REACTION-DIFFUSION EQUATIONS FOR POPULATION DYNAMICS WITH FORCED SPEED II - CYLINDRICAL-TYPE DOMAINS

. This work is the continuation of our previous paper [6]. There, we dealt with the reaction-diﬀusion equation where e ∈ S N − 1 and c > 0 are given and f ( x,s ) satisﬁes some usual assumptions in population dynamics, together with f s ( x, 0) < 0 for | x | large. The interest for such equation comes from an ecological model introduced in [1] describing the eﬀects of global warming on biological species. In [6], we proved that existence and uniqueness of travelling wave solutions of the type u ( x,t ) = U ( x − cte ) and the large time behaviour of solutions with arbitrary nonnegative bounded initial datum depend on the sign of the generalized prin- cipal eigenvalue in R N of an associated linear operator. Here, we establish analogous results for the Neumann problem in domains which are asymptoti- cally cylindrical, as well as for the problem in the whole space with f periodic in some space variables, orthogonal to the direction of the shift e . The L 1 convergence of solution u ( t,x ) as t → ∞ is established next. In this paper, we also show that a bifurcation from the zero solution takes place as the principal eigenvalue crosses 0. We are able to describe the shape of solutions close to extinction thus answering a question raised by M. Mimura. These two results are new even in the framework considered in [6]. Another type of problem is obtained by adding to the previous one a term g ( x − c ′ te,u ) periodic in x in the direction e . Such a model arises when considering environmental change on two diﬀerent scales. Lastly, we also solve the case of an equation when f ( t,x,s ) is periodic in t . This for instance represents the seasonal dependence of f . In both cases, we obtain a necessary and suﬃcient condition for the existence, uniqueness and stability of pulsating travelling waves, which are solutions with a proﬁle which is periodic in time.


Introduction
In a recent paper [1], a model to study the impact of climate change (global warming) on the survival and dynamics of species was proposed. This model involves a reaction-diffusion equation on the real line In our previous paper [6], we extended the results of [1] to arbitrary dimension N : with c > 0 and e ∈ S N −1 given. The function f (x, s) : R N × R → R considered in [6] (which is slightly more general than in [1]) satisfies some usual assumptions in population dynamics, together with (2) lim sup |x|→∞ f s (x, 0) < 0.
In the ecological model, this assumption describes the fact that the favourable habitat is bounded. We proved in [6] that (1) admits a unique travelling wave solution, that is, a positive bounded solution of the form U (x − cte), if and only if the generalized principal eigenvalue λ 1 of an associated linear elliptic operator in the whole space is negative. Then, we were able to characterize the large time behaviour of any solution u of (1) with nonnegative bounded and not identically equal to zero initial datum. We showed that (i) if λ 1 ≥ 0 then u(t, x) → 0 as t → ∞, uniformly in x ∈ R N ; (ii) if λ 1 < 0 then (u(t, x) − U (x − cte)) → 0 as t → ∞, uniformly in x ∈ R N . We further considered the "two-speeds problem", obtained by adding a term g(x − c ′ te, u) to the "pure shift problem" (1), with x → g(x, s) periodic in the direction e. We derived analogous results to the previous ones, by replacing travelling waves with pulsating travelling waves.
Here, we deal with the same reaction-diffusion equation as in [6], but in different geometries.
We also investigate here the behaviour of travelling wave solutions near the critical threshold. This topic was not discussed in [6]. We prove that, when c crosses a critical value c 0 , a bifurcation takes place: stable travelling wave solutions U disappear and the trivial solution u ≡ 0 becomes stable. We characterize the shape of U near c 0 . Another type of results we derive here concerns the behaviour of the solution u(t, x) as t → ∞ in terms of the L 1 norm. This is done for problem (3) in the straight infinite cylinder as well as for problem (1) in the whole space treated in [6].
Finally, we consider the following problem: (6) ∂ t u = ∆u + f (t, x 1 − ct, y, u), t > 0, x 1 ∈ R, y ∈ ω ∂ ν u(t, x 1 , y) = 0, t > 0, x 1 ∈ R, y ∈ ∂ω, with f periodic in the first variable t. This equation serves as a model for instance to describe the situation in which the climate conditions in the "normal regime" (that is, in the absence of global warming) are affected by seasonal changes. The methods used to solve (6) also apply to the two-speeds problem (7) ∂ t u = ∆u + f (x 1 − ct, y, u) + g(x 1 − c ′ t, y, u), t > 0, x 1 ∈ R, y ∈ ω ∂ ν u(t, x 1 , y) = 0, t > 0, x 1 ∈ R, y ∈ ∂ω, with c ′ = c and g periodic in the x 1 variable. The term g enables one to describe situations in which some characteristics of the habitat -such as the availability of nutrient -are affected by the climate change on a time scale different from that of the overall change. One may also consider the case in which they are not affected at all: c ′ = 0 (mixed periodic/shift problem). However, the case of two or more cohabiting species is not treated here. One then has to consider systems of evolution equations (see e. g. [9], [16] and [11], where segregation phenomena are also described). This extension is still open.

Statement of the main results
2.1. Straight infinite cylinder. Let us list the assumptions on the function f (x, s) in the case of problem (3). We will sometimes denote the generic point x ∈ Ω by (x 1 , y) ∈ R × ω and we set ∂ 1 := ∂ ∂x 1 . We will always assume that f (x, s) : Ω × [0, +∞) → R is a Carathéodory function such that (8) s → f (x, s) is locally Lipschitz continuous, uniformly for a. e. x ∈ Ω, ∃ δ > 0 such that s → f (x, s) ∈ C 1 ([0, δ]), uniformly for a. e. x ∈ Ω.
Moreover, we will require the following assumptions which are typical in population dynamics: (9) f (x, 0) = 0 for a. e. x ∈ Ω, (10) ∃ S > 0 such that f (x, s) ≤ 0 for s ≥ S and for a. e. x ∈ Ω, (11) s → f (x, s) s is nonincreasing for a. e. x ∈ Ω and it is strictly decreasing for a. e. x ∈ D ⊂ Ω, with |D| > 0.
A travelling wave solution for problem (3) is a positive bounded solution of the form u(t, x 1 , y) = U (x 1 − ct, y). The problem for U reads (13)        ∆U + c∂ 1 U + f (x, U ) = 0 for a. e. x ∈ Ω ∂ ν U = 0 on ∂Ω U > 0 in Ω U is bounded.
In the literature, such kind of solutions are also called pulses. If f satisfies (9), then the linearized operator about 0 associated with the elliptic equation in (13) is Our main results in the pure shift case depend on the stability of the solution w ≡ 0 for the Neumann problem Lw = 0 in Ω, ∂ ν w = 0 on ∂Ω, that is, on the sign of the generalized Neumann principal eigenvalue λ 1,N (−L, Ω). For a given operator L in the form L = ∆ + β(x) · ∇ + γ(x), with β and γ bounded, we define the quantity λ 1,N (−L, Ω) by λ 1,N (−L, Ω) := sup{λ ∈ R : ∃ φ > 0, (L + λ)φ ≤ 0 a. e. in Ω, ∂ ν φ ≥ 0 on ∂Ω}. (14) This definition of the generalized principal eigenvalue for the Neumann problem is in the same spirit as the one in [4] for the Dirichlet boundary condition case. In (14), the function φ is understood to belong to W 2,p ((−r, r) × ω) for some p > N and every r > 0. Thus, ∂ ν φ has the classical meaning. We will set for brief λ 1,N := λ 1,N (−L, Ω).

2.2.
General cylindrical-type domains. The large time behaviour of solutions to the pure shift problem either in the semi-infinite cylinder Ω + , as well as in the asymptotically cylindrical domain Ω ′ , is characterized by the sign of the generalized Neumann principal eigenvalue λ 1,N = λ 1,N (−L, Ω) in the straight infinite cylinder, as defined in (14).
In the first case, in order to give sense to problem (4), the function f (·, s) has to be defined in the whole straight infinite cylinder Ω. We will always require that f satisfies (8). In (4), the function σ, which defines the Dirichlet condition at the "bottom" of the cylinder, is assumed to be of class W 2,∞ (R + × ω) and to satisfy (15) Here is the result for the half cylinder.
