François Hamel

Bistable transition fronts in R N

This paper is chiefly concerned with qualitative properties of some reaction-diffusion fronts. The recently defined notions of transition fronts generalize the standard notions of traveling fronts. In this paper, we show the existence and the uniqueness of the global mean speed of bistable transition fronts in R N . This speed is proved to be independent of the shape of the level sets of the fronts. The planar fronts are also characterized in the more general class of almost-planar fronts with any number of transition layers. These qualitative properties show the robustness of the notions of transition fronts. But we also prove the existence of new types of transition fronts in R N that are not standard traveling fronts, thus showing that the notions of transition fronts are broad enough to include other relevant propagating solutions.

Introduction

This paper is concerned with some existence results and qualitative properties of generalized fronts, and with some estimates of their propagation speeds, for semilinear parabolic equations of the type

u t = ∆u + f (u), (t, x) ∈ R × R N , (1.1)
where u t = ∂u ∂t and ∆ denotes the Laplace operator with respect to the space variables x ∈ R N . Throughout the paper, the reaction term f : [0, 1] → R is a C 1 function such that f (0) = f (1) = 0, f ′ (0) < 0 and f ′ (1) < 0.

(

Both zeroes 0 and 1 of f are then stable. We define θ -= min s ∈ (0, 1); f (s) = 0 and θ + = max s ∈ (0, 1); f (s) = 0 .

(

1.3)

There holds 0 < θ -≤ θ + < 1. A particular important case corresponds to bistable nonlinearities f which, in addition to (1.2), satisfy θ -= θ + , that is ∃ θ ∈ (0, 1), f < 0 on (0, θ) and f > 0 on (θ, 1).

(1.4)

A typical example of a function f satisfying (1.4) is the cubic nonlinearity f θ (s) = s(1-s)(s-θ) with 0 < θ < 1, and (1.1) is then often referred to as Nagumo's or Huxley's equation. When θ = 1/2, then equation (1.1) with the function f (s) = 8f 1/2 ((s + 1)/2) = ss 3 with s ∈ [-1, 1] corresponds to the celebrated Allen-Cahn equation arising in material sciences. More generally speaking, equation (1.1) is also one of the most common reaction-diffusion equations arising in various mathematical models in biology or ecology.

In one of the main results of this paper (Theorem 2.8 below), assumption (1.4) will be made together with (1.2). In all other results, the function f is assumed to fulfil (1.2) only and can then be more general than the specific bistable type (1.4).

The solution u : R × R N → [0, 1] typically stands for a normalized density and it is understood as a classical solution of (1.1). From the strong maximum principle, either u is identically equal to 0, or it is identically equal to 1, or it ranges in the open interval (0, 1). We only consider this last situation in the paper.

One of the most important aspects of these equations, which accounted for their success, is the description of propagation phenomena. By that we understand front-like time-global (also called entire, or eternal) solutions u of (1.1) which connect the two stable stationary states 0 and 1 and which, in general, move as time runs (see precise definitions later). These solutions are an important class of solutions in that they usually describe the large-time behavior of the solutions of the associated Cauchy problems (some important large-time dynamics and stability results will actually be used in the proofs of the present paper). Much work has been devoted in the last decades to the study of standard front-like solutions for equations of the type (1.1), and some of the main results in this very active field will be recalled below. On the other hand, new more general notions of propagation speeds and transition fronts, including the standard traveling fronts, have been introduced recently.

In this paper, we prove the existence and the uniqueness of the speed of any front among the class of transition fronts connecting 0 and 1 for problem (1.1). We also establish some one-dimensional symmetry properties and various classification results related to the shape of the level sets of the fronts. All these qualitative properties show the robustness of the notions of transition fronts.

Furthermore, we prove the existence of new types of transition fronts in that are not known standard traveling fronts, thus showing that the notions of transition fronts are broad enough to include other relevant propagating solutions.

Before doing so, we first review the main existence and qualitative results for the standard traveling fronts.

Standard traveling fronts One-dimensional traveling fronts

On the one-dimensional real line, standard traveling fronts are solutions of the type

u(t, x) = φ f (x -c f t),
where c f ∈ R is the propagation speed and φ f : R → [0, 1] is the propagation profile, such that

φ ′′ f + c f φ ′ f + f (φ f ) = 0 in R, φ f (-∞) = 1 and φ f (+∞) = 0.
(1.5)

The profile φ f is then a heteroclinic connection between the stable states 0 and 1. Such solutions u(t, x) = φ f (xc f t) move with constant speed c f and they are invariant in the moving frame with speed c f . If f satisfies (1.4) in addition to (1.2), then such fronts exist, see [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][START_REF] Ya | Stabilization of solution of the Cauchy problem for equations encountred in combustion theory[END_REF]. Under the sole condition (1.2), such fronts do not exist in general but further more precise conditions for the existence and non-existence have been given by Fife and McLeod in [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF]. For instance, if f satisfies (1.2) and if there are some real numbers 0 < θ -< µ < θ + < 1 such that f (θ -) = f (µ) = f (θ + ) = 0, f ′ (µ) < 0, f < 0 on (0, θ -) ∪ (µ, θ + ) and f > 0 on (θ -, µ) ∪ (θ + , 1), then there some fronts (c - f , φ - f ) and (c + f , φ + f ) connecting 0 and µ, and µ and 1 respectively, since the function f is of the bistable type on each of the subintervals [0, µ] and [µ, 1]; furthermore, in this case, it has been proved in [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that a front (c f , φ f ) solving (1.5) exists if and only if c -< c + , and then the inequalities c -< c f < c + hold necessarily.

Coming back to the general case (1.2), if a front (c f , φ f ) solving (1.5) exists, then the speed c f is unique, it only depends on f and it has the sign of 1 0 f , see [START_REF] Fife | Mathematical aspects of reacting and diffusing systems[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF]. In particular, if f is balanced, that is 1 0 f = 0, then c f = 0. Moreover, the profile φ f , if it exists, is unique up to shifts, it is such that φ ′ f < 0 in R and it can be assumed to be fixed with the normalization φ f (0) = 1/2. Lastly, when they exist, these fronts are globally stable in the sense that any solution of the Cauchy problem u t = u xx + f (u) for t > 0 with an initial condition u(0, •) : R → [0, 1] such that lim inf x→-∞ u(0, x) > θ + ≥ θ -> lim sup x→+∞ u(0, x) converges to the traveling front φ f (xc f t + ξ) uniformly in x ∈ R as t → +∞, where ξ is a real number which only depends on u(0, •) and f , see [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF][START_REF] Gallay | A variational proof of global stability for bistable travelling waves[END_REF]. Let us mention here that the uniqueness of the speed c f is in sharp contrast with the case of positive nonlinearities f > 0 on (0, 1), for which the set of admissible speeds is a continuum [c * f , +∞) with c * f > 0, see e.g. [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF].

Throughout the paper (except in the specific Proposition 3.3 in Section 3.2 below), we assume (1.2) and the existence (and then the uniqueness) of a planar front (c f , φ f ) solving (1.5).

In particular, we insist on the fact that all results of this paper hold if f is of the important bistable type (1.4). In one of the main results (Theorem 2.8), we actually assume additionally that f has the bistable profile (1.4).

Standard planar traveling fronts in R N with N ≥ 1

In any dimension N ≥ 1, planar fronts u(t, x) = φ f (x • ec f t), if any, are unique up to shifts, for any given unit vector e of R N , where the one-dimensional profile φ f is as above. The level sets of such traveling fronts are parallel hyperplanes which are orthogonal to the direction of propagation e. These fronts are invariant in the moving frame with speed c f in the direction e and the unique speed c f , if any, can then be referred in the sequel as the speed of planar fronts connecting 0 and 1 for problem (1.1). Lastly, when f is of the bistable type (1.4), these planar fronts, which exist, are known to be stable with respect to some natural classes of perturbations, see [START_REF] Levermore | Multidimensional stability of traveling waves in a bistable reaction-diffusion equation[END_REF][START_REF] Matano | Large time behavior of disturbed planar fronts in the Allen-Cahn equation[END_REF][START_REF] Matano | Stability of planar waves in the Allen-Cahn equation[END_REF][START_REF] Xin | Multidimensional stability of travelling waves in a bistable reaction-diffusion equation[END_REF].

Standard non-planar traveling fronts in R N with N ≥ 2 When N ≥ 2 and f fulfills (1.4) with, say, c f > 0, there are other traveling fronts, which have non-planar level sets. That is, there are fronts whose profiles are still invariant in a moving frame with constant speed, but whose level sets are not hyperplanes anymore. Namely, taking x N as the direction of propagation without loss of generality, calling x ′ = (x 1 , . . . , x N -1 ) 1 + • • • + x 2 N -1 ) 1/2 and letting α ∈ (0, π/2) be any given angle, equation (1.1) admits "conical-shaped" axisymmetric non-planar fronts of the type

u(t, x) = φ(|x ′ |, x N -ct)
such that φ(r, z) → 1 (resp. 0) unif. as zψ(r) → -∞ (resp. + ∞),

c = c f sin α and ψ ′ (+∞) = cot α, (1.6)
for some C 1 function ψ : [0, +∞) → R, see [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF][START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF] and the joint figure. When c f < 0, the same result holds after changing the roles of the limits 0 and 1. Fronts u(t, x) = φ(x ′ , x Nct) with non-axisymmetric shape, such as pyramidal fronts, are also known to exist, see [START_REF] Taniguchi | Traveling fronts of pyramidal shapes in the Allen-Cahn equation[END_REF][START_REF] Taniguchi | Multi-dimensional traveling fronts in bistable reaction-diffusion equations[END_REF]. A large literature has been devoted to the study of these axisymmetric and non-axisymmetric fronts in the recent years. For symmetry, uniqueness, stability and further qualitative properties of these traveling fronts, we refer to [START_REF] Hamel | Solutions of semilinear elliptic equations in R N with conical-shaped level sets[END_REF][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] Roquejoffre | Nontrivial large-time behaviour in bistable reaction-diffusion equations[END_REF][START_REF] Taniguchi | The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations[END_REF][START_REF] Taniguchi | Multi-dimensional traveling fronts in bistable reaction-diffusion equations[END_REF] (see also [START_REF] Haragus | Corner defects in almost planar interface propagation[END_REF] for the existence of conical-shaped fronts for some systems of reaction-diffusion equations and for angles α close to π/2).

When N ≥ 2 and f fulfills (1.4) with c f = 0, fronts u(t, x) = φ(|x ′ |, x Nct) with conicalshaped level sets cannot exist anymore, see [START_REF] Hamel | Solutions of semilinear elliptic equations in R N with conical-shaped level sets[END_REF]. Nevertheless, for every c = 0, there exist some fronts u(t, x) = φ(|x ′ |, x Nct) such that φ(r, z) → 1 (resp. 0) as z → -∞ (resp. +∞) for every r ≥ 0 and whose level sets have an exponential shape (if N = 2) or a parabolic shape (if N ≥ 3), see [START_REF] Chen | Traveling waves with paraboloid like interfaces for balanced bistable dynamics[END_REF]. The axisymmetry, up to shifts, of these fronts in dimension N = 2 has been proved in [START_REF] Gui | Symmetry of traveling wave solutions to the Allen-Cahn equation in R 2[END_REF]. Furthermore, when N ≥ 3 and |c| = 0 is small enough, (1.1) also admits fronts u(t, x) = φ(|x ′ |, x Nct) such that φ(r, ±∞) = 0 for every r ≥ 0, sup x∈R N φ(|x ′ |, x N ) = 1 and the level set

E = x ∈ R N ; φ(|x ′ |, x N ) = 1
2 is made of two non-Lipschitz graphs, see [START_REF] Del Pino | Traveling waves with multiple and non-convex fronts for a bistable semilinear parabolic equation[END_REF] (other axisymmetric fronts exist for which E has only one connected component and has the shape of a catenoid, see [START_REF] Del Pino | Traveling waves with multiple and non-convex fronts for a bistable semilinear parabolic equation[END_REF]). On the other hand, for any N ≥ 2 (still with c f = 0), planar stationary fronts u(t, x) = φ f (x•e) obviously still exist, for any unit vector e. Stationary solutions u(x) of (1.1) such that u x N < 0 and u(x ′ , x N ) → 1 (resp. 0) as x N → -∞ (resp. +∞) are necessarily planar -hence of the type φ f (x • e) up to shifts-if N ≤ 8 [START_REF] Ambrosio | Entire solutions of semilinear elliptic equations in R 3 and a conjecture of De Giorgi[END_REF][START_REF] Ghoussoub | On a conjecture of De Giorgi and some related problems[END_REF][START_REF] Savin | Regularity of flat level sets in phase transitions[END_REF] whereas they are not always planar when N ≥ 9 [START_REF] Del Pino | On the De Giorgi conjecture in dimensions N ≥ 9[END_REF] (this problem is related to a celebrated conjecture by De Giorgi [START_REF] Giorgi | Convergence problems for functionals and operators[END_REF] for f (s) = s(1s)(s -1/2)). Other non-monotone stationary saddle-shaped solutions or solutions whose level sets have multiple ends are also known to exist for some balanced bistable functions f , see [START_REF] Alessio | Saddle-type solutions for a class of semilinear elliptic equations[END_REF][START_REF] Cabré | Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation[END_REF][START_REF] Cabré | Saddle-shaped solutions of bistable diffusion equations in all of R 2m[END_REF][START_REF] Cabré | Qualitative properties of saddle-shaped solutions to bistable diffusion equations[END_REF][START_REF] Dang | Saddle solutions of the bistable diffusion equation[END_REF][START_REF] Del Pino | Multiple-end solutions to the Allen-Cahn equation in R 2[END_REF].

Notions of transition fronts and global mean speed

The above examples show that equation (1.1) admits many types of traveling fronts. For all of them, the solutions u converge to the stable states 0 or 1 far away from their moving or stationary level sets, uniformly in time. This observation will be the key-point of the more general notion of transition fronts given in Definition 1.1 below. Furthermore, another common property fulfilled by all the standard traveling fronts is that their level sets move at the global mean speed |c f |, only depending on f , in a sense to be made more precise below. One of the main goals of the present paper is actually to prove that this property is shared by all transition fronts.

Let us now describe the general notions of transition fronts and global mean speed for problem (1.1). First, for any two subsets A and B of R N and for x ∈ R N , we set

d(A, B) = inf |x -y|, (x, y) ∈ A × B (1.7)
and d(x, A) = d({x}, A). The notions of transition fronts and global mean speeds are borrowed from [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] and are adapted here to the case of connections between the constant stationary states 0 and 1 in R N . They involve two families (Ω - t ) t∈R and (

Ω + t ) t∈R of open nonempty subsets of R N such that ∀ t ∈ R,          Ω - t ∩ Ω + t = ∅, ∂Ω - t = ∂Ω + t =: Γ t , Ω - t ∪ Γ t ∪ Ω + t = R N , sup d(x, Γ t ); x ∈ Ω + t = sup d(x, Γ t ); x ∈ Ω - t = +∞ (1.8) and    inf sup d(y, Γ t ); y ∈ Ω + t , |y -x| ≤ r ; t ∈ R, x ∈ Γ t → +∞ inf sup d(y, Γ t ); y ∈ Ω - t , |y -x| ≤ r ; t ∈ R, x ∈ Γ t → +∞ as r → +∞. 1 (1.9)
Notice that the condition (1.8) implies in particular that the interface Γ t is not empty for every t ∈ R. As far as condition (1.9) is concerned, it is illustrated in the joint figure. Roughly speaking, it means that, when r is large, for every t ∈ R and every point x ∈ Γ t , there are some points y + r,t,x and y - r,t,x in both Ω + t and Ω - t which are far from Γ t and are at a distance at most r from x (in particular, the points y ± r,t,x belong to the geodesic tubular neighborhoods of width r of the sets Γ t ). Moreover, the sets Γ t are assumed to be made of a finite number of graphs: there is an integer n ≥ 1 such that, for each t ∈ R, there are n open subsets

ω i,t ⊂ R N -1 (for 1 ≤ i ≤ n), n continuous maps ψ i,t : ω i,t → R and n rotations R i,t of R N , such that Γ t ⊂ 1≤i≤n R i,t x ∈ R N ; x ′ ∈ ω i,t , x N = ψ i,t (x ′ ) . (1.10) When N = 1, (1.10) means that Γ t has cardinal at most n, that is Γ t = {x 1,t , • • • , x n,t }, for each t ∈ R.
Definition 1.1 [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] For problem (1.1), a transition front connecting 0 and 1 is a classical solution u : R × R N → (0, 1) for which there exist some sets (Ω ± t ) t∈R and (Γ t ) t∈R satisfying (1.8), (1.9) and (1.10), and, for every ε > 0, there exists M ≥ 0 such that

∀ t ∈ R, ∀ x ∈ Ω + t , d(x, Γ t ) ≥ M =⇒ u(t, x) ≥ 1 -ε , ∀ t ∈ R, ∀ x ∈ Ω - t , d(x, Γ t ) ≥ M =⇒ u(t, x) ≤ ε . (1.11)
Furthermore, u is said to have a global mean speed γ (≥ 0) if

d(Γ t , Γ s ) |t -s| → γ as |t -s| → +∞. (1.12)
Before stating the main results of this paper, let us comment the notions given in Definition 1.1 and let us connect them with the standard notions of traveling fronts. Firstly, it is easy to check that all moving or stationary fronts mentioned in Section 1.1 are transition fronts connecting 0 and 1 in the sense of Definition 1.1, for some suitable choices of sets (Ω ± t ) t∈R . For instance, when f fulfils (1.4) with c f = 0, the conical-shaped fronts u(t, x) = φ(|x ′ |, x Nct) satisfying (1.6) are transition fronts connecting 0 and 1 with, say,

Ω + t = x ∈ R N ; x N < ψ(|x ′ |) + ct , Ω - t = x ∈ R N ; x N > ψ(|x ′ |) + ct and Γ t = {x ∈ R N ; x N = ψ(|x ′ |)
+ ct} for every t ∈ R. However, Definition 1.1 covers other transition fronts than the one described in Section 1.1 (see Theorem 2.8 below). Definition 1.1 was actually given in [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] for more general domains, equations or limiting states (instead of 0 and 1). In the recent years, many papers have been devoted to the existence and stability of transition fronts for equations of the type (1.1) in R or in infinite cylinders with x-dependent [START_REF] Mellet | Stability of generalized transition fronts[END_REF][START_REF] Mellet | Generalized fronts for one-dimensional reaction-diffusion equations[END_REF][START_REF] Nolen | Existence and non-existence of Fisher-KPP transition fronts[END_REF][START_REF] Nolen | Traveling waves in a one-dimensional heterogeneous medium[END_REF][START_REF] Zlatoš | Generalized traveling waves in disordered media: existence, uniqueness, and stability[END_REF][START_REF] Zlatoš | Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations[END_REF] and t-or (t, x)-dependent [START_REF] Berestycki | Generalized transition waves and their properties[END_REF][START_REF] Nadin | Propagation phenomena for time heterogeneous KPP reaction-diffusion equations[END_REF][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 | Dynamical systems and traveling waves in almost periodic structures[END_REF][START_REF] Shen | Traveling waves in diffusive random media[END_REF][START_REF] Shen | Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models[END_REF][START_REF] Shen | Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations[END_REF] bistable, combustion or monostable nonlinearities f , as well as for (1.1) in exterior [START_REF] Berestycki | Bistable travelling waves around an obstacle[END_REF] or cylindrical-type domains (see [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][START_REF] Matano | Talks at various conferences[END_REF][START_REF] Roques | A population facing climate change: joint influences of Allee effects and environmental boundary geometry[END_REF], where blocking phenomena are also shown). We also mention [START_REF] Alikakos | Periodic traveling waves and locating oscillating patterns in multidimensional domains[END_REF][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][START_REF] Heinze | Wave solutions to reaction-diffusion systems in perforated domains[END_REF][START_REF] Papanicolaou | Reaction-diffusion fronts in periodically layered media[END_REF][START_REF] Wang | Periodic traveling curved fronts in reaction-diffusion equation with bistable timeperiodic nonlinearity[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 | Analysis and modeling of front propagation in heterogeneous media[END_REF] for the existence and qualitative properties of bistable pulsating fronts in periodic media. In a subsequent paper [START_REF] Hamel | On the mean speed of bistable fronts in heterogeneous media[END_REF], we establish some bounds and estimates for the mean speed of transition fronts in heterogeneous media or in more general domains. In the present paper, for the sake of clarity and homogeneity of the presentation, we only focus on the case of the homogeneous equation (1.1) in the whole space R N but we prove that even this simple-looking problem already has many deep properties: new classification results and general estimates shared by all transition fronts are shown, not to mention the existence of new transition fronts. For a given transition front u, the sets (Ω ± t ) t∈R and (Γ t ) t∈R are not uniquely determined, in the sense that two families (Ω ± t ) t∈R (resp. (Γ t ) t∈R ) and ( Ω ± t ) t∈R (resp. ( Γ t ) t∈R ) may be associated to the same transition font u. Nevertheless, due to the key uniformity property in (1.11), the sets Γ t are located at a uniformly bounded distance of any given level set of u and this boundedness property is intrinsic. Namely, under the assumption that sup d(x, Γ t-τ ); t ∈ R, x ∈ Γ t < +∞ for some τ > 0 (the interfaces Γ t and Γ t-τ are in some sense not too far from each other), it follows from Theorem 1.2 of [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] that

∀ λ ∈ (0, 1), sup d(x, Γ t ); u(t, x) = λ, (t, x) ∈ R × R N < +∞ and, for every C ≥ 0, there is η > 0 such that η ≤ u(t, x) ≤ 1 -η for all (t, x) ∈ R × R N with d(x, Γ t ) ≤ C.
Roughly speaking, the transition zone between 0 and 1 is thus a neighborhood with uniformly bounded width of the (possibly moving) interfaces Γ t (notice however that the transition zone between 0 and 1 can also be made of several possibly disconnected transition zones, each of them being a neighborhood of a graph, see for instance the aforementioned examples from [START_REF] Del Pino | Traveling waves with multiple and non-convex fronts for a bistable semilinear parabolic equation[END_REF] and another example after Theorem 2.5 below). The word transition in Definition 1.1 thus corresponds to the intuitive idea of a spatial transition (we refer to [START_REF] Nadin | Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations[END_REF] for the related but different notion of critical transition).

