Weiwei Ding

François Hamel

Xiao-Qiang Zhao

Transition fronts for periodic bistable reaction-diffusion equations

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Transition fronts for periodic bistable reaction-diffusion equations

Introduction and main results

This paper is devoted to the study of existence and qualitative properties of generalized fronts of one-dimensional spatially periodic reaction-diffusion equations with bistable nonlinearities. It is a follow-up to the paper [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF] which dealt with the case of pulsating fronts. In [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF] and the present paper, we consider periodic reaction-diffusion equations of the type

u t = (a L (x)u x ) x + f L (x, u), t ∈ R, x ∈ R (1.1)
with L > 0, where u t stands for u t = ∂ t u(t, x) = ∂u/∂t(t, x), u x stands for u x = ∂ x u(t, x) = ∂u/∂x(t, x) and (a L (x)u x ) x stands for (a L (x)u x ) x = ∂ x (a L u x )(t, x) = ∂(a L ∂u/∂x)/∂x(t, x). The diffusion and reaction coefficients a L and f L are given by

a L (x) = a x L and f L (x, u) = f x L , u ,
where the function a : R → R is positive, of class C 1,α (R) (with 0 < α < 1) and 1-periodic, that is, a(x + 1) = a(x) for all x ∈ R. Throughout the paper, the function f : R × [0, 1] → R, (x, u) → f (x, u) is continuous, 1-periodic in x, of class C 0,α in x uniformly in u ∈ [0, 1], and of class C 1,1 in u uniformly in x ∈ R. One also assumes that 0 and 1 are uniformly (in x) stable zeroes of f (x, •), in the sense that there exist γ > 0 and δ ∈ (0, 1/2) such that

     f (x, 0) = f (x, 1) = 0 for all x ∈ R, f (x, u) ≤ -γu for all (x, u) ∈ R × [0, δ], f (x, u) ≥ γ(1 -u) for all (x, u) ∈ R × [1 -δ, 1]. (1.2)
Notice that this implies in particular that max ∂ u f (x, 0), ∂ u f (x, 1) ≤ -γ for all x ∈ R.

A particular case of such a function f satisfying (1.2) is the cubic nonlinearity

f (x, u) = u(1 -u)(u -θ x ), (1.3)
where x → θ x is a 1-periodic C 0,α (R) function ranging in (0, 1). Notice that in (1.3) the intermediate zero θ x of f (x, •) is not assumed to be constant. More generally speaking, under the assumption (1.2), the function f (x, •) may have several zeroes in the open interval (0, 1) and these zeroes may not be constant or of constant number. However, in one result of the paper, namely Theorem 1.7 below, we will assume that for every x ∈ R, the function f (x, •) has exactly one zero in the interval (0, 1). The bistable equation (1.1) arises naturally in modeling a variety of physical and biological phenomena, such as phase field models of solidification, signal propagation along bistable transmission lines, propagation of nerve pulses and population biology, see, e.g., [START_REF] Alikakos | Periodic traveling waves and locating oscillating patterns in multidimensional domains[END_REF][START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Bates | Traveling waves in a convolution model for phase transitions[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF][START_REF] Pauwelussen | Nerve impulse propagation in a branching nerve system: A simple model[END_REF][START_REF] Shen | Traveling waves in time almost periodic structures governed by bistable nonlinearities, I. Stability and uniqueness[END_REF][START_REF] Xin | Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media[END_REF][START_REF] Xin | Front propagation in heterogeneous media[END_REF][START_REF] Zhao | Dynamical Systems in Population Biology[END_REF]. In particular, in the context of population dynamics, the quantity u(t, x) represents the population density at location x and at time t, the coefficient a L (x) is the diffusion rate at location x and the reaction term f L (x, u) measures the growth rate of the population density u at location x. The dependency of a L (x) and f L (x, u) on x allows to model the effects of the features of the habitat on the population density. The periodicity condition on a L (x) and f L (x, u) with respect to x indicates a typical spatial heterogeneity of the habitat. The bistable assumption (1.2) means that the growth rate f L (x, u) is negative at low densities, which refers to a strong Allee effect [START_REF] Lewis | Allee dynamics and the spread of invading organisms[END_REF][START_REF] Roques | Modèles de Réaction-Diffusion pour l' Écologie Spatiale[END_REF][START_REF] Turchin | Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants[END_REF].

In the sequel, for mathematical purposes, the function f is extended in R × (R\[0, 1]) as follows: f (x, u) = ∂ u f (x, 0)u for (x, u) ∈ R × (-∞, 0) and f (x, u) = ∂ u f (x, 1)(u -1) for (x, u) ∈ R × (1, +∞). Thus, f is continuous in R × R, 1-periodic in x, min x∈R f (x, u) > 0 for all u < 0 and max x∈R f (x, u) < 0 for all u > 1, while f (x, u) and ∂ u f (x, u) are globally Lipschitzcontinuous in u uniformly in x ∈ R.

Transition fronts

The notion of transition fronts was introduced by Berestycki and Hamel [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Berestycki | Generalized transition waves and their properties[END_REF] to describe a general class of front-like solutions for reaction-diffusion equations in unstructured heterogeneous media. For one-dimensional equations such as (1.1), the notion of transition fronts can be presented precisely as follows (see also the notion of wave-like one-dimensional solutions defined by Shen [START_REF] Shen | Traveling waves in diffusive random media[END_REF]). Definition 1.1. For problem (1.1), a transition front connecting 0 and 1 is a time-global solution u : R × R → [0, 1] for which there exists a family (ξ t ) t∈R of real numbers such that u(t, x + ξ t ) → 1 as x → -∞, u(t, x + ξ t ) → 0 as x → +∞, uniformly in t ∈ R.

(1.4)

Clearly, from the strong maximum principle, any transition front u connecting 0 and 1 for (1.1) is such that 0 < u(t, x) < 1 for all (t, x) ∈ R 2 . We point out that the uniformity in t is essential in this definition. As a matter of fact, there are solutions satisfying the limits of (1.4) pointwise in t ∈ R, but not uniformly. Consider for instance the case of (1.1) when both a and f do not depend on x (whence θ x = θ ∈ (0, 1) is independent of x): this homogeneous equation admits some time-global solutions u such that 0 < u(t, x) < 1 in R 2 , u(t, -∞) = 1 and u(t, +∞) = 0 for every t ∈ R, while u(t, x) is close to θ on some unbounded intervals as t → -∞, see [START_REF] Morita | Entire solutions with merging fronts to reaction-diffusion equations[END_REF]. Thus, these solutions are not transition fronts connecting 0 and 1 in the sense of Definition 1.1.

For a given transition front u of problem (1.1), the real numbers ξ t reflect the positions of u as time runs. These real numbers ξ t are not uniquely defined since, for any bounded function t → ξ t , the family (ξ t +ξ t ) t∈R is also associated with u in the sense of (1.4). However, it follows straightforwardly from Definition 1.1 that the family (ξ t ) t∈R is unique up to an additive bounded function. Furthermore, the distance between ξ t and any level set of u(t, •) is uniformly bounded in t: indeed, for any real numbers b 1 and b 2 with 0 < b 1 ≤ b 2 < 1, there is a constant C = C(u, b 1 , b 2 ) ≥ 0 such that x ∈ R ; b 1 ≤ u(t, x) ≤ b 2 ⊂ ξ t -C, ξ t + C for every t ∈ R.

Thus, an equivalent definition to (1.4) is that a transition front connecting 0 and 1 is a time-global solution u : R × R → [0, 1] that converges to the steady states 0 (on the right) and 1 (on the left) far away from any of its level sets, uniformly in t.

An important notion attached to a transition front is the limiting average speed, if any, of the distance between the positions ξ t . Definition 1.2. We say that a transition front connecting 0 and 1 for (1.1) admits a global mean speed c ≥ 0 if

|ξ t -ξ s | |t -s| → c as |t -s| → +∞.
For a given transition front u, the global mean speed, if it exists, is uniquely determined and does not depend on the special choice of the positions (ξ t ) t∈R , since they are defined up to a bounded function (see also [START_REF] Berestycki | Generalized transition waves and their properties[END_REF]Theorem 1.7]). Thus, the notion of global mean speed is meaningful.

A simple example of a transition front connecting 0 and 1 for (1.1) is a pulsating front, that is, a global solution u : R × R → [0, 1] such that there exist a real number c L (the average speed) and a function φ : R × R → [0, 1] satisfying      u(t, x) = φ(x -c L t, x/L) for all (t, x) ∈ R × R, φ(ξ, y) is 1-periodic in y, φ(-∞, y) = 1, φ(+∞, y) = 0 uniformly in y ∈ R.

(1.5)

Clearly, any pulsating front with average speed c L ∈ R is a transition front with positions ξ t = c L t for all t ∈ R, whence |c L | is the global mean speed. If c L = 0, then the map (t, x) → (x -c L t, x/L) is a bijection from R × R to R × R and φ is uniquely determined by u. In this case, (1.5) is equivalent to the following definition u(t + L/c L , x) = u(t, x -L) for all (t, x) ∈ R 2 , u(t, -∞) = 1, u(t, +∞) = 0 locally in t ∈ R.

(1.6)

As a matter of fact, definition (1.6) was first given in [START_REF] Shigesada | Traveling periodic waves in heterogeneous environments[END_REF] to denote pulsating fronts with nonzero speed. On the other hand, a pulsating front with speed c L = 0 simply means a steady solution u(t, x) = φ 0 (x) of (1.1) such that φ 0 : R → [0, 1], φ 0 (-∞) = 1 and φ 0 (+∞) = 0. Throughout this paper, such steady solutions are called stationary fronts.

In this paper, we will first establish some qualitative properties of transition fronts connecting 0 and 1 for problem (1.1) and some uniform bounds on their rate of propagation. With the a priori existence of a pulsating front of the type (1.6), we will show that any transition front is equal to this pulsating front up to shift in time. We will finally prove the existence of new types of transition fronts, which are not pulsating fronts satisfying (1.5). In particular, that will show the broadness of the notion of transition fronts and the necessity to introduce and use it even in periodic media.

Some known results

Before going further on, let us recall some known existence results of transition fronts for problem (1.1). We first mention the important case where the function a is equal to a positive constant d, and the function f does not depend on x. In this case, equation (1.1) can be reduced to the homogeneous equation

u t = du xx + f (u). (1.7)
Under the additional assumption that the function f

: [0, 1] → R is such that f (0) = f (θ) = f (1) = 0 for some θ ∈ (0, 1), f > 0 on (0, θ), f > 0 on (θ, 1), f ′ (0) < 0, f ′ (1) < 0, (1.8)
it is well known [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that there exists a unique speed c ∈ R and a unique (up to shifts in x) front

u(t, x) = φ(x -ct) such that 0 < φ < 1 in R, φ(-∞) = 1 and φ(+∞) = 0.
Moreover, φ is decreasing in x and the speed c has the same sign as the integral 1 0 f . Such a standard traveling front can be viewed as a pulsating front with average speed c, and a transition front with global mean speed |c| as well. Furthermore, it has been shown recently in [START_REF] Hamel | Bistable transition fronts in R N[END_REF] that transition fronts connecting 0 and 1 for problem (1.7) are nothing else but the standard traveling fronts φ(x -ct). We point out that this uniqueness result for the one-dimensional equation (1.7) is not valid for multidimensional equations. As a matter of fact, for bistable homogeneous equations in R N with N ≥ 2, the class of transition fronts is larger than that of usual planar traveling fronts: there exist non-planar traveling fronts which are invariant in a moving frame with constant speed [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF][START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF][START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF][START_REF] Ninomiya | Global stability of traveling curved fronts in the Allen-Cahn equations[END_REF][START_REF] Taniguchi | Traveling fronts of pyramidal shapes in the Allen-Cahn equation[END_REF][START_REF] Taniguchi | The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations[END_REF], as well as some transition fronts which are not invariant in any moving frame as time runs [START_REF] Hamel | Bistable transition fronts in R N[END_REF]. We also mention [START_REF] Berestycki | Bistable traveling waves passing an obstacle[END_REF] for the existence of bistable transition fronts for multi-dimensional homogeneous equations in exterior domains.

For time heterogeneous and space homogeneous bistable equations, the existence and qualitative properties of standard traveling, pulsating or transition fronts have been investigated by Alikakos, Bates and Chen [START_REF] Alikakos | Periodic traveling waves and locating oscillating patterns in multidimensional domains[END_REF], Fang and Zhao [START_REF] Fang | Bistable traveling waves for monotone semiflows with applications[END_REF], and Shen [START_REF] Shen | Traveling waves in time almost periodic structures governed by bistable nonlinearities, I. Stability and uniqueness[END_REF][START_REF] Shen | Traveling waves in time almost periodic structures governed by bistable nonlinearities, II. Existence[END_REF][START_REF] Shen | Traveling waves in diffusive random media[END_REF][START_REF] Shen | Traveling waves in time dependent bistable equations[END_REF]. But, for space heterogeneous bistable equations, there is not so much work on transition fronts. In periodic media, the existence of pulsating fronts for (1.1), under the assumption (1.2), have just been discussed in our previous paper [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]. More precisely, under various additional assumptions on the reaction terms f (x, u) and by using different types of arguments, we proved several existence results of pulsating fronts with nonzero or zero speeds when the spatial period L is small or large (we will come back to the precise assumptions in the comments following Theorem 1.5 below). We also established some properties of the set of periods for which there exist pulsating fronts with nonzero speeds. However, in a given periodic medium, finding a general necessary and sufficient condition for the existence of pulsating fronts with nonzero speeds is still unclear in general. We point out that the existence result is known to hold for all L > 0 in some particular cases where f = f (u) does not depend on x and a L is close to a constant in some norms, see [START_REF] Fang | Bistable traveling waves for monotone semiflows with applications[END_REF][START_REF] Xin | Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity[END_REF][START_REF] Xin | Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media[END_REF][START_REF] Xin | Front propagation in heterogeneous media[END_REF], or under various more abstract conditions, see [START_REF] Ducrot | Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations[END_REF][START_REF] Fang | Bistable traveling waves for monotone semiflows with applications[END_REF]. Apart from the existence results of pulsating fronts, no other existence results of transition fronts have been known for the spatially periodic bistable equation (1.1).

Therefore, it is of particular interest to investigate whether transition fronts of (1.1) are all identically equal to pulsating fronts, or whether there exist transition fronts which are not pulsating fronts. Under the bistability assumption (1.2), the change of sign of f (x, •) in (0, 1) and the fact that the roots of f (x, •) in (0, 1) depend on x in general may yield the existence of multiple ordered steady states, and thus makes these questions very difficult.

