, and then a.e. in R N \Ω and ϕ ∞ L 2 (Ω∩C 0 ) = 1. Furthermore, since ϕ ∞ ∈ H 1 per (R N ), one gets that the restriction of ϕ ∞ to Ω belongs to H 1 0,per (Ω), that is the space of periodic H 1 loc (Ω) functions whose trace is equal to 0 on ∂Ω. Lastly, observe that

Ω∩C 0 A∇ϕ n • ∇ϕ n ≤ C 0 A∇ϕ n • ∇ϕ n = λ 1,n + C 0 ζ n ϕ 2 n ≤ λ 1,∞ + C 0 ζ 0 ϕ 2 n , while C 0 ζ 0 ϕ 2 n → C 0 ζ 0 ϕ 2 ∞ = Ω∩C 0 ζϕ 2 ∞ as n → +∞. Therefore, Ω∩C 0 A∇ϕ ∞ • ∇ϕ ∞ ≤ lim inf n→+∞ Ω∩C 0 A∇ϕ n • ∇ϕ n ≤ λ 1,∞ + Ω∩C 0 ζϕ 2 ∞ , that is R ∞ [ϕ ∞ ] ≤ λ 1,∞ ≤ λ 1,D , where the functional R ∞ is defined by R ∞ [φ] = Ω∩C 0 A∇φ • ∇φ -ζφ 2 Ω∩C 0 φ 2 for all φ ∈ H 1 0,per (Ω)\{0}. But since min φ∈H 1 0,per (Ω)\{0} R ∞ [φ] = λ 1,D , one concludes that λ 1,∞ = λ 1,D . In other words, λ 1,n → λ 1,D as n → +∞.
In the sequel, assume now that λ 1,D < 0. Consequently, for each n ∈ N, one has λ 1,n < λ 1,D < 0 and, from Proposition 1.1, there exists a minimal periodic solution p n of

-∇ • (A(x)∇p n ) = f n (x, p n ) in R N , 0 < p n ≤ M in R N .
Fix any two integers n ≤ m. Since

-∇ • (A(x)∇p n ) -f m (x, p n ) = f n (x, p n ) -f m (x, p n ) ≥ 0 in R N ,
the function p n is a supersolution for the equation satisfied by p m . From the proof of Proposition 1.1, there exists ε m > 0 such that all functions εϕ m with ε ∈ (0, ε m ] are subsolutions of (1.7) with the nonlinearity f m . Since min R N p n > 0, there exists ε ∈ (0, ε m ] such that εϕ m ≤ p n in R N . Hence, the maximum principle implies that

v(t, x) ≤ p n (x) for all (t, x) ∈ R + × R N ,
where v is the solution of the Cauchy problem (1.5) with the nonlinearity f m and initial datum v 0 = εϕ m . But v is nondecreasing in t and converges as t → +∞ to a solution q of (1.7) with nonlinearity f m , such that 0 < q ≤ p n in R N . By minimality of p m (from Proposition 1.1), one gets that p m ≤ q in R N , whence

p m ≤ p n in R N .
In other words, the sequence of functions (p n ) n∈N is nonincreasing and then converges pointwise to a periodic function p ∞ (x) ranging in [0, M].

Let us now show that p ∞ = 0 in R N \Ω. By multiplying by p n the equation (1.7) with the nonlinearity f n , that is -∇ • (A(x)∇p n ) = f n (x, p n ), and by integrating by parts over the cell C 0 , it follows that