The global mean speed γ of a transition front, if any, corresponds to the limiting average speed of the minimal distance between the interfaces Γ t and can then be viewed as a mean smallest normal speed of the interfaces Γ t . Since the sets Γ t are not uniquely determined, the notion of instantaneous normal speed of Γ t has no sense. However, due to the remarks of the previous paragraph, the notion of global mean speed γ given in (1.12) is meaningful. As a matter of fact, it is essential to say that, for a given transition front u, the global mean speed γ, if any, is uniquely determined and does not depend on the specific choice of the sets (Ω ± t ) t∈R and (Γ t ) t∈R , see Theorem 1.7 in [START_REF] Berestycki | Generalized transition waves and their properties[END_REF]. For instance, when f fulfils (1.4) with c f = 0, the conical-shaped fronts u(t, x) = φ(|x ′ |, x Nct) satisfying (1.6) have a global mean speed γ and γ = |c f |, whatever the angle α ∈ (0, π/2) may be. For any such front u, the speed c = c f / sin α is the speed in the vertical direction x N of the frame in which the front is invariant, but the asymptotical smallest normal speed of the level sets of u is equal to |c f |. It is also straightforward to check that, when f fulfils (1.4) with c f = 0, the fronts mentioned in Section 1.1 have global mean speed γ = 0: this fact is obvious when the fronts are stationary, since the Γ t can all be chosen as any given (time-independent) level set, but this property also holds good for the exponentially-shaped or parabolic-shaped fronts since the level sets have an infinite slope in the (x ′ , x N ) coordinates as |x ′ | → +∞. These exponentially-shaped or parabolic-shaped fronts have zero global mean speed γ = 0, but they are not stationary.

What is much stronger and not trivial at all is to show that, whatever the shape of the fronts and the value of c f may be, all transition fronts for (1.1) have a global mean speed and this speed is equal to |c f |: this will be one of the main results of this paper, see Theorem 2.6 below.

Main results

The first main results are concerned with some qualitative geometrical properties of the transition fronts connecting 0 and 1 for problem (1.1), including some new classification Liouvilletype results, and with some estimates of their global mean speed. More precisely, in the following subsections, we first give a characterization of the planar fronts among the more general class of almost-planar transition fronts. We then give a characterization of the mean speed of all transition fronts. Lastly, we deal with the existence of new non-standard transition fronts.

Almost-planar and planar fronts

Planar fronts connecting 0 and 1 for (1.1) are solutions of the type φ(x•e-ct) with φ(-∞) = 1 and φ(+∞) = 0. As recalled in Section 1.1, the function φ = φ f , if any, is unique up to shifts and the speed c = c f , if any, is unique. These fronts have planar level sets and they are monotone with respect to the direction of propagation, at each time t. In particular, they fall within the more general class of almost-planar fronts introduced in [START_REF] Berestycki | Generalized transition waves and their properties[END_REF], and defined as follows.

Definition 2.1 A transition front u in the sense of Definition 1.1 is called almost-planar if, for every t ∈ R, the set Γ t can be chosen as the hyperplane

Γ t = x ∈ R N ; x • e t = ξ t
for some vector e t of the unit sphere S N -1 and some real number ξ t .

In other words, the level sets of almost-planar fronts are in some sense close to hyperplanes, even if they are not a priori assumed to be planar. In [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF], we gave a characterization of the almost-planar fronts for which e t = e is a given constant vector and for which there exists γ ≥ 0 such that |ξ tξ t | -γ|t -s| is bounded uniformly with respect to (t, s) ∈ R 2 : such fronts have to be planar fronts φ f (±x • ec f t), up to shifts, and γ = |c f | (see Theorem 3.1 in [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF]).

In this paper, we first give a more general characterization of the planar fronts φ f (x•e-c f t) for problem (1.1), without assuming that the directions e t are a priori constant and without assuming any a priori bound on the positions ξ t . Proposition 2.2 For problem (1.1), any almost-planar transition front u connecting 0 and 1 is planar, that is there exist a unit vector e of R N and a real number ξ such that

u(t, x) = φ f (x • e -c f t + ξ) for all (t, x) ∈ R × R N .
(2.1) The first step in the proof of Proposition 2.2 is to show that the directions e t appearing in Definition 2.1 are equal to a constant vector e independent of time. As a consequence, u converges to 0 and 1 as x • eξ t → ±∞, uniformly in t and in the spatial variables orthogonal to e. These properties have some similarities with the Gibbons conjecture about the onedimensional symmetry of solutions 0 ≤ v ≤ 1 of elliptic equations ∆v + f (v) = 0 in R N with f satisfying (1.2) and v(x) → 0 and 1 as x•e → ±∞ uniformly in the variables orthogonal to e: for this latter problem, the solutions are proved to depend on x•e only (and, necessarily, [START_REF] Barlow | The Liouville property and a conjecture of De Giorgi[END_REF][START_REF] Berestycki | One-dimensional symmetry of bounded entire solutions of some elliptic equations[END_REF][START_REF] Farina | Symmetry for solutions of semilinear elliptic equations in R n and related conjectures[END_REF][START_REF] Farina | The state of the art for a conjecture of De Giorgi and related problems[END_REF]. As for our parabolic problem (1.1), the difference is that nothing is imposed a priori on the function ξ t : to get the conclusion (2.1), the one-dimensional stability of the planar front φ f (see Fife and McLeod [29]) is used to get the boundedness of t → ξ tc f t, together with the aforementioned parabolic Liouville type result of Berestycki and the author (Theorem 3.1 in [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF]).

1 0 f = 0), see
Remark 2.3 As a matter of fact, the existence of the planar front (c f , φ f ), which is always assumed by default throughout the paper, can almost be dropped in Proposition 2.2. Namely, there is a C 1 ([0, 1]) dense set of functions f satisfying (1.2) such that the existence of an almostplanar transition front connecting 0 and 1 for problem (1.1) in R N implies the existence of a planar one-dimensional front (c f , φ f ), and then the conclusion (2.1): in other words, for these functions f , either there is a planar front (c f , φ f ) connecting 0 and 1, or there is no almostplanar transition front connecting 0 and 1 in R N . We refer to Section 3.2 and Proposition 3.3 below for more details. However, for the sake of the unity of the presentation, in Proposition 2.2 as well as in all other results, we have chosen to keep the default assumption of the existence of a planar front (c f , φ f ). Lastly, we repeat that the assumption is fulfilled automatically if f is of the bistable type (1.4) Indeed, the boundedness of diam(Γ t ) in dimension N = 1 implies that Γ t can be reduced to a singleton without loss of generality, that is there is only one interface between the limiting values 0 and 1. Notice that the boundedness of the width of the transition between 0 and 1, which is one of the key-properties in Definition 1.1, is necessary for the conclusion of Proposition 2.2 to hold in general, even in dimension N = 1. For instance, if f is of the bistable type (1.4), there exist some solutions u of (1.1) in R such that 0 < u(t, x) < 1 in R 2 , u(t, -∞) = 1 and u(t, +∞) = 0 for every t ∈ R and for which (1.11) is not satisfied for any families (Ω ± t ) t∈R and (Γ t ) t∈R satisfying (1.8), (1.9) and (1.10), see [START_REF] Morita | Entire solutions with merging fronts to reaction-diffusion equations[END_REF]. These solutions are indeed constructed in such a way that they are close to θ on very large intervals as t → -∞. Thus they cannot be transition fronts in the sense of Definition 1.1.

Coming back to the transition fronts in R N for any dimension N ≥ 1, the conclusion of Proposition 2.2 still holds when, at each time t, the transition between 0 and 1 is made of a finite number of bounded parallel strips, under the additional condition that the planar speed c f is not zero. More precisely, the following result holds.

Theorem 2.5 For problem (1.1), let u be a transition front connecting 0 and 1 such that, for every t ∈ R, there are e t in S N -1 and ξ 1,t , . . . , ξ n,t in R such that

Γ t = 1≤i≤n x ∈ R N ; x • e t = ξ i,t . (2.2)
If c f = 0, then u is a planar front of the type (2.1).

The condition that c f is not zero is actually necessary. Indeed, for some nonlinearities f such that c f = 0, there are transition fronts connecting 0 and 1 in R such that, say, Γ t = ξ 1,t , ξ 2,t , ξ 3,t for all t < 0, with ξ 1,t < ξ 2,t < ξ 3,t for every t < 0, ξ 1,t → -∞, ξ 3,t → +∞, |ξ 1,t | = o(|t|), ξ 3,t = o(|t|) as t → -∞, and Γ t = 0 for all t ≥ 0.

These transition fronts, which can be derived from [START_REF] Eckmann | Coarsening by Ginzburg-Landau dynamics[END_REF][START_REF] Ei | The motion of weakly interacting pulses in reaction-diffusion systems[END_REF], describe the slow dynamics of some almost-stationary fronts. They have three interfaces as t → -∞, the leftmost and rightmost ones move toward the origin and disappear in finite time, and only one remains as t → +∞, in the sense that the solution converges to a finite shift of the stationary front φ f (±x) as t → +∞. Notice that such solutions have global mean speed γ = c f = 0 in the sense of Definition 1.1, and that such fronts of course exist in any dimension N ≥ 2, by extending them in a trivial manner in the variables x 2 , . . . , x N .

Existence and uniqueness of the global mean speed among all transition fronts

Once we have characterized the (almost-)planar fronts for equation (1.1), we now consider the general case of transition fronts whose level sets have arbitrary shapes. For this problem, as mentioned in Section 1, the planar fronts u(t, x) = φ f (x•e-c f t), the conical-shaped, pyramidal, exponentially-shaped or parabolic-shaped fronts u(t, x) = φ(x ′ , x Nct) with c f = 0 or c f = 0, as well as the stationary fronts when c f = 0, share a common property: they all have a global mean speed and this mean speed is equal to γ = |c f |, which depends on f only. The goal of the next theorem is to show both the existence and the uniqueness of the global mean speed of any transition front, whatever the shape of the level sets of the fronts may be and whatever the value of the planar speed c f may be.

Theorem 2.6 For problem (1.1), any transition front connecting 0 and 1 has a global mean speed γ. Furthermore, this global mean speed γ is equal to |c f |.

We point out the difference between this result and Theorem 1.7 of [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] recalled in Section 1.2. Theorem 1.7 of [START_REF] Berestycki | Generalized transition waves and their properties[END_REF] was concerned with the uniqueness of the global mean speed, if any, of a given transition front u. Theorem 2.6 of the present paper not only shows the existence of a global mean speed for any transition front connecting 0 and 1, but it also shows the uniqueness of this global mean speed among all transition fronts. Notice that this existence and uniqueness result is in sharp contrast with the case of transition fronts of (1.1) with other nonlinearities f . For instance, if f is positive and concave on (0, 1), then not only the admissible speeds of standard traveling fronts are not unique [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF], but there are also some transition fronts connecting 0 and 1 which do not have any global mean speed, even in dimension N = 1, see [START_REF] Hamel | Travelling waves and entire solutions of the Fisher-KPP equation in R N[END_REF].

d(Γ t , Γ s ) |t -s| → |c f | as |t -s| → +∞. (2.4)
For instance, for all the usual traveling fronts u(t, x) = φ(x ′ , x Nct) mentioned in Section 1 with conical-shaped, pyramidal, exponential or parabolic level sets, the global mean speed defined by (2.4) exists and is equal to |c f |, as for (1.12). On the other hand, all these fronts are invariant in the moving frame with speed c in the direction x N . The vertical speed of this specific frame can also be viewed as the asymptotic speed of the tip of the fronts, in the sense that

d(Γ t , Γ s ) |t -s| → |c| as |t -s| → +∞.
Therefore, the Hausdorff distance gives rise to different global mean speeds, which provide another type of information about the evolution of the level sets but depend on the given transition front (remember that |c| can take all values in the interval [|c f |, +∞) under assumption (1.4), whatever the value of c f may be). We think that the most natural notion of distance is the one defined in (1.12): it corresponds to the asymptotic smallest normal speed of the level sets. Moreover, as d in (2.4), it has the advantage of depending only on f and thus being independent of the transition front, as shown in Theorem 2.6.

Existence of non-standard transition fronts

The previous qualitative properties showed the strength of Definition 1.1, since the solutions of (1.1) in the large class of transition fronts are proved to share some common features (existence and uniqueness of the global mean speed) as well as some further strong qualitative symmetry properties under some additional geometrical conditions. As far as the standard traveling fronts are concerned, all these well-known fronts u(t, x) = φ(x ′ , x Nct), which were mentioned in Section 1.1 and which exist under the bistable condition (1.4), share another simple property, in addition to the existence and uniqueness of the global mean speed |c f |.

Namely, as already emphasized, they are invariant in the frame moving with the speed c in the direction x N . However, what is even more intriguing is that there exist other transition fronts, which are not usual traveling fronts in the sense that there is no frame in which they are invariant as time runs.

Theorem 2.8 Let N ≥ 2 and assume that f is of the bistable type (1.4) with c f > 0. Then problem (1.1) admits transition fronts u connecting 0 and 1 which satisfy the following property: there is no function Φ : R N → (0, 1) (independent of t) for which there would be some families (R t ) t∈R and (x t ) t∈R of rotations and points in

R N such that u(t, x) = Φ R t (x -x t ) for all (t, x) ∈ R × R N .
Theorem 2.8 means that the class of transition fronts includes new types of solutions, even in the homogeneous space R N , thus showing the broadness of Definition 1.1. In dimension N = 2, the new transition fronts u described in Theorem 2.8 are constructed by mixing three planar fronts moving in three different directions: say, the direction x 2 and two directions which are symmetric with respect to the vertical axis x 2 (see the joint figure). As t → -∞, the level sets of these solutions look like two symmetric oblique half-lines moving in the x 2 direction and separated by a larger and larger segment parallel to the x 1 axis. Then, as time increases, the medium segment disappears and, finally, the constructed solutions converge as t → +∞ to a conical-shaped usual traveling front φ(x 1 , x 2ct). This scheme leads to the desired conclusion in dimension N = 2 and then immediately in all dimensions N ≥ 3 by trivially extending the two-dimensional solutions in the variables x 3 , . . . , x N . Outline of the paper. In the remaining sections, we perform the proof of the results. Section 3 is devoted to the Liouville-type results related to the characterization of planar fronts among the larger class of almost-planar fronts or fronts having finitely many parallel transition layers, that is we do the proof of Proposition 2.2 and Theorem 2.5. Section 4 is devoted to the proof of Theorem 2.6, that is we show the existence, the uniqueness and the characterization of the global mean speed among all transition fronts. Lastly, Section 5 is devoted to the proof of Theorem 2.8, that is the construction of transition fronts which are not standard traveling fronts.

Characterization of planar fronts

This section is devoted to the proof of Proposition 2.2 and Theorem 2.5. That is, we characterize the planar fronts among the larger class of almost-planar fronts with a single or a finite number of interfaces between the limiting values 0 and 1. In other words, we prove the uniqueness of the transition fronts in this class. Firstly, we prove that, for the transition fronts whose level sets are almost-planar and orthogonal to the directions e t in the sense of Proposition 2.2 and Theorem 2.5, the directions e t = e must be constant. Once this is done, the stability of the one-dimensional fronts and the one-dimensional symmetry of the almost-planar fronts moving with constant speed in a constant direction will lead to the conclusion of Proposition 2.2. For Theorem 2.5, one needs to exclude the case of transition fronts with 2 or more oscillations in the direction e. The proof will use the fact that the planar speed c f is assumed to be nonzero and that initial conditions which are above θ + on a large set spread with speed c f at large times (if c f > 0).

Proof of Proposition 2.2

We first begin with the following elementary lemma.

Lemma 3.1 Let u : R × R N → [0, 1
] be a solution of (1.1) for which there are a real num-ber t 0 ∈ R and a unit vector e ∈ S N -1 such that

inf x∈R N , x•e≤-A u(t 0 , x) → 1 resp. sup x∈R N , x•e≥A u(t 0 , x) → 0 as A → +∞. (3.1)
Then property (3.1) holds at every time t 1 > t 0 with the same vector e.

