Thomas Giletti

Luca Rossi

Pulsating solutions for multidimensional bistable and multistable equations

We investigate the existence of pulsating front-like solutions for spatially periodic heterogeneous reaction-diusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sucient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a nite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.

Main results Before stating our theorems, let us also recall a result by Weinberger.

). Let p > q be two periodic steady states of (1.1), and assume that any periodic function u0 ∈ C(R N ) satisfying q ≤ u0 ≤ p, u0 ≡ q, lies in the basin of attraction of p. Then, for any e ∈ S N -1 , there is some c * ∈ R such that a pulsating travelling front in the direction e with speed c connecting p to q exist if and only if c ≥ c * .

Assumptions 1.1 or 1.2 allow us to apply this theorem around any given unstable periodic state q between 0 and p. To check the hypothesis of Theorem 1.3, x x0 ∈ R N and let p+ > q be a stable state realizing the following minimum: min{p(x0) : q < p ≤ p and p is a stable periodic steady state}.

Assumption 1.1 and the elliptic strong maximum principle imply that the unique periodic steady state q satisfying q < q ≤ p is q ≡ p, hence p+ ≡ p in such case. Under Assumption 1.2, the denition of p+ is well posed because there is a nite number of stable periodic steady states. Moreover, there are no other periodic steady states between q and p+, because otherwise the last condition in Assumption 1.2 (and the strong maximum principle) would contradict the denition of p+. We now apply the order interval trichotomy Theorem A.1 to p ≡ q and p ≡ p+. We have seen that the case (a) cannot occur under both Assumptions 1.1 or 1.2. The case (c) is also ruled out, by the stability of p+. It remains the case (b), namely, there exists a spatially periodic solution u of (1.1) such that u(k, •) → q as k → -∞ and u(k, •) → p+ as k → +∞. By comparison principles, this implies that any periodic initial datum q ≤ u0 ≤ p+ with u0 ≡ q lies in the basin of attraction of p+. We can therefore apply Theorem 1.3 and nd a minimal speed cq of fronts in a given direction e ∈ S N -1 connecting p+ to q. Applying the same arguments to (1.1) with f (x, u) replaced by -f (x, -u), we nd a minimal speed c q of fronts Ũ in the direction -e connecting -pto -q, where pis the largest stable periodic steady state lying below q. Hence, c q := -c q is the maximal speed of fronts U (x, z) := -Ũ (x, -z) for (1.1) in the direction e connecting q to p-.

After these considerations, we are in a position to state our last assumption.

Assumption 1.3. For any unstable periodic steady state q between 0 and p and any e ∈ S N -1 , there holds that cq > c q ,

Introduction

In this work we consider the reaction-diusion equation ∂tu = div(A(x)∇u) + f (x, u), t ∈ R, x ∈ R N , (1.1) where N ≥ 1 is the space dimension. The diusion matrix eld A = (Ai,j) 1≤i,j≤N is always assumed to be smooth and to satisfy the ellipticity condition ∃C1, C2 ∈ (0, ∞), ∀x, ξ ∈ R N , C1|ξ| 2 ≤ i,j Ai,j(x)ξiξj ≤ C2|ξ| 2 .

(1.2)

As far as the regularity of the reaction term f (x, u) is concerned, we assume that it is at least globally Lipschitz continuous (a stronger hypothesis will be made in the general multistable case; see below).

Equation (1.1) is spatially heterogeneous. As our goal is to construct travelling fronts, i.e., self-similar propagating solutions, we impose a spatial structure on the heterogeneity. More precisely, we assume that the terms in the equation are all periodic in space, with the same period. For simplicity and without loss of generality up to some change of variables, we choose the periodicity cell to be [0, 1] N , that is,

∀L ∈ Z N , A(• + L) ≡ A(•), f (• + L, •) ≡ f (•, •). (1.3)
From now on, when we say that a function is periodic, we always understand that its period is (1, . . . , 1).

In the spatially periodic case, one can consider the notion of pulsating travelling front, which we shall recall precisely below. Roughly, these are entire in time solutions which connect periodic steady states of the parabolic equation (1.1). The existence of such solutions is therefore deeply related to the underlying structure of (1.1) and its steady states.

In this paper, we shall always assume that (1.1) admits at least two spatially periodic steady states: the constant 0 and a positive state p(x). Namely, we assume that f (•, 0) ≡ 0, 1 as well as div(A(x)∇ p) + f (x, p) = 0, ∀L ∈ Z N , p(• + L) ≡ p > 0.

We shall restrict ourselves to solutions u(t, x) of (1.1) that satisfy the inequality 0 ≤ u ≤ p.

Notice that, as far as the Cauchy problem is concerned, owing to the parabolic comparison principle, it is sucient to assume that the above property is fullled by the initial datum (we restrict ourselves to bounded solutions, avoiding in this way situations where the comparison principle fails). Let us also mention that 0 could be replaced by a spatially periodic steady state; we make this choice to keep the presentation simpler.

The steady states 0 and p will be assumed to be asymptotically stable; we shall recall what this means in a moment. Then we distinguish the situation where these are the unique periodic steady states (bistable case) to that where there is a nite number of intermediate stable states (multistable case). In the latter, we will strengthen the stability condition.

Assumption 1.1 (Bistable case). The functions 0 and p are the only asymptotically stable periodic steady states of (1.1). Furthermore, there does not exist any pair q, q of periodic steady states of (1.1) such that 0 < q < q < p. Assumption 1.2 (Multistable case). The function ∂uf (x, u) is well-dened and continuous. There is a nite number of asymptotically stable periodic steady states, which include 0 and p, and they are all linearly stable. Furthermore, for any pair of ordered periodic steady states q < q, there is a linearly stable periodic steady state p such that q ≤ p ≤ q.

The main dierence between these two assumptions is that only the latter allows the existence of intermediate stable steady states. As we shall see, the presence of such intermediate states might prevent the existence of a pulsating travelling front connecting directly the two extremal steady states 0 and p. More complicated dynamics involving a family of travelling fronts, which we refer to as a propagating terrace, may instead occur.

We emphasize that the stable states in Assumption 1.2 are not necessarily ordered.

Let us recall the dierent notions of stability. A steady state p is said to be asymptotically stable if its basin of attraction contains an open neighbourhood of p in the L ∞ (R N ) topology; the basin of attraction of p refers to the set of initial data for which the solution of the Cauchy problem associated with (1.1) converges uniformly to p as t → +∞.

A periodic state p is said to be linearly stable (resp. unstable) if the linearized operator around p, i.e., Lpw := div(A(x)∇w) + ∂uf (x, p(x))w, has a negative (resp. positive) principal eigenvalue in the space of periodic functions.

Owing to the regularity of f from Assumption 1.2, it is rather standard to construct sub-and supersolutions using the principal eigenfunction and to use them to show that linear stability implies asymptotic stability. The converse is not true in general; this is why the bistable Assumption 1.1 is not a particular case of Assumption 1.2.

We also point out that the second part of Assumption 1.1 automatically prevents the existence of intermediate asymptotically stable steady states, thanks to a crucial result in dynamical systems due to Dancer and Hess [START_REF] Dancer | Stability of xed points for order-preserving discrete-time dynamical systems[END_REF] known as order interval trichotomy; see also [START_REF] Matano | Existence of nontrivial unstable sets for equilibriums of strongly ordered-preserving systems[END_REF]. We recall such result in Theorem A.1 in the Appendix.

Remark 1. In the case of the spatially-invariant equation ∂tu = ∆u + f (u), t ∈ R, x ∈ R N , (1.4) Assumption 1.1 is fullled if and only if p is constant, say, equal to 1, and there exists θ ∈ (0, 1) such that f (u) < 0 for u ∈ (0, θ) and f (u) > 0 for u ∈ (θ, 1). This is shown in Lemma 7.2 below and the subsequent remark. Then, with the same arguments, one can readily check that Assumption 1.2 is equivalent to require that f ∈ C 1 ([0, 1]) and that it has an odd (nite) number of zeroes such that, counting from smallest to largest, the odd ones (which include 0, 1) satisfy f < 0 (these are the only stable periodic steady states).

With a slight abuse of terminology, in the sequel we shall simply refer to the asymptotic stability as stability. Then a solution will be said to be unstable if it is not (asymptotically) stable.

The notion of pulsating fronts and terraces Let us rst recall the notion of pulsating travelling front, which is the extension to the periodic framework of the usual notion of travelling front. We refer to [START_REF] Xin | Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity[END_REF] for an early introduction of this concept. Denition 1.1. A pulsating travelling front for (1.1) is an entire in time solution of the type u(t, x) = U (x, x • e -ct), where c ∈ R, e ∈ S N -1 , the function U (x, z) is periodic in the x-variable and satises

U (•, -∞) ≡ q1(•) > U (•, •) > U (•, +∞) ≡ q2(•).
Furthermore, we call c the speed of the front, the vector e its direction, and we say that U connects q1 to q2.

Remark 2. The functions q1, q2 in the above denition are necessarily two steady states of (1.1). Let us also point out that the change of variables (t, x) → (x, x • e -ct) is only invertible when c = 0, so that one should a priori carefully distinguish both functions u and U .

In the bistable case, our goal is to construct a pulsating front connecting p to 0. Let us reclaim a few earlier results. In [START_REF] Xin | Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity[END_REF], a pulsating front was already constructed in the special case where coecients are close to constants. Yet dealing with more general heterogeneities turned out to be much more dicult, and only recently a pulsating front was constructed in [START_REF] Fang | Bistable traveling waves for monotone semiows with applications[END_REF] for the one-dimensional case, through an abstract framework which is similar to the one considered in the present work. Higher dimensions were tackled in [START_REF] Ducrot | A multi-dimensional bistable nonlinear diusion equation in a periodic medium[END_REF] under an additional nondegeneracy assumption and with a more PDEoriented approach in the spirit of [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF].

However, as mentioned before, the notion of pulsating travelling front does not suce to describe the dynamics in the more general multistable case. The good notion in such case is that of a propagating terrace, as dened in [START_REF] Ducrot | Existence and convergence to a propagating terrace in one-dimensional reaction-diusion equations[END_REF][START_REF] Giletti | Existence and uniqueness of propagating terraces[END_REF]. An earlier equivalent notion, called minimal decomposition, was introduced in [START_REF] Fife | The approach of solutions of nonlinear diusion equations to traveling front solutions[END_REF] in the homogeneous case. Denition 1.2. A propagating terrace connecting p to 0 in the direction e ∈ S N -1 is a couple of two nite sequences (qj) 0≤j≤J and (Uj) 1≤j≤J such that:

• the functions qj are periodic steady states of (1.1) and satisfy p ≡ q0 > q1 > • • • > qJ ≡ 0;

• for any 1 ≤ j ≤ J, the function Uj is a pulsating travelling front of (1.1) connecting qj-1 to qj with speed cj ∈ R and direction e;

• the sequence (cj) 1≤j≤J satises c1 ≤ c2 ≤ • • • ≤ cJ .
Roughly speaking, a propagating terrace is a superposition of pulsating travelling fronts spanning the whole range from 0 to p. We emphasize that the ordering of the speeds of the fronts involved in a propagating terrace is essential. Indeed, while there may exist many families of steady states and fronts satisfying the rst two conditions in Denition 1.2, only terraces can be expected to describe the large-time behaviour of solutions of the Cauchy problem associated to (1.1), see [START_REF] Ducrot | Existence and convergence to a propagating terrace in one-dimensional reaction-diusion equations[END_REF], which makes them more meaningful. We also refer to [START_REF] Polá£ik | Propagating terraces and the dynamics of front-like solutions of reaction-diusion equations on R[END_REF] which deals with the homogeneous equation under weaker assumptions on f . where cq and c q are dened above.

Notice that under the bistable Assumption 1.1, clearly p+ ≡ p and p-≡ 0. Therefore, in that case, Assumption 1.3 means that pulsating fronts connecting p to an intermediate state q have to be strictly faster than pulsating fronts connecting q to 0.

We point out that this hypothesis, though implicit, was already crucial in the earlier existence results for bistable pulsating fronts; see [START_REF] Ducrot | A multi-dimensional bistable nonlinear diusion equation in a periodic medium[END_REF][START_REF] Fang | Bistable traveling waves for monotone semiows with applications[END_REF] where it was referred to as the counter-propagation assumption.

When u → f (x, u) is C 1 , a sucient condition ensuring Assumption 1.3 is that q is linearly unstable. In such a case there holds that cq > 0 > c q , as shown in Proposition A.2 in the Appendix. We also show there for completeness that if q is just unstable then cq ≥ 0 ≥ c q . The fact that the minimal speed in a monostable problem cannot be 0 seems to be a natural property. Besides the non-degenerate (q linearly unstable) case, it is known to hold for homogeneous equations as well as for some special (and more explicit) bistable equations, c.f. [START_REF] Ding | Bistable pulsating fronts for reaction-diusion equations in a periodic habitat[END_REF][START_REF] Fang | Bistable traveling waves for monotone semiows with applications[END_REF] and the references therein. However, as far as we know, it remains an open problem in general.

Our rst main result concerns the bistable cases.

Theorem 1.4 (Bistable case). If Assumptions 1.1 and 1.3 are satised, then for any e ∈ S N -1 , there exists a monotonic in time pulsating travelling front connecting p to 0 in the direction e with some speed c(e) ∈ R.

Remark 3. Our previous theorem includes the possibility of a front with zero speed. However, there does not seem to be a unique denition of a pulsating front with zero speed in the literature, mainly because the change of variables (t, x) → (x, x • e -ct) is not invertible when c = 0. Here, by Denition 1.1 a front with zero speed is simply a stationary solution u(x) with asymptotics u(x) -q1,2 → 0 as x • e → ∓∞. As a matter of fact, in the zero speed case our approach provides the additional property that there exists a function U as in Denition 1.1, such that u(t, x) = U (x, x • e + z) solves (1.1) for any z ∈ R. However, this function U lacks any regularity, so that in particular it is not a standing pulsating wave in the sense of [START_REF] Ducrot | A multi-dimensional bistable nonlinear diusion equation in a periodic medium[END_REF].

Theorem 1.5 (Multistable case). If Assumptions 1.2 and 1.3 are satised, then for any e ∈ S N -1 , there exists a propagating terrace ((qj)j, (Uj)j) connecting p to 0 in the direction e.

Furthermore, all the qj are stable steady states and all the fronts Uj are monotonic in time.

Earlier existence results for propagating terraces dealt only with the one-dimensional case, where a Sturm-Liouville zero number and steepness argument is available [START_REF] Ducrot | Existence and convergence to a propagating terrace in one-dimensional reaction-diusion equations[END_REF][START_REF] Giletti | Existence and uniqueness of propagating terraces[END_REF].

We also refer to [START_REF] Risler | Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure[END_REF] where a similar phenomenon is studied by an energy method in the framework of systems with a gradient structure. As far as we know, this result is completely new in the heterogeneous and higher dimensional case.

The stability of these pulsating fronts and terraces will be the subject of a forthcoming work. Let us point out that, quite intriguingly, the shape of the terrace may vary depending on the direction. More precisely, for dierent choices of the vector e, the terrace may involve dierent intermediate states (qj)j; it is even possible that the number of such states varies, as we state in the next proposition. Proposition 1.6. There exists an equation (1.1) in dimension N = 2 for which Assumptions 1.2, 1.3 hold and moreover:

• in the direction (1, 0), there exists a unique propagating terrace connecting p to 0, and it consists of exactly two travelling fronts;

• in the direction (0, 1), there exists a unique propagating terrace connecting p to 0, and it consists of a single travelling front.

Uniqueness here is understood up to shifts in time of the fronts. It will be especially interesting to study how this non-symmetric phenomenon aects the large-time dynamics of solutions of the Cauchy problem.

Plan of the paper We start in the next section with a sketch of our argument in the homogeneous case, to explain the main ingredients of our method. This relies on a time discretization, in the spirit of Weinberger [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], and on the study of an associated notion of a discrete travelling front. For the sake of completeness, some of the arguments of [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF] will be reclaimed along the proofs. We also point out here that the resulting discrete problem shares similarities with the abstract bistable framework considered in [START_REF] Fang | Bistable traveling waves for monotone semiows with applications[END_REF], though we shall use a dierent method to tackle bistable and multistable equations without distinction.

The proof of the general case is carried out in several steps:

1. Introduction of the iterative scheme (Sections 3.1, 3.2).

2. Denition of the speed of the front (Section 3.3).

3.

Capturing the iteration at the good moment and position (Section 4.1).

Derivation of the travelling front properties (Section 4.2).

At this stage we shall have constructed a discrete pulsating travelling front connecting p to some stable periodic steady state 0 ≤ p < p. In the bistable case, one necessarily has that p ≡ 0 and then it only remains to prove that the front is actually a continuous front. For the multistable case, we shall iterate our construction getting a family of travelling fronts. In order to conclude that this is a propagating terrace, we need to show that their speeds are ordered; this is the only point which requires the linear stability in Assumption 1.2. Summing up, the method proceeds as follows:

5. Construction of the (discrete) pulsating terrace (Section 5).

6. Passing to the continuous limit (Section 6).

Finally, Section 7 is dedicated to the proof of Proposition 1.6, which provides an example where the shape of the propagating terrace strongly depends on its direction. To achieve this, we shall exhibit a bistable equation for which pulsating fronts have dierent speeds, depending on their direction, see Proposition 7.1 below.

