Weiwei Ding

François Hamel
email: francois.hamel@univ-amu.fr

Xing Liang
email: xliang@ustc.edu.cn

Bistable pulsating fronts in slowly oscillating environments *

come

Introduction and main results

In this paper, we study the following reaction-diffusion equation

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

a L (x) = a x L and f L (x, u) = f x L , u ,
where the function a : R → R is positive, of class C 2 (R), 1-periodic, that is, a(x + 1) = a(x) for all x ∈ R, and the function f : R × [0, 1] → R, (x, u) → f (x, u) is of class C 1 , 1-periodic in x, and (x, u) → ∂ u f (x, u) is Lipschitz-continuous in u ∈ [0, 1] uniformly with respect to x ∈ R. We also assume the following conditions on f : 1 (A1) there is a continuous function b : R → (0, 1), x → b(x), such that for every x ∈ R, the profile f (x, •) is of the bistable type in the following sense f (x, 0) = f (x, 1) = f (x, b(x)) = 0, f (x, •) < 0 on (0, b(x)), f (x, •) > 0 on (b(x), 1);

(A2) 0 and 1 are uniformly (in x) stable zeroes of f (x, •), in the sense that there exist γ 0 > 0 and δ 0 ∈ (0, 1/2) such that f (x, u) ≤ -γ 0 u for all (x, u) ∈ R × [0, δ 0 ],

f (x, u) ≥ γ 0 (1 -u) for all (x, u) ∈ R × [1 -δ 0 , 1], (1.2)
and ∂ u f (•, 0) and ∂ u f (•, 1) are assumed to be of class C 1 in R.

Notice that the assumptions (A1)-(A2) imply in particular that max(∂ u f (x, 0), ∂ u f (x, 1)) ≤ -γ 0 for all x ∈ R. Hence, 0 and 1 are two linearly stable L-periodic steady states of (1.1).

Here, an L-periodic steady state ū is said to be linearly stable (resp. linearly unstable) if λ 1 (ū) < 0 (resp. λ 1 (ū) > 0), where λ 1 (ū) is the principal eigenvalue of the operator ϕ → Lϕ := ∂ x (a L (x)∂ x ϕ) + ∂ u f L (x, ū(x))ϕ in the space of L-periodic functions ϕ ∈ C 2 (R).

In the paper, for mathematical convenience, we extend f in R × (R\[0, 1]) as follows:

f (x, u) = ∂ u f (x, 0)u for (x, u) ∈ R × (-∞, 0), f (x, u) = ∂ u f (x, 1)(u -1) for (x, u) ∈ R × (1, +∞). (1.3)
Thus, it is clear that f is of class C 1 in R 2 , 1-periodic in x, min x∈R f (x, u) > 0 for all u < 0 and max x∈R f (x, u) < 0 for all u > 1, and f (x, u), ∂ u f (x, u) are globally Lipschitz-continuous in u uniformly in x ∈ R. It is also easily seen that this extension does not affect the behavior of the pulsating front connecting 0 and 1 defined below. By a pulsating front of (1.1) connecting 0 and 1, we mean a classical entire solution U L : R × R → (0, 1) of (1.1) for which there exist a real number c L and a function φ L : R × R → (0, 1) satisfying

       U L (t, x) = φ L x -c L t, x L for all (t, x) ∈ R × R, φ L (ξ, y) is 1-periodic in y,
φ L (-∞, y) = 1, φ L (+∞, y) = 0 uniformly in y ∈ R.

(1.4)

The constant c L is called the front speed and φ L is the front profile. Clearly, if c L = 0, then

U L (t, x) = U L t + L c L , x + L for all (t, x) ∈ R × R, (1.5)
that is, the spatial profiles x → U L (t, x + c L t) of the solution in the frame moving with speed c L repeat periodically in time with period L/c L . In the case c L = 0, (1.4) simply means that U L does not depend on t (that is, it is a steady solution) and U L (-∞) = 1, U L (+∞) = 0. Throughout this paper, when we refer to a pulsating front for (1.1), we mean a solution U L : R × R → (0, 1) satisfying (1.4) or, equivalently, a pair (φ L , c L ) given as in (1.4), with φ L : R×R → (0, 1) and U L solving (1.1).

Fronts are of great importance in reaction-diffusion equations, which are the most-used equations in population dynamics models in biology and ecology, as fronts are the main mathematical notion for the description of the invasion of a state (here, the trivial state 0) by another one (here, the constant state 1). The notion of pulsating front in spatially periodic media was first introduced in [START_REF] Shigesada | Traveling periodic waves in heterogeneous environment[END_REF][START_REF] Xin | Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity[END_REF]. It is a natural generalization of the classical notion of traveling front in spatially homogeneous media where the coefficients a and f are independent of x. In that case, the function x → b(x) is a constant which is the only unstable zero of f in (0, 1). For the homogeneous bistable equation

u t = au xx + f (u),
it is well known [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that there exist a unique speed c ∈ R and a unique traveling front u(t, x) = φ(x -ct) with 0 < φ < 1 in R and φ(-∞) = 1, φ(+∞) = 0. The speed c has the sign of the integral 1 0 f , and the front profile φ is decreasing, unique up to shifts, and globally asymptotically stable, as in Theorem 1.1 (iii) below. The global stability of the traveling fronts reinforces their fundamental role in reaction-diffusion equations.

