Juliette Bouhours

Grégoire Nadin

A variational approach to reaction diffusion equations with forced speed in dimension 1

Keywords: 2010 Mathematics Subject Classification: 35B40, 35C07, 35J20, 35K10, 35K57, 92D52 Reaction diffusion equations, travelling waves, forced speed, energy functional, long time behaviour

We investigate in this paper a scalar reaction diffusion equation with a nonlinear reaction term depending on xct. Here, c is a prescribed parameter modelling the speed of climate change and we wonder whether a population will survive or not, that is, we want to determine the large-time behaviour of the associated solution. This problem has been solved recently when the nonlinearity is of KPP type. We consider in the present paper general reaction terms, that are only assumed to be negative at infinity. Using a variational approach, we construct two thresholds 0 < c ≤ c < ∞ determining the existence and the non-existence of travelling waves. Numerics support the conjecture c = c. We then prove that any solution of the initial-value problem converges at large times, either to 0 or to a travelling wave. In the case of bistable nonlinearities, where the steady state 0 is assumed to be stable, our results lead to constrasting phenomena with respect to the KPP framework. Lastly, we illustrate our results and discuss several open questions through numerics.

1 Introduction and main results

Motivation: models on climate change

Reaction diffusion problems are often used to model the evolution of biological species. In 1937, Kolmogorov, Petrovskii and Piskunov in [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de la matière et son application à un problème biologique[END_REF], Fisher in [START_REF] Fisher | The advance of advantageous genes[END_REF] used reaction diffusion to investigate the propagation of a favourable gene in a population. One of the main notions introduced in [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de la matière et son application à un problème biologique[END_REF][START_REF] Fisher | The advance of advantageous genes[END_REF] is the notion of travelling waves, i.e solution of the form u(t, x) = U (x -ct) for x ∈ R, t > 0 and some constant c ∈ R. Since then a lot of papers have been dedicated to reaction diffusion equations and travelling waves in settings modelling all sorts of phenomena in biology. In this paper we are interested in the following problem,

u t -u xx = f (x -ct, u), x ∈ R, t > 0, u(0, x) = u 0 (x), x ∈ R, (P)
where u 0 is bounded, nonnegative and compactly supported. This problem has been proposed in [START_REF] Berestycki | Can a species keep pace with a shifting climate?[END_REF] to model the effect of climate change on biological species. In this setting u is the density of a biological population that is sensitive to climate change. We assume that the North Pole is found at +∞ whereas the equator is at -∞, which gives a good framework to study the effect of global warming on the distribution of the population. The dependence on z in the reaction term takes into account the notion of favourable/unfavourable area depending on the latitude for populations which are sensitive to the climate/temperature of the environment. The constant c can be seen as the speed of the climate change. In such a setting, one will be interested to know when the population can keep track with its favourable environment despite the climate change and thus persists at large times. In [START_REF] Berestycki | Can a species keep pace with a shifting climate?[END_REF] Berestycki et al studied the existence of non trivial travelling wave solutions converging to 0 at infinity in dimension 1 when f satisfies the KPP property: s ∈ R + → f (z, s)/s is decreasing for all z ∈ R. As they converge to 0 as z → ±∞, such homoclinic solutions are sometimes called travelling pulses in the literature, in contrast with heteroclinic travelling waves. We will use in this paper the same terminology as in [START_REF] Berestycki | Can a species keep pace with a shifting climate?[END_REF] and call travelling waves such homoclinic solutions (see equation (S) below for a precise definition).

Berestycki et al proved that in this framework, the persistence of the population depends on the sign of the principal eigenvalue of the linearized equation around the trivial steady state 0. Their results have been extended by Berestycki and Rossi to R N in [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space[END_REF] and to infinite cylinders in [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed. II. Cylindrical-type domains[END_REF]. In [START_REF] Vo | Traveling fronts for equations with forced speed in mixed environments[END_REF] Vo studies the same type of problem with more general classes of unfavourable media toward infinity.

A similar model was developed by Potapov and Lewis in [START_REF] Potapov | Climate and competition: the effect of moving range boundaries on habitat invasibility[END_REF] and by Berestycki, Desvillettes and Diekmann in [START_REF] Berestycki | Can climate change lead to gap formation? in preparation[END_REF] in order to investigate a two-species competition system facing climate change. These papers studied the effect of the speed of the climate change on the coexistence between the competing species. In [START_REF] Berestycki | Can climate change lead to gap formation? in preparation[END_REF] the authors pointed out the formation of a spatial gap between the two species when one is forced to move forward to keep up with the climate change and the other has limited invasion speed. The persistence of a specie facing climate change was also investigated mathematically through an integrodifference model by Zhou and Kot in [START_REF] Zhou | Discrete-time growth-dispersal models with shifting species ranges[END_REF].

The particularity of all these papers is the KPP assumption for the reaction term, where the linearized equation around 0 determines the behaviour of the solution of the nonlinear equation. As far as we know, such questions were only investigated numerically for other types of nonlinearities by Roques et al in [START_REF] Roques | A population facing climate change: joint influences of allee effects and environmental boundary geometry[END_REF], where the authors were mainly interested in the effects of the geometry of the domain (in dimension 2) on the persistence of the population considering KPP and bistable nonlinearities.

Framework

In this paper we are interested in this persistence question, when the evolution of the density of the population is modelled by a reaction diffusion equation, with more general hypotheses on the nonlinearity f . Indeed we point out that we consider general nonlinearities f , without assuming f to satisfy the KPP property. We will assume that f is a Carathéodory function, i.e s → f (z, s) is continuous for all z ∈ R, z → f (z, s) is measurable for all s ∈ R, satisfying the following hypotheses, f (z, 0) = 0, (

s → f (z, s) is Lipschitz-continuous and of class C 1 uniformly with respect to z ∈ R, (

∃M > 0 , f (z, s) ≤ 0, ∀s ∈ R\(0, M ) and z ∈ R,

∃R > 0, δ > 0, f (z, s) ≤ -δs, ∀|z| > R, s ∈ (0, M ).

(1.4) Assumption (1.1) means than when the population vanishes then no reaction takes place, i.e 0 is a steady state of the problem which corresponds to the extinction of the population. Hypothesis (1.3) models some overcrowding effect: the resources being limited, the environment becomes unfavourable when the population grows above some threshold M > 0.

Assumption (1.4) gives information on the boundedness of the favourable environment and postulates that outside a bounded region the environment is strictly unfavourable. Lastly, hypothesis (1.5) will imply that the equation with c = 0 admits a non-trivial stationary solution which is more stable, in a sense, than the equilibrium u ≡ 0.

We will use the following weighted spaces throughout this paper:

L 2 c (R) := L 2 (R, e cz dz), H 1 c (R) := H 1 (R, e cz dz), H 2 c (R) := H 2 (R, e cz dz).

Main results

Up to a change of variable (z := x -ct) Problem (P) is equivalent to

u t -u zz -cu z = f (z, u), z ∈ R, t > 0, u(0, z) = u 0 (z), z ∈ R, ( P )
In our paper we investigate the existence of travelling wave solutions of (P), i.e nonnegative solution of the form u(t, x) = U (x -ct) for all x ∈ R, t > 0 with U ≡ 0, U (±∞) = 0 . This particular solutions are non trivial solutions of the following stationary problem

     -U zz -cU z = f (z, U ), z ∈ R, U (z) ≥ 0, z ∈ R, U (±∞) = 0. (S)
Solutions of (S) are also the stationary solutions of Problem ( P ) and notice that 0 is a solution of (S) but not a travelling wave solution. We have the following theorem, Theorem 1.1 Under hypotheses (1.1)-(1.5), there exist c ≥ c > 0, such that • for all c ∈ (0, c), (P) has a travelling wave solution

U c ∈ H 1 c (R),
• For all c > c, (P) has no travelling wave solution, that is 0 is the only solution of (S).

The proof of Theorem 1.1 is based on a variational approach, used in [START_REF] Lucia | Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders[END_REF] to prove the existence of travelling front for gradient like systems of equations. We use the same variational formula but in the case of scalar equations and when f depends on z = x -ct. Namely we introduce an energy E c defined by (2.1) such that for all c > 0 if there exists u ∈ H 1 c (R) with E c [u] < 0, then there exists a travelling wave solution constructed as a minimiser of E c . Theorem 1.1 is derived from a continuity argument using assumption (1.5).

Then we will be interested in the convergence of the Cauchy problem.

Theorem 1.2 Let u 0 ∈ H 2 (R) and u 0 non-negative and compactly supported. Then the unique solution u of (P)

satisfies u ∈ L 2 ([0, T [, H 1 c (R)), u t ∈ L 2 ([0, T [, L 2 c (R))
, for all T > 0, and t → u(t, • -ct) converges to a solution of (S) in H 2 c (R) as t → +∞.

Note that the limit of u in the previous theorem could be either the trivial solution 0 or a travelling wave. And if t → u(t, • -ct) converges to 0 as t → +∞ this implies that the population goes exctinct, whereas if t → u(t, • -ct) converges to U c > 0 non trivial solution of (S) as t → +∞ this means that there is persistence of the population and convergence to a travelling wave solution.

After proving these two main theorems, we study the existence of travelling wave solutions and the behaviour of the solution of the Cauchy problem ( P ) depending on the linear stability of 0. Then we study the solution u of ( P ) for particular f , δ and c.

• We prove that, as in the KPP framework, when 0 is linearly unstable the solution u of ( P ) converges to a travelling wave solution. We also exhibit some bistable-like frameworks, that is, frameworks where 0 is linearly stable, admitting travelling wave solutions (see Section 4.2) but for which solutions could converge to 0 or a travelling wave depending on the initial datum. This emphasises the particularity of the KPP framework where the linearized equation near u ≡ 0 determines the existence of travelling wave solutions, which is not true in the general framework.

• In the last section we first study numerically the existence of a speed threshold such that the population survives if c is below this threshold, i.e the solution of the Cauchy problem ( P ) converges toward travelling waves for large times, while the population dies if the speed c is above this threshold, i.e the solution of the Cauchy problem ( P ) converges to 0 for large times, for f KPP, monostable and bistable in the favourable area. For such nonlinearities, we believe that we chose our initial data in such a way that they lie in the basin of attraction of a travelling wave solution, when it exists. Thus in view of the numerical results, we state the following conjecture:

Conjecture 1 Let c ≤ c be defined by Theorem 1.1 then c = c.

We also plot the shape of the profile for different values of the parameter δ and f bistable.

Then we give an example of nonlinearity f such that there exist several locally stable travelling wave solutions in Proposition 5.1 and illustrate this result with numerical simulations displaying the shape of the profile for different initial conditions.

Organisation of the paper

Theorem 1.1 concerning the stationary framework is proved using a variational method in section 2. Sections 3 and 4 are devoted to the study of the Cauchy problem (P). We prove Theorem 1.2 in section 3 and discuss the linear stability of 0 and its consequences on the convergence of the Cauchy problem in section 4. We give some examples and discuss possible improvements of our results with numerical insight in section 5.