C 0 A∇p n • ∇p n = C 0 f n (x, p n ) p n ≤ C 0 f 0 (x, p n ) p n ≤ M × max R N ×[0,M ] |f 0 |, whence the sequence (p n ) n∈N is bounded in H 1 per (R N ). Since it converges monotonically to p ∞ , one infers that p ∞ ∈ H 1 per (R N ) and p n → p ∞ as n → +∞ weakly in H 1 per (R N ) and strongly in L 2 per (R N ). For any compact set K such that K ⊂ (R N \Ω) ∩ C 0 , one has -max K×[0,M ] g n K p 2 n ≤ - K g n (x, p n ) p 2 n = - K f n (x, p n )p n = - C 0 A∇p n • ∇p n + C 0 \K f n (x, p n ) p n ≤ C 0 \K f 0 (x, p n ) p n ≤ M × max R N ×[0,M ] |f 0 |.
The assumption (1.16) yields max K×[0,M ] g n → -∞ as n → +∞, whence p ∞ = 0 a.e. in any such compact K. Finally, p ∞ = 0 a.e. in R N \Ω. Therefore, the restriction of p ∞ on Ω is in H 1 0,per (Ω). Furthermore, since

-∇ • (A(x)∇p n ) = F (x, p n ) in Ω, (3.1)
the function p ∞ is a solution of the same equation in Ω in the weak H 1 0,per (Ω) sense. The elliptic regularity theory then implies that, up a negligible set, p ∞ is actually a C 2,α (Ω) solution of (1.18) and the convergence p n → p ∞ holds at least in the C 2 loc (Ω) sense. Lastly, let us show that p ∞ ≥ p in Ω. Since p ∞ is nonnegative and p = 0 in all Ω i with i ∈ I -, one only needs to prove that p n ≥ p in Ω i for all i ∈ I -and for all n ∈ N. For any n ∈ N and i ∈ I -, observe that the function p n is a supersolution of (2.3) in Ω i , because it solves (3.1) in Ω i and p n > 0 on ∂Ω i . Since min

Ω i p n ≥ min R N p n > 0, there is ε > 0 such that ε ϕ i ≤ p n in Ω i ,
where ϕ i solves (2.4). Since λ 1,Ω i ,D < 0, one can even assume without loss of generality that ε ϕ i is a subsolution of (2.3), in the sense of (2.5). Therefore,

w(t, x) ≤ p n (x) for all (t, x) ∈ R + × Ω i ,
where w denotes the solution of the Cauchy problem (2.6) in Ω i with initial datum ε ϕ i . Since w is nondecreasing in t, it converges as t → +∞ to a solution w ∞ of (2.3) such that 0 < ε ϕ i ≤ w ∞ ≤ p n in Ω i . From the construction of p in Theorem 1.2 and its minimality, one infers that p ≤ w ∞ in Ω i , whence

p ≤ p n in Ω i .
As a conclusion, p ≤ p n in Ω for all n ∈ N, whence p ≤ p ∞ in Ω. The proof of Theorem 1.3 is thereby complete.

Remark 3.1 We first show in this remark that, if F fulfills the KPP condition (1.4) in Ω, that is if s → F (x, s) s is decreasing in s > 0 for all x ∈ Ω, (3.2)
then p ∞ = p in Ω. Consider first i ∈ I -and let us prove that the function p solving (2.3) in Ω i is unique. The proof is similar to the ones used for instance in [START_REF] Berestycki | Le nombre de solutions de certains problèmes semi-linéaires elliptiques[END_REF][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] and it is just sketched. Let q be any periodic solution of (2.3) in Ω i . From the proof of Theorem 1.2, one knows that q ≥ p in Ω i . But εq ≤ p in Ω i for ε > 0 small enough, from the Hopf lemma applied to p. Therefore, the quantity

ε * = sup ε > 0, εq ≤ p in Ω i is a positive real number. If ε * < 1, then -∇ • (A(x)∇(ε * q)) -F (x, ε * q) < 0 in Ω i , from (3.2)
. The strong maximum principle and Hopf lemma then imply that ε * q < p in Ω i and even (ε * + η)q < p in Ω i for all η ∈ [0, η 0 ] and for some η 0 > 0. This contradicts the maximality of ε * . Consequently, ε * ≥ 1, whence q ≤ p in Ω i and finally q = p in Ω i . Actually, with the same arguments as those used in the proof of Theorem 1.2, the same conclusion holds even if q is not assumed to be periodic. Now, if i ∈ I -, then we prove that there does not exist any solution q of (2.3) that is positive in Ω i (or in any of its connected components). Indeed, since F (x, s) < ζ(x)s for all x ∈ Ω i and s > 0 from (3.2), there holds