Similar problems have been addressed recently for other types of nonlinearities f , such as ignition type, and Fisher-KPP nonlinearities. Consider for instance an ignition nonlinearity of the type f (x, u) = g(x)f 0 (u) for which g is continuous, 1-periodic and positive in R, and f 0 ≥ 0 in [0, 1], f 0 = 0 in [0, θ] ∪ {1} for some θ ∈ (0, 1) and f 0 is nonincreasing in [1 -δ, 1] for some δ > 0. It is known that for each L > 0, (1.1) has a unique (up to shift in t) transition front connecting 0 and 1 in the sense of Definition 1.1, see [START_REF] Mellet | Ryzhik Stability of generalized transition fronts[END_REF][START_REF] Mellet | Existence of Generalized Transition Fronts in reaction-diffusion equations[END_REF][START_REF] Nolen | Traveling waves in a one-dimensional random medium[END_REF]. Furthermore, this front is actually identically equal to a pulsating front (up to shift in t) with a positive speed, since pulsating fronts are known to exist (and to be unique), see [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]. Similar results have been obtained for multidimensional ignition-type equations, see [START_REF] Zlatoš | Generalized travelling waves in disordered media: Existence, uniqueness, and stability[END_REF]. On the other hand, for a Fisher-KPP nonlinearity f with f (x, 0) = f (x, 1) = 0 and 0 < f (x, u) ≤ ∂ u f (x, 0)u for (x, u) ∈ R × (0, 1), and for each L > 0, there exists a positive minimal speed c * L in the following sense: (1.1) admits a pulsating front with a speed c if c ≥ c * L , while it does not admit any transition front with a global mean speed c if c < c * L , see [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Nadin | Transition waves for Fisher-KPP equations with general time-heterogeneous and spaceperiodic coefficients[END_REF][START_REF] Rossi | Transition waves for a class of space-time dependent monostable equations[END_REF][START_REF] Shen | Existence of generalized traveling waves in time recurrent and space periodic monostable equations[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF]. However, there exist other types of transition fronts which are not standard pulsating fronts (even in the homogeneous media, Fisher-KPP equations with concaves reaction f admit transition fronts which are not of the type φ(x -ct), see [START_REF] Hamel | Travelling waves and entire solutions of the Fisher-KPP equation in R N[END_REF][START_REF] Hamel | Transition fronts for the Fisher-KPP equation[END_REF] and additional comments after Theorem 1.4 below).

General properties of transition fronts

In this subsection, we present some qualitative properties shared by the solutions of (1.1) in the class of transition fronts, thus showing the robustness of the notion of transition fronts. Proposition 1.3. Let u be a transition front connecting 0 and 1 for problem (1.1). Then

∀ C ≥ 0, 0 < inf t∈R, |x|≤C u(t, x + ξ t ) ≤ sup t∈R, |x|≤C u(t, x + ξ t ) < 1.
(1.9)

Furthermore, there is k 0 ∈ N such that u is kL-decreasing in x for all integers k ≥ k 0 , that is, u(t, x) > u(t, x + kL) for all t ∈ R, x ∈ R and k ≥ k 0 .
The first property in Proposition 1.3, namely (1.9), shows that in the neighborhood of the positions ξ t , any transition front is uniformly bounded away from 0 and 1. We refer to [START_REF] Hamel | Transition fronts for the Fisher-KPP equation[END_REF][START_REF] Hamel | Admissible speeds of transition fronts for time-dependent KPP equations[END_REF] for the same type of result for homogeneous or time-dependent monostable equations. Indeed, (1.2) is not needed in the proof of (1.9). Property (1.9) is another way of seeing immediately that the positions (ξ t ) t∈R of a given transition front u are unique up to an additive bounded function: if ( ξ t ) t∈R is another family associated with u in the sense of (1.4), then it follows from (1.4) applied to (ξ t ) t∈R and from (1.9) applied to ( ξ t ) t∈R that sup t∈R |ξ t -ξ t | < +∞.

The second property reveals the effect of the periodicity of the coefficients a L and f L on the solutions of (1.1) in the class of transition fronts. In particular, if the transition front u is a pulsating front of type (1.6), then it is L-decreasing in x (that is, one can take k 0 = 1) since c L × u is increasing in t (see [START_REF] Berestycki | Generalized transition waves and their properties[END_REF][START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]). In addition, all transition fronts are then equal to this pulsating front up to shift in time (see Theorem 1.5 below), and thus they are all L-decreasing in x. Nevertheless, whether all transition fronts for equation (1.1) be L-decreasing in x is not clear in general.

Next, we establish a uniform bound for the propagation rates of all transition fronts of (1.1).

Theorem 1.4. There is a constant C depending only on the functions f and a such that for any L > 0 and any transition front u connecting 0 and 1 for equation (1.1) and associated with (ξ t ) t∈R , there holds lim sup

|t-s|→+∞ |ξ t -ξ s | |t -s| ≤ C. (1.10)
Several comments are in order on Theorem 1.4. Firstly, we point out that the bound C in Theorem 1.4 is uniform with respect to all transition fronts for equation (1.1). This uniform boundedness is in sharp contrast with the case of Fisher-KPP nonlinearities, for which there is some minimal speed c * L > 0 such that for any c ≥ c * L , equation (1.1) has a pulsating front with average speed c, see [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration models in a periodic habitat[END_REF].

We also note that the bound C in Theorem 1.4 is independent of the spatial period L. In particular, if equation (1.1) for a given L > 0 has a transition front with a global mean speed γ L , then γ L has a uniform bound independent of L. This global boundedness is more general than the local boundedness of the speeds of pulsating fronts we have established in [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]Lemma 3.4].

Finally, we remark that although Theorem 1.4 provides a uniform bound for the global mean speeds of transition fronts, the question of the existence of a global mean speed for a given transition front is still open. However, there is no example showing that equation (1.1), under the assumption (1.2), has a transition front without a global mean speed. On the other hand, in the case of Fisher-KPP nonlinearities, there are examples of transition fronts which do not admit any global mean speed. Consider for instance the homogeneous equation u t = u xx + g(u) where g : [0, 1] → R is of class C 2 and concave, g(0) = g(1) = 0 and g(u) > 0 for all u ∈ (0, 1). This equation admits some transition fronts with positions (ξ t ) t∈R satisfying ξ t /t → c -as t → -∞ and ξ t /t → c + as t → +∞, where c -and c + are any two given real numbers such that 2 g ′ (0) ≤ c -< c + < +∞, see [START_REF] Hamel | Travelling waves and entire solutions of the Fisher-KPP equation in R N[END_REF][START_REF] Hamel | Transition fronts for the Fisher-KPP equation[END_REF] (see also [START_REF] Hamel | Admissible speeds of transition fronts for time-dependent KPP equations[END_REF][START_REF] Zlatoš | Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations[END_REF] for further results in time or space dependent media).

Sufficient conditions for the uniqueness of transition fronts

For the homogeneous equation (1.7) with (1.8), it was proved in [START_REF] Hamel | Bistable transition fronts in R N[END_REF] that the transition fronts are unique up to shifts and equal to the standard traveling front φ(x -ct), whether c be zero or not. For the periodic equation (1.1), we showed in [16, Theorem 1.1] that for any given period L > 0, the speed of pulsating fronts for (1.1) is unique and that, if the speed is not zero, pulsating fronts are unique up to shift in time. For (1.1), it is therefore natural to wonder whether transition fronts can be reduced to stationary or non-stationary pulsating fronts. One of the striking results of this paper is to show that the answer to this question can be yes (see Theorem 1.5 below) or no (see Theorem 1.7).

In this subsection, we provide some sufficient conditions for the uniqueness of solutions in the (a priori larger) class of transition fronts. These conditions also lead to the existence and uniqueness of the global mean speeds. We first give a uniqueness result of Liouville-type under the a priori existence of a pulsating front with nonzero speed. Theorem 1.5. If (1.1) admits a pulsating front with speed c L = 0, then any transition front connecting 0 and 1 is equal to this pulsating front up to shift in time. In particular, any transition front has a global mean speed, equal to |c L |.

As a consequence of Theorem 1.5, we get the uniqueness of transition fronts of equation (1.1) when the spatial period L is small or large, under various assumptions on f . More precisely, if, in addition to (1.2), one assumes that for every x ∈ R, there exists θ x ∈ (0, 1) such that f (x, θ x ) = 0, f (x, •) < 0 on (0, θ x ) and f (x, •) > 0 on (θ x , 1), (1.11) and that there exists θ ∈ (0, 1) such that f < 0 on (0, θ), f > 0 on (θ, 1), f ′ (θ) > 0, and

1 0 f (u) du = 0, (1.12)
where

f (u) = 1 0 f (x, u) dx for u ∈ [0, 1], then from [16, Theorems 1.2, 1.4],
there is L * > 0 such that for every 0 < L < L * , equation (1.1) admits a pulsating front with nonzero speed, whence any transition front is equal to this pulsating front up to shift in time. In a similar way, it follows from [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]Theorem 1.5] that if, in addition to (1.2) and (1.11), the function f is such that 1 0 f (x, u) du > 0 (resp. < 0) and ∂f /∂u(x, θ x ) > 0 for all x ∈ R, then there is L * > 0 such that, for every L > L * , any transition front of equation (1.1) is equal to a pulsating front with positive (resp. negative) speed, and is unique up to shift in time.

In the following theorem, we give a characterization of the pulsating fronts in the class of transition fronts, without assuming a priori the existence of a pulsating front with nonzero speed. then there exists a pulsating front with speed c L = 0, and any transition front is then equal to this pulsating front up to shift in time.

Clearly, Theorem 1.6 is more general than Theorem 1.5. Under their assumptions, Theorems 1.5 and 1.6 both imply in particular the existence and uniqueness of the global mean speed in the class of transition front. As a matter of fact, if a transition front u admits a positive global mean speed c > 0 (as does a pulsating front with a nonzero speed), then

lim |t-s|→+∞ |ξ t -ξ s | = +∞.
(1.14)

Theorem 1.6 says that the existence of a transition front u satisfying the rough condition (1.14) (or even a weaker one obtained by passing to the limit along a sequence of time) is actually sufficient for the existence of a pulsating front with nonzero speed. Under this condition, any transition front u associated with the positions ( ξ t ) t∈R is then equal to this pulsating front up to shift in time and then satisfies | ξ t -ξ s | ∼ c|t -s| as |t -s| → +∞ for some c > 0. In particular, any transition front u which admits a global mean speed c ≥ 0 is either a pulsating front with nonzero speed or a transition front with null global mean speed and such that lim inf |t-s|→+∞ |ξ t -ξ s | < +∞. We point out that transition fronts with null global mean speed may not be stationary fronts in general, and they may be not unique up to shift in time (see Theorem 1.7 below).

Transition fronts which are not pulsating fronts

In periodic media, pulsating fronts are the natural extension of the standard traveling fronts φ(x-ct) in homogeneous media. But the notion of pulsating fronts is not broad enough to describe all the transition fronts connecting the steady states 0 and 1. In this subsection, we show indeed that the notion of transition fronts is truly needed, even in periodic media.

To do so, we investigate wave-blocking phenomena for the bistable equation (1.1). Wave-blocking refers to the fact that stationary fronts block the propagation. Some results on wave-blocking phenomena have been obtained in the specific case where the diffusions a L are not too close to their average and the reactions f L are x-independent with f L (x, u) = f (u) = u(1 -u)(u -θ) for some θ ≃ 1/2, see [START_REF] Xin | Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media[END_REF][START_REF] Xin | Quenching and propagation of bistable reaction-diffusion fronts in multidimensional periodic media[END_REF] (see also [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF] for some blocking phenomena in the case where f (x, u) is compared to a homogeneous bistable function g with nonzero average). Moreover, wave-blocking phenomena have been extensively investigated for various other bistable models, see, e.g., [START_REF] Afraimovich | Stability of stationary sates and topological spatial chaos in multidimensional lattice dynamical systems[END_REF][START_REF] Bell | Threshold behavior and propagation for nonlinear differential-difference systems motivated by modelling myelineated axons[END_REF][START_REF] Cahn | Travelling wave solutions for systems of ODE's on a twodimensional spatial lattice[END_REF][START_REF] Chow | Dynamics in a discrete Nagumo equation: spatial topological chaos[END_REF][START_REF] Keener | Propagation and its failure in coupled systems of discrete excitable cells[END_REF][START_REF] Mallet-Paret | The global structure of traveling waves in spatially discrete dynamical systems[END_REF] for spatially discrete models, [START_REF] Aronson | Wave propagation and blocking in inhomogeneous media[END_REF][START_REF] Lewis | Wave-block in excitable media due to regions of depressed excitability[END_REF][START_REF] Nadin | Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations[END_REF][START_REF] Pauwelussen | Nerve impulse propagation in a branching nerve system: A simple model[END_REF] for some non-periodic equations, and [START_REF] Berestycki | Front blocking and propagation in cylinders with varying cross section[END_REF][START_REF] Chapuisat | Existence and non-existence of progressive wave solutions for a bistable reactiondiffusion equation in an infinite cylinder whose diameter is suddenly increased[END_REF] for some higher-dimensional equations. In the present paper, for constant diffusions a L and some x-dependent reactions f L , we prove that wave-blocking occurs when the period L is large, and furthermore we obtain the existence of transition fronts connecting 0 and 1 between two ordered stationary fronts. We state the main result as follows.

Theorem 1.7. Assume, in addition to (1.2), that f satisfies (1.11), the functions a, ∂ u f (•, 0) and ∂ u f (•, 1) are constants, and

min x∈R 1 0 f (x, u)du < 0 and max x∈R 1 0 f (x, u)du > 0.
(1.15)

0 1 u u v v v(t,x) u(t,x) Figure 1: Stationary fronts 0 < u -(x) < u + (x) ≤ v -(x) < v + (x) < 1 and transition fronts u and v such that u -(x) < u(t, x) < u + (x) and v -(x) < v(t, x) < v + (x).
Then there is L * > 0 such that for any L > L * , equation (1.1) with the period L admits both some stationary fronts u ± and v

± satisfying 0 < u -(x) < u + (x) ≤ v -(x) < v + (x) < 1 for all x ∈ R
, as well as some transition fronts u and v connecting 0 and 1 and such that

u(-∞, x) = u + (x), u(+∞, x) = u -(x), v(-∞, x) = v -(x), v(-∞, x) = v + (x), uniformly in x ∈ R.
Furthermore, u(t, x) is decreasing in t, while v(t, x) is increasing in t. Lastly, for any transition front u associated with positions

(ξ t ) t∈R , there is M ≥ 0 such that |ξ t -ξ s | ≤ M for all (t, s) ∈ R 2 .
In particular, all transition fronts have zero global mean speed and there are transition fronts which are not pulsating fronts.