A variational approach to travelling waves

The variational structure of travelling wave solutions of homogeneous reaction-diffusion equations is known since the pioneering work of Fife and McLeod [START_REF] Paul | The approach of solutions of nonlinear diffusion equations to travelling front solutions[END_REF]. However, this structure has only been fully exploited quite recently in order to derive existence and stability results for travelling waves in bistable equations in parallel by Heinze [START_REF] Heinze | A variational approach to traveling waves[END_REF], Lucia, Muratov and Novaga [START_REF] Muratov | A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type[END_REF][START_REF] Lucia | Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium[END_REF][START_REF] Lucia | Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders[END_REF] and then by Risler [START_REF] Risler | Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure[END_REF] for gradient systems (see also [START_REF] Gallay | A variational proof of global stability for bistable travelling waves[END_REF][START_REF] Gallay | Global stability of travelling fronts for a damped wave equation with bistable nonlinearity[END_REF] for various other applications). The situation we consider in the present paper is different. First, we deal with heterogeneous reaction-diffusion equations. The homogeneity was indeed a difficulty in earlier works, since the invariance by translation caused a lack of compactness. Here, the behaviour of the nonlinearity at infinity will somehow trap minimising sequences in the favourable habitat where f is positive. Second, we consider general nonlinearities, including monostable ones. The variational approach is not a relevant tool in order to investigate such equations when the coefficients are homogeneous since travelling waves do not decrease sufficiently fast at infinity and thus have an infinite energy. Here, again, the behaviour of the nonlinearity at infinity forces an admissible exponential decay and we could thus define an energy and make use of it.

We are interested in the existence of travelling wave solution of equation (P), i.e u(t, x) = U (x -ct) = U (z), and U is a solution of the ordinary differential equation

     -U ′′ -cU ′ = f (z, U ), z ∈ R, U > 0 in R, U (z) → 0 as |z| → +∞.
To study existence of non trivial travelling waves, we introduce the energy functional defined as follow

E c [u] = R e cz u 2 z 2 -F (z, u) dz, ∀u ∈ H 1 c (R), (2.1)
where

H 1 c (R) = H 1 (R, e cx dx) and
F (z, s) = s 0 f (z, t)dt.
One can notice that (1.1) and (1.2) ensure that R F (z, u)e cz dz is well defined for all u ∈ H 1 c (R). We start by proving the first part of the Theorem and by pointing out the link between solutions of (S) and the functional E c .

Lemma 2.1 Consider u ∈ H 1 c (R) a nonnegative function. Then u is a critical point of the energy functional E c if and only if u is a solution of (S). Moreover u ∈ W 2,p loc (R), for all p > 1.

Proof: The first part of the proof is classical. Let u be a critical point of E c . Standard arguments yield that E c is Gâteaux-differentiable and that its differential at u is given, for all w ∈ H 1 c (R), by

dE c [u](w) = R e cz {u z w z -f (z, u)w} dz. (2.2)
This quantity is clearly continuous with respect to u ∈ H 1 c (R) as a linear form over H 1 c (R) since f is Lipschitz-continuous by hypothesis (1.2). Moreover letting v(z) := u(z)e cz 2 for all z ∈ R and w ∈ C ∞ compactly supported, then

v ′′ = c 2 4 v -f (z, e -cz 2 v)e cz 2 , in R and v ′′ (z) v(z) ≥ δ + c 2 4 , if z ≤ -R.
As u ∈ H 1 c (R), v(z) → 0 as z → -∞ and we can apply [4, Lemma 2.2] we have that

v(z)e -c 2 4 +δz → z→-∞ 0, which implies that u(z) ≤ e γz , ∀z ≤ R -,
for some R -< -R and γ > 0. This implies that u(z) → 0 as z → -∞ and as u ∈ H 1 c (R), u(z) → 0 as z → +∞ and u is a solution of (S). Now let u ∈ H 1 c (R) be a solution of (S) then u is a critical point of E c , as u(±∞) = 0. Thus u ∈ H 1 c (R) is a critical point of E c iff u is a weak solution of (S). Moreover u is in H 1 loc (R), and thus continuous, and so is v. It follows from the equation on v that it is C 2 , from which the conclusion follows.

Let us state a Poincaré type inequality that will be useful in the sequel, which is due to [START_REF] Lucia | Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders[END_REF].

Lemma 2.2 (Lemma 2.1 in [START_REF] Lucia | Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders[END_REF]) For all u ∈ H 1 c (R),

c 2 4 R e cz u 2 dz ≤ R e cz u 2 z dz. (2.3)
Now notice that we can always assume that a global minimiser of E c is bounded and nonnegative.

Remark 2.3 Considering u = min {u, M }, we have

F (z, u) -F (z, u) = u u f (z, s)ds =    0 if u < M, u M f (z, s)ds otherwise, ≤ 0. Thus E c [u] ≥ R e cz u 2 z 2 -F (z, u) dz ≥ R e cz u 2 z 2 -F (z, u) dz = E c [ u].
As we want to minimise the energy functional, u will always be a better candidate than u.

Similarly, taking u = max{0, u} instead of u gives a lower energy.

Hypothesis (1.1) ensures that E c (0) = 0 and thus inf

u∈H 1 c (R) E c [u] ≤ 0.
Moreover, the following lemma yields that inf

u∈H 1 c (R) E c [u] > -∞.
Lemma 2.4 For all c > 0, there exists

C > 0 such that for all u ∈ H 1 c (R), E c [u] ≥ -C.
Proof: We can assume that 0 ≤ u ≤ M using Remark 2.3. For all u ∈ H 1 c (R), using assumption (1.4),

E c [u] ≥ R -R e cz u 2 z 2 -F (z, u) dz + R\(-R,R) e cz u 2 z 2 + δu 2 2 dz ≥ E R c [u],
where

E R c [u] = R -R e cz u 2 z 2 -F (z, u) dz. This implies that inf u∈H 1 c (R) E c [u] ≥ inf u∈H 1 c (-R,R) E R c [u].
Using the assumptions on f , there exists C 0 > 0 such that -F (z, s) > -C 0 for all z ∈ (-R, R) and s ∈ [0, M ]. Thus there exists C > 0 such that E R c [u] ≥ -C for all u ∈ H 1 (-R, R) and then inf

u∈H 1 c (R) E c [u] ≥ -C. Proposition 2.5 There exists u ∞ ∈ H 1 c (R) such that E c [u ∞ ] = min u∈H 1 c (R) E c [u].
To prove Proposition 2.5 we consider (u n ) n a minimising sequence of

E c in H 1 c (R), i.e such that E c [u n ] → inf u∈H 1 c (R) E c [u] > -∞ as n → +∞.
In view of Remark 2.3 we can assume that u n is bounded for n large enough.

Lemma 2.6 There exist N ∈ N, C 1 > 0, locally bounded with respect to c, such that for all n > N ,

u n 2 H 1 c (R) = R e cz (u n ) 2 z + u 2 n dz ≤ 1 + C 1 min{ 1 2 , δ 2 } . Proof of Lemma 2.6: For all u ∈ H 1 c (R), bounded, E c [u] ≥ R -R e cz u 2 z 2 -F (z, u) dz + R\(-R,R) e cz u 2 z 2 + δu 2 2 dz = R -R e cz -F (z, u) - δu 2 2 dz + R e cz u 2 z 2 + δu 2 2 dz ≥ -C 1 + min{ 1 2 , δ 2 } u 2 H 1 c (R) ,
where

C 1 = -1 c (C 0 + δM 2 2
)(e cR -e -cR ), with C 0 as in the proof of Lemma 2.4. Moreover as

E c [u n ] → inf u∈H 1 c (R) E c [u] ≤ 0 = E c [0], there exists N ∈ N such that for all n > N , E c [u n ] ≤ 1.
Then using the previous computation we obtain the Lemma.

One can now prove Proposition 2.5.

Proof of Proposition 2.5:

From Lemma 2.6, if (u n ) is a minimising sequence of E c in H 1 c (R) then (u n ) is bounded in H 1 c (R). Thus up to a subsequence (u n ) converges weakly to some u ∞ ∈ H 1 c (R). One has: R e cz (u ∞ ) 2 z dz ≤ lim inf n→+∞ R e cz (u n ) 2 z dz. (2.4)
Moreover as u n ∈ (0, M ), classical Sobolev injections yield that

u n → u ∞ in C loc (R) as n → +∞. (2.5)
As F is bounded, the dominated convergence theorem gives, for all

T ∈ R, T -∞ e cz F (z, u n )dz → T -∞ e cz F (z, u ∞ )dz as n → +∞. (2.6)
Thus, as -

+∞ T e cz F (z, u n )dz ≥ 0, for all T > R, lim inf n→+∞ E c [u n ] ≥ R e cz (u ∞ ) 2 z 2 dz + T -∞ -e cz F (z, u ∞ )dz = E c [u ∞ ] + +∞ T e cz F (z, u ∞ )dz ≥ E c [u ∞ ] - +∞ T Ce cz u 2 ∞ dz,
the last inequality following from (1.2). As, for all ε > 0, there exists T > R such that

+∞ T Ce cz u 2 ∞ dz < ε, since u ∞ ∈ H 1 c , we have E c [u ∞ ] ≤ lim inf n→+∞ E c [u n ] = inf u∈H 1 c (R) E c [u],
and the Proposition is proved.

We have proved that the minimum is reached in H 1 c . This implies that there exists a solution

U of (S) such that E c [U ] = inf u∈H 1 c E c [u]. Proposition 2.7 The function c ≥ 0 → inf u∈H 1 c (R) E c [u] is continuous. Proof: Let (c n ) n be a sequence in R such that c n → c as n → +∞. From Proposition 2.5
we know that for all n ∈ N there exists

u n ∈ H 1 cn (R) such that inf u∈H 1 cn (R) E cn [u] = E cn [u n ]. Let v n := e cnz 2 u n ∈ H 1 (R) and notice that E cn [u n ] = R (v n ) 2 z 2 + c 2 n 8 v 2 n -e cnz F (z, e -cnz 2 v n ) dz =: E cn [v n ]. Moreover the sequence (v n ) n is uniformly bounded in H 1 (R) by Lemma 2.6, as (c n ) n is uniformly bounded, thus up to a subsequence, v n ⇀ v ∞ weakly in H 1 (R) as n → +∞.
Moreover, as f is Lipschitz-continuous and f (z, 0) = 0 for all z ∈ R, there exists a constant C > 0 such that |f (z, s)| ≤ Cs for all s ≥ 0 and z ∈ R. Integrating this inequality, one gets

e cnz |F (z, e -cnz 2 v)| ≤ C v 2 2 for all z ∈ R, v ∈ H 1 (R)
and n ∈ N. It follows from the dominated convergence theorem and the continuity of

f in s that lim n→+∞ R e cnz F (z, e -cnz 2 v)dz = R e cz F (z, e -cz 2 v)dz for all v ∈ H 1 (R). Hence, E cn [v] → E c [v] as n → +∞. As v n is a minimiser, for all v ∈ H 1 (R), E cn [v n ] ≤ E cn [v].
(2.7)

Passing to the limit and using the same arguments as in the proof of Proposition 2.5 we obtain

E c [v ∞ ] ≤ lim inf n→+∞ E cn [v n ] ≤ E c [v], ∀v ∈ H 1 (R). (2.8) This implies that E c [v ∞ ] = inf v∈H 1 (R) E c [v],
and letting u ∞ = e -cz 2 v ∞ we get the Proposition.

By the continuity of c → inf

u∈H 1 c (R)
E c [u] and using Proposition 2.5, the first part of Theorem 1.1 is proved.

This proposition will prove the second part of the Theorem Proposition 2.8 There exists c > 0 such that for all c > c, 0 is the only solution of equation (S).

Proof: Define

g(z, u) = sup s≥u f (z, s) s × u, ∀z ∈ R, u ∈ R + .
(2.9)

Then g satisfies the following assumptions:

g(z, 0) = 0, ∀z ∈ R, (2.10) u → g(z, u) is Lipschitz-continuous uniformly with respect to z ∈ R, (2.11) g(z, s) ≤ 0, ∀z ∈ R, s ≥ M, (2.12
)

∀z ∈ R, u → g(z, u) u is decreasing , (2.13)
g(z, u) ≤ -δu, ∀|z| > R.