Media are however rarely homogeneous, and reaction-diffusion equations with spatially periodic coefficients, such as (1.1), are some of the most important classes of non-homogeneous equations. Their study has attracted much attention in the mathematical literature. One of the reasons for that interest is that the question of the existence of pulsating fronts is rather subtle. Indeed, the possible presence of multiple ordered L-periodic steady states may prevent the existence of pulsating fronts connecting the extremal steady states 0 and 1, see [START_REF] Ding | Admissible speeds in spatially periodic bistable reaction-diffusion equations[END_REF][START_REF] Ducrot | Existence and convergence to a propagating terrace in onedimensional reaction-diffusion equations[END_REF][START_REF] Giletti | Existence and uniqueness of propagating terraces[END_REF][START_REF] Giletti | Pulsating solutions for multidimensional bistable and multistable equations[END_REF]. On the other hand, it is known from [START_REF] Ducrot | Existence and convergence to a propagating terrace in onedimensional reaction-diffusion equations[END_REF][START_REF] Fang | Bistable traveling waves for monotone semiflows with applications[END_REF] that such a possibility does not happen provided that equation (1.1) admits a bistable structure in the sense that any L-periodic steady state strictly between 0 and 1 is linearly unstable (we recall that 0 and 1 are linearly stable steady states). With such a bistable structure, pulsating fronts have been proved to exist by using different approaches [START_REF] Ducrot | A multi-dimensional bistable nonlinear diffusion equation in a periodic medium[END_REF][START_REF] Ducrot | Existence and convergence to a propagating terrace in onedimensional reaction-diffusion equations[END_REF][START_REF] Fang | Bistable traveling waves for monotone semiflows with applications[END_REF]. Furthermore, in an earlier paper by the first two authors [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF], the bistable structure was verified under various explicit conditions on the functions a(x) and f (x, u) when the spatial period L is large or small. In particular, if in addition to (A1)-(A2) the function f is assumed to satisfy (A3) 1 0 f (x, u) du > 0 and ∂ u f (x, b(x)) > 0 for all x ∈ R, 1then a pulsating front with a positive speed c L exists when L is large (i.e., the medium oscillates slowly). On the other hand, if the function x → 1 0 f (x, u)du is not of a constant sign, it was shown in [START_REF] Ding | Propagation phenomena for periodic bistable reaction-diffusion equations[END_REF] that, when L is large, non-stationary pulsating fronts do not exist, while multiple stationary fronts do exist (see also [START_REF] Ding | Propagation phenomena for periodic bistable reaction-diffusion equations[END_REF][START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF][START_REF] Keener | Homogenization and propagation in the bistable equation[END_REF][START_REF] Xin | Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media[END_REF][START_REF] Xin | Quenching and propagation of bistable reaction-diffusion fronts in multidimensional periodic media[END_REF] for further results on the existence of stationary fronts for bistable equations of the type (1.1)). We point out that, apart from the aforementioned works, the existence of pulsating fronts was investigated in [START_REF] Matano | Traveling waves in spatially random media[END_REF][START_REF] Nolen | Traveling waves in a one-dimensional heterogeneous medium[END_REF][START_REF] Xin | Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity[END_REF][START_REF] Xin | Front propagation in heterogeneous media[END_REF][START_REF] Zlatoš | Existence and non-existence of transition fronts for bistable and ignition reactions[END_REF]. References [START_REF] Matano | Traveling waves in spatially random media[END_REF][START_REF] Nolen | Traveling waves in a one-dimensional heterogeneous medium[END_REF][START_REF] Zlatoš | Existence and non-existence of transition fronts for bistable and ignition reactions[END_REF] were also concerned with generalized transition fronts in the sense of [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Berestycki | Generalized transition waves and their properties[END_REF], for which the media can be non-periodic (see [START_REF] Aronson | Wave propagation and blocking in inhomogeneous media[END_REF][START_REF] Caputo | Reaction-diffusion front crossing a local defect[END_REF][START_REF] Dowdall | Invasion pinning in a periodically fragmented habitat[END_REF][START_REF] Eberle | Front blocking in the presence of gradient drift[END_REF][START_REF] Eberle | Front blocking versus propagation in the presence of drift term varying in the direction of propagation[END_REF][START_REF] Lewis | Wave-block in excitable media due to regions of depressed excitability[END_REF][START_REF] Nadin | Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations[END_REF][START_REF] Pauwelussen | Nerve impulse propagation in a branching nerve system: a simple model[END_REF][START_REF] Scheel | Depinning asymptotics in ergodic media, Patterns of Dynamics[END_REF][START_REF] Sneyd | On the propagation of calcium waves in an inhomogeneous medium[END_REF] for further propagation or blocking results in non-periodic bistable media).

For clarity, we collect some known results from [START_REF] Ding | Propagation phenomena for periodic bistable reaction-diffusion equations[END_REF][START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF] concerning the existence and qualitative properties of pulsating front which will be frequently used in the present work.