-∇ • (A(x)∇(ε ϕ i )) -F (x, ε ϕ i ) > -ε∇ • (A(x)∇ ϕ i ) -ζ(x)ε ϕ i = λ 1,Ω i ,D ε ϕ i ≥ 0 in Ω i
for all ε > 0, where ϕ i solves (2.4) in Ω i , with λ 1,Ω i ,D ≥ 0. In other words, ε ϕ i is a strict supersolution of (2.3) for all ε > 0. It follows with the same arguments as above or as in the proof of Theorem 1.2 that q ≤ ε ϕ i for all ε > 0, for any solution q of (2.3). Therefore, a positive periodic solution of (2.3) cannot exist, which implies that p ∞ = p = 0 in Ω i for all i ∈ I -. As a conclusion, the condition (3.2) implies that

p ∞ = p in Ω. (3.3)
On the other hand, we can construct examples for which (3.3) does not hold. It is indeed possible to construct a situation for which λ 1,D < 0 and there exist an index j ∈ {1, . . . , m} and s 0 ∈ (0, M) such that F (x, s) = λs + s 2 for all x ∈ Ω j and s ∈ [0, s 0 ], where λ > 0 denotes the principal periodic eigenvalue of the operator -∇ • (A(x)∇) in Ω j with zero Dirichlet boundary condition on ∂Ω j . Thus, λ 1,Ω j ,D = 0 and j ∈ I -. Let ϕ j be the principal periodic eigenfunction of (2.4) in Ω j with ζ = λ in Ω j , such that max Ω j ϕ j = 1. For any ε ∈ (0, s 0 ], there holds

-∇ • (A(x)∇(ε ϕ j )) -F (x, ε ϕ j ) = -ε∇ • (A(x)∇ ϕ j ) -λε ϕ j -ε 2 ϕ 2 j = -ε 2 ϕ 2 j < 0 in Ω j .
As above, it follows from the strong maximum principle that ε ϕ j ≤ p n in Ω j for all ε ∈ (0, s 0 ] and for all n ∈ N. In particular, 0 < s 0 ϕ j ≤ p ∞ in Ω j , whereas p = 0 in Ω j by definition.

Pulsating travelling fronts and limiting minimal speed

In this section, we give the proof of Theorem 1.4. We establish the relationship between the pulsating travelling fronts for the problems (1.1) in R N and (1.8) in Ω when the nonlinearity F is approximated with nonlinearities f n which are very negative in R N \Ω, in the sense of (1.16).

We also prove that the minimal speeds of the fronts in R N converge monotonically to a quantity which is equal to 0 in a direction e when the connected components of Ω are bounded with respect to e. We use especially some bounds for the minimal speeds, which involve some linear eigenvalue problems. Throughout this section, we assume that λ 1,D < 0 and e is any given unit vector of R N . The functions F and f n are assumed to fulfill (1.9) and (1.16). For each n ∈ N, one has λ 1,n < 0 from Theorem 1.3. The functions p n denote the minimal solutions of (1.7) with the nonlinearities f n , given by Proposition 1.1, and the speeds c * n (e) > 0 denote the minimal speeds of pulsating fronts φ n (x • ect, x) connecting 0 to p n for problems (1.1) in R N with the nonlinearities f n .

Proof of part a) of Theorem 1.4. Fix any two integers n ≤ m and let us show that c * m (e) ≤ c * n (e). First, remember that 0 < p m ≤ p n ≤ M (from Proposition 1.1 and Theorem 1.3) and that both functions p m and p n are periodic in R N . Let η > 0 be such that 0 < η < min R N p m and u 0 : R → [0, M] be defined by

u 0 (x) = 0 if x • e > 0, η if x • e ≤ 0.
Let v n and v m denote the solutions of the Cauchy problems (1.5) with initial datum u 0 and nonlinearities f n and f m respectively. Since f m ≤ f n , the maximum principle yields 0 < v m (t, x) ≤ v n (t, x) < M for all t > 0 and x ∈ R N .