(2.14)

Hence we know from [4, Theorem 3.2] that there exists c > 0 such that if v is a solution of

-v zz -cv z = g(z, v) in R, (2.15)
for c > c, then v ≡ 0. Moreover for all z ∈ R, s ∈ R, g(z, s) ≥ f (z, s). Take c > c and let u a solution of (S), then u is a subsolution of the associated KPP equation, i.e

-u ′′ -cu ′ ≤ g(z, u) in R.
Let M > 0 be as in condition (1.3), then w(z) = M for all z ∈ R is a super solution of the associated KPP problem, i.e

-w ′′ -cw ′ ≥ g(z, w) in R,
and we can take M large enough such that u ≤ M in R. Thus there exists v a solution of the KPP problem (2.15), such that u(z) ≤ v(z) ≤ M for all z ∈ R. But as c > c, v ≡ 0, which implies that (S) has no positive solution as soon as c > c and the Proposition is proved.

Convergence of the Cauchy problem

In this section we come back to the parabolic problem (P), that we remind below

u t -u xx = f (x -ct, u) x ∈ R, t > 0, u(0, x) = u 0 (x) x ∈ R,
where u 0 ∈ H 2 (R) is nonnegative, bounded and compactly supported.

Letting z := x -ct, u satisfies the following problem

u t -u zz -cu z = f (z, u) ∀z ∈ R, t > 0, u(0, z) = u 0 (z), for all z ∈ R. ( P )
We know that such a u exists, using sub-and super-solution arguments with assumptions (1.1) and (1.3). As s → f (•, s) is Lipschitz-continuous, applying the maximum principle we have that u is unique. Defining v(t, z) = u(t, z)e c 2 z for all t > 0, z ∈ R, then v satisfies the following equation

v t -v zz + c 2 4 v = e c 2 z f (z, e -c 2 z v).
Multiplying the previous equation by v and using (1.4) we have

d dt R v 2 2 dz+ R |v z | 2 dz+(δ+ c 2 4 ) R v 2 dz ≤ R -R ( f (t, e -c 2 z v) e -c 2 z v +δ)v 2 dz ≤ (||f || lip +δ)M 2 e cR ×2R,
using (1.2) and the fact that v(•, z) → 0 as z → ±∞, as v 0 is bounded and compactly supported and v satisfies v t -v zz + ( c 2 4 + δ)v ≤ 0 when |z| > R. Proceeding as in [START_REF] Evans | Partial differential equations[END_REF] for example, as v(0, •) ∈ H 2 (R), we get that v ∈ L 2 ((0, T ), H 2 (R)) and v t ∈ L 2 ((0, T ), L 2 (R)), for all T > 0. And thus as soon as

u 0 ∈ H 2 c (R), there exists a unique u ∈ L 2 ([0, T [, H 2 c (R)), with u t ∈ L 2 ((0, T ), L 2 c (R)
) for all T > 0, solution of ( P ). Moreover u(t, z) > 0 for all t > 0, z ∈ R. We will now prove Theorem 1.2 on the convergence of solution of ( P ) as t → +∞. In [START_REF] Matano | Convergence of solutions of one-dimensional semilinear parabolic equations[END_REF] Matano proves the convergence of solutions of one dimensional semilinear parabolic equations in bounded domains using a geometric argument and the maximum principle and Du and Matano extended this result in [START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF] to unbounded domains for homogeneous f . Their method relies on classification of solutions for homogeneous problems and uses a reflexion principle which cannot be applied in our case. An alternative proof of this result was first given by Zelenyak in [START_REF] Zelenyak | Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable[END_REF] using a variational approach. In [START_REF] Hale | Convergence in gradient-like systems with applications to PDE[END_REF] Hale and Raugel proved an abstract convergence result in gradient like systems which might apply in the present framework. It roughly states that if the kernel of the linearized equation near any equilibrium has dimension 0 or 1, then the solution of the Cauchy problem converges. We prove such an intermediate step in Lemma 3.5. We chose to prove directly the convergence of the Cauchy problem in section 3.2 using arguments inspired from Zelenyak's paper [START_REF] Zelenyak | Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable[END_REF]. But we had to deal with some additional difficulties coming from the fact that our equation is set in R, which induced a lack of compactness and the necessity of finding some controls at infinity. All of this is detailed in section 3.2. In the next section we start by pointing out the convergence up to a subsequence of the solution u of ( P ).

Convergence up to a subsequence

Proposition 3.1 Let u ∈ L 2 ([0, T [, H 1 c (R)
) for all T > 0, be the solution of ( P ). Then there exists a sequence (t n ) n that goes to infinity as n → +∞, such that u(t n , z) converges to a solution of (S) as n → +∞ locally in z ∈ R.

Proof of Proposition 3.1: As u(t, •) ∈ H 2 c (R) and u t (t, •) ∈ L 2 c (R) for all t > 0, standard arguments show that t → E c [u(t, •)] is C 1 and d dt E c [u(t, •)] = R e cz {u zt u z -f (z, u)u t } dz = R (e cz u z )u tz dz - R e cz f (z, u)u t dz = - R (cu z + u zz )e cz u t dz - R e cz f (z, u)u t dz = R (-cu z -u zz -f (z, u))e cz u t dz = R -(u t ) 2 e cz dz ≤ 0.
We know from Proposition 2.4 that E c [u] is bounded from below. It implies that

E c [u] → C as t → +∞, and there exists (t n ) n , such that t n → +∞ and d dt E c [u](t n ) → 0 as n → +∞, i.e u t (t n , •) L 2 c (R)
→ 0 as n → +∞, which implies from standard arguments, that up to extraction u t (t n , z) → 0 as n → +∞ for almost every z ∈ R. Using Schauder Theory, we have that (u(t n , z)) n converges toward u ∞ a stationary solution of ( P ), i.e a solution of (S), up to extraction. Now we investigate the uniqueness of the limit u ∞ .

Uniqueness of the limit

We want to prove that, considering compactly supported initial data u 0 , the solution of our parabolic problem ( P ) admits a unique limit. Define the ω-limit set:

Ω(u 0 ) = ∩ t>0 {u(τ, •), τ ≥ t}.
The closure is taken with respect to the topology of H 2 c (R). We first prove the following Lemma, Lemma 3.2 If w ∈ Ω(u 0 ), then w is a solution of the stationary equation

-w zz -cw z = f (z, w) in R.
Proof : If w ∈ Ω(u 0 ), then there exists a sequence (t n ) n≥1 that converges to +∞ as n → +∞ such that u(t n , z) → w(z) in H 2 c (R) as n → +∞. Let u n (t, z) = u(t + t n , z) for all t > 0 and z ∈ R, then using parabolic estimates, u n → w as n → +∞ (up to a subsequence) with w solution of ( P ) such that w(0, z) = w(z) for all z ∈ R. Moreover as E c [u] is decreasing in t and bounded from below E c [u n (t, •)] → C as n → +∞ and thus E c [ w] = C for all t > 0. We have

d dt E c [ w] = 0,
this implies that R e cz ( wt ) 2 dz = 0. We thus obtain that w = w is a stationary solution of ( P ), i.e a solution of (S) and we have proved the Lemma.

We will need to prove some Lemmas before starting the proof of the Theorem 1.2. Take w ∈ Ω(u 0 ). Let B :

H 2 c (R) → L 2 c (R), w → w ′′ + cw ′ + f (z, w). (3.1)
We know that w is a stationary solution of (S), in other words, B(w) = 0. Define the linear operator: z+R) for all z ≤ -R and w(z) ≤ w(R)e λ -(z-R) for all z > R, where λ -< 0 < λ + are the solutions of λ 2 + λc = δ.

L w := DB(w) : H 2 c (R) → L 2 c (R), h → h ′′ + ch ′ + f ′ u z, w(z) h. Lemma 3.3 Assume that w ∈ H 2 c (R) is a non negative, bounded solution of -w ′′ -cw ′ ≤ -δw on R\(-R, R) such that w(±∞) = 0, then w(z) ≤ w(-R)e λ + (
Proof. Define φ -(z) := w(-R)e λ + (z+R) , ∀ z < -R, h(z) := w(z) -φ(z), ∀ z < -R.
Then h is solution of

-h ′′ -ch ′ + δh ≤ 0 for all z < -R, h(-∞) = 0, h(-R) ≤ 0.
Assume that h achieves a maximum at z 0 ∈ (-∞, -R). This would imply that h(z 0 ) ≤ 0 and thus h ≤ 0 in (-∞, -R]. Otherwise either h admits a minimum in (-∞, -R) or is monotone on (-∞, -R), which also implies that h ≤ 0 in (-∞, -R] and the first inequality is proved. The inequality on [R, ∞) is proved similarly.

Lemma 3.4 There exists z -∈ R such that, if w 1 , w 2 ∈ H 2 c (R) are two positive, bounded, solutions of w ′′ + cw ′ + f (z, w) = 0 over R with w 1 (z) = w 2 (z) for some z ≤ z -, then w 1 ≡ w 2 .
Proof. Let u := (w 1 -w 2 ) 2 . This function satisfies

u ′′ + cu ′ = 2(w ′ 1 -w ′ 2 ) 2 + 2 -f (z, w 1 ) + f (z, w 2 ) (w 1 -w 2 ) ≥ -2f ′ u (z, 0)u -2 -f (z, w 1 ) + f (z, w 2 ) -f ′ u (z, 0)(w 2 -w 1 ) |w 1 -w 2 |.
On the other hand, Lemma 3.3 and the C 1 smoothness of f (z, s) with respect to s yields that there exists z -such that

∀z ≤ z -, |f (z, w 2 ) -f (z, w 1 ) -f ′ u (z, 0)(w 2 -w 1 )| ≤ δ 2 |w 2 -w 1 |
where δ is the constant defined by (1.4). We thus get

∀z ≤ z -, u ′′ + cu ′ ≥ -2f ′ u (z, 0)u -δu ≥ δu
decreasing z -once more if necessary. It now follows from this inequation that u cannot reach any local maximum over (-∞, z -). As u(-∞) = 0 and u ≥ 0, it implies that u is nondecreasing. Lastly, if w 1 (z) = w 2 (z) for some z ≤ z -, then u(z) = 0 and thus u ≡ 0, meaning that w 1 ≡ w 2 . Lemma 3.5 dimKerL w ∈ {0, 1}.

Proof. The Cauchy theorem yields that

KerL w = {h ∈ H 2 c (R), h ′′ + ch ′ + f ′ u z, w(z) h = 0}
has at most dimension 2. If it has dimension 2, then it would mean that for all z 0 ∈ R and for all couple (h 0 , h 1 ), the solution of

h ′′ + ch ′ + f ′ u z, w(z) h = 0, h(z 0 ) = h 0 , h ′ (z 0 ) = h 1 belongs to H 2 c (R).
In particular h(+∞) = 0. But now the same arguments as in the proof of Lemma 3.4 yields that h 2 is nonincreasing over (z + , +∞) and thus one reaches a contradiction by taking z 0 > z + and (h 0 , h 1 ) such that h 0 h 1 > 0.

Lemma 3.6 Assume that dimKerL w = 1. Then there exists a constant

C = C(w) such that for all g ∈ L 2 c (R), if u ∈ H 2 c (R) satisfies L w u = g in R and R e cz u(z)v(z)dz = 0 for all v ∈ KerL w , then u H 2 c (R) ≤ C g L 2 c (R) . Moreover, if W is a family of solutions w ∈ H 1 c (R) of (S)
such that dimKerL w = 1 for all w ∈ W and sup w∈W w H 1 c (R) < ∞, then the constant C can be chosen to be the same for all w ∈ W .