Let us first explain the phenomenon behind Theorem 1.7. On the one hand, when the period L is large, thanks to (1.15), there are large intervals in, say, the positive direction (in x) where the nonlinearity f L (x, •) is locally close to a homogeneous bistable nonlinearity with a negative integral over [0, 1]. Roughly speaking, this means that around such large intervals, the speed of any solution of the associated Cauchy problem with a front-like initial condition is (strictly) negative, at least on some interval of time. In other words, any solution is blocked by a supersolution moving leftwards and located far on the right. Similarly, any solution is blocked by a subsolution moving rightwards and located far on the left. The supersolution and the subsolution are ordered and move towards each other as time runs. Therefore, some ordered stationary fronts exist. On the other hand, the supersolution (resp. subsolution) only blocks the convergence to 1 (resp. 0) as t → +∞ of the solution, but not the convergence to some stationary fronts. Thus, there exist time-monotone solutions which connect two ordered stationary fronts as t → ±∞ (see Figure 1). Such solutions converge to the stable steady states 0 and 1 as x → ±∞ uniformly in t and therefore they are non-stationary transition fronts with null global mean speed. Furthermore, notice that, since (1.1) is L-periodic in x, Theorem 1.7 actually provides the existence of two sequences of stationary fronts (u

± n ) n∈Z = (u ± (nL + •)) n∈Z and (v ± n ) n∈Z = (v ± (nL + •)) n∈Z and two sequences of time-monotone non-stationary transitions fronts (u n ) n∈Z = (u(•, nL + •)) n∈Z and (v n ) n∈Z = (v(•, nL + •)) n∈Z converging to the stationary fronts u ±
n and v ± n as t → ±∞. Theorem 1.7 actually shows the broadness of Definition 1.1, since the class of transition fronts includes time-global solutions which connect 0 and 1 in space and connect two strictly ordered stationary fronts in time. Notice that this type of solutions cannot be described by non-stationary pulsating fronts, since any such pulsating front connects 0 and 1 both in space and in time. It is also worth pointing out that in the homogeneous case where a and f do not depend on x, the transition fronts are noting else but the standard traveling fronts, see [START_REF] Hamel | Bistable transition fronts in R N[END_REF]. As far as we know, Theorem 1.7 provides the first example of transition fronts in periodic media, which are neither stationary nor non-stationary pulsating fronts. It shows in particular the larger complexity of the dynamics in periodic media than in homogeneous media.

Theorem 1.7 also provides some examples of transition fronts which are not unique up to shift in time. Indeed, under the assumptions of Theorem 1.7, for any L > L * , some transition fronts of (1.1) are increasing in time, others are decreasing and other are stationary. But whether all the transition fronts are monotone in time is still unclear. Moreover, some other natural questions arise: for a given L > L * , does any transition front converge to some stationary fronts as t → ±∞ ? Under the assumptions in Theorem 1.7 and the assumption (1.12), is it true that L * = L * , if L * denotes the smallest such real number for which the conclusion of Theorem 1.7 holds and L * denotes the largest real number for which equation (1.1) admits a pulsating front with nonzero speed ?

Generally speaking, for diffusion and reaction coefficients a(x) and f (x, u) satisfying (1.2), the questions of the existence of front-like solutions connecting 0 and 1 and of the existence of propagation speeds are still open, even under the additional assumption (1.11). Based on the work in this present paper and the companion paper [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF], one can conclude that the spatial period L plays an important role in answering these questions. Namely, under various assumptions, there are transition fronts propagating with nonzero speeds (pulsating fronts with nonzero speeds) when L is small, while there are both stationary and non-stationary transition fronts propagating with zero speed when L is large. However, so far there is no explicit condition to guarantee the existence of transition fronts in general, and no example is known to show their non-existence.

Outline of the paper. Section 2 is devoted to the proof of the general properties of transition fronts, that is, Proposition 1.3 and Theorem 1.4. In Section 3, we prove Theorems 1.5 and 1.6 on the uniqueness of transition fronts. Lastly, Section 4 is devoted to the proof of Theorem 1.7, that is, the existence of stationary fronts and non-stationary transition fronts which are not standard pulsating fronts.

General properties of transition fronts

This section is devoted to the proofs of Proposition 1.3 and Theorem 1.4 on the kL-spatial monotonicity of the transition fronts and the boundedness of the propagation rates.

Spatial monotonicity: proof of Proposition 1.3

We first do the proof of Proposition 1.3. The first statement will follow easily from Definition 1.1, that is, u converges to the steady states 0 or 1 far away from the positions ξ t uniformly in t. The proof of the second statement of Proposition 1.3 is more involved. We will slide u with respect to the x-variable on the set LZ. Before doing so, we first present a result on the uniform boundedness of the local oscillations of (ξ t ) t∈R .

Lemma 2.1. For any transition front u connecting 0 and 1 for problem (1.1), there holds

∀ τ > 0, sup (t,s)∈R 2 , |t-s|≤τ |ξ t -ξ s | < +∞. (2.1)
Proof. The proof is an immediate application of [25, Proposition 4.1], and we omit the details.

This property holds for more general heterogeneous one-dimensional equations with other types of nonlinearities, and it has its own interest. As an application of Lemma 2.1, by applying recursively (2.1) with τ = 1, one infers that lim sup t→±∞ |ξ t /t| < +∞. Namely, the propagation speeds of any transition front are asymptotically bounded as t → ±∞.

Proof of Proposition 1.3. We only show the first inequality in (1.9), since the last one follows similar lines, and the second one is obvious. Assume, by contradiction, that there exists a se- ), and set

quence (t n , x n ) n∈N in R 2 such that (x n -ξ tn ) n∈R is bounded and u(t n , x n ) → 0 as n → +∞. Write x n = x ′ n + x ′′ n with x ′ n ∈ LZ and x ′′ n ∈ [0, L
u n (t, x) = u(t + t n , x + x ′ n ) for (t, x) ∈ R 2 and n ∈ N.
Since the functions a L and f L are independent of t and L-periodic in x, each function u n obeys equation (1.1). Up to extraction of some subsequence, one can assume that

x ′′ n → x ∞ ∈ [0, L] as n → +∞ and that, from standard parabolic estimates, u n (t, x) → u ∞ (t, x) as n → +∞ locally uniformly in R 2 , where 0 ≤ u ∞ ≤ 1 solves (1.1). Furthermore, since u(t n , x n ) → 0 as n → +∞, one has u ∞ (0, x ∞ ) = lim n→+∞ u n (0, x ′′ n ) = lim n→+∞ u(t n , x n ) = 0.
It then follows from the strong maximum principle that u ∞ ≡ 0. On the other hand, from Definition 1.1, there exists M > 0 such that u(t, x) ≥ 1/2 for all x -ξ t ≤ -M and t ∈ R. One then infers that

u n (0, x) ≥ 1 2 for all x + x n -x ′′ n -ξ tn ≤ -M and t ∈ R.
Remember that the sequences (x n -ξ tn ) n∈R and (x ′′ n ) n∈N are bounded. Thus, there exists M 1 ∈ R such that u ∞ (0, M 1 ) ≥ 1/2, which is impossible. Hence, the proof of the first inequality in (1.9) is complete.

Let us now turn to the proof of the second assertion of Proposition 1.3. From Definition 1.1, there is B > 0 such that

∀ (t, x) ∈ R 2 , x -ξ t ≥ B =⇒ 0 < u(t, x) ≤ δ, x -ξ t ≤ -B =⇒ 1 -δ ≤ u(t, x) < 1, (2.2)
where δ

∈ (0, 1/2) is the constant in (1.2). Since ∂ u f is continuous in R × [0, 1] and periodic in x (∂ u f is thus uniformly continuous in R × [0, 1]
), one can also assume without loss of generality, even if it means decreasing δ > 0, that, for every x ∈ R, f (x, •) is decreasing in [0, δ] and [1 -δ, 1], and then in (-∞, δ] and [1 -δ, +∞) owing to the definition of the extension of f on R × (R\[0, 1]). Choose k 0 ∈ N large enough such that k 0 L ≥ 2B and fix any k ∈ N with k ≥ k 0 . One then sees from the second inequality of (2.2) that

v(t, x) := u(t, x -kL) ≥ 1 -δ ≥ δ for all x -ξ t ≤ B. (2.3)
This together with the first inequality of (2.2) yields

v(t, x) ≥ u(t, x) when x -ξ t = B. (2.4)
Since the function u is bounded, the nonnegative real number

ε * := inf ε > 0 ; v(t, x) ≥ u(t, x) -ε for all (t, x) ∈ R 2 is well-defined. It is obvious that u(t, x -kL) = v(t, x) ≥ u(t, x) -ε * for all (t, x) ∈ R 2 .
(2.5)

In order to prove that u(t, x) is kL-decreasing in x, it is sufficient to show ε * = 0. Indeed, if ε * = 0, then u(t, x-kL) ≥ u(t, x) for all (t, x) ∈ R 2 and since both functons u(t, x-kL) and u(t, x) satisfy the same equation (1.1) (because a L and f L are L-periodic in x), the strong maximum principle implies that either u(t, x -kL) > u(t, x) for all (t,

x) ∈ R 2 , or u(t, x -kL) = u(t, x) for all (t, x) ∈ R 2 . The latter is impossible since k ≥ k 0 > 0 and u(t, -∞) = 1 > 0 = u(t, +∞) for all t ∈ R.
Assume, by contradiction, that ε * > 0. Then there exist a sequence (ε n ) n∈N of positive real numbers and a sequence (

t n , x n ) in R 2 such that ε n → ε * as n → +∞ and v(t n , x n ) < u(t n , x n ) -ε n for all n ∈ N. (2.6)
We first claim that the sequence (x n -ξ tn ) n∈N is bounded. Otherwise, up to extraction of some subsequence, it would converge to +∞ or -∞ as n → +∞. If

x n -ξ tn → +∞ as n → +∞, then, from Definition 1.1, one has u(t n , x n ) → 0 and v(t n , x n ) = u(t n , x n -kL) → 0 as n → +∞. Thus, v(t n , x n )-u(t n , x n ) → 0 as n → +∞, which contradicts (2.6
) and the positivity of ε * . Similarly, the case lim n→+∞ (x n -ξ tn ) = -∞ leads to a contradiction. Therefore, the sequence (x n -ξ tn ) n∈N is bounded.

Next, we set

w(t, x) := v(t, x) -(u(t, x) -ε * ) = u(t, x -kL) -u(t, x) + ε * for (t, x) ∈ R 2 .
Clearly, w ≥ 0 in R 2 from (2.5). Define now E 1 := (t, x) ∈ R 2 ; x -ξ t ≥ B and E 2 := (t, x) ∈ R 2 ; x -ξ t ≤ B . As a consequence of (2.2) and (2.3), one has u(t, x) -ε * < u(t, x) ≤ δ for all (t, x) ∈ E 1 , and v(t, x)

+ ε * > v(t, x) ≥ 1 -δ for all (t, x) ∈ E 2 . Since f L (x, •) is decreasing in (-∞, δ], one infers that (u -ε * ) t = (a L (x)(u -ε * ) x ) x + f L (x, u) ≤ (a L (x)(u -ε * ) x ) x + f L (x, u -ε * ) for all (t, x) ∈ E 1 .
Namely, the function u -ε * is a subsolution of (1.1) in the set E 1 . Since v solves (1.1), it follows that

w t ≥ (a L (x)w x ) x + f L (x, v) -f L (x, u -ε * ) v -(u -ε * ) w in E 1 ,
where the quotient is defined as

∂ u f L (x, v(t, x)) for any point (t, x) ∈ E 1 such that v(t, x) = u(t, x)-ε * .
Similarly, the function v + ε * is a supersolution of (1.1) in the set E 2 , and then w satisfies

w t ≥ (a L (x)w x ) x + f L (x, v + ε * ) -f L (x, u) (v + ε * ) -u w in E 2 .
Finally, since ∂ u f (x, u) is bounded in R 2 , the function w satisfies an inequation of the type

w t ≥ (a L (x)w x ) x + b(t, x)w for all (t, x) ∈ R 2
for some bounded function b : R 2 → R. Furthermore, from standard parabolic estimates, the function w = v -u + ε * has bounded first-order derivatives. Since the sequence (x n -ξ tn ) n∈N is bounded, it follows from Lemma 2.1 that, for any τ > 0, the sequence (x n -ξ tn-τ -B) n∈N is bounded. Since w is nonnegative in R 2 and w(t n , x n ) → 0 as n → +∞ from (2.5) and (2.6), one then concludes from Krylov-Safonov-Harnack type inequalities that, for any τ > 0, w(t n -τ, ξ tn-τ + B) → 0 as n → +∞, which contradicts (2.4) and the positivity of ε * . Hence, ε * = 0 and the proof of Proposition 1.3 is thereby complete.

2.2 Global boundedness of the propagation rates: proof of Theorem 1.4

This section is devoted to the proof of Theorem 1.4 on global bounds for the propagation rates of any transition front connecting 0 and 1 for (1.1). The general idea can be summarized as follows. From the first part of Proposition 1.3, any transition front u ranges in (ε 0 , 1-ε 0 ) on the set (t, ξ t ); t ∈ R , for some ε 0 ∈ (0, 1/2). We will show that there exists a constant C > 0, independent of u and of L, such that for any fixed t 0 ∈ R, u(t, x) is less than ε 0 when t -t 0 > > 1 and x -ξ t 0 > > C(t -t 0 ), and is larger than 1 -ε 0 when t -t 0 > > 1 and x -ξ t 0 < < -C(t -t 0 ). This guarantees that, up to some bounded shifts, the positions ξ t stay in the expanding interval [ξ t 0 -C(t-t 0 ), ξ t 0 +C(t-t 0 )] as t-t 0 is large. In other words, C gives an upper bound for the propagation speed of the positions ξ t , while -C provides a lower bound. To get the constant C, we will construct, for every L > 0, a supersolution (resp. subsolution) for (1.1), which approaches the limiting state 0 (resp. 1) exponentially fast as t → +∞ and x → +∞ (resp. x → -∞), the exponential rates being independent of L > 0. Before doing so, let us first recall some properties of the principal eigenvalue of some linear second-order differential operators. For every L > 0 and µ ∈ R, let T L,µ be the linear operator defined on

C 2 L := ψ ∈ C 2 (R) ; ψ(x + L) = ψ(x) for all x ∈ R by T L,µ [ψ] = (a L ψ ′ ) ′ + 2µa L ψ ′ + (µ 2 a L + µa ′ L )ψ.
The Krein-Rutman theory provides the existence and uniqueness of the principal eigenvalue λ(L, µ) of T L,µ , associated with a (unique up to multiplication) positive eigenfunction ψ L,µ ∈ C 2 L . One can normalize ψ L,µ in such a way that ψ L,µ = 1 with the norm • := • L ∞ (R) . Furthermore, for any fixed µ ∈ R, it is known from [START_REF] Nadin | The effect of Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF] that λ(L, µ) is nondecreasing in L > 0. On the other hand, integrating both sides of

T L,µ [ψ L,µ ] = λ(L, µ)ψ L,µ over [0, L] yields λ(L, µ) ≤ a µ 2 + a ′ |µ|/L for all L > 0 and µ ∈ R. As a consequence, λ(L, µ) ≤ lim L ′ →+∞ λ(L ′ , µ) ≤ a µ 2 for all L > 0 and µ ∈ R.
(2.7)

Next we construct a supersolution for equation (1.1).

Lemma 2.2.