Theorem 1.1. [START_REF] Ding | Propagation phenomena for periodic bistable reaction-diffusion equations[END_REF][START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF] Let (A1)-(A3) hold. Then there exists L * > 0 such that, for any L ≥ L * , problem (1.1) admits a pulsating front U L (t, x) = φ L (x -c L t, x/L) with speed c L . Furthermore:

(i) c L > 0, c L is the unique front speed and the map L → c L is bounded in [L * , +∞);

(ii) the front U L is unique up to shifts in time t, and ∂ t U L (t, x) > 0 for all (t, x) ∈ R 2 ;

(iii) the front U L is globally asymptotic stable in the sense that, for any u 0 ∈ C(R) ∩ L ∞ (R) with lim inf x→-∞ u 0 (x) > 1 -δ 0 and lim sup x→+∞ u 0 (x) < δ 0 , (1.6) where δ 0 > 0 is the constant given in (A2), we have

lim t→+∞ u(t, •; u 0 )-U L (t+τ 0 , •) L ∞ (R) = lim t→+∞ u(t, •; u 0 )-φ L (• -c L t-c L τ 0 , •/L) L ∞ (R) = 0,
where (t, x) → u(t, x; u 0 ) is the solution of the Cauchy problem for (1.1) with initial condition u 0 , and τ 0 ∈ R is a constant depending on u 0 ;

(iv) the front speed c L and the front profile φ L are continuous with respect to L ≥ L * in the sense that c L → c L 0 as L → L 0 in [L * , +∞) and, up to translations of φ L in such a way that φ L (0, 0) = φ L 0 (0, 0), there holds φ L -φ L 0 → 0 in H 1 (R × (0, 1)) and

U L → U L 0 , φ L → φ L 0 locally uniformly in R 2 .
More precisely, we refer to [18, Theorems 1.1, 1.5, 1.9] for the existence, uniqueness, monotonicity and global stability of pulsating fronts as well as the sign property of c L . The continuity of (c L , φ L , U L ) with respect to L follows directly from [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]Theorem 1.8] and its proof, together with the uniform boundedness of

∂ t U L L ∞ (R 2 ) + ∂ x U L L ∞ (R 2 )
with respect to L ∈ [L * , +∞) (which itself follows from standard parabolic estimates). The boundedness of c L is an easy consequence of [START_REF] Ding | Propagation phenomena for periodic bistable reaction-diffusion equations[END_REF]Theorem 1.4] which established a uniform bound for the propagation rates of all generalized transition fronts.

In this paper, we are chiefly interested in the asymptotic behavior of the front speeds c L and the front profiles φ L as L → +∞. Before stating our main results, let us first recall some related results in the spatially periodic Fisher-KPP case,2 i.e., when, instead of (A1)-(A2), f (x, u) is periodic in x with f (•, 0) = f (•, 1) = 0 in R and 0 < f (x, u) ≤ ∂ u f (x, 0)u for all (x, u) ∈ R × (0, 1).

(1.7)

In this case, for each L > 0, there is a minimal front speed c * L of pulsating fronts [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] (namely, pulsating fronts exist for all speeds c ≥ c * L , and do not exist if c < c * L ), this minimal speed c * L is positive and non-decreasing with respect to L > 0 [START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF], the limit of c * L as L → +∞ was determined in [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF][START_REF] Hamel | A viscosity solution method for the spreading speed formula in slowly varying media[END_REF], 3 and the homogenization limit of c * L as L → 0 was characterized in [START_REF] Smaily | Homogenization and influence of fragmentation in a biological invasion model[END_REF][START_REF] Kinezaki | Spatial dynamics of invasion in sinusoidally varying environments[END_REF][START_REF] Shigesada | Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution[END_REF]. The proofs in the aforementioned papers strongly rely on the fact that the minimal speed is linearly determined and has a variational characterization in terms of principal eigenvalues of parameterized elliptic operators [START_REF] Berestycki | The speed of propagation for KPP type problems. I -Periodic framework[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model : II -Biological invasions and pulsating travelling fronts[END_REF][START_REF] Liang | Asymptotic speeds of spread and traveling waves for monotone semiflows with applications[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF]. However, for equation (1.1) with the bistable assumptions (A1)-(A2) or (A1)-(A3), the unique front speed c L is not linearly determined and it is not monotone in L in general. More precisely, first of all, the limit of c L as L → 0 was determined in [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF][START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF] under certain assumptions (see also Remark 1.5 below). Moreover, in the case where the diffusion rate a is a constant and b in (A1) is piecewise constant, numerical results [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF] showed that c L can in some cases be increasing in small L and then decreasing for larger L. On the other hand, if a(x) truly depends on x while the reaction is homogeneous and has the special form f (u) = u(1 -u)(u -b) with 0 < b < 1/2, it was proved in [START_REF] Dkhil | Traveling wave speeds in rapidly oscillating media[END_REF] that the speed c L for L 0 is smaller than its limit as L → 0.

In short, the question whether, in the bistable case (A1)-(A3), the speeds c L of the pulsating fronts of (1.1) (which exist for all large L by Theorem 1.1) converge as L → +∞ is the main remaining open problem in the theory of bistable pulsating fronts. This question, which is definitely more challenging than in the Fisher-KPP case (1.7), is solved in this paper (see the main Theorem 1.2 below). Furthermore, a formula for the limit speed is established, as well as further properties on the front profiles as L → +∞.