Proof. Clearly the operator

T : (KerL w ) ⊥ → ImL w h → L w h
where (KerL w ) ⊥ is with respect to the scalar product of L 2 c . T is invertible and continuous. Hence the bounded inverse theorem yields that its inverse is continuous. Taking C its continuity constant, this means that for all g ∈ L 2 c (R) such that there exists u ∈ (KerL w ) ⊥ satisfying L w u = g, one has u H 2 c (R) ≤ C g L 2 c (R) and the result follows. Next, we first prove that there exists C > 0 such that if W is a family of solutions w ∈ H 1 c (R) of (S) such that KerL w = {0} for all w ∈ W and sup w∈W w

H 1 c (R) < ∞, then u ′ L 2 c (R) ≤ C g L 2 c (R)
. Assume that this is not true, there would exist a sequence (w n ) n of solutions of (S), bounded in H 1 c (R), such that KerL wn = {0} for all n and the associated constants C n = C(w n ) converge to +∞ as n → +∞. In other words, there exist v n ∈ KerL wn for all n and two sequences

(u n ) n in H 2 c (R) and (g n ) n in L 2 c (R) such that L wn u n = g n in R, R e cz u n (z)v n (z)dz = 0, u ′ n L 2 c (R) = 1 for all n and lim n→+∞ g n L 2 c (R) = 0. Up to multiplication, we can assume that v ′ n L 2 c (R) = 1. As (w n ) n is bounded in H 1 c (R)
, we can assume, up to extraction, that it converges locally uniformly to some function w ∞ ∈ H 1 c (R). Similarly, the Poincaré inequality stated in Lemma 2.2 yields that (u n ) n and (v n ) n are indeed bounded in H 1 c (R) and we can thus define their weak limits

u ∞ and v ∞ in H 1 c (R). As u ′′ n = -cu ′ n -f ′ u z, w n (z) u n + g n , multiplying by u n e cz and integrating over R, as u ′ n L 2 c(R) = 1, we get 1 - R e cz f ′ u (z, w n )u 2 n dz = - R e cz u n g n dz.
As u n converge weakly in L 2 c and g n → 0 in L 2 c as n → +∞, the right-hand side converges to 0 as n → +∞. Assuming u n ⇀ 0 in L 2 c yields a contradiction. Indeed, using Lemma 3.3 for all n, for all ε > 0 there exists r > 0 such that w n (z) < ε for all |z| > r. As f (z, •) is C 1 , for ε small enough, f ′ u (z, w n ) < 0 for all |z| > r. And we obtain

1 - r -r e cz f ′ u (z, w n )u 2 n dz ≤ - R e cz u n g n dz,
which yields a contradiction when we let n → +∞, as u n → 0, strongly in L 2 c ([-r, r]). This implies that u ∞ ≡ 0. Using the same arguments with v n , as

v ′ n L 2 c (R) = 1 for all n, we have that 1 - R e cz f ′ u (z, w n )v 2 n dz = 0.
On the other hand, it follows from Lemma 3.3 that one can apply the dominated convergence theorem using the bounds v n (z) ≤ M for all z < R and v n (z) ≤ M e λ -(z-R) for all z > R, since c < -2λ -. We thus obtain

1 - R e cz f ′ u (z, w ∞ )v 2 ∞ dz = 0.
Moreover, classical elliptic regularity estimates yield that v ∞ satisfies L w∞ v ∞ = 0 in R.

Integrating by parts, we get

R e cz (v ′ ∞ ) 2 -f ′ u (z, w ∞ )v 2 ∞ dz = 0. We thus conclude that v ′ ∞ L 2 c (R) = 1. As L 2 c (R) is an Hilbert space, this indeed implies that (v ′ n ) n converges strongly to v ′ ∞ in L 2 c (R) as n → +∞.
Using the Poincaré type inequality given in Lemma 2.2 we have that v n → v ∞ in L 2 c as n → +∞. This implies that R e cz u ∞ (z)v ∞ (z)dz = 0, L w∞ u ∞ = 0 and L w∞ v ∞ = 0 over R. Hence, dimKerL w∞ = 2, which contradicts Lemma 3.5.

Thus there exists C > 0 such that if W is a family of solutions w ∈ H 1 c (R) of (S) such that KerL w = {0} for all w ∈ W and sup w∈W w

H 1 c (R) < ∞, then u ′ L 2 c (R) ≤ C g L 2 c (R)
. Now to prove the last assertion of the Lemma we just apply Lemma 2.2 to get a bound on ||u|| L 2 c (R) and the bound on ||u ′′ || L 2 c (R) follows from the equation. Lemma 3.7 Assume that for some T > 0, there exist two constants K, C > 0 such that for all t ∈ [0, T ],

∞ t R e cz u 2 t (s, z)dsdz ≤ Ke -Ct .
Then for all 0 ≤ t ≤ τ ≤ T , one has:

||u(t, •) -u(τ, •)|| L 2 c (R) ≤ √ K 1 -e -C/2 e -Ct/2 .
Proof: This Lemma is similar to Lemma 4 in Zelenyak paper [24, Lemma 4]. As our solutions are defined on the full line R instead of a segment, we obtain a control in

L 2 instead of L 1 . Assume first that |t -τ | ≤ 1. Then ||u(t, •) -u(τ, •)|| 2 L 2 c (R) = R e cz τ t u t (s, z)ds 2 dz ≤ R τ t (τ -t)e cz u 2 t (s, z)dsdz ≤ R ∞ t e cz u 2 t (s, z)dsdz ≤ Ke -Ct . Next, if |t -τ | > 1, let N = [τ -t]
be the integer part of τ -t. We compute:

||u(t, •) -u(τ, •)|| L 2 c (R) ≤ N -1 n=0 ||u(t + n, •) -u(t + n + 1, •)|| L 2 c (R) + ||u(t + N, •) -u(τ, •)|| L 2 c (R) ≤ N -1 n=0 √ Ke -C(t+n)/2 + √ Ke -C(t+N )/2 ≤ √ K 1 -e -C/2 e -Ct/2 ,
which ends the proof.

Proof of Theorem 1.2: Assume that Ω(u 0 ) is not an isolated point. Using Lemma 3.2 and Lemma 3.4 we can choose R large enough such that Ω(u 0 ) is parametrised by the value of the function at -R, i.e Ω(u 0 ) = {w(α, •), w(α, -R) = α and w is a stationary solution}. As u is bounded, the quantities 0 ≤ α 1 = lim inf t→+∞ u(t, -R) < α 2 = lim sup t→+∞ u(t, -R) are well-defined and classical connectedness and compactness arguments yield that for all α ∈ [α 1 , α 2 ], as t → u(t, -R) is continuous, by Schauder parabolic regularity estimates, there exists (t n ) n such that u(t n , -R) → α as n → +∞. Hence Ω(u 0 ) is the curve {w(α, •), α ∈ [α 1 , α 2 ]}.

For each w(α, •) ∈ Ω(u 0 ), we have that α → w(α, •) is continuous with respect to the L ∞ (R) norm, using the uniqueness property proved in Lemma 3.4 and the elliptic regularity. Define

v ε := w(α + ε, •) -w(α, •) ε , it satisfies -v ′′ ε -cv ′ ε -(f ′ u (z, w(α, •)) + R ε )v ε = 0, with R ε L ∞ (R) → 0 as ε → 0 since α → w(α, •) is continuous and s → f (•, s) is C 1 .
Then applying Lemma 3.3 to the nonnegative and nonpositive parts of v ε , we get that

|v ε (z)| ≤ |v ε (-R)|e λ + (z+R) = e λ + (z+R) , ∀ z ∈ (-∞, -R),
where we have used that

v ε (-R) = w(α + ε, -R) -w(α, -R) ε = α + ε -α ε .
This implies for all h > 0:

e -λ + h -1 h ≤ | v ε (-R -h) -v ε (α, -R) -h | ≤ -e -λ + h + 1 h .
Letting h → 0 we get that v ′ ε (-R) is bounded uniformly with respect to ε. The continuity with respect to the initial condition and parameters for ordinary differential equations yields that (v ε ) ε is bounded in C 2 loc and using Lemma 3.3 again

|v ε (z)| ≤ |v ε (R)|e -λ -(z-R) , ∀ z ∈ (R, +∞).
Up to a subsequence v ε → v in C 2 loc by elliptic regularity, and v ∈ L 2 c (R) using the uniform exponential convergence. We have thus proved that v = ∂w ∂α is well-defined and is a solution of v(-R) = 1 and

L w v = v ′′ + cv ′ + f ′ u z, w(α, •) v = 0 over R.
Lastly, multiplying the equation by ve cz , integrating over R we have that

R (v z ) 2 e cz dz = R f ′ u (z, w(α, •))v 2 e cz dz,
integrating by part and using the exponential convergence of v. As f (•, s) is non positive outside (-R, R), we get

R (v z ) 2 e cz dz ≤ C R -R
v 2 e cz dz.

Hence v ∈ H 1 c (R). We have v ≡ 0 in R. Now we define for fixed t > 0,

α(t) := arg inf ||u(t, •) -w(α, •)|| L 2 c (R) , α ∈ [α 1 , α 2 ] . For each t > 0, if the inf is attained at an interior point α(t) ∈ (α 1 , α 2 ), then ∂ ∂α ||u(t, •) -w(α, •)|| 2 L 2 c (R) α=α(t)
= 0, and thus R e cz u(t, z) -w(α, z) ∂w ∂α α=α(t) dz = 0.

We thus have for all t > 0 such that α(t) ∈ (α 1 , α 2 ):

L w(α(t),•) v = 0, R e cz (u -w)v| α=α(t) dz = 0 and L w(α(t),•) (u -w) = g, with g(t, z) := u t (t, z) + b(t, z)(u(t, z) -w α(t), z) , b(t, z) := f ′ u z, w(α(t), z) - f z, u(t, z) -f z, w(α(t), z) u(t, z) -w(α(t), z) .
Lemma 3.6 thus applies and gives

||u(t, •) -w(α(t), •)|| H 2 c (R) ≤ C||u t (t, •)|| L 2 c (R) + C||b(t, •)|| L 2 c (R) ||u(t, •) -w(α(t), •)|| L 2 c (R) , for all t > 0 such that α(t) ∈ (α 1 , α 2 ). But as f = f (z, u) is of class C 1 with respect to u uniformly in z and as lim t→+∞ ||u(t, •) -w(α(t), •)|| L 2 c (R) = 0, one has ||b(t, •)|| L 2 c (R)
→ 0 as t → +∞ and it thus follows that, even if it means increasing C, for all admissible t > 0, one has

||u(t, •) -w(α(t), •)|| H 2 c (R) ≤ C||u t (t, •)|| L 2 c (R) . and C is bounded independently of α(t) ∈ (α 1 , α 2 ).
Now ending the proof as in Zelenyak [START_REF] Zelenyak | Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable[END_REF], we have that for all t > 0 and any w ∈ Ω(u 0 ), the solution u of our parabolic problem satisfies

E c [u(t, •)] -E c [w] = 1 2 R e cz u 2 z (t, z) -w 2 z (z) dz - R e cz F (z, u(t, z)) -F (z, w(z)) dz = 1 2 R e cz (u z -w z ) 2 dz + R e cz (u z -w z )w z dz - R e cz f z, w(z) (u(t, z) -w(z))dz + R e cz σ(t, z)(u(t, z) -w(z)) 2 dz,
where σ = σ(t, z) is a bounded and measurable function since f = f (z, u) is of class C 1 with respect to u, uniformly in z. As w is a stationary solution of (S), integrating by parts, we get