Proof. For any ϑ ∈ (0, 1), let v ϑ and v ϑ be the solutions of the one-dimensional Cauchy problem

v t = v yy + f (v), t > 0, y ∈ R with initial conditions v ϑ (0, y) = ϑ if y ≤ 0, 0 if y > 0, and v ϑ (0, y) = 1 if y ≤ 0, ϑ if y > 0. (3.2)
Let ̺ ϑ : R → (0, 1) denote the solution of the equation

̺ ′ ϑ (t) = f (̺ ϑ (t)) with initial con- dition ̺ ϑ (0) = ϑ.
It follows from the maximum principle and standard parabolic estimates that, for each

t > 0, v ϑ (t, •) and v ϑ (t, •) are decreasing in R and that v ϑ (t, -∞) = ̺ ϑ (t), v ϑ (t, +∞) = 0, v ϑ (t, -∞) = 1 and v ϑ (t, +∞) = ̺ ϑ (t).
Let now u, e and t 0 < t 1 be as in the lemma. We assume first that inf

x∈R N , x•e≤-A u(t 0 , x) → 1 as A → +∞.
Let ε ∈ (0, 1) be arbitrary. There is M ∈ R such that

u(t 0 , x) ≥ v 1-ε (0, x • e + M) for all x ∈ R N , whence u(t 1 , x) ≥ v 1-ε (t 1 -t 0 , x•e+M) for all x ∈ R N from the maximum principle. Therefore, lim inf A→+∞ inf x∈R N , x•e≤-A u(t 1 , x) ≥ v 1-ε (t 1 -t 0 , -∞) = ̺ 1-ε (t 1 -t 0 ).
Since this holds for all ε > 0 small enough, since ̺ 1-ε (t 1t 0 ) → 1 as ε → 0 and since u ranges in (0, 1), it follows that inf

x∈R N , x•e≤-A u(t 1 , x) → 1 as A → +∞.
Similarly, if sup

x∈R N , x•e≥A u(t 0 , x) → 0 as A → +∞, one gets that lim sup A→+∞ sup x∈R N , x•e≥A u(t 1 , x) ≤ v ε (t 1 -t 0 , +∞) = ̺ ε (t 1 -t 0 )
for all ε > 0 small enough, and the conclusion follows.

From the previous lemma, the next result follows immediately.

Corollary 3.2 Let u : R × R N → (0, 1) be a solution of (1.1) such that, for every time t ∈ R, there is a unit vector e t ∈ S N -1 such that

inf x∈R N , x•et≤-A u(t, x) → 1 and sup x∈R N , x•et≥A u(t, x) → 0 as A → +∞. (3.3)
Then e t = e is independent of time t.

Proof of Proposition 2.2. Let u be an almost-planar transition front connecting 0 and 1, in the sense of Definition 2.1, for problem (1.1). That is, there exist some families (e t ) t∈R in S N -1 and (ξ t ) t∈R in R such that

Γ t = x ∈ R N ; x • e t = ξ t
for every t ∈ R. Up to changing e t into -e t , it follows from (1.8) and Definition 1.1 that (3.3) holds for every t ∈ R. Corollary 3.2 implies that e t = e is a constant vector, whence

Ω + t = x ∈ R N ; x • e < ξ t and Ω - t = x ∈ R N ; x • e > ξ t (3.4)
for all t ∈ R.

We shall now prove that the function R ∋ t → ξ tc f t is bounded. To do so, we use the stability result of planar fronts of Fife and McLeod [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that we recalled in Section 1.1. The last step of the proof will be based on a Liouville-type result of Berestycki and the author [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF] on the characterization of the almost-planar transition fronts with constant direction and position of the order c f t.

Let α and β be two given real numbers such that

0 < α < θ -≤ θ + < β < 1, (3.5)
where we recall that θ ± are defined in (1.3). From the previous observations and from Definition 1.1, there is M ≥ 0 such that

∀ (t, x) ∈ R × R N , x • e -ξ t ≤ -M =⇒ β ≤ u(t, x) < 1, x • e -ξ t ≥ M =⇒ 0 < u(t, x) ≤ α. (3.6)
Therefore, with the notations used in the proof of Lemma 3.1, one infers that, for every

t 0 ∈ R and x ∈ R N , v β (0, x • e -ξ t 0 + M) ≤ u(t 0 , x) ≤ v α (0, x • e -ξ t 0 -M). Thus, v β (t -t 0 , x • e -ξ t 0 + M) ≤ u(t, x) ≤ v α (t -t 0 , x • e -ξ t 0 -M) (3.7)
for all t > t 0 and x ∈ R N , from the maximum principle.

On the other hand, from (1.2) and the existence of a planar front (c f , φ f ) solving (1.5), it follows from [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that there exist two real numbers ξ = ξ(f, β) and ξ = ξ(f, α) depending only on f , α and β, such that

sup y∈R v β (s, y) -φ f (y -c f s + ξ) + sup y∈R v α (s, y) -φ f (y -c f s + ξ) → 0 as s → +∞. (3.8)
In particular, since φ f (-∞) = 1 and φ f (+∞) = 0, there exist T > 0 and A > 0 such that, for all s ≥ T ,

v β (s, y) > α if y ≤ c f s -A, v α (s, y) < β if y ≥ c f s + A.
Together with (3.7), it follows that, for all t 0 < t 0 + T ≤ t,

u(t, x) > α if x • e -ξ t 0 + M ≤ c f (t -t 0 ) -A, u(t, x) < β if x • e -ξ t 0 -M ≥ c f (t -t 0 ) + A. (3.9)
Properties (3.6) and (3.9) imply that, for all t 0 < t 0 + T ≤ t,

ξ t 0 -M + c f (t -t 0 ) -A < ξ t + M and ξ t 0 + M + c f (t -t 0 ) + A > ξ t -M. (3.10)
By fixing t = 0, one gets that lim sup

t 0 →-∞ |ξ t 0 -c f t 0 | ≤ |ξ 0 |+2M +A. For any arbitrary t ∈ R, letting t 0 → -∞ in (3.10) then leads to |ξ t -c f t| ≤ |ξ 0 | + 4M + 2A.
Therefore, Definition 1.1 together with (3.4) implies that our solution u : R × R N → (0, 1) of (1.1) satisfies inf

(t,x)∈R×R N , x•e-c f t≤-A u(t, x) → 1 and sup (t,x)∈R×R N , x•e-c f t≥A u(t, x) → 0 as A → +∞.
It follows finally from Theorem 3.1 of [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF] and the uniqueness of the planar fronts that there exists ξ ∈ R such that u(t, x) = φ f (x • ec f t + ξ) for all (t, x) ∈ R × R N . The proof of Proposition 2.2 is thereby complete.

Planar and almost-planar transition fronts

In this section, we show that, without assuming a priori the existence of a one-dimensional planar front (c f , φ f ) solving (1.5), the existence of an almost-planar transition front connecting 0 and 1 for problem (1.1) in R N actually implies (and is then equivalent to) the existence of such a one-dimensional planar front (c f , φ f ), at least for a C 1 ([0, 1]) dense set of functions f . Namely, we prove the following result.

Proposition 3.3 For any C 1 ([0, 1]) function f satisfying (1.2) and for any ε > 0, there is a C 1 ([0, 1]) function f ε satisfying (1.2) such that f -f ε C 1 ([0, 1
]) ≤ ε and for which the following holds: if (and only if) there exists an almost-planar transition front u connecting 0 and 1 for problem (1.1) with f ε , then there exists a planar front (c fε , φ fε ) solving (1.5), and then u is a planar front of the type (2.1), with f ε instead of f .

Proof. Let f be any given C 1 ([0, 1]) function satisfying (1.2) and let ε > 0 be arbitrary. Firstly, it is straightforward to check that there is a C 1 ([0, 1]) function g satisfying (1.2) and

f -g C 1 ([0,1]) ≤ ε 2 ,
and for which there exist k ∈ N and some real numbers

0 = θ 0 < θ 1 < • • • < θ 2k-1 < θ 2k = 1 with                g(θ i ) = 0 for all 0 ≤ i ≤ 2k, g < 0 on (θ 2i , θ 2i+1 ) for all 0 ≤ i ≤ k -1, g > 0 on (θ 2i+1 , θ 2i+2 ) for all 0 ≤ i ≤ k -1, g ′ (θ 2i ) < 0 for all 0 ≤ i ≤ k, g ′ (θ 2i+1 ) > 0 for all 0 ≤ i ≤ k -1. (3.11)
In particular, the restriction of the function g on each interval [θ 2i , θ 2i+2 ] is of the bistable type. Notice also that θ -= θ 1 and θ + = θ 2k-1 (3.12) for this function g, with the notation (1.3).

As far as the existence of a planar connection (c g , φ g ) between 0 and 1 for this function g is concerned, two and only two cases occur, as follows from Fife and McLeod [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF]:

• (a) either there is a (unique) planar front (c g , φ g ) solving (1.5) with g in place of f ,

• (b) or the pair (c g , φ g ) does not exist and there exist some integers l ∈ {2, • • • , k} and 0 = i 0 < i 1 < • • • < i l = k such that there exists a (unique) planar front (γ j , ϕ j ) connecting θ 2i j-1 and θ 2i j for g and for every 1 ≤ j ≤ l, with

γ 1 ≥ γ 2 ≥ • • • ≥ γ l-1 ≥ γ l .
The fact that ϕ j connects θ 2i j-1 and θ 2i j with the speed γ j for the function g means that the pair (γ j , ϕ j ) solves (1.5) with the limits ϕ j (-∞) = θ 2i j and ϕ j (+∞) = θ 2i j-1 , and with (γ j , ϕ j , g) in place of (c f , φ f , f ).

In case (a), we simply set f ε = g and the conclusion of Proposition 3.3 follows trivially, since there is a planar connection (c fε , φ fε ) solving (1.5) with the function f ε .

Consider now the case (b). As it follows from [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF][START_REF] Taniguchi | Traveling fronts in perturbed multistable reaction-diffusion equations[END_REF], for every 1 ≤ j ≤ l, there is a real number η j > 0 such that for every

g ∈ B j := g ∈ C 1 ([θ 2i j-1 , θ 2i j ]); g -g C 1 ([θ 2i j-1 ,θ 2i j ]) ≤ η j , g(θ 2i j-1 ) = g(θ 2i j ) = 0
there is a unique real number γ which is the speed of a planar connection between θ 2i j-1 and θ 2i j for the function g; furthermore, the map g → γ is continuous in B j endowed with the C 1 ([θ 2i j-1 , θ 2i j ]) norm and, if g and g belong to B j and satisfy g ≤ ≡ g in [θ 2i j-1 , θ 2i j ], then the corresponding speeds satisfy γ < γ. Pick some arbitrary points x j ∈ (θ 2i j-1 , θ 2i j ) for every 1 ≤ j ≤ l. By slightly changing the function g locally around the points x j (namely by adding or subtracting some small nonnegative C 1 functions supported in some small neighborhoods of x j ), one infers straightforwardly that there is a C 1 ([0, 1]) function h satisfying (1.2) with h instead of f , satisfying (3.11) with h instead of g, such that

g -h C 1 ([0,1]) ≤ ε 2 ,
and for which there are some planar fronts ( γ j , ϕ j ) connecting θ 2i j-1 and θ 2i j for the function h and for every 1 ≤ j ≤ l, with

γ 1 > γ 2 > • • • > γ l-1 > γ l .
Finally, we will see that this last situation is incompatible with the existence of an almostplanar transition front connecting 0 and 1 for problem (1.1) in R N with h in place of f . This will lead to the desired conclusion of Proposition 3.3. Assume on the opposite that there is such an almost-planar transition front u in the sense of Definition 2.1. Let 0 < α < β < 1 be given as in (3.5) with the notation (3.12), and let v β and v α be the same as in the proof of Proposition 2.2 with the function h instead of f . Observe now that the proof of Lemma 3.1 does not use the existence of a planar front solving (1.5). Hence, Lemma 3.1 and Corollary 3.2 hold, that is e t = e is a constant vector, and one can also assume without loss of generality that all properties (3.3), (3.4), (3.6) and (3.7) hold with the functions v β and v α . On the other hand, it follows from [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that, instead of (3.8),2 there exist some real numbers ξ j and ξ j for every 1

≤ j ≤ l, such that              sup y≤( γ l + γ l-1 )s/2 v β (s, y) -ϕ l (y -γ l s + ξ l ) -→ s→+∞ 0, sup ( γ j + γ j+1 )s/2≤y≤( γ j + γ j-1 )s/2 v β (s, y) -ϕ j (y -γ j s + ξ j ) -→ s→+∞ 0 for every 2 ≤ j ≤ l -1, sup y≥( γ 1 + γ 2 )s/2 v β (s, y) -ϕ 1 (y -γ 1 s + ξ 1 ) -→ s→+∞ 0 and              sup y≤( γ l + γ l-1 )s/2 v α (s, y) -ϕ l (y -γ l s + ξ l ) -→ s→+∞ 0, sup ( γ j + γ j+1 )s/2≤y≤( γ j + γ j-1 )s/2 v α (s, y) -ϕ j (y -γ j s + ξ j ) -→ s→+∞ 0 for every 2 ≤ j ≤ l -1, sup y≥( γ 1 + γ 2 )s/2 v α (s, y) -ϕ 1 (y -γ 1 s + ξ 1 ) -→ s→+∞ 0.
In other words, the one-dimensional functions v β and v α expand as an ordered family (or a terrace, with the terminology used in a more general framework in [START_REF] Ducrot | Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations[END_REF]) of traveling fronts with ordered speeds. Since u is trapped between some finite shifts of v β and v α from (3.7), this will be in contradiction with the uniform boundedness of the transition zone where u is, say, between α and β. Indeed, owing to the aforementioned properties of the fronts ( γ j , ϕ j ), there exist then some real numbers T > 0 and A > 0 such that, for all s ≥ T ,

v β (s, y) > θ 1 = θ -> α if y ≤ γ 1 s -A, v α (s, y) < θ 2k-1 = θ + < β if y ≥ γ l s + A.
Together with property (3.7) applied with t 0 = 0, one infers that, for all t ≥ T ,

u(t, x) > α if x • e -ξ 0 + M ≤ γ 1 t -A, u(t, x) < β if x • e -ξ 0 -M ≥ γ l t + A. With (3.6), it follows that ξ 0 -M + γ 1 t -A < ξ t + M and ξ 0 + M + γ l t + A > ξ t -M
for all t ≥ T . This leads to a contradiction as t → +∞, since γ 1 > γ l .

Therefore, in case (b) for g, we set f ε = h and the conclusion of Proposition 3.3 follows since in this case there is no almost-planar transition front connecting 0 and 1 for problem (1.1) with the function f ε .

Proof of Theorem 2.5

Although the statement of Theorem 2.5 looks similar to that of Proposition 2.2, the proof is much more involved, not to mention that it requires necessarily that the planar speed c f be nonzero, as emphasized in Section 2.1. The proof is based on a series of auxiliary lemmas establishing some bounds on the largest and/or smallest positive and/or negative parts of the positions ξ i,t given in (2.2) of the interfaces along the direction e t = e (the direction e t is easily seen to be independent of time). The bounds on the positions ξ i,t rely crucially on the fact that the one-dimensional solutions of the Cauchy problem associated to (1.1) in R with compactly supported initial conditions being above θ + on a large set spread at the speed c f if c f > 0.

Let u be as in the statement of Theorem 2.5 and assume that c f = 0. Let u, g and φ g be the functions defined by

u(t, x) = 1 -u(t, x) for all (t, x) ∈ R × R N , g(s) = -f (1 -s) for all s ∈ [0, 1]
and φ g (x) = 1φ f (-x) for all x ∈ R. The function u obeys the equation (1.1) with g instead of f , while φ g (-∞) = 1 > φ g (x) > φ g (+∞) = 0 for all x ∈ R and

φ ′′ g + c g φ ′ g + g(φ g ) = 0 in R
with c g = -c f . Therefore, even if it means replacing u by u, f by g and c f by -c f = c g , one can assume without loss of generality that c f > 0. Definition 1.1 and the assumptions made in Theorem 2.5 imply that, for every t ∈ R, both sets Ω + t and Ω - t contain a half-space. Therefore, up to changing e t into -e t , one can assume without loss of generality that condition (3.3) is fulfilled for every t ∈ R. It follows then from Corollary 3.2 that e t = e is a constant vector and that (3.3) holds with e t = e for every t ∈ R.

Even if it means reordering the real numbers ξ i,t given in (2.2), we denote, for every t ∈ R,

Γ t = nt i=1 x ∈ R N ; x • e = ξ i,t , (3.13)
with n t ∈ N and ξ 1,t < • • • < ξ nt,t . In particular,

Ω + t ⊃ x ∈ R N ; x • e < ξ 1,t and Ω - t ⊃ x ∈ R N ; x • e > ξ nt,t . (3.14)
In the following lemmas, we show some estimates for the positions (along the vector e) of the leftmost and rightmost interfaces ξ 1,t and ξ nt,t , as well as ξ ± t defined in (3.19) below, as t → -∞. These estimates are based in particular on the spreading properties of the solutions of the Cauchy problem associated to (1.1) in R with sufficiently large initial conditions. They also use the fact that, from Definition 1.1, in some not-too-far neighborhoods of the hyperplanes Γ t , there are big regions where u is close to 0 and other ones where u is close to 1. Finally, putting together all the estimates, we show that the solution u has essentially one interface, that has to move with the speed c f , and then the solution u has to be a planar front. Lemma 3.4 There holds

lim inf t→-∞ ξ nt,t -c f t > -∞ and lim sup t→-∞ ξ 1,t -c f t < +∞.
Proof. Assume that the first conclusion is not satisfied. Then there is a sequence (t k ) k∈N of real numbers such that

t k → -∞ and ξ nt k ,t k -c f t k → -∞ as k → +∞.
Let α ∈ (0, θ -) be given, where we recall that θ -is defined in (1.3). From Definition 1.1 and (3.14), there is

M α ≥ 0 such that u(t, x) ≤ α for all (t, x) ∈ R × R N with x • e -ξ nt,t ≥ M α . Therefore, definition (3.2) of v α yields u(t k , x) ≤ v α (0, x • e -ξ nt k ,t k -M α ) for all x ∈ R N , whence u(t, x) ≤ v α (t -t k , x • e -ξ nt k ,t k -M α ) for all x ∈ R N
and for all t > t k . For any fixed (t, x) ∈ R × R N , since

lim k→+∞ t k = lim k→+∞ ξ nt k ,t k -c f t k = -∞,
it follows then from the existence of a planar front (c f , φ f ) and from (3.8) that, for some ξ = ξ(f, α) ∈ R,

u(t, x) ≤ lim sup k→+∞ φ f x • e -ξ nt k ,t k -M α -c f (t -t k ) + ξ = φ f (+∞) = 0.
This is impossible, since, as already emphasized in the introduction, u is assumed to range in the open interval (0, 1). Similarly, if there is a sequence (t k ) k∈N of real numbers such that t k → -∞ and ξ 1,t kc f t k → +∞ as k → +∞, one would get that u(t, x) ≥ 1 for all (t, x) ∈ R × R N , which is ruled out. Finally, lim sup t→-∞ ξ 1,tc f t < +∞ and the proof of Lemma 3.4 is thereby complete.

Let us now introduce a few additional useful notations. Let β be any given real number such that

θ + < β < 1,
where we recall that θ + is defined in (1.3). Due to the existence of a planar front (c f , φ f ) solving (1.5) with c f > 0, it follows from [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that there are A > 0 and σ ∈ R such that the solution w of the Cauchy problem

w t = w yy + f (w), t > 0, y ∈ R, with initial condition w(0, y) = β if |y| ≤ A, 0 if |y| > A, (3.15)
is such that w(t, y) → 1 as t → +∞ locally uniformly in y ∈ R, and moreover

sup y∈R w(s, y) -φ f (y -c f s + σ) -φ f (-y -c f s + σ) + 1 → 0 as s → +∞. (3.16)
From Definition 1.1, there is M β ≥ 0 such that

∀ (t, x) ∈ R × R N , x ∈ Ω + t and d(x, Γ t ) ≥ M β =⇒ β ≤ u(t, x) < 1 . (3.17)
Without loss of generality, one can assume that M β ≥ A. From condition (1.9), there is then r > 0 such that

∀ t ∈ R, ∀ x ∈ Γ t , ∃ y ∈ Ω + t , |y -x| ≤ r, d(y, Γ t ) ≥ 2M β . (3.18)
Lastly, let ξ ± t be defined in [-∞, +∞] as

ξ - t = sup x • e; x ∈ Γ t , x • e ≤ 0 and ξ + t = inf x • e; x ∈ Γ t , x • e ≥ 0 . (3.19)
If ξ 1,t ≤ 0 (this holds for t negative enough from the previous lemma), then ξ - t is a real number and

ξ - t = ξ n - t ,t for some 1 ≤ n - t ≤ n t , otherwise ξ - t = -∞. If ξ nt,t ≥ 0, then ξ + t is a real number and ξ + t = ξ n + t ,t for some 1 ≤ n + t ≤ n t , otherwise ξ + t = +∞. Lemma 3.5 One has lim sup t→-∞ ξ - t -c f t) < +∞, lim inf t→-∞ ξ + t + c f t) > -∞ (3.20)
and there is T 1 ∈ R such that

Ω - t ⊃ x ∈ R N ; ξ - t < x • e < ξ + t =: E t (3.21)
for all t ≤ T 1 .