The 1-D homogeneous case

In order to illustrate our approach, let us consider the simpler (and, as far as travelling fronts are concerned, already well-understood [START_REF] Aronson | Multidimensional nonlinear diusion arising in population genetics[END_REF]) bistable homogeneous equation

∂tu = ∂xxu + f (u), t ∈ R, x ∈ R, (2.1) with f ∈ C 1 ([0, 1]) satisfying f (0) = f (1) = 0, f < 0 in (0, θ), f > 0 in (θ, 1).
In this framework, pulsating fronts simply reduce to planar fronts, i.e., entire solutions of the form U (x -ct).

The hypotheses on f guarantee that Assumption 1.1 is fullled with p ≡ 1. They also entail the counter-propagation property, Assumption 1.3, because in the homogeneous monostable case travelling fronts have positive speeds, see [START_REF] Aronson | Multidimensional nonlinear diusion arising in population genetics[END_REF]. Namely, fronts connecting 1 to θ exist for speeds c larger than some c > 0, whereas fronts connecting θ to 0 exist for speeds c smaller than some c < 0 (the latter property is derived from [START_REF] Aronson | Multidimensional nonlinear diusion arising in population genetics[END_REF] by considering fronts moving leftward for the equation for θ -u).

The equation in the frame moving rightward with speed c ∈ R reads

∂tu = ∂xxu + c∂xu + f (u), t ∈ R, x ∈ R. (2.2)

The dynamical system

We start by placing ourselves in a more abstract framework which we shall use to dene a candidate front speed c * , in the same way as in [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF]. We shall then turn to the construction of a travelling front connecting 1 to 0. We point out that in [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], such a travelling front was only shown to exist in the monostable case, and that a dierent argument is needed to deal with bistable or more complicated situations.

For any given c ∈ R, we call Fc the evolution operator after time 1 associated with (2.2). Namely, Fc[φ](x) := v(1, x), where v is the solution of (2.2) emerging from the initial datum v(0, x) = φ(x). It follows from the parabolic strong maximum principle that the operator Fc is increasing.

Let us already point out that the prole U of a usual travelling front U (x -ct) for (2.1) is a stationary solution of (2.2) and thus a xed point for the operator Fc.

As a matter of fact, in the homogeneous case the converse is also true (this follows for instance from a uniqueness result for almost planar fronts derived in [START_REF] Berestycki | Generalized transition waves and their properties[END_REF]). Therefore, our goal in this section will be to construct such a xed point.

Consider a function φ ∈ W 1,∞ (R) satisfying φ is nonincreasing, φ(-∞) ∈ (θ, 1), φ = 0 in [0, +∞). (2.3)
We then dene a sequence (ac,n) n∈N through the following iterative procedure:

ac,0 := φ, ac,n+1 := max{φ, Fc[ac,n]},
where the maximum is taken at each x ∈ R.

It follows from the monotonicity of φ and Fc (the latter being strict) that ac,n(x) is nondecreasing with respect to n and nonincreasing with respect to x, and that it satises 0 < ac,n < 1. Then, observing that

Fc[V ] = F0[V ](• + c), (2.4)
for any function V , we deduce that ac,n is nonincreasing with respect to c. One also checks by iteration that ac,n(+∞) = 0, thanks to standard parabolic arguments. All these properties are summarized in the following.

Lemma 2.1. The sequence (ac,n) n∈N is nondecreasing and satises 0 < ac,n < 1 and ac,n(+∞) = 0 for all n ≥ 1. Moreover, ac,n(x) is nonincreasing with respect to both c and x, the latter monotonicity being strict in the set where ac,n > φ.

Lemma 2.1 implies that (ac,n) n∈N converges pointwise to some nonincreasing function φ ≤ ac ≤ 1. The convergence actually holds locally uniformly in R, because the ac,n are equi-uniformly Lipschitz-continuous, due to parabolic estimates. We also know that the ac are nonincreasing with respect to c.

We then introduce c * := sup{c ∈ R : ac ≡ 1}.

One may check that c * is indeed a well-dened real number. Without going into the details (this a particular case of either Section 3 or [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF]), we simply point out that this can be proved using some super-and subsolutions which exist thanks to the Lipschitz continuity of f as well as to the choice of φ(-∞) in the basin of attraction of 1.

We further see that the denition of c * does not depend on the particular choice of the initialising function φ. Indeed, if φ satisfying (2.3) is the initialisation of another sequence, then for c < c * there holds that ac,n > φ for n suciently large. From this and the monotonicity of Fc one deduces by iteration that the value of c * obtained starting from φ is larger than or equal to the one provided by φ. Equality follows by exchanging the roles of φ and φ.

We shall also use the fact that ac * ≡ 1.

(2.5) This comes from the openness of the set {c ∈ R : ac ≡ 1}, which is established in either Section 3 or [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF] in the more general periodic case. Let us briey sketch a more direct proof. Let c ∈ R be such that ac ≡ 1. We can nd n such that ac,n(1) > φ(-∞).

Arguing by induction and exploiting (2.4), one sees that ∀δ > 0, n ∈ N, x ∈ R, a c+δ,n (x) ≥ ac,n(x + nδ).

Thus, a c+ 1 n ,n (0) > φ(-∞) which implies that a c+ 1 n ,n > φ because a c+ 1 n ,n and φ are nonincreasing and φ is supported in (-∞, 0]. Using the next result we eventually deduce that a c ≡ 1 for all c in some neighborhood of c, and thus c * > c.

Lemma 2.2. Let c ∈ R and n ∈ N be such that a c ,n > φ. Then a c ≡ 1 for all c < c .

Proof. The monotonicities provided by Lemma 2.1 yield a c ,n+m > φ for all c ≤ c and m ∈ N, which, recalling the denition of the sequences (ac,n) n∈N , implies in turn that a c ,n+m = (F c ) m [a c ,n ]. Then, taking c < c and exploiting (2.4), we get

∀m ∈ N, x ∈ R, a c ,n+m (x) = (F c ) m [a c ,n ](x) = (F c ) m [a c ,n ](x -(c -c )m) ≥ (F c ) m [a c ,n ](x -(c -c )m) = a c ,n+m (x -(c -c )m).
Passing to the limit as m → +∞ (and using again the monotonicity of the sequence) we nd that a c (x) ≥ a c ,n (-∞) for all x ∈ R and n ∈ N. Observe that (a c ,n (-∞)) n∈N is the solution of the ODE U = f (U ) computed on the integers and starting from φ(-∞) > θ, whence it converges to 1. This shows that a c ≡ 1.

Capturing the sequence at the good moment and position

From here we diverge from Weinberger's scheme which, as we mentioned above, does provide a front in the monostable case but not in the bistable one. Consider c < c * . Because ac ≡ 1, we have seen before that we can nd n(c) such that a c,n(c)+m > φ for m ∈ N. This means that, starting from n(c), the sequence (ac,n) n∈N is simply given by iterations of Fc, namely, ∀m ∈ N, a c,n(c)+m = (Fc) m [a c,n(c) ].

(2.6) Fix θ ∈ (θ, φ(-∞)) and, for n ≥ n(c), dene the point x(c, n) through the relation ac,n(x(c, n)) = θ .

Note that x(c, n) exists because ac,n(-∞) ≥ φ(-∞) > θ and ac,n(+∞) = 0 by Lemma 2.1. Moreover we claim that, by construction of c * , there holds that

lim sup n→∞ x(c, n) n ≤ c * -c.
(2.7)

Let us postpone the proof of this for a moment and continue with our construction. By

(2.7), one readily sees that, up to increasing n(c) if need be, the following holds:

∀0 ≤ m ≤ 1/ √ c * -c, x(c, n(c) + m) -x(c, n(c)) ≤ 2 √ c * -c. ( 2
∀n ∈ N, uc(n + 1, •) ≡ a c,n(c)+n+1 ≥ a c,n(c)+n ≡ uc(n, •),
that is, the sequence (uc(n, •)) n∈N is nondecreasing. Furthermore, the function uc inherits the monotonicity in x of the initial datum, which is strict by Lemma 2.1 because ac,n > φ.

We nally consider the translation uc(t, x + x(c, n(c))) of uc. By parabolic estimates up to t = 0, we have that (up to subsequences) uc(t, x + x(c, n(c))) → a * (t, x) as c c * , locally uniformly in (t, x) ∈ [0, +∞) × R, where a * (t, x) satises the equation (2.2) with c = c * . We further know that a * (0, 0) = θ and that a * (n, x) is nondecreasing in n ∈ N and nonincreasing in x ∈ R.

Let us now prove (2.7). First, the function φ being nonincreasing, for any c < c * we deduce from (2.4) that

ac,1 = max{φ, Fc[φ]} ≤ max{φ(• + (c -c * )), Fc * [φ](• + (c -c * ))} = ac * ,1(• + (c -c * )).
An iterative argument then shows that ∀n ∈ N, ac,n ≤ ac * ,n(• + n(c -c * )).

(2.9)

Now it follows from (2.5) that inf ac * ≤ θ. Indeed, assume by contradiction that inf ac * > θ.

X θ ∈ R such that ∀n ∈ N, θ > ac * (X θ ) ≥ ac * ,n(X θ ) ≥ ac,n(X θ + n(c * -c)),
where the last inequality follows from (2.9). This means that ∀n ∈ N, x(c, n) < X θ + n(c * -c), from which (2.7) immediately follows.

The function a * converges to the prole of a front

We recall that, by construction, the sequence a * (n, •) is nondecreasing with respect to n ∈ N. In particular, we can dene

U * (t, x) := lim n→+∞ a * (t + n, x),
By parabolic estimates, the above limit exists (up to subsequences) locally uniformly in (t, x) ∈ R 2 and U * is a periodic in time solution of (2.2) with c = c * . Moreover, U * satises U * (0, 0) ≥ θ and inherits from a * that it is nonincreasing with respect to x.

Let us check that it is actually a travelling front.

Using parabolic estimates and the monotonicity with respect to x, we see that the sequences (U * (t, x ± n)) n∈N converge locally uniformly in (t, x) ∈ R 2 (up to subsequences) to two steady states U * ± of the same ODE U = f (U ) (here we used that this ODE does not admit non-trivial periodic solutions), i.e., U * ± are constantly equal to 0, θ or 1. The fact that U * (0, 0) ≥ θ > θ and the monotonicity in x then imply that

U * -= U * (•, -∞) ≡ 1.
Next, we claim that U * -≡ 0. Once this claim is proved, one may show by a sliding argument as in [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] that U * is actually independent of t, and thus it is the prole of a front moving with speed c * . Therefore, in order to conclude this preliminary section, we need to rule out the cases U * + ≡ θ and U * + ≡ 1. Condition (2.8) is specically devised to prevent the latter possibility. Indeed, it yields

∀0 ≤ m ≤ 1/ √ c * -c, uc(m, x(c, n(c)) + 2 √ c * -c) ≤ uc(m, x(c, n(c) + m)) = θ .
Passing to the limit as c c * in this inequality we get ∀m ∈ N, a * (m, 0) ≤ θ , whence U * (0, 0) ≤ θ . By the monotonicity in x, we then derive

U * + = U * (•, +∞) < 1.
It remains to rule out the case U * + ≡ θ. To achieve this, we shall compare c * with the spreading speeds associated with the restrictions of f to [0, θ] and [θ, 1] respectively, which are of the well-known (even in the periodic and multidimensional case) monostable type. This is where the counter-propagation property comes into play. We recall that such a property is guaranteed in the homogeneous case we are considering now, but should be imposed in general through Assumption 1.3.

We proceed by contradiction and suppose that U * + ≡ θ. Thus U * (0, •) ≥ θ, as well as U * (0, •) ≥ u0 dened by u0 = θ 1 (-∞,0] + θ 1 (0,+∞) .

Consider now the solution u of (2.1) with initial datum u0. Since θ is an unstable steady state, we can use the well-known result about the spreading speed for solutions of the monostable equation from [START_REF] Aronson | Multidimensional nonlinear diusion arising in population genetics[END_REF]. Namely, we nd a speed c > 0 such that ∀c < c, u(t, ct) → 1 as t → +∞, ∀c > c, u(t, ct) → θ as t → +∞.

It is also proved in [START_REF] Aronson | Multidimensional nonlinear diusion arising in population genetics[END_REF] that c coincides with the minimal speed of fronts, c.f. Theorem 1.3, that is, using the same notation as in the introduction, there holds that c = c θ . Since U * (t, x -c * t) satises (2.1) and U * (0, •) ≥ u0, we infer by comparison that for all c < c, there holds U * (t, (c -c * )t) → 1 as t → +∞. Recalling that U * is periodic in time and that we are assuming that U * + ≡ θ, we eventually nd that c * ≥ c > 0. Let us go back now to the construction of a * , U * . We have that, up to a subsequence,

U * (0, x) = lim k→+∞ lim c c * a c,n(c)+k x + x(c, n(c)) .
In particular, one can take a sequence c k c * such that, locally uniformly in x,

U * (0, x) = lim k→+∞ a c k ,n(c k )+k (x + x(c k , n(c k ))). (2.10) Now for any c < c * and n ∈ N, let x (c, n) be such that ac,n(x (c, n)) = θ 2 .
Let us extract another subsequence so that the solution of (2.2) with initial datum

a c k ,n(c k )+k (x + x (c k , n(c k ) + k)) converges locally uniformly in (t, x) ∈ R 2 to some V * (t, x), which is an entire solution of (2.2) with c = c * . Moreover, V * (n, x) is nondecreasing in n ∈ Z, nonincreasing in x ∈ R, and satises V * (0, 0) = θ/2. One can further see that V * (0, •) ≤ θ; this follows from the fact that x (c k , n(c k ) + k) -x(c k , n(c k )) → +∞, which, in turn, is a consequence of (2.10
) and of the contradictory assumption U * + ≡ θ. In particular, we have that V * (0, •) ≤ u 0 dened by

u 0 = θ 1 (-∞,0] + θ 2 1 (0,+∞) .
Owing again to the spreading result for the monostable equation, there exists a speed c < 0 such that the solution u of (2.1) emerging from u 0 satises

∀c < c, u(t, ct) → θ as t → +∞, ∀c > c, u(t, ct) → 0 as t → +∞.
On one hand, by comparison we get that V * (t, x -c * t) ≤ u(t, x). On the other hand, by monotonicity we know that V * (n, x) ≥ θ 2 for all n ∈ N, x ≤ 0. One then easily infers that c * ≤ c < 0. We have nally reached a contradiction. [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] The iterative scheme in the periodic, N -dimensional case

We now turn to the general periodic case in arbitrary dimension. Because the equation is no longer invariant by any space translation, we need to introduce a more complicated operator involving also a somewhat articial variable. This makes things more technical, though the overall strategy remains the same.

A time discretization

The main ingredient of our proofs is inspired by Weinberger [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], and consists in looking for travelling fronts as xed points of an appropriate family of mappings issued from a time discretization of (1.1).

First, we use the notation

v(t, y; x → v0(x))
to indicate the solution to (1.1) with initial datum v0, evaluated at (t, y). In the sequel, we shall often omit to write x → and we shall just use x as the variable involved in the initial datum.

Let us now recall (see Denition 1.1) that a pulsating travelling front in a direction e ∈ S N -1 is a solution of (1.1) of the form

u(t, x) = U (x, x • e -ct)
with U (x, z) periodic in the x-variable and converging to two distinct steady states as z → ±∞. In particular, one may look at a travelling front as a family (U (x, z)) z∈R , using the second variable as an index.

Let us translate the notion of pulsating travelling front to the discrete setting. Denition 3.1. A discrete travelling front in a direction e ∈ S N -1 with speed c ∈ R is a function U (y, z) which is periodic in its rst variable, satises

∀(y, z) ∈ R N +1 , v(1, y; x → U (x, z + x • e)) ≡ U (y, z + y • e -c),
and connects two steady states q1 and q2, i.e.,

U (•, -∞) ≡ q1(•) > U (•, •) > U (•, +∞) ≡ q2(•),
where convergences are understood to be uniform.

Clearly, if u(t, x) = U (x, x • e -ct) is a (continuous) pulsating travelling front then U (x, z) is a discrete travelling front, at least if c = 0 so that the change of variables (t, x) → (x, x • e -ct) is invertible.
The converse is a priori not obvious: we immediately deduce from Denition 3.1 that, for every τ ∈ R, the function U (x, x • e -ct) coincides with a solution uτ of the parabolic equation (1.1) on the 1-time-step set ({τ }+Z)×R N , but to recover a pulsating front we should have that the uτ are time-translations of the same solution. This diculty will be overcome by instead considering dierent discretizations with time steps converging to 0.