On the other hand, it follows from the results of Weinberger [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] that

∀ c < c * m (e), sup x∈R N , x•e≤ct |v m (t, x) -p m (x)| → 0 as t → +∞, while ∀ c > c * n (e), sup x∈R N , x•e≥ct v n (t, x) → 0 as t → +∞.
One infers that c * m (e) ≤ c * n (e). Consequently, the sequence (c * n (e)) n∈N is nonincreasing and it converges to a real number c * (e) ≥ 0.

From the assumptions (1.9) and (1.16) and the regularity of F and f n , there exist a function F : (x, u) → F (x, u) and a sequence of functions (f n ) n∈N such that: i) the function F is defined and continuous in Ω × R + , of class C 0,α with respect to x ∈ Ω locally uniformly in u ∈ R + , of class C 1 with respect to u with ζ := ∂F ∂u (•, 0) ∈ C 0,α (Ω), periodic with respect to x ∈ Ω and F satisfies (1.9); ii) each function f n is defined and continuous in R N ×R + , of class C 0,α with respect to x ∈ R N locally uniformly in u ∈ R + , of class C 1 with respect to u with ζ n := ∂f n ∂u (•, 0) ∈ C 0,α (R N ), periodic with respect to x ∈ R N and f n satisfies (1.3); iii) the functions f n satisfy (1.16) with F instead of F and

g n (x, u) = f n (x, u)/u if u > 0, g n (x, 0) = ζ n (x); iv) the function F satisfies F (x, u) ≤ F (x, u) for all (x, u) ∈ Ω × R +
and F (x, u)/u is decreasing with respect to u > 0 for all x ∈ Ω; v) the functions f n satisfy f n (x, u) ≤ f n (x, u) for all (x, u) ∈ R N × R + and f n (x, u)/u is decreasing with respect to u > 0 for all x ∈ R N .

Let λ 1,n and λ 1,D be the principal periodic eigenvalues of problems (1.17 with the nonlinearity f n , given by Proposition 1.1. Actually, the function p n is unique from property v) above and from [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF], and it is such that

p n ≥ p n in R N since f n ≥ f n in R N × R + ,
from the proof of Theorem 1.3. Let c * n (e) > 0 be the minimal speed of pulsating travelling fronts φ n (x • ect, x) connecting 0 to p n for problem (1.1) with the nonlinearity f n , that is φ n is periodic with respect to x ∈ R N 0 < φ n (s, x) < p n (x) and φ n (-∞, x) = p n (x), φ n (+∞, x) = 0. As in the beginning of the proof of this theorem, there holds

0 < c * n (e) ≤ c * n (e), (4.1)
since f n ≤ f n . Furthermore, it follows from [START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] that c * n (e) is given by

c * n (e) = min λ>0 -k e,λ,n λ , (4.2)
where k e,λ,n denotes the principal periodic eigenvalue of the operator

L e,λ,n := -∇ • (A∇) + 2λ Ae • ∇ + λ ∇ • (Ae) -λ 2 Ae • e -ζ n in R N .
Let us now show that, for every λ ∈ R, one has k e,λ,n → k e,λ,D as n → +∞, where k e,λ,D is the principal periodic eigenvalue of the operator

L e,λ,Ω := -∇ • (A∇) + 2λ Ae • ∇ + λ ∇ • (Ae) -λ 2 Ae • e -ζ in Ω
with zero Dirichlet boundary condition on ∂Ω. The proof starts as in the proof of the convergence λ 1,n → λ 1,D in Theorem 1.3. First, it follows as in the proof of Theorem 1.3 that the sequence (k e,λ,n ) n∈N is nondecreasing and that k e,λ,n < k e,λ,D for all n ∈ N. Let ϕ n be a principal periodic eigenfunction of L e,λ,n , that is L e,λ,n ϕ n = k e,λ,n ϕ n and ϕ n > 0 in R N .