There is a constant C > 0, depending only on a and f , such that for every L > 0, the function u defined by

u(t, x) = min e -(x-Ct) ψ L,-1 (x) + δ 2 e -γt , 1 for t ≥ 0 and x ∈ R, (2.8)
is a supersolution of equation (1.1), where ψ L,-1 is the principal eigenfunction of T L,-1 , and δ ∈ (0, 1/2) and γ > 0 are given in (1.2).

Proof. Set K = max (x,u)∈R×[0,1] |∂ u f (x, u)| and let C > 0 be given by

C = a + γ + 2K (2.9)
(remember that max(∂ u f (x, 0), ∂ u (x, 1)) ≤ -γ < 0 for all x ∈ R, whence γ ≤ K and the computations below would also work with C = a + 3K). Now, let us check that

N (t, x) := u t (t, x) -(a L (x)u x (t, x)) x -f L (x, u(t, x)) ≥ 0
for all t ≥ 0 and x ∈ R such that u(t, x) < 1. This will be sufficient to ensure that u is a supersolution, since f L (•, 1) = 0. From (2.7) and the definition of λ(L, -1), it is straightforward to check that, when u(t, x) < 1,

N (t, x) = -(a L ψ ′ L,-1 ) ′ (x) -2a L (x)ψ ′ L,-1 (x) + (a L (x) -a ′ L (x))ψ L,-1 (x) e -(x-Ct) + C e -(x-Ct) ψ L,-1 (x) - δγ 2 e -γt -f L (x, u(t, x)) = -λ(L, -1) -C e -(x-Ct) ψ L,-1 (x) - δγ 2 e -γt -f L (x, u(t, x)) ≥ -a -C e -(x-Ct) ψ L,-1 (x) - δγ 2 e -γt -f L (x, u(t, x)).
(2.10)

If 0 < u(t, x) ≤ δ, then f L (x, u(t, x)) ≤ -γu(t, x) from (1.
2), whence

N (t, x) ≥ -a -C -γ e -(x-Ct) ψ L,-1 (x) ≥ 0 from (2.9
) and the positivity of γ and K. On the other hand, if δ ≤ u(t, x) < 1, then

f L (x, u(t, x)) = f L (x, u(t, x)) -f L (x, 0) ≤ Ku(t, x). It then follows that δ ≤ u(t, x) = e -(x-Ct) ψ L,-1 (x) + δ 2 e -γt ≤ e -(x-Ct) ψ L,-1 (x) + δ 2 ,
whence (δ/2) e -γt ≤ δ/2 ≤ e -(x-Ct) ψ L,-1 (x). Thus, if δ ≤ u(t, x) < 1, then (2.10) yields

N (t, x) ≥ -a -C + K e -(x-Ct) ψ L,-1 (x) - δ 2 (γ + K) e -γt , whence N (t, x) ≥ -a -C + γ + 2K e -(x-Ct) ψ L,-1 (x) = 0 from (2.9)
. Hence, u is a supersolution of (1.1). The proof of Lemma 2.2 is thereby complete.

The supersolution u depends on period L, but its exponential decay rates as t → +∞ and x → +∞ are independent of L. In a similar way, we can construct a subsolution. Lemma 2.3. For every L > 0, the function u defined by

u(t, x) = max 1 -e x+Ct ψ L,1 (x) - δ 2 e -γt , 0 for t ≥ 0 and x ∈ R, (2.11)
is a subsolution of equation (1.1), where ψ L,1 is the principal eigenfunction of T L,1 , δ ∈ (0, 1/2) and γ > 0 are given in (1.2), and C > 0 is given in (2.9).

Proof. The proof of this lemma is similar to that of Lemma 2.2. For the sake of completeness, we include the details here. To prove that u is a subsolution of equation (1.1), one only needs to check that

N (t, x) := u t (t, x) -(a L (x)u x (t, x)) x -f L (x, u(t, x)) ≤ 0
for all t ≥ 0 and x ∈ R such that u(t, x) > 0, since f L (•, 0) = 0. From (2.7) and the definition of λ(L, 1), it is straightforward to check that

N (t, x) = λ(L, 1) -C e x+Ct ψ L,1 (x) + δγ 2 e -γt -f L (x, u(t, x)) ≤ a 2 -C e x+Ct ψ L,1 (x) + δγ 2 e -γt -f L (x, u(t, x)). If 1 -δ ≤ u(t, x) < 1, then f L (x, u(t, x)) ≥ γ(1 -u(t, x)) from (1.
2), whence

N (t, x) ≤ a 2 -C -γ e x+Ct ψ L,1 (x) ≤ 0.
On the other hand, if 0

< u(t, x) ≤ 1-δ, then f L (x, u(t, x)) = f L (x, u(t, x))-f L (x, 1) ≥ -K(1-u(t, x)). In that case, 1 -e x+Ct ψ L,1 (x) -δ/2 ≤ 1 -e x+Ct ψ L,1 (x) -(δ/2)e -γt = u(t, x) ≤ 1 -δ, whence (δ/2) e -γt ≤ δ/2 ≤ e x+Ct ψ L,1 (x). Thus, if 0 < u(t, x) ≤ 1 -δ, then N (t, x) ≤ a 2 -C + K e x+Ct ψ L,1 (x) + δ 2 (γ + K) e -γt ≤ a 2 -C + γ + 2K e x+Ct ψ L,1 (x) = 0.
Hence, u is a subsolution of (1.1) and the proof of Lemma 2.3 is thereby complete.

Given the previous two lemmas, we are now ready to prove Theorem 1.4.

Proof of Theorem 1.4. Let u(t, x) be a transition front connecting 0 and 1 for problem (1.1) with period L > 0, associated with positions (ξ t ) t∈R satisfying (1.4). We will prove the inequality (1.10) with the constant C given in (2.9) and used in Lemmas 2.2 and 2.3. To do so, we shall show an upper bound for lim sup |t-s|→+∞ (ξ t -ξ s )/(t-s) and a lower bound for lim inf |t-s|→+∞ (ξ t -ξ s )/(t-s). First of all, for each k ∈ Z, there is a unique n k ∈ Z such that n k L ≤ ξ k < (n k + 1)L. For all k ∈ Z and t ∈ [k, k + 1), by defining ξ t = n k L, it follows from (2.1) applied with τ = 1 that sup t∈R |ξ t -ξ t | < +∞, whence property (1.4) in Definition 1.1 still holds with the family ( ξ t ) t∈R . Therefore, even if it means redefining ξ t , one can assume without loss of generality that ξ t ∈ LZ for all t ∈ R (and even that ξ t is constant on every interval [k, k + 1) with k ∈ Z).

Step 1: the upper estimate. Let u be the function given in (2.8) with the constant C defined in (2.9). We shall show that lim sup

|t-s|→+∞ ξ t -ξ s t -s ≤ C.
(2.12)

From Definition 1.1, there is

C 1 > 0 such that u(t, x + ξ t ) ≤ δ 2 for all x ≥ C 1 and t ∈ R. (2.13)
Owing to the definition of u in (2.8), there is x 0 < 0 such that u(0, x) = 1 for all x ≤ x 0 . Let M 1 ∈ LZ be such that M 1 ≥ C 1 -x 0 . Then 0 < u(t, x + ξ t ) < 1 = u(0, x -M 1 ) for all x ≤ C 1 and t ∈ R.

On the other hand, it follows from (2.13) that 0 < u(t, x + ξ t ) ≤ δ/2 < u(0, x -M 1 ) for all x ≥ C 1 and t ∈ R. Therefore, u(t, x + ξ t ) < u(0, x -M 1 ) for all x ∈ R and t ∈ R. Since equation (1.1) is L-periodic in x and since ξ t (for every t ∈ R) and M 1 belong to LZ, it then follows from Lemma 2.2 and the maximum principle that 0 < u(s, x + ξ t ) < u(s -t, x -M 1 ) for all t ≤ s and x ∈ R.

(2.14)

Now let us assume by contradiction that there are a real number c 1 ∈ (C, +∞) and two sequences (t k ) k∈N and (s k ) k∈N of real numbers such that |t k -s k | → +∞ as k → +∞ and

ξ t k -ξ s k t k -s k > c 1 for all k ∈ N. (2.15)
Without loss of generality, one can assume that t k < s k for all k ∈ N. It then follows from (2.14

) that 0 < u(s k , x + ξ t k ) < u(s k -t k , x -M 1 ) for all k ∈ N and x ∈ R. Notice that lim k→+∞ u(s k -t k , c 1 (s k -t k ) -M 1 ) = 0 by definition of u in (2.8), since c 1 > C. This yields lim k→+∞ u(s k , c 1 (s k -t k ) + ξ t k ) = 0 Set ε 1 := inf (s,x)∈R×(-∞,0] u(s, x + ξ s ). Clearly, ε 1 > 0 from Definition 1.1 and Proposition 1.3. Since there is k 1 ∈ N such that u(s k , c 1 (s k -t k ) + ξ t k ) < ε 1 for all k ≥ k 1 , one finally gets that c 1 (s k -t k )+ξ t k ≥ ξ s k for all k ≥ k 1 , which is a contradiction with (2.15) (remember that t k < s k ).
Hence, the proof of inequality (2.12) is finished.

Step 2: the lower estimate. Let u be the function given in (2.11) with the constant C defined in (2.9). Here we show that lim inf

|t-s|→+∞ ξ t -ξ s t -s ≥ -C. (2.16)
The proof is similar to that of (2.12). Namely, from Definition 1.1, there is C 2 > 0 such that 1 > u(t, x + ξ t ) ≥ 1 -δ/2 for all x ≤ -C 2 and t ∈ R. Then owing to the definition of u, there is M 2 ∈ LZ such that 1 > u(t, x + ξ t ) > u(0, x + M 2 ) for all x ∈ R and t ∈ R. It follows then from Lemma 2.2 and the maximum principle that 1 > u(s, x + ξ s ) > u(s -t, x + M 2 ) for all t ≤ s and x ∈ R.

(2.17)

Let us assume by contradiction that there are a positive constant c 2 > C and two sequences (

t ′ k ) k∈N and (s ′ k ) k∈N of real numbers such that |t ′ k -s ′ k | → +∞ as k → +∞ and ξ t ′ k -ξ s ′ k t ′ k -s ′ k < -c 2 for all k ∈ N. (2.18)
Without loss of generality, one can assume that

t ′ k < s ′ k for all k ∈ N. It then follows from (2.17) that 1 > u(s ′ k , x + ξ t ′ k ) > u(s ′ k -t ′ k , x + M 2 ) for all k ∈ N and x ∈ R. Notice that lim k→+∞ u(s ′ k -t ′ k , -c 2 (s ′ k -t ′ k ) + M 2 ) = 1 by definition of u in (2.11), and since c 2 > C. Therefore, lim k→+∞ u(s ′ k , -c 2 (s ′ k -t ′ k ) + ξ t ′ k ) = 1. Set ε 2 := inf (s,x)∈R×[0,+∞) (1 -u(s, x + ξ s )) > 0. Then there is k 2 ∈ N such that u(s ′ k , -c 2 (s ′ k -t ′ k ) + ξ t ′ k ) > 1 -ε 2 for all k ≥ k 2 . One infers that -c 2 (s ′ k -t ′ k ) + ξ t ′ k ≤ ξ s ′ k for all k ≥ k 2 , which contradicts (2.18), since t ′ k < s ′ k .
Hence, the proof of the lower bound (2.16) is complete.

Consequently, lim sup |t-s|→+∞ |ξ t -ξ s |/|t -s| ≤ C, where the constant C defined in (2.9) depends only on the functions a and f . The proof of Theorem 1.4 is thereby complete.

Uniqueness in the class of transition fronts

In this section, we prove Theorem 1.5 and Theorem 1.6 on the uniqueness of transition fronts for problem (1.1). Applying [10, Theorem 1.14 [START_REF] Afraimovich | Stability of stationary sates and topological spatial chaos in multidimensional lattice dynamical systems[END_REF]] to the one-dimensional equation (1.1), one infers that a transition front u with positions (ξ t ) t∈R is reduced to a pulsating front provided that there is a real number c > 0 such that sup (t,s)∈R 2 |ξ t -ξ s | -c|t -s| < +∞. Therefore, under the assumption of existence of a pulsating front with nonzero speed c L , the conclusion of Theorem 1.5 follows easily by checking that for a given transition front u, the function t → ξ t -c L t is bounded. On the other hand, for the proof of Theorem 1.6, one cannot apply [10, Theorem 1.14 (1)] directly, since the existence of a pulsating front is not assumed a priori. Instead, we will use the sliding method with respect to the time variable to get the conclusion under the weaker assumption (1.13).

Proof of Theorem 1.5

The strategy for the proof of Theorem 1.5 is similar to that used in Proposition 2.2 of [START_REF] Hamel | Bistable transition fronts in R N[END_REF] for homogeneous equations. The main difference is that we will use the global stability and uniqueness of pulsating fronts for the spatially periodic equation (1.1), instead of these properties for standard traveling waves solving homogeneous equations. For the sake of completeness, we include all the details as follows.

Proof of Theorem 1.5. Let u L (t, x) = φ L (x -c L t, x/L) be a pulsating front of equation (1.1), in the sense of (1.5), with speed c L = 0. Let u(t, x) be a transition front connecting 0 and 1 for problem (1.1). That is, there is a family (ξ t ) t∈R of real numbers such that u converges to the steady states 0 and 1 as x → ±∞ in the sense of (1.4). As in the proof of Theorem 1.4, one can assume without loss of generality that ξ t ∈ LZ for all t ∈ R. Now we prove that the function t → ξ t -c L t is bounded. Let v and v be the solutions of the Cauchy problem associated to (1.1), with initial conditions

v(0, x) = 1 if x ≤ 0, δ/2 if x > 0, and v(0, x) = 1 -δ/2 if x ≤ 0, 0 if x > 0,
where δ ∈ (0, 1/2) is the constant given in (1.2). Since the pulsating front φ L is globally stable from Theorem 1.12 in [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF], it follows that there exist two real numbers η and η such that

sup y∈R v(s, y) -φ L (y -c L s + η, y) + sup y∈R v(s, y) -φ L (y -c L s + η, y) → 0 as s → +∞
(notice that only the assumption (1.2) is used in the proof of that stability result, while the assumption (1.11) on the profile of f (x, •) is not needed, see Proposition 4.4 in [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]). Therefore, since φ L (-∞, y) = 1 and φ L (+∞, y) = 0 uniformly in y ∈ R, there are T > 0 and M > 0 such that, for all s ≥ T , v(s, y

) > δ/2 if y ≤ c L s -M, v(s, y) < 1 -δ/2 if y ≥ c L s + M. (3.1)
On the other hand, from Definition 1.1, there is B ∈ LN such that

∀ (t, x) ∈ R 2 , x -ξ t ≥ B =⇒ 0 < u(t, x) ≤ δ/2, x -ξ t ≤ -B =⇒ 1 -δ/2 ≤ u(t, x) < 1, (3.2)
whence v(0, x -ξ t 0 + B) ≤ u(t 0 , x) ≤ v(0, x -ξ t 0 -B) for all x ∈ R and t 0 ∈ R. Then, since B and ξ t 0 belong to LZ for all t 0 ∈ R and since (1.1) is L-periodic in x, the maximum principle yields v(t -t 0 , x -ξ t 0 + B) ≤ u(t, x) ≤ v(t -t 0 , x -ξ t 0 -B) for all t ≥ t 0 and x ∈ R.