Convergence of the front speeds c L as L → +∞

In this section, we state our main result on the convergence of c L as L → +∞. To do so, we need some notations. For each y ∈ R, let u(t, x) = ψ(x -c(y)t, y) be the traveling front connecting 0 and 1, with speed c(y), of the following equation with coefficients frozen at y:

u t = a(y)u xx + f (y, u) for t ∈ R and x ∈ R. (1.8)
Namely, (ψ(ξ, y), c(y)) satisfies

a(y)∂ ξξ ψ(ξ, y) + c(y)∂ ξ ψ(ξ, y) + f (y, ψ(ξ, y)) = 0 for ξ ∈ R, ψ(-∞, y) = 1, ψ(+∞, y) = 0. (1.9)
In (1.8)-(1.9), y is viewed as a parameter, since the derivatives only concern the variables (t, x) or ξ. Under conditions (A1)-(A3), it is well known [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] that the function c : y → c(y) is 1-periodic and positive, and that for each y ∈ R, the wave profile ψ(ξ, y) is unique up to shifts in ξ (see Section 4.1 below for more properties of the speed c(y) and the wave profile ψ(ξ, y), in particular the C 1 smoothness of y → c(y)). Throughout this paper, we normalize ψ(ξ, y) uniquely in such a way that

ψ(0, y) = 1 2 for each y ∈ R, (1.10)
and we denote by c * the harmonic mean of the 1-periodic function y → c(y), that is,

c * = 1 0 c -1 (y)dy -1
.

(1.11)

Theorem 1.2. Let (A1)-(A3) hold, and for any L ≥ L * (with L * > 0 as in Theorem 1.1), let c L be the unique front speed of the pulsating front of (1.1). Then, c L → c * as L → +∞.

Several comments are in order. Let us first point out that Theorem 1.2 coincides with the known result for the minimal front speeds c * L of equations with Fisher-KPP reactions (1.7) in the special case where the function r := ∂ u f (x, 0) is a constant, see [START_REF] Hamel | A viscosity solution method for the spreading speed formula in slowly varying media[END_REF]Corollary 2.4]. Indeed, in the latter case, the limit of the minimal front speeds c * L as L → +∞ is given by

2 √ r 1 0 (a(y)) -1/2 dy -1
, which is equal to the harmonic mean of the 1-periodic function y → w * (y) := 2 r a(y), where, for each y ∈ R, 2 r a(y) is the minimal front speed of the associated homogeneous equation (1.8).

In the case where ∂ u f (x, 0) truly depends on x, the limit of the minimal front speeds c * L has a variational formula in terms of the underlying parameters (see [START_REF] Hamel | A viscosity solution method for the spreading speed formula in slowly varying media[END_REF]Theorem 2.3]), but it may be not equal to the harmonic mean of the function w * (see [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF] for the case where ∂ u f (x, 0) is a piecewise constant function). It is noteworthy that, in our bistable case (A1)-(A3), Theorem 1.2 shows that the unique wave speed c L converges to the harmonic mean c * of the function y → c(y), no matter whether f depends on x or not. Theorem 1.2 is proved by taking the large-time and large-space scaling, i.e., t → Lt, x → Lx, and determining the singular limit of the Cauchy problem of the rescaled equation

∂ t u L = L -1 ∂ x (a(x)∂ x u L ) + Lf (x, u L ), t ∈ R, x ∈ R.
(1.12)

For a given initial condition independent of L, the singular limit of the solutions u L of (1.12) as L → +∞ has been widely investigated, even in the general case where the coefficients are allowed to be non-periodic. Due to the smallness of the diffusion and the largeness of the reaction in (1.12), an interface separating the regions where u L (t, x) → 1 and u L (t, x) → 0 is developed in a short time. This property has been proved by different approaches, see e.g. [START_REF] Gärtner | Bistable reaction-diffusion equations and excitable media[END_REF] for a probabilistic approach and [START_REF] Barles | Front propagation for reaction-diffusion equations of a bistable type[END_REF][START_REF] Barles | Front propagation and phase field theory[END_REF][START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF][START_REF] De Zan | Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity[END_REF] for a viscosity solution approach. Particularly, in our spatially periodic case, Gärtner's result [START_REF] Gärtner | Bistable reaction-diffusion equations and excitable media[END_REF] suggested that, as L → +∞, the interface propagates at a mean speed equal to c * . Thanks to this observation and further crucial uniform estimates on the boundedness of the wave length (more precisely, see Theorem 1.7 below), we show that c * is the limit of c L by constructing suitable sub-and super-solutions of (1.12). Notice that, if the assumption

1 0 f (x, u)du > 0 in (A3) if replaced by 1 0 f (x, u)du < 0, equation (1.
1) admits a pulsating front with negative speed c L < 0 for all large L, see [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]. In this situation, the quantity c * is negative, and similarly to Theorem 1.2, we can prove that c L → c * as L → +∞. Furthermore, as an immediate consequence of this observation and [17, Theorem 1.7], we have the following classification of the asymptotic behavior of c L as L → +∞.

min x∈R 1 0 f (x, u)du < 0 < max x∈R 1 0 f (x, u)du,
then (1.1) admits pulsating fronts for all large L, with necessarily c L = 0, that is, the fronts are stationary.