For the next result, let us now make precise what we mean by Ω ′ being an asymptotically cylindrical domain. We assume that Ω ′ is uniformly smooth and that there exists a C 2 diffeomorphism Ψ : where I denotes the identity map from R N into itself. We define the family of sets (ω ′ (x 1 )) x1∈R in R N −1 by the equality Note that by (16) the ω ′ (x 1 ) are (uniformly) smooth, bounded and connected for x 1 large enough. In order to make sense of (5), the function f (·, s) has to be defined in the set Clearly, one has that Ω ⊂ Ω. Besides the regularity assumptions (8) on f , where Ω is replaced by Ω, we further require that f and f s (x, 0) are Hölder continuous 1 in x: In this setting, hypotheses (9)-(12) are understood to hold with Ω replaced by Ω, except for the condition D ⊂ Ω in (11) which is unchanged.
Theorem 2.4. Let u(t, x) be the solution of (5) with an initial condition u(0, x) = u 0 (x) ∈ L ∞ (Ω ′ ) which is nonnegative and not identically equal to zero. Under assumptions (9)- (12), (17) the following properties hold: 1 which is also understood to imply that they are bounded. Precisely, for k ∈ N and α ∈ (0, 1), C k+α (O) denotes the space of functions φ ∈ C k (O) whose derivatives up to order k are bounded and uniformly Hölder continuous with exponent α in O.
Actually, the results of Theorem 2.4 hold under more general boundary conditions than those considered in (5). In fact, it is only needed that they coincide with Neumann boundary conditions for x 1 large (and that they imply the existence of a unique solution of the evolution problem for any given initial datum, as well as the validity of the comparison principle). Since Ω + is a particular case of asymptotically cylindrical domain (with Ψ ≡ 1 and h = 0), Theorem 2.3 is actually contained in Theorem 2.4. However, we treat it separately because the proof is much simpler.
2.3. Lateral-periodic conditions. In the last case considered for the pure shift problem, we deal with problem (1) with c > 0 and e ∈ S N −1 given and with f periodic in the last P variables, 1 ≤ P ≤ N − 1. That is, there exist P positive constants l 1 , · · · , l P such that where {e 1 , · · · , e N } denotes the canonical basis of R N . We assume that the shift direction e ∈ S N −1 is orthogonal to the directions in which f is periodic: e · e i = 0 for i = N − P + 1, · · · , N . We set M := N − P and we will sometimes denote the generic point x ∈ R N by x = (z, y) ∈ R M × R P , in order to distinguish the periodic directions y from the others. Henceforth, we say that a function φ : R N → R is lateral-periodic (with period (l 1 , · · · , l P )) if φ(x + l i e M+i ) = φ(x) for i = 1, · · · , P and a. e. x ∈ R N .
Besides the regularity assumptions (8) (with Ω now replaced by R N ) we require that f satisfies (19) f (x, 0) = 0 for a. e. x ∈ R N , (20) ∃ S > 0 such that f (x, s) ≤ 0 for s ≥ S and for a. e. x ∈ R N , and it is strictly decreasing for a. e. x ∈ D ⊂ R N , with |D| > 0, The problem for travelling wave solutions u(t, The associated linearized operator L about 0 is the same as before but in R N . We consider the generalized principal eigenvalue of a linear elliptic operator −L in a domain O ⊂ R N , as defined in [4]: In the sequel, we will set λ 1 := λ 1 (−L, R N ). We now state our main results for the lateral periodic (pure shift) problem. uniformly with respect to y ∈ R P . Theorem 2.6. Let u(t, x) be the solution of (1) with an initial condition u(0, x) = u 0 (x) ∈ L ∞ (R N ) which is nonnegative and not identically equal to zero. Under assumptions (18)-(22) the following properties hold: globally uniformly with respect to z ∈ R M and locally uniformly with respect to y ∈ R P , where U is the unique solution of (23). If, in addition, u 0 is either lateral-periodic or satisfies then the previous limit holds globally uniformly also with respect to y ∈ R P .
It is easy to see that, in general, the convergence of u(t, z, y) to U ((z, y) − cte) is not uniform globally with respect to y. For instance, if the initial datum u 0 has compact support, then, for all fixed t > 0, u(t, x) → 0 as |x| → ∞.

2.4.
Behaviour near critical value. The next result is to answer a question that was raised by Professor Mimura to one of the authors regarding the behaviour of the solutions near the extinction limit. We show here that a simple bifurcation takes place when the generalized principal eigenvalue becomes nonnegative (or, in other terms, when the speed c crosses a critical value c 0 ). For simplicity, we only state the result in the case of pure shift problem (3) in the straight infinite cylinder, but it also holds in the whole space case (1), either under the hypotheses of the lateral periodic framework, as well as under condition (2) considered in [6].
We assume that f and Ω in (3) are such that c 0 > 0, where c 0 is the critical speed defined in Section 3.2, i. e. that λ 1,N < 0 when c = 0. Below, for any 0 < c < c 0 , U c denotes the unique (stable) solution of (13) given by Theorem 2.1.
Theorem 2.7. Assume that (9)-(12) hold. Then, the following properties hold: uniformly with respect to x ∈ Ω, where ϕ is the unique positive solution of It should be noted that the uniqueness of the solution to (26) is a remarkable property which does not hold in general for positive solutions of linear equations in unbounded domains.
2.5. L 1 convergence. We still consider the case of straight infinite cylinder. Starting from the pointwise convergence of the solution u(t, x) of (3) as t → ∞, we are able to show that the convergence also holds in L 1 (Ω). This is interesting from the point of view of biological models, as u(t, ·) L 1 (Ω) represents the total population at time t.
Theorem 2.8. Consider problem (3) in the straight infinite cylinder Ω. The convergences in Theorem 2.2 also hold in the L 1 sense, provided the initial datum u 0 belongs to L 1 (Ω).
An analogous result holds true for the pure shift problem in the whole space considered in [6] which we now state. Theorem 2.9. Let u(t, x) be the solution of (1) in all of space with an initial condition u(0, x) = u 0 (x) ∈ L ∞ (R N ) ∩ L 1 (R N ) which is nonnegative and not identically equal to zero. Under assumptions (19)-(21) and (2) the following properties hold: where U is the unique solution of (23).
2.6. Seasonal dependence. We consider problem (6) with f (t, x, s) : R × Ω × [0, +∞) → R periodic in t, with period T > 0: As in the case of asymptotically cylindrical domains, besides conditions (8), which are now required uniformly in t ∈ R, we need some Hölder continuity assumptions on f for some α ∈ (0, 1): , with I ⊂ R and O ⊂ R N , denotes the space of functions φ(t, x) such that φ(·, x) ∈ C α 2 (I) and φ(t, ·) ∈ C α (O) uniformly with respect to x ∈ O and t ∈ I respectively. The other assumptions on f are: and it is strictly decreasing for some t ∈ R, x ∈ Ω.
The analogue of condition (12) is required uniformly in t, that is, The notion of travelling wave is replaced in this framework by that of pulsating travelling wave, that is, a solution u to (6) such that U (t, x 1 , y) := u(t, x 1 + ct, y) is periodic in t with period T . Thus, U satisfies (32) where U is extended by periodicity for t < 0. We denote by P the linearized operator about the steady state w ≡ 0 associated with the parabolic equation in (32): By analogy to (24), we define the generalized T -periodic Neumann principal eigenvalue of the parabolic operator P in R × Ω in the following way: uniformly with respect to t ∈ R and y ∈ ω. Theorem 2.11. Let u(t, x) be the solution of (6) with an initial condition u(0, x) = u 0 (x) ∈ L ∞ (Ω) which is nonnegative and not identically equal to zero. Under assumptions (28)-(31) the following properties hold: uniformly with respect to (x 1 , y) ∈ Ω, where U is the unique solution of (32).