Remark 4. This part of the argument, about going from discrete to continuous travelling fronts, was actually omitted by Weinberger in the paper [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF] that we refer to in Theorem 1.3 above. A proof in the homogeneous case can be found in [START_REF] Li | Spreading speeds as slowest wave speeds for cooperative systems[END_REF]. However this does not seem to raise signicant diculties in the periodic case. Let us also mention that one can see that a discrete travelling front gives rise to an almost planar generalized transition front in the sense of Berestycki and Hamel [START_REF] Berestycki | Generalized transition waves and their properties[END_REF]. Then, in some situations (typically under some strong stability assumptions and provided also that the front speed is not zero), it is shown in [START_REF] Berestycki | Generalized transition waves and their properties[END_REF]Theorem 1.14] that an almost planar transition front is also a travelling front in a usual sense. Denition 3.1 leads us to consider the mappings Fc :

L ∞ (R N +1 ) → L ∞ (R N +1 ) dened by Fc[V ](y, z) := v(1, y; V (x, z + x • e -y • e + c)). (3.1)
They depend on c ∈ R and e ∈ S N -1 , but to keep the notation lighter, we omit to write the latter because it is xed once and for all. Rewriting the mapping Fc as

Fc[V ](y, z + y • e -c) = v(1, y; V (x, z + x • e)), (3.2)
we see that the discrete travelling fronts are given by the xed points of Fc. Formula (3.2) also allows one to use parabolic estimates to obtain regularity with respect to y → (y, z + y • e).

Notice that any spatially periodic stationary state p(y) of (1.1) is a z-independent xed point of Fc for any c and e. The converse is also true, as shown below. Proposition 3.2. Let u(t, x) be a 1-periodic in time solution of (1.1) which is also periodic in space. Then u is actually stationary in time.

Proof. Let us rst introduce the energy

E(w) := [0,1] N A|∇w| 2 2 -F (x, w) dx, for any periodic function w ∈ C 1 (R N ), where F (x, s) := s 0 f (x, σ)dσ.
Then one may check that the solution u(t, x) of (1.1) satises

∂tE(u(t, •)) = - [0,1] N |∂tu| 2 dx ≤ 0.
On the other hand, the mapping t → E(u(t, •)) is 1-periodic, whence it is necessarily constant. This implies that ∂tu ≡ 0.

We now derive several properties of the mapping Fc that will be useful in the sequel.

Proposition 3.3. For given e ∈ S N -1 and c ∈ R, the mapping Fc satises the following properties.

(i) Periodicity: if V (y, z) is periodic with respect to y ∈ R N then this holds true for

Fc[V ](y, z). (ii) Monotonicity: if V1 ≤ V2 then Fc[V1] ≤ Fc[V2]; if in addition sup y∈R N (V2 -V1)(y, z + y • e) > 0 for all z ∈ R, then Fc[V1] < Fc[V2]. (iii) Continuity: if Vn(y, z + y • e) → V∞(y, z + y • e) as n → +∞ locally uniformly in y ∈ R N , for some z ∈ R, then Fc[Vn](y, z + y • e -c) → Fc[V∞](y, z + y • e -c) as n → +∞ locally uniformly in y ∈ R N . (iv) Compactness: for any sequence (Vn) n∈N bounded in L ∞ (R N +1
) and any z ∈ R, there exists a subsequence (depending on z) along which the function

y → Fc[Vn](y, z + y • e) converges in L ∞ loc (R N ) as n → +∞. Proof. Let V (y, z) be a periodic function in its rst variable. Then for any y ∈ R N , z ∈ R and L ∈ Z N , the periodicity of equation (1.1) yields Fc[V ](y + L, z) = v(1, y + L; V (x, z + x • e -y • e -L • e + c)) = v(1, y; V (x + L, z + x • e -y • e + c)) = Fc[V ](y, z).

This proves (i).

Statement (ii) simply follows from (3.2) and the parabolic weak and strong comparison principles.

The continuity property follows from standard parabolic estimates. Indeed, take a sequence (Vn(y, z + y • e)) n∈N converging locally uniformly in y and for some z ∈ R to V∞(y, z + y • e). Then the functions (wn) n∈N dened by

wn(t, y) := v(t, y; Vn(x, z + x • e)) -v(t, y; V∞(x, z + x • e))
solve, for any xed z ∈ R, a linear parabolic equation of the type ∂twn = div(A(y)∇wn) + g z,n (t, y)wn, with |g z,n | less than or equal to the Lipschitz constant of f , together with the initial condition Vn(x, z + x • e) -V∞(x, z + x • e). It follows from the comparison principle and parabolic estimates that (wn) n∈N converges to 0 locally uniformly with respect to t > 0, y ∈ R N . In particular, y → v(1, y; Vn(x, z + x • e)) converges locally uniformly as n → +∞ to v(1, y; V∞(x, z + x • e)), which owing to (3.2) translates into the desired property.

The last statement (iv) is an immediate consequence of the parabolic estimates.

Let us point out that the operators Fc were initially introduced by Weinberger in [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], who exhibited the existence of a spreading speed of solutions in a rather general context, but only proved the existence of pulsating fronts in the monostable case.

These operators also fall into the scope of [START_REF] Fang | Bistable traveling waves for monotone semiows with applications[END_REF] (though they lack the compactness property required in some of their results). In particular, though one may proceed as in the aforementioned paper at least in the bistable case, we suggest here a slightly dierent approach. In some sense, our method is actually closer to the initial argument of Weinberger in [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], and though we do not address this issue here, it also seems well-suited to check that the speed of the pulsating front (or the speeds of the propagating terrace) also determines the spreading speed of solutions of the Cauchy problem associated with (1.1).

Basic properties of the iterative scheme

From this point until the end of Section 4, we assume that the following holds.

Assumption 3.1. The equation (1.1) admits a nite number of asymptotically stable steady states, among which 0 and p. Furthermore, for any pair of ordered periodic steady states q < q, there is an asymptotically stable steady state p such that q ≤ p ≤ q.

This hypothesis is guaranteed by both the bistable Assumption 1.1 and the multis-

table Assumption 1.2.
For the sake of completeness as well as for convenience (several of the following properties will play an important role here), we repeat some of the arguments of [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF].

In particular, we start by reproducing how to dene the speed c * (depending on the direction e ∈ S N -1 ) which was shown in [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF] to be the spreading speed for planar like solutions of the Cauchy problem. Roughly, for any c < c * we construct a time increasing solution of the parabolic equation in the moving frame with speed c in the direction e. Later we shall turn to a new construction of a pulsating travelling front connecting p to a stable periodic steady state p < p with speed c * .

The construction starts with an L ∞ function φ satisfying the following:

                     φ(y, z) is periodic in y ∈ R N , and nonincreasing in z ∈ R, φ(y, z) is uniformly continuous in (y, z) ∈ R N +1 , φ(y, z) = 0 for y ∈ R N , z ≥ 0, φ(y, -∞) < p(y),
∃δ > 0 such that φ(y, -∞) -δ lies in the basin of attraction of p.

(3.3)

Observe that the limit φ(y, -∞) exists uniformly with respect to y, and thus it is continuous (and periodic). The last condition is possible due to the (asymptotic) stability of p. Owing to the comparison principle, it implies that φ(y, -∞) lies in the basin of attraction of p too. Then, for any e ∈ S N -1 and c ∈ R, we dene the sequence (ac,n) n∈N by ac,0 := φ, ac,n+1 := max{φ, Fc[ac,n]},

where Fc was dened in (3.1). The maximum is to be taken at each point (y, z).

Lemma 3.4. The sequence (ac,n) n∈N dened by (3.4) is nondecreasing and satises 0 < ac,n < p for n ≥ 1. Moreover, ac,n(y, z) is periodic in y, nonincreasing with respect to c and z and satises ac,n(y, +∞) ≡ 0 uniformly with respect to y. Lastly, ac,n(y, z + y • e) is uniformly continuous in y ∈ R N , uniformly with respect to z ∈ R, n ∈ N and c ∈ R.

Proof. Firstly, recall from Proposition 3.3(ii) that the operator Fc is order-preserving.

By recursion, one readily checks that the sequence (ac,n) n∈N is nondecreasing. Moreover, 0 < ac,n < p for n ≥ 1, always by Proposition 3.3(ii). Another consequence of (3.2) and the comparison principle is that if V (y, z) is monotone in z then so is Fc[V ](y, z); whence the monotonicity of ac,n(y, z) with respect to z.

Let us now investigate the monotonicity with respect to c. We derive it by noting that if c1 < c2, then (3.2) yields

Fc 1 [V ](y, z + y • e -c1) = Fc 2 [V ](y, z + y • e -c2), (3.5)
for any function V . If furthermore V (y, z) is nonincreasing in its second variable, then so is Fc 2 [V ] and we deduce that

Fc 1 [V ] ≥ Fc 2 [V ].
Thus, owing to the monotonicity of the Fc, the monotonicity of ac,n with respect to c follows by iteration.

Next, we want to show that ac,n(y, +∞) = 0. This is an easy consequence of the same property for φ, but we now derive a quantitative estimate which will prove useful in the sequel. For this, we observe that, for any xed λ > 0, there exists a supersolution of (1.1) of the type e -λ(x•e-ct) , provided c is suciently large. Namely, by bounding f (x, u) by a linear function Ku and also using the boundedness of the components of the diusion matrix and their derivatives, we can nd c such that e -λ(x•e-ct)

satises ∂tu ≥ div(A(x)∇u) + Ku, t ∈ R, x ∈ R N . Let us show that if V and C > 0 satisfy ∀(y, z) ∈ R N +1 , V (y, z) ≤ Ce -λz , then there holds ∀(y, z) ∈ R N +1 , Fc[V ](y, z) ≤ Ce λ(c-c) e -λz .
Indeed, we have that

V (x, z + x • e -y • e + c) ≤ Ce -λ(z-y•e+c) e -λx•e , whence Fc[V ](y, z) = v(1, y; V (x, z + x • e -y • e + c)) ≤ Ce -λ(z+c-c) .
Up to increasing c, we can assume without loss of generality that c ≥ c. Now, for any C ≥ max p, we have that

∀(y, z) ∈ R N +1 , φ(y, z) ≤ Ce -λz . As a consequence ∀(y, z) ∈ R N +1 , ac,1(y, z) = max{φ, Fc[φ]} ≤ Ce λ(c-c) e -λz ,
and therefore, by iteration,

∀n ∈ N, ∀(y, z) ∈ R N +1 , ac,n(y, z) ≤ Ce nλ(c-c) e -λz . (3.6)
In particular ac,n(y, +∞) = 0 uniformly with respect to y; however, this limit may not be uniform with respect to c nor to n.

Finally, we point out that the uniform continuity in the crossed variables follows from our choice of φ and parabolic estimates. Indeed, the function

y → Fc[ac,n-1](y, z + y • e) = v(1, y; ac,n-1(x, z + x • e + c))
is not only uniformly continuous but also C 2 , and its derivatives are uniformly bounded by some constant which only depends on the terms in the equation (1.1) as well as max p. Recalling that ac,n is the maximum of Fc[ac,n-1] and φ, the latter being also uniformly continuous, we reach the desired conclusion.

From Lemma 3.4 and the fact that the mapping Fc preserves spatial periodicity, one readily infers the following. Lemma 3.5. The pointwise limit ac(y, z) := lim n→+∞ ac,n(y, z), is well-dened, fulls φ ≤ ac ≤ p and ac(y, z) is periodic in y and nonincreasing with respect to both z and c.

Moreover, the convergence ac,n(y, z + y • e) → ac(y, z + y • e) as n → +∞ holds locally uniformly in y ∈ R N , but still pointwise in z ∈ R N .

We emphasize that no regularity properties could be expected for ac with respect to the second variable. Let us further note that, as a byproduct of the proof of Lemma 3.4, and more specically of (3.5), we deduce by iteration that

∀c < c , n ∈ N, ac,n(•, • + n(c -c)) ≤ a c ,n . (3.7)
This will be used in later arguments, in particular in the proof of Lemma 4.2 below.

Dening c *

We want to dene c * as the largest c such that ac ≡ p, where ac comes from Lemma 3.5. This is the purpose of the following lemma. Lemma 3.6. For any c ∈ R, the function ac satises ac(y, -∞) = p(y) uniformly with respect to y ∈ [0, 1] N . Moreover, (i) ac ≡ p for -c large enough;

(ii) ac ≡ p for c large enough. In particular, the following is a well-dened real number:

c * := sup{c ∈ R : ac(y, z) ≡ p(y)}.
Proof. We rst prove that, for -c large enough,

(Fc) n [φ](y, z) → p(y) as n → +∞, (3.8)
uniformly with respect to y ∈ [0, 1] N and z ∈ (-∞, Z0], for any Z0 ∈ R. In particular, because ac,n ≥ (Fc) n [φ] by the monotonicity of Fc, this will yield statement (i) of the lemma.

In order to show (3.8), we rst introduce, in a similar fashion as in the proof of Lemma 3.4, two real numbers λ > 0 and c large enough such that the function e λ(x•e+ct)

satises the parabolic inequality ∂tu ≥ div(A∇u) + Ku.

Here K is the supremum with respect to x of the Lipschitz constants of u → f (x, u).

Next, we let ψ(t, x) be the solution of (1.1) emerging from the initial datum φ(x, -∞)δ, where δ is the positive constant in condition (3.3), that is, such that φ(x, -∞) -δ lies in the basin of attraction of p. Hence ψ(t, •) → p uniformly as t → +∞. The choice of λ and c imply that, for any γ > 0, the function

uγ(t, x) := ψ(t, x) -γe λ(x•e+ct)
is a subsolution of (1.1). Let us now pick C large enough such that ∀(y, z) ∈ R N +1 , φ(y, z) ≥ φ(y, -∞) -δ -Ce λz , and thus, for any given c ∈ R,

φ(x, z + x • e) ≥ φ(x, -∞) -δ -Ce λ(z+x•e) = u Ce λz (0, x). Now, iterating (3.2) one gets ∀n ∈ N, (Fc) n [V ](y, z + y • e -nc) = v(n, y; V (x, z + x • e)).
It then follows from the comparison principle that (Fc

) n [φ](y, z -nc) ≥ u Ce λz (n, y), that is, (Fc) n [φ](y, z) ≥ ψ(n, y) -Ce λ[z+y•e+n(c+c)] . (3.9)
From one hand, this inequality implies that if c < -c then (3.8) holds uniformly with respect to y ∈ [0, 1] N and z ∈ (-∞, Z0], for any Z0 ∈ R, whence statement (i) of the lemma. On the other hand, if c ≥ -c we derive ac,n(y, -2n(c

+ c + 1)) ≥ ψ(n, y) -Ce -nλ(c+c+2)+λy•e ≥ ψ(n, y) -Ce λ(-2n+y•e) .
Because the sequence (ac,n) n∈N is nondecreasing and converges to ac, we get that ac(y, -2n(c + c + 1)) ≥ ψ(n, y) -Ce λ(-2n+y•e) , for any n ∈ N. Passing to the limit as n → +∞ and recalling that ac is monotone with respect to its second variable, we infer that ac(y, -∞) = p(y) uniformly with respect to y ∈ [0, 1] N .

It remains to prove statement (ii). Fix λ > 0. Because φ satises (3.3), for C := max p there holds that φ(y, z) ≤ Ce -λz for all (y, z) ∈ R N +1 . As seen in the proof of Lemma 3.4, this implies that (3.6) holds for all c smaller than or equal a suitable value c, and then in particular for c = c, i.e., ac,n(y, z) ≤ Ce -λz for all n ∈ N. As a consequence, ac ≡ p and, by monotonicity with respect to c, we also have that ac ≡ p if c ≥ c.

We see now that, while c * is the supremum of the speeds c such that ac ≡ p, it actually holds that ac * ≡ p. This will be crucial for the construction of the front. Lemma 3.7. The following properties are equivalent:

(i) c < c * , (ii) ac ≡ p, (iii) ∃n0 ∈ N, ∃z0 > 0, ∀y ∈ [0, 1] N , ac,n 0 (y, z0) > φ(y, -∞).
In particular, in the case c = c * , we have that for all n ∈ N and z > 0, there exists y ∈ [0, 1] N such that ac * ,n(y, z) ≤ φ(y, -∞).

Proof. By denition of c * and monotonicity of ac with respect to c, we already know that (i) implies (ii). We also immediately see that (ii) implies (iii), using the fact that ac,n(y, z) is nonincreasing in z and ac,n(y, z + y • e) → ac(y, z + y • e) as n → +∞ uniformly with respect to y ∈ [0, 1] N (see Lemma 3.5).

It remains to prove that (iii) implies (i). We assume that (iii) holds and we start by showing (ii), which will serve as an intermediate step. Thanks to the monotonicity with respect to z and the fact that ac,n 0 > 0 and φ(•, z) = 0 for z ≥ 0, we get ∀n ≥ n0, ∀(y, z) ∈ R N +1 , ac,n(y, z + z0) > φ(y, z).