Remark 1.4. By [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF], the conditions (A1)-(A3) not only imply the existence of pulsating fronts of (1.1) satisfying (1.4), but they also provide the existence of pulsating fronts propagating in the opposite direction for the same range of periods L. Namely, for any L ≥ L * , (1.1) admits a pulsating front of the type U L (t, x) = φ L (x + c L t, x/L), where φ L is 1-periodic in the second variable, and satisfies the reversed limiting conditions:

φ L (-∞, y) = 0, φ L (+∞, y) = 1 uniformly in y ∈ R.
(1.13)

Moreover, both c L and c L are positive. It is worthy pointing out that if the functions a(x) and f (x, u) are not even in x, the speeds c L and c L may be dramatically different [START_REF] Ding | Admissible speeds in spatially periodic bistable reaction-diffusion equations[END_REF]. Nevertheless, since the function y → c(y) given in (1.9) is 1-periodic and c * in (1.11) can be expressed as

c * = 1/2 -1/2 c -1 (y)dy -1 = 1/2 -1/2 c -1 (-y)dy -1
, it follows from Theorem 1.2 that c L and c L have the same limit c * as L → +∞.

Remark 1.5. For Fisher-KPP functions f satisfying (1.7), for each L > 0, pulsating fronts of (1.1) exist, with speeds not smaller than a minimal speed c * L > 0, and the map L → c * L is nondecreasing [START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF], hence lim L→0 c * L ≤ lim L→+∞ c * L (and the inequality is strict if the coefficients a(x) and ∂ u f (x, 0) are not constant in x).

In the bistable case with assumptions (A1)-(A3), the situation is in general different and more complicated. Consider for instance a function f defined by

f (x, u) = u(1 -u)(u -b(x)) for (x, u) ∈ R × [0, 1], where b : R → (0, 1/2) is a given 1-periodic function of class C 1 . Such a function f satisfies (A1)-(A3).
On the one hand, if a is constant, then it follows from [18, Theorems 1.2 and 1.4] that equation (1.1) admits a pulsating front with speed c L for every small L > 0, and that c L → c 0 as L → 0, where c 0 = √ 2a (1/2 -b) is the unique speed of a traveling front connecting 0 and 1 for the homogeneous equation

u t = au xx + u(1 -u)(u -b), with b = 1 0 b(x) dx ∈ (0, 1/2). With the notations (1.9), one has c(y) = √ 2a (1/2 -b(y)) for each y ∈ R, and Theorem 1.2 implies that c L → c * = √ 2a 1 0 (1/2 -b(y)) -1 dy -1 as L → +∞.
c L → c * = √ 2 (1/2 -b) 1 0
a(y) -1 dy -1 as L → +∞. Therefore, the Cauchy-Schwarz inequality yields c 0 ≤ c * and even c 0 < c * as soon as a is not constant.

To sum up, the influence of the oscillations of the heterogeneities in (1.1) under the bistable assumptions (A1)-(A3) turns out to be more complicated than under the Fisher-KPP assumption (1.7). Moreover, the oscillations in the diffusion and reaction terms can lead to opposite effects on the bistable front speed.

Convergence of the front profiles φ L as L → +∞

The second main result of the present paper is concerned with the convergence as L → +∞ of the front profiles φ L (ξ, y) given in (1.4). It is difficult to address this problem by considering the singular equation (1.12), since the solutions φ L (L(x-c L t), x) of (1.12) converge to a step function as L → +∞ for any t ∈ R (this will actually be a consequence of Theorem 1.8 below), hence the information of φ L (ξ, y) is lost when passing to the limit as L → +∞. We instead show the convergence of φ L (ξ, y) by constructing sub-and super-solutions for the original equation (1.1) when L is large (as a matter of fact, this approach also provides the convergence of c L as L → +∞, as will follow from the proof of Theorem 1.6 below).

Before stating the result, we point out that, since equation (1.1) is invariant by translation in time t, we have to suitably shift the fronts in order to pass to the limit as L → +∞. Namely, with L * > 0 as in Theorem 1.1, since for each L ≥ L * the pulsating front U L is unique up shifts in time, with U L (-∞, x) = 0 and U L (+∞, x) = 1 for all x ∈ R, we normalize for definiteness U L by requiring

U L (0, 0) = 1 2 . (1.14)
Clearly, under such a normalization, U L (t, x) is uniquely determined, and the front profile φ L (ξ, y) is uniquely determined and satisfies φ L (0, 0) = 1/2. Now, for each L ≥ L * , since ∂ ξ φ L < 0 in R 2 and φ L satisfies (1.4) and is at least of class C 1 (R 2 ), the implicit function theorem yields the existence of a uniquely determined

C 1 (R) function y → ζ L (y) such that φ L (ζ L (y), y) = 1 2 . (1.15)
Clearly, ζ L (0) = 0 and ζ L (y) is 1-periodic in y ∈ R. Moreover, it follows from Theorem 1.1 (ii) and (iv) that the function (L, y)

→ ζ L (y) is continuous in [L * , +∞) × R.
For the convergence of the front profiles as L → +∞, in addition to (A1)-(A3), the following condition will also be assumed:

(A4) the function a is a constant, and there exists a constant δ 0 ∈ (0, 1/2) such that ∂ x f (x, u) = 0 for all x ∈ R and u ∈ [0, δ 0 ]∪[1-δ 0 , 1], and then for all x ∈ R and u ∈ [-δ 0 , δ 0 ]∪[1-δ 0 , 1+δ 0 ] by (1.3).