Indeed, if u is a solution of (7) thenũ(t, x 1 , y) := u(t, x 1 + ct, y) satisfies where the function As a consequence, the problem of pulsating travelling wave solutions u to (7) such that U (t, x 1 , y) := u(t, x 1 + ct, y) is l/(c − c ′ )-periodic in t is given by (32) with f replaced by h and T = l/(c − c ′ ). Furthermore, as the transformationũ(t, x 1 , y) := u(t, x 1 + ct, y) reduces (6) and (7) to the same kind of problem, Theorem 2.11 holds with (6) replaced by (7), f by h and T = l/(c − c ′ ).
3. The pure shift problem: Straight infinite cylinder Let us recall the notation used in this framework: To prove the existence and uniqueness of travelling wave solutions to (3), Theorem 2.1, we use the same method as in [6]. The only difference is that here we take into account the Neumann boundary conditions in the definition of the generalized principal eigenvalue λ 1,N . This leads us to consider eigenvalue problems in the finite cylinders (−r, r) × ω, with mixed Dirichlet-Neumann boundary conditions, for which we need some regularity results up to the corners {±r} × ∂ω presented in the appendix. Properties of this eigenvalue are described in Section 3.1. Next, we reduce the elliptic equation in (13) to an equation with self-adjoint linear term via a Liouville transformation. This will allow us to define the critical speed c 0 as well as to derive the exponential decay of solutions to (13). Using this result we prove a comparison principle for (13) which yields the uniqueness and the necessary condition for the existence of travelling wave solutions. The sufficient condition will be seen to follow from the properties of λ 1,N and a sub and supersolution argument. Thanks to Theorem 2.1, we will derive a result about entire solutions to (3) which is useful in completing the proof of Theorem 2.2.
3.1. Properties of λ 1,N . We derive some results concerning the generalized Neumann principal eigenvalue λ 1,N (−L, Ω) that will be needed in the sequel. Here, L is an operator of the type with β = (β 1 , · · · , β N ) and γ bounded.
We first introduce the principal eigenvalues in the finite cylinders Ω r , with Neumann boundary conditions on the "sides" (−r, r) × ∂ω and Dirichlet boundary conditions on the "bases" {±r} × ω. The existence of such eigenvalues follows from the Krein-Rutman theory, as for the principal eigenvalues in bounded smooth domains with either Dirichlet or Neumann boundary conditions. Some technical difficulties arise due to the non-smoothness of Ω r on the "corners" {±r} × ∂ω. This problem can be handled by extending the solutions outside Ω r by reflection. Since such an argument is quite classical and technical, we postpone the proof of the next result to Appendix A. The quantity λ(r) and the function ϕ r in the previous theorem are respectively called principal eigenvalue and eigenfunction of −L in Ω r (with mixed Dirichlet/Neumann boundary conditions). Furthermore, there exists a generalized Neumann principal eigenfunction of −L in Ω, that is, a positive function ϕ ∈ W 2,p (Ω r ), for any p > 1 and r > 0, such that Proof. Let 0 < r 1 < r 2 and assume, by way of contradiction, that λ 1,N (r 1 ) ≤ λ 1,N (r 2 ). Consider the associated principal eigenfunctions ϕ r1 and ϕ r2 of −L in Ω r1 and Ω r2 respectively. Note that the Hopf lemma yields ϕ r2 > 0 on (−r 2 , r 2 ) × ∂ω. Set k := max Clearly, k > 0 and the function w := kϕ r2 − ϕ r1 is nonnegative, vanishes at some point x 0 ∈ Ω r1 and satisfies in Ω r1 .
Since ϕ r1 = 0 on {±r 1 } × ω, the point x 0 must belong to (−r 1 , r 1 ) × ω. If x 0 ∈ Ω r1 then the strong maximum principle yields w ≡ 0, which is impossible. As a consequence, it is necessarily the case that x 0 ∈ (−r 1 , r 1 ) × ∂ω. But this leads to another contradiction in view of Hopf's lemma: Hence, the function λ(r) : R + → R is decreasing. Let us show that the quantity λ 1,N (−L, Ω) is well defined and satisfies Taking φ ≡ 1 in (14) shows that λ 1,N (−L, Ω) ≥ − sup Ω γ. If (35) does not hold then there exists R > 0 such that λ(R) < λ 1,N (−L, Ω). By definition (14), we can find a constant λ > λ(R) and a positive function φ ∈ W 2,N +1 (Ω r ), for any r > 0, such that A contradiction follows by arguing as before, with ϕ r1 and ϕ r2 replaced by ϕ R and φ respectively. Consequently, To prove equality, consider the sequence of generalized principal eigenfunctions (ϕ n ) n∈N , normalized by ϕ n (x 0 ) = 1, where x 0 is fixed, say in Ω 1 . Extending by reflection the functions ϕ n to larger cylinders, as done in Appendix A, and using the Harnack inequality, we see that, for m ∈ N, the (ϕ n ) n>m are uniformly bounded in Ω m . Hence, by standard elliptic estimates and embedding theorems, there exists a subsequence (ϕ n k ) k∈N converging in C 1 (Ω ρ ) and weakly in W 2,p (Ω ρ ), for any ρ > 0 and p > 1, to some nonnegative function ϕ satisfying −Lϕ =λϕ a. e. in Ω ∂ ν ϕ = 0 on ∂Ω.
In what follows, λ(r) and ϕ r will always denote respectively the principal eigenvalue and eigenfunction of −L in Ω r . We will further denote by ϕ a generalized Neumann principal eigenfunction of −L in Ω, given by Proposition 1.
In order to define the critical speed c 0 , we introduce the linear operator Proposition 2. Define the critical speed as Proof. This simply follows from the fact that

Exponential decay of travelling waves.
Owing to the results of Section 3.1, the exponential decay of solutions to (13) follows essentially as in [6]. However, for the sake of completeness, we include the proofs here.
Proof. By the hypotheses on V , there exist ε, R > 0 such that ∆V ≥ (γ+ε)V a. e. in Since V is a subsolution of the above problem and V ≤ θ a on {±R, ±(R + a)} × ω, the comparison principle yields V ≤ θ a in Ω R+a \Ω R , for any a > 0. Therefore, for |x 1 | > R and y ∈ ω we get which concludes the proof.

Comparison principle.
The following is a comparison principle which contains, as a particular case, the uniqueness of solutions to (13) vanishing at infinity. Theorem 3.3. Assume that (9), (11), (12) hold. Let U, U ∈ W 2,p (Ω r ), for some p > N and every r > 0, be two nonnegative functions satisfying Then U ≤ U in Ω.
Then, there exists a sequence (ε n ) n∈N in R + such that From (38) it follows that k * < ∞ and that the function W := k * U −U is nonnegative and vanishes at (ξ, η). Also, since k * > 1, condition (11) yields with strict inequality a. e. in D. Therefore, thanks to the Lipschitz continuity of f in the second variable, W is a supersolution of a linear elliptic equation in Ω. Since W is nonnegative in Ω, vanishes at (ξ, η) and ∂ ν W = 0 on ∂Ω, the strong maximum principle and the Hopf lemma yield W ≡ 0. This is a contradiction because W is a strict supersolution in D.
We proceed exactly as in [2]. By Proposition 1 there exists R > 0 such that λ(R) < 0. Define the function where κ > 0 will be chosen appropriately small later. We see that ∂ ν U = 0 on ∂Ω and that Hence, since f (x, 0) = 0 by (9) and in Ω R . One can then readily check that On the other hand, the function U (x) ≡ S -where S is the constant in (10) -is a supersolution to (40). Also, we can chose κ small enough in such a way that U ≤ U . Consequently, using a classical iterative scheme (see e. g. [3]) we can find a function U ∈ W 2,p (Ω r ), for any p > 1 and r > 0, satisfying (40) and U ≤ U ≤ U in Ω. The strong maximum principle implies that U is strictly positive and then it solves (13). Case 2: λ 1,N ≥ 0. Assume by contradiction that (13) admits a solution U . Let ϕ be a generalized Neumann principal eigenfunction of −L in Ω (cf. Proposition 1), normalized in such a way that 0 < ϕ(x 0 ) < U (x 0 ), for some x 0 ∈ Ω. Then, ϕ satisfies ∂ ν ϕ = 0 on ∂Ω and, by (9), (11), Therefore, since by Proposition 3 lim |x1|→∞ U (x 1 , y) = 0 uniformly in y ∈ ω, we can apply Theorem 3.3 with U = U and U = ϕ and infer that U ≤ ϕ: contradiction.