E c [u(t, •)] -E c [w] = 1 2 R e cz (u z -w z ) 2 dz + R e cz σ(t, z)(u(t, z) -w(z)) 2 dz ≤ sup{ 1 2 , σ L ∞ (R) }||u(t, •) -w|| 2 H 1 c (R)
. Next, we have shown in the proof of Proposition 3.1 that the energy is decreasing and bounded from below, so E ∞ c := lim t→+∞ E c [u(t, •)] is well-defined. For all t > 0 such that α(t) ∈ (α 1 , α 2 ), gathering the previous inequalities, one gets

d dt (E c (u(t, •)-E ∞ c ) = -||u t (t, •)|| 2 L 2 c (R) ≤ -C -1 ||u(t, •)-w(α(t), •)|| 2 H 2 c (R) ≤ -K(E c [u(t, •)]-E ∞ c ), (3.2) where K is an explicit constant and E ∞ c = lim t→+∞ E c [u(t, •)] is equal to E c [w] for all w ∈ Ω[u 0 ]. Now let α 0 ∈ (α 1 , α 2
) and take a sequence (t n ) n such that lim n→+∞ t n = +∞ and lim n→+∞ u(t n , z) = w(α 0 , z). There exists η > 0 such that

||w(α 0 , •) -w(α 1 , •)|| L 2 c (R) > η and ||w(α 0 , •) -w(α 2 , •)|| L 2 c (R) > η. Choose N large enough such that ||u(t N , •) -w(α 0 , •)|| L 2 c (R) ≤ η 8 and for all t ≥ t N E c [u(t, •)] -E ∞ c ≤ (1 -e -C/2 ) η 8 .
We set

t = inf t ≥ t N , ||u(t, •)-w(α 0 , •)|| L 2 c (R) ≥ min{||u(t, •)-w(α 1 , •)|| L 2 c (R) , ||u(t, •)-w(α 2 , •)|| L 2 c (R) } .
Clearly α(t) = α 1 and α(t) = α 2 , that is, α(t) is an interior point, for all t ∈ [t N , t). Hence, inequality (3.2) holds for all t ∈ [t N , t), i.e

E c [u(t, •)] -E ∞ c ≤ (E c [u(t N , •)] -E ∞ c ) e -C(t-t N )
. By Lemma 3.7, one has for all t N ≤ t ≤ τ ≤ t:

||u(t, z) -u(τ, z)|| L 2 c (R) ≤ E c [u(t N , •)] -E ∞ c 1 -e -C/2 e -C(t-t N )/2 ≤ η 8 e -C(t-t N )/2 . (3.3)
If t is finite then from the previous inequality we obtain that

||u( t, •) -w(α 0 , •)|| L 2 c (R) ≤ ||u( t, •) -u(t N , •)|| L 2 c (R) + ||u(t N , •) -w(α 0 , •)|| L 2 c (R) ≤ η 4 (3.4)
and, for k = 1 and k = 2:

||u( t, •) -w(α k , •)|| L 2 c (R) ≥ ||w(α k , •) -w(α 0 , •)|| L 2 c (R) -||u( t, •) -w(α 0 , •)|| L 2 c (R) ≥ η - η 4 = 3 4 η.
(3.5) Comparing (3.4) and (3.5) we conclude that inf ||u( t, •) -w(α, •)|| L 2 c (R) cannot be attained for α = α k ,(k = 1, 2), and thus t = ∞. We thus conclude that (3.3) holds for all τ ≥ t ≥ t N which proves that t → u(t, •) converges strongly in L 2 c . This contradicts the assumption that Ω(u 0 ) is not an isolated point and concludes the proof.

4 On the stability of the trivial steady state 0

In this section we discuss the different behaviours of the solution of ( P ) depending on the stability of 0 and the initial condition u 0 . We first define what we mean by stability of the trivial steady state 0.

Let L be the linearized operator around 0:

-Lu := -u ′′ -cu ′ -f s (z, 0)u, defined for all u ∈ H 1 (R). It is easy to check (using Lemma 3.3) that the operator L admits a principal eigenfunction in

H 1 c (R), that is there exist (λ c , φ) such that      -Lφ = λ c φ in R, φ > 0 in R, φ ∈ H 1 c (R). (4.1)
This eigenvalue λ c is also characterized as the generalized eigenvalue of L:

λ c (-L, R) := sup λ ∈ R, ∃φ ∈ W 2,1 loc (R), φ > 0, (L + λ)φ ≤ 0 a.e in R . (4.2)
One can look at [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space[END_REF] and references therein for more details about generalized eigenvalue. We know from [4, Proposition 1 -section 2] that, if we denote by λ(r) the principal eigenvalue of our problem on B r with Dirichlet boundary condition, then λ(r) → λ c as r → +∞ and there exists φ c ∈ W 2,p loc (R), 1 ≤ p < +∞, the principal eigenfunction solution of (4.1).

Letting v(z) = u(z)e cz 2 , then

-Lu = 0 ⇐⇒ -Lv = -v ′′ + c 2 4 v -f s (z, 0)v = 0,
where L is self adjoint. From [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space[END_REF][START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]

λ c (-L, R) = λ c (-L, R) = inf φ∈H 1 (R),φ ≡0 R φ ′ (z) 2 + ( c 2 4 -f s (z, 0))φ(z) 2 dz R φ(z) 2 dz . (4.3)
If we define λ 0 as the generalized eigenvalue corresponding to c = 0, i.e when the medium does not move with time, then we have that

λ c = λ 0 + c 2 4 .
We will say that 0 is linearly stable (respectively unstable) if λ c > 0 (respectively λ c < 0). Let us notice that if 0 is stable in the steady frame, i.e λ 0 > 0, then 0 is necessarily stable in the moving frame.

Convergence to a travelling wave solution when 0 is linearly unstable

In this section we want to prove that when 0 is linearly unstable, i.e λ 0 < 0 and c < 2 √ -λ 0 , for u 0 ≡ 0 nonnegative initial condition, the solution u of ( P ) converges to a non trivial travelling wave solution as time goes to infinity. Proposition 4.1 Let us assume that λ 0 < 0 and that f satisfies (1.1)-(1.4), then for all c < 2 -λ 0 , inf

u∈H 1 c (R) E c [u] < 0,
i.e there exists a non trivial solution of (S)

If f is of KPP-type, that is u → f (z,u)
u is decreasing, this result is optimal, i.e inf

u∈H 1 c (R) E c [u] ≥ 0
and travelling wave solutions do not exist if c ≥ 2 √ -λ 0 (cf [START_REF] Berestycki | Can a species keep pace with a shifting climate?[END_REF]). This is not true for general f (see the next section).

Proof of Proposition 4.1: Take λ such that λ 0 < λ < -c 2 /4. It follows from (4.3) that there exists

φ 0 ∈ H 1 (R) such that R φ ′ 0 (z) 2 -f s (z, 0)φ 2 0 (z) dz ≤ λ R φ 2 0 (z)dz.
Let

φ n (z) = φ 0 (z)e -c 2 z n ∀z ∈ R.
Then we have the following computation:

E c [φ n ] = R |(φ 0 (z)e -c 2 z ) z | 2 2n 2 -F z, φ 0 (z)e -c 2 z n e cz dz = R (φ ′ 0 (z)) 2 2n 2 + c 2 4 (φ 0 (z)) 2 2n 2 -F (z, 0) + F s (z, 0) φ 0 (z)e -c 2 z n + F ss (z, 0) (φ 0 (z)e -c 2 z ) 2 2n 2 + o (φ 0 (z)e -c 2 z ) 2 n 2 e cz dz = R (φ ′ 0 (z)) 2 2n 2 + c 2 4 (φ 0 (z)) 2 2n 2 -f (z, 0) φ 0 (z)e -c 2 z n + f s (z, 0) (φ 0 (z)e -c 2 z ) 2 2n 2 + o (φ 0 (z)e -c 2 z ) 2 n 2 e cz dz ≤ R (λ + c 2 4 ) (φ 0 (z)) 2 2n 2 dz + o( 1 n 2 ).
This implies that min

u∈H 1 c (R) E c [u] ≤ (λ + c 2 4 ) R (φ 0 (z)) 2 2n 2 dz + o( 1 n 2 ) < 0,
for n large enough. The Proposition is proved.

And we have the following Proposition to characterize the behaviour of u as time goes to infinity.

Proposition 4.2 If λ 0 < 0, for all c < 2 √ -λ 0 , the solution u of ( P ) converges to a non trivial solution of (S) as t → +∞.

Proof of Proposition 4.2: We will use the same argument as in [4, section 2.4]. We know that λ(R) → λ c as R → +∞, and λ c < 0 thus for R large enough λ(R) < 0 and let φ R > 0 be the principal eigenfunction. Define

U = κφ R in (-R, R), 0 otherwise, (4.4)
Then for κ small U is a subsolution of ( P ) and U ≤ u(τ, •) in R for some τ > 0 small, U ≡ M ≥ u 0 in R and is a super solution. Then the solution u of ( P ) is greater than U for all t > 0 and x ∈ R. Moreover using Theorem 1.2 we know that u converges to u ∞ ≥ U as t → +∞. And thus u converges to a non trivial travelling wave solution as t → +∞.

Remark 4.3 Notice that Proposition 4.2 contains Proposition 4.1 but the proof of Proposition 4.1 is interesting as it exhibits the link between the energy and the principal eigenvalue.

Existence of a travelling wave with positive energy when 0 is linearly stable

In this section we use the same notations than in the previous one and assume now that λ c > 0 and inf

u∈H 1 c (R) E c [u] < 0. (4.5)
In this framework, we show that the Mountain Pass Theorem applies and gives the existence of a travelling wave solution with positive energy. This provides a class of examples for which travelling wave solutions are not unique. Moreover, we construct in section 5.2 an example where uniqueness does not hold even in the class of stable travelling wave solutions with negative energy.

We also exhibit at the end of this section the dependence of the asymptotic limit on the initial condition and we show in Proposition 4.8 that, depending on the initial condition, we can converge either to 0 or to a travelling wave solution.

Proposition 4.4 Assume that λ c > 0, if min

u∈H 1 c (R)
E c [u] < 0 then there exists at least two non trivial travelling wave solution of (S) and one of them has a positive energy.

An easy application of this proposition is the following Corollary.

Corollary 4.5 Let f (z, u) = f 0 (u) if |z| < R, -δu otherwise,
where R, δ > 0, f 0 is a bistable function., i.e

There exists θ ∈ (0, 1) such that f 0 (0) = f 0 (θ) = f 0 (1) = 0, and f ′ 0 (0) < 0, f ′ 0 (1) < 0, f 0 (s) < 0 for all s ∈ (0, θ), f 0 (s) > 0 for all s ∈ (θ, 1), with positive mass:

1 0 f 0 (τ )dτ > 0. ( 4.6)
Then for R sufficiently large, there exists u, u

∈ H 1 c (R) solution of (S) such that E c [u] < 0 and E c [u] > 0.
Let us highlight this corollary which is totally different from what is known when f satisfies the KPP property. Indeed in the present framework 0 is linearly stable, nevertheless we still have the existence of travelling wave solutions.

Proof of Corollary 4.5. As f s (z, 0) = f ′ 0 (0) < 0 if |z| < R, -δ < 0 otherwise, one has λ 0 > 0 and thus λ c = λ 0 + c 2 /4 > 0.

Moreover, as f 0 has a positive mass, taking

w ∈ H 1 c (R), w ≡ 0, d 2 E c [0](w, w) > 0, with d 2 E c [0](w, w) = R e cz w 2 z -f s (z, 0)w 2 dz.
Using the equalities in (4.3) with φ(z) = e cz/2 w(z), we get,

d 2 E c [0](w, w) ≥ λ c w 2 H 1 c (R)
, for all w ∈ H 1 c (R), which proves the Lemma, as λ c is assumed to be positive. Now to prove Proposition 4.4, we want to use the Mountain Pass Theorem, so we need to prove that our energy functional satisfies the Palais-Smale Condition.