This together with (3.1) implies that, for all t 0 < t 0 + T ≤ t,

u(t, x) > δ/2 if x -ξ t 0 + B ≤ c L (t -t 0 ) -M, u(t, x) < 1 -δ/2 if x -ξ t 0 -B ≥ c L (t -t 0 ) + M.
Thus, from (3.2), one obtains that, for all t 0 < t 0 + T ≤ t,

ξ t 0 -B + c L (t -t 0 ) -M < ξ t + B and ξ t 0 + B + c L (t -t 0 ) + M > ξ t -B. (3.3)
By fixing t = 0, one gets lim sup

t 0 →-∞ |ξ t 0 -c L t 0 | ≤ |ξ 0 | + 2B + M . For any arbitrary t ∈ R, letting t 0 → -∞ in (3.3) leads to |ξ t -c L t| ≤ |ξ 0 | + 4B + 2M . Therefore, the function t → ξ t -c L t is bounded.
Since the family of positions (ξ t ) t∈R is defined up to an additive bounded function, without loss of generality, one can finally assume that ξ t = c L t for all t ∈ R. That is, ξ t is monotone in t, and |ξ t -ξ s | -|c L ||t -s| = 0 for all (t, s) ∈ R 2 . It finally follows from [10, Theorem 1.14 [START_REF] Afraimovich | Stability of stationary sates and topological spatial chaos in multidimensional lattice dynamical systems[END_REF]] that u is a pulsating front and that it is equal to u L up to shift in time. The proof of Theorem 1.5 is thereby complete.

Proof of Theorem 1.6

Let us now turn to the proof of Theorem 1.6. Due to the uniqueness result in Theorem 1.5, it is sufficient to prove the existence of a pulsating front with nonzero speed for equation (1.1). To do so, we first show that equation (1.1) admits a transition front v associated with a family (ζ t ) t∈R satisfying lim |t-s|→+∞ |ζ t -ζ s | = +∞. More precisely, we have the following result. Lemma 3.1. Under the assumptions of Theorem 1.6, equation (1.1) admits a transition front 0 < v < 1 connecting 0 and 1 associated with a family (ζ t ) t∈R in the sense of (1.4), such that lim |t-s|→+∞ |ζ t -ζ s | = +∞. Furthermore, either there is a positive integer K 1 such that

ζ (k+1)K 1 > ζ kK 1 + 1 for all k ∈ Z, (3.4)
or there is a positive integer K 2 such that

ζ (k+1)K 2 < ζ kK 2 -1 for all k ∈ Z. (3.5)
Proof. Throughout the proof, we assume the existence of a transition front u connecting 0 and 1 for (1.1), associated with a family (ξ t ) t∈R satisfying (1.4). We are also given a sequence (t n ) n∈N satisfying (1.13). Firstly, as in the proof of Theorem 1.4, one can assume without loss of generality that ξ t ∈ LZ for all t ∈ R and that ξ t is constant on [m, m + 1) for all m ∈ Z. Set

u n (t, x) = u(t + t n , x + ξ tn ) for (t, x) ∈ R 2 and n ∈ N.
For each n ∈ N, the function u n satisfies (1.1). Furthermore, from Proposition 1.3, there holds

0 < inf n∈N u n (0, 0) ≤ sup n∈N u n (0, 0) < 1. (3.6)
Up to extraction of some subsequence, one can assume, from standard parabolic estimates, that u n (t, x) → v(t, x) as n → +∞ locally uniformly in (t, x) ∈ R 2 , where 0 ≤ v ≤ 1 is a classical solution of (1.1). It follows from (3.6) that 0 < v(0, 0) < 1, whence 0 < v(t, x) < 1 for all (t, x) ∈ R 2 by the strong maximum principle. On the other hand, for each n ∈ N, the function t → ξ t+tn -ξ tn ranges in LZ and is constant on [m -t n , m + 1 -t n ) for every m ∈ Z. Furthermore, Lemma 2.1 implies that, for each C ≥ 0, sup n∈N, |t|≤C |ξ t+tn -ξ tn | < +∞. Therefore, up to extraction of a subsequence, there exists a function t → ξ t such that ξ t ∈ LZ for all t ∈ R and ξ t+tn -ξ tn → ξ t as n → +∞ for every t ∈ R.

Let us now show that v is a transition front connecting 0 and 1 for (1.1) with this family ( ξ t ) t∈R . To do so, let ε > 0 be any arbitrary positive real number. Let M ≥ 0 be such that u(t, x + ξ t ) ≥ 1 -ε for all t ∈ R and x ≤ -M , and u(t, x+ξ t ) ≤ ε for all t ∈ R and x ≥ M . For any t ∈ R and x < ξ t -M , one has

x + ξ tn -ξ t+tn -→ n→+∞

x -ξ t < -M, whence x + ξ tn < ξ t+tn -M for n large and u n (t, x) = u(t + t n , x + ξ tn ) ≥ 1 -ε for n large. As a consequence, v(t, x) ≥ 1 -ε for all t ∈ R and x < ξ t -M . Similarly, one can show that v(t, x) ≤ ε for all t ∈ R and x > ξ t + M . In other words, v is a transition front connecting 0 and 1, with the family ( ξ t ) t∈R satisfying (1.4). Furthermore, by writing ξ t+tn -ξ s+tn = (ξ t+tn -ξ tn ) -(ξ s+tn -ξ tn ), it follows from assumption (1.13) that lim |t-s|→+∞

| ξ t -ξ s | = +∞. (3.7)
Let us set from (3.7). In particular, there is a positive integer K 0 > 0 such that

ζ t = ξ k + ( ξ k+1 -ξ k )(t -k) for t ∈ [k, k + 1)
|ζ K 0 i -ζ K 0 j | > 1 for all i, j ∈ Z with i = j. (3.9)
Next, we claim that either the set

E + := k ∈ N ; ζ K 0 k > 0 ∪ k ∈ Z\N ; ζ K 0 k < 0 or the set E -:= k ∈ N ; ζ K 0 k < 0 ∪ k ∈ Z\N ; ζ K 0 k > 0
is finite. Suppose on the contrary that both E + and E -are infinite. Since E + is infinite, let us assume here that the set k ∈ N ; ζ K 0 k > 0 is infinite (the case where the set k ∈ Z\N ; ζ K 0 k < 0 is infinite can be handled similarly). Since the set E -is infinite, two cases may then occur. Case 1: the set k ∈ N ; ζ K 0 k < 0 is infinite. From (3.8), there exist two sequences of positive integers (i k ) k∈N and (j k ) k∈N such that

0 < i 0 < j 0 < i 1 < j 1 < • • • < i k < j k < • • • and ζ K 0 j k+1 < ζ K 0 j k < 0 < ζ K 0 i k < ζ K 0 i k+1 for all k ∈ N.
By continuity of the function t → ζ t , there exists a sequence of real numbers (s k ) k∈N such that K 0 i k < s k < K 0 j k and ζ s k = 0 for all k ∈ N. The facts that lim k→+∞ s k = +∞ and ζ s k = 0 contradict (3.8). Therefore, case 1 is ruled out.

Case 2: the set k ∈ Z\N ; ζ K 0 k > 0 is infinite. From (3.8), there exist then two sequences

(i ′ k ) k∈N in N and (j ′ k ) k∈N in Z\N such that • • • < j ′ k+1 < j ′ k < • • • < j ′ 0 < 0 < i ′ 0 < • • • < i ′ k < i ′ k+1 < • • • and ζ K 0 i ′ k < ζ K 0 j ′ k < ζ K 0 i ′ k+1 < ζ K 0 j ′ k+1 for all k ∈ N. By continuity of t → ζ t , there exists a sequence of real numbers (s ′ k ) k∈N such that K 0 i ′ k < s ′ k < K 0 i ′ k+1 and ζ s ′ k = ζ K 0 j ′ k for all k ∈ N. But s ′ k → +∞ while K 0 j ′ k → -∞ as k → +∞, which contradicts (3.8
). Thus, case 2 is ruled out too. The proof of our claim is thereby complete, that is, either E + or E -is a finite set. Finally, we prove that, if E -is a finite set, then there is K 1 ∈ K 0 N such that (3.4) holds, while there is K 2 ∈ K 0 N such that (3.5) holds if E + is a finite set. We will just consider the case where the set E -is finite, the proof for the case where the set E + is finite being similar. So, let us assume that the set E -is finite. Even if it means increasing K 0 , one can then assume without loss of generality that ζ K 0 k ≥ 0 for all k ∈ N\{0} and ζ K 0 k ≤ 0 for all k ∈ Z\N. As a consequence of (3.8), one infers that ζ K 0 k → ±∞ as k → ±∞ (and even ζ t → ±∞ as t → ±∞ by Lemma 2.1 applied to v and (ζ t ) t∈R ). Let us now assume by contradiction that (3.4) does not hold. In particular, for each m ∈ N\{0}, there is Consequently, (3.4) holds if E -is a finite set. Similarly, (3.5) holds if E + is a finite set. Since either E -or E + is a finite set, it follows that either (3.4) or (3.5) holds and the proof of Lemma 3.1 is thereby complete. Let 0 < v < 1 be the transition front given in Lemma 3.1 and associated with positions (ζ t ) t∈R . By sliding v with respect to t, we will show that equation (1.1) admits a pulsating front with positive speed if (ζ t ) t∈R satisfies (3.4), while it admits a pulsating front with negative speed if (ζ t ) t∈R satisfies (3.5). Lemma 3.2. Assume that (ζ t ) t∈R satisfies (3.4). Then there is a real number

k m ∈ Z such that ζ (km+1)K 0 m ≤ ζ kmK 0 m + 1, whence ζ (km+1)K 0 m < ζ kmK 0 m -1 from (3.9). Since t → ζ t is
τ 0 > 0 such that v(t + τ, x + L) ≥ v(t, x) for all τ ≥ τ 0 and (t, x) ∈ R 2 . Proof. From Definition 1.1, there is B > 0 such that ∀ (t, x) ∈ R 2 , x -ζ t ≥ B =⇒ 0 < v(t, x) ≤ δ, x -ζ t ≤ -B =⇒ 1 -δ/2 ≤ v(t, x) < 1, (3.10)
where δ ∈ (0, 1/2) is the constant given in (1.2). As in the proof of Proposition 1.3, one can also assume that f (x, •) is (strictly) decreasing in (-∞, δ] and [1-δ, +∞) for every x ∈ R. Because of (3.4) and Lemma 2.1 applied to v and (ζ t ) t∈R , there is a real number τ 0 > 0 such that ζ t+τ -ζ t ≥ 2B + L for all t ∈ R and τ ≥ τ 0 . It then follows that, for any τ ≥ τ 0 ,

v(t + τ, x + L) ≥ 1 -δ for all (t, x) ∈ R 2 such that x -ζ t ≤ B. (3.11)
Together with (3.10) and 1 -δ ≥ δ, one gets that

v(t + τ, x + L) ≥ v(t, x) for all (t, x) ∈ R 2 with x -ζ t = B. (3.12)
Fix any τ ∈ [τ 0 , +∞), define

ε * := inf ε > 0 ; v(t + τ, x + L) + ε ≥ v(t, x) for all (t, x) ∈ R 2 with x -ζ t ≤ B
and let us show that ε * = 0. Assume by contradiction that ε * > 0. There are then a sequence (ε n ) n∈N of positive real numbers and a sequence (t n , x n ) n∈N in R 2 such that ε n → ε * as n → +∞ and

v(t n + τ, x n + L) + ε n < v(t n , x n ) and x n -ζ tn ≤ B for all n ∈ N. ( 3
> 0 such that ∀ n ∈ N, ∀ (t, x) ∈ R 2 , |t -t n | + |x -x n | < ρ =⇒ x -ζ t < B . (3.14) Write x n = x ′ n + x ′′ n with x ′ n ∈ LZ, x ′′ n ∈ [0, L) and assume that, up to extraction of a subsequence, x ′′ n → x ∞ ∈ [0, L] as n → +∞. Define v n (t, x) = v(t + t n , x + x ′ n ) for n ∈ N and (t, x) ∈ R 2 .
Up to extraction of a subsequence, the functions v n converge locally uniformly in R 2 to a solution 0 ≤ v ∞ ≤ 1 of (1.1). Furthermore, for any (t, x) ∈ R 2 with |t| + |x| < ρ, there holds x + x n -ζ t+tn < B for all n ∈ N by (3.14), whence

v n (t + τ, x + x ′′ n + L) = v(t + t n + τ, x + x n + L) ≥ 1 -δ by (3.11) and v n (t + τ, x + x ′′ n + L) + ε * = v(t + t n + τ, x + x n + L) + ε * ≥ v(t + t n , x + x n ) = v n (t, x + x ′′ n )
by definition of ε * . Therefore, for any (t, x) ∈ R 2 with |t| + |x| < ρ, one has

v ∞ (t + τ, x + x ∞ + L) ≥ 1 -δ and v ∞ (t + τ, x + x ∞ + L) + ε * ≥ v ∞ (t, x + x ∞ ).
On the other hand, (3.13) implies that v n (τ,

x ′′ n + L) + ε n < v n (0, x ′′ n ) for every n ∈ N, whence v ∞ (τ, x ∞ + L) + ε * ≤ v ∞ (0, x ∞ ) and finally v ∞ (τ, x ∞ + L) + ε * = v ∞ (0, x ∞ ). Finally, de- fine z(t, x) = v ∞ (t + τ, x + L) -v ∞ (t, x) for (t, x) ∈ R 2 . There holds z(t, x) ≥ -ε * for all (t, x) ∈ R 2 such that |t| + |x -x ∞ | < ρ, while z(0, x ∞ ) = -ε * . Since z satisfies the equa- tion z t -(a L z x ) x = f (x/L, v ∞ (t + τ, x + L)) -f (x/L, v ∞ (t, x)) in R 2 , one gets at (0, x ∞ ) that f x ∞ L , v ∞ (τ, x ∞ + L) -f x ∞ L , v ∞ (0, x ∞ ) ≤ 0. (3.15) However, v ∞ (0, x ∞ ) = v ∞ (τ, x ∞ + L) + ε * > v ∞ (τ, x ∞ + L) ≥ 1 -δ and f (x ∞ /L, •) is (strictly) decreasing in [1 -δ, +∞). Therefore, (3.15) is impossible, whence ε * = 0 and v(t + τ, x + L) ≥ v(t, x) for all (t, x) ∈ R 2 with x -ζ t ≤ B.
Similarly, by using the first property in (3.10) and (3.12), one gets that v(t+τ, x+L) ≥ v(t, x) for all τ ≥ τ 0 and for all (t, x) ∈ R 2 with x -ζ t ≥ B. The proof of Lemma 3.2 is thereby complete.