The uniqueness result immediately follows from Proposition 3 and Theorem 3.3.
3.6. Large time behaviour. We will make use of a result concerning entire solutions (that is, solutions for all t ∈ R) of the evolution problem associated with (13): Lemma 3.4. Let u * be a nonnegative bounded solution of (41). Under assumptions (9)-(12) the following properties hold: (ii) if λ 1,N < 0 and there exist a sequence (t n ) n∈N in R and a point x 0 ∈ Ω such that where U is the unique solution of (13).
Proof. Let S be the positive constant in (10). Set S * := max{S, u * L ∞ (R×Ω) } and let w be the solution to (41) for t > 0, with initial condition w(0, x) = S * . Since the constant function S * is a stationary supersolution to (41), the parabolic comparison principle implies that w is nonincreasing in t (and it is nonnegative). Consequently, as t → +∞, w(t, x) converges pointwise in x ∈ Ω to a function W (x). Using standard parabolic estimates up to the boundary, together with compact injection results, one sees that this convergence is actually uniform in Ω ρ , for any ρ > 0, and that W solves (40). For any h ∈ R the function w h (t, Let us consider separately the two different cases. (i) λ 1,N ≥ 0. Due to Theorem 2.1, the function W cannot be strictly positive in Ω. Thus, W vanishes somewhere in Ω and then the elliptic strong maximum principle yields W ≡ 0. The statement then follows from (43).
(ii) λ 1,N < 0 and (42) holds for some (t n ) n∈N in R and x 0 ∈ Ω. We claim that condition (42) yields Let us postpone for a moment the proof of (44). By Proposition 1, there exists R > 0 such that λ(R) < 0. Consider the same function U as in the proof of Theorem 2.1: otherwise, We know that, for κ small enough, U is a subsolution to (40). Moreover, owing to (44), κ can be chosen in such a way that U (x) ≤ u * (t n , x) for n large enough and x ∈ Ω. Let v be the solution to (41) for t > 0, with initial condition v(0, x) = U (x). By comparison, we know that the function v is nondecreasing in t and it is bounded from above by S * . Then, as t goes to infinity, v(t, x) converges locally uniformly to the unique solution U to (13) (the strict positivity follows from the elliptic strong maximum principle). For n large enough the function v n (t, Hence, the parabolic comparison principle yields Combining the above inequality with (43) we obtain This shows that W is positive and then it is a solution to (13). The uniqueness result of Theorem 2.1 then yields u * ≡ U .
To conclude the proof, it only remains to show (44). Assume by contradiction that there exists r > 0 such that the inequality does not hold. Then, there exists a sequence ((x n 1 , y n )) n∈N in Ω r such that (up to subsequences) lim n→∞ u * (t n , x n 1 , y n ) = 0.
It is not restrictive to assume that (x n 1 , y n ) converges to some (ξ, η) ∈ Ω r as n goes to infinity. Parabolic estimates and embedding theorems imply that the sequence of functions u * n (t, x 1 , y) := u * (t + t n , x 1 , y) converges (up to subsequences) in (−ρ, ρ) × Ω ρ , for any ρ > 0, to a nonnegative solution u * ∞ of (41) satisfying u * ∞ (0, ξ, η) = 0. If u * ∞ was smooth then the parabolic strong maximum principle and Hopf's lemma would imply u * ∞ (t, x) = 0, for t ≤ 0 and x ∈ Ω, which is impossible because u * ∞ (0, x 0 ) > 0 by (42). To handle the case where u * ∞ is only a weak solution of (41), one can extend u * ∞ to a nonnegative solution of a parabolic equation in R × R ×ω, with ω ⊂⊂ω, as done in the appendix. Hence, one gets a contradiction by applying the strong maximum principle.
Step 1: the functionũ satisfies Let (t k ) k∈N be a sequence in R satisfying lim k→∞ t k = +∞. Then, parabolic estimates and embedding theorems imply that (up to subsequences) the functions u(t + t k , x) converge as k → ∞, uniformly in (−ρ, ρ) × Ω ρ , for any ρ > 0, to some function u * (t, x) which is a nonnegative bounded solution to (41). If λ 1,N ≥ 0 then u * ≡ 0 by Lemma 3.4. Therefore, owing to the arbitrariness of the sequence (t k ) k∈N , (46) holds in this case. Consider now the case λ 1,N < 0. Set x 0 := (0, y 0 ), where y 0 is an arbitrary point in ω. Let us show that hypothesis (42) in Lemma 3.4 holds for any sequence (t n ) n∈N tending to −∞. By Proposition 1, there exists R > 0 such that λ(R) < 0. Arguing as in the proof of Theorem 2.1, we can choose κ > 0 small enough in such a way that the function U := κϕ R satisfies U (x) ≤ũ(1, x) and is a subsolution to the elliptic equation of (13) in Ω R . Hence, (t, x) → U (x) is a subsolution to (45) in R × Ω R and satisfies U (±R, y) = 0 ≤ũ(t, ±R, y) for t > 0, y ∈ ω. The parabolic comparison principle yields U (x) ≤ũ(t + 1, x) for t > 0 and x ∈ Ω R . As a consequence, We can then apply Lemma 3.4 and derive u * ≡ U . Thus, (46) holds.
Remark 3. The results of Theorems 2.1 and 2.2 also hold if one considers Dirichlet boundary condition u(t, x) = 0 on R + × ∂Ω in (3). In this case, the existence, uniqueness and stability of travelling waves depend on the sign of the generalized principal eigenvalue λ 1 (−L, Ω) defined by (24). The proofs are easier than in the Neumann case considered here. In particular, one can consider an increasing sequence of bounded smooth domains converging to Ω instead of the Ω r . This avoids any difficulty due to the lack of smoothness of the boundary in the definition of the principal eigenvalues. Robin boundary conditions are also allowed.

Large time behaviour in general cylindrical-type domains
In this section, we use the same notation as in Section 3: The basic idea to prove Theorems 2.3 and 2.4 is to show that, as τ → ∞, the functionũ(t + τ, x) (whereũ(t, x 1 , y) := u(t, x 1 + ct, y)) converges locally uniformly (up to subsequences) to an entire solution u * (t, x) in the straight infinite cylinder Ω. Thus, owing to Lemma 3.4, the convergence results of statements (i) and (ii) hold locally uniformly provided u * satisfies (42). In the case of semi-infinite cylinder, condition (42) is derived by comparingũ with the principal eigenfunction ϕ R of −L in Ω R , as done in the proof of Theorem 2.2. The case of asymptotically cylindrical domain is actually much more delicate, becauseũ and ϕ R do not satisfy the same boundary conditions and therefore cannot be compared. We overcome this difficulty by replacing ϕ R with a suitable "generalized" strict subsolution which is compactly supported. Then, we can conclude using the fact that, essentially, the problem satisfied byũ "approaches" locally uniformly the Neumann problem in the straight cylinder as t → ∞.

4.1.
Straight semi-infinite cylinder. We start by considering here problem (4) which is set in a straight semi-infinite cylinder Ω + = R + × ω.
Proof of Theorem 2.3. Set where S is the constant in (10). Since 0 and S ′ are respectively a sub and a supersolution of (4), the same arguments as in the proof of Theorem 2.2 show that the unique solution u to (4) with initial condition u(0, x) = u 0 (x) satisfies 0 < u ≤ S ′ in R + × R + × ω. The function defined byũ(t, x 1 , y) := u(t, x 1 + ct, y) satisfies the following equation and boundary conditions: with initial conditionũ(0, x) = u 0 (x) for x ∈ Ω + . For the rest of the proof, U denotes the unique solution to (13) if λ 1,N < 0, while U ≡ 0 if λ 1,N ≥ 0. We first derive the local convergence ofũ to U .