Since the operator Fc is order preserving, we also get that

∀(y, z) ∈ R N +1 , ac,n 0 +1(y, z + z0) ≥ Fc[ac,n 0 ](y, z + z0) ≥ Fc[φ](y, z).
It follows from the two inequalities above that ∀(y, z) ∈ R N +1 , ac,n 0 +1(y, z + z0) ≥ ac,1(y, z).

A straightforward induction leads to ∀m ≥ 0, ∀(y, z) ∈ R N +1 , ac,n 0 +m(y, z + z0) ≥ ac,m(y, z).

Passing to the limit m → +∞ on both sides, we infer that ∀(y, z) ∈ R N +1 , ac(y, z + z0) ≥ ac(y, z).

Recalling that z0 > 0 and that ac is nonincreasing with respect to z, we nd that ac(y, z) = ac(y) does not depend on z. Since we know by Lemma 3.6 that ac(•, -∞) ≡ p(•), we conclude that ac ≡ p. We have shown that (iii) implies (ii).

Next we show that the set of values of c such that (iii) holds is open. Using (3.2), it is readily seen by iteration that, for any xed n ∈ N, the function ac,n inherits from φ the continuity with respect to the variable (y, z) (though this is not uniform with respect to n ∈ N). From this, by another iterative argument and (3.5), one deduces that ac,n(y, z) → ac 0 ,n(y, z) locally uniformly in (y, z) as c → c0, for every n ∈ N. Openness follows.

We are now in the position to conclude the proof of Lemma 3.7. Assume that (iii) holds for some c. From what we have just proved, we know that (iii) holds true for some c > c, and thus (ii) holds for c too. By the denition of c * , we have that c * ≥ c > c, that is, (i) holds.

Before proceeding we have to check that c * is intrinsic to (1.1) and does not depend on φ. This will be useful later on, when going back to the continuous case and more specically to check that the speed of the discrete front we shall obtain does not depend on the choice of the time step of the discretization.

√ N + z) ≥ max{ φ, Fc[ac,n](•, • - √ N + z)} ≥ max{ φ, Fc[ φ]} = âc,1.
By iteration we eventually infer that ac,n+m(•, • -√ N + z) ≥ âc,m for all m ∈ N. This implies that c * ≥ ĉ * . Switching the roles of φ and φ we get the reverse inequality. [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF] A discrete travelling front with speed c * Under the Assumption 3.1, we have constructed in the previous section a candidate speed c * for the existence of a pulsating travelling front. In the current one we show that there exists a discrete travelling front in the direction e with speed c * connecting p to some stable periodic steady state (in the sense of Denition 3.1). To derive the stability of the latter we will make use of the additional Assumption 1.3. We recall that in order to dene the minimal speeds cq and c q appearing in Assumption 1.3, we have shown after the statement of Theorem 1.3 that the hypothesis there is guaranteed by Assumption 1.2. However, this was achieved without using the linear stability hypothesis in Assumption 1.2 and therefore cq and c q are well dened under Assumption 3.1 too. The strategy is as follows. For c < c * , Lemma 3.7 implies that ac,n > φ for n suciently large. We deduce that the nondecreasing sequence (ac,n) n∈N is eventually given by the recursion ac,n = Fc[ac,n-1]. Roughly speaking, we have constructed a solution of (1.1) which is non-decreasing with respect to 1-time steps in the frame moving with speed c in the direction e. We now want to pass to the limit as c c * in order to get a xed point for Fc * and, ultimately, a pulsating travelling front in the direction e. To achieve this, we shall need to capture such solutions at a suitable time step, and suitably translated.

Remark 5. The equivalent argument in the continuous case of what we are doing here is to construct a family of functions Uc such that Uc(x, x • e -ct) is a subsolution of (1.1), and to use this family and a limit argument to nd a pulsating front. Notice that an inherent diculty in such an argument is that a subsolution does not satisfy regularity estimates in general. We face a similar diculty in the discrete framework.

Choosing a diagonal sequence as c c *

Consider the function φ satisfying (3.3) from which we initialize the construction of the sequence (ac,n) n∈N .

The rst step in order to pass to the limit as c c * is to capture the sequence at a suitable iteration, and roughly at the point where it `crosses' the limit φ(•, -∞), which, we recall, lies in the basin of attraction of p. Lemma 4.1. For c < c * , there exists n(c) ∈ N such that, for all n ≥ n(c), the quantity zc,n := sup{z : ac,n(y, z + y • e) > φ(y, -∞) for all y ∈ [0, 1] N } is a well-dened real number. In addition, there holds

∀m ≥ 0, a c,n(c)+m = (Fc) m [a c,n(c) ] ≥ a c,n(c)+m-1 , (4.1) ∀0 ≤ m ≤ 1/ √ c * -c, 0 ≤ z c,n(c)+m -z c,n(c) ≤ 2 √ c * -c. (4.2)
While property (4.1) holds for any c < c * provided n(c) is suciently large, the same is not true for (4.2). The latter will play a crucial role for getting a travelling front in the limit. Loosely speaking, it guarantees that, as c c * , there exists an index n(c) starting from which the crossing point zc,n moves very little along an arbitrary large number of iterations.

Proof of Lemma 4.1. Fix c < c * . First of all, from the equivalence between (i) and (iii) in Lemma 3.7, we know that there exists n(c) such that ac,n > φ for n ≥ n(c). We deduce that the nondecreasing sequence (ac,n) n≥n(c) is simply given by the recursion ac,n = Fc[ac,n-1], that is property (4.1). Now, Lemma 3.4 implies that the set {z : ac,n(y, z + y • e) > φ(y, -∞) for all y ∈ [0, 1] N } is either a left half-line or the empty set, while Lemmas 3.5-3.6 show that it is nonempty for n suciently large. As a consequence, up to increasing n(c) if need be, its supremum zc,n is well-dened and nite for n ≥ n(c).

It remains to prove (4.2), for which we can assume that c ≥ c * -1. We claim that lim sup n→+∞ zc,n n ≤ c * -c. Hence if (4.3) does not hold, we would nd a large n contradicting the last statement of Lemma 3.7.

Next, let N (c) ≥ 1 be the integer part of 1/ √ c * -c. Owing to (4.3), we can further increase n(c) to ensure that

z c,n(c)+N (c) -z c,n(c) ≤ 2N (c)(c * -c).
Moreover, we know that zc,n+1 ≥ zc,n for all c and n, due to the monotonicity of ac,n with respect to n. In particular, for any integer 0 ≤ m ≤ N (c), we also have that

0 ≤ z c,n(c)+m -z c,n(c) ≤ 2N (c)(c * -c),
from which we deduce (4.2).

In the next lemma, we state what we obtain when passing to the limit as c c * .

Lemma 4.2. There exists a lower semicontinuous function a * (y, z) satisfying the following properties:

(i) a * (y, z + y • e) is uniformly continuous in y ∈ R N , uniformly with respect to z ∈ R;

(ii) a * (y, z) is periodic in y and nonincreasing in z;

(iii) (Fc * ) n [a * ] is nondecreasing with respect to n;

(iv) limn→+∞ max y∈[0,1] N p(y) -(Fc * ) n [a * ](y, y • e) > 0; (v) (Fc * ) n [a * ](•, -∞) p uniformly as n → +∞; (vi) (Fc * ) n [a * ](•, +∞)
p uniformly as n → +∞, where 0 ≤ p < p is a periodic steady state of (1.1).

Thanks to our previous results, we know that the properties (i)-(iii) are fullled with c * and a * replaced respectively by any c < c * and ac,n with n suciently large. In order to get (iv)-(vi) we need to pass to the limit c c * by picking the ac,n at a suitable iteration n. The choice will be n = n(c) given by Lemma 4.1, which fulls the key property (4.2). When passing to the limit, we shall face the problem of the lack of regularity in the z-variable. This will be handled by considering the following relaxed notion of limit. Lemma 4.3. Let (αn) n∈N be a bounded sequence of functions from R N × R to R such that αn(y, z) is periodic in y and nonincreasing in z, and αn(y, z+y•e) is uniformly continuous in y ∈ R N , uniformly with respect to z and n. Then there exists a subsequence (αn k ) k∈N such that the following double limit exists locally uniformly in y ∈ R N :

β(y, z) := lim Q ζ→z + lim k→+∞ αn k (y, ζ + y • e) .
Furthermore, β(y, z) is uniformly continuous in y ∈ R N uniformly with respect to z ∈ R. Finally, the function α * (y, z) := β(y, z -y • e) is periodic in y and nonincreasing and lower semicontinuous in z.

Proof. Using a diagonal method, we can nd a subsequence αn k (y, ζ + y • e) converging locally uniformly in y ∈ R to some function β(y, ζ) for all ζ ∈ Q. The function β(y, ζ) is uniformly continuous in y uniformly with respect to ζ ∈ Q. We then dene β : R N × R → R by setting β(y, z) := lim

Q ζ→z + β(y, ζ).
This limit exists thanks to the monotonicity with respect to z, and it is locally uniform with respect to y by equicontinuity. We point out that β ≤ β on R N × Q, but equality may fail. We also see that β(y, z) is uniformly continuous in y ∈ R N uniformly with respect to z ∈ R, and it is nonincreasing and lower semicontinuous in z.

It remains to show that the function α * (y, z) := β(y, z -y •e) is periodic with respect to y. That is, α * (y, z + y • e) ≥ α * (y + L, z + y • e). Because y and z are arbitrary, this means that α * ≥ α * (• + L, •) for all L ∈ Z N , i.e., α * is periodic in its rst variable.

Proof of Lemma 4.2. Consider the family of functions (a c,n(c) (y, z + z c,n(c) ))c<c * , with n(c), z c,n(c) given by Lemma 4.1. From Lemma 3.4, we know that this family is uniformly bounded by 0 and max p, and that any element a c,n(c) is periodic in the rst variable and nonincreasing in the second one. Moreover, the functions a c,n(c) (y, z +y •e) are uniformly continuous in y ∈ R N , uniformly with respect to z ∈ R and c ∈ R, due to Lemma 3.4. In particular, any sequence extracted from this family fulls the hypotheses of Lemma 4.3. Then, there exists a sequence c k c * such that the following limits exist locally uniformly in y ∈ R N : a * (y, z + y • e) := lim We further know that the function a * satises the desired properties (i)-(ii). The denition of z c,n(c) translates into the following normalization conditions:

Q ζ→z + lim k→+∞ a c k ,n(c k ) (y, ζ + z c k ,n(c k ) + y • e) .
∀z < 0, min y∈[0,1] N a * (y, z + y • e) -φ(y, -∞) ≥ 0, (4.5)
min

y∈[0,1] N a * (y, y • e) -φ(y, -∞) ≤ 0, (4.6)
where we have used the monotonicity in z and for the second one also the locally uniform convergence with respect to y.

Let us check property (iii). Using the continuity property of Proposition 3.3 together with (3.5) we obtain

Fc * [a * ](y, z + y • e -c * ) = lim Q ζ→(z-c * ) + lim k→+∞ Fc * [a c k ,n(c k ) ](y, ζ + z c k ,n(c k ) + y • e) = lim Q ζ→(z-c * ) + lim k→+∞ F c k [a c k ,n(c k ) ](y, ζ + z c k ,n(c k ) + y • e + c * -c k ) .
We now use property (4.1) to deduce that the latter term is larger than or equal to lim sup

Q ζ→(z-c * ) + lim sup k→+∞ a c k ,n(c k ) (y, ζ + z c k ,n(c k ) + y • e + c * -c k ) ,
a c k ,n(c k )+m (y, ζ + z c k ,n(c k ) + y • e) = (F c k ) m [a c k ,n(c k ) ](y, ζ + z c k ,n(c k ) + y • e) ≥ (Fc * ) m [a c k ,n(c k ) ](y, ζ + z c k ,n(c k ) + y • e). Let k large enough so that 1/ √ c * -c k ≥ m and 2 √ c * -c k < ζ. We deduce from (4.2) that ζ + z c k ,n(c k ) > z c k ,n(c k )+m
and thus, by the denition of the latter,

min y∈[0,1] N (Fc * ) m [a c k ,n(c k ) ](y, ζ + z c k ,n(c k ) + y • e) -φ(y, -∞) ≤ 0.
Letting now k → +∞ and next ζ → 0 + and using the continuity of Fc (hence of (Fc) m ) in the locally uniform topology, we eventually obtain ∀m ∈ N, min

y∈[0,1] N (Fc * ) m [a * ](y, y • e) -φ(y, -∞) ≤ 0,
from which property (iv) readily follows.

It remains to look into the asymptotics of a * as z → ±∞. We dene the left limit

a (y) := lim z→-∞ a * (y, z + y • e),
which exists by the monotonicity of a * (y, z) with respect to z, and it is locally uniform in y. Then, by monotonicity and periodicity, we deduce that the limit a * (y, -∞) = a (y) holds uniformly in y. The function a is continuous and periodic. Moreover, the normalization condition (4.5) yields a (y) ≥ φ(y, -∞). This means that the sequence ((Fc * ) n [a ]) n∈N is nondecreasing. Because a is independent of z, by the denition (3.1) we see that (Fc * ) n [a ] reduces to (F0) n [a ], that is, to the solution of (1.1) with initial datum a computed at time t = n. Then, because a ≥ φ(y, -∞) and recalling that the latter lies in the basin of attraction of p, we infer that (F0) n [a ] → p as n → +∞, and the limit is uniform thanks to Proposition 3.3(iv).

In a similar fashion, we dene the (locally uniform) right limit ar(y) := lim z→+∞ a * (y, z + y • e).

As before, we see that the limit a * (y, +∞) = ar(y) is uniform in y, it is continuous, periodic and the sequence ((F0) n [ar]) n∈N is nondecreasing. Therefore, (F0) n [ar](y) converges uniformly as n → +∞ to a xed point p(y) of F0. This means that the solution u of (1.1) with initial datum p is 1-periodic in time and periodic in space and therefore, by Proposition 3.2, it is actually stationary. Observe that

(F0) n [ar](y) = (Fc * ) n [ar](y) ≤ (Fc * ) n [a * ](y, y • e),
whence p ≡ p owing to (iv). We conclude that (vi) holds, completing the proof of the lemma.

The uppermost pulsating front

From now on, a * will denote the function provided by Lemma 4.2 and more specically dened by (4.4) for a suitable sequence c k c * . Next we show that the discrete front is given by the limit of the iterations (Fc * ) n [a * ]. We shall further show that its limit state as z → +∞ is stable. Lemma 4.4. There holds that

(Fc * ) n [a * ](y, z + y • e) → U * (y, z + y • e) as n → +∞,
locally uniformly in y and pointwise in z, where U * (x, z) is nonincreasing in z and it is a discrete travelling front connecting p to some stable periodic steady state p * < p, in the sense of Denition 3.1.

Proof. Let us observe that, because ((Fc * ) n [a * ]) n∈N is a nondecreasing sequence, it is already clear that it converges pointwise to some function U * (y, z) which is periodic in y and nonincreasing in z. By writing

(Fc * ) n+1 [a * ](y, z + y • e) = Fc * • (Fc * ) n [a * ](y, z + y • e),
we deduce from Proposition 3.3(iv) that (Fc * ) n [a * ](y, z + y • e) converges as n → +∞ locally uniformly in y, for any z ∈ R. In particular, we can pass to the limit n → +∞ in the above equation and conclude that U * is a xed point for Fc * .

Let us now turn to the asymptotics as z → ±∞. We know from Lemma 4.2(v) that (Fc * ) n [a * ](•, -∞) → p as n → +∞. We can easily invert these limits using the continuity of (Fc * ) n and the uniformity of the limit a * (•, -∞), together with the monotonicity of U * in the second variable. This yields U * (•, -∞) ≡ p.

Next, property (iv) of Lemma 4.2 implies that max

y∈[0,1] N p(y) -U * (y, y • e) > 0.
Writing U * (y, +∞) = limz→+∞ U * (y, z + y • e), we deduce that the limit U * (•, +∞) is uniform and therefore F0[U * ](•, +∞) ≡ Fc * [U * ](•, +∞) ≡ U * (•, +∞). We also deduce from the previous inequality that U * (•, +∞) ≡ p. As seen in Proposition 3.2, any solution of (1.1) that is periodic in both time and space is actually constant in time. Thus, U * (•, +∞) is a periodic steady state of (1.1), denoted by p * , that satises 0 ≤ p * < p,

where the second inequality is strict due to the elliptic strong maximum principle.