Similarly, if (ζ t ) t∈R satisfies property (3.5), then there is a real number τ ′ 0 > 0 such that v(t + τ, x) ≤ v(t, x + L) for all (t, x) ∈ R 2 and τ ≥ τ ′ 0 . Based on the previous properties, we are now ready to prove Theorem 1.6.

Proof of Theorem 1.6. Firstly, assume that the family (ζ t ) t∈R satisfies (3.4). Then Lemma 3.2 yields the existence of τ 0 > 0 such that v(t + τ, x + L) ≥ v(t, x) for all (t, x) ∈ R 2 and τ ≥ τ 0 . Define

τ * := inf τ > 0 ; v(t + τ ′ , x + L) ≥ v(t, x) for all (t, x) ∈ R 2 and τ ′ ≥ τ .
One then has 0 ≤ τ * ≤ τ 0 and

v(t + τ * , x + L) ≥ v(t, x) for all (t, x) ∈ R 2 .
(3.16)

Now we claim that τ * > 0. Otherwise, v(t, x + L) ≥ v(t, x) for all (t, x) ∈ R 2 , whence v(t, x + iL) ≥ v(t, x) for all (t, x) ∈ R 2 and i ∈ N. Definition 1.1 applied to v implies in particular that v(0, 0) ≤ lim i→+∞ v(0, iL) = 0, which contradicts the positivity of v. Therefore, τ * > 0.

Next, we prove that

ω := inf v(t + τ * , x + L) -v(t, x) ; |x -ζ t | ≤ B = 0, (3.17)
where B > 0 is as in (3.10). Assume, by contradiction, that ω > 0. Notice that the derivative v t is globally bounded from standard parabolic theory. Then there exists η 0 ∈ (0, τ * ) such that

v(t + τ * -η, x + L) ≥ v(t, x) for all η ∈ [0, η 0 ] and |x -ζ t | ≤ B. (3.18)
For each η ∈ [0, η 0 ], one then has v(t

+ τ * -η, x + L) ≥ v(t, x) for all (t, x) ∈ R 2 with x -ζ t = B, while 0 < v(t, x) ≤ δ for all (t, x) ∈ R 2 with x -ζ t ≥ B from (3.10
). Therefore, with the same arguments as in the proof of Lemma 3.2, one infers that

v(t + τ * -η, x + L) ≥ v(t, x) for all η ∈ [0, η 0 ] and x -ζ t ≥ B. (3.19)
On the other hand, since v(t + τ * , x + L) ≥ v(t, x) ≥ 1 -δ/2 for all x -ζ t ≤ -B by (3.10), one can assume without loss of generality that η 0 > 0 is small enough so that

v(t + τ * -η, x + L) ≥ 1 -δ for all η ∈ [0, η 0 ] and x -ζ t ≤ -B. 1 Since v(t + τ * -η, x + L) ≥ v(t, x) for all (t, x) ∈ R 2 with x -ζ t = -B from (3.18), one concludes as in Lemma 3.2 that v(t + τ * -η, x + L) ≥ v(t, x) for all η ∈ [0, η 0 ] and x -ζ t ≤ -B.
This, together with

(3.18)-(3.19), implies that v(t + τ * -η, x + L) ≥ v(t, x) for all η ∈ [0, η 0 ] and (t, x) ∈ R 2
, contradicting the minimality of τ * . Thus, the proof of (3.17) is finished. We finally show that equation (1.1) admits a pulsating front with speed c L = L/τ * > 0. From (3.17), there exists a sequence (

s n , x n ) n∈N in R 2 such that v(s n + τ * , x n + L) -v(s n , x n ) → 0 as n → +∞ and |x n -ζ sn | ≤ B for all n ∈ N. (3.20)
Let us now write

x n = x ′ n + x ′′ n with x ′ n ∈ LZ and x ′′ n ∈ [0, L) and set v n (t, x) = v(t + s n , x + x ′ n ) for (t, x) ∈ R 2 and n ∈ N.
For each n ∈ N, the function v n satisfies (1.1). Up to extraction of some subsequence, one can assume that x ′′ n → x ∞ ∈ [0, L] as n → +∞ and that, from standard parabolic estimates, (3.20). Since v ∞ (t + τ * , x + L) is also a solution to equation (1.1), it follows from the strong parabolic maximum principle that

v n (t, x) → v ∞ (t, x) as n → +∞ locally uniformly in R 2 , where 0 ≤ v ∞ ≤ 1 solves (1.1). Fur- thermore, v ∞ (t+τ * , x+L) ≥ v ∞ (t, x) for all (t, x) ∈ R 2 from (3.16), and v ∞ (τ * , x ∞ +L) = v ∞ (0, x ∞ ) from
v ∞ (t + τ * , x + L) = v ∞ (t, x) for all (t, x) ∈ R 2 .
On the other hand, from Definition 1.1, one has

v n (t, x) = v(t + s n , x + x ′ n ) → 0 as x + x ′ n -ζ sn+t → +∞ uniformly with respect to n ∈ N and (t, x) ∈ R 2 . Write x + x ′ n -ζ sn+t = (x -ζ t ) + (x n -ζ sn ) -x ′′ n + (ζ t + ζ sn -ζ t+sn )
and remember that the sequences (x ′′ n ) n∈N and (x n -ζ sn ) n∈N are bounded. Notice that, thanks to Lemma 2.1, the quantities ζ t and ζ t + ζ sn -ζ t+sn are bounded locally in t ∈ R and independently of n ∈ N. It then follows that

v ∞ (t, x) → 0 as x → +∞ locally in t ∈ R. Similar arguments imply that v ∞ (t, x) → 1 as x → -∞ locally in t ∈ R.
Therefore, v ∞ is a pulsating front with positive speed c L = L/τ * . Thus, from Theorem 1.5, any transition front connecting 0 and 1 for equation (1.1) is equal to v ∞ up to shift in time.

Similarly, if (ζ t ) t∈R satisfies property (3.5), then one can conclude that equation (1.1) admits a pulsating front with speed c L = -L/τ * < 0, where τ * is the positive number defined by

τ * = inf τ > 0 ; v(t + τ ′ , x) ≤ v(t, x + L) for all (t, x) ∈ R 2 and τ ′ ≥ τ .
Furthermore, any transition front connecting 0 and 1 is equal to this pulsating front up to shift in time. The proof of Theorem 1.6 is thereby complete.

Transition fronts which are not pulsating fronts

This section is devoted to the proof of Theorem 1.7 on the existence of a new type of transition fronts, which are not pulsating fronts. Let us explain the general strategy before going into the details. The first step consists in constructing super-and subsolutions for equation (1.1) when the period L is large. The construction will use the properties of the standard traveling wave for the homogeneous equation

(u y ) t (t, x) = a (u y ) xx (t, x) + f y (u y (t, x))
with 0 < u y (t, x) < 1 for all (t, x) ∈ R 2 , where y is a fixed real number and

f y = f (y, •) in [0, 1]
(remember that a is a constant in Theorem 1.7). As a matter of fact, for large L, by choosing some fixed suitable real numbers y > y, we will get a supersolution u L and a subsolution u L of (1.1) which are located around the positions Ly and Ly respectively, which are strictly ordered and which move towards each other as time runs. As a consequence, there exist stationary fronts for equation (1.1) at large L, which emerge from these super-and subsolutions. By shifting suitably in space the subsolution (or the supersolution) of the first step, we can then construct a pair of strictly ordered stationary fronts such that the lower one is stable from above, and the upper one is stable from below. Once this is done, the Dancer-Hess connecting orbit lemma will lead to the existence of timemonotone transition fronts between these two stationary fronts. Actually, we will see that there exist at least two transition fronts: the first one is decreasing in time and the second one is increasing. Finally, we will prove that all transition fronts have zero global mean speed. In Section 4.1, we provide a series of auxiliary lemmas for the proof of Theorem 1.7, which is carried out in Section 4.2.

Some auxiliary lemmas

We recall that a, ∂ u f (•, 0) and

∂ u f (•, 1) are constants. Denote µ + = ∂ u f (•, 0) and µ -= ∂ u f (•, 1
), and remember that max(µ

+ , µ -) ≤ -γ < 0 where γ > 0 is given in (1.2). Since the function f (x, u) is continuous in R × [0, 1] and 1-periodic in x, one can choose x ∈ [0, 1] such that 1 0 f (x, u)du = min x∈R 1 0 f (x, u)du.
Due to the assumptions (1.2) and (1.11), the profile f (x, •) : [0, 1] → R is bistable in the sense of (1.8). It then follows from [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that the homogeneous equation

u t (t, x) = a u xx (t, x) + f (x, u(t, x)) (4.1)
admits a standard traveling front φ(x -ct) such that 0 < φ < 1 in R, φ(-∞) = 1 and φ(+∞) = 0. The front φ(x) is decreasing in x and is unique up to shift in x (one can normalize φ in such a way that φ(0) = 1/2). Furthermore, the speed c is unique and it has the sign of the integral 1 0 f (x, u)du, that is, c < 0 thanks to the assumption (1.15). Moreover, φ(x) approaches the limits 1 and 0 as x → ±∞ exponentially, namely there are two positive constants A ± such that

φ(x) ∼ A + e -λ + x as x → +∞, 1-φ(x) ∼ A -e λ -x as x → -∞, λ ± = ±c + c 2 -4aµ ± 2a > 0. (4.2) Lastly, φ ′ (x)/φ(x) → -λ + as x → +∞ and φ ′ (x)/(1 -φ(x)) → -λ -as x → -∞. Lemma 4.1.
There is L 1 > 0 large enough such that for every L > L 1 , the function

u L (x) = φ(x -Lx)
is a strict supersolution of (1.1).

Proof. Fix any real number ε such that 0

< ε < min(-c λ + /3, -c λ -/3) (remember that c < 0). Since f (•, 0) = f (•, 1) = 0 and since ∂ u f is continuous in R × [0, 1] and 1-periodic in x with µ + = ∂ u f (•, 0) and µ -= ∂ u f (•, 1
) being constant, it follows that there exists a number C > 0 (independent of L > 0) such that

(µ + -ε)φ(x) ≤ f (y, φ(x)) ≤ (µ + + ε)φ(x) for all x ≥ C and y ∈ R, (4.3)
-(µ -+ ε)(1 -φ(x)) ≤ f (y, φ(x)) ≤ -(µ --ε)(1 -φ(x)) for all x ≤ -C and y ∈ R, (4.4)
and

c φ ′ (x) φ(x) + c λ + ≥ -ε for all x ≥ C, (4.5)
c φ ′ (x) 1 -φ(x) + c λ -≥ -ε for all x ≤ -C. (4.6)
Let us now check that N L (x) := -a u ′′ L (x) -f L (x, u L (x)) > 0 in R for all L > 0 large enough. Since φ(x -ct) is a solution to (4.1), one infers that, for all x ∈ R,

N L (x) = c φ ′ (x -Lx) + f x, φ(x -Lx) -f L x, φ(x -Lx) . (4.7)
If x ≥ C + Lx, then from (4.3), (4.5) and the choice of ε, one has

N L (x) ≥ c φ ′ (x -Lx) + (µ + -ε)φ(x -Lx) -(µ + + ε)φ(x -Lx) = φ(x -Lx) c φ ′ (x -Lx) φ(x -Lx) -2ε ≥ φ(x -Lx)(-c λ + -3ε) > 0.
On the other hand, if x ≤ -C + Lx, then, from (4.4), (4.6) and the choice of ε, there holds

N L (x) ≥ c φ ′ (x -Lx) -(µ -+ ε) 1 -φ(x -Lx) + (µ --ε) 1 -φ(x -Lx) = 1 -φ(x -Lx) c φ ′ (x -Lx) 1 -φ(x -Lx) -2ε ≥ 1 -φ(x -Lx) (-c λ --3ε) > 0.
Finally, for -C + Lx ≤ x ≤ C + Lx, due to the continuity and negativity of φ ′ , one gets that φ ′ (x -Lx) ≤ -β < 0 for some positive real number β := -max [-C,C] φ ′ > 0 which is independent of L > 0. Since the function f (x, u) is of class C 0,α in x uniformly in u ∈ [0, 1], one concludes that there is a positive real number κ independent of L > 0 such that, for all -C +Lx ≤ x ≤ C +Lx,

f x, φ(x -Lx) -f L x, φ(x -Lx) ≤ κ x - x L α ≤ κ C α L α ,
whence N L (x) ≥ -cβ -κC α /L α from (4.7) and c < 0. Therefore, if L > L 1 := (κC α /(|c|β)) 1/α > 0, then N L (x) > 0 for all -C + Lx ≤ x ≤ C + Lx. Thus, for every L > L 1 , one has N L (x) > 0 for all x ∈ R. Namely, u L is a strict supersolution of equation (1.1). The proof of Lemma 4.1 is thereby compete.

A subsolution for equation (1.1) at large L can be constructed in a similar way. More precisely, choose x ∈ [-1, 0] such that

1 0 f (x, u)du = max x∈R 1 0 f (x, u)du.
It follows from (1.11) and (1.15) that, for the homogeneous equation v t = a v xx + f (x, v), there are a unique c > 0 and a standard traveling front v(t, x) = φ(x -ct) such that 0 < φ < 1 in R, φ(-∞) = 1 and φ(+∞) = 0. The front φ is decreasing in R and is unique up to shifts. Furthermore, there are two positive constants B ± such that

φ(x) ∼ B + e -λ + x as x → +∞, 1-φ(x) ∼ B -e λ -x as x → -∞, λ ± = ±c + c 2 -4aµ ± 2a > 0. (4.8) Lemma 4.2.
There is L 2 > 0 large enough such that for every L > L 2 , the function

u L (x) = φ(x -Lx)
is a strict subsolution of equation (1.1).

Proof. The proof is similar to that of Lemma 4.1, and we omit the details.

Furthermore, when L is large enough, the following comparison holds.

Lemma 4.3. There is L * > 0 large enough such that for every L > L * , 0 < u L < u L < 1 in R.

Proof. From (4.2), (4.8) and c < 0 < c, one infers that 0 < λ + < λ + and λ -> λ -> 0, whence φ(x)/φ(x) → +∞ as x → +∞ and (1 -φ(x))/(1 -φ(x)) → 0 as x → -∞. It then follows that there is C ′ > 0 such that φ(x) > φ(x) for all x ∈ R\[-C ′ , C ′ ]. Remember that φ and φ are decreasing in R and that -1 ≤ x ≤ 0 ≤ x ≤ 1 with x = x, whence x < x. Consequently, there is L * > 0 large enough such that, for every L > L * and x ∈ R, 1 > φ(x -Lx) > φ(x -Lx) > 0. The proof of Lemma 4.3 is thereby complete.