Step 1: the functionũ satisfies (46). Let (t k ) k∈N be a sequence such that lim k→∞ t k = +∞. By standard arguments we see that, as k → ∞ and up to subsequences, the functionsũ(t + t k , x) converge locally uniformly in R × Ω to a solution u * of (41). Owing to Lemma 3.4, we only need to show that if λ 1 < 0, then (42) holds. By Proposition 1, there exists R > 0 such that λ(R) < 0. As we have seen in the proof of Theorem 2.1, for κ > 0 small enough the function U(x) := κϕ R (x) is a subsolution to (40) in Ω R . Set t R := R/c + 1. The functionũ is well defined and strictly positive in [t R , +∞) × Ω R . Hence, up to decreasing κ if need be, we can assume that U (x) ≤ũ(t R , x) for x ∈ Ω R . Since (t, x) → U(x) is a subsolution to (48) in R × Ω R and ∀ t > t R , y ∈ ω, U (±R, y) = 0 <ũ(t, ±R, y), the comparison principle yields U (x) ≤ũ(t, x) for t > t R , x ∈ Ω R . Therefore, for any that is, (42) holds for any sequence (t n ) n∈N tending to −∞.
Step 2: conclusion of the proof. Argue by contradiction and assume that there exist ε > 0 and some sequences (t n ) n∈N in R + and ((x n 1 , y n )) n∈N in Ω + such that lim We may assume that y n converges to some η ∈ ω. By step 1 we know that the sequence (x n 1 − ct n ) n∈N cannot be bounded. Since U (·, y) vanishes at infinity, we get in particular that whatever the sign of λ 1,N is. Suppose for a moment that (x n 1 ) n∈N is unbounded. Then, by parabolic estimates and embedding theorems, the functions u n (t, x 1 , y) := u(t + t n , x 1 + x n 1 , y) converge, as n → ∞ and up to subsequences, uniformly in (−ρ, ρ) × Ω ρ , for any ρ > 0, to a nonnegative function u ∞ satisfying and, by (49), u ∞ (0, 0, η) ≥ ε. We then get a contradiction by comparing u ∞ with θ h (t, x) := S ′ e −ζ(t+h) in (−h, +∞) × Ω and letting h go to infinity, as done at the end of the proof of Theorem 2.2. It remains to consider the case when (x n 1 ) n∈N is bounded. For n ∈ N define u n (t, x 1 , y) := u(t+t n , x 1 , y). Using L p estimates up to the boundary for u, ∂ t u, ∆u (which hold good here owing to the compatibility condition ∂ ν σ = 0 on R + × ∂ω, see e. g. [14], [15]) we infer that (a subsequence of) (u n ) n∈N converges uniformly in (−ρ, ρ) × (0, ρ) × ω, for any ρ > 0, to a function u ∞ satisfying (50) for x 1 > 0, together with u ∞ (t, 0, y) = 0 for t ∈ R, y ∈ ω. Moreover, (49) yields u ∞ (0, ξ, η) ≥ ε, where ξ is the limit of a subsequence of (x n 1 ) n∈N . A contradiction follows exactly as before, by comparison with the functions θ h (t, x) := S ′ e −ζ(t+h) .

4.2.
Asymptotically cylindrical domain. As in the case of the straight cylinder that we considered in the previous section, the large time behaviour of u rests on proving thatũ(t, x) := u(t, x 1 + ct, y) does not converge to 0 as t → ∞ when λ 1,N < 0. With respects to the straight cylinder, the difficulty here is that the condition λ 1,N allows one to construct a compactly supported stationary subsolution of the Neumann problem in the straight cylinder, but not in the time-dependent domain whereũ is defined. Thus, the proof becomes technically more involved. Let us sketch our strategy to prove this result. Through the mapping Ψ we can transformũ into a functionṽ solution of an oblique derivative problem with a modified operator but in the straight cylinder. The transformed problem converges, in some sense, to the Neumann problem (45) as t → ∞. Thus, for t large enough, it is possible to derive a positive lower bound forṽ by the same comparison argument as in the previous sections, provided that (45) admits some kind of compactly supported stationary strict subsolution. Actually, we construct a generalized strict subsolution V in the sense of [3]: V is the supremum of two strict subsolutions. The precise properties of V are stated in the next lemma, which is proved at the end of the section.
In the sequel, we will make use of the following fact, which is a consequence of (16): locally uniformly with respect to (x 1 , y) ∈ ∂Ω. Note that the right hand side does not depend on x 1 .
Step 1: the functionũ satisfies where U ≡ 0 if λ 1,N ≥ 0, while U is the unique solution of (13) if λ 1,N < 0. Let (t k ) k∈N be a sequence in R such that lim k→∞ t k = +∞. From parabolic estimates it follows that the functionsũ(t + t k , x) converge as k → ∞ (up to subsequences) locally uniformly in R × Ω to some function u * (t, x) which is a nonnegative bounded solution of the parabolic equation in (41). Moreover, using (51) and estimates up to the boundary of Ω ′ , one can check that u * satisfies also the boundary condition of (41). Hence, if λ 1,N ≥ 0, Lemma 3.4 yields u * ≡ 0, that is, (53) holds. In the case λ 1,N < 0, we want to show that (42) holds. To do this, we consider the domains O 1 , O 2 , O, the constant κ and the functions V 1 , V 2 , V given by Lemma 4.1. We set e 1 := (1, 0, · · · , 0) ∈ R N . By (16) there exists t 0 > 0 such that Take t 1 > t 0 large enough in such a way that, for σ ∈ {1, 2}, the following inequalities hold in (t 1 , +∞) × O σ : Here, the last inequality is a consequence of (16) and the uniform continuity of f s (x, 0). Moreover, up to increasing t 1 , it is seen that Therefore, as f (x, 0) = 0 and s → f s (x, s) ∈ C 1 ([0, δ]), uniformly in x, there exists k σ > 0 such that for any k ∈ (0, k σ ] the function kV σ is a strict subsolution of the problem solved byṽ in (t 1 , +∞) × O σ . Let τ > t 1 be such that the matrix field A(t, x) is uniformly elliptic for t > τ and x ∈ O and the vector field β(t, x) points outside Ω for t > τ and x ∈ ∂O ∩ ∂Ω. Let k < min(k 1 , k 2 ) be such that the function U := kV satisfies ∀ x ∈ O, U (x) <ṽ(τ, x).
Assume by contradiction that the above property does not hold for some (t n ) n∈N and ((x n 1 , y n )) n∈N . Hence, setting ξ n := x n 1 − ct n we get lim sup n→∞ |ũ(t n , ξ n , y n ) − U (ξ n , y n )| > 0.
This contradicts (53). The case of (ξ n ) n∈N unbounded can be handled exactly as in the second step of the proof of Theorem 2.3.
Step 4: conclusion of the proof. Note that if λ 1,N ≥ 0 then condition (16) and the uniform continuity of u imply that the result of step 2 holds even if we drop the assumption (x n 1 , y n ) ∈ Ω. Therefore, Theorem 2.4 part (i) follows from steps 2 and 3. Assume by contradiction that statement (ii) does not hold. Then, there exist ε > 0, (t n ) n∈N in R + , ((x n 1 , y n )) n∈N in Ω ′ such that lim n→∞ t n = ∞ and either ∀ n ∈ N, y n ∈ ω, |u(t n , x n 1 , y n ) − U (x n 1 − ct n , y n )| ≥ ε, or ∃ γ < c, ∀ n ∈ N, x n 1 < γt n , u(t n , x n 1 , y n ) ≥ ε. The first case is ruled out because the sequence (x n 1 ) n∈N is not bounded from above -by step 3 and the last statement of Theorem 2.1 -nor unbounded from aboveby step 2. In the second case, step 3 implies that x n 1 → +∞ as n → ∞. Hence, owing to the uniform continuity of u, we can assume without loss of generality that (x n 1 , y n ) ∈ Ω ′ ∩ Ω for n large enough. As a consequence, since x n 1 − ct n → −∞ as n → ∞, we derive lim inf n→∞ (u(t n , x n 1 , y n ) − U (x n 1 − ct n , y n )) = lim inf n→∞ u(t n , x n 1 , y n ) ≥ ε, which is in contradiction with step 2.