It remains to check that p * is stable. We shall do this using Assumption 1.3. Proceed by contradiction and assume that p * is unstable. As seen after the statement of Theorem 1.3, Assumption 3.1 guarantees the existence of a minimal (resp. maximal) stable periodic steady state above (resp. below) p * , denoted by p+ (resp. p-), and also that (1.1) is of the monostable type between pand p * , as well as between p * and p+. As a consequence, Theorem 1.3 provides two minimal speeds of fronts cp * and c p * connecting p+ to p * and p * to prespectively. Our Assumption 1.3 states that c p * < cp * . According to Weinberger [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], these quantities coincide with the spreading speeds for (1.1) in the ranges between p * and p+ and between pand p * respectively. Namely, taking a constant δ > 0 such that p * + δ < p+, and considering the Heaviside-type function

H(y, z) := p * (y) + δ if z < - √ N , p * (y) if z ≥ - √ N ,
we have that for any Z ∈ R, the solution v(t, y; H(x, Z + x • e)) of (1.1) spreads with speed cp * in the following sense: for any ε > 0, we nd that U * (y, y • e -nc * ) > p+(y) -δ for n suciently large and for all y such that y • e ≤ (c * + ε)n. Taking for instance y = (c * + ε)n e and passing to the limit as n → +∞ yields p * (y∞) ≥ p+(y∞) -δ, where y∞ is the limit of (c * + ε)n e (up to subsequences and modulo the periodicity; recall that the limit U * (•, +∞) is uniform). This is impossible because δ was chosen in such a way that p * + δ < p+. As announced, there holds c * ≥ cp * .

Let us now show that c * ≤ c p * . The strategy is to follow a level set between pand p * of a suitable iteration (Fc * ) n [a c,n(c) ] and to pass again to the (relaxed) limit as c c * . Notice that, in the situation where p (coming from Lemma 4.2(vi)) satises p < p * , then it would be sucient to consider the sequence ((Fc * ) n [a * ])n to capture such a level set; however it may happen that p ≡ p * and for this reason we need to come back to the family a c,n(c) .

For k ∈ N, we can nd n k ∈ N such that the following properties hold:

max y∈[0,1] N (Fc * ) n k [a * ](y, k + y • e) -U * (y, k + y • e) < 1 k , max y∈[0,1] N (Fc * ) n k [a * ](y, 2k + y • e) -U * (y, 2k + y • e) < 1 k .
Then, recalling the denition (4.4) of a * and up to extracting a subsequence of the sequence c k c * appearing there, we nd that for every k ∈ N, there holds

max y∈[0,1] N (Fc * ) n k [a c k ,n(c k ) ](y, k + 1 + z c k ,n(c k ) + y • e) -U * (y, k + y • e) < 2 k , (4.8) min y∈[0,1] N (Fc * ) n k [a c k ,n(c k ) ](y, 2k -1 + z c k ,n(c k ) + y • e) -U * (y, 2k + y • e) > - 2 k . (4.9)
Notice that in (4.8) and (4.9) we have translated by

z c k ,n(c k ) ± 1 instead of z c k ,n(c k )
because of the `relaxed' limit in (4.4). In order to pick the desired level set, take a constant δ > 0 small enough so that p-+ δ < p * . We then dene

ẑk := inf{z : (Fc * ) n k [a c k ,n(c k ) ](y, z + y • e) -p-(y) ≤ δ for all y ∈ [0, 1] N }.
Observe that ẑk ∈ R and actually ẑk ≥ z c k ,n(c k ) , as a consequence of the denition of zc,n in Lemma 4.1 and the fact that ϕ(•, -∞) > p * , since it lies in the basin of attraction of p. Because U * (y, +∞) = p * (y) > p-(y) + δ uniformly in y, we deduce from (4.9) that, for k large enough,

min y∈[0,1] N (Fc * ) n k [a c k ,n(c k ) ](y, 2k -1 + z c k ,n(c k ) + y • e) -p-(y) > δ, whence ẑk ≥ 2k -1 + z c k ,n(c k )
. It then follows from (4.8) that, for k suciently large, Finally, property (4.10) and the monotonicity in z yield α * ≤ p * . We are now in a position to prove that c * ≤ c p * . We again use a comparison argument with an Heaviside-type function. Indeed, from the above, we know that α * ≤ Ĥ, where

max y∈[0,1] N (Fc * ) n k [a c k ,n(c k ) ](y, ẑk -k + 2 + y • e) -U * (y, k + y • e) < 2 k .
F c k • (Fc * ) n k [a c k ,n(c k ) ] ≥ (Fc * ) n k [a c k ,n(c k ) ] that (Fc * ) n [ α * ]
Ĥ(y, z) := p * (y) if z ≤ √ N , p-(y) + δ if z > √ N .
According to Weinberger's spreading result in [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], the solution v(t, y; Ĥ(x, x•e)) of (1.1) spreads with speed c p * , which implies in particular that for any ε > 0, By periodicity, we can drop the ξn in the above expression. We eventually deduce from (4.11) that c p * + ε -c * ≥ 0, that is, c p * ≥ c * due to the arbitrariness of ε > 0.

In the end, we have shown that cp * ≤ c * ≤ c p * , which directly contradicts Assumption 1.3. Lemma 4.4 is thereby proved.

Remark 6. Under the bistable Assumption 1.1, obviously p * has to be 0, and therefore we have constructed a discrete travelling front connecting p to 0. In order to conclude the proof of Theorem 1.4, one may directly skip to Section 6.

A (discrete) propagating terrace

At this stage we have constructed the `highest oor' of the terrace. Then in the bistable case we are done. In the multistable case it remains to construct the lower oors, and thus we place ourselves under the pair of Assumptions 1.2 and 1.3. To proceed, we iterate the previous argument to the restriction of (1.1) to the `interval' [0, p * ], with p * given by Lemma 4.4, and we nd a second travelling front connecting p * to another stable state smaller than p * . For this the stability of p * is crucial. The iteration ends as soon as we reach the 0 state, which happens in a nite number of steps because there is a nite number of stable periodic steady states.

This procedure provides us with some nite sequences (qj) 0≤j≤J and (Uj) 1≤j≤J , where the qj are linearly stable periodic steady states and the Uj are discrete travelling fronts connecting qj-1 to qj. We need to show that the speeds are ordered, so that the family of travelling fronts we construct is a (at this point, discrete) propagating terrace. It is here that we use the linear stability hypothesis in Assumption 1.2. As we mentioned in the introduction, the order of the speeds is a crucial property of the terrace, which is not a mere collection of unrelated fronts but what should actually emerge in the large-time limit of solutions of the Cauchy problem.

Proposition 5.1. Under Assumptions 1.2 and 1.3, the speeds cj of the fronts Uj are ordered:

c1 ≤ c2 ≤ • • • ≤ cJ .
Proof. We only consider the two uppermost travelling fronts U1 and U2 and we show that c1 ≤ c2. The same argument applies for the subsequent speeds. We rst come back to the family (ac,n)c,n and the function a * used to construct the front U1 connecting q0 ≡ p to q1. The main idea is that, capturing another level set between q2 and q1, we should obtain a solution moving with a speed larger than or equal to c1, but which takes values below q1. Then, comparing it with the second front U2, we expect to recover the desired inequality c1 ≤ c2.

In the proof of Lemma 4.4, we have constructed two sequences (n k ) k∈N , (c k ) k∈N , with c k c1, such that (4.8), (4.9) hold with c * = c1 and U * = U1. Take a small positive constant δ so that qj ± δ lie in the basin of attraction of qj, for j = 1, 2, and moreover min(q1 -q2) ≥ 2δ. Then dene

ẑk := inf{z : (Fc 1 ) n k [a c k ,n(c k ) ](y, z + y • e) -q2(y) ≤ δ for all y ∈ [0, 1] N
The inequality (4.9) implies that, for y ∈ [0, 1] N and z ≤ z c k ,n(c k ) + 2k -1, there holds

(Fc 1 ) n k [a c k ,n(c k ) ](y, z + y • e) > U1(y, 2k + y • e) - 2 k → q1(y) as k → +∞.
Because q1 > q2 + δ, we infer that, for k large enough,

ẑk ≥ z c k ,n(c k ) + 2k -1, whence, by (4.8), max y∈[0,1]
N (Fc 1 ) n k [a c k ,n(c k ) ](y, ẑk + 2 -k + y • e) -U1(y, k + y • e) < 2 k .
(

We now consider the sequence of functions (Fc 1 ) n k [a c k ,n(c k ) ](y, z + ẑk + y • e) k∈N and apply Lemma 4.3. We obtain a function α(y, z) which is periodic in y, nonincreasing in z. Moreover, it is such that α(y, z + y • e) is uniformly continuous in y, uniformly with respect to z, and (Fc 1 ) n [ α] is nondecreasing with respect to n. Our choice of ẑk further implies ∀z < 0, max

y∈[0,1] N α(y, z + y • e) -q2(y) ≥ δ, (5.2)
∀y ∈ [0, 1] N , α(y, z + y • e) ≤ q2(y) + δ.

(

The latter property, together with the facts that (Fc 1 ) n [ α](•, +∞) is nondecreasing in n ∈ N and that q2 + δ lies in the basin of attraction of q2, yield α(•, +∞) ≤ q2.

On the other hand, using (5.1) one infers that α(•, -∞) ≤ q1.

Our aim is to compare α with U2 using the sliding method. To this end, we shall increase U2 a bit without aecting its asymptotical dynamics, exploiting the linear stability of q1, q2. Let ϕq 1 and ϕq 2 denote the periodic principal eigenfunctions associated with the linearization of (1.1) around q1 and q2 respectively, normalized by max ϕq 1 = max ϕq 2 = 1. Then consider a smooth, positive function Φ = Φ(y, z) which is periodic in y and satises

Φ(y, z) = ϕq 1 (y) if z ≤ -1, ϕq 2 (y) if z ≥ 1,
and dene, for ε ∈ (0, δ), U2,ε(y, z) := U2(y, z) + ε Φ(y, z).

Now, because the limits as z → ±∞ satisfy the following inequalities uniformly in y:

U2,ε(y, -∞) > q1(y) ≥ α(y, -∞), U2,ε(y, +∞) > q2(y) ≥ α(y, +∞), and using also (5.2), we can dene the following real number:

Zε := sup Z : U2,ε(y, Z + z + y • e) > α(y, z + y • e) for all (y, z) ∈ R N +1 .
Let us assume by way of contradiction that the speed of U2 satises c2 < c1. Then if we x

Zε ∈ (Zε -c1 + c2, Zε),

we can nd (yε, zε) ∈ R N +1 such that U2,ε(yε, Zε + c1 -c2 + zε + yε • e) ≤ α(yε, zε + yε • e).
(5.4)

Consider the following functions:

uε(t, := v(t, y; α(x, c1 + zε + x • e)),
wε(t, y) := v(t, y; U2(x, Zε + c1

+ zε + x • e)) + ε Φ(y, Zε + c1 -c2t + zε + y • e).
We nd from one hand that ∀y ∈ R N , wε(0, y) -uε(0, y) = U2,ε(y, Zε + c1

+ zε + y • e) -α(y, c1 + zε + y • e) > 0
because Zε < Zε. Hence, recalling that U2,ε, α are periodic in y and satisfy U2,ε(y, ±∞) > α(y, ±∞), which yields lim inf

y•e→±∞ (wε -uε)(0, y) ≥ ε min {min ϕq 1 , min ϕq 2 } > 0, (5.5)
we infer that infy(wε(0, y) -uε(0, y)) > 0. Then, by uniform continuity, wε > uε for t > 0 small enough. On the other hand, using the fact that, for all m ∈ N,

uε(m, y) = Fc 1 [ α] m (y, c1 + zε + y • e -mc1) ≥ α(y, c1 + zε + y • e -mc1), (5.6)
wε(m, y) = U2,ε(y, Zε + c1 + zε + y • e -mc2),

we derive

wε(1, yε) -uε(1, yε) ≤ U2,ε(yε, Zε + c1 -c2 + zε + yε • e) -α(yε, zε + yε • e),
which is nonpositive by (5.4). Let us point out that, if wε was a supersolution on the whole domain, this would contradict the comparison principle; unfortunately we shall see below that we only know it to be a supersolution in some subdomains. Therefore we shall rst use a limiting argument as ε → 0 to nd that α also lies below a shift of U2 itself, so that the comparison principle will become available.

From the above we deduce the existence of a time Tε ∈ (0, 1] such that wε > uε for t ∈ [0, Tε) and infy(wε -uε)(Tε, y) = 0. There exists then a sequence (y n ε ) n∈N satisfying (wε -uε)(Tε, y n ε ) → 0 as n → +∞. We observe that the sequence (y n ε • e) n∈N is necessarily bounded because the inequalities (5.5) hold true for all times, as a consequence of the fact that, for solutions of parabolic equations such as (1.1), the property of being bounded from one side by a steady state at the limit in a given direction is preserved along evolution.

The linear stability of q1 and q2 means that the periodic principal eigenvalues λq 1 , λq 2 of the associated linearized operators are negative. Then, for a given solution u to (1.1), the function u + εϕq j , with ε > 0 and j = 1, 2, satises for t > 0, x ∈ R N , ∂t(u + εϕq j )div(A(x)∇(u + εϕq j )) = f (x, u) + (fu(x, qj) -λq j )εϕq j = f (x, u + εϕq j ) + (fu(x, qj) -fu(x, s) -λq j )εϕq j , for some u(t, x) < s < u(t, x) + εϕq j (x). Thus, because λq j < 0, the regularity of fu allows us to nd γ > 0 such that u+εϕq j is a supersolution to (1.1) whenever |u-qj| < γ and ε ∈ (0, γ). From now on, we restrict to ε ∈ (0, γ). Take Z ≥ 1 in such a way that

U2(•, z) > q1 -γ if z ≤ -Z, U2(•, z) < q2 + γ if z ≥ Z,
as well as, for all 0 ≤ t ≤ 1, v(t, y; U2(x, Zε + c1

+ zε + x • e)) > q1 -γ if y • e ≤ -Z -Zε -c1 + c2t -zε, v(t, y; U2(x, Zε + c1 + zε + x • e)) < q2 + γ if y • e ≥ Z -Zε -c1 + c2t -zε.
We have just seen that these conditions imply the property that wε is a supersolution to (1.1) in corresponding subdomains. We claim that this implies that

lim inf n→+∞ | Zε + c1 + zε + y n ε • e| ≤ Z + |c2| + 3 √ N , (5.8)
which will in turn guarantee that functions uε and wε do not become trivial as ε → 0.

To prove (5.8), consider

(k n ) n∈N in Z N such that y n ε -k n ∈ [0, 1] N . Clearly, (k n • e) n∈N is bounded because (y n ε • e) n∈N is. Let y ∞
ε be the limit of (a subsequence of )

(y n ε -k n ) n∈N .
The functions wε(t, y + k n ) and uε(t, y + k n ) converge as n → +∞ (up to subsequences) locally uniformly in [0, 1) × R N to some functions wε, ũε satisfying min

[0,Tε]×R N ( wε -ũε) = ( wε -ũε)(Tε, y ∞ ε ) = 0.
The function ũε is a solution to (1.1). Instead, wε is a supersolution to (1.1) for t ∈ (0, Tε] and y • e < 2 √ N or y • e > -2 √ N if respectively one or the other of the following inequalities holds for innite values of n:

Zε + c1 + zε + k n • e < -Z -|c2| -2 √ N , Zε + c1 + zε + k n • e > Z + |c2| + 2 √ N .
Hence if (5.8) does not hold we have that wε is a supersolution of (1.1) in a half-space orthogonal to e containing the point y ∞ ε , and thus the parabolic strong maximum principle yields wε ≡ ũε in such half-space for t ≤ Tε. This is impossible because, by the boundedness of (k n • e) n∈N , the property (5.5) holds true with wε -uε replaced by wε -ũε. This proves (5.8). Using (5.8) we can nd a family (ỹε) ε∈(0,γ) such that ( Zε + c1 + zε + ỹε • e) ε∈(0,γ) is bounded and (wε -uε)(Tε, ỹε) → 0 as ε → 0. Arguing as before, by considering the translations uε(t, y + kε), wε(t, y + kε) with kε ∈ Z N such that ỹε -kε ∈ [0, 1] N , we obtain at the limit ε 0 (up to some subsequences) two functions ũ and w which are now both solutions to (1.1) and satisfy

min y∈R N ( w -ũ)( T , y) = ( w -ũ)( T , ỹ) = 0,
where T = limε→0 Tε and ỹ = limε→0(ỹε -kε). If T > 0 then w ≡ ũ, otherwise we can only infer that w ≥ ũ for all times and that ( w -ũ)(0, ỹ) = 0. In both cases, roughly the spreading speed of ũ has to be less than that of w, which ultimately will contradict the inequality c2 < c1.

More precisely, since ( Zε + c1 + zε + kε • e) ε∈(0,γ) is bounded, we derive

ũ(0, ỹ) = w(0, ỹ) = lim ε→0 wε(0, ỹ + kε) = lim ε→0 U2(ỹ, Zε + c1 + zε + kε • e + ỹ • e),
and thus q2(ỹ) < ũ(0, ỹ) < q1(ỹ) because q2 < U2 < q1 thanks to Proposition 3.3(ii). Next, x c ∈ (c2, c1) and consider a sequence (hm) m∈N satisfying c m < hm • e < c1m for m larger than some m0. From one hand, using (5.6) and the monotonicity of α with respect to its second variable, we get

∀m ≥ m0, y ∈ R N , uε(m, y + hm) ≥ α(y, c1 + zε + (y + hm) • e -mc1) ≥ uε(0, y),
from which we deduce ũ(m, ỹ + hm) ≥ ũ(0, ỹ) > q2(ỹ).