Without loss of generality, one can assume that L * > max(L 1 , L 2 ), where L 1 and L 2 are given in Lemmas 4.1 and 4.2. Thus, for any given L > L * , 0 < u L < u L < 1 are strictly ordered sub-and supersolutions for equation (1.1), which both converge to 0 and 1 as x → ±∞. As a consequence of the maximum principle, as will be explained in the proof of Theorem 1.7 below, stationary fronts for (1.1) will then exist, that is, stationary solutions 0 < p(x) < 1 such that p(-∞) = 1 and p(+∞) = 0.

Before going further on, we need to prove a useful property, which actually holds for any L > 0, on the asymptotic behavior at ±∞ of any stationary front of (1.1). Lemma 4.4. Let L > 0 be arbitrary and let 0 < p(x) < 1 be a stationary solution of equation (1.1) such that p(-∞) = 1 and p(+∞) = 0. Then there are two positive constants M ± such that p(x) ∼ M + e -λ + x as x → +∞, 1 -p(x) ∼ M -e λ -x as x → -∞, λ ± = -µ ± a > 0. (4.9)

Proof. We will just show the first property of (4.9), that is, p(x) decays exponentially to 0 as x → +∞, the proof of the behavior as x → -∞ being similar. First of all, because of (1.2), there is x ∈ R such that f L (x, p(x)) < 0 for all x ≥ x, whence p ′′ (x) > 0 for all x ≥ x. Thus, p ′ is increasing in [ x, +∞) and since p(+∞) = 0, one infers that p ′ (+∞) = 0 and p ′ < 0 in [ x, +∞). On the other hand, from standard elliptic interior estimates and from Harnack inequality, the function p ). Without loss of generality, one can assume that x ′ n ≥ x for all n ∈ N and that x ′′ n → y ∈ [0, L] as n → +∞. From the boundedness of p ′ /p, the functions

′ /p is bounded in R. Hence λ := lim sup x→+∞ p ′ (x)/p(x) ∈ (-∞, 0]. Let (x n ) n∈N be a sequence in R such that x n → +∞ and p ′ (x n )/p(x n ) → λ as n → +∞. Write x n = x ′ n + x ′′ n with x ′ n ∈ LZ and x ′′ n ∈ [0, L
x → p n (x) := p(x + x ′ n ) p(x ′ n ) > 0
are locally bounded (that is, sup n∈N p n L ∞ (K) < +∞ for any compact subset K ⊂ R). Furthermore, since equation (1.1) is L-periodic in x, the functions p n obey a p

′′ n + f L (x, p(x ′ n )p n )/p(x ′ n ) = 0 in R.
Notice that p(x ′ n ) → 0 as n → +∞ since x ′ n → +∞, and remember that ∂ u f is uniformly continuous in R × [0, 1] and that ∂ u f (•, 0) = µ + < 0 is constant. From standard elliptic estimates, there is then a nonnegative function p ∞ such that, up to extraction of a subsequence, p n → p ∞ in C 1 loc (R) and p ∞ is a classical solution of a p ′′ ∞ + µ + p ∞ = 0 in R. Furthermore, p ∞ (0) = 1, whence p ∞ > 0 in R from the strong maximum principle. On the other hand, since x ′ n ≥ x for all n ∈ N and p ′ < 0 on [ x, +∞), there holds p ′ n < 0 on [0, +∞), whence p ′ ∞ ≤ 0 in [0, +∞) (actually, p ′ ∞ ≤ 0 in R since x ′ n → +∞ as n → +∞). Therefore, p ∞ (x) = e -λ + x for all x ∈ R, with λ + = -µ + /a > 0. Finally, since p

′ n (x ′′ n )/p n (x ′′ n ) = p ′ (x n )/p(x n ) → λ as n → +∞, one infers that p ′ ∞ (y)/p ∞ (y) = λ, whence λ = -λ + .
One has then proved that lim sup x→+∞ p ′ (x)/p(x) = -λ + . With similar arguments, one can get that lim inf x→+∞ p ′ (x)/p(x) = -λ + , and finally p ′ (x)/p(x) → -λ + as x → +∞. Then the asymptotic stability theory for ordinal differential equations (see, e.g. [15, Chapter 13, Theorem 4.5]) implies that p(x) ∼ M + e -λ + x as x → +∞ for some constant M + > 0.

Similarly, applying the above analysis to the function 1-p, one concludes that 1-p(x) ∼ M -e λ -x as x → -∞ for some constant M -> 0. The proof of Lemma 4.4 is thereby complete.

In the following lemma, we show that when L ≥ L * , any transition front connecting 0 and 1 for problem (1.1) can be bounded from above by a translate of u L and from below by a translate of u L at large time, up to some exponentially small terms. As a matter of fact, this property will be used in Theorem 1.7 to prove that all transition fronts have zero global mean speed. Lemma 4.5. Even if it means increasing L * > 0, the following property holds: for every L > L * and every transition front 0 < u < 1 connecting 0 and 1 for problem (1.1), there are some real numbers x * , x * , q 0 > 0 and γ 0 > 0 such that max u L (x -x * ) -q 0 e -γ 0 t , 0 ≤ u(s + t, x + ξ s ) ≤ min u L (x -x * ) + q 0 e -γ 0 t , 1 (4.10)

for all x ∈ R, s ∈ R and t ≥ 0. Furthermore, q 0 and γ 0 can be chosen independently of L and u.

Proof. We will only prove the upper inequality of (4.10), since the proof of the lower one is similar. We are given a transition front u connecting 0 and 1 for (1.1), for some L > 0 (that will be large at the end of the proof), associated with a family (ξ t ) t∈R satisfying (1.4). First of all, as already emphasized, one can assume without loss of generality that ξ s ∈ LZ for all s ∈ R. We first claim that, for any real number q 0 ∈ (0, 1), there exists a η 0 ∈ L(N\{0}) such that

u(s, x + ξ s ) ≤ u L (x -η 0 ) + q 0 for all (s, x) ∈ R 2 . (4.11)
Indeed, from Definition 1.1, for any q 0 ∈ (0, 1), there is M > 0 such that u(s, x + ξ s ) ≤ q 0 for all x ≥ M and s ∈ R. Since lim x→-∞ u L (x) = 1 and 0 < u < 1 in R 2 , there exists η 0 ∈ L(N\{0}) large enough such that u(s + ξ s , x) ≤ u L (x -η 0 ) + q 0 for all x ≤ M and s ∈ R. Therefore, the inequality (4.11) is proved. Next, we set v L (t, x) := min u L (x -η(t)) + q(t), 1 > 0 for t > 0 and x ∈ R, where η and q are C 1 ([0, +∞)) functions such that η(0) = η 0 , η ′ (t) > 0 for all t ≥ 0, q(0) = q 0 and 0 < q(t) ≤ q 0 for all t ≥ 0. By choosing later some appropriate functions η(t) and q(t), we will the proof of Lemma 4.1. More precisely, since u L (x -η(t)) = φ(x -η(t) -Lx), it follows that, for all (t, x) ∈ (0, +∞) × R such that v L (t, x) < 1,

N 2 (t, x) = c φ ′ (x -η(t) -Lx) + f x, φ(x -η(t) -Lx) -f L x, φ(x -η(t) -Lx) ,
that is, N 2 (t, x) is nothing but the expression (4.7) with x -η(t) instead of x. As a consequence, as in the proof of Lemma 4.1, there exists a positive number C (independent of L and u) such that N 2 (t, x) > 0 for all (t, x) ∈ (0, +∞) × R with |x -η(t) -Lx| ≥ C and v L (t, x) < 1. On the other hand, there is a constant β > 0 (independent of L and u) such that φ ′ (x -η(t) -Lx) ≤ -β for all (t, x) ∈ (0, +∞) × R with |x -η(t) -Lx| ≤ C and v L (t, x) < 1. Moreover, in this case, since

x -η 0 -Lx = (x -η(t) -Lx) + (η(t) -η 0 )
and since the function t → η(t)-η 0 is bounded and independent of L and u, there exists a positive real number C 1 (independent of L, u and (t, x))

such that |x-η 0 -Lx| ≤ C 1 if |x-η(t)-Lx| ≤ C. Remem- ber that f (x, u) is 1-periodic in x, that f (x, u) is of class C 0,α in x uniformly in u ∈ [0, 1]
, that η 0 ∈ LZ and that c < 0. One then concludes that, for all (t, x) ∈ (0, +∞) × R such that |x -η(t) -Lx| ≤ C and v L (t, x) < 1, there holds

N 2 (t, x) = c φ ′ (x -η(t) -xL) + f x, φ(x -η(t) -xL) -f x -η 0 L , φ(x -η(t) -xL) ≥ -c β - C 2 C α 1 L α ,
for some positive number C 2 independent of L, u and (t, x). Thus, even if it means increasing L * (but still independently of u and (t, x)), one has N 2 (t, x) ≥ 0 for all L > L * and (t, x) ∈ (0, +∞) × R with |x -η(t) -Lx| ≤ C and v L (t, x) < 1. Combining the above properties, one has N 2 (t, x) ≥ 0 for all L > L * and (t, x) ∈ (0, +∞) × R with v L (t, x) < 1. By choosing q(t) and η(t) as in (4.14), we then get the inequality (4.11) and N (t, x) ≥ 0 for all (t, x) ∈ (0, +∞) × R such that v L (t, x) < 1. Since ξ s ∈ LZ for all s ∈ R and since u < 1 in R 2 , the maximum principle implies that u(s + t, x + ξ s ) ≤ min u L (x -η(t)) + q(t), 1 for all t ≥ 0, s ∈ R and x ∈ R.

Set x * = sup t≥0 η(t) = η 0 + q 0 (2K + γ)/(γρ) and γ 0 = γ/2. Since u L is decreasing in R, the second inequality of (4.10) follows immediately.

As already emphasized, the proof of the first inequality follows the same scheme. Hence, the proof of Lemma 4.5 is complete.

Proof of Theorem 1.7

Based on the above preparations, we are in a position to carry out the proof of Theorem 1.7. Throughout the proof, we fix a period L > L * , where L * > 0 is given in Lemmas 4.3 and 4.5. The proof will be divided into five main steps.

Step 1: some useful notations. Let E = BU C(R, R) be the Banach space of bounded uniformly continuous functions from R to R, endowed with the

L ∞ norm • E = • L ∞ (R) and the usual order in C(R, R), that is, u ≤ E v if u(x) ≤ v(x) for all x ∈ R and u < E v if u(x) ≤, ≡ v(x) in R. For u ∈ E and σ > 0, B E (u, σ) = v ∈ E ; v -u E < σ
is the open ball with center u and radius σ. For any u 0 ∈ E, let (t, x) → u(t, x; u 0 ) denote the unique solution of (1.1) in (0, +∞) × R with initial value u(0, •; u 0 ) = u 0 . This solution exists for all t > 0 since ∂ u f is bounded in R 2 . For any t ≥ 0, set

S t [u 0 ] = u(t, •; u 0 ).
Then (S t ) t>0 is a continuous semiflow on E, and the strong parabolic maximum principle implies that (S t ) t>0 is strictly order-preserving on E in the sense that if u 0 and v 0 belong to E and u 0 ≤, ≡ v 0 in R, then S t [u 0 ] < S t [v 0 ] in R for every t > 0. An equilibrium point p ∈ E for the semiflow (S t ) t>0 is nothing but a stationary solution of (1.1). Lastly, given two equilibrium points p ± in E, an entire orbit connecting p -to p + is a continuous map Γ : R → E such that Γ(t) = S t-s [Γ(s)] for all s < t in R and Γ(t) → p ± in E as t → ±∞.

Step 2: the existence of some ordered stationary fronts. From Lemmas 4.1 and 4.2, 0 < u L < 1 is a strict subsolution and 0 < u L < 1 is a strict subsolution of (1.1). Hence, from the parabolic maximum principle, u(t, x; u L ) is increasing in t > 0, while u(t, x; u L ) is decreasing in t > 0. From standard parabolic estimates, u(t, x; u L ) converges as t → +∞ to a stationary solution 0 ≤ u 1 (x) ≤ 1 of (1.1) and u(t, x; u L ) converges as t → +∞ to a stationary solution 0 ≤ u 2 (x) ≤ 1 of (1.1), locally uniformly in x ∈ R. Furthermore, since 0 < u L < u L < 1 in R from Lemma 4.3, there holds 0 < u L (x) < u(t, x; u L ) < u(t, x; u L ) < u L (x) < 1 for all x ∈ R and t > 0, from the strong maximum principle, whence

0 < u L < u 1 ≤ u 2 < u L < 1 in R. (4.15)
On the other hand, since u

L (-∞) = u L (-∞) = 1 and u L (+∞) = u L (+∞) = 0, one has u 1 (-∞) = u 2 (-∞) = 1, u 1 (+∞) = u 2 (+∞) = 0, while u(t, -∞; u L ) = u(t, -∞; u L ) = 1 and u(t, +∞; u L ) = u(t, +∞; u L ) = 0 uniformly in t > 0. Therefore, u(t, •; u L ) → u 1 and u(t, •; u L ) → u 2 as t → +∞ uniformly in R, that is, S t [u L ] → u 1 and S t [u L ] → u 2 in E as t → +∞.
The functions u 1 and u 2 are thus ordered stationary fronts connecting 0 and 1 for (1.1). The lower front u 1 is approached from below by a solution of the Cauchy problem associated to (1.1), while the upper front u 2 is approached from above. For the conclusion of Theorem 1.7, we would need the opposite situation, namely a lower front which is approached from above and an upper front which is approached from below. To do so, we will, say, keep u 2 as such and shift u 1 to the right in order to make it larger than u 2 . Namely, since both u 1 and u 2 have the same exponential convergence rates λ ± at ±∞ in the sense of (4.9) in Lemma 4.4, and since both u 1 and u 2 are continuous and range in (0, 1), it is elementary to check that there is k ∈ N such that 0 < u 2 (x) < u 3 (x) := u 1 (x -kL) < 1 for all x ∈ R.