Proof. Assume first that lim sup t→-∞ ξ - tc f t) = +∞. Then there is a sequence (t k ) k∈N of real numbers such that

t k → -∞ and ξ - t k -c f t k → +∞ as k → +∞. Since ξ - t k ∈ [-∞, 0] for all k ∈ N and since ξ - t k -c f t k → +∞ as k → +∞, one can assume without loss of generality that ξ - t k ∈ R and x ∈ R N ; x • e = ξ - t k ⊂ Γ t k for all k ∈ N.
For every k ∈ N, let now x k = ξ - t k e ∈ Γ t k and, from (3.18), let

y k ∈ Ω + t k such that |y k -x k | ≤ r and d(y k , Γ t k ) ≥ 2M β . Set ω k = y k • e. There holds ξ - t k -r ≤ ω k ≤ ξ - t k + r for all k ∈ N. (3.22) Furthermore, since B(y k , 2M β ) ⊂ Ω + t k and Γ t k = 1≤i≤nt k x ∈ R N ; x • e = ξ i,t k , it follows that x ∈ R N ; ω k -2M β < x • e < ω k + 2M β ⊂ Ω + t k for all k ∈ N. Property (3.17) implies that, for every k ∈ N, u(t k , x) ≥ β for all x ∈ R N such that ω k -M β ≤ x • e ≤ ω k + M β . (3.23)
The definition of w(0, •) in (3.15) and the inequality

M β ≥ A yield u(t k , x) ≥ w(0, x • e -ω k ) for all x ∈ R N , whence u(t, x) ≥ w(t -t k , x • e -ω k ) for all t > t k and x ∈ R N , for every k ∈ N. For any fixed (t, x) ∈ R × R N , one gets from (3.16) that u(t, x) ≥ lim sup k→+∞ φ f (x • e -ω k -c f (t -t k ) + σ) + φ f (-x • e + ω k -c f (t -t k ) + σ) -1 .
Notice now that ω kc f t k → +∞ as k → +∞, since ξ - t kc f t k → +∞ as k → +∞ by assumption and since the sequence (ω kξ - t k ) k∈N is bounded from (3.22). On the other hand, since ξ - t k ≤ 0 < c f and t k → -∞ as k → +∞, one gets that ξ -

t k + c f t k → -∞ and ω k + c f t k → -∞ as k → +∞. Finally, φ f (x • e -ω k -c f (t -t k ) + σ) → φ(-∞) = 1 as k → +∞ and φ f (-x • e + ω k -c f (t -t k ) + σ) → φ(-∞) = 1 as k → +∞, whence u(t, x) ≥ 1 for all (t, x) ∈ R × R N .
One has then reached a contradiction. Therefore, lim sup t→-∞ ξ - tc f t) < +∞. Similarly, if one assumes that there is a sequence (t k ) k∈N of real numbers such that t k → -∞ and ξ + t k + c f t k → -∞ as k → +∞ (which would imply in particular that 0 ≤ ξ + t k < +∞ for k large enough), one would get a similar contradiction (one can also apply the previous result to the function (t, x) → u(t, x), where x = x -2(x • e)e denotes the image of a point x by the orthogonal symmetry with respect to the hyperplane orthogonal to e and containing the origin).

From Lemma 3.4, it follows in particular that ξ 1,t → -∞ as t → -∞. Hence, there is T 0 ∈ R such that, for all t ≤ T 0 , ξ - t ∈ (-∞, 0] and either E t ⊂ Ω + t or E t ⊂ Ω - t , under the notation given in (3.21). Assume now by contradiction that there is a sequence (t k ) k∈N of real numbers such that t k → -∞ as k → +∞ and E t k ⊂ Ω + t k for all k ∈ N. Then (3.20) and Definition 1.1 imply in particular that inf

x∈R N , |x•e|≤A u(t k , x) → 1 as k → +∞.
Therefore, for k large enough, u(t k , x) ≥ w(0, x•e) for all x ∈ R N , whence u(t, x) ≥ w(t-t k , x•e) for all t > t k and x ∈ R N . Finally, for any fixed (t, x) ∈ R × R N , one would get that

u(t, x) ≥ lim sup k→+∞ φ f (x • e -c f (t -t k ) + σ) + φ f (-x • e -c f (t -t k ) + σ) -1 = 1 since t k → -∞ as k → +∞.
A contradiction has been reached, and the desired conclusion (3.21) follows for -t large enough.

The following lemma is one of the key-points in the proof of Theorem 2.5. It states that it the positions ξ + t are far on the right of the (very positive) position c f |t| along the direction e, at least for a sequence of times t k converging to -∞, then they are actually pushed to +∞ and the solution u has no interface far on the right of the (very negative) position c f t as t → -∞.

Lemma 3.6 If lim sup t→-∞ ξ + t + c f t = +∞, then there are T 2 ∈ R, δ > 0 and B ∈ R such that u(t, x) ≤ δ e -δ(x•e-c f t-B) for all t ≤ T 2 and x • e ≥ c f t + B. (3.24)
Remark 3.7 With the same arguments as in the proof of Lemma 3.6, one can get the following result, which we state in a remark since it will actually not be used in the sequel:

if lim inf t→-∞ ξ - t -c f t = -∞, then there are T ′ 2 ∈ R, δ ′ > 0 and B ′ ∈ R such that u(t, x) ≤ δ ′ e δ ′ (x•e+c f t+B ′ ) for all t ≤ T ′ 2 and x • e ≤ -c f t -B ′ .
The proof of Lemma 3.6 is quite lengthy and is postponed in Section 3.4. We prefer to go on the proof of Theorem 2.5 with the following lemma.

Lemma 3.8 If there is a sequence (t k ) k∈N of real numbers such that t k → -∞ as k → +∞ and lim inf k→+∞ ξ - t k -c f t k > -∞ (resp. lim sup k→+∞ ξ + t k +c f t k < +∞), then there is η ∈ R such that u(t, x) ≥ φ f (x • e -c f t + η) resp. u(t, x) ≥ φ f (-x • e -c f t + η) for all (t, x) ∈ R × R N .
Since φ f (-∞) = 1 and u(t, x) → 0 as x • e → +∞ for every t ∈ R by (3.14), the following corollary follows immediately. Corollary 3.9 There holds ξ + t + c f t → +∞ as t → -∞.

Proof of Lemma 3.8. Let (t k ) k∈N be a sequence of real numbers such that t k → -∞ as k → +∞ and lim inf k→+∞ ξ - t kc f t k > -∞. It follows then from Lemma 3.5 that the sequence (ξ - t kc f t k ) k∈N is bounded. As in the proof of Lemma 3.5, there are r > 0 and a sequence (ω k ) k∈N in R such that (3.22) and (3.23) hold, whence u(t k , x) ≥ w(0, x • eω k ) and u(t, x) ≥ w(t-t k , x•e-ω k ) for all k ∈ N, t > t k and x ∈ R N . For any fixed (t, x) ∈ R×R N , one infers that

u(t, x) ≥ lim sup k→+∞ φ f (x • e -ω k -c f (t -t k ) + σ) + φ f (-x • e + ω k -c f (t -t k ) + σ) -1 .
Since the sequences (ξ - t k -c f t k ) k∈N and (ω k -ξ - t k ) k∈N are bounded, so is the sequence (ω k -c f t k ) k∈N and one can assume, up to extraction of a sequence, that there is

σ ∈ R such that ω k -c f t k → σ as k → +∞. On the other hand, ω k + c f t k = ω k -c f t k + 2c f t k → -∞ since t k → -∞ as k → +∞. Finally, one concludes that, for any fixed (t, x) ∈ R × R N , u(t, x) ≥ φ f (x • e -c f t + σ -σ) + φ f (-∞) -1 = φ f (x • e -c f t + σ -σ),
which gives the desired result with η = σσ.

Similar arguments imply that, if lim sup k→+∞ ξ

+ t k + c f t k < +∞ for some sequence (t k ) k∈N converging to -∞, then u(t, x) ≥ φ f (-x • e -c f t + η) in R × R N for some η ∈ R.
With all the previous lemmas, we are now ready to finish the proof of Theorem 2.5.

End of the proof of Theorem 2.5. First, Lemma 3.6 and Corollary 3.9 provide the existence of some T 2 ∈ R, δ > 0 and B ∈ R such that (3.24) is satisfied.

We shall now prove that lim sup

t→-∞ ξ nt,t -c f t < +∞. ( 3

.25)

Assume not. There is then a sequence (t k ) k∈N of real numbers such that t k → -∞ and ξ nt k ,t kc f t k → +∞ as k → +∞. Denote x k = ξ nt k ,t k e. As in the proof of Lemma 3.5, there are r > 0 and a sequence (y k ) k∈N in R N such that

y k ∈ Ω + t k , |y k -x k | ≤ r and u(t k , y k ) ≥ β for all k ∈ N. But the sequence (y k • e -ξ nt k ,t k ) k∈N is bounded, whence y k • e -c f t k → +∞ as k → +∞. Therefore, t k ≤ T 2 and y k • e ≥ c f t k + B for k large enough. Finally, (3.24) yields u(t k , y k ) ≤ δ e -δ(y k •e-c f t k -B)
for k large enough. The right-hand side converges to 0 as k → +∞, whereas the left-hand side is bounded from below by β > 0. One is led to a contradiction, and (3.25) is proved. Notice that (3.25) implies in particular that ξ nt,t < 0, ξ + t = +∞ and ξ - t = ξ nt,t for t negative enough. Furthermore, together with Lemma 3.4, one obtains that lim sup

t→-∞ ξ - t -c f t = lim sup t→-∞ ξ nt,t -c f t < +∞. (3.26)
Lemma 3.8 provides then the existence of a real number η such that

u(t, x) ≥ φ f (x • e -c f t + η) for all (t, x) ∈ R × R N . (3.27)
On the other hand, (3.26) also yields the existence of a real number ξ and a sequence of real numbers (t k ) k∈N such that t k → -∞ and ξ nt k ,t kc f t k → ξ as k → +∞. As in the proof of Lemma 3.4, there are then α ∈ (0, θ -) and M α ≥ 0 such that

u(t k , x) ≤ v α 0, x • e -ξ nt k ,t k -M α whence u(t, x) ≤ v α (t -t k , x • e -ξ nt k ,t k -M α ) for all k ∈ N, t > t k and x ∈ R N . Since v α (s, y) -φ f (y -c f s + ξ) → 0 as s → +∞ uniformly in y ∈ R, for some ξ ∈ R, one infers that, for any fixed (t, x) ∈ R × R N , u(t, x) ≤ lim sup k→+∞ φ f (x • e -ξ nt k ,t k -M α -c f (t -t k ) + ξ) = φ f (x • e -c f t -ξ -M α + ξ).
As a conclusion,

φ f (x • e -c f t + η) ≤ u(t, x) ≤ φ f (x • e -c f t + η) for all (t, x) ∈ R × R N , with η = ξ -ξ -M α .
As in the end of the proof of Proposition 2.2, Theorem 3.1 of [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF] yields the existence of a real number ξ such that

u(t, x) = φ f (x • e -c f t + ξ) for all (t, x) ∈ R × R N .
The proof of Theorem 2.5 is thereby complete. Remark 3.10 It is immediate to see that the conclusion of Theorem 2.5 still holds even if the family of integers (n t ) t∈R appearing in (3.13) is not bounded. In other words, if c f = 0 and if u satisfies all assumptions of Theorem 2.5 with the exception of (1.10) and the boundedness of

n t in Γ t = 1≤i≤nt x ∈ R N ; x • e t = ξ i,t ,
then u is still a planar front of the type u(t, x) = φ f (x • ec f t + ξ) for all (t, x) ∈ R × R N , for some unit vector e of R N and some real number ξ.

Proof of Lemma 3.6

Let (t k ) k∈N be a sequence of real numbers such that

t k → -∞ and ξ + t k + c f t k → +∞ as k → +∞. (3.28)
Without loss of generality, one can assume in particular that

ξ + t k + c f t k ≥ 0 for all k ∈ N. (3.29)
The strategy of the proof consists in constructing a sequence of supersolutions of (1.1) which are approximately of the type δ e -δ(x•e-c f t-B) + φ f (-x • ec f (tt k ) + ξ + t k ) (plus some shifts and some small exponential terms, as in the original proof of Fife and McLeod [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF]) for t k ≤ t ≤ T 2 and x•e-c f t ≥ B. The parameters T 2 ∈ R, δ > 0 and B ∈ R will be chosen independently of k.

The passage to the limit as k → +∞ in the supersolution will provide the desired conclusion, since ξ + t k + c f t k → +∞. Let us first choose some parameters. Remember that f ′ (0) and f ′ (1) are negative. Let δ > 0 be such that

0 < δ < min 1, |f ′ (0)| 2 , |f ′ (1)| 2 , f ′ ≤ f ′ (0) 2 on [0, 3δ] and f ′ ≤ f ′ (1) 2 on [1 -δ, 1]. (3.30)
Let C > 0 be such that

φ f ≥ 1 -δ on (-∞, -C] and φ f ≤ δ on [C, +∞). (3.31)
Since φ ′ f is negative and continuous on R, there is κ > 0 such that

-φ ′ f ≥ κ > 0 on [-C, C]. (3.32) Set L = max [0,1] |f ′ | and let ω > 0 such that κ ω ≥ L + δ and κ ω c f ≥ L + δ 2 (3.33)
(remember that c f is assumed to be positive in the proof of Theorem 2.5, without loss of generality). From Definition 1.1 and Lemma 3.5, there are T 2 ≤ 0 and B > 0 such that

c f t + B < -c f t -2B < ξ + t -B for all t ≤ T 2 , u(t, x) ≤ δ for all t ≤ T 2 and c f t + B ≤ x • e ≤ ξ + t -B, (3.34)
and

c f t + 2(ω + B + C) ≤ 0 for all t ≤ T 2 . (3.35)
Without loss of generality, one can assume that t k < T 2 for all k ∈ N. For all k ∈ N and (t, x) ∈ R × R N , let us now set

u k (t, x) = min φ f ζ k (t, x) + δ e -δ(x•e-c f t-B) + δ e -δ(t-t k ) , 1 , where ζ k (t, x) = -x • e -c f (t -t k ) + ξ + t k + ω e -δ(t-t k ) -ω -ω e δc f t -B -C, under the convention that ζ k (t, x) = +∞ and u k (t, x) = min δ e -δ(x•e-c f t-B) + δ e -δ(t-t k ) , 1 if ξ + t k = +∞. Let us check that u k is a supersolution of the equation (1.1) satisfied by u, in the set E k = (t, x) ∈ R × R N ; t k ≤ t ≤ T 2 , x • e ≥ c f t + B .
In what follows, k denotes an arbitrary integer. At the time t k , if follows from (3.34) and the definition of u k that

u(t k , x) ≤ δ ≤ u k (t k , x) for all x ∈ R N such that c f t k + B ≤ x • e ≤ ξ + t k -B.
On the other hand, if

x • e ≥ ξ + t k -B, then ζ k (t k , x) ≤ -ω e δc f t k -C ≤ -C, whence u k (t k , x) ≥ min φ f (ζ k (t k , x)) + δ, 1 ≥ min (1 -δ) + δ, 1 = 1 ≥ u(t k , x)
due to the first part of (3.31

N k (t, x) := u k (t, x) -∆u k (t, x) -f (u k (t, x)) ≥ 0 for all (t, x) ∈ E k such that u k (t, x) < 1.
This will be sufficient to ensure that u k is a supersolution of (1.1) in E k , since f (1) = 0. In this paragraph, (t, x) denotes any point in E k such that u k (t, x) < 1. From (1.5), it is straightforward to check that

N k (t, x) = f (φ f (ζ k (t, x))) -f (u k (t, x)) -ω δ e -δ(t-t k ) + c f e δc f t φ ′ f (ζ k (t, x)) +δ 2 (c f -δ) e -δ(x•e-c f t-B) -δ 2 e -δ(t-t k ) ≥ f (φ f (ζ k (t, x))) -f (u k (t, x)) -ω δ e -δ(t-t k ) + c f e δc f t φ ′ f (ζ k (t, x)) -δ 3 e -δ(x•e-c f t-B) -δ 2 e -δ(t-t k ) , whether ξ + t k and ζ k (t, x) be real numbers or equal to +∞. If ζ k (t, x) ≤ -C, then (3.31) implies that 1 -δ ≤ φ f (ζ k (t, x)) ≤ u k (t, x) < 1, whence f (φ f (ζ k (t, x))) -f (u k (t, x)) ≥ -f ′ (1) 2 δ e -δ(x•e-c f t-B) + δ e -δ(t-t k )
from (3.30), and thus

N k (t, x) ≥ δ -f ′ (1) 2 -δ 2 e -δ(x•e-c f t-B) + δ -f ′ (1) 2
δ e -δ(t-t k ) ≥ 0 from (3.30) and the negativity of

φ ′ f . Similarly, if ζ k (t, x) ≥ C, then φ f (ζ k (t, x)) ≤ δ. Thus, 0 < φ f (ζ k (t, x)) ≤ u k (t, x) ≤ 3δ and f (φ f (ζ k (t, x))) -f (u k (t, x)) ≥ -f ′ (0) 2 δ e -δ(x•e-c f t-B) + δ e -δ(t-t k )
from (3.30), whence

N k (t, x) ≥ δ -f ′ (0) 2 -δ 2 e -δ(x•e-c f t-B) + δ -f ′ (0) 2 -δ e -δ(t-t k ) ≥ 0,
again from (3.30) and the negativity of

φ ′ f . Finally, if (t, x) ∈ E k is such that u k (t, x) < 1 and -C ≤ ζ k (t, x) ≤ C, then (3.32) yields -φ ′ f (ζ k (t, x)) ≥ κ > 0, while f (φ f (ζ k (t, x))) -f (u k (t, x)) ≥ -L δ e -δ(x•e-c f t-B) + δ e -δ(t-t k ) .
Therefore,

N k (t, x) ≥ δ (κ ω -L -δ) e -δ(t-t k ) + δ κ ω c f e δc f t -(L + δ 2 ) e -δ(x•e-c f t-B) ≥ δ κ ω c f e δc f t -(L + δ 2 ) e -δ(x•e-c f t-B)
from (3.33). On the other hand, the inequality

ζ k (t, x) ≤ C implies that x • e -c f t -B ≥ -2 c f t + c f t k + ξ + t k + ω e -δ(t-t k ) -ω -ω e δc f t -2B -2C ≥ -2 c f t -2 (ω + B + C)
from (3.29) and the fact that t ≤ T 2 ≤ 0. Thus,

-δ x • e -c f t -B) ≤ 2 δ c f t + 2 δ (ω + B + C) ≤ δ c f t from (3.35), whence N k (t, x) ≥ δ (κ ω c f -L -δ 2 ) e δc f t ≥ 0 from (3.33).
As a conclusion, for every k ∈ N, the function u k is a supersolution, in the set E k , of the equation (1.1) satisfied by u. The maximum principle implies that

u(t, x) ≤ u k (t, x) ≤ φ f (ζ k (t, x)) + δ e -δ(x•e-c f t-B) + δ e -δ(t-t k ) (3.36)
for all k ∈ N and (t, x) ∈ E k . As a consequence, for any fixed (t, x) ∈ R × R N such that t ≤ T 2 and x • e ≥ c f + B, one has (t, x) ∈ E k for k large enough, while t k → -∞ and ζ k (t, x) → +∞ as k → +∞ from our assumption (3.28). Passing to the limit as k → +∞ in (3.36) gives

u(t, x) ≤ δ e -δ(x•e-c f t-B)
for all t ≤ T 2 and x • e ≥ c f + B. The proof of Lemma 3.6 is thereby complete.