(5.9)

On the other hand, (5.7) yields ∀m ≥ m0, wε(m, ỹ + hm + kε) ≤ U2,ε(ỹ, Zε + c1

+ zε + (kε + ỹ) • e + m(c -c2)),
whence, letting L > 0 be such that Zε + c1 + zε + kε • e ≥ -L for all ε ∈ (0, γ), we nd that w(m, ỹ + hm) ≤ U2(ỹ, -L + ỹ • e + m(c -c2)).

The above right-hand side converges to q2(ỹ) as m → +∞, and therefore, by (5.9), we derive for m suciently large, w(m, ỹ + hm) < ũ(0, ỹ) ≤ ũ(m(z), ỹ + hm).

This contradicts the inequality w ≥ ũ, concluding the proof of the proposition. [START_REF] Ducrot | A multi-dimensional bistable nonlinear diusion equation in a periodic medium[END_REF] To the continuous case

In this section, we place ourselves under Assumption 1.3 and either Assumption 1.1 or 1.2. In both situations, we have constructed in the previous sections a `discrete' travelling front or terrace (i.e., a nite and appropriately ordered sequence of discrete travelling fronts) in the sense of Denition 3.1. Clearly our argument may be performed with any positive time step (not necessarily equal to 1), and thus we can consider a sequence of `discrete' terraces associated with the time steps 2 -k , k ∈ N. By passing to the limit as k → +∞, we expect to recover an actual propagating terrace in the sense of Denition 1.2.

Remark 7. As we mentioned earlier, in some cases this limiting argument is not needed. Indeed, it is rather straightforward to show that a discrete travelling front, regardless of the time step, is also a generalized transition front in the sense of Berestycki and Hamel [START_REF] Berestycki | Generalized transition waves and their properties[END_REF]; without going into the details, we recall that a transition front is an entire solution whose level sets remain at a bounded distance uniformly with respect to time. Under an additional monotonicity assumption on the neighborhood of limiting stable steady states, and provided that the speed is not 0, they have proved that any almost planar transition front is also a pulsating travelling front. However, this is not true in general, therefore we proceed with a dierent approach.

For any direction e ∈ S N -1 and any k ∈ N, the discrete terrace associated with the time step 2 -k consists of a nite sequence of ordered stable steady states

p ≡ q 0,k > q 1,k > • • • > q J(k),k ≡ 0,
and a nite sequence of discrete travelling fronts connecting these steady states with nondecreasing speeds. Because the (q j,k ) 0≤j≤J(k) belong to the nite set of periodic stable steady states of (1.1), we can extract from the sequence of time steps (2 -k ) k∈N a subsequence (τ k ) k∈N along which the family (qj,τ k ) 0≤j≤J(τ k ) does not actually depend on k. Therefore, we simply denote it by (qj) 0≤j≤J . Let (U j,k ) 0≤j≤J, k∈N be the corresponding fronts, i.e., the U j,k (y, z) are periodic in y, nonincreasing in z and satisfy

∀(y, z) ∈ R N +1 , v(τ k , y; U j,k (x, z + x • e)) = U j,k (y, z + y • e -c j,k ), with c 1,k ≤ c 2,k ≤ • • • ≤ c J,k , as well as U j,k (•, -∞) ≡ qj-1, U j,k (•, +∞) ≡ qj.
As a matter of fact, the speeds c j,k are proportional to the time step τ k , by a factor depending on j. This is the subject of the next lemma, whose proof exploits the link between the front and the spreading speed, which is the heart of the method developed by Weinberger in [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF] and used in the present paper. Lemma 6.1. There exists a sequence

c1 ≤ c2 ≤ • • • ≤ cJ such that ∀j ∈ {1, . . . , J}, k ∈ N, c j,k = τ k cj.
Proof. The proof amounts to showing that ∀j ∈ {1, . . . , J}, k ∈ N, c j,k+1 c j,k = τ j,k+1 τ j,k .

We do it for j = 1. Then, since the intermediate states qi do not depend on k, and because the subsequent speeds were constructed in a similar fashion, the case j > 1 is analogously derived.

Let us rst show that

c 1,k+1 c 1,k ≥ τ 1,k+1 τ 1,k
. This easily follows from our earlier construction. Let us consider the shifted evolution operators associated with the time steps (τ k ) k∈N . In analogy with (3.2), these are dened by

F c,k [V ](y, z + y • e -c) := v(τ k , y; V (x, z + x • e)).
Then, for φ satisfying (3.3), we dene the sequence (a k c,n ) n∈N through (3.4) with Fc replaced by F c,k . Fix k ∈ N and call ρ :

= τ k τ k+1 ∈ N * . Because (F c,k+1 ) ρ [V ](y, z + y • e) = v(ρτ k+1 , y; V (x, z + x • e + ρc)) = F ρc,k [V ](y, z + y • e),
we nd that a k+1 c,ρ ≥ a k ρc,1 , where the inequality comes from the fact that in the time step τ k the sequence (a k+1 c,n ) n∈N is `boosted' ρ times by the function φ, while (a k ρc,n ) n∈N only once. We can readily iterate this argument to get that a k+1 c,ρn ≥ a k ρc,n for all n ∈ N. It then follows from Lemma 3.7 that if ρc < c 1,k then c < c 1,k+1 This means that ρc 1,k+1 ≥ c 1,k , which is the rst desired inequality.

To prove the reverse inequality, we shall use Lemma 3.8 which asserts that c 1,k does not depend on the choice of φ satisfying (3.3). Then we choose the function generating the sequences (a k c,n ) n∈N , (a k+1 c,n ) n∈N of a particular form. Namely, we consider a solution u of the Cauchy problem associated with (1.1), with a continuous periodic initial datum u0 < p such that u0 -δ lies in the basin of attraction of p, for some constant δ > 0. In particular, there exists T > 0 such that u(t, •) > u0 for t ≥ T . We then initialize (a k c,n ) n∈N with a function φ satisfying φ(y, -∞) = u(T, y). It follows that v(t, y; φ(x, -∞)) > u0(y) for all t ≥ 0, and thus, by parabolic estimates,

∀0 ≤ t ≤ τ k , y ∈ [0, 1] N , v(t, y; φ(x, z + x • e)) > u0(y),
provided z is smaller than some Z. Then, by the periodicity of φ and u0, we get

∀0 ≤ t ≤ τ k , z + y • e ≤ Z - √
N , v(t, y; φ(x, z + x • e)) > u0(y).

We now initialize (a k+1 c,n ) n∈N with a function φ satisfying

φ (y, -∞) = u(0, y), φ (y, z) = 0 for z ≥ Z - √ N -ρ|c|.
We deduce that ∀j = 1, . . . , ρ, (y, z) ∈ R N +1 , φ (y, z + y • e -ρ|c|) ≤ v(jτ k+1 , y; φ(x, z + x • e)).

We claim that a k+1 c,ρn ≤ a k ρc,n for all n ∈ N. This property holds for n = 0. Suppose that it holds for some n ∈ N. Using the property of φ , and recalling that φ ≤ a k ρc,n , we nd that a k+1 c,ρn+1 (y, z + y • e -c) = max{φ (y, z + y • e -c), v(τ k+1 , y; a k+1 c,ρn (x, z + x • e))} ≤ v(τ k+1 , y; a k ρc,n (x, z + x • e)).

Iterating ρ times we get a k+1 c,ρ(n+1) (y, z + y • e -ρc) = max{φ (y, z + y • e -ρc), v(τ k+1 , y; a k+1 c,ρn+ρ-1 (x, z

+ x • e -(ρ -1)c))} ≤ v(ρτ k+1 , y; a k ρc,n (x, z + x • e)) = F ρc,k [a k ρc,n ](y, z + y • e -ρc) ≤ a k ρc,n+1 (y, z + y • e -ρc).
The claim a k+1 c,ρn ≤ a k ρc,n is thereby proved for all n ∈ N. Then, as before, owing to Lemma 3.7 we conclude that c 1,k ≥ ρc 1,k+1 .

We are now in a position to conclude the proofs of Theorems 1.4 and 1.5. Namely, in the next lemma we show that for each level 1 ≤ j ≤ J of the discrete propagating terrace one can nd a continuous propagating terrace whose fronts have the same speed cj from Lemma 6.1. Then, by `merging' the so obtained J terraces, one gets a propagating terrace of (1.1) connecting p to 0. In the bistable case, the terrace reduces to a single pulsating travelling front, thanks to Assumptions 1.1 and 1.3. Instead, in the multistable case, our construction allows the possibility that the continuous propagating terrace contains more fronts than the discrete terraces did. This is actually not true in typical situations (such as the already mentioned ones where the argument of Berestycki and Hamel [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] applies), but it remains unclear whether this can happen in general. Lemma 6.2. For any 1 ≤ j ≤ J, there exists a propagating terrace connecting qj-1 to qj in the sense of Denition 1.2. Moreover, all the fronts in this terrace have the speed cj.

Proof. The aim is to pass to the limit as k → +∞ in the sequence of discrete terraces associated to the time steps (τ k ) k∈N . The rst step consists in showing that the proles U j,k converge as k → +∞. Due to the lack of regularity with respect to the second variable, the limit will be taken in the relaxed sense of Lemma 4.3.

As usual, the argument is the same regardless of the choice of j and then for simplicity of notation we take j = 1. Beforehand, we shift U 1,k so that ∀z < 0, min

y∈[0,1] N U 1,k (y, z + y • e) -η(y) ≥ 0, (6.1)
min

y∈[0,1] N U 1,k (y, y • e) -η(y) ≤ 0, (6.2)
where q1 < η < p is a given function lying in the basin of attraction of p. We know that U 1,k is a xed point for F c 1,k ,k by construction, that is, it is a xed point for F τ k c 1 ,k owing to the previous lemma. Then, for k < k, observing that

F τ k c 1 ,k = (F τ k c 1 ,k ) τ k τ k ,
where τ k τ k ∈ N, we see that it is a xed point for F τ k c 1 ,k too. We now apply Lemma 4.3 to the sequence (U 1,k ) k∈N . We point out that the hypothesis there that U 1,k (y, z + y • e) is uniformly continuous in y ∈ R N , uniformly with respect to z and k, follows from parabolic estimates due to the fact that all the U 1,k are xed points of Fτ 1 c 1 ,1. We obtain in the relaxed limit (up to subsequences) a function U1(y, z) which is periodic in y, nonincreasing in z and such that U1(y, z + y • e) is uniformly continuous in y, uniformly with respect to z. Moreover, U1 satises the normalization (6.1)-(6.2). Finally, by the above consideration, it also follows from By continuity of the solution of (1.1) with respect to time, as well as the monotonicity of U1 with respect to its second variable, we immediately extend this inequality to all positive times, i.e., u(t, y) = U1(y, y • e -c1t).

In particular, U1(y, y • e -c1t) solves (1.1) for positive times in the whole space; by periodicity in the rst variable, it is straightforward to check that it solves (1.1) for negative times too.

Remark 8. We have shown above that U1 is continuous with respect to both its variables, on the condition that c1 = 0.

To show that U1(y, y • e -c1t) is a pulsating travelling front in the sense of Denition 1.1, it only remains to check that it satises the appropriate asymptotic. By monotonicity in the second variable, we already know that U1(•, ±∞) exist, and moreover these limits are periodic steady states of (1.1). We further have that U1(•, -∞) ≥ η and U1(•, +∞) ≡ p, because U1 satises (6.1)-(6.2). Recalling that η lies in the basin of attraction of p, we nd that U1(•, -∞) ≡ p.

Let us now deal with the limit as z → +∞. Let us call p * := U1(•, +∞). This is a periodic steady state satisfying q1 ≤ p * < p; however it could happen that the rst inequality is strict too. We claim that p * is stable. In which case, changing the normalization (6.1)-(6.2) by taking η < p * in the basin of attraction of p * , and then passing to the limit as before, we end up with a new function U2. Because of this normalization, together with the fact that U1(•, +∞) = p * , it turns out that U2 connects p * to another steady state p * 2 ≥ q1. Then, by iteration, we eventually construct a terrace connecting p to q1.

It remains to show that p * is stable. We proceed by contradiction and assume that this is not the case. In particular, p * > q1. Let p+, pdenote respectively the smallest stable periodic steady state above p * and the largest stable periodic steady state below p * , and let cp * and c p * be the minimal speeds of fronts connecting p+ to p * and p * to prespectively. By the same comparison argument as in the proof of Lemma 4.4 one readily sees that the speed c1 of U1 satises c1 ≥ cp * .

We recall that the argument exploits Weinberger's result in [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF] which asserts that cp * coincides with the spreading speed for solutions between p * and p+. Next, one shows that c1 ≤ c p * . This is achieved by choosing the normalization ∀z < 0, max

y∈[0,1] N U 1,k (y, z + y • e) -η * (y) ≥ 0, max y∈[0,1] N U 1,k (y, y • e) -η * (y) ≤ 0,
with η * between pand p * and in the basin of attraction of p-, which is possible because U 1,k (•, +∞) ≡ q1 ≤ p-. One gets in the (relaxed) limit a solution U * (y, y • e -c1t) satisfying U * (•, -∞) ≤ p * (because compared with U1, the function U * is obtained as the limit of an innite shift of the sequence U 1,k ), as well as U * (•, +∞) ≤ η * . Then the desired inequality follows again from the spreading result. Finally, combining the previous two inequalities one gets cp * ≤ c p * , which contradicts our Assumption 1.3.

This concludes the proof.

Remark 9. As pointed out in Remark 2, in general it is not equivalent to nd a function U1 as above and a pulsating front solution. The function U1 constructed above actually gives rise to a whole family of pulsating fronts U1(x, x • e + z -c1t). In the case when c1 = 0, then this family merely reduces to the time shifts of a single front.

In the case when c1 = 0, however, it is much less clear how these fronts are related to each other: as observed earlier the function U1(x, z) may be discontinuous with respect to z, hence the resulting family may not be a continuum of fronts (in general, it is not).

Highly non-symmetric phenomena

It is clear that, because equation (1.1) is heterogeneous, the terrace ((qj)j, (Uj)j) provided by Theorem 1.5 depends in general on the direction e. In this section, we shall go further and exhibit an example where not only the fronts Uj, but also the intermediate states qj and even their number, i.e., the number of `oors' of the terrace, change when e varies. Obviously this cannot happen in the bistable case where the stable steady states reduce to p, 0. Namely, we prove Proposition 1.6.

The main idea is to stack a heterogeneous bistable problem below an homogeneous one. Then in each direction there exists an ordered pair of pulsating travelling fronts.

Whether this pair forms a propagating terrace depends on the order of their speeds.

If the latter is admissible for a terrace, that is, if the uppermost front is not faster than the lowermost, then the terrace will consists of the two fronts, otherwise it will reduce to a single front. Since those speeds are given respectively by a function S N -1 e → c(e) and by a constant c, and since the heterogeneity should make such a function nonconstant, it should be possible to end up with a case where the number of fronts of the terrace is nonconstant too.

Owing to the above consideration, the construction essentially amounts to nding a heterogeneous bistable problem for which the speed of the pulsating travelling front c(e) is nonconstant in e. While such property should be satised by a broad class of problems (perhaps even generically), getting it in the context of a bistable equation (in respect to xi we nd that ∂iq is a sign-changing eigenfunction of the linearized operator around q, with eigenvalue 0. It follows that the principal eigenvalue λq of such operator in the space of periodic functions (which is maximal, simple and associated with a positive eigenfunction) is positive, that is, q is linearly unstable. We prove statement (iii) by contradiction. Assume that (7.3) admits a pair of periodic steady states 0 < q < q < 1. We know from (i)-(ii) that such solutions are linearly unstable. Then, calling ϕq the principal eigenfunction associated with λq, one readily checks that for ε > 0 suciently small, q + εϕq is a stationary strict subsolution of (7.3). Take ε > 0 such that the above holds and in addition q + εϕq < q. It follows from the parabolic comparison principle that the solution with initial datum q + εϕq is strictly increasing in time and then it converges as t → +∞ to a steady state q satisfying q < q ≤ q. This is impossible, because q is linearly unstable by (i)-(ii) and then its basin of attraction cannot contain the function q + εϕq.

Remark 10. Consider the homogeneous equation (1.4) with a general reaction term f = f (u). Statement (ii) of Lemma 7.2 holds true in such case, because its proof only relies on the spatial-invariance of the equation. Thus, if Assumption 1.1 holds, the uppermost steady state p must be constant. One then nds that Assumption 1.1

necessarily implies that ∃θ ∈ (0, p), f (0) = f (θ) = f (p) = 0, f < 0 in (0, θ), f > 0 in (θ, p).
As a matter of fact, these conditions are equivalent to Assumption 1.1. Indeed, even though the constant state θ may not be linearly unstable (if f (θ) = 0), one sees that θ is unstable in a strong sense: θ + ε belongs to the basin of attraction of 0 if ε < 0 and of p if ε > 0. This is enough for the proof of Lemma 7.2 (iii) to work.