Lemma 4.7 If (u n ) n is a sequence in H 1 c (R) such that E c [u n ] ≤ C for all n ∈ N and dE c [u n ] → 0 as n → +∞ in (H 1 c ) * , in the sense that dE c [u n ] (H 1 c
) * → 0 as n → +∞, then there exists a subsequence, that we still call (u n ) n , which converges strongly in H 1 c (R) toward a solution u of dE c [u] = 0.

Here, (H 1 c ) * denotes the dual of the space H 1 c (R) for the extension of the L 2 c (R) scalar product. Proof of Lemma 4.7: As E c [u n ] ≤ C for all n ∈ N and using Lemma 2.4, we have

u n 2 H 1 c ≤ C + C 1 min{1, δ} ,
which implies that, up to a subsequence, (u n ) converges weakly to

u ∈ H 1 c (R). Moreover for all w ∈ H 1 c (R), dE c [u n ](w) → 0 as n → +∞, so 0 = lim n→+∞ dE c [u n ](w) = lim n→+∞ R e cz {(u n ) z w z -f (z, u n )w} dz = R e cz {u z w z -f (z, u)w} dz.
Hence dE c [u] ≡ 0. Now let us prove that (u n ) converges strongly to u in H 1 c (R) as n → +∞. We just need to prove that

u n H 1 c (R) → u H 1 c (R) as n → +∞, since H 1 c (R) is a Hilbert space. Taking w = u n we get R e cz (u n ) 2 z -f (z, u n )u n dz = dE c [u n ], u n (H 1 c ) * ,H 1 c (4.8) And dE c [u n ], u n (H 1 c ) * ,H 1 c ≤ dE c [u n ] (H 1 c ) * u n H 1 c = o(1), since (u n ) is bounded in H 1 c (R). Hence, dE c [u n ], u n → 0 as n → +∞. As dE c [u], u = 0, we have that R e cz u 2 z ds = R e cz f (z, u)udz.
Using the same arguments as in Proposition 2.5 we have that for all ε > 0,

lim n→+∞ R e cz f (z, u n )u n dz ≤ R e cz f (z, u)udz + ε.
This inequality and (4.8) implies that

u H 1 c (R) ≤ lim inf n→+∞ u n H 1 c (R) ≤ lim sup n→+∞ u n H 1 c (R) ≤ u H 1 c (R) + ε,
for all ε > 0. One has proved the Lemma.

Proof of Proposition 4.4: As assumed in the Proposition min We want to prove that we can always find non trivial initial conditions u 0 ≡ 0 such that u solutions of ( P ) converge to 0 and to a travelling wave solution.

Proposition 4.8 Assume that (4.5) holds, then

• there exists u 0 ≡ 0, compactly supported, such that the solution u of ( P ) converges to 0 as t → +∞,

• there exists u 0 ≡ 0, compactly supported, such that the solution u of ( P ) converges to a travelling wave solution as t → +∞.

Proof : We noticed in the previous section that λ c = λ 0 + c 2 4 and if λ 0 > 0, then λ c > 0. We know that there exists a positive function φ ∈ W 2,p loc (R), for any 1 ≤ p < +∞, such that

-φ ′′ -cφ ′ -f s (z, 0)φ = λ c φ in R.
Let w(t, z) := κφ(z)e -δt for all t ≥ 0, z ∈ R, κ > 0, δ > 0 some constants that we specify later. Then w satisfies the following equation

w t -w zz -cw z = (f s (z, 0) + λ c -δ)w.
As λ c > 0, choosing δ = λ c 2 , there exists κ > 0 small enough such that w t -w zz -cw z ≥ f (z, w).

Thus if u 0 ≤ κφ in R, using the weak parabolic maximum principle we have that for all t ≥ 0, z ∈ R, u(t, z) ≤ κφ(z)e -δt , for some constants κ > 0, δ > 0 small enough. Now we prove the second assertion. As inf

u∈H 1 c (R) E c [u] < 0 there exists u 0 ≡ 0, compactly supported such that E c [ u 0 ] < 0. As t → E c [
u] is decreasing, its limit is negative. Thus u converges to a solution of (S) with negative energy, i.e u converges to a travelling wave solution. This proves Proposition 4.8.

Examples and discussion

Numerical simulations

In this section we illustrate the behaviour of the solution of the parabolic problem considering different types of reaction terms f , different values of δ and c. We solve numerically the following problem

           ∂ t u -∂ zz u -c∂ z u = f (z, u), for t ∈ [0, T ], z ∈ [0, L], u(0, z) = e -( z-L/2 l ) 2 , for z ∈ (0, L), u(t, 0) = 0, for t ∈ [0, T ], u(t, L) = 0, for t ∈ [0, T ], (5.1)
where

f (z, u) = f 0 (u) if L 2 -l 2 < z < L 2 + l 2 , -δu otherwise, (5.2)
with L = 300, T = 150, l = 30. We compute this problem using FreeFem++ with ∆x = 10 -1 and ∆t = 10 -1 . As our solution converges to 0 as z → ±∞ and the initial condition in (5.1) is approximatively equal to 1.4 × 10 -11 on the boundary of the domain, we approximate our problem ( P ) by a Dirichlet boundary value problem with L large enough.

Existence of a critical speed

We consider three types of reaction function f 0 : the KPP case, the monostable case and the bistable case (see figure 1). We restrict our analysis to [0, T ] × [0, L] and take T and L large enough to act as if it was +∞.

(u) = u(1 -u), Monostable nonlinearity: f 0 (u) = u 2 (1 -u) and Bistable nonlinearity: f 0 (u) = u(1 -u)(u -0.2).
In [START_REF] Berestycki | Can a species keep pace with a shifting climate?[END_REF] and [START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space[END_REF] the authors studied the asymptotic behaviour of the parabolic solution and more precisely the existence of non trivial travelling wave solution in the KPP case, i.e f 0 (u) u is maximal when u = 0. The authors proved that there exist travelling wave solutions if and only if λ 0 < 0 and c < 2 √ -λ 0 , where λ 0 is the generalized eigenvalue when c = 0. In other words there exists a critical speed c lin = 2 -λ 0 such that c = c = c lin in Theorem 1.1.

In our paper we consider more general nonlinearities f and do not assume that f satisfies the KPP property. We proved in Theorem 1.1 that there exists c ≤ c such that there exist travelling wave solutions for all c < c and the only solution of (S) is 0 for all c > c. We wonder if in this general framework, there still exists a critical speed, that is, c = c. We investigate this conjecture numerically in the monostable and bistable case. As the initial data gathers a lot of mass in the favourable area [ L 2 -l 2 ; L 2 + l 2 ], while it is small in the unfavourable environment, we believe that the solution will converge to a travelling wave solution when it exists for reasonable nonlinearities. The existence of a critical speed has already been introduced in [START_REF] Roques | A population facing climate change: joint influences of allee effects and environmental boundary geometry[END_REF], where the authors highlight some monotonicity of the global population with respect to the speed c. Figure 2 displays the behaviour proved analytically in [START_REF] Berestycki | Can a species keep pace with a shifting climate?[END_REF][START_REF] Berestycki | Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space[END_REF]: there exists a critical speed c (around 2) such that for c < c the population survives whereas for c > c the population dies. In Figure 3 and 4 one can observe the same phenomenon but for lower critical speeds.

Let us also notice that, as proved in Corollary 4.5, we still have existence of travelling waves when λ c > 0 in the bistable case (Figure 4 for c ∈ [0, 0.4]). We now investigate the shape of the front when δ varies and f is bistable, i.e f 0 (u) = u(1 -u)(u -0.2). When δ is small (figure 5), a tail grows at the bottom of the front whereas the transition at the front edge of the front stays sharp when the speed c > 0 is small enough for the population to survive, as it was already observed by Berestycki et al [START_REF] Berestycki | Can a species keep pace with a shifting climate?[END_REF] for KPP nonlinearity. This tail is created by the movement of the favourable environment, indeed the death rate δ is too small to kill the population which reproduced quickly in the favourable zone. When the speed is too large the population can not keep tracks with its favourable environment and slowly converges to 0. On the other hand when c=0, both edges of the front become less and less sharp. Then we see that when c > 0 (small enough for the population to survive), both edges of the front become sharper and sharper as δ increases (Figures 5, 6 and7).

Non uniqueness of stable travelling waves

We can also build f such that (S) has more than one stable solution with negative energies in the sense that the solutions are local minimisers of the energy functional. Let f be as follow

f (z, u) = f 0 (u) if |z| < R, -δu otherwise, (5.3)
where f 0 is a multistable function, i.e there exist 0 < θ 0 < 1 < θ 1 < C such that

f (0) = f (θ 0 ) = f (1) = f (θ 1 ) = f (C) = 0, f (
s) < 0, for s ∈ (0, θ 0 ) ∪ (1, θ 1 ), f (s) > 0 for s ∈ (θ 0 , 1) ∪ (θ 1 , C), 1 0 f 0 (s)ds > 0 and C 0 f 0 (s)ds > 1 0 f 0 (s)ds (one can look at Figure 8 for an example of f 0 ), and δ > 0. Such multistable functions have already been used in other frameworks in order to construct multiple stable solutions of semilinear problems (see [START_REF] Brown | On the existence of positive solutions for a class of semilinear elliptic boundary value problems[END_REF] for example).

We start with the proof of the following Lemma.

Using Proposition 2.5 we know that there exists u * , travelling travelling wave solution of (S) with f * for some c > 0 such that min We have proved the Lemma.

Proof of Propostion 5.1: Now let us prove that there exists v * ∈ H 1 c (R) solution of (S) such that E c [v * ] < E c [u * ] < 0. Let u 3 be as follow,

u 3 (z) = C if |z| < R, 0 if |z| > R + ε, (5.6)
such that u 3 ∈ H 1 c (R). Then We now illustrate the previous results. Choosing a specific reaction term

f 0 (u) = u(1 -u)(u -0.2)(1.1 -u)(1.5 -u)
and an appropriate initial condition we get different convergence results as one can see in Figures 9 and10. We computed the same problem (5.1) that in section 5.1.2, with δ = 1 and f 0 (u) = u(1 -u)(u -0.2)(1.1 -u)(1.5 -u). In the first figure (Figure 9), one can see that depending on the initial condition, we get two different fronts but with a similar shape with sharp edge on both sides. On the other hand when c > 0 the front edge takes the shape of a stairs, indeed in the favourable environment the population moves rapidly to 1 but need more time to grow from 1 to 1.5.

∃u ∈ H 1 ( 2 z 2 - 0 f

 1220 R), E 0 [u] := R u F (z, u) dz < 0, with F (z, s) := s (z, t)dt, (1.5)

u

  min (z) = 1 for all |z| < R, 0 for all |z| > R + 1, (4.7) such that u min ∈ H 1 c (R), one can check that for R large enough E c [u min ] < 0 and u min H 1 c > r. Proposition 4.4 applies and gives the conclusion. To prove Proposition 4.4 we start with the following Lemma. Lemma 4.6 For all r > 0 small enough, one has inf u H 1 c (R) =r E c [u] > 0. Proof of Lemma 4.6: To prove this Lemma, we just need to prove that 0 achieves a strict local minimum, i.e dE c [0] ≡ 0 and d 2 E c [0] > 0 in the sense that for all

u∈H 1 c

 1 (R) E c [u] < 0. By Proposition 2.5, this minimum is reached for some u min ∈ H 1 c , and one has u min ≡ 0 since E c [0] = 0. Lemma 4.6 yields that for all r > 0 sufficiently small, one has inf u H 1 c (R) =r E c [u] > 0. Choose r small enough such that u min H 1 c > r. Then using the Mountain Pass Theorem, there exists u ∈ H 1 c such that dE c [ u] ≡ 0 and E c [ u] ≥ γ. We have proved Proposition 4.4.