Up to normalization, one can assume that ϕ n L 2 (C 0 ) = 1. By multiplying the above equation by ϕ n , by integrating by parts over C 0 and by using Young's inequality, it follows that the sequence (ϕ n ) n∈N is bounded in H 1 per (R N ). Up to extraction of a subsequence, it converges weakly in H 1 per (R N ) and strongly in

L 2 per (R N ) to a nonnegative function ϕ ∞ ∈ H 1 per (R N ) such that ϕ ∞ L 2 (C 0 ) = 1. Furthermore, since the sequence (k e,λ,n ) n∈N is bounded and ζ n → -∞ as n → +∞ locally uniformly in R N \Ω, one infers as in the proof of Theorem 1.3 that ϕ ∞ = 0 a.e. in R N \Ω. The restriction of ϕ ∞ to Ω is then a C 2,α (Ω) periodic function such that L e,λ,Ω ϕ ∞ = k e,λ,∞ ϕ ∞ in Ω with ϕ ∞ = 0 on ∂Ω,
where lim n→+∞ k e,λ,n = k e,λ,∞ ≤ k e,λ,D . Since the function ϕ ∞ is periodic, nonnegative and nontrivial, it follows that it is positive in Ω i for some i ∈ {1, . . . , m}, that is k e,λ,∞ is equal to the principal periodic eigenvalue k e,λ,Ω i ,D of the operator L e,λ,Ω in Ω i with zero Dirichlet boundary condition on ∂Ω i . But since k e,λ,Ω i ,D ≥ k e,λ,D (≥ k e,λ,∞ ), one concludes eventually that k e,λ,∞ = k e,λ,D , that is k e,λ,n → k e,λ,D as n → +∞.

Assume now that all connected components of Ω are bounded in the direction e, in the sense of (1.19). Let us show that c * (e) = 0. First, it follows from (4.1), (4.2) and (4. On the other hand, for every λ > 0, there is an index i ∈ {1, . . . , m}, which may depend on λ, such that k e,λ,D = k e,λ,Ω i ,D and thus there is a periodic function ϕ defined in Ω i such that L e,λ,Ω ϕ = k e,λ,D ϕ in Ω i with ϕ > 0 in Ω i and ϕ = 0 on ∂Ω i . The function

ψ = e -λ(x•e) ϕ satisfies -∇ • (A(x)∇ψ) -ζ(x)ψ = k e,λ,D ψ in Ω i (4.5) with ψ > 0 in Ω i and ψ = 0 on ∂Ω i . Let C be any connected component of Ω i , that is C = ω i + k for some k ∈ L 1 Z × • • • × L N Z.
= φ n (+∞, x) < φ n (s, x) < φ n (-∞, x) = p n (x) ≤ M for all (s, x) ∈ R × R N .
Actually, from [START_REF] Hamel | Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity[END_REF], each solution

u n satisfies (u n ) t > 0 in R × R N . On the one hand, since 0 < u n (t, x) < p n (x) in R × R N , Theorem 1.3 implies that u n → 0 in L 1 loc (R × (R N \Ω)).
On the other hand, since f n (x, s) = F (x, s) for all (x, s) ∈ Ω × R + , it follows from standard parabolic estimates that there exists a function u : R ×