We now turn to the proof of Lemma 4.1. Let is first describe the ideas before giving the technical details of the construction. We first define the function V 1 as the principal eigenfunction of −L in some bounded smooth domain O under boundary condition of Robin type. The advantage of taking Robin boundary conditions is that we obtain a function with negative normal derivative, which is useful for comparison purposes. Using the fact that λ 1,N < 0, we are able to choose O and the boundary condition in such a way that the associated principal eigenvalue λ is strictly negative. Hence, in the set where V 1 is bounded away from zero, we can take κ small enough such that The above inequality may fail when V 1 approaches 0, and this is why we introduce the function V 2 . We want V 2 to be positive in a bounded domain O ⊃ O, to vanish on ∂O ∩ Ω and to satisfy the above inequality together with ∂ ν V 2 < 0 at least in the set where it is small. The differential inequality is obviously fulfilled by taking a function of exponential type. The boundary condition is less easy to obtain because it implies that at the "corners " ∂O ∩ Ω ∩ ∂Ω the vector field −ν has to point inside O (hence, we cannot take O = Ω r for some r > 0). This is achieved by taking O to be the straight cylinder truncated by two "caps" -see Figure 1 -obtained as the graph of a function ξ satisfying ξ = 0 and ∂ νω ξ < 0 on ∂ω (∂ νω denoting the exterior normal derivative to ω). A simple way to find such a function ξ is by solving the Dirichlet problem −∆ξ = 1 in ω, ξ = 0 on ∂ω. The functions that will be used to define V 1 and V 2 are constructed in Lemma 4.2 and Lemma 4.3 respectively.
Hence, the strong maximum principle implies that x * ∈ ∂ O and ϑ(x * 1 ) < 1. As a consequence, which is in contradiction with the Hopf lemma. Therefore, the λ ε are bounded from below by λ(R + 2γ). A direct application of the strong maximum principle shows that they are bounded from above by the Dirichlet principal eigenvalue of −L in any domain A ⊂⊂ O. Hence, from any positive sequence (ε n ) n∈N converging to 0 one can extract a subsequence (ε n k ) k∈N such that (λ εn k ) k∈N converges to some λ * ∈ R. Using Schauder's estimates up to the boundary and the Arzela Ascoli theorem we see that (up to subsequences) the φ εn k converge as k → ∞ in C 2 ( O) to a non-negative nontrivial solution φ * of Thus, φ * > 0 in O by the strong maximum principle and then the uniqueness of the principal eigenvalue of −L in O under Robin boundary condition yields λ * = λ 0 . This shows that the λ ε converge to λ 0 as ε → 0 + . To check that λ 0 < λ(R) one uses the same contradictory argument as before: suppose that λ 0 ≥ λ(R) and set w : Note that ∂ ν φ 0 = 0 on [−R, R] × ∂ω and then φ 0 > 0 in Ω R by Hopf's lemma. The points where w vanishes do not lie neither on {±R} × ω, because ϕ R = 0 there, nor in Ω R due to the strong maximum principle. Neither do they lie on (−R, R) × ∂ω due to Hopf's lemma. This yields a contradiction and the claim is then proved. Thus, we can chose ε > 0 small enough in such a way that the function φ := φ ε satisfies −Lφ = −hφ in O, where h := −λ ε > 0. The Hopf lemma implies that φ > 0 in Ω R . Hence, it only remains to check that φ satisfies the desired boundary conditions. The negativity of ∂ ν φ(x 1 , y) for (x 1 , y) ∈ ∂ O ∩∂Ω follows from the Hopf lemma, if φ(x 1 , y) = 0, and from equality If (x 1 , y) ∈ ∂ O\∂Ω then, necessarily, |x 1 | ≥ R + γ/2. Consequently, ϑ(x 1 ) = 1 and then φ(x 1 , y) = 0.
Proof of Lemma 4.1. Consider the functions ξ, χ and the constant ε given by Lemma 4.3. There exists γ > 0 such that ∂ ν χ < 0 on [−2γ, 0] × ∂ω. Let R, h, O and φ be the constants, the domain and the function given by Lemma 4.2 associated with γ. We define O in the following way: Take k > 0 small enough in such a way that Then, we define V 1 := φ in O, extended by 0 outside O, It is then possible to find a positive constant κ < ε such that The proof is thereby complete.

The lateral-periodic case
Henceforth, for every Q ∈ N and r > 0, B Q r stands for the ball in R Q centred at the origin with radius r, and B r := B N r . Other notations used in this section are: Lw = ∆w + ce · ∇w + f s (x, 0)w, In Section 5.1, we introduce the lateral-periodic principal eigenvalues λ 1,l (r) of an elliptic operator −L in the domains O r , under Dirichlet boundary condition on ∂O r and periodicity condition in the last P variables. Then, we show that as r → ∞ the λ 1,l (r) converge to a quantity that we call λ 1,l (−L, R N ).
Let us explain why we need to consider both λ 1 and λ 1,l := λ 1,l (−L, R N ). The negativity of λ 1,l yields the existence of a lateral-periodic subsolution V to (23) which is as small as we want. This function allows one to prove the existence of a travelling wave, but not to derive the large time behaviour of solutions u to (1), because we cannot put V below u (1, x). Instead, the subsolution U one can construct when λ 1 < 0 is compactly supported and then we can put it below u(1, x) and derive Theorem 2.6 part (ii). For similar reasons, we use λ 1 instead of λ 1,l to prove the uniqueness of travelling wave solutions. On the other hand, we make use of the lateral periodic principal eigenfunction χ associated with λ 1,l to derive the nonexistence result for travelling waves when λ 1,l ≥ 0, because it satisfies the needed property inf Or χ > 0 for any r > 0, while the principal eigenfunction associated with λ 1 does not. Thus, a crucial point to prove our main results consists in showing that λ 1 and λ 1,l have the same sign. Actually, using a general result for self-adjoint operators quoted from [7], we will show that they coincide. 5.1. The lateral-periodic principal eigenvalue. Here, L denotes an elliptic operator of the form where (a ij ) ij is an elliptic and symmetric matrix field with Lipschitz continuous entries and β i , γ are bounded. We further require that a ij , β i , γ are lateral-periodic, that is, they are periodic in the last P variables, with the same period (l 1 , · · · , l P ). We remark that, through a regularizing argument, one can prove that the results of this section hold for more general elliptic operators in non-divergence form.
First of all, we reclaim some properties of λ 1 . A basic result of [4] is that if O is bounded and smooth then Next, we consider the eigenvalue problem with mixed Dirichlet/periodic conditions.
Theorem 5.1. For any r > 0 there exists a unique number λ 1,l (r) such that the eigenvalue problem    −Lχ r = λ 1,l (r)χ r a. e. in O r χ r = 0 on ∂O r χ r is lateral-periodic admits a positive solution. We call λ 1,l (r) and χ r (which is unique up to a multiplicative constant) respectively the lateral-periodic principal eigenvalue and eigenfunction of −L in O r .
Proof. Define the Banach space equipped with the W 1,∞ (O r ) norm. Set M := L − d, with d large enough such that the associated bilinear form is coercive on the space of lateral-periodic functions φ ∈ H 1 (B M r ×(0, l 1 )×· · ·×(0, l P )) satisfying φ = 0 in H 1/2 (∂B M r ×(0, l 1 )×· · ·×(0, l P )). Then, the result follows from the Krein-Rutman theorem (as in the proof of Theorem 3.1 in the appendix, but now we do not have the problem of non-smoothness of the boundary).
Proposition 5. The map r → λ 1,l (r) is decreasing and, as r goes to infinity, λ 1,l (r) converges to a quantity that we call λ 1,l (−L, R N ).
Furthermore, there exists a lateral-periodic principal eigenfunction associated with λ 1,l (−L, R N ), that is, a lateral-periodic positive function χ such that Proof. We follow the same arguments as in the proof of Proposition 5 in [2]. Let 0 < r 1 < r 2 . Owing to the lateral-periodicity of the principal eigenfunctions χ r1 and χ r2 , there exists k > 0 such that kχ r2 touches from above χ r1 at some point in O r1 . If λ 1,l (r 1 ) ≤ λ 1,l (r 2 ) then the function w := kχ r2 − χ r1 satisfies −Lw ≥ λ 1,l (r 1 )w a. e. in O r1 .