Characterization of the global mean speed of transition fronts

This section is devoted to the proof of Theorem 2.6 on the existence and the uniqueness of the global mean speed of the transition fronts connecting 0 and 1 for problem (1.1). The general idea can be summarized as follows. Any such transition front u is above θ + (resp. below θ -) on big sets in some neighborhoods of Γ t , thanks to (1.9). We recall that 0 < θ -≤ θ + < 1 are given in (1.3). When c f > 0, a solution of the Cauchy problem associated to (1.1) with a compactly supported initial condition above some β > θ + on a large ball spreads at the speed c f in all directions. On the other hand, when the initial condition is below some α < θ - on a large ball and is, say, equal to 1 outside, then the region where the solution is above α cannot move towards the center of the ball too much faster than c f . These two key-properties, which are stated in Lemmas 4.1 and 4.2 below, will force the interfaces Γ t of the transition front u to move at the global mean speed γ = |c f |, in the sense of (1.12).

Two key-lemmas

In the sequel, we fix to real numbers α and β as in (3.5), that is

0 < α < θ -≤ θ + < β < 1, (4.1)
where θ ± are defined in (1.3). For any R > 0, let v R and w R denote the solutions of the Cauchy problems

(v R ) t = ∆v R + f (v R ), t > 0, x ∈ R N , (w R ) t = ∆w R + f (w R ), t > 0, x ∈ R N , (4.2) with initial conditions v R (0, x) = β if |x| < R, 0 if |x| ≥ R (4.3) and w R (0, x) = α if |x| < R, 1 if |x| ≥ R (4.4) Lemma 4.1 Assume that c f > 0.
There is R > 0 such that the following holds: for all ε ∈ (0, c f ], there is

T ε > 0 such that v R (t, x) ≥ β for all t ≥ T ε and |x| ≤ (c f -ε)t (4.5)
and, in fact,

v R (t, •) → 1 uniformly in x ∈ R N ; |x| ≤ (c f -ε)t as t → +∞. (4.6)
In the proof of Theorem 2.6, Lemma 4.1 will provide a sharp lower bound for the speed of the interfaces Γ t of any transition front of (1.1) connecting 0 and 1.

The following lemma, which is a kind of counterpart of Lemma 4.1, will give the upper bound. That is, if c f ≥ 0 and if the initial condition w R (0, •) given in (4.4) is equal to (it could also be less than) α on a very large ball with radius R and equal to 1 outside, then the region where w R is above α is not filled at a speed too much larger than c f on some interval of time whose size is related to the initial radius R. Lemma 4.2 Assume that c f ≥ 0. Then, for any ε > 0, there are some real numbers

T ε > 0 and R ε ≥ (c f + ε)T ε > 0 such that for all R ≥ R ε , the solution w R of (4.2) and (4.4) satisfies w R (t, x) ≤ α for all T ε ≤ t ≤ R c f + ε and |x| ≤ R -(c f + ε)t. (4.7)
Before doing the proof of these two lemmas, let us first comment and compare them with some existing results of the literature. Lemma 4.1 could be viewed as an analog of the onedimensional propagation result of Fife and McLeod [29] used in (3.16) in the proof of Theorem 2.5. Actually, (3.16) is much more precise than Lemma 4.1 since (3.16) implies in particular that the position of any given level set of the considered solution is ±c f t + O(1) as t → +∞. Such an estimate cannot hold in higher dimension due to the curvature effects. Actually, the position of the level sets of the solutions v R of (4.2-4.3) could likely be estimated more precisely, but the conclusion of Lemma 4.1 will be sufficient for the proof of Theorem 2.6.

Notice also that if f is of the bistable type (1.4) and if β = β f < 1 is sufficiently close to 1, then Lemma 4.1 follows directly from Theorem 6.2 of Aronson and Weinberger [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF]. The proof of Lemma 4.1 is inspired by [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF] but the subsolutions used in the proof of Lemma 4.1 are different and, as such, the result is new.

Lemmas 4.1 and 4.2 also have some similarities with Theorem 2 of Chen [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF]. In [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF], some problems with small diffusion have been considered, for bistable functions f of the type (1.4) with f ′ (θ) > 0. In this case, after scaling and coming back to (1.1), it follows from [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF]

that, if c f > 0 and if R > 0 is large, then v R (t, x) is larger than 1 -O(R -k ) (where k > 0 is given) for t ≥ O(ln R) and |x| ≤ R + (c f -O(R -1 ))t -O(1)
. This latter estimate is quantitatively more precise than (4.5) for R ≥ O(ε -1 ). In Lemma 4.1 of the present paper, R can be chosen independently of ε and we only need v R to be not too far from 1, without any precise rate of convergence, in some balls expanding with a speed close to c f . The proof of [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF] is based on the construction of subsolutions with nonlinearities of the type fλ. Even if the ideas of [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF] could likely be adapted here to the case of functions f satisfying only (1.2) together with the existence of a planar front (c f , φ f ), we are going to work directly with the function f in the proof of Lemma 4.1. As far as Lemma 4.2 is concerned, it also follows from [START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF] that, if c f > 0 with (1.4) and f ′ (θ) > 0, and if

R > 0 is large, then w R (t, x) is smaller than O(R -k ) for O(ln R) ≤ t ≤ R/(2c f ) and |x| ≤ R -c f t -O(ln R).
The above pointwise estimate (4.7), that is w R (t, x) ≤ α, is less precise. However it holds until a time of the order R/c f (instead of R/(2c f )). In the proof of Theorem 2.6, we will actually need the estimate (4.7) on a time interval of the order R/c f , in order to show that the global mean speed of any transition front for problem (1.1) exists and is exactly equal to c f . Finally, even if Lemmas 4.1 and 4.2 have some similarities with [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Chen | Generation and propagation of interfaces for reaction-diffusion equations[END_REF], the assumptions and conclusions are different and, as such, the statements and the proofs are new to the best of our knowledge. Let us now turn to the proofs.

Proof of Lemma 4.1. We assume here that c f > 0. Observe first that (4.6) is clearly stronger than (4.5). The strategy to prove (4.6) is to construct some radially symmetric subsolutions in R N of the type φ f (|x| -(c fε/2)t -R) plus some exponentially decaying terms, for 0 < ε ≤ c f . In doing so, the solution v R will be close to 1 inside the balls of radii (c fε)t as t is large.

Step 1: choice of some parameters which are independent of ε. As in the proof of Lemma 3.6, we first introduce some parameters which are independent of ε. Let δ > 0 be chosen such that

0 < δ < min 1, |f ′ (0)| 2 , |f ′ (1)| 2 , f ′ ≤ f ′ (0) 2 on [0, δ] and f ′ ≤ f ′ (1) 2 on [1 -2δ, 1]. (4.8) Since φ ′′ f (s) ∼ -σ e µs as s → -∞ with σ > 0 and µ = (-c f + (c 2 f -4f ′ (1)) 1/2 )/2 > 0, one can choose C > 0 so that (3.31) holds together with φ ′′ f ≤ 0 on (-∞, -C], that is φ f ≥ 1 -δ on (-∞, -C], φ ′′ f ≤ 0 on (-∞, -C] and φ f ≤ δ on [C, +∞). (4.9)
Let κ > 0 be chosen as in (3.32), that is -φ ′ f ≥ κ on [-C, C], and let ω > 0 be such that κ ω ≥ L + δ, (4.10) where

L = max [0,1] |f ′ |.
Let ̺ β be the solution of the ordinary differential equation ̺ ′ β (t) = f (̺ β (t)) with initial condition ̺ β (0) = β. Since β ∈ (θ + , 1), there holds ̺ β (t) → 1 as t → +∞, and there is T > 0 such that ̺ β (T ) ≥ 1δ/2. It follows from the maximum principle that

0 ≤ ̺ β (T ) -v R (T, x) ≤ e LT (4πT ) N/2 |y|≥R e -|x-y| 2 4T dy for all R > 0 and x ∈ R N . Therefore, if 0 < B ≤ R and |x| ≤ R -B, one infers that 0 ≤ ̺ β (T ) -v R (T, x) ≤ e LT (4πT ) N/2 |z|≥B e -|z| 2
4T dz.

Thus, there exists B > 0 such that, for all R ≥ B and

|x| ≤ R -B, ̺ β (T ) -v R (T, x) ≤ δ/2, whence v R (T, x) ≥ ̺ β (T ) - δ 2 ≥ 1 -δ for all R ≥ B and |x| ≤ R -B. (4.11)
Step 2: choice of some functions h ε which depend on ε. It is elementary to check that, for every ε > 0, there is a C 2 function h ε : [0, +∞) → R satisfying the following properties:

             0 ≤ h ′ ε ≤ 1 on [0, +∞), h ′ ε = 0 on a neighborhood of 0, h ε (r) = r on [H ε , +∞) for some H ε > 0, (N -1)h ′ ε (r) r + h ′′ ε (r) ≤ ε 2 on [0, +∞).
(4.12)

Notice in particular that, necessarily, r ≤ h ε (r) ≤ r + h ε (0) for all r ≥ 0. (4.13)

Step 3: proof of (4.6) when c f /2 ≤ ε ≤ c f . To do so, it is sufficient to show that (4.6) holds with ε = ε 0 := c f /2 > 0, for some R > 0. Let us set

R = B + H ε 0 + ω + 2 C > B > 0. (4.14)
We will see that the conclusion of Lemma 4.1 holds for all 0 < ε ≤ c f with this value of R (notice that R is independent of ε). Let us show in this step that (4.6) holds with ε = ε 0 . For all (t, x) ∈ R × R N , we set

v(t, x) = max φ f ζ(t, x) -δ e -δ(t-T ) , 0 , (4.15)
where

ζ(t, x) = h ε 0 (|x|) -c f - ε 0 2 (t -T ) -ω e -δ(t-T ) -H ε 0 -C. (4.16)
Let us then check that v is a subsolution for the problem satisfied by v R , for t ≥ T and x ∈ R N . First, at the time T , it follows from (4.11), (4.14) and the definition of

v that v R (T, x) ≥ 1 -δ ≥ v(T, x) for all |x| ≤ R -B. On the other hand, if |x| ≥ R -B, then h ε 0 (|x|) ≥ |x| ≥ R -B = H ε 0 + ω + 2C
v(T, x) = max φ f (ζ(T, x)) -δ, 0 ≤ max δ -δ, 0 = 0 ≤ v R (T, x)
from (4.9) and the fact that v R ≥ 0 in (0, +∞) × R N . Thus,

v R (T, x) ≥ v(T, x) for all x ∈ R N .
Let us now check that

N(t, x) = v t (t, x) -∆v(t, x) -f (v(t, x)) ≤ 0 (4.17)
for all t ≥ T and x ∈ R N such that v(t, x) > 0. This will be sufficient to ensure that v is a subsolution, since

f (0) = 0 (notice that v(t, x) = φ f (ζ(t, x)) -δe -δ(t-T ) is of class C 2 in
the set where it is positive, since φ f is of class C 2 and h vanishes in a neighbourhood of 0). In this paragraph, let (t, x) be any point in [T, +∞) × R N such that v(t, x) > 0. Since φ f obeys (1.5), there holds

N (t, x) = f (φ f (ζ(t, x))) -f (v(t, x)) + ω δ e -δ(t-T ) φ ′ f (ζ(t, x)) + δ 2 e -δ(t-T ) + ε 0 2 - (N -1) h ′ ε 0 (|x|) |x| -h ′′ ε 0 (|x|) φ ′ f (ζ(t, x)) + 1 -(h ′ ε 0 (|x|)) 2 φ ′′ f (ζ(t, x)) ≤ f (φ f (ζ(t, x))) -f (v(t, x)) + ω δ e -δ(t-T ) φ ′ f (ζ(t, x)) + δ 2 e -δ(t-T ) + 1 -(h ′ ε 0 (|x|)) 2 φ ′′ f (ζ(t, x))
from (4.12) and φ

′ f ≤ 0. If ζ(t, x) ≤ -C, then 1 -δ ≤ φ f (ζ(t, x)) < 1 from (4.9), whence 1 -2δ ≤ v(t, x) ≤ φ f (ζ(t, x)) < 1 and f (φ f (ζ(t, x))) -f (v(t, x)) ≤ f ′ (1) 2 δ e -δ(t-T ) from (4.8). Furthermore, φ ′′ f (ζ(t, x)) ≤ 0 from (4.9), while 0 ≤ h ′ ε 0 (|x|) ≤ 1 from (4.12). Therefore, if ζ(t, x) ≤ -C, then N (t, x) ≤ δ f ′ (1) 2 + δ e -δ(t-T ) + ω δ e -δ(t-T ) φ ′ f (ζ(t, x)) ≤ 0 from (4.8
) and the negativity of φ ′ f . On the other hand, if

ζ(t, x) ≥ C, then φ f (ζ(t, x)) ≤ δ, whence 0 < v(t, x) ≤ φ f (ζ(t, x)) ≤ δ and f (φ f (ζ(t, x))) -f (v(t, x)) ≤ f ′ (0) 2 δ e -δ(t-T ) , again from (4.8). The inequality ζ(t, x) ≥ C also implies that h ε 0 (|x|) ≥ 2 C + H ε 0 ≥ H ε 0 , whence h ′ ε 0 (|x|) = 1 from (4.12). Thus, if ζ(t, x) ≥ C, then N (t, x) ≤ δ f ′ (0) 2 + δ e -δ(t-T ) + ω δ e -δ(t-T ) φ ′ f (ζ(t, x)) ≤ 0 from (4.8) and the negativity of φ ′ f . Finally, if -C ≤ ζ(t, x) ≤ C, then f (φ f (ζ(t, x))) -f (v(t, x)) ≤ L δ e -δ(t-T ) ,
where we recall that (3.32) and (4.10).

L = max [0,1] |f ′ |, while ζ(t, x) ≥ -C yields h ε 0 (|x|) ≥ H ε 0 and h ′ ε 0 (|x|) = 1. Hence, if -C ≤ ζ(t, x) ≤ C, then N(t, x) ≤ δ (L -κ ω + δ) e -δ(t-T ) ≤ 0 from
As a conclusion, the maximum principle implies that, for all t ≥ T and

x ∈ R N , 1 ≥ v R (t, x) ≥ v(t, x) ≥ φ f ζ(t, x) -δ e -δ(t-T ) . (4.18) But max |x|≤(c f -ε 0 )t ζ(t, x) ≤ (c f -ε 0 ) t + h ε 0 (0) -c f - ε 0 2 (t -T ) → -∞ as t → +∞, from (4.13)
, (4.16) and the positivity of

ε 0 , ω, H ε 0 and C. Since φ f (-∞) = 1, it follows from (4.18) that v R (t, •) → 1 uniformly in x ∈ R N ; |x| ≤ (c f -ε 0 )t as t → +∞. (4.19)
Step 4: conclusion and proof of (4.6) for all 0 < ε ≤ c f . Property (4.6) has already been proved for (4.19). Let now ε be arbitrary in (0, ε 0 ). With the notations used in Steps 1 and 2, set

ε 0 = c f /2 ≤ ε ≤ c f , from
R ε = H ε + ω + 2 C > 0 (4.20)
and, from (4.19), let

T ε ≥ T such that v R (T ε , x) ≥ 1 -δ for all |x| ≤ R ε .
Let v and ζ be defined as in (4.15) and (4.16) where T and ε 0 are replaced by T ε and ε. The same calculations as in Step 3 show that (4.17) holds for all (t, x)

∈ [T ε , +∞) × R N such that v(t, x) > 0. The only difference now is the comparison of v R and v at time T ε . If |x| ≤ R ε , then v R (T ε , x) ≥ 1 -δ ≥ v(T ε , x). If |x| ≥ R ε , then ζ(T ε , x) = h ε (|x|) -ω -H ε -C ≥ R ε -ω -H ε -C = C from (4.13) and (4.20), whence φ f (ζ(T ε , x)) ≤ δ from (4.9) and v(T ε , x) = 0 ≤ v R (T ε , x). Finally, v R (T ε , x) ≥ v(T ε , x) for all x ∈ R N .
Therefore, it follows from the maximum principle that

v R (t, x) ≥ v(t, x) ≥ φ f (ζ(t, x)) -δ e -δ(t-T ε ) for all t ≥ T ε and x ∈ R N .
As in Step 3, this leads to (4.6). The proof of Lemma 4.1 is thereby complete.

Let us now turn to the proof of Lemma 4.2. The strategy used is similar to that used in the proof of Lemma 4.1, with the construction of suitable supersolutions instead of subsolutions. However, one needs to be more careful with the application of the maximum principle, since the estimates (4.7) only hold in bounded time intervals. Notice also that Lemma 4.2 is not an immediate consequence of Lemma 4.1 after changing v into 1v and f (s) into -f (1s), since this operation would change the sign of the speed c f (however, Corollary 4.3 below can be deduced from Lemma 4.2 thanks to this operation).

Proof of Lemma 4.2. The strategy is to construct some radially symmetric supersolutions in R N of the type φ f (-(c f + ε/2)t + R -|x|) plus some exponentially decaying terms. In doing so, the solution w R will be small inside the balls of radii R -

(c f + ε)t.
As in the proofs of Lemmas 3.6 and 4.1, we first introduce some parameters which are independent of ε. Let δ > 0 be chosen so that

0 < δ < min 1, |f ′ (0)| 2 , |f ′ (1)| 2 , f ′ ≤ f ′ (0) 2 on [0, 2δ] and f ′ ≤ f ′ (1) 2 on [1 -δ, 1]. (4.21) Since φ ′′ f (s) ∼ ν e -λs as s → +∞ with ν > 0 and λ = c f + (c 2 f -4f ′ (0)) 1/2 /2, one can choose C > 0 so that (3.31) holds together with φ ′′ f ≥ 0 and φ f ≤ α/2 on [C, +∞), that is φ f ≥ 1 -δ on (-∞, -C], φ f ≤ min δ, α 2 on [C, +∞) and φ ′′ f ≥ 0 on [C, +∞). (4.22)
Let κ > 0 and ω > 0 be chosen as in (3.32) and (4.10), that is

-φ ′ f ≥ κ > 0 on [-C, C] and κ ω ≥ L + δ. (4.23)
Let ̺ α be the solution of the ordinary differential equation ̺ ′ α (t) = f (̺ α (t)) with initial condition ̺ α (0) = α. Since α ∈ (0, θ -), there holds ̺ α (t) → 0 as t → +∞, and there is T > 0 such that ̺ α (T ) ≤ δ/2. As in Step 1 of the proof of Lemma 4.1, it follows from the maximum principle that there exists B > 0 such that 0

≤ w R (T, x) -̺ α (T ) ≤ δ/2 for all R ≥ B and |x| ≤ R -B, whence w R (T, x) ≤ δ for all R ≥ B and |x| ≤ R -B. (4.24)
Now, we pick an arbitrary ε > 0 and we introduce some quantities which depend on ε. Let the function h ε be as in (4.12) and (4.13). Then, we choose

T ε ≥ T (> 0) such that δ e -δ(t-T ) ≤ α 2 and ε t 2 ≥ h ε (0) + ω + B + 2C for all t ≥ T ε , (4.25)
and

R ε > 0 such that R ε ≥ max B, (c f + ε)T ε and ε R ε 2(c f + ε) ≥ ω + B + 2C + H ε . (4.26)
We shall now prove that the conclusion of Lemma 4.2 holds with these choices of T ε and R ε .