Ω → [0, M] such that, up to extraction of a subsequence, u n → u as n → +∞ in C 1 t and C 2 x in R × Ω, where u obeys u t -∇ • (A(x)∇u) = F (x, u) in R × Ω and 0 ≤ u(t, x) ≤ p ∞ (x)
≤ M for all (t, x) ∈ R × Ω, under the notation of Theorem 1.3. In particular, the function u can be extended continuously by 0 on R × ∂Ω and, from parabolic regularity, the function u is a classical solution of (1.8) in R × Ω (of course, one could also extend u by 0 in R × (R N \Ω) and u would then be continuous in R × R N ). Moreover, the equalities

u n t + k • e c n , x = u n (t, x -k) in R × R N for all k ∈ L 1 Z × • • • × L N Z carry over at the limit, whence u(t + (k • e)/c, x) = u(t, x -k) in R × Ω for all k ∈ L 1 Z × • • • × L N Z.
In other words, the function u can be written as u(t, x) = φ(x • ect, x) in R × Ω where φ : R × Ω → [0, M] is such that φ(s, •) is periodic in Ω for all s ∈ R. Lastly, since all functions u n are increasing in time in R × R N , the function u is such that u t ≥ 0 in R × Ω. From the previous observations and parabolic regularity, there are then two periodic functions u ± defined in Ω such that 0 ≤ u

-≤ u + ≤ p ∞ in Ω, u(t, x) → u ± (x) as t → ±∞ in C 2 loc (Ω) and u ± obey -∇ • (A(x)∇u ± ) = F (x, u ± ) in Ω, u ± = 0 on ∂Ω.
Let now any index i ∈ I -, that is λ 1,Ω i ,D < 0 in the sense of (1.12). From the proof of Theorem 1.2, there is a minimal periodic solution p i of (2.3). Furthermore, in Ω i , there holds p i = p ≤ p ∞ ≤ p n for all n ∈ N, under the notation of Theorem 1.3. Therefore, one can always shift in time the functions u n so that, say,

C 0 ∩Ω i u n (0, x) dx = 1 2 C 0 ∩Ω i p i (x) dx,
where we recall that

C 0 = [0, L 1 ] × • • • × [0, L N ]
. From Lebesgue's dominated convergence theorem, the function u satisfies the same equality at the limit, whence

0 ≤ C 0 ∩Ω i u -(x) dx ≤ 1 2 C 0 ∩Ω i p i (x) dx ≤ C 0 ∩Ω i u + (x) dx
by monotonicity of u with respect to t. The minimality of p i and the strong maximum principle imply that u -= 0 in Ω i , while u + > 0 in Ω i , again from the strong maximum principle.

If we further assume that F satisfies the KPP assumption (3.2) in Ω (or just in Ω i ), then it follows from Remark 3.1 that the solution of (2.3) is actually unique, whence u + = p i in Ω i in this case.

Proof of part c) of Theorem 1.4. Firstly, it follows from [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] Furthermore, the maps λ → -k e,λ,n are all convex and their derivatives at λ = 0 are all equal to 0, see [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating travelling fronts[END_REF]. In particular, for every n ∈ N, -k e,λ,n is nondecreasing with respect to λ ≥ 0 and -k e,λ,n ≥ -k e,0,n = -λ 1,n for all λ ∈ R. By passing to the limit as n → +∞ pointwise in λ, one gets that the map λ → -k e,λ,D is convex in R, nondecreasing in R + , and there holds -k e,λ,D ≥ -k e,0,D = -λ 1,D > 0 for all λ ∈ R. Notice here that, if assumption (1. [START_REF] Kinezaki | Spatial dynamics of invasion in sinusoidally varying environments[END_REF]) is made, then -k e,λ,D ≤ max Ω ζ for all λ, under the notation used in the proof of part a). Therefore, the infimum in (4.7) is not reached in general.