Thus, the strong maximum principle yields w ≡ 0, which is impossible. Again by the strong maximum principle, we immediately see that λ 1,l (r) > − sup Or γ, for any r > 0. Hence, the quantity is a well defined real number.
Let us show the existence of a lateral-periodic principal eigenfunction associated with λ 1,l (−L, R N ). By Harnack's inequality, the family (χ r ) r>0 , normalized by χ r (0) = 1, is uniformly bounded in any compact subset of R N . Then, interior elliptic estimates and embedding theorems imply that, up to subsequences, the χ r converge as r → ∞, locally uniformly in R N , to a function χ satisfying −Lχ = λ 1,l (−L, R N )χ a. e. in R N . Moreover, χ is lateral-periodic, satisfies χ(0) = 1 and it is strictly positive by the strong maximum principle.
As for the Neumann principal eigenfunction in Proposition 1, the function χ is not unique a priori.
In order to compare λ 1 and λ 1,l , we consider another notion of generalized principal eigenvalue of −L in a domain O: We quote from [7] the following result about self-adjoint operators: Proof. Let r > 0. Taking φ = χ r in (24) and (56) we see that . Hence, Theorem 5.2 yields λ 1,l (r) = λ 1 (−L, O r ). The statement then follows from Propositions 4 and 5.
We can now derive the comparison principle. and for any ρ > 0 there exists C ρ > 0 such that Proof. First note that, by the embedding theorem, condition (57) yields U , U ∈ C 0 (R N ) ∩ L ∞ (O r ), for any r > 0. For ε > 0 define (the above set is nonempty by the hypotheses on U and U ). Clearly, ε → k(ε) is nonincreasing. Furthermore, for ε ∈ (0, sup U), the function W ε := k(ε)U − U + ε is nonnegative and there exist a bounded sequence (z ε n ) n∈N in R M and a sequence (y ε n ) n∈N in R P such that lim n→∞ W ε (z ε n , y ε n ) = 0. We use the lateral periodicity of f and condition (57) to reduce to the case where the minimizing sequence is bounded: let (q ε n ) n∈N be the sequence in Zl 1 × · · · × Zl P such that η ε n := y ε n − q ε n belongs to [0, l 1 ) × · · · × [0, l P ). For n ∈ N define U n (z, y) := U(z, y + q n ) and U n (z, y) := U (z, y + q n ). As f is lateral-periodic, these functions satisfy the same differential inequalities as U and U respectively. By (57), as n → ∞ and up to subsequences, U n → U ∞ and U n → U ∞ locally uniformly in R N , where U ∞ and U ∞ satisfy the same hypotheses as U and U respectively. Therefore, denoting (z(ε), y(ε)) the limit of (a subsequence of) ((z ε n , η ε n )) n∈N , we find that the function W ε ∞ := k(ε)U ∞ − U ∞ + ε is nonnegative and vanishes at (z(ε), y(ε)). Note that y(ε) are bounded with respect to ε. The result then follows exactly as in the proof of Theorem 3.3.
Proof of Theorem 2.5.
Step 1: existence. If λ 1 < 0 then by Proposition 4 there exists R > 0 large enough such that λ 1 (−L, B R ) < 0. We recall that, as B R is bounded and smooth, λ 1 (−L, B R ) coincides with the Dirichlet principal eigenvalue of −L in B R . That is, there exists a function φ R which is positive in B R and satisfies For κ ∈ R and for a. e. x ∈ B R we see that Then, owing to the C 1 regularity of f (x, ·), there exists κ 0 > 0 such that for any 0 < κ ≤ κ 0 the function κφ R is a subsolution to Hence, the function U equal to κ 0 φ R in B R and extended by 0 outside B R is a generalized subsolution of the elliptic equation in (23). Since by (20) the function is a supersolution of the same equation, a standard iterative method implies the existence of a solution U ≤ U ≤ U . The function U is strictly positive by the strong maximum principle and then it solves (23). Assume by contradiction that λ 1 ≥ 0 and (23) admits a solution U . Let χ be a lateral-periodic principal eigenfunction associated with λ 1,l (cf. Proposition 5) normalized in such a way that χ(0) < U (0).
If we show that the hypotheses of Theorem 5.4 are satisfied by U := U and U := χ, we would get the following contradiction: U ≤ χ. Proposition 8 yields lim |z|→∞ U (z, y) = 0 uniformly in y ∈ R P , while λ 1,l ≥ 0 and condition (21) imply that χ is a supersolution of (23). The other hypotheses are immediate to check.
Step 2: uniqueness. It follows from the comparison principle, Theorem 5.4, provided that we show that any solution U to (23) satisfies the hypotheses on both U and U there. All conditions are immediate to check (the decay of U (z, y) with respect to z is given by Proposition 8), except the following one: (note indeed that we do not assume a priory that U is lateral-periodic). The existence result implies that if (23) admits a solution U then λ 1 < 0. In order to prove that U satisfies (59), fix r > 0 and consider the same constants R, κ 0 and function φ R as in the first step. It is not restrictive to assume that Hence, κ(q)φ R (z, y) ≤ U (z, y + q) for (z, y) ∈ B R and, as φ R = 0 on ∂B R , there exists (z q , y q ) ∈ B R such that κ(q)φ R (z q , y q ) = U (z q , y q + q). If κ(q) ≤ κ 0 for some q ∈ Zl 1 × · · · × Zl P , then U (z, y + q) and κ(q)φ R (z, y) would be respectively a solution and a subsolution of (58) and then they would coincide in B R by the strong maximum principle. This is impossible because φ R = 0 on ∂B R . Therefore, ∀ q ∈ Zl 1 × · · · × Zl P , (z, y) ∈ B R , U (z, y + q) ≥ κ(q)φ R (z, y) > κ 0 φ R (z, y).
Since φ R has a positive minimum on B M r × [0, l 1 ] × · · · × [0, l P ] ⊂ B R , (59) follows. The lateral-periodicity of the solution to (23) follows from the uniqueness result. 5.3. Large time behaviour. Once we have proved Theorem 2.5, Theorem 2.6 follows essentially from the same ideas as Theorem 2.2. Thus, we will skip some details.
Proof of Theorem 2.6. The functionũ(t, x) := u(t, x + cte) satisfies where S is the positive constant in (20), and solves (60) with initial conditionũ(0, x) = u 0 (x). Let w be the solution to (60) with initial condition w(0, x) = S ′ . The comparison principle implies that w satisfiesũ ≤ w ≤ S ′ , is nonincreasing in t and, as t → ∞, converges locally uniformly in R N to a nonnegative bounded solution W of (61) ∆U + ce · ∇U + f (x, U ) = 0 a. e. in R N .
Since w(t, x) is lateral-periodic in x by uniqueness, it follows that W is lateralperiodic too and that Step 1: the functionũ satisfies lim min(t,|z|)→∞ũ (t, z, y) = 0 uniformly in y ∈ R P .
Asũ ≤ w, it is sufficient to show that the above property is satisfied by w. The advantage is that w is lateral-periodic. Suppose that there exist ε > 0, (t n ) n∈N in R + and ((z n , y n )) n∈N in R N such that lim n→∞ t n = lim n→∞ |z n | = ∞, ∀ n ∈ N, w(t n , z n , y n ) > ε.
It is not restrictive to assume that (y n ) n∈N is bounded. Thus, we get a contradiction by arguing as in the step 2 of the proof of Theorem 2.2.
Step 2: conclusion of the proof. In the case λ 1 ≥ 0, the function W can not be strictly positive by Theorem 2.5. Hence, the strong maximum principle yields W ≡ 0 and then statement (i) follows from (62) and step 1. Consider the case λ 1 < 0. We know that, for R large enough and κ small enough, the function κφ R is a subsolution to (58) (see the proof of Theorem 2.5 above). Hence, for κ small the function is a subsolution of (61) and satisfies U( is nondecreasing in t and, as t → ∞, converges locally uniformly in R N to a nonnegative bounded solution V of (61) satisfying U ≤ V ≤ W . Therefore, the strong maximum principle yields 0 < V ≤ W and then both V and W coincide with the unique solution U to (23). By (62) we then infer that, as t → ∞,ũ(t, x) converges to U locally uniformly in x ∈ R N . Assume by contradiction that there exist ε > 0, (t n ) n∈N in R + and (z n , y n ) n∈N in R N such that (y n ) n∈N is bounded, lim n→∞ t n = ∞, ∀ n ∈ N, |ũ(t n , z n , y n ) − U (z n , y n )| ≥ ε.