In the sequel, R is an arbitrary real number such that

R ≥ R ε . (4.27)
For all (t, x) ∈ R × R N , we set

w(t, x) = min φ f ζ(t, x) + δ e -δ(t-T ) , 1 , where ζ(t, x) = -h ε (|x|) -c f + ε 2 (t -T ) + ω e -δ(t-T ) -ω + R -B -C.
Let us then check that w is a supersolution for the problem satisfied by w R , in the set

E = T, R c f + ε × R N .
At the time T , it follows from (4.24), (4.26), (4.27) and the definition of w that

w R (T, x) ≤ δ ≤ w(T, x) for all |x| ≤ R -B. On the other hand, if |x| ≥ R-B, then h ε (|x|) ≥ |x| ≥ R-B from (4.13), whence ζ(T, x) ≤ -C and w(T, x) = min φ f (ζ(T, x)) + δ, 1 ≥ min (1 -δ) + δ, 1 = 1 ≥ w R (T, x) from (4.22) and the fact that w R ≤ 1 in (0, +∞) × R N . Thus, w R (T, x) ≤ w(T, x) for all x ∈ R N .
Let us now check that

N (t, x) = w t (t, x) -∆w(t, x) -f (w(t, x)) ≥ 0 for all (t, x) ∈ E such that w(t, x) < 1.
This will be sufficient to ensure that w is a supersolution, since f (1) = 0 (notice that, as for v in Lemma 4.1, w is of class C 2 in the set where it is less than 1). In this paragraph, let (t, x) be any point in E such that w(t, x) < 1. Since φ f obeys (1.5), there holds

N(t, x) = f (φ f (ζ(t, x))) -f (w(t, x)) -ω δ e -δ(t-T ) φ ′ f (ζ(t, x)) -δ 2 e -δ(t-T ) - ε 2 - (N -1) h ′ ε (|x|) |x| -h ′′ ε (|x|) φ ′ f (ζ(t, x)) + 1 -(h ′ ε (|x|)) 2 φ ′′ f (ζ(t, x)) ≥ f (φ f (ζ(t, x))) -f (w(t, x)) -ω δ e -δ(t-T ) φ ′ f (ζ(t, x)) -δ 2 e -δ(t-T ) + 1 -(h ′ ε (|x|)) 2 φ ′′ f (ζ(t, x)) from (4.12) and φ ′ f ≤ 0. If ζ(t, x) ≤ -C, then 1 -δ ≤ φ f (ζ(t, x)) ≤ w(t, x) < 1 from (4.22), whence f (φ f (ζ(t, x))) -f (w(t, x)) ≥ -f ′ (1) 2 δ e -δ(t-T ) from (4.21). Furthermore, the inequalities ζ(t, x) ≤ -C and 0 < T ≤ t ≤ R/(c f + ε) yield h ε (|x|) ≥ -c f + ε 2 (t -T ) -ω + R -B ≥ εR 2(c f + ε) -ω -B ≥ H ε
because of (4.26) and (4.27) (notice that the term -2C in (4.26) is not needed here, but it will be later). The inequality h ε (|x|) ≥ H ε implies that h ′ ε (|x|) = 1, due to the properties (4.12). Therefore, if ζ(t, x) ≤ -C, then

N (t, x) ≥ δ -f ′ (1) 2 -δ e -δ(t-T ) -ω δ e -δ(t-T ) φ ′ f (ζ(t, x)) ≥ 0 from (4.21
) and the negativity of φ ′ f . On the other hand, if

ζ(t, x) ≥ C, then φ f (ζ(t, x)) ≤ δ from (4.22), whence 0 < φ f (ζ(t, x)) ≤ w(t, x) ≤ 2δ and f (φ f (ζ(t, x))) -f (w(t, x)) ≥ -f ′ (0) 2 δ e -δ(t-T ) from (4.21). Since φ ′′ f ≥ 0 on [C, +∞) from (4.22), since 0 ≤ h ′ ε ≤ 1 on [0, +∞) and since φ ′ f ≤ 0 on R, one gets that, if ζ(t, x) ≥ C, then N (t, x) ≥ δ -f ′ (0) 2 -δ e -δ(t-T ) -ω δ e -δ(t-T ) φ ′ f (ζ(t, x)) ≥ 0 from (4.21). Lastly, if -C ≤ ζ(t, x) ≤ C, then f (φ f (ζ(t, x))) -f (w(t, x)) ≥ -L δ e -δ(t-T ) , while ζ(t, x) ≤ C yields h ε (|x|) ≥ -c f + ε 2 (t -T ) -ω + R -B -2C ≥ εR 2(c f + ε) -ω -B -2C ≥ H ε from (4.26) and (4.27). Thus, h ′ ε (|x|) = 1 and N(t, x) ≥ δ (-L + κ ω -δ) e -δ(t-T ) ≥ 0 from (4.23).
As a conclusion, the maximum principle implies that, for all T ≤ t ≤ R/(c f +ε) and

x ∈ R N , w R (t, x) ≤ w(t, x) ≤ φ f ζ(t, x) + δ e -δ(t-T ) . For all T ε ≤ t ≤ R/(c f + ε) and |x| ≤ R -(c f + ε)t, there holds δ e -δ(t-T ) ≤ α/2 from (4.25), while h ε (|x|) ≤ |x| + h ε (0) ≤ R -(c f + ε)t + h ε (0) and ζ(t, x) ≥ -R + (c f + ε)t -h ε (0) -c f + ε 2 (t -T ) + ω e -δ(t-T ) -ω + R -B -C ≥ ε t 2 -h ε (0) -ω -B -C ≥ C from (4.25). Thus, φ f (ζ(t, x)) ≤ α/2 from (4.22). Finally, if T ε ≤ t ≤ R/(c f + ε) and |x| ≤ R -(c f + ε)t, then w R (t, x) ≤ φ f ζ(t, x) + δ e -δ(t-T ) ≤ α 2 + α 2 = α.
The proof of Lemma 4.2 is thereby complete.

By changing f (s) into -f (1 -s), c f into -c f , w R into 1 -v R , α into
1β and β into 1α, the following result holds: Corollary 4.3 Assume that c f ≤ 0. Then, for any ε > 0, there are some real numbers

T ε > 0 and R ε ≥ (|c f | + ε) T ε > 0 such that for all R ≥ R ε , the solution v R of (4.2) and (4.3) satisfies v R (t, x) ≥ β for all T ε ≤ t ≤ R |c f | + ε and |x| ≤ R -(|c f | + ε)t. (4.28)
4.2 Proof of Theorem 2.6

Let u be any transition front connecting 0 and 1 for problem (1.1). As in the proof of Theorem 2.5, even if it means changing u(t, x) into u(t, x) = 1u(t, x), f (s) into g(s) = -f (1s) and c f into -c f , one can assume without loss of generality that

c f ≥ 0.
In order to prove that

d(Γ t , Γ s ) |t -s| → c f as |t -s| → +∞,
we prove one inequality for the lim inf and another one for the lim sup.

Step 1: the lower estimate. We first claim that lim inf

|t-s|→+∞ d(Γ t , Γ s ) |t -s| ≥ c f . (4.29)
Since there is nothing to prove when c f = 0, we only consider the case c f > 0. Let α and β be given as in (4.1). From Definition 1.1, there is M ≥ 0 such that

∀ t ∈ R, ∀ x ∈ Ω + t , d(x, Γ t ) ≥ M =⇒ β ≤ u(t, x) < 1 , ∀ t ∈ R, ∀ x ∈ Ω - t , d(x, Γ t ) ≥ M =⇒ 0 < u(t, x) ≤ α . (4.30)
Let R > 0 be as in Lemma 4.1. Since the functions v ρ are nondecreasing with respect to the parameter ρ > 0, one can assume without loss of generality that R ≥ M. From (1.9), there is r > 0 such that

∀ t ∈ R, ∀ x ∈ Γ t , ∃ y ± t ∈ Ω ± t , |x -y ± | ≤ r and d(y ± , Γ t ) ≥ 2R. (4.31)
Let now ε ∈ (0, c f ] and let T ε > 0 be as in Lemma 4.1. Let us assume by contradiction that lim inf

|t-s|→+∞ d(Γ t , Γ s ) |t -s| < c f -2 ε. (4.32)
There are then two sequences (t k ) k∈N and (s k ) k∈N of real numbers such that

|t k -s k | → +∞ as k → +∞ and d(Γ t k , Γ s k ) < (c f -2 ε) |t k -s k | for all k ∈ N.
Without loss of generality, one can assume that t k < s k for all k ∈ N. By definition of the distance d(Γ t k , Γ s k ), there are then two sequences (x k ) k∈N and (z k ) k∈N in R N such that

x k ∈ Γ t k , z k ∈ Γ s k and |x k -z k | < (c f -2ε) (s k -t k ) for all k ∈ N.
On the one hand, it follows from (4.31) that there exists a sequence (y

+ k ) k∈N of points in R N such that y + k ∈ Ω + t k , |x k -y + k | ≤
y - k ∈ Ω - s k , |z k -y - k | ≤ r and d(y - k , Γ s k ) ≥ 2R (≥ M) for all k ∈ N.
In particular,

u(s k , y - k ) ≤ α for all k ∈ N, (4.34)
due to (4.30).

Let us now check that one can choose t = s k and x = y - k in (4.33) for k large enough. Indeed, s k ≥ t k + T ε for k large enough since s kt k → +∞ as k → +∞. Furthermore,

|y - k -y + k | ≤ |y - k -z k | + |z k -x k | + |x k -y + k | ≤ r + (c f -2ε) (s k -t k ) + r
for all k ∈ N, whence

|y - k -y + k | ≤ (c f -ε) (s k -t k )
d(Γ t , Γ s ) |t -s| ≥ c f -2 ε.
Since ε > 0 can be arbitrarily small, the claim (4.29) follows.

Step 2: the upper estimate. Let us here show that lim sup

|t-s|→+∞ d(Γ t , Γ s ) |t -s| ≤ c f . (4.35)
Let first α and β be fixed as in (4.1), let M ≥ 0 be as in (4.30), and let r ≥ 0 be such that

∀ t ∈ R, ∀ x ∈ Γ t , ∃ y ± t ∈ Ω ± t , |x -y ± | ≤
(Γ t k , Γ s k ) > (c f + 3 ε) |t k -s k | for all k ∈ N.
Without loss of generality, one can assume that t k < s k for all k ∈ N. For each k ∈ N, pick a point z k on Γ s k . From (4.36), there are two sequences (y ± k ) k∈N of points in R N such that

y ± k ∈ Ω ± s k , |z k -y ± k | ≤ r and d(y ± k , Γ s k ) ≥ M for all k ∈ N.
Thus, (4.30) implies that

0 < u(s k , y - k ) ≤ α < β ≤ u(s k , y + k ) < 1 for all k ∈ N. (4.38)
On the other hand, since

d(z k , Γ t k ) > (c f + 3 ε) (s k -t k ) > 0, there holds either B z k , (c f + 3 ε) (s k -t k ) ⊂ Ω + t k or B z k , (c f + 3 ε) (s k -t k ) ⊂ Ω - t k .
Let us assume by contradiction that, up to extraction of a subsequence,

B z k , (c f + 3 ε) (s k -t k ) ⊂ Ω + t k for all k ∈ N. (4.39)
Two cases shall be considered: c f > 0 and c f = 0. Consider first the former. Let R > 0 be given as in Lemma 4.1. Since s kt k → +∞ as k → +∞, there holds, for k large enough, B(z k , R) ⊂ Ω + t k together with d(y, Γ t k ) ≥ M for all y ∈ B(z k , R). Thus, it follows from (4.30) that, for k large enough, u(t k , y) ≥ β for all y ∈ B(z k , R),

whence u(t k , x) ≥ v R (0, x -z k ) for all x ∈ R N and u(t, x) ≥ v R (t -t k , x -z k ) for all t > t k and x ∈ R N (4.40)
from the maximum principle. Let T ε ′ > 0 be given by Lemma 4.1 with ε ′ = c f /2 ∈ (0, c f ]. It follows from (4.40) and Lemma 4.1 that, for k large enough,

u(t, x) ≥ β for all t ≥ t k + T ε ′ and |x -z k | ≤ (c f -ε ′ ) (t -t k ) = c f 2 (t -t k ).
Since c f > 0 and s kt k → +∞ as k → +∞, there holds

s k ≥ t k + T ε ′ and |y - k -z k | ≤ r ≤ (c f /2) (s k -t k )
for k large enough. Therefore, u(s k , y - k ) ≥ β for k large enough, which is in contradiction with (4.38). As a consequence, the assumption (4.39) is ruled out if c f > 0.

Consider now the case c f = 0. Since s kt k → +∞ as k → +∞, assumption (4.39) implies that, for k large enough,

B(z k , 2 ε (s k -t k )) ⊂ Ω + t k together with d(y, Γ t k ) ≥ M for all y ∈ B(z k , 2 ε (s k -t k )). It follows from (4.30) that, for k large enough, u(t k , y) ≥ β for all y ∈ B(z k , 2 ε (s k -t k )). Hence, for k large enough, u(t k , x) ≥ v 2ε(s k -t k ) (0, x -z k ) for all x ∈ R N , whence u(t, x) ≥ v 2ε(s k -t k ) (t -t k , x -z k ) for all t > t k and x ∈ R N (4.41)
from the maximum principle. On the other hand, Corollary 4.3 provides the existence of T ε and R ε ≥ ε T ε > 0 such that (4.28) holds with c f = 0 for all R ≥ R ε . In particular, since s kt k → +∞ as k → +∞, there holds, for k large enough, 2 ε (s

k -t k ) ≥ R ε and    T ε ≤ s k -t k ≤ 2 (s k -t k ) = 2 ε (s k -t k ) ε , |y - k -z k | ≤ r ≤ ε (s k -t k ) = 2 ε (s k -t k ) -ε (s k -t k ).
Therefore, the conclusion (4.28) can be applied with

c f = 0, R = 2 ε (s k -t k ), t = s k -t k and x = y - k -z k ,
u(s k , y - k ) ≥ v 2ε(s k -t k ) (s k -t k , y - k -z k ) ≥ β
for k large enough, which is in contradiction with (4.38). As a conclusion, the assumption (4.39) is impossible (even for a subsequence). Hence,

B z k , (c f + 3 ε) (s k -t k ) ⊂ Ω - t k for k large enough. Since s k -t k → +∞ as k → +∞, it follows that, for k large enough, B z k , (c f + 2ε)(s k -t k ) ⊂ Ω - t k and d(y, Γ t k ) ≥ M for all y ∈ B z k , (c f + 2ε)(s k -t k ) . Therefore, u(t k , y) ≤ α for all y ∈ B z k , (c f + 2ε)(s k -t k )
, for k large enough, due to (4.30). Hence, for k large enough, u(t k , x) ≤ w (c f +2ε)(s k -t k ) (0, xz k ) for all x ∈ R N , and u(t, x) ≤ w (c f +2ε)(s k -t k ) (tt k , xz k ) for all t > t k and x ∈ R N from the maximum principle. Let now T ε > 0 and R ε ≥ (c f + ε)T ε > 0 be given by Lemma 4.2, so that (4.7) is valid for all R ≥ R ε . In particular, since s kt k → +∞ as k → +∞, there holds, for k large enough, (c

f + 2ε)(s k -t k ) ≥ R ε and    T ε ≤ s k -t k ≤ (c f + 2ε) (s k -t k ) c f + ε , |y + k -z k | ≤ r ≤ ε (s k -t k ) = (c f + 2ε) (s k -t k ) -(c f + ε) (s k -t k ).
Therefore, the conclusion (4.7) can be applied with R = (c f + 2ε) (s kt k ), t = s kt k and x = y + kz k , for k large enough. Finally, there holds

u(s k , y + k ) ≤ w (c f +2ε)(s k -t k ) (s k -t k , y + k -z k ) ≤ α
for k large enough, which is in contradiction with (4.38).

To sum up, we have shown that the assumption (4.37) is impossible. Since ε > 0 can be arbitrarily small, the conclusion (4.35) follows. The proof of Theorem 2.6 is thereby complete. for any transition front u connecting 0 and 1 for equation (1.1). This follows immediately from Step 2 of the previous proof when

c f ≥ 0. If c f < 0, one can change u(t, x) into u(t, x) = 1 -u(t, x), f (s) into g(s) = -f (1 -s), c f into -c f , Ω ± t into Ω ∓ t ,

Existence of non-standard transition fronts

This section is devoted to the proof of Theorem 2.8 on the existence of non-standard transition fronts for problem (1.1), that is transition fronts which are not invariant in any moving frame. We first consider the two-dimensional case R 2 . Let us explain heuristically the general strategy before going into the details of the proof. For a better understanding, we refer to the figure shown in Section 2 after the statement of Theorem 2.8. The first key-ingredient, from [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF][START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF], is the existence, under the condition (1.4) with c f > 0, of two-dimensional traveling fronts of the type φ(x 1 , x 2ct) whose level sets are asymptotic to two half-lines having an angle α ∈ (0, π/2) with respect to the x 2 -axis, see the figure in Section 1.1. Consider such a front with c = c f / sin α and π/4 < α < π/2 and rotate it of angle π/2α clockwise. Namely, we consider the front φ(R -1 (x 1 , x 2ct)) where R denotes the rotation of angle π/2α clockwise. As far as the level sets of this new front are concerned, the right asymptotic half-line becomes parallel to the x 1 -axis, the other one being then very far away from the x 2 -axis at very negative times.

The next step is to take the restriction of this front on the half-plane H = {x 1 < 0}. One will check that this "left" traveling front is almost a solution of the same equation (1.1) in H with Neumann boundary conditions on ∂H for very negative times. One will then solve this Neumann boundary value problem and symmetrize the solution with respect to ∂H. Finally, the obtained solution is shown to behave as three moving planar fronts at very negative times, and then as a V -shaped classical traveling front φ(x 1 , x 2ct) made of two moving planar fronts for very positive times.

Proof of Theorem 2.8

In this section, we carry out the proof of Theorem 2.8. We leave the proof of some auxiliary lemmas in Section 5.2. Throughout the proof of Theorem 2.8, we assume that f satisfies (1.4) with c f > 0, in addition to (1.2). We repeat that the existence (and uniqueness) of c f is guaranteed by (1.4). We first consider the case N = 2 and construct two-dimensional transition fronts satisfying the conclusion of Theorem 2.8. The conclusion in higher dimensions will be then obtained immediately by trivially extending the constructed two-dimensional fronts in the variables x 3 , . . . , x N .

Step 1: an auxiliary V -shaped front. Fix an angle α such that π 4 < α < π 2 .