Theorem 1.6. Let (A1)-(A4) hold. For each L ≥ L * , let U L (t, x) = φ L (x -c L t, x/L) be the pulsating front of (1.1) given by Theorem 1.1, and let ζ L be the function given by (1.15). Let also (ξ, y) → ψ(ξ, y) and y → c(y) be given in (1.9)-(1.10). Then,

sup y∈R, ξ∈R φ L ξ + ζ L (y), ξ L + y -ψ(ξ, y) -→ L→+∞ 0. (1.16) Moreover, sup y∈R, x∈[-A,A] ζ L y + x L -ζ L (y) -L y+x/L y 1 - c L c(s) ds -→ L→+∞ 0 (1.17)
for every A > 0, and

sup L≥L * , y∈R, x∈[-L,L] ζ L y + x L -ζ L (y) -L y+x/L y 1 - c L c(s) ds < +∞. (1.18)
Several remarks are in order. First of all, by choosing y = 0 and x = y 0 L for some y 0 ∈ (0, 1] in (1.18), one has sup

L≥L * ζ L (y 0 ) -L y 0 0 1 - c L c(s) ds < +∞.
In the special case y 0 = 1, since ζ L (1) = ζ L (0) = 0, we rediscover that c L → c * = ( 1 0 c(s) -1 ds) -1 as L → +∞, here under the additional assumption (A4). Furthermore, since the function y → c(y) is in general not constant even under the assumption (A4), it then follows that ζ L (y 0 ) is unbounded in L ≥ L * for each y 0 ∈ (0, 1) such that y 0 1 0 c(s) -1 ds = y 0 0 c(s) -1 ds. On the other hand, (1.18) implies that ζ L (y 0 ) = O(L) as L → +∞ for all y 0 ∈ R (that will coincide with the observation in Theorem 1.8 (iii) below).

Secondly, we point out that (1.16) can be equivalently rephrased as sup y∈R, ξ∈R

U L Ly -ζ L (y) c L , ξ + Ly -ψ(ξ, y) -→ L→+∞ 0.
As we will see, this last formulation implies that, as L → +∞, the spatial profiles U L (t, •) look, at any time t and uniformly in space, like shifts of the profiles ψ(•, y) of traveling fronts of the homogeneous equations (1.9) for some values of y. Indeed, by (1.18) and lim L→+∞ c L = c * , the continuous functions z → z -ζ L (z)/L converge as L → +∞ locally uniformly in z ∈ R to the function z → c * z 0 c(s) -1 ds (by applying (1.18) with y = 0 and x = Lz). Therefore, for every L > 0 large enough and for every t

∈ [0, L/c L ], there is z L,t ∈ R such that z L,t -ζ L (z L,t )/L = c L t/L. Hence U L (t, •) = U L ((Lz L,t -ζ L (z L,t ))/c L , •) and U L (t, •) -ψ(• -Lz L,t , z L,t ) L ∞ (R) → 0 as L → +∞ uniformly in t ∈ [0, L/c L ]. Since the profiles x → U L (t, x + c L t)
in the frames moving with speeds c L are periodic in time with period L/c L , one finally gets that

sup t∈R d U L (t, •), ψ(• + a, y) : a ∈ R, y ∈ R -→ L→+∞ 0, (1.19)
where d is the distance associated with the norm in L ∞ (R). Actually, the convergence of the profiles of the pulsating fronts to profiles of homogeneous traveling fronts as L → +∞ is quite natural since equation (1.1) looks like, as L → +∞, families of locally spatially homogeneous equations. But the difficulty in Theorem 1.6 and in the consequent formula (1.19) is to show rigorously this convergence and its uniformity with respect to time and space. The strategy of the proof of Theorem 1.6 can be briefly summarized as follows. First, by virtue of Theorem 1.1 (iii) on the global stability of the pulsating front φ L , we show that, up to some shift in ξ, φ L (•, y) is close to ψ(•, y) when L is sufficiently large. Next, we use Theorem 1.8 below on the uniform boundedness of the front width to determine the limit shifts appearing in (1.16). The assumption (A4) is used in the construction of suitable sub-and super-solutions (see Lemmas 4.2-4.3 below), which is the key point of the proof of Theorem 1.6. It is unclear whether this assumption can be relaxed.

Uniform boundedness of front width

As a matter of fact, Theorems 1.2 and 1.6 on the limit speeds and profiles of the pulsating fronts as L → +∞ are based on the uniform boundedness of the width of the front profiles, in a sense to be made more precise below. Under conditions (A1)-(A3), with L * > 0 as in Theorem 1.1, it follows from the definition (1.4) of pulsating fronts that, for any δ ∈ (0, 1/2] and L ≥ L * , there exists a constant C > 0 such that

sup x∈R diam {t ∈ R : δ ≤ U L (t, x) ≤ 1 -δ} ≤ C (1.20)
and sup

t∈R diam {x ∈ R : δ ≤ U L (t, x) ≤ 1 -δ} ≤ C, (1.21)
where diam(E) denotes the diameter of a set E ⊂ R. Notice that, for each δ ∈ (0, 1/2], L ≥ L * and x ∈ R, the set {t ∈ R : δ ≤ U L (t, x) ≤ 1 -δ} is a compact interval, from (1.4) and the continuity of U L and its monotonicity with respect to t. Hence the diameter of the set in (1.20) is equal to its length. We also point out that property (1.21) could be rephrased equivalently by saying that, for any λ ∈ (0, 1), the distance between c L t and the level set {x ∈ R : U L (t, x) = λ} is uniformly bounded in t.