Figure 2 :Figure 3 :

 23 Figure 2: Average of the population P (t) = L 0 u(t, x)dx for c ∈ [0, 3] in the KPP case (L=300)

Figure 4 :

 4 Figure 4: Average of the population P (t) = L 0 u(t, x)dx for c ∈ [0, 3] in the bistable case (L=300)

Figure 5 :

 5 Figure 5: Solution of (5.1) for δ = 0.001 and c = 0; 0.4 and 0.8 (from left to right) for t=150. For c = 0.8, the figure on the right displays an intermediate state slowly converging to 0.

Figure 6 :

 6 Figure 6: Solution of (5.1) for δ = 1 and c = 0.4 for t=150.

Figure 7 :Proposition 5 . 1

 751 Figure 7: Solution of (5.1) for δ = 10 and c = 0.4 for t=150.

  29

Figure 8 : f 0 a multistable function such that 1 0 1 0

 811 Figure 8: f 0 a multistable function such that 1 0f 0 (s)ds > 0, there exist 0 < θ 0 < 1 < θ 1 < C such that f (0) = f (θ 0 ) = f (1) = f (θ 1 ) = f (C) = 0, f (s) < 0, for s ∈ (0, θ 0 ) ∪ (1, θ 1 ) and f (s) > 0 for s ∈ (θ 0 , 1) ∪ (θ 1 , C) with C 0 f 0 (s)ds > 1 0 f 0 (s)ds.

1 c 1 ce

 11 ] = E * c [u * ], where E * c is the energy functional associated with f * . We know that u * ≤ 1 in R by Remark 2.3. Thus u * satisfies the following equation-(u * ) ′′ -c(u * ) ′ = f (z, u * ), and E c [u * ] = min u∈H 1 c E * c [u].Takingu min (z) = 1 for all |z| < R, 0 for all |z| > R + 1,(5.5)such that u min ∈ H 1 c (R), one can check that for R large enough E * c [u min ] < 0, which implies that E c [u * ] = min u∈H 1 c E * c [u] < 0.We have proved that there exists a solution u * ∈ H 1 c (R) of (S), such that 0 < u * in R and E c [u * ] < 0. Now let us prove that u * is a local minimiser. Using classical Sobolev injections, there exists ρ > 0 small enough, such thatu -u * H (R) < ρ =⇒ u -u * L ∞ (-R,R) ≤ θ 1 -1.Now let us prove that as soon asu -u * H (R) < ρ, then E c [u] ≥ E c [u * ]. E c [u] = R e cz (u ′ ) 2 2 -F (z, u) dz, cz {F * (z, u) -F (z, u)} dz. As u -u * L ∞ (-R,R) ≤ θ 1 -1, f * (z, u) ≥ f (z, u) for all z ∈ (-R, R), thus R -Re cz {F * (z, u) -F (z, u)} dz ≥ 0.

E c [u 3 2 e

 32 cz dz.Thus choosing C close enough to 1 and f 0 >> 0 in (θ 1 + η, C -η) for some η > 0, small, we haveE c [u 3 ] < E c [u * ].Using Proposition 2.5, we know that there existsv * ∈ H 1 c (R) such that E c [v * ] = min u∈H 1 c (R) E c [u] ≤ E c [u 3 ].One has proved Proposition 5.1.

Figure 9 : 2 l) 2 2 l) 2 Figure 10 : 2 l) 2 .

 922221022 Figure 9: Solution of (5.1) for c = 0 and t = 300 with u 0 (x) = e -( z-L/2 l ) 2 for the figure on

Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement n.321186 -ReaDi-Reaction-Diffusion Equations, Propagation and Modelling. Juliette Bouhours is funded by a PhD fellowship "Bourse hors DIM" of the "Région Ile de France".


Juliette Bouhours
email: juliette.bouhours@ljll.math.upmc.fr

Robustness for a Liouville type theorem in exterior domains

Keywords: 2010 Mathematics Subject Classification: 35K57, 35B51, 35B53 elliptic equation, Liouville type result, obstacle, maximum principle

We are interested in the robustness of a Liouville type theorem for a reaction diffusion equation in exterior domains. Indeed H. Berestycki, F. Hamel and H. Matano (2009) proved such a result as soon as the domain satisfies some geometric properties. We investigate here whether their result holds for perturbations of the domain. We prove that as soon as our perturbation is close to the initial domain in the C 2,α topology the result remains true while it does not if the perturbation is not smooth enough.

1 Introduction and main results

Problem and motivations

This paper investigates the exterior domain problem:

         -∆u = f (u) in R N \K, ∂ ν u = 0 on ∂K, 0 < u ≤ 1 in R N \K, u(x) → 1 as |x| → +∞ uniformly in x ∈ R N \K, (1.1)
where K is a compact set of R N , f is a bistable non-linearity. This problem is motivated by the construction of generalized transition fronts for the associated parabolic problem

u t -∆u = f (u) for all t ∈ R, x ∈ R N \K, ∂ ν u = 0 for all t ∈ R, x ∈ ∂K, (1.2)
such that sup

x∈R N \K |u(t, x) -φ(x 1 -ct)| → 0 as t → -∞,
where φ is a planar traveling wave connecting 1 to 0, i.e

-φ ′′ -cφ ′ = f (φ) in R, φ(-∞) = 1, φ(+∞) = 0. (1.
3)

It is proved in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] that the unique solution of (1.2) converges toward a solution of (1.1) as t → +∞. Thus problem (1.1) determines whether there is a complete invasion or not, that is whether u(t, x) → 1 as t → +∞ for all x ∈ R N \K. More precisely, complete invasion is shown to hold if and only if (1.1) has no solution different from 1. In [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF], Berestycki, Hamel and Matano have shown that if K is star-shaped or directionally convex the unique solution of (1.1) is 1 (see at the end of this section for precise definitions of star-shaped or directionally convex domain). The present paper examines under which conditions this Liouville type theorem is robust under perturbations of the domain. This is shown here to strongly depend on the smoothness of the perturbations that are considered. Indeed our main result is to show that it is true for C 2,α perturbations but not for C 0 ones. This is stated precisely in the next section. We leave as an open problem to determine what is the optimal space of regularity of the perturbation for which the result remains true.

In this paper f is assumed to be a C 1,1 ([0, 1]) function such that

f (0) = f (1) = 0, f ′ (0) < 0, f ′ (1) < 0, (1.4a)
and there exists θ ∈ (0, 1) such that, f (s) < 0 ∀s ∈ (0, θ), f (s) > 0 ∀s ∈ (θ, 1).

(1.4b)

Moreover we suppose that f satisfies the following positive mass property,

1 0 f (τ )dτ > 0. (1.5)
Before stating the main results, let explain what we mean by star-shaped or directionally convex obstacles.

Definition 1.1 K is called star-shaped, if either K = ∅, or there is x ∈ • K such that,
for all y ∈ ∂K and t ∈ [0, 1), the point x + t(yx) lies in • K and ν K (y) • (yx) ≥ 0, where ν K (y) denotes the outward unit normal to K at y. Definition 1.2 K is called directionally convex with respect to a hyperplane P if there exists a hyperplane P = {x ∈ R N , x • e = a} where e is a unit vector and a is some real number, such that • for every line Σ parallel to e the set K ∩ Σ is either a segment or empty,

• K ∩ P = π(K) where π(K) is the orthogonal projection of K onto P .

Main results

Our main result is the following Theorem Theorem 1.3 Let (K ε ) 0<ε≤1 be a family of C 2,α compact sets of R N , for some α > 0. Assume that K ε → K for the C 2,α topology as ε → 0, and K is either star-shaped or directionally convex with respect to some hyperplane P. Then there exists ε 0 > 0 such that for all 0 < ε < ε 0 , the unique solution of (1.1) is u ε ≡ 1 This theorem means that for obstacles that are compact sets in R N and close enough (in the C 2,α sense) to some star-shaped or directionally convex domains, the unique solution of (1.1) is the constant 1. And thus a sufficient condition for the Liouville theorem to be robust under perturbation is the C 2,α convergence. On the other hand one can prove that the C 0 convergence of the perturbation is not enough for the result to stay true. This is stated in the Theorem below.

Theorem 1.4 There exists (K ε ) ε a family of compact manifolds of R N such that K ε → B R 0 for the C 0 topology as ε → 0, and for all ε > 0 there exists a solution u ε of (1.1) such that

0 < u ε < 1 in R N \K ε . Notations We denote by B R 0 the ball of radius R 0 centred at 0 in R N , i.e B R 0 := x ∈ R N , |x| < R 0 ,
and by B r (x 0 ) the ball of radius r centred at x 0 in R N , i.e

B r (x 0 ) := x ∈ R N , |x -x 0 | < r ,
Remark 1.5 (C 0 or C 2,α convergence) When we write K ε → K for the X topology we mean that for each x 0 in ∂K, and for some r > 0 such that ∂K ε ∩ B r (x 0 ) = ∅ there exists a couple of functions ψ ε and ψ defined on B r (x 0 ), parametrization of K ε and K, such that,

ψ ε ∈ X(B r (x 0 )) and ψ ∈ X(B r (x 0 )) with ψ ε -ψ X(Br(x 0 )) → 0 as ε → 0.
For more details about the C 2,α topology one can look at [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], chapter 6.

Before proving the previous statements, let give some examples of domains (K ε ) ε and K to illustrate our results.

Examples of domains

We assume that N = 2 and we construct two families of obstacles; one which converges to a star-shaped domain and the other which converges to a directionally convex domain. The black plain line represents the limit K and the thin parts represent the small perturbations (of order ε). The long-dashed lines are used during the construction of K. For the star-shaped domain, it is on this line that we could find the center(s) of the domain (i.e the point x in Definition 1.1). For the directionally convex domain it represents the hyperplane P . We can clearly see that for all ε > 0, K ε is not star-shaped for figure 1 and not directionally convex for figure 2. One needs to be careful on the shape of the perturbations. Indeed in figure 3 below K ε converges to an ellipse as ε → 0 but the convergence of K ε is not C 2,α (see section 3 for more details) but only C 0 which is not enough for the Liouville theorem to remain valid. 2 Robustness of the result for C 2,α perturbations

In this section we prove the robustness of the Liouville result when the perturbation is close to a star-shaped or directionally convex domain in the C 2,α topology. To prove Theorem 1.3, we will need the following Proposition:

Proposition 2.1 For all 0 < δ < 1, there exists R = R δ such that, if u ε is a solution of (1.1) with K ε , then u ε (x) ≥ 1 -δ for all |x| ≥ R and for all 0 < ε < 1.
This proposition means that u ε converges toward 1 as |x| → +∞ uniformly in ε. Let first admit this result and prove Theorem 1.3.