We can now derive the bistability character of (7.1). Lemma 7.3. Consider the equation (7.1) with f1 dened by (7.2). The following properties hold:

(i) any periodic steady state 0 < q < 1 is linearly unstable; (ii) there does not exist any pair 0 < q < q < 1 of periodic steady states. Proof. The proof is achieved in several steps.

Step 1: any periodic steady state which is not x-independent is linearly unstable.

Because the equation (7.1) is invariant by translation in the x-variable, we can proceed exactly as in the proof of Lemma 7.2 (i).

Step 2: if 0 ≤ q < 1 is a periodic steady state which is x-independent then q ≤ S.

We recall that S is dened by S 0 f0 = 0. Suppose that q = q(y) is not constant, otherwise it is identically equal to 0 or 1 2 < S. Consider an arbitrary η ∈ R with q (η) > 0. Let a < η < b be such that q (a) = q (b) = 0 and q > 0 in (a, b). Multiplying the inequality -q = f1(y, q) ≥ f0(q) by q and integrating on (a, b) we get 0 ≥ b a f0(q(y))q (y)dy = q(b) q(a) f0(u)du.

This implies rst that q(a) ≤ 1 2 , and then that q(b) 0 f0(u)du ≤ 0. Recalling the denition of S, we nd that q(b) ≤ S, whence q(η) < S. We have thereby shown that q(η) < S whenever q (η) > 0, and therefore that q ≤ S.

Step 3: if (7.1) admits a pair of periodic steady states 0 < q < q < 1, then there exists a periodic steady state q ≤ q ≤ S which is not linearly unstable.

If q is not linearly unstable then the Steps 1-2 imply that q ≤ S, which means that the conclusion holds with q = q in such case. Suppose now that q is linearly unstable. It follows from the same argument as in the proof of Lemma 7.2 (iii) that for ε > 0 suciently small, the function q -εϕq is a supersolution of (7.1), which is larger than q, where ϕq is the principal eigenfunction of the linearized operator around q.

The comparison principle then implies that the solution of (7.1) with initial datum q -εϕq is strictly decreasing in time and then it converges as t → +∞ to a steady state q satisfying q ≤ q < q -εϕq. Such state cannot be linearly unstable, because its basin of attraction contains the function q -εϕq. Then, as before, q ≤ S by the Steps 1-2.

Step 4: conclusion.

Assume by contradiction that there is a periodic steady state 0 < q < 1 which is not linearly unstable. From the Steps 1-2 we deduce that q ≤ S. This means that q is a stationary solution of (7.3). Lemma 7.2 then implies that q is linearly unstable for (7.3), and thus for (7.1) too. This is a contradiction. We have thereby proved (i). Suppose now (7.1) admits a pair of periodic steady states 0 < q < q < 1. Then Step 3 provides us with a periodic steady state 0 < q ≤ S which is not linearly unstable, contradicting (i).

Let f1 be dened by (7.2), with L, M > 0 still to be chosen. Lemma 7.3 implies that the equation (7.1) is bistable in the sense of Assumption 1.1 with p ≡ 1. Moreover, thanks to Proposition A.2 in the Appendix, it also entails Assumption 1.3. We can thus apply Theorem 1.4, which provides us with a monotonic in time pulsating travelling front connecting 1 to 0, for any given direction e ∈ S 1 . Let c(e) be the associated speed. Before showing that c(e1) > c(e2), let us derive the uniqueness of the pulsating travelling front and the positivity of its speed. Lemma 7.4. The equation (7.1) with f1 dened by (7.2) admits a unique (up to shifts in time) pulsating travelling front connecting 1 to 0 for any given direction e ∈ S 1 .

Furthermore, the front is strictly increasing in time and its speed c(e) is positive.

Proof. Firstly, the positivity of the speed of any front connecting 1 to 0 is an immediate consequence of the facts that f1 ≥ f0 and that equation (7.3) admits solutions with compactly supported initial data which spread with a positive speed [START_REF] Aronson | Multidimensional nonlinear diusion arising in population genetics[END_REF].

Next, the fronts provided by Theorem 1.4 are monotonic in time. Applying the strong maximum principle to their temporal derivative (which satises a linear parabolic equation) we infer that the monotonicity is strict, unless they are constant in time. The positivity of their speed then implies that they are necessarily strictly increasing in time.

Hence, the second part of the lemma holds for the fronts given by Theorem 1.4. If we show that such fronts are the only ones existing we are done.

Throughout this proof, we use the notation x to indicate a point in R 2 . Let ui(t, x) = Ui(x, x • e -cit), i = 1, 2, be two pulsating travelling fronts for (7.1) connecting 1 to 0 in a given direction e ∈ S 1 . We have seen before that necessarily ci > 0. This means that the transformation (x, t) → (x, x • e -cit) is invertible and thus Ui(x, z) enjoys the regularity in (x, z) coming from the parabolic regularity for ui (at least C 1 , with bounded derivatives). Let us suppose to x the ideas that c1 ≥ c2. We shall also assume that either U1 or U2 is the front provided by Theorem 1.4, so that we further know that it is decreasing in z.

We use a sliding method. The conditions Ui(•, -∞) = 1 and Ui(•, +∞) = 0 imply that for any ε ∈ (0, 1), the following property holds for -k > 0 suciently large (depending on ε): ∀x ∈ R 2 , z ∈ R, U1(x, z) < U2(x, z + k) + ε.

The above property clearly fails for k > 0 large, thus we can dene k ε ∈ R as the supremum for which it is fullled. Call U ε 2 (x, z) := U2(x, z + k ε ) and u ε 2 (t, x) := U ε 2 (x, x • e -c2t). Observe that u ε 2 is just a temporal translation of u2, because c2 = 0, whence it is still a solution of (7.1). We see that sup(U1 -U ε 2 ) = ε.

Using again Ui(•, -∞) = 1 and Ui(•, +∞) = 0 one infers that a maximizing sequence (xn, zn) n∈N for U1 -U ε 2 has necessarily (zn) n∈N bounded. By periodicity, we can assume that the sequence (xn) n∈N is contained in [0, L] 2 . Hence, there exists (x ε , z ε ) such that (U1 -U ε 2 )(x ε , z ε ) = max(U1 -U ε 2 ) = ε.

It follows that u1(t ε 1 , x ε ) = u ε 2 (t ε 2 , x ε ) + ε, with t ε i :=

x ε • e -z ε ci .

for a suitable choice of M . Indeed, for any M ≥ 0, (7.5) admits a unique planar front, see [START_REF] Aronson | Multidimensional nonlinear diusion arising in population genetics[END_REF], and it is not hard to check that its speed c M depends continuously on M . To conclude, we observe that c0 = 0 [START_REF] Aronson | Multidimensional nonlinear diusion arising in population genetics[END_REF] and that c M → +∞ as M → +∞, as we have shown in the Step 1 of the proof of Proposition 7.1. We point out that the proof of Lemma 7.4 still works for the homogeneous equation (7.5). Namely, the planar front is the unique pulsating travelling front for (7.5) (up to shift in time or space).

We can now dene the reaction f as follows:

f (y, u) := f1(y, u)

if 0 ≤ u ≤ 1, f2(u -1) if 1 < u ≤ 2.
This function is of class C 1 because, we recall, ∂uf1(y, 1) = f 0 (1) = -1 = f 2 (0).

Moreover, it is a superposition of two reaction terms which are bistable in the sense of Assumption 1.1, due to Lemmas 7.2, 7.3. Let us show that f satises Assumption 1.2 with I 2 and p0 ≡ p ≡ 2, p1 ≡ 1, p2 ≡ 0.

We claim that any periodic steady state q satisfying 0 < q < 2 and q ≡ 1 is linearly unstable. By Lemmas 7.2 and 7.3, we only need to consider the case when min q < 1 < max q. Assume by contradiction that such a q is not linearly unstable. Because the equation is invariant in the direction e1, the Step 1 of the proof of Lemma 7.3 implies that q is x-independent, i.e., q = q(y). On the level set q = 1 we necessarily have that q = 0, because otherwise q ≡ 1. Then, the function q being periodic, there exists η ∈ R such that q(η) = 1 and q (η) > 0. Let a < η be such that q (a) = 0 and q > 0 in (a, η). Then in (a, η) there holds that -q = f1(y, q) ≥ f0(q). Multiplying this inequality by q and integrating on (a, η) we get -1 2 (q ) 2 (η) ≥ η a f0(q(y))q (y)dy = 1 q(a) f0(u)du.

This is impossible, because 1 s f0 > 0 for any s ∈ [0, 1], by denition of the function f0.

The claim is proved.

Summing up, we know that all periodic steady states of (1.1) are linearly unstable, excepted for the constant states 0, 1, 2 which are linearly stable. As shown in the proof of Lemma 7.2, between any pair of linearly unstable periodic steady states q < q there must exists a periodic steady state which is not linearly unstable. This implies that Assumption 1.2 holds, as announced. It entails Assumption 1.3 too, owing to Proposition A.2 in the Appendix.

We are in the position to apply Theorem 1.5. This provides us with a propagating terrace in any direction e ∈ S 1 . Two situations may occur: either the terrace reduces to one single front connecting 2 to 0, or it consists of two fronts, one connecting 2 to 1 and the other connecting 1 to 0. In the latter case, we have by uniqueness that the two fronts are respectively given (up to translation in time) by the unique planar front for (7.5) increased by 1, which has speed c, and by the unique pulsating front of Proposition 7.1, having speed c(e). This case is ruled out if c > c(e) because this violates the condition on the order of the speeds of the propagating terrace, see Denition 1.2. Therefore, when c > c(e) the terrace consists of a single front connecting 2 to 0, and proceeding as in the proof of Lemma 7.4, one can show that this front is unique up to time shift.

Conversely, let us show that if c ≤ c(e) then the case of a single front is forbidden. Suppose that there exists a pulsating travelling front ũ connecting 2 to 0 in the direction e with some speed c. Observe that the argument for the uniqueness result in the proof of Lemma 7.4 still works if U2(•, -∞) ≥ 1 or if U1(•, +∞) ≤ 0. Hence, on one hand, applying this argument with u1 equal to the front connecting 1 to 0 and with u2 = ũ we get c > c(e). On the other hand, taking u1 = ũ -1 and u2 equal to the planar front for (7.5) yields c < c. We eventually infer that c > c(e), a contradiction. Therefore, when c ≤ c(e), a terrace necessarily consists of two fronts, and as we pointed out above each of them is unique up to time shift.

We have proved that there exists a unique propagating terrace in any given direction e ∈ S 1 and that it consists of two fronts if and only if c ≤ c(e). This concludes the proof of the proposition because c(e2) < c < c(e1).

Lemma 3 . 8 .

 38 The speed c * does not depend on the choice of φ satisfying the properties(3.3).Proof. Consider two admissible functions φ and φ for the conditions(3.3). Let ac,n, âc,n and c * , ĉ * denote the functions and constants constructed as above, starting from φ, φ respectively. Take an arbitrary c ∈ R. Using the rst part of Lemma 3.6 and the fact that ac,n(y, z + y • e) → ac(y, z + y • e) locally uniformly in y as n → +∞, we can nd z < 0 and n ∈ N such that inf y∈[0,1] N ac,n(y, z + y • e) -φ(y, -∞) > 0. Because |y•e| ≤ √ N if y ∈ [0, 1] N , one readily deduces that ac,n(y, z-√ N + z) > φ(y, z) for all (y, z) ∈ R N +1 , whence ac,n(•, • -√ N + z) > φ for all n ≥ n by the monotonicity in n. It follows that ac,n+1(•, • -

(4. 3 )

 3 Indeed by the denition of zc,n and (3.7), for n ≥ n(c) we get0 < min y∈[0,1] N ac,n(y, zc,n -1 + y • e) -φ(y, -∞) ≤ min y∈[0,1] N ac * ,n(y, zc,n + n(c -c * ) + y • e -1) -φ(y, -∞) .

  Fix (y, z) ∈ R N × R and L ∈ Z N . Then, using the periodicity of αn, for every ζ, ζ ∈ Q satisfying ζ < z and ζ + L • e > ζ, we get αn(y, ζ + y • e) ≥ αn(y + L, ζ + (y + L) • e). Passing to the limit along the subsequence αn k we deduce β(y, ζ) ≥ β(y + L, ζ ). Now we let Q ζ → z + and Q ζ → (z -L • e) + and we derive β(y, z) ≥ β(y + L, z -L • e).

  which, in turn, is larger than or equal tolim k→+∞ a c k ,n(c k ) (y, ζ + z c k ,n(c k ) + y • e), for any rational ζ > z -c * . Letting Q ζ → (z -c * ) + , we eventually conclude that Fc * [a * ](y, z + y • e -c * ) ≥ a * (y, z + y • e -c * ).Property (iii) then follows by iteration.Next, x m ∈ N and a positive ζ ∈ Q. We know by (4.1) that, for every k ∈ N and y ∈ R N ,

  Finally, by the continuity of Fc * we get, for all y ∈ [0, 1] N , Fc * [a ](y) = lim z→-∞ Fc * [a * ](y, z + y • e) ≥ a (y).

  (c p * -ε)t |v(t, y; H(x, Z + x • e)) -p+(y)| = 0, lim t→+∞ sup y•e≥(c p * +ε)t |v(t, y; H(x, Z + x • e)) -p * (y)| = 0. A similar result holds when looking at solutions between pand p * . Let us show that c * ≥ cp * . Since U * (•, -∞) ≡ p ≥ p+ and U * ≥ p * , we can choose Z > 0 large enough so that U * ≥ H(•, • + Z). Now we argue by contradiction and assume that c * < cp * . Then, calling ε := (cp *c * )/2, we have that cp * -ε = c * + ε and thus, by comparison, lim inf n→+∞ inf y•e≤(c * +ε)n v(n, y; U * (x, x • e)) -p+(y) ≥ 0.

  , y; U * (x, x • e)) = (Fc * ) n [U * ](y, y • e -nc * ) = U * (y, y • e -nc * ),

(4. 10 )

 10 We now apply Lemma 4.3 to the sequence (Fc * ) n k [a c k ,n(c k ) ](y, z + ẑk + y • e) k∈N . This provides us with a function α * (y, z) periodic in y and nonincreasing in z and such that α * (y, z + y • e) is uniformly continuous in y ∈ R N , uniformly with respect to z ∈ R. Moreover, proceeding exactly as in the proof of Lemma 4.2, we deduce from the inequality

  is nondecreasing with respect to n. The choice of ẑk further implies that ∀z < 0, max y∈[0,1] N α * (y, z + y • e) -p-(y) ≥ δ, y, y • e) -p-(y) ≤ δ.

  (c p * +ε)t |v(t, y; Ĥ(x, x • e)) -p-(y)| = 0. By comparison we obtain lim sup n→+∞ sup y•e≥(c p * +ε)n v(n, y; α * (x, x • e)) -p-(y) ≤ 0. However, because (Fc * ) n [ α * ] is nondecreasing in n, we have that α * (y, y • e -nc * ) ≤ (Fc * ) n [ α * ](y, y • e -nc * ) = v(n, y; α * (x, x + •e)), (c p * +ε)n α * (y, y • e -nc * ) -p-(y) ≤ 0. For y ∈ [0, 1] N and ξn := [(c p * + ε)n + √ N ]e there holds (y + ξn) • e ≥ (c p * + ε)n, whence lim sup n→+∞ max y∈[0,1] N α * (y + ξn, (c p * + ε -c * )n) -p-(y + ξn) ≤ 0.

Lemma 4 .

 4 3 and the continuity of the operators F τ k c 1 ,k in the locally uniform topology, that U1 fulls ∀k ∈ N, F τ k c 1 ,k [U1] ≡ U1. Let u(t, y) denote the solution of the problem (1.1) with initial datum U1(y, y • e). Then for any k, m ∈ N, we have that u(mτ k , y) = (F τ k c 1 ,k ) m [U1](y, y • e -mτ k c1) = U1(y, y • e -mτ k c1).

  (notice that uc(t, x) satises parabolic estimates up to time t = 0 because a c,n(c) = Fc[a c,n(c)-1 ]). Property (2.6) and the monotonicity of (ac,n) n∈N imply that

	.8)
	Conditions (2.6),(2.8) determine our choice of the diagonal sequence (a c,n(c) )c<c * .
	Let uc(t, x) denote the solution of the Cauchy problem for (2.2) with initial da-
	tum a c,n(c)

This theorem slightly improves the existence result of[START_REF] Ducrot | A multi-dimensional bistable nonlinear diusion equation in a periodic medium[END_REF], which additionally requires the stability or instability of the steady states to be linear. However, we emphasize that our argument is completely dierent: while in[START_REF] Ducrot | A multi-dimensional bistable nonlinear diusion equation in a periodic medium[END_REF] the proof relies on an elliptic regularization technique, here we proceed through a time discretization and a dynamical system approach.

the sense of Assumption 1.1) is rather delicate. We were not able to nd an example of this type in the literature.