From [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], there exists a unique traveling front φ(x 1 , x 2ct) of (1.1) in R 2 satisfying the following properties:

0 < φ < 1 in R 2 , φ is of class C 2 (R 2 ), c = c f / sin α,      lim inf A→+∞ inf x 2 ≤|x 1 | cot α-A φ(x 1 , x 2 ) = 1, lim sup A→+∞ sup x 2 ≥|x 1 | cot α+A φ(x 1 , x 2 ) = 0 (5.1)
and φ is asymptotically planar along the directions (± sin α, cos α) in the sense that there exist some positive constants ρ 1 and ω 1 such that

0 ≤ φ(x 1 , x 2 ) -max φ f (x 1 cos α + x 2 sin α), φ f (-x 1 cos α + x 2 sin α) ≤ ρ 1 e -ω 1 √ x 2 1 +x 2 2 (5.2)
for all (x 1 , x 2 ) ∈ R 2 . Since φ f (s) converges exponentially fast to 0 and 1 as s → ±∞, the function φ converges then exponentially fast to 0 and 1 as x 2 -|x 1 | cot α → ±∞. From the Schauder interior estimates, it follows then that there exist some positive constants ρ 2 and ω 2 such that ∇φ

(x 1 , x 2 ) ≤ ρ 2 e -ω 2 |x 2 -|x 1 | cot α| for all (x 1 , x 2 ) ∈ R 2 . ( 5.3)
Similar arguments yield the existence of some positive constants ρ 3 and ω 3 such that

∇φ(x 1 , x 2 ) -∇ φ f (-x 1 cos α + x 2 sin α) ≤ ρ 3 e -ω 3 √ x 2 1 +x 2 2 for all x 1 ≥ 0, x 2 ∈ R, whence φ x 1 (x 1 , x 2 )+cos α φ ′ f (-x 1 cos α+x 2 sin α) ≤ ρ 3 e -ω 3 √ x 2 1 +x 2 2 φ x 2 (x 1 , x 2 )-sin α φ ′ f (-x 1 cos α+x 2 sin α) ≤ ρ 3 e -ω 3 √ x 2 1 +x 2 2 for all x 1 ≥ 0, x 2 ∈ R. (5.4)
Lastly, it follows from [START_REF] Hamel | Solutions of semilinear elliptic equations in R N with conical-shaped level sets[END_REF] that

∀ A ≥ 0, sup -A≤x 2 -|x 1 | cot α≤A φ x 2 (x 1 , x 2 ) < 0 (5.5)
and that φ is decreasing in any direction (cos ϕ, sin ϕ) such that π/2α < ϕ < π/2 + α. In particular, the function φ is nonincreasing along the directions (± sin α, cos α).

Step 2: the rotated V -shaped front. Let us now rotate the function φ with angle απ/2 clockwise. Namely, we define

ψ(x 1 , x 2 ) = φ(x 1 sin α -x 2 cos α, x 1 cos α + x 2 sin α) (5.6) for all (x 1 , x 2 ) ∈ R 2 .
The function ψ is decreasing in any direction (cos ϕ, sin ϕ) with 0 < ϕ < 2α. In particular, ψ is nonincreasing in the horizontal direction (1, 0) and it converges to the planar front φ f (x 2 ) along this direction. In other words, one of the asymptotic branches of the level sets of ψ corresponds to the half-line R + (1, 0). The other branch is the half R + (cos(2α), sin(2α)) and it belongs to the left half-plane {x

1 ≤ 0} since α is chosen such that π/4 < α < π/2. Since φ(x 1 , x 2 -ct) solves (1.1) in R 2 , the C 2 (R × R 2 ) function v defined in R × R 2 by v(t, x 1 , x 2 ) = ψ(x 1 -ct cos α, x 2 -ct sin α) = φ(x 1 sin α -x 2 cos α, x 1 cos α + x 2 sin α -ct) (5.7)
satisfies (1.1) in R 2 too. The function v is a traveling front which is invariant in the moving frame with speed c in the direction (cos α, sin α), in the sense that

v(t + τ, x 1 + cτ cos α, x 2 + cτ sin α) = v(t, x 1 , x 2 ) for all (t, x 1 , x 2 ) ∈ R × R 2 and for all τ ∈ R. At any time t ∈ R, any level set of v, that is the set (x 1 , x 2 ) ∈ R 2 , v(t, x 1 , x 2 ) = λ
for a given value λ ∈ (0, 1) and at a given time t ∈ R, is asymptotic to some finite shifts of the two half-lines (ct cos α, ct sin α) + R + (cos(2α), sin(2α)) and (ct cos α, ct sin α) + R + (1, 0), and the first one (i.e. the left one) is very far from the x 2 -axis for very negative times. More precisely, (5.2) implies that

0 ≤ v(t, x 1 , x 2 ) -max φ f (x 1 sin(2α) -x 2 cos(2α) -ct sin α), φ f (x 2 -ct sin α) ≤ ρ 1 e -ω 1 √ (x 1 sin α-x 2 cos α) 2 +(x 1 cos α+x 2 sin α-ct) 2 for all (t, x 1 , x 2 ) ∈ R × R 2 .
Step 3: a Neumann boundary value problem in a half-space and construction of some suband supersolutions. In the remaining steps, the strategy consists in constructing a solution of (1.1) in R 2 which looks like the function v(t, x 1 , x 2 ) for very negative times in the half-plane

H = (x 1 , x 2 ) ∈ R 2 , x 1 < 0 .
To do so, we will work in the half-plane H with Neumann boundary conditions on ∂H and we will then extend the constructed solution by orthogonal symmetry with respect to ∂H.

Let us then consider the problem

v t = ∆v + f (v), (t, x 1 , x 2 ) ∈ R × H, v x 1 = 0, (t, x 1 , x 2 ) = (t, 0, x 2 ) ∈ R × ∂H. (5.8)
Remember that the function ψ defined in (5.6) is nonincreasing along the direction (1, 0), that is ψ

x 1 (x 1 , x 2 ) ≤ 0 in R 2 . Therefore, v x 1 (t, x 1 , x 2 ) ≤ 0 in R × R 2 .
In particular, the function v is a subsolution of (5.8).

Problem (5.8) also admits a supersolution which looks like the function v for very negative times, up to some exponentially small terms, as shown in the following lemma. Lemma 5.1 There exist some constants σ > 0, δ > 0 and T < 0 such that the function v defined in R × H by v(t, x 1 , x 2 ) = min v(t + σ e δt , x 1 , x 2 ) + δ e δ(x 1 +t) , 1

(5.9)

is a supersolution of (5.8) for t ≤ T .

In order not to lengthen the proof of Theorem 2.8, the proof of Lemma 5.1 is postponed in Section 5.2.

Step 4: construction of a solution v of (5.8) in H. Observe first that

v t (t, x 1 , x 2 ) = -c φ x 2 (x 1 sin α -x 2 cos α, x 1 cos α + x 2 sin α -ct) > 0 (5.10) for all (t, x 1 , x 2 ) ∈ R×R 2 , whence v(t, x 1 , x 2 ) < v(t, x 1 , x 2 ) in R×H (remember also that v < 1 in R × R 2 )
. For any n ∈ N such that n > |T |, let v n be the solution of the Cauchy problem associated to (5.8) for times t > -n, with initial condition

v n (-n, x 1 , x 2 ) = v(-n, x 1 , x 2 ) for all (x 1 , x 2 ) ∈ H at time t = -n. From
Step 3 and the above observations, the maximum principle implies that

0 < v(t, x 1 , x 2 ) ≤ v n (t, x 1 , x 2 ) ≤ v(t,
n (-n + h, x 1 , x 2 ) ≥ v(-n + h, x 1 , x 2 ) > v(-n, x 1 , x 2 ) = v n (-n, x 1 , x 2 )
for all n > |T |, h > 0 and (x 1 , x 2 ) ∈ H, whence v n is increasing with respect to time t in H, from the maximum principle.

From monotone convergence and standard parabolic estimates up to the boundary, the functions v n converge then as n

→ +∞ in C 1,2 loc (R × H) to a solution v of (5.8) such that 0 < v(t, x 1 , x 2 ) ≤ v(t, x 1 , x 2 ) ≤ v(t, x 1 , x 2 ) ≤ 1 for all t ≤ T and (x 1 , x 2 ) ∈ H and 0 < v ≤ v ≤ 1 in R × H (the strong maximum principle also yields 0 < v < 1 in R × H, since v(t, x 1 , x 2 ) → 0 < 1 as t → -∞ for each fixed (x 1 , x 2 ) ∈ H). Moreover, v t ≥ 0 in R × H
with even the strict inequality v t > 0 in R × H from the strong maximum principle, since the Step 7: the solution u satisfies the conclusion of Theorem 2.8. Assume by contradiction that there exist a function Φ : R 2 → (0, 1) and some families (R t ) t∈R and (X t ) t∈R = (x 1,t , x 2,t ) t∈R of rotations and points in R 2 such that

u(t, x 1 , x 2 ) = Φ(R t (x 1 -x 1,t , x 2 -x 2,t )) for all (t, x 1 , x 2 ) ∈ R × R 2 .
It would then follow from Lemma 5.2 that there is M ≥ 0 such that

R t (Γ t -X t ) ⊂ (x 1 , x 2 ) ∈ R 2 ; d((x 1 , x 2 ), R s (Γ t -X s )) ≤ M for all (t, s) ∈ R 2 .
This is clearly in contradiction with the definitions (5.17) and (5.18) of the sets Γ t . Therefore, the function u satisfies all properties of Theorem 2.8 in R 2 .

Step 8: the case of the space R N with N ≥ 3. We start from the solution u of (1.1) in R 2 constructed in the previous steps and we extend it trivially in R N as

u(t, x 1 , • • • , x N ) = u(t, x 1 , x 2 ) for all (t, x 1 , • • • , x N ) ∈ R × R N .
From the previous steps, the function u is obviously a transition front for problem (1.1) in R N with the sets

Ω ± t = (x 1 , • • • , x N ) ∈ R N ; (x 1 , x 2 ) ∈ Ω ± t
for all t ∈ R. This solution fulfills the desired conclusion and the proof of Theorem 2.8 is thereby complete.

that is v(t + σe δt , x 1 , x 2 ) = φ(ζ 1 (x 1 , x 2 ), ζ 2 (t, x 1 , x 2 )). If ζ 2 (t, x 1 , x 2 ) ≤ |ζ 1 (x 1 , x 2 )| cot α -C, then 1 > v(t, x 1 , x 2 ) > v(t + σe δt , x 1 , x 2 ) = φ(ζ 1 (x 1 , x 2 ), ζ 2 (t, x 1 , x 2 )) ≥ 1 -δ from (5.22), whence N (t, x 1 , x 2 ) ≥ f v(t + σe δt , x 1 , x 2 ) -f (v(t, x 1 , x 2
)) + σδv t (t + σe δt , x 1 , x 2 ) e δt ≥ 0 (5.31) from (5.21), (5.30) and the positivity of v t in (5.10

). If ζ 2 (t, x 1 , x 2 ) ≥ |ζ 1 (x 1 , x 2 )| cot α + C, then (5.22) yields 0 < v(t + σe δt , x 1 , x 2 ) ≤ δ, whence 0 < v(t + σe δt , x 1 , x 2 ) < v(t, x 1 , x 2 ) = v(t + σe δt , x 1 , x 2 ) + δe δ(x 1 +t) ≤ 2δ
since x 1 ≤ 0 and t ≤ T < 0. Thus, as above, (5.31) holds from (5.21), (5.30) and the positivity of

v t . Lastly, if -C ≤ ζ 2 (t, x 1 , x 2 ) -|ζ 1 (x 1 , x 2 )| cot α ≤ C, then f v(t + σe δt , x 1 , x 2 ) -f (v(t, x 1 , x 2 )) = f v(t + σe δt , x 1 , x 2 ) -f (v(t + σe δt , x 1 , x 2 ) + δe δ(x 1 +t) ) ≥ -L δ e δ(x 1 +t) since L = max [0,1] |f ′ |, while v t (t + σe δt , x 1 , x 2 ) = -c φ x 2 (ζ 1 (x 1 , x 2 ), ζ 2 (t, x 1 , x 2
)) ≥ cκ from (5.23). Therefore, N (t, x 1 , x 2 ) ≥ -L δ e δ(x 1 +t) + σ δ c κ e δt ≥ δ (σ c κ -L) e δt ≥ 0 from (5.24), (5.30) and the fact that x 1 ≤ 0. As a conclusion, N (t, x 1 , x 2 ) ≥ 0 for all (t, x 1 , x 2 ) ∈ (-∞, T ] × H such that v(t, x 1 , x 2 ) < 1. The proof of Lemma 5.1 is thereby complete.

Proof of Lemma 5.2. We have to show that u(t, x 1 , x 2 ) → 1 (resp. 0) as d((x 1 , x 2 ), Γ t ) → +∞ with (x 1 , x 2 ) ∈ Ω + t (resp. (x 1 , x 2 ) ∈ Ω - t ), uniformly in t.

Step 1: convergence to 1 in Ω + t . The inequality (5.14) and the fact that φ f is decreasing imply that max φ f (-x 1 sin(2α)-x 2 cos(2α)-c f t), φ f (x 1 sin(2α)-x 2 cos(2α)-c f t), φ f (x 2 -c f t) ≤ u(t, x 1 , x 2 ) (5.32) for all (t, x 1 , x 2 ) ∈ R × R 2 . It immediately follows from the definitions (5.17), (5.18) and (5.19) of Γ t and Ω + t , and from the fact that φ f (-∞) = 1, that lim M →+∞ inf t∈R, (x 1 ,x 2 )∈Ω + t , d((x 1 ,x 2 ),Γt)≥M u(t, x 1 , x 2 ) = 1.

(5.33)

Step 2: convergence to 0 in Ω - t for negative enough times. As far as the behavior of u in Ω - t far away from Γ t is concerned, we will consider three cases: when t is very negative, when t is very positive and when t is in some bounded interval. Let us first consider the case when t is very negative. Let ε > 0 be arbitrary. Remember that T < 0 is given in Lemma 5.1. Since φ f (+∞) = 0, it follows from the definitions (5.17) and ( 5 In other words, owing to the definitions (5.16) of the points P l t and P r t for t ≤ 0, this means that the sequence (d((x 1,n , x 2,n ), {P l tn , P r tn })) n∈N is bounded. But the points P l tn and P r tn lie on Γ tn for all n ∈ N (since t n ≤ T ≤ 0). As a consequence, the sequence (d((x 1,n , x 2,n ), Γ tn )) n∈N is bounded, contradicting (5.37). Finally, (5.36) holds for some M 2 > 0 and it follows from (5.15), (5.34), (5.35) for all n ∈ N, t ≤ T and (x 1 , x 2 ) ∈ R 2 . Since φ f (+∞) = 0, π/4 < α < π/2, and x 2,n -| tan(2α)| |x 1,n | → +∞ as n → +∞, the first term of the right-hand side converges to 0 as n → +∞ locally uniformly in (t, x 1 , x 2 ) ∈ (-∞, T ] × R 2 . The second-term also converges to 0 as in the proof of (5.36). Therefore, by passing to the limit as n → +∞, one infers that u ∞ (t, x 1 , x 2 ) ≤ δ e δt for all t ≤ T and (x 1 , x 2 ) ∈ R 2 .

Let η 0 > 0 be such that f ≤ 0 on [0, η 0 ]. For any η ∈ (0, η 0 ], there is t 0 ≤ T such that 0 ≤ u ∞ (s, x 1 , x 2 ) ≤ δ e δs ≤ η for all s ≤ t 0 and (x 1 , x 2 ) ∈ R 2 , whence 0 ≤ u ∞ (t, x 1 , x 2 ) ≤ η for all (t, x 1 , x 2 ) ∈ [s, +∞) × R 2 from the maximum principle, and then for all (t, x 1 , x 2 ) ∈ R × R 2 since s can be arbitrarily negative. Since η > 0 can be arbitrarily small, it follows that u ∞ ≡ 0 in R × R 2 , which contradicts u ∞ (t ∞ , 0, 0) > 0. As a consequence, (5.39) is proved.

Step 4: convergence to 0 in Ω - t for large enough times. In the beginning of the proof of Theorem 2.8, we introduced a V -shaped front φ(x 1 , x 2ct) solving (1.1) in R 2 with c = c f / sin α. Similarly, since 2απ/2 ∈ (0, π/2), it follows from [START_REF] Hamel | Existence and qualitative properties of multidimensional conical bistable fronts[END_REF][START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF] that there is a unique V -shaped front φ(x 1 , x 2ct) solving (1.1) in R for all (x 1 , x 2 ) ∈ R 2 . The goal of this step is to show that u(t, x 1 , x 2 )φ(x 1 , x 2ct) → 0 as t → +∞ uniformly in (x 1 , x 2 ) ∈ R 2 .

(5.41)

Observe first that (5.32) implies that u(0, x 1 , x 2 ) ≥ max φ f (-x 1 sin(2α)x 2 cos(2α)), φ f (x 1 sin(2α)x 2 cos(2α)) =: u 0 (x 1 , x 2 ) for all (x 1 , x 2 ) ∈ R 2 . Let u be the solution of the Cauchy problem associated to (1.1) in R 2 with initial condition u 0 at time t = 0. It follows from the maximum principle that u(t, x 1 , x 2 ) ≥ u(t, x 1 , x 2 ) for all (t, x 1 , x 2 ) ∈ R + × R 2 and from [START_REF] Hamel | Stability of conical fronts in a combustion model[END_REF][START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF] that u(t, x 1 , x 2 )φ(x 1 , x 2ct) → 0 as t → +∞ uniformly in (x 1 , x 2 ) ∈ R 2 .

As a consequence, lim inf t→+∞ inf (x 1 ,x 2 )∈R 2 (u(t, x 1 , x 2 )φ(x 1 , x 2ct)) ≥ 0.

(5.42)

Let now t 0 ∈ (-∞, T ] be arbitrary. Since u ≤ 1, there holds u(t 0 , x 1 , x 2 ) ≤ min φ f (-|x 1 | sin(2α)-x 2 cos(2α)-c f t 0 -c f σe δt 0 )+ς(x 1 , x 2 ), 1 =: u 0 (x 1 , x 2 ) (5. On the other hand, it follows from (5.15), from the fact that φ f is decreasing and from the condition t 0 ≤ T that ς(x 1 , x 2 ) ≤ max φ f (-|x 1 | sin(2α)-x 2 cos(2α)-c f t 0 -c f σe δt 0 ), φ f (x 2 -c f t 0 -c f σe δt 0 ) -φ f (-|x 1 | sin(2α)-x 2 cos(2α)-c f t 0 -c f σe δt 0 ) +ρ 1 e -ω 1 √ (|x 1 | sin α+x 2 cos α) 2 +(|x 1 | cos α-x 2 sin α+ct 0 +cσe δt 0 ) 2 + δe δ(t 0 -|x 1 |) ≤ φ f (x 2 -c f t 0 -c f σe δt 0 ) +ρ 1 e -ω 1 √ (|x 1 | sin α+x 2 cos α) 2 +(|x 1 | cos α-x 2 sin α+ct 0 +cσe δt 0 ) 2 + δe δ(t 0 -|x 1 |) for all (x 1 , x 2 ) ∈ R 2 . Therefore, there holds (u(t, x 1 , x 2 )φ(x 1 , x 2ct)) ≤ 0.

∀ A ≥ 0,
Together with (5.42), the desired claim (5.41) follows.