(4.16)

Step 3: the stability from above and below of u 2 and u 3 . Let us define the order interval

I := [u 2 , u 3 ] E = u ∈ E ; u 2 ≤ E u ≤ E u 3 = u ∈ BU C(R, R) ; u 2 ≤ u ≤ u 3 in R
and let us show in this step that u 2 is stable from above in I for the semiflow (S t ) t>0 and that u 3 is stable from below in I for the semiflow (S t ) t>0 , in the sense that there is σ > 0 such that

∀ u ∈ I ∩ B E (u 2 , σ), S t [u] → u 2 in E as t → +∞, ∀ u ∈ I ∩ B E (u 3 , σ), S t [u] → u 3 in E as t → +∞.
Indeed, first of all, it follows from (4.15) and (4.16) that u 2 < min(u 3 , u L ) ≤ u 3 in R. Observe from (4.2), (4.9) and the negativity of c that 0 < λ + < λ + , 0 < λ -< λ -and that

lim x→+∞ u 3 (x) u L (x) = 0 and lim x→-∞ 1 -u 3 (x) 1 -u L (x) = +∞.
Therefore, there is C > 0 such that u 3 (x) < u L (x) for all |x| ≥ C, whence min(u 3 , u L ) = u 3 in R\(-C, C). On the other hand, by continuity of u 2 , u 3 and u L and the inequality u 2 < min(u 3 , u L ) in R, there is σ > 0 such that u 2 (x) + σ ≤ min(u 3 (x), u L (x)) for all |x| ≤ C. As a consequence, for any u

∈ I ∩ B E (u 2 , σ) = [u 2 , u 3 ] E ∩ B E (u 2 , σ), one has u 2 (x) ≤ u(x) ≤ u 3 (x) = min(u 3 (x), u L (x)) for all |x| ≥ C, u 2 (x) ≤ u(x) < u 2 (x) + σ ≤ min(u 3 (x), u L (x)) for all |x| ≤ C, (4.17)
whence Similarly, it follows from (4.8) and (4.9) that 0 < λ + < λ + , 0 < λ -< λ -and that max(u 2 (x), u L (x -kL)) = u 2 (x) for all |x| ≥ C ′ , for some C ′ > 0. On the other hand, u L (• -kL) ≤ max(u 2 , u L (• -kL)) < u 3 in R from (4.15) and (4.16). Hence, as in the previous paragraph, there is σ ′ > 0 such that, for all u ∈ I ∩ B E (u 3 , σ ′ ), there holds

u 2 ≤ u ≤ min(u 3 , u L ) ≤ u L in R.
u L (• -kL) ≤ max(u 2 , u L (• -kL)) ≤ u ≤ u 3 in R, whence S t [u] → u 3 in E as t → +∞ (notice that S t [u L (• -kL)] = S t [u L ](• -kL) → u 1 (• -kL) = u 3
in E as t → +∞, since (1.1) is L-periodic in x). Therefore, u 3 is stable from below in I for the semiflow (S t ) t>0 .

Step 4: the existence of transition fronts which are not pulsating fronts. From the previous steps, u 2 < u 3 are ordered equilibria for the continuous and strictly order-preserving semiflow (S t ) t>0 , and u 2 and u 3 are stable respectively from above and below in I = [u 2 , u 3 ] E for (S t ) t>0 . Furthermore, for any t > 0, the set S t [I] is precompact with respect to the compact open topology, from standard parabolic estimates. Since u 2 (-∞) = u 3 (-∞) = 1 and u 2 (+∞) = u 3 (+∞) = 0, and since the semiflow is order-preserving, it follows then that S t [I] is precompact in E with respect to the L ∞norm for every t > 0.

It follows then from the Dancer-Hess connecting orbit lemma (see, e.g., [27, Proposition 9.1]) that the semiflow (S t ) t>0 has an equilibrium u * in I\{u 2 , u 3 }. From the strong elliptic maximum principle, u * is a stationary solution of (1.1) such that u 2 < u * < u 3 in R.

Let us now define the set E = p ∈ [u 2 , u * ] E \{u 2 } ; p is an equilibrium of (S t ) t>0 , that is, E is the set of stationary solutions p of (1.1) such that u 2 ≤, ≡ p ≤ u * in R. We will apply Zorn lemma to show the existence of a minimal element in E. First of all, notice that E is not empty, since u * ∈ E. Consider now any non-empty and totally ordered subset F of E and let us show that there is p ∈ E such that p ≤ E q for all q ∈ F. Namely, define p(x) = inf q(x) ; q ∈ F .

One immediately has 0 < u 2 ≤ p ≤ q ≤ u * ≤ u 3 < 1 in R for all q ∈ F.

In particular, p ∈ [u 2 , u * ] E . One shall show that p ∈ E, that is, p ≡ u 2 in R and p is a stationary solution of (1.1). First of all, we claim that, for any q ∈ F, there is x q ∈ R such that q(x q ) > min(u 3 (x q ), u L (x q )).

Otherwise, (u 2 ≤) q ≤ min(u 3 , u L ) (≤ u L ) in R and q = S t [q] → u 2 in E as t → +∞, whence q = u 2 , contradicting q ∈ E. Thus, q(x q ) > min(u 3 (x q ), u L (x q )) for some x q ∈ R. Furthermore, q ≤ u * ≤ u 3 in R and min(u 3 (x), u L (x)) = u 3 (x) for all |x| ≥ C, where C > 0 independent of q is as in (4.17). Therefore |x q | < C, whence q(x q ) > min(u 3 (x q ), u L (x q )) > u 2 (x q ) + σ, where σ > 0 independent of q is as in (4.17). Since q ≥ u 2 are two stationary solutions of (1.1), it follows then from Harnack inequality that there is a constant ω > 0, independent of q, such that min [-C,C] (q -u 2 ) ≥ ω > 0. As a consequence, p(x) ≥ u 2 (x) + ω for all |x| ≤ C. In particular, p ≡ u 2 in R.

Let us then show that p is a stationary solution of (1.1). First of all, since F is totally ordered and consists of stationary solutions of (1.1), the strong maximum principle implies that, for any q, q ∈ F, there holds either q < q in R, or q > q in R, or q = q in R. By definition of p, there is a sequence (p n ) n∈N in F such that p n (0) → p(0) as n → +∞.

Since p n (0) ≥ p(0) for all n ∈ N, one can assume without loss of generality that p n (0) ≥ p n+1 (0) for all n ∈ N, whence p n ≥ p n+1 in R for all n ∈ N. Since q(0) ≥ p(0) for all q ∈ F by definition of p, two cases may then occur: either there is q 0 ∈ F such that q 0 (0) = p(0), or q(0) > p(0) for all q ∈ F.

In the first case, one infers that q 0 (0) = p(0) ≤ q(0) for all q ∈ F, whence (p ≤) q 0 ≤ q in R for all q ∈ F; finally q 0 = p and thus p ∈ E. In the second case, one has q(0) > p(0) for all q ∈ F. Fix any x ∈ R and ε > 0. By definition of p, there is q ∈ F such that p(x) ≤ q(x) ≤ p(x) + ε. Since p(0) < q(0), there is n 0 ∈ N such that p(0) ≤ p n (0) < q(0) for all n ≥ n 0 , whence p n < q in R and p(x) ≤ p n (x) < q(x) ≤ p(x) + ε for all n ≥ n 0 . Therefore, p n (x) → p(x) as n → +∞ for all x ∈ R. Since 0 < u 2 ≤ p n ≤ u * < u 3 < 1 in R and each p n is a stationary solution of (1.1), standard elliptic estimates imply then that p n → p in C 2 loc (R) and p is thus a stationary solution of (1.1).

Finally, in all cases, one has shown that p ∈ [u 2 , u * ] E \{u 2 } is an equilibrium of the semiflow (S t ) t>0 such that p ≤ q in R for all q ∈ F. In other words, p is a minorant of F which belongs to E. Therefore, Zorn lemma provides the existence of a minimal element u 4 in E, the minimality meaning that there is no element p ∈ E\{u 4 } such that p ≤ u 4 . In other words, u 4 is a stationary solution of (1.1) such that u 2 < u 4 ≤ u * (from the strong maximum principle) and the semiflow (S t ) t>0 has no equilibrium in [u 2 , u 4 ] E \{u 2 , u 4 }. It follows then from the Dancer-Hess connecting orbit lemma and from the stability of u 2 from above in I that there is a time-decreasing entire orbit connecting u 4 to u 2 , that is, a solution u : R 2 → R of (1.1) such that u(t, x) is time-decreasing, with u(t, x) → u 4 (x) as t → -∞ and u(t, x) → u 2 (x) as t → +∞ uniformly in x ∈ R. In particular, 0 < u 2 (x) < u(t, x) < u 4 (x) ≤ u * (x) < u 3 (x) < 1 for all (t, x) ∈ R 2 and since u 2 (-∞) = u 3 (-∞) = 1 and u 2 (-∞) = u 3 (-∞) = 0, one gets that u is a transition front connecting 0 and 1 for (1.1), with for instance ξ t = 0 for all t ∈ R.

Similarly, one can show the existence of a stationary solution u 5 and a time-increasing solution v(t, x) of (1.1) such that 0 < u 2 (x) < u * (x) ≤ u 5 (x) < v(t, x) < u 3 (x) < 1 for all (t, x) ∈ R 2 , with v(t, x) → u 5 (x) as t → -∞ and v(t, x) → u 3 (x) as t → +∞ uniformly in x ∈ R. Furthermore, v is a transition front connecting 0 and 1 for (1.1). Finally, 0 < u 2 < u 4 ≤ u * ≤ u 5 < u 3 in R and the desired conclusion of Theorem 1.7 follows with u -= u 2 , u + = u 4 , v -= u 5 and v + = u 3 .

Step 5: the characterization of global mean speeds. Lastly, we show that for any transition front u connecting 0 and 1 for (1.1) and associated with positions (ξ t ) t∈R , there is M ≥ 0 such that |ξ t -ξ s | ≤ M for all (t, s) ∈ R 2 . Assume not. By Lemma 2.1, there are then two sequences (t k ) k∈N and (s k ) k∈N of real numbers such that |t k -s k | → +∞ and |ξ t k -ξ s k | → +∞ as k → +∞. Without loss of generality, one can assume that s k < t k for all k ∈ N.

Suppose firstly that, up to extraction of some subsequence, ξ t k -ξ s k → +∞ as k → +∞. From Definition 1.1, one then infers that, for any x ∈ R, lim k→+∞ u(t k , x + ξ s k ) = 1. On the other hand, it follows from Lemma 4.5, there are x * ∈ R, γ 0 > 0 and q 0 > 0 such that u(t k , x + ξ s k ) ≤ min u L (x -x * ) + q 0 e -γ 0 (t k -s k ) , 1 for all k ∈ N and x ∈ R.

Notice that, for any x ∈ R, lim k→+∞ (u L (x -x * ) + q 0 e -γ 0 (t k -s k ) ) = u L (x -x * ) < 1. Therefore, for every x ∈ R, lim sup k→+∞ u(t k , x + ξ s k ) < 1. One has then reached a contradiction. Secondly, in the case where, up to extraction of some subsequence, ξ t k -ξ s k → -∞ as k → +∞, one can derive a similar contradiction by using the lower bound for u in Lemma 4.5.

Therefore, there is M > 0 such that |ξ t -ξ s | ≤ M for all (t, s) ∈ R 2 . In particular, every transition front connecting 0 and 1 for (1.1) has zero global mean speed. The proof of Theorem 1.7 is thereby complete.

Theorem 1 . 6 .

 16 If there exist a transition front u connecting 0 and 1 and a sequence (t n ) n∈N in R such that lim |t-s|→+∞ lim inf n→+∞ |ξ t+tn -ξ s+tn | = +∞, (1.13)

  continuous and ζ t → +∞ as t → +∞, there are a sequence of real numbers (τ m ) m∈N\{0} such that τ m ≥ (k m + 1)K 0 m and ζ τm = ζ kmK 0 m for all m ∈ N\{0}. Since τ m -k m K 0 m → +∞ as m → +∞, one gets a contradiction with (3.8).

  and k ∈ N. The family (ζ t ) t∈R is also associated to v in the sense of (1.4), since the function t → ζ t -ξ t is bounded by Lemma 2.1 applied to v and ( ξ t ) t∈R . Moreover, by using again Lemma 2.1, the function t → ζ t is actually Lipschitz continuous, and lim

	|t-s|→+∞	|ζ t -ζ s | = +∞	(3.8)

  .[START_REF] Chapuisat | Existence and non-existence of progressive wave solutions for a bistable reactiondiffusion equation in an infinite cylinder whose diameter is suddenly increased[END_REF]) By(3.12) and the uniform continuity of v in R 2 , one infers that sup n∈N (x n -ζ tn ) < B. Furthermore, since the map t → ζ t is Lipschitz-continuous (that is true owing to the construction of ζ t , but this property can always be assumed without loss of generality by Lemma 2.1), there is then ρ

  Since u 2 is a stationary solution of (1.1) and since S t [u L ] → u 2 in E as t → +∞, one concludes from the maximum principle that S t [u] → u 2 in E as t → +∞ for every u ∈ I ∩ B E (u 2 , σ), namely u 2 is stable from above in I for the semiflow (S t ) t>0 .

Notice that this is the place where we use the choice of δ/2 in the second property of (3.10).

* This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government programme managed by the French National Research Agency (ANR). The research leading to these results has also received funding from the ANR within the project NONLOCAL ANR-14-CE25-0013 and from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 -ReaDi -Reaction-Diffusion Equations, Propagation and Modelling. Part of this work was carried out during visits by F. Hamel to the Department of Mathematics and Statistics of Memorial University of Newfoundland, whose hospitality is thankfully acknowledged. X.-Q. Zhao's research is supported in part by the NSERC of Canada. W. Ding is supported by the China Scholarship Council for 2 years of study at Aix-Marseille Université.

show that v L (t, x) is a supersolution of equation (1.1). To do so, for all (t, x) ∈ (0, +∞) × R such that v L (t, x) < 1, define

A straightforward calculation gives, for all (t, x) ∈ (0, +∞) × R such that v L (t, x) < 1,

Since u L (x) = φ(x -Lx) and φ(x -ct) is a traveling front of equation (4.1), it follows that, for all (t, x) ∈ (0, +∞) × R such that v L (t, x) < 1,

For all such (t, x), define

In this paragraph, we will choose suitable functions q(t) and η(t

and 1-periodic in x, there exists a real number δ 0 ∈ (0, 1) (which depends only on f ) such that, if (q 0 , t, x) ∈ (0,

On the other hand, due to the monotonicity of φ, there is a constant ρ = -max δ 0 ≤φ(y)≤1-δ 0 φ ′ (y) > 0 (which actually depends only on φ and δ 0 , that is, only on f

where K = max (x,u)∈R×[0,1] |∂ u f (x, u)|. Now, let us choose q 0 = δ 0 (depending only on f ). There exists η 0 ∈ L(N\{0}) (depending on L and u) such that (4.11) holds. Let us then choose q(t) and η(t) such that q(0) = q 0 , q ′ (t) = -γq(t) 2 for all t ≥ 0, η(0) = η 0 and η ′ (t) = 2K + γ 2ρ q(t) for all t ≥ 0. (4.14)

Namely, q(t) = q 0 e -γt/2 and η(t) = η 0 + q 0 (2K + γ)(1 -e -γt/2 )/(γρ) for all t ≥ 0. It is easy to check from (4.12)-(4.14) and the negativity of u ′ L that q(t) and η(t) are such that N 1 (t, x) ≥ 0 for all (t, x) ∈ (0, +∞) × R such that v L (t, x) < 1. Notice also that the functions q and η -η 0 depend only on f and are thus independent of L and u (but η 0 and η(t) depend on L and u)

Now, for those chosen functions q(t) and η(t), we show that N 2 (t, x) ≥ 0 for all (t, x) ∈ (0, +∞)×R such that v L (t, x) < 1, as soon as L is large enough. The arguments are similar to those used in