Our next result, which is a key stone in determining the convergences of front profiles φ L and front speeds c L as L → +∞, and which has its own interest, shows that the constants C in (1.20)-(1.21) can be taken independently of L ≥ L * .

Theorem 1.7. Let (A1)-(A3) hold. For each L ≥ L * , with L * as in Theorem 1.1, let U L be a pulsating front of (1.1). Then, for any δ ∈ (0, 1/2], there exist constants C > 0 and β > 0 (both are independent of L) such that (1.20) and (1.21) hold true for all L ≥ L * , and

∂ t U L (t, x) ≥ β for all L ≥ L * and (t, x) ∈ R 2 such that δ ≤ U L (t, x) ≤ 1 -δ. (1.22)
The following theorem is a consequence of Theorem 1.7. (ii) for any δ ∈ (0, 1/2], there exists β > 0 (independent of L) such that

∂ ξ φ L (ξ, y) ≤ -β for all L ≥ L * and (ξ, y) ∈ R 2 such that δ ≤ φ L (ξ, y) ≤ 1 -δ; (iii) the function (L, y) → ζ L (y)/L is continuous and bounded in [L * , +∞) × R.
Notice that the continuity of (L, y) → ζ L (y)/L in [L * , +∞) × R follows from the comments after (1.15), while the quantities ζ L (y) are in general unbounded with respect to L ≥ L * , from the discussion after Theorem 1.6.

Theorems 1.7 and 1.8 mean that the fronts do not flatten as L → +∞. Their proofs are based on some intersection number arguments inspired by [START_REF] Angenent | The zero set of a solution of a parabolic equation[END_REF] and from the comparison of the pulsating fronts with some well chosen stationary solutions.

Outline of the paper

Section 2 is devoted to the proof of Theorems 1.7 and 1.8 on the uniform boundedness of the wave lengths of the pulsating fronts U L with respect to L ≥ L * . The proof of Theorem 1.2 on the limit of the speeds c L as L → +∞ is done in Section 3, and that of Theorem 1.6 on the limit of the profiles is done in Section 4. Lastly, Section 5 is an appendix on some properties of the profiles ψ(ξ, y) and speeds c(y) of the family of homogeneous equations (1.8)-(1.9).

2 Uniform boundedness of front width: proofs of Theorems 1.7 and 1.8

This section is devoted to the proof of Theorems 1.7 and 1.8. Our proof strongly relies on a zero number argument, and the classification of all solutions to the ordinary differential equations

(a L w ) + f L (x, w) = 0 in R, w(0) ∈ (0, 1), (2.23)
according to the number of intersections of their graphs with those of the constant solutions 0 and 1. First, after recalling in Section 2.1 some properties on intersection number arguments, we investigate in Section 2.2, for any given L ≥ L * with L * > 0 as in Theorem 1.1 and for any time t ∈ R, the number and type of intersection points of the spatial profiles U L (t, •) of the pulsating fronts U L with all solutions to (2.23). Then, using a passage to the limit as L → +∞ and some properties of the stationary solutions to the y-frozen equation (1.8), we prove Theorem 1.7 by contradiction in Section 2.3. Lastly, we show in Section 2.4 that Theorem 1.7 implies Theorem 1.8.

Zero number properties

We begin with the collection of some zero number properties which will be used later. We refer the reader to [START_REF] Angenent | The zero set of a solution of a parabolic equation[END_REF][START_REF] Du | Convergence and sharp thresholds for propagation in nonlinear diffusion problems[END_REF][START_REF] Ducrot | Existence and convergence to a propagating terrace in onedimensional reaction-diffusion equations[END_REF][START_REF] Poláčik | Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R[END_REF] for a more detailed overview of the general arguments and their applications.

Let us first introduce some fundamental notions. For any function w : R → R, let Z[w] denote the number of sign changes of w, namely, the supremum of all k ∈ N such that there exist real numbers

x 1 < x 2 < • • • < x k+1 with w(x i ) • w(x i+1 ) < 0 for all i = 1, 2, • • • , k,
and we set Z[w] = 0 if w has a constant sign and is not identically 0, and For any two words A, B consisting of + and -, we write A B (or equivalently, B A) if B is a subword of A.

Z[w] = -1 if w ≡ 0 in R. Furthermore, if 0 < Z[w] < +∞, we use SGN [w] to denote the (k + 1)-uple of the type [ε 1 , • • • , ε k+1 ] consisting of + and -, which stand for the signs of w(x 1 ), w(x 2 ), • • • , w(x k+1 ). When Z[w] = 0,
It follows from the above definitions that Z is semi-continuous with respect to the pointwise convergence, as stated below.