Step 5: convergence to 0 in Ω - t . We here put together the conclusions of the steps 2, 3 and 4. Let ε > 0 and let T 1 ≤ T , M 1 > 0 and M 2 > 0 be as in (5.38). From (5.40), (5.41) and from the definitions (5.18) and (5.19) of Γ t and Ω - t for t ≥ 0, there are T 2 > 0 and M 3 > 0 such that ∀ t ≥ T 2 , ∀ (x 1 , x 2 ) ∈ Ω - t , d((x 1 , x 2 ), Γ t ) ≥ M 3 =⇒ u(t, x 1 , x 2 ) ≤ ε .

Lastly, it follows from (5.39) and from the definitions (5.17) and (5.18) of Γ t that there exists M 4 > 0 such that

∀ T 1 ≤ t ≤ T 2 , ∀ (x 1 , x 2 ) ∈ Ω - t , d((x 1 , x 2 ), Γ t ) ≥ M 4 =⇒ u(t, x 1 , x 2 ) ≤ ε .
With (5.38), one infers that

∀ t ∈ R, ∀ (x 1 , x 2 ) ∈ Ω - t , d((x 1 , x 2 ), Γ t ) ≥ max(M 1 , M 2 , M 3 , M 4 ) =⇒ u(t, x 1 , x 2 ) ≤ ε .
This, together with (5.33) and the inequality 0 ≤ u ≤ 1, means that u is a transition front connecting 0 and 1 for problem (1.1) in R 2 . The proof of Lemma 5.2 is thereby complete.

Figure 1 :

 1 Figure 1: Level sets of a conical-shaped curved front

Figure 2 :

 2 Figure 2: Geometrical interpretation of the condition (1.9)

1 Figure 3 :

 13 Figure 3: Almost-planar fronts

Remark 2 . 7

 27 Other notions of distance could be used. For any two subsets A and B of R N , the quantity d(A, B) defined by (1.7) is the smallest geodesic distance between pairs of points in A and B. Other notions are the distance d and the Hausdorff distance d defined by d(A, B) = min sup d(x, B); x ∈ A , sup d(y, A); y ∈ B (2.3) and d(A, B) = max sup d(x, B); x ∈ A , sup d(y, A); y ∈ B . There holds d(A, B) ≤ d(A, B) ≤ d(A, B). It follows from the proof of Theorem 2.6 (see Remark 4.4 below for the details) that, under the same assumptions, any transition front connecting 0 and 1 for equation (1.1) has a global mean speed for the distance d and this global mean speed is equal to |c f |, in the sense that

Figure 4 :

 4 Figure 4: Example of a transition front given in Theorem 2.8

Remark 4 . 4

 44 The second part of the proof of Theorem 2.6 actually shows that lim sup s-t→+∞ sup x∈Γs d(x, Γ t ) st ≤ |c f | (4.42)

7 Figure 5 :

 75 Figure 5: Profiles of the sets Γ t given in (5.17) and (5.18) for t ≤ 0 and t > 0 Lemma 5.2 The function u is a transition front connecting 0 and 1 for problem (1.1) in R 2 with this choice of sets (Ω ± t ) t∈R and (Γ t ) t∈R . We point out that, in the course of the proof of Lemma 5.2, the following interesting property is shown: the solution u converges uniformly in R 2 as t → +∞ to a traveling front of the type φ(x 1 , x 2ct) solving (1.1) and (5.40) below, with vertical speed c = c f / | cos(2α)|.

  ) n∈N := (|x 1,n | cos α-x 2,n sin α+ct n +cσe δtn ) n∈N are bounded. Therefore, |x 1,n | = z n cos α + y n sin αct n cos α -(cσ cos α)e δtn = -ct n cos α + O(1) x 2,n = -|x 1,n | tan α + y n cos α = ct n sin α + O(1) as n → +∞.

2 1 , x 2 ) 2 1 + x 2 2 →

 21222 with vertical speedc = c f sin(2απ/2) = c f | cos(2α)| , such that the function φ is of class C 2 (R 2 ), 0 < φ < 1 in R 2 , max φ f (-x 1 sin(2α)x 2 cos(2α)), φ f (x 1 sin(2α)x 2 cos(2α)) → 0 as x +∞ with φ(x 1 , x 2 )max φ f (-x 1 sin(2α)x 2 cos(2α)), φ f (x 1 sin(2α)x 2 cos(2α)) ≥ 0

  [START_REF] Matano | Talks at various conferences[END_REF] for all (x 1 , x 2 ) ∈ R 2 , whereς(x 1 , x 2 ) = u(t 0 , x 1 , x 2 )φ f (-|x 1 | sin(2α)-x 2 cos(2α)-c f t 0 -c f σe δt 0 )+ and s + = max(s, 0) denotes the positive part of any real number s. Since φ f (-∞) = 1, φ f (+∞) = 0 and since 0 ≤ u ≤ 1 satisfies (5.32) and (5.39), one infers that limA→+∞ sup | x 2 -| tan(2α)| |x 1 | |≥A ς(x 1 , x 2 ) = 0.(5.44)

  . Let u be any transition front connecting 0 and 1 for problem(1.1) in R. If sup t∈R diam(Γ t ) < +∞, then u is a classical traveling front u(t, x) = φ f (±xc f t + ξ)for some ξ ∈ R. In particular, if u is almost-planar in the sense of Definition 2.1, then the same conclusion holds.

	Corollary 2.4
	It follows in particular from Proposition 2.2 that the almost-planar fronts in any dimen-
	sion N ≥ 1 have a (global mean) speed γ = |c f |. Another immediate consequence of Propo-sition 2.2 is a classification result in dimension N = 1. For any non-empty set E ⊂ R N , let
	diam(E) = sup (x,y)∈E×E	|x -y|
	denote the Euclidean diameter of E.

  ). The above calculation holds whether ξ + t k and ζ k (t, x) be real numbers or equal to +∞. As far as the boundary condition is concerned, if t k ≤ t ≤ T 2 and x • e = c f t + B, then u(t, x) ≤ δ ≤ u k (t, x) from (3.34) and the definition of u k . Let us then check that

  from (4.13) and (4.14), whence ζ(T, x) ≥ C and

  r and d(y + k , Γ t k ) ≥ 2R for all k ∈ N. Property (4.30) implies then that, for every k ∈ N and every y ∈ B(y+ k , R), one has y ∈ Ω + t k and d(y, Γ t k ) ≥ R ≥ M, whence u(t k , y) ≥ β. Therefore, u(t k , x) ≥ v R (0, xy + k ) and u(t, x) ≥ v R (tt k , xy + k ) for all k ∈ N, t > t k and x ∈ R Nfrom the maximum principle. The conclusion of Lemma 4.1 implies that, for every k ∈ N,u(t, x) ≥ β for all t ≥ t k + T ε and |xy + k | ≤ (c fε) (tt k ).

	(4.33)
	On the other hand, (4.31) provides the existence of a sequence (y -k ) k∈N of points in R N
	such that

  for k large enough, since s kt k → +∞ as k → +∞ and ε > 0. Finally, one can apply (4.33) with t = s k and x = y - k for k large enough. Thus, u(s k , y - k ) ≥ β for k large enough. This is in contradiction with (4.34), since α < β. Finally, our assumption (4.32) cannot hold. In other words,

	lim inf |t-s|→+∞

  r and d(y ± , Γ t ) ≥ M.There are then two sequences (t k ) k∈N and (s k ) k∈N of real numbers such that |t ks k | → +∞ as k → +∞ and d

	Let ε > 0 be an arbitrary positive real number. Assume by contradiction that
	lim sup |t-s|→+∞	d(Γ t , Γ s ) |t -s|	> c f + 3 ε.	(4.37)
				(4.36)

  for k large enough. Finally, it follows from (4.28) and (4.41) that

  but one can keep the interfaces Γ t for the front u, so (4.42) is still valid. On the other hand, there holds has been defined in(2.3). Since d(A, B) ≤ d(A, B) for any two subsets A and B of R N , it finally follows from (4.42-4.43) and from the conclusion of Theorem 2.6 that any transition front u connecting 0 and 1 for equation (1.1) satisfiesd(Γ t , Γ s ) |t -s| → |c f | as |t -s| → +∞,that is it has a global mean speed equal to |c f | for the distance d.

	lim sup |t-s|→+∞	d(Γ t , Γ s ) |t -s|	≤ lim sup s-t→+∞	sup x∈Γs	d(x, Γ t ) s -t	,	(4.43)
	where d(A, B)

  .[START_REF] Del Pino | Multiple-end solutions to the Allen-Cahn equation in R 2[END_REF]) of Γ t and Ω - t for t ≤ 0 that there is M 1 > 0 such that∀ t ≤ T, ∀ (x 1 , x 2 ) ∈ Ω - t , d((x 1 , x 2 ), Γ t ) ≥ M 1 =⇒ max φ f (-|x 1 | sin(2α)-x 2 cos(2α)-c f t-c f σe δt ), φ f (x 2 -c f t-c f σe δt ) ≤ Obviously, there is T 1 ≤ T such that δ e δ(t-|x 1 |) ≤ ε 3 for all t ≤ T 1 and (x 1 , x 2 ) ∈ R 2 .(5.35)We now claim that there is M 2 > 0 such that∀ t ≤ T, ∀ (x 1 , x 2 ) ∈ Ω - t , d((x 1 , x 2 ), Γ t ) ≥ M 2 =⇒ ρ 1 e -ω 1 √ (|x 1 | sin α+x 2 cos α) 2 +(|x 1 | cos α-x 2 sin α+ct+cσe δt ) 2 ≤ ε 3 .(5.36)Otherwise, there would exist a sequence (t n , x 1,n , x 2,n ) n∈N in R × R 2 such that t n ≤ T, (x 1,n , x 2,n ) ∈ Ω - tn , d((x 1,n , x 2,n ), Γ tn ) ≥ n for all n ∈ N (5.37) and the sequences (y n ) n∈N := (|x 1,n | sin α+x 2,n cos α) n∈N and (z n

	ε 3	.	(5.34)

  and (5.36) that∀ t ≤ T 1 , ∀ (x 1 , x 2 ) ∈ Ω - t , d((x 1 , x 2 ), Γ t ) ≥ max(M 1 , M 2 ) =⇒ u(t, x 1 , x 2 ) ≤ ε . (5.38)Step 3: convergence to 0 in Ω - t for bounded time intervals. We show in this step that Indeed, if this were not true, there would exist a sequence (t n , x 1,n , x 2,n ) n∈N in R × R 2 such that (t n ) n∈N is bounded, limUp to extraction of a subsequence, one can assume that (t n ) n∈N converges to t ∞ ∈ R. From standard parabolic estimates, the functions u n defined byu n (t, x 1 , x 2 ) = u(t, x 1 + x 1,n , x 2 + x 2,n ) for all (t, x 1 , x 2 ) ∈ R × R 2converge, up to extraction of a subsequence, to a solution 0 ≤ u ∞ ≤ 1 of (1.1) in R 2 such that u ∞ (t ∞ , 0, 0) > 0. On the other hand, (5.15) implies thatu n (t, x 1 , x 2 ) ≤ max φ f (-|x 1 +x 1,n | sin(2α)-(x 2 +x 2,n ) cos(2α)-c f t-c f σe δt ), φ f (x 2 +x 2,n -c f t-c f σe δt ) +ρ 1 e -ω 1 √ (|x 1 +x 1,n | sin α+(x 2 +x 2,n ) cos α) 2 +(|x 1 +x 1,n | cos α-(x 2 +x 2,n ) sin α+ct+cσe δt ) 2 + δ e δ(t-|x 1 +x 1,n |)

	∀ τ > 0, lim A→+∞	|t|≤τ, x 2 ≥| tan(2α)| |x 1 |+A sup	u(t, x 1 , x 2 ) = 0.	(5.39)

n→+∞ x 2,n -| tan(2α)| |x 1,n | = +∞ and lim inf n→+∞ u(t n , x 1,n , x 2,n ) > 0.

  |≤A, x 2 1 +x 2 2 ≥ρ 2 ς(x 1 , x 2 ) = 0, since φ f (+∞) = 0. Together with(5.44), this implies that lim Let now u be the solution of the Cauchy problem associated to (1.1) in R 2 with initial condition u 0 at time t = t 0 . It follows from the maximum principle that u(t,x 1 , x 2 ) ≤ u(t, x 1 , x 2 ) for all (t, x 1 , x 2 ) ∈ [t 0 , +∞) × R 2 .It also follows from the nonnegativity of ς and from (5.45) thatu(t, x 1 , x 2 )φ x 1 , x 2ct -c f σe δt 0 | cos(2α)| → 0 as t → +∞, uniformly in (x 1 , x 2 ) ∈ R 2 ,see[START_REF] Hamel | Stability of conical fronts in a combustion model[END_REF][START_REF] Ninomiya | Existence and global stability of traveling curved fronts in the Allen-Cahn equations[END_REF]. As a consequence, denotingL = sup R 2 | φ x 2 |, there holds lim sup t→+∞ sup (x 1 ,x 2 )∈R 2 (u(t, x 1 , x 2 )φ(x 1 , x 2ct)) ≤ Lc f σe δt 0 | cos(2α)| .Since t 0 ≤ T can be arbitrarily negative, one gets that lim sup

	ρ→+∞	x 2 1 +x 2 2 ≥ρ 2 sup	ς(x 1 , x 2 ) = 0.	(5.45)

lim ρ→+∞ sup -A≤x 2 -| tan(2α)| |x 1 t→+∞ sup (x 1 ,x 2 )∈R 2

In[START_REF] Berestycki | Generalized transition waves and their properties[END_REF], the condition |y -x| = r was used instead of |y -x| ≤ r (more precisely, the condition d Ω (y, x) = r was used, where d Ω denotes the geodesic distance in a domain Ω ⊂ R N ). In the case of the whole space R N , the condition |y -x| ≤ r in (1.9) of the present paper is broader than just |y -x| = r and in some sense more natural. However, it is straightforward to check that all qualitative properties stated in[START_REF] Berestycki | Generalized transition waves and their properties[END_REF] still hold with the condition |y -x| ≤ r or with d Ω (y, x) ≤ r in a general domain Ω ⊂ R N , instead of the corresponding condition (1.5) of[START_REF] Berestycki | Generalized transition waves and their properties[END_REF].

This was the place where the existence of (c f , φ f ) played a role in the proof of Proposition 2.2.

The research leading to these results has received funding from the French ANR within the project PRE-FERED 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 also carried out during visits by the author to the Departments of Mathematics of the University of California, Berkeley and of Stanford University, the hospitality of which is thankfully acknowledged.

previous inequalities prevent v from being independent of time.

Step 5: construction of a solution u of (1.1) in R 2 . Define u in R × R 2 as u(t, x 1 , x 2 ) = v(t, x 1 , x 2 ) for all t ∈ R, x 1 ≤ 0, x 2 ∈ R, v(t, -x 1 , x 2 ) for all t ∈ R, x 1 > 0, x 2 ∈ R.

Since v satisfies (5.8) in the half-plane H with Neumann boundary conditions, it follows that u is a classical time-global solution of (1.1) in the whole plane R 2 . Furthermore, 0 < u < 1 in R × R 2 , together with

(5.12)

and

(5.13) Therefore, it follows from (5.2), (5.7), (5.12) and the equality c

and from (5.2), (5.7), (5.9) and (5.13) that

for all t ≤ T and (x 1 , x 2 ) ∈ R 2 .

(5.15)

Step 6: the solution u is a transition front connecting 0 and 1. To show this property, we need to introduce some families (Ω ± t ) t∈R and (Γ t ) t∈R , drawn on the joint figure below, satisfying the properties of Definition 1.1. For t ≤ 0, set

where the superscript l (resp. r) stands for left (resp. right). Define

for all t > 0.

(5.18)

Therefore, for every t ∈ R, Γ t can be written as a graph Γ t = {(x 1 , x 2 ) ∈ R 2 ; x 2 = ϕ t (x 1 )} for some Lipschitz-continuous function ϕ t : R → R. We finally define, for all t ∈ R,

It is immediate to see that the families (Ω ± t ) t∈R and (Γ t ) t∈R satisfy the general properties (1.8), (1.9) and (1.10).

Proof of the auxiliary lemmas

In this section, we do the proof of Lemmas 5.1 and 5.2 stated in the previous section. We begin with Lemma 5.1.

Proof of Lemma 5.1. As in the paper of Fife and McLeod [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] and as in the lemmas of Section 4.1, the general idea is to slightly perturb the function v by some exponentially small terms in order to make it a supersolution of (5.8). Here, we shall also deal with the boundary conditions on ∂H, that is we have to prove that v x 1 (t, 0, x 2 ) ≥ 0 for all t ≤ T and x 2 ∈ R. We shall make use of the stability of the limiting states 0 and 1, as well as the uniform strict monotonicity (5.5) of the V -shaped front φ along its level sets. Remember that f (1) = 0. Thus, in order to prove that the function v defined in Lemma 5.1 is a supersolution of (5.8) for t ≤ T , it is sufficient to show that

Let us first choose some parameters. Remember that the positive real numbers ρ 2 , ω 2 , ρ 3 and ω 3 are given in (5.3) and (5.4). Call

We fix a real number δ > 0 such that 0 < δ < min(1, ω 4 ) and f ′ ≤ 0 on [0, 2δ] and [1δ, 1].

(5.21)

From (5.1), let C > 0 be such that

From (5.5), let κ > 0 be such that sup

and choose σ > 0 so that σcκ ≥ L = max

(5.24)

Call ρ 4 = (sin α + cos α) max ρ 2 e ω 2 cσ , ρ 3 > 0.

(5.25)

Let finally T < 0 be such that T ≤ -2σ < 0 and δ 2 e δt ≥ ρ 4 e ω 4 t for all t ≤ T.

(5.26)

Let us now estimate v x 1 on the boundary ∂H. In this paragraph, we fix a point (t, 0, x 2 ) on (-∞, T ] × ∂H such that v(t, 0, x 2 ) < 1. From (5.7) and (5.9), there holds

(5.27)

We shall estimate this quantity when

Consider first the case when |x 2c f t| ≥ (c f /2)|t| and x 2 ≤ 0. There holds

)t ≤ 0 and (5.4) and (5.27) yield

≥ -(sin α + cos α) ρ 3 e -ω 3 |x 2 | cos α + δ 2 e δt ≥ -(sin α + cos α) ρ 3 e (ω 3 c f t cos α)/2 + δ 2 e δt .

(5.29)

Finally, for all (t, 0, x 2 ) ∈ (-∞, T ] × ∂H such that v(t, 0, x 2 ) < 1, there holds v x 1 (t, 0, x 2 ) ≥ -ρ 4 e ω 4 t + δ 2 e δt ≥ 0, from (5.20), (5.25), (5.26), (5.28) and (5.29). As a last step, let us check that v is a supersolution of the parabolic equation (5.8) inside H. In this paragraph, (t, x 1 , x 2 ) denotes a point in R × H such that v(t, x 1 , x 2 ) < 1. Since v satisfies (1.1) in R 2 and since δ < 1, one gets from (5.9) that N (t, x 1 , x 2 ) := v t (t, x 1 , x 2 ) -∆v(t, x 1 , x 2 )f (v(t, x 1 , x 2 )) = v t (t + σe δt , x 1 , x 2 ) + σδv t (t + σe δt , x 1 , x 2 ) e δt + δ 2 e δ(x 1 +t) -∆v(t + σe δt , x 1 , x 2 )δ 3 e δ(x 1 +t)f (v(t, x 1 , x 2 )) ≥ f v(t + σe δt , x 1 , x 2 )f (v(t, x 1 , x 2 )) + σδv t (t + σe δt , x 1 , x 2 ) e δt .

(5.30) Call ζ 1 (x 1 , x 2 ) = x 1 sin αx 2 cos α and ζ 2 (t, x 1 , x 2 ) = x 1 cos α + x 2 sin αctcσe δt ,