Because of (4.7), the inequality -k e,λ,D ≥ -λ 1,D > 0 and the limit lim n→+∞ k e,λ,n = k e,λ,D for all λ, it follows that, in order to show the positivity of c * (e), it is sufficient to prove that there exist Λ > 0 and α > 0 such that -k e,λ,n ≥ α λ 2 for all λ ≥ Λ and for all n ∈ N. Owing to the definition of ψ λ and the uniform positivity of φ n , the quantity ε * is a positive real number. Furthermore, ε * ψ λ ≤ φ n in S e ′ ,a,b and there is a sequence (x m ) m∈N of points in S e ′ ,a,b such that lim inf m→+∞ d(x m , ∂S e ′ ,a,b ) > 0, lim m→+∞ (ε * ψ λ (x m )-φ n (x m )) = 0 and lim inf m→+∞ L e,λ,n (ε * ψ λφ n )(x m ) ≥ 0. Since there holds L e,λ,n φ n (x m ) = k e,λ,n φ n (x m ) and L e,λ,n ψ λ (x m ) ≤ -αλ 2 ψ λ (x m ) for every m ∈ N, one concludes that k e,λ,n ≤ -αλ 2 , that is (4.8). This yields the desired inequality c * (e) > 0, as already emphasized. Assume now that there exist a unit vector e ′ , a point x 0 ∈ R N and a positive real number r such that e ′ is an eigenvector of A with e ′ • e = 0 and (1.21) holds. Let β > 0 be such that Ae ′ = βe ′ . Since the matrix A is symmetric, there is an orthonormal family of eigenvectors e ′ 1 , . . . , e ′ N -1 of A in R N such that e ′ i • e ′ = 0 for all 1 ≤ i ≤ N -1. Even if it means decreasing r > 0 in (1.21), one can assume without loss of generality that Ω ⊃ C e ′ ,r := x ∈ R N , |(xx 0 ) • e ′ i | < r for all 1 ≤ i ≤ N -1 .

For any λ > 0, let ψ λ be the function defined in C e ′ ,r by

ψ λ (x) = 1≤i≤N -1 e λ ′ i (x•e ′ i ) cos π(x -x 0 ) • e ′ i ) 2 r
,

where λ ′ i = λ (Ae • e ′ i )/(Ae ′ i • e ′ i )

  then c * (e) = 0. b) For any c ≥ c * (e) with c > 0 and for any sequence (c n ) n∈N such that c n → c as n → +∞ and c n ≥ c * n (e), the pulsating travelling fronts u

  ) and (1.11) with coefficients ζ n and ζ instead of ζ n and ζ, respectively. Since ζ n ≥ ζ n in R N and ζ ≥ ζ in Ω, there holds λ 1,n ≤ λ 1,n and λ 1,D ≤ λ 1,D , while λ 1,n < λ 1,D and λ 1,n → λ 1,D as n → +∞ monotonically, from Theorem 1.3. In particular, λ 1,n < λ 1,D < 0 for all n ∈ N. Let p n be the minimal periodic solution of (1.7)

  The function ψ is positive and bounded in C because of(1.19) and since ϕ is bounded. It follows then from Hopf lemma and the smoothness of ∂C that there exist r > 0 and a sequence (x n ) n∈N in C such that B(x n , r) ⊂ C for all n ∈ N and ψ(x n ) → sup C ψ as n → +∞. By using the standard elliptic estimates and passing to the limit in (4.5) in B(x n , r), up to extraction of a subsequence, one infers thatk e,λ,D ≥ lim inf n→+∞ -ζ(x n ) ≥max Ω i ζ.Finally, k e,λ,D ≥max Ω ζ for all λ > 0, whence c * (e) = 0 from (4.4). Proof of part b) of Theorem 1.4. Let c be any positive real number such that c ≥ c * (e) and let (c n ) n∈N be any sequence such that c n → c as n → +∞ and c n ≥ c * n (e) for all n ∈ N. Let u n (t, x) = φ n (x • ec n t, x) be pulsating travelling fronts for (1.1) in R N with nonlinearity f n , such that 0

-

  that, for each n ∈ N, c * n (e) ≥ min λ>0 -k e,λ,n λ = -k e,λn,n λ nfor some λ n > 0, where k e,λ,n denotes the principal periodic eigenvalue of the operatorL e,λ,n := -∇ • (A∇) + 2λ Ae • ∇ + λ ∇ • (Ae)λ 2 Ae • eζ n in R N . (4.6) Since, as above, k e,λ,n → k e,λ,D as n → +∞ nondecreasingly for every λ ∈ R, where k e,λ,D is the principal periodic eigenvalue of the operator L e,λ,Ω := -∇ • (A∇) + 2λ Ae • ∇ + λ ∇ • (Ae)λ 2 Ae • eζ in Ω with zero Dirichlet boundary condition on ∂Ω, it follows that c * n (e) ≥ -k e,λn,n λ n ≥ -k e,λn,D λ n ≥ inf λ>0 -k e,λ,D λ for all n ∈ N, whence c * (e) ≥ inf λ>0