We place ourselves in dimension N = 2 and denote a generic point in R 2 by (x, y), as well as e1 := (1, 0), e2 := (0, 1). We derive the following. Proposition 7.1. There exists a function f1 = f1(y, u) which is periodic in the variable y ∈ R, satises Assumptions 1.1, 1.3 with p ≡ 1, and for which the equation ∂tu = ∆u + f1(y, u), t ∈ R, (x, y) ∈ R 2 , (7.1) admits a unique (up to shifts in time) pulsating travelling front connecting 1 to 0 for any given direction e ∈ S 1 .

Furthermore, the corresponding speeds c(e) satisfy c(e1) > c(e2) > 0.

The function f1(y, u) we construct will be periodic in y with some positive period, which one can then reduce to 1 (to be coherent with the rest of the paper) by simply rescaling the spatial variables.

We rst introduce a smooth function f0 : [0, 1] → R with the following properties:

We let 1 2 < S < 1 be the quantity identied by the relation

Next, we consider two smooth functions χi : R → R, i = 1, 2, satisfying χi ≥ 0, ≡ 0 and supp χ1 ⊂ (0, 1), χ1 = 1 on 1 4 , 3 4 , supp χ2 ⊂ (S, 1),

where supp denotes the closed support. We then set ∀u ∈ R, y ∈ [0, 2L], f1(y, u) := f0(u) + M χ1 y L χ2(u),

L, M being positive constants that will be chosen later. We nally extend f1(y, u) to R 2 by periodicity in the y-variable, with period 2L. Observe that f1(y, u) ≥ f0(u), and that equality holds for y ∈ [(2j -1)L, 2jL], j ∈ Z. Until the end of the proof of Proposition 7.1, when we say that a function is periodic we mean that its period is 2L.

Let us show that the equation (7.1) is bistable in the sense of Assumption 1.1. We shall also check that it fulls Assumption 1.3, for which, owing to Proposition A.2 in the Appendix, it is sucient to show that any intermediate state is linearly unstable. We shall need the following observations about the periodic steady states of the homogeneous equation.

Lemma 7.2. For the equation

the following properties hold:

(i) the constant steady states 0, 1 are linearly stable, whereas 1 2 is linearly unstable; (ii) any periodic steady state which is not identically constant is linearly unstable; (iii) there does not exist any pair 0 < q < q < 1 of periodic steady states. Proof. Statement (i) is trivial, because the principal eigenvalue of the linearized operator around the constant states q1 ≡ 0, q2 ≡ 0, q3 ≡ 1 2 is equal to f 0 (qi). Statement (ii) is a consequence of the invariance of the equation by spatial translation. Indeed, if q is a steady state which is not identically constant then it admits a partial derivative ∂iq which is not identically equal to 0; if in addition q is periodic then ∂iq must change sign. Then, dierentiating the equation ∆q + f0(q) = 0 with Next, if U1(x, z) is the front decreasing in z provided by Theorem 1.4 then for t ≤ 0 we nd that

where we have used the equality c1t ε 1 = c2t ε 2 . Similarly, if U2(x, z) is decreasing in z then, for t ≤ 0, we get

Namely, in any case, u1(t ε 1 + t, x) lies below u ε 2 (t ε 2 + t, x) + ε until t = 0, when the two functions touch. Both u1(t ε 1 + t, x) and u ε 2 (t ε 2 + t, x) are solutions of (7.1). Moreover, because f1 = f0 for u close to 0 and 1, and f 0 (0), f 0 (1) < 0, one readily checks that the function u ε 2 (t ε 2 + t, x) + ε is a supersolution of (7.1) in the regions where it is smaller than δ or larger than 1 -δ + ε, for some small δ depending on f0 and S. If the contact point x ε were in one of such regions, the parabolic strong maximum principle would imply that u1(t ε 1 + t, x) ≡ u ε 2 (t ε 2 + t, x) + ε there, for t ≤ 0, which is impossible because Ui(•, -∞) = 1 and Ui(•, +∞) = 0. Therefore, we have that

, the above bounds imply that both t ε 1 and t ε 2 -k ε c 2 stay bounded as ε 0. Calling x, t1, t2 the limits as ε 0 of (some converging subsequences of )

The parabolic strong maximum principle nally yields u1( t1 + t, x) ≡ u2( t2 + t, x) for t ∈ R, x ∈ R 2 . This concludes the proof of the lemma.

Proof of Proposition 7.1. We need to show that c(e1) > c(e2) for a suitable choice of L, M > 0. The proof is divided into several parts.

Step 1: for L > 8 there holds that c(e1) → +∞ as M → +∞.

Fix an arbitrary c > 0. We want to construct a subsolution u of (7.1) of the form

χ2 > 0.

We then dene w as follows:

which is negative for M large. This implies that, for M suciently large (depending on c), there exists RM > ρ such that w > 0 in (0, RM ), w (RM ) = 0.

It also yields that RM ρ as M → +∞. Thus, for M large enough there holds that σ2 > R c+3 M > w(RM ). From now on we restrict ourselves to such values of M .

Direct computation shows that the function u satises (in the weak sense)

Consider rst the case 0 < γ < ρ (< 1). We see that

This means that u is a subsolution of (7.1) in the region 0 < γ < ρ.

Instead, if ρ < γ < RM , there holds that σ1 < w • γ < σ2 and thus

Then, under such condition, it turns out that u is a subsolution of (7.1) in the region ρ < γ < RM too.

We nally extend w to 0 on (-∞, 0) and we change it into the constant w(RM ) (< σ2) on [RM , +∞). This is still of class W 2,∞ and, for L > 8 and M large enough, the function u := w • γ is a generalized subsolution of (7.1) in the whole space.

Notice that u shifts in the direction e1 with speed c. Moreover, for xed time, it is compactly supported and bounded from above by σ2. It follows that, up to translation in time, it can be placed below the pulsating travelling front in the direction e1. This readily implies by comparison that the speed of the latter satises c(e1) ≥ c. Step 1 is thereby proved due to the arbitrariness of c.

Step 2: for L > ln 4, there exists τ > 0, depending on L but not on M , such that c(e2) ≤ 2L/τ .

We introduce the following function:

This is a strict supersolution of (7.3). Indeed, we have that

where the last inequality holds because f0(ψ

In order to have τ > 0 we impose L > ln 4. We nally dene ∀j ∈ N, t ∈ (0, τ ], y ∈ R, ū(jτ + t, y) := ψ(jL + t, y).

The function ū(t, y) is increasing and lower semicontinuous in t, because L > τ . Consider now a pulsating travelling front u(t, x, y) = U (x, y, y -c(e2)t) for (7.1) in the direction e2 connecting 1 to 0. The functions u and U are periodic in the x variable. Moreover, there exists k ∈ N such that ū(kτ, y) > U (x, y, y) = u(0, x, y) for all (x, y) ∈ R 2 . Assume by contradiction that the inequality ū(kτ +t, y) > u(t, x, y) fails for some positive time t and let T ≥ 0 be the inmum of such times. Then, because ū is increasing in the rst variable and u is continuous, we have that ū(kτ + t, y) ≥ u(t, x, y) for all t ∈ [0, T ]. Moreover, there exist some sequences tn T and ((xn, yn)) n∈N such that ū(kτ + tn, yn) ≤ u(tn, xn, yn) for all n ∈ N. By the periodicity of u in x, it is not restrictive to assume that the sequence (xn) n∈N is bounded. The sequence (yn) n∈N is also bounded, because from one hand ū(kτ + tn, y) ≥ ū(kτ + T, y) ≥ ψ(T, y) > e 2T -y-L , which is larger than 1 = sup u if y < 2T -L, while on the other hand u(t, x, y) = U (x, y, y -c(e2)t) which converges to 0 < inf ū as y → +∞, uniformly in x and locally uniformly in t. Let (x, ȳ) be the limit of (a converging subsequence) of ((xn, yn)) n∈N . The continuity of u and the lower semicontinuity of ū yield ū(kτ + T, ȳ) ≤ u(T, x, ȳ), whence in particular T > 0. Summing up, we have that min 0≤t≤T (x,y)∈R 2 ū(kτ + t, y) -u(t, x, y) = 0 = ū(kτ + T, ȳ) -u(T, x, ȳ). (7.4) Let j ∈ N be such that kτ + T ∈ (jτ, (j + 1)τ ]. Using the inequalities

we nd that ȳ > (2j -1)L.

We claim that f1(y, ū(kτ + t, y)) = f0(ū(kτ + t, y)) for t ≤ T and y > (2j -1)L. Clearly, the claim holds if (2j -1)L < y < 2jL, because f0 and f1 coincide there. Take t ≤ T and y ≥ 2jL. We see that

where the last equality follows from the denition of τ . In particular, ū(kτ + t, y) < S and therefore f1(y, ū(kτ + t, y)) = f0(ū(kτ + t, y)). This proves the claim. Thus, the function ψ being a strict supersolution of (7.3), as seen before, we deduce that ū is a (continuous) strict supersolution of (7.1) for t ∈ (jτ, kτ + T ], x ∈ R, y > (2j -1)L.

Recalling that (7.4) holds with ȳ > (2j -1)L, a contradiction follows from the parabolic strong maximum principle.

We have thereby shown that u(t, x, y) < ū(kτ + t, y) for all t ≥ 0, (x, y) ∈ R 2 . Now, the function ū satises, for j ∈ N, j ≥ k,

(recall that L > ln 4). From this and the fact that u(t, x, y) < ū(kτ + t, y) for t > 0, one easily infers that the speed of u satises c(e2) ≤ lim

Step 3: there exist L, M > 0 such that c(e1) > c(e2).

Take L > 8, so that the conclusions of the Steps 1-2 hold. Hence we can choose M large enough in such a way that c(e1) is larger than the upper bound 2L/τ provided by the Step 2. It follows that c(e1) > c(e2).

Proof of Proposition 1.6. Let f1 = f1(y, u) be the function provided by Proposition 7.1 and let c(e) be the speed of the unique (up to shifts in time) pulsating travelling front connecting 1 to 0 in the direction e ∈ S 1 . We know that c(e1) > c(e2) > 0. Fix c(e2) < c < c(e1). We claim that there exists a bistable reaction term f2 = f2(u) satisfying f 2 (0) = -1 and such that the homogeneous equation

admits a (unique up to shift) planar front with a speed equal to c. Such a reaction term can be obtained under the form

Appendix

Here we recall the order interval trichotomy of Dancer and Hess [START_REF] Dancer | Stability of xed points for order-preserving discrete-time dynamical systems[END_REF]; see also [START_REF] Matano | Existence of nontrivial unstable sets for equilibriums of strongly ordered-preserving systems[END_REF].

Theorem A.1 ([5]). Let p < p be two periodic steady states of (1.1). Then one of the following situations occurs:

(a) there is a periodic steady state p satisfying p < p < p , (b) there exists an entire solution u to (1.1) such that (u(k, •)) k∈Z is an increasing family of periodic functions satisfying

there exists an entire solution u to (1.1) such that (u(k, •)) k∈Z is a decreasing family of periodic functions satisfying

This trichotomy plays a crucial role in our proofs, as it allows us to look at multistable equations as juxtapositions of monostable problems. Owing to Theorem 1.3 quoted from Weinberger [START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF], we infer the existence of the minimal speeds of fronts above and below any unstable steady state q. In Assumption 1.3 we require that such speeds are strictly ordered. In the next proposition we show that a sucient condition guaranteering this hypothesis is that q is linearly unstable. We also point out for completeness that the order between the speeds is always true in the large sense.

Under either Assumption 1.1 or 1.2, and with the notation of Assumption 1.3, for any unstable periodic steady state q between 0 and p and any e ∈ S N -1 , there holds that

Moreover, if q is linearly unstable, then cq > 0 > c q .

Proof. We show the inequalities for cq, the ones for c q follow by considering the nonlinear term -f (x, -u) and the direction -e.

We recall that cq is the minimal speed of fronts in the direction e connecting pi 1 to q, where pi 1 is the smallest stable periodic steady state lying above q. Let λ0 denote the periodic principal eigenvalue of the linearized operator L0w := div(A(x)∇w) + ∂uf (x, q(x))w.

The instability of q implies that λ0 ≥ 0. We distinguish two cases.

Linearly unstable case: λ0 > 0.

Because the operator L0 is self-adjoint, it is well-known that λ0 can be approximated by the Dirichlet principal eigenvalue of L0 in a large ball (see, e.g., [START_REF] Berestycki | Analysis of the periodically fragmented environment model. I. Species persistence[END_REF]Lemma 3.6]). Namely, calling λ(r) the principal eigenvalue of L0 in Br with Dirichlet boundary condition, there holds that λ(r) → λ0 as r → +∞. Then we can nd r large enough so that λ(r) > 0. Let ϕ be the associated principal eigenfunction. The function ψ dened by ψ(t, x) := q(x) + ϕ(x)e 1 2 λ(r)t ,

Hence, by the C 1 regularity of u → f (x, u), there exists T ∈ R such that ψ is a subsolution of (1.1) for t ≤ T , x ∈ Br. Up to reducing T , we further have that ψ < pi 1 for all t ≤ T . Assume by way of contradiction that (1.1) admits a pulsating front U (x, x • e -ct) connecting pi 1 to q with a speed c ≤ 0. Let ξ ∈ Z N be such that U (ξ, ξ•e-cT ) < ψ(T, 0).

Observe that U (x, x•e-ct) is bounded from below away from q for t ≤ T and x ∈ Br(ξ), because c ≤ 0 and U (•, -∞) ≡ pi 1 > q. We can then nd T < T such that ∀x ∈ Br(ξ), U (x, x • e -cT ) > ψ(T , x -ξ).

Because ψ(t, x -ξ) is a subsolution of (1.1) for t < T and x ∈ Br(ξ), which is equal to q(x) for x ∈ ∂Br(ξ), the comparison principle eventually yields ∀ T ≤ t ≤ T, x ∈ Br(ξ), U (x, x • e -ct) > ψ(t, x -ξ), contradicting U (ξ, ξ • e -cT ) < ψ(T, 0). This shows that cq > 0 in this case.

Case λ0 = 0.

The denition of pi 1 , together with either Assumption 1.1 or 1.2, imply that the case (b) is the only possible one in Theorem A.1 with p = q and p = pi 1 . Let u be the corresponding entire solution. For σ ∈ R, let λσ and ϕσ denote the periodic principal eigenvalue and eigenfunction of the operator Lσw := div(A(x)∇w) + 2σeA(x)∇w + σ 2 eA(x)e + σdiv(A(x)e) + ∂uf (x, q(x)) w.

Fix ε > 0. We dene the following function:

For σ > 0, there exists δ > 0 depending on ε, σ such that q + δ < pi 1 and moreover, for

Then take k ∈ Z, also depending on ε, σ, in such a way that ∀t ≤ k, x ∈ [0, 1] N , u(t, x) ≤ q(x) + δ.

We deduce that, for t < k and x ∈ R N such that ψ(t, x) > q(x), the following holds:

∂tψdiv(A(x)∇ψ) ≤ f (x, ψ) -1 2 σε -λσ ϕσ(x)e σ(x•e+εt) .

Now, for r > 0, call as before λ(r) and ϕ the Dirichlet principal eigenvalue and eigenfunction of L0 in Br. Direct computation shows that for σ ∈ R, ϕ(x)e -σx•e is the Dirichlet principal eigenfunction of Lσ in Br, with eigenvalue λ(r). It follows that λ(r) < λσ, because otherwise ϕσ would contradict the properties of this principal eigenvalue. Because λ(r) → λ0 = 0 as r → +∞, we deduce that λσ ≥ λ0 = 0. Namely, σ → λσ attains its minimal value 0 at σ = 0 and thus, being regular (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]) it satises λσ ≤ Cσ 2 for some C > 0 and, say, |σ| ≤ 1 (this inequality can also be derived using the min-max formula of [START_REF] Nadin | The eect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF]Theorem 2.1]). As a consequence, taking σ = ε 2 we nd that, for ε smaller than some ε0, the function ψ is a subsolution of (1.1) for the values (t, x) such that t < k and ψ(t, x) > q(x).

Assume now by contradiction that there is a pulsating front U (x, x • e -ct) connecting pi 1 to q with a speed c < -ε and ε < ε0. Up to translation in time, it is not restrictive to assume that U (0, -ck) < u(k, 0). Let R ∈ R be such that U (x, z) > q + δ for x ∈ R N and z ≤ R. It follows that U (x, x•e-ct) ≥ ψ(t, x) for t ≤ k and x•e-ct ≤ R.

On the other hand, we see that ∀t < k, x • e -ct ≥ R, ψ(t, x) ≤ δ -(min ϕσ) e ε 2 (R+ct+εt) .

The right-hand side goes to -∞ as t → -∞ because c + ε < 0. We can then nd T < k such that U (x, x • e -ct) ≥ ψ(t, x) for all t ≤ T and x ∈ R N . Hence, because U > q, we can apply the comparison principle and infer that U (0, -ck) ≥ u(k, 0), which is a contradiction. We have shown that fronts cannot have a speed smaller than -ε, for ε suciently small, whence cq ≥ 0.