Proof of Theorem 1.3

As u ε is uniformly bounded for all ε > 0, using Schauder estimates, we know that up to a subsequence u εn → u 0 in C 2 loc as n → +∞, in the sense that for all r > 0,

u εn -u 0 C 2 (Br\(Kε n ∪K)) → 0 as n → +∞
and u 0 satisfies:

∆u 0 + f (u 0 ) = 0 in R N \K, ν • ∇u 0 = 0 on ∂K. (2.1)
Using Proposition 2.1 we get lim |x|→+∞ u 0 (x) = 1. And K is either star-shaped or directionally convex. We now recall the following results from [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF]:

Theorem 1 (Theorem 6.1 and 6.4 in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF]) Let f be a Lipschitz-continuous function in [0, 1] such that f (0) = f (1) = 0 and f is nonincreasing in [1δ, 1] for some δ > 0. Assume that

∀ 0 ≤ s < 1, 1 s f (τ )dτ > 0. (2.2)
Let Ω be a smooth, open, connected subset of R N (with N ≥ 2) with outward unit normal ν, and assume that

K = R N \Ω is compact. Let 0 ≤ u ≤ 1 be a classical solution of      -∆u = f (u) in Ω, ν • ∇u = 0 on ∂Ω, u(x) → 1
as |x| → +∞.

(2.3)

If K is star-shaped or directionally convex, then u ≡ 1 in Ω. (2.4)
It thus follows that u 0 ≡ 1. It also proves that the limit u 0 is unique and thus u ε → u 0 as ε → 0 in C 2 loc (and not only along a subsequence). Now we need to prove that there exists ε 0 > 0 such that u ε ≡ 1 for all 0 < ε < ε 0 . Let assume that for all ε > 0, u ε ≡ 1 in Ω ε = R N \K ε . Then there exists x 0 ∈ Ω ε such that u ε (x 0 ) = min x∈Ωε u ε (x) < 1 and x 0 , which depends on ε, is uniformly bounded with respect to ε (using Proposition 2.1). Assume that u ε (x 0 ) > θ, as u ε is a solution of (1.1), the Hopf lemma yields that,

if x 0 ∈ ∂K ε then δu ε δν (x 0 ) < 0,
which is impossible due to Neumann boundary conditions. Hence x 0 ∈ Ω ε and

-∆u ε (x 0 ) = f (u ε (x 0 )) > 0,
which is impossible since x 0 is a minimizer. So, for all 0 < ε < 1,

0 ≤ min x∈Ωε u ε (x) ≤ θ,

But min

x∈Ωε u ε → 1 as ε → 0 by Proposition 2.1 and the local uniform convergence to u 0 ≡ 1, which is a contradiction. Thus there exists ε 0 such that for all ε < ε 0 , u ε ≡ 1.

Proof of Proposition 2.1

We will now prove Proposition 2.1, using the following lemma:

Lemma 2.2 There exists ω = ω(r) with r ∈ R + such that

           -ω ′′ (r) = f (ω(r)), ∀r ∈ R + * , ω(0) = 0, ω ′ (0) > 0, ω ′ > 0, 0 < ω < 1 in R * + , lim r→+∞ ω(r) = 1.
(2.5) This is a well known result. In deed, by a shooting argument, if ω is a solution of the initial value problem

     -ω ′′ = f (ω) in (0, +∞), ω(0) = 0, ω ′ (0) = 2F (1),
where F (1) = 1 0 f (s)ds, it is easily seen that ω is also a solution of (2.5). Proof of Proposition 2.1. Now we introduce a function f δ defined in [0, 1 -δ 2 ], satisfying the same bistability hypothesis as f but such that

• f δ ≤ f in [0, 1 -δ 2 ], • f δ = f in [0, 1 -δ], • f δ (1 -δ 2 ) = 0.
Notice that

1-δ 2 0
f δ (z)dz > 0 for δ small. Using the same arguments than in Lemma 2.2 there exists ω = ω δ such that

         -ω ′′ δ (x) = f δ (ω δ (x)) in (0, +∞), ω δ (0) = 0, ω δ (+∞) = 1 -δ 2 , 0 < ω δ < 1 -δ 2 in (0, +∞), ω ′ δ > 0 in (0, +∞).
(2.6)

As K ε is a compact set of R N converging to a fix compact set K, there exists R 0 such that K ε ⊂ B R 0 for all ε > 0.
Next, for any R > R 0 let consider z(x) = ω δ (|x| -R), for every |x| ≥ R. One gets:

-∆z < f (z) in R N \B R . (2.7)
We want to prove that

ω δ (|x| -R 0 ) < u ε (x), ∀x ∈ R N , |x| ≥ R 0 .
We know from (1.1) that u ε (x) → 1 as |x| → +∞. Hence there exists

A = A(ε) > 0 such that u ε (x) ≥ 1 - δ 3 > ω δ (|x| -A), for all |x| ≥ A. Consider R = inf {R ≥ R 0 ; u ε (x) > ω δ (|x| -R), for all |x| ≥ R} . (2.8) As R ≥ R 0 and K ε ⊂ B R 0 , u ε is always defined in |x| > R . One will prove that R = R 0 .
As ω δ is increasing, we know that

∀R ≥ A u ε (x) ≥ ω δ (|x| -R), ∀|x| ≥ R. Hence R ≤ A.
Assume that R > R 0 . Then there are two cases to study:

• either inf |x|>R u ε (x) -ω δ (|x| -R) > 0, (1)
• or inf

|x|>R u ε (x) -ω δ (|x| -R) = 0. ( 2
)
In the first case (1), one gets u ε (x) > ω δ (|x|-R) for all |x| > R. As ∇u ε and ω ′ δ are bounded, there exists R * < R such that u ε (x) ≥ ω δ (|x| -R * ) for all |x| > R * . This contradicts the optimality of R.

In the second case (2), there necessarily exists

x 0 with |x 0 | > R such that u ε (x 0 ) = ω δ (|x 0 | - R). Let v(x) = u ε (x) -ω δ (|x| -R), for all |x| > R.
As u ε is a solution of (1.1) and using (2.7), v satisfies:

-∆v > c(x)v in |x| > R , v > 0 on |x| = R , (2.9)
where c is a bounded function. From the definition of R, v(x) ≥ 0, for all |x| ≥ R. But there exists x 0 such that |x 0 | > R and v(x 0 ) = 0 which implies that v(•) ≡ 0. This is impossible because v(•) > 0, for all |x| = R.

Then R = R 0 which does not depend on ε and

∀|x| ≥ R 0 u ε (x) ≥ ω δ (|x| -R 0 ).
As ω δ (x) → 1-δ 2 as |x| → +∞, there exists R, independent of ε, such that for all |x| > R+R 0 , u ε (x) > ω δ (|x| -R 0 ) ≥ 1δ. One has proved Proposition 2.1.

3 Counter example in the case of C 0 perturbations Until now we have assumed that K ε → K in C 2,α , in order to use the Schauder estimates and ensure the convergence of u ε as ε → 0 . One can wonder if we can weaken this hypothesis, i.e would the C 0 or C 1 convergence be enough? We prove that C 0 perturbations are not smooth enough for the Liouville result to remain true.

Construction of a particular family of C 0 perturbations

In this subsection we construct a family of obstacles that are neither star-shaped nor directionally convex but converges uniformly to B R 0 which is convex. We want to prove that for all ε ∈]0, 1] there exists a solution of (1.1) which is not identically equal to 1. To do so we will use the counterexample of section 6.3 in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF]. We consider an obstacle K 1 = K η 1 (see figure 4 and5), such that:

K η ε K = B R0 R 0
1 R 1 R 2 A β 1 x 0 2η 4η x B R0
       A ∩ {x; x 1 ≤ x 0 1 } ∪ B R 0 ∪ C 1 ⊂ K η 1 , A ∩ x; x 1 > x 0 1 , |x ′ | > 2η ⊂ K η 1 , K η 1 ⊂ A ∩ x; x 1 > x 0 1 , |x ′ | > η ∪ B R 0 ∪ A ∩ x; x 1 ≤ x 0 1 ∪ C 1 . (3.1)
The proof of this corollary relies on the existence of a channel of width of order η > 0 opening on the interior of the annular region A (third assumption in (3.6)). This condition cannot be satisfied if the convergence of the obstacle is C 1 (see Lemma 3.1). The functional J B R \K η ε admits a local minimum in the ball of radius δ around v 0 in H 1 (B R \K η ε ) ∩ {v = 0 on ∂B R }. This yields a (stable) solution v of (3.5) for small enough η > 0. Furthermore, provided that δ is chosen small enough, this solution does not coincide either with 1 or with 0 in B R \K η ε . We have proved that for all ε ∈]0, 1], 0 < u ε < 1.

One has proved that C 0 convergence of the domain is not sufficient and thus Theorem 1.4.

One can conclude that if the perturbation is smooth in the C 2,α topology, we still have a Liouville type result for reaction diffusion equation in exterior domain. Whereas one can construct a counterexample of this Liouville result for C 0 perturbations. One question that is still open is thus the optimal space of regularity of the perturbation for the results to remain true under perturbation. For instance is the C 1 convergence of the perturbation enough to get the result?

Figure 1 :

 1 Figure 1: Obstacle that converges toward a star-shaped domain

Figure 2 :

 2 Figure 2: Obstacle that converges toward a directionally convex domain

Figure 3 :

 3 Figure 3: Obstacles converging only in the C 0 topology

Figure 4 :

 4 Figure 4: Liouville counterexample Zooming on the dashed part:

C

Figure 5 :

 5 Figure 5: Zoom on the perturbation

Acknowledgments

The author thanks Henri Berestycki, Grégoire Nadin and François Hamel for their fruitful discussions and encouragements, Sylvain Arguillère for his geometric insights. This study was supported by the French "Agence Nationale de la Recherche" through the project PREF-ERED (ANR 08-BLAN-0313) and by the French "Région Ile de France" through the fellowship "Bourse Hors DIM".

where x ′ = (x 2 , ..., x N ) and A = {x : R 1 ≤ |xx 0 | ≤ R 2 }, R 0 , R 1 < R 2 , are three positive constants, x 0 = (x 0 1 , 0, 0, ..., 0) is the center of the annular region A with x 0 1 = R 0 + R 2 + β 1 , C 1 is some corridor that links smoothly A and B R 0 which length is β 1 and η > 0, small enough. The family (K ε ) is constructed by downsizing K 1 such that for all 0 < ε < 1, A ε stays an annular region, with

• β ε converging to 0 as ε → 0.

We have the following lemma.

Lemma 3.1 K ε → K for the C 0 topology as ε → 0 but not for the C 1 topology.

This Lemma is easily proved using smooth parametrization of B R 0 and K 1 and noticing that for all ε > 0 there exists a point on the boundary of the perturbation that has an outward unit normal orthogonal to e 1 = (1, 0, ..., 0).

Existence of a non constant solution u ε of (1.1)

We want to prove that for all 0 < ε < 1 there exists a solution 0 < u ε < 1 of

We will follow the same steps as in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF], section 6. First, let notice that it is enough to find

for some R > 0 large enough such that K η ε ⊂ B R . Indeed then ω extended by 1 outside B R is a supersolution of (3.2) and one can define:

where U : R + → (0, 1) satisfies

Hence there exists a solution ψ < u ε < ω of (3.2). If we prove that ω ≡ 1 then 0 < u ε < 1 (with the maximum principle).

Now consider our problem (3.3) and replace ω by

(3.5)

Using exactly the same arguments as in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] one proves that, if we consider:

δ and v is a local minimizer of the associated energy functional in H 1 0 . For more clarity we will give the main steps of the proof but for details and proofs see [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF], section 6.3.

We introduce the energy functional in a domain D:

defined for functions of H 1 (D), where

g defined in (3.5). Using Proposition 6.6 in [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] one gets the following Corollary

More precisely, there exist α > 0 and δ > 0 for which

And then using Proposition 6.8 of [START_REF] Berestycki | Bistable traveling waves around an obstacle[END_REF] and Corollary 3.2 one gets the following Corollary Corollary 3.3 There exist γ > 0 and η 0 > 0 (which depend on ε) such that for all 0 < η < η 0 and v ∈ H