(4. 8 )+ b 2 ,

 82 Of course, from the proof of part a), this cannot be always true. However, assuming from now on that A is constant, we shall now show that (4.8) holds under conditions(1.20) or(1.21). Assume first that there exist a unit vector e ′ = ±e and two real numbers a < b such that (1.20) is fulfilled, that isΩ ⊃ S e ′ ,a,b := x ∈ R N , a < x • e ′ < b .For any λ > 0, let ψ λ be the function defined in S e ′ ,a,b byψ λ (x) = e λ ′ (x•e ′ ) cos π ba × x • e ′ -a where λ ′ = λ (Ae • e ′ )/(Ae ′ • e ′ ).The function ψ λ is bounded and of class C ∞ S e ′ ,a,b , it is positive in S e ′ ,a,b and vanishes on ∂S e ′ ,a,b . Furthermore, sinceζ n = ζ in Ω ⊃ S e ′ ,a,b , it is straightforward to check that L e,λ,n ψ λ = π 2 (Ae ′ • e ′ ) (ba) 2ζ(x) -2 α λ 2 ψ λ in S e ′ ,a,b for all n ∈ N, where α = (Ae•e)/2-(Ae•e ′ ) 2 /(2 Ae ′ •e ′ ) > 0 from Cauchy-Schwarz inequality, since the unit vectors e and e ′ are not parallel. Since ζ is bounded in Ω, it follows that there exists Λ > 0 such that L e,λ,n ψ λ ≤ -α λ 2 ψ λ in S e ′ ,a,b for all λ ≥ Λ and n ∈ N. This inequality yields (4.8), as in the course of the proof of Proposition 1.1. We just sketch the proof here. Fix any λ ≥ Λ and n ∈ N and let φ n be a principal periodic eigenfunction of the operator L e,λ,n . Namely, L e,λ,n φ n = k e,λ,n φ n and φ n is periodic and positive in R N . Define ε * = sup ε > 0, εψ λ ≤ φ n in S e ′ ,a,b .

1 π 2 Remark 4 . 1

 1241 . The function ψ λ is bounded and of class C ∞ C e ′ ,r , it is positive in C e ′ ,r and vanishes on ∂C e ′ ,r . Furthermore, sinceζ n = ζ in Ω ⊃ C e ′ ,r , it is straightforward to check that L e,λ,n ψ λ = 1≤i≤N -(Ae ′ i • e ′ i ) 4r 2 ζ(x) -2 α λ 2 ψ λ in C e ′ ,rfor all n ∈ N, where α = β(e • e ′ ) 2 /2 > 0 since β > 0 and e • e ′ = 0 by assumption. Thus, one concludes as above that there is Λ > 0 such that L e,λ,n ψ λ ≤ -α λ 2 ψ λ in C e ′ ,r for all λ ≥ Λ and n ∈ N. This yields (4.8) and finally c * (e) > 0. The proof of Theorem 1.4 is thereby complete. In the case when the functions f n fulfill the KPP assumption (1.4), then c * (e) is given by an explicit variational formula. Namely, under assumption (1.4) for the functions f n , it follows from the proof of part a) of Theorem 1.4 with the choices f n = f n and F = F that c * (e) ≤ inf λ>0 -k e,λ,D /λ, because of (4.4). On the other hand, the reverse inequality (4.7) always holds, from the proof of part c) of Theorem 1.4. As a conclusion, the assumption (1.4) for the functions f n yields c * (e) = inf λ>0 -k e,λ,D λ .