Owing to the local uniform convergence ofũ, we necessarily have that the sequence (z n ) n∈N diverges. Hence, step 1 and Proposition 8 yield a contradiction. It only remains to show that if u 0 is either lateral-periodic or it satisfies (25) then By Propositions 5 and 7 there exists ρ > 0 such that λ 1,l (ρ) < 0 (we recall that λ 1,l (ρ) denotes the lateral-periodic principal eigenvalue of −L in O ρ , and χ ρ the associated eigenfunction). With usual arguments, one sees that the functioñ is a subsolution to (61) for κ small enough. Moreover, if (25) holds then we can chose κ in such a way thatŨ ≤ u 0 . On the other hand, if u 0 is lateral-periodic theñ u(t, x) is lateral-periodic in x and then, as it is positive for t > 0,Ũ (x) ≤ũ(1, x) for κ small enough. In the first case we defineṽ as the solution to (60) satisfying v(0, x) =Ũ (x), while in the second as the solution to (60) for t > 1 satisfying v(1, x) =Ũ (x). In both cases, the maximum principle implies thatṽ(t, x) ≤ũ(t, x) for t ≥ 1, x ∈ R N and thatṽ is nondecreasing in t and lateral-periodic in x. Then, as t → ∞ it converges to the unique solution U ≡ W to (23) uniformly in O r , for any r > 0. Therefore, (62) yields uniformly in x ∈ O r , for any r > 0.
Step 1 and the decay of U then imply (63).

Behaviour near critical value
Bifurcation results of the type of Theorem 2.7 have been proved by Crandall and Rabinowitz [8] in very general frameworks. However, we will not make use of the abstract result of [8], but rather give a direct proof. Indeed, to check that its hypotheses are satisfied in our case requires essentially the same work as the direct derivation of Theorem 2.7 which we give here.
In order to prove statement (ii) of Theorem 2.7 we make use of the fact that the generalized Neumann principal eigenvalue λ 1,N is simple when c = c 0 (i. e. when λ 1,N = 0). This type of property, which follows directly from the Krein-Rutman theory in the case of the principal eigenvalue of an operator in a bounded smooth domain, is not true in general for unbounded domains. Thus, this part is rather delicate. It holds here because of the additional property that the zero order term of L is negative at infinity, cf. condition (12). We prove this result in [7] by first showing that there exists a generalized principal eigenvalue which vanishes at infinity and then using a comparison result of the same type as Theorem 3.3 here.  The reader is referred to [7] for the details of the proof.
Proof of Theorem 2.7. (i) Assume by contradiction that there exist ε > 0 and two sequences (c n ) n∈N in (0, c 0 ) and ((x n 1 , y n )) n∈N in Ω such that lim n→∞ c n = c 0 , U cn (x n 1 , y n ) ≥ ε.
We know that 0 < U cn ≤ S, where the second inequality -with S given by (10) -follows from Theorem 3.3. By elliptic estimates and embedding theorems (a subsequence of) the sequence (U cn ) n∈N converges uniformly in Ω r , for any r > 0, to a nonnegative bounded solution U * of Since U * is not strictly positive by Proposition 2 and Theorem 2.1, the strong maximum principle yields U * ≡ 0. Hence, the sequence (x n 1 ) n∈N has to be divergent. It is not restrictive to assume that ∀ n ∈ N, U cn (x n 1 , y n ) ≥ U cn L ∞ (Ω) −

L 1 convergence
We first derive the following result for linear parabolic problems.
Remark 6. If the initial datum u 0 does not belong to L 1 (Ω) then the convergences in Theorem 2.2 do not hold in general in the L 1 sense. As an example, the function is a solution of (3) with f (x, s) = −s − s 2 and initial datum u 0 ≡ 1. As t → ∞, u(t, ·) converges to 0 in L ∞ (Ω) but not in L 1 (Ω).
Proof of Theorem 2.9. The result follows from the same ideas as before, with some minor changes that we briefly outline here. Indeed, owing to the uniform convergence of u(t, x) to 0 as t → ∞ given by Theorem 1.3 in [6], one can prove Theorem 2.9 by establishing an analogous result to Lemma 7.1.
In the whole space, the analogue of Lemma 7.1 is obtained by replacing Ω by R N and assuming that ξ satisfies lim r→∞ sup t>0 |x|>r ξ(t, x) < 0.
To prove it, one uses again the superposition principle, writing w = w 1 + w 2 , but then considers a different function v than that one introduced in the proof of Lemma 7.1: where ε is chosen in such a way that P v ≥ g in R N \B R (recall that g(t, x) ≤ 0 for |x| ≥ R). Hence, by comparison, w 2 ≤ v and then the Lebesgue theorem yields lim t→∞ w + 2 (t, ·) = 0 in L 1 (R N ). Let us mention that the arguments in the proofs of Theorems 2.8 and 2.9 allow one to prove that, in the lateral periodic case, the convergences of u given by Theorem 2.6 also hold in L 1 (R M × K), for any K ⊂⊂ R P .

Seasonal dependence
We only outline the proofs of Theorems 2.10 and 2.11. Essentially, these results are obtained by using the same ideas as in Section 3 and Appendix A and following the strategy of [6] Section 3, where the two-speeds problem in the whole space is treated.
First, one shows the existence of the time periodic principal eigenvalue of P in the finite cylinders Ω r , with mixed Dirichlet/Neumann boundary conditions, that is, the unique real number µ(r) such that the eigenvalue problem        Pψ = µ(r)ψ in R × Ω r ∂ ν ψ(t, x) = 0 on R × (−r, r) × ∂ω ψ = 0 on R × {±r} × ω ψ is T -periodic in t admits a positive solution ψ. The arguments of Appendix A, which enable one to apply the Krein-Rutman theory and find the µ(r), also work in this framework thanks to the Hölder continuity of f s (t, x, 0). Then, proceeding as in the proof of Proposition 4, one shows that lim r→∞ µ(r) = µ 1,N .
The following result is proved in [6] in the case Ω = R N , but it also holds for general domains.
If µ 1,N < 0 then one can find R > 0 large enough such that µ(R) < 0. Hence, the principal eigenfunction ψ associated with µ(R) -suitably normalized and extended by 0 in Ω\Ω R -is a T -periodic in t subsolution to (65). Applying Theorem 8.1 with v = ψ we then find a T -periodic in t solution U ≥ ψ to (65). Consequently, as U > 0 by the strong maximum principle, the sufficient condition of Theorem 2.10 for the existence of pulsating travelling waves is proved. To derive the necessary condition and the uniqueness result one proceeds as in Section 3, by establishing the exponential decay of solutions and a comparison principle analogous to Theorem 3.3 (see Proposition 9 and Theorem 3.3 in [6]).
Theorem 8.1 also allows one to prove that the convergences in Theorem 2.11 hold locally uniformly in Ω. We recall that to prove Theorem 2.2 we used the property that any solution of (45) coinciding with a subsolution or a supersolution of the stationary problem at the initial time is monotone in t. This is no longer true for (65) because the terms in the equation depend on time. However, owing to Theorem 8.1, one can derive the locally uniform convergence as in the case of (45) by considering solutions of (65) coinciding with a subsolution and a supersolution which is T -periodic in t. The uniform convergence then follows by arguing exactly as in the step 2 of the proof of Theorem 2.2.
Hopf lemma, one can check that it is also strictly positive, that is, T (C\{0}) is contained in the interior of C, where C denotes the closed positive cone of nonnegative functions of X r . It is at this stage that the W 1,∞ norm is required for the space X r . Then, from the Krein-Rutman theory (see [13] and [12]) it follows that T admits a unique eigenvalue λ (> 0) with associated positive eigenfunction ϕ r ∈ X r (unique up to a multiplicative constant). Therefore, the constant λ(r) := 1 λ − d satisfies the desired property.