Lemma 2.1. Let (w n ) n∈N be a sequence of real-valued functions converging to w pointwise on R. Then,

Z[w] ≤ lim inf n→+∞ Z[w n ], and SGN [w] lim inf n→+∞ SGN [w n ].
The following lemma is an easy application of the zero number properties for solutions to a linear parabolic equation of the form

∂ t w = a(t, x)∂ xx w + b(t, x)∂ x w + c(t, x)w in (t 1 , t 2 ) × R, (2.24)
where

-∞ ≤ t 1 < t 2 ≤ +∞ and the coefficients a > 0, a -1 , a t , a x , a xx , b, b t , b x , c belong to L ∞ ((t 1 , t 2 ) × R). Lemma 2.2. [2, 21, 23] Let L > 0, let u 1 (t,
x) be a bounded solution of the Cauchy problem associated with (1.1) in (0, +∞)×R, with a piecewise continuous bounded initial condition u 1 (0, •), and let u 2 (x) be a C 2 (R) stationary solution of (1.1). Assume also that u 1 (0, •) -u 2 changes sign at most finitely many times on R, that is,

Z[u 1 (0, •) -u 2 ] < +∞. Then, (i) For any 0 ≤ t < t < +∞, we have Z[u 1 (t, •) -u 2 ] ≥ Z[u 1 (t , •) -u 2 ]
and

SGN [u 1 (t, •) -u 2 ] SGN [u 1 (t , •) -u 2 ]. (ii) If u 1 (t * , x * ) = u 2 (x * ) and ∂ x u 1 (t * , x * ) = u 2 (x * ) for some t * > 0 and x * ∈ R, and if u 1 (t, x) ≡ u 2 (x) in (0, +∞) × R, then Z[u 1 (t, •) -u 2 ] -2 ≥ Z[u 1 (s, •) -u 2 ] ≥ 0 for all t ∈ (0, t * ) and s ∈ (t * , +∞). Since the function (x, u) → f (x, u) is globally Lipschitz-continuous with respect to u ∈ R uniformly in x ∈ R, it is easily seen that w(t, x) := u 1 (t, x) -u 2 (x) satisfies an equation of the form (2.24) in (0, +∞) × R with a bounded continuous coefficient c defined by c(t, x) := (f L (x, u 1 (t, x))-f L (x, u 2 (x)))/(u 1 (t, x)-u 2 (x)) if u 1 (t, x) = u 2 (x) and c(t, x) := ∂ u f L (x, u 1 (t, x)) if u 1 (t, x) = u 2 (x).
Then, the proof of Lemma 2.2 follows from that of [START_REF] Ducrot | Existence and convergence to a propagating terrace in onedimensional reaction-diffusion equations[END_REF]Lemma 2.4] with some obvious modifications; therefore, we omit the details.

Intersections of pulsating fronts with stationary solutions

Now, for any given

L ≥ L * , with L * > 0 as in Theorem 1.1, we investigate Z[U L (t, •) -w]
and SGN [U L (t, •) -w] at any time t ∈ R, where U L : R 2 → (0, 1) is the pulsating front of (1.1) and w is an arbitrary C 2 (R) solution to (2.23). Let us first classify the solutions of (2.23) as follows.

Lemma 2.3. Let L ≥ L * be fixed, and let w ∈ C 2 (R) be a solution of (2.23). Then, w must be one of the following types:

(a) Z[w -1] = 1 and Z[w] = 1; (b) Z[w -1] = 0, Z[w] = 0, and 0 < w < 1 in R; (c) Z[w -1] = 0, Z[w] = 1, w < 1 in R, and SGN [w] = [-, +]; (d) Z[w -1] = 0, Z[w] = 1, w < 1 in R, and SGN [w] = [+, -]; (e) Z[w -1] = 0, Z[w] = 2, w < 1 in R, and SGN [w] = [-, +, -].
Proof. First of all, since w(0) ∈ (0, 1), we have Z[w -1] ≥ 0 and Z[w] ≥ 0. Suppose now that there exists some x 1 ∈ R such that w(x 1 ) = 1. It is then clear that w (x 1 ) = 0 (otherwise, since f L (•, 1) ≡ 0, the Cauchy-Lipschitz theorem would imply that w ≡ 1 in R, a contradiction with w(0) ∈ (0, 1)).

Then, we observe that, if w (x 1 ) > 0, then w(x) > 1 for all x > x 1 . Indeed, otherwise, there would exist some

x 2 > x 1 such that w(x 2 ) = 1, w (x 2 ) < 0 and w(x) > 1 for all x ∈ (x 1 , x 2 ). Since max R f L (•, u) < 0 for all u > 1, one would have (a L w ) > 0 in (x 1 , x 2 ), and hence, a L (x 2 )w (x 2 ) > a L (x 1 )w (x 1 ) > 0.
Since a is a positive function, there holds w (x 2 ) > 0, which is a contradiction. Similarly, we can conclude that, if w (x 1 ) < 0, then w(x) > 1 for all x < x 1 . As a consequence, we obtain that

Z[w -1] ≤ 2, and if Z[w -1] = 2, then SGN [w -1] = [+, -, +].
Since min R f L (•, u) > 0 for all u < 0, with similar arguments as above, one gets that Z[w] ≤ 2, and if

Z[w] = 2, then SGN [w] = [-, +, -].
Consequently, we always have

Z[w -1] + Z[w] ≤ 2.
Observe that the Cauchy-Lipschitz theorem also implies that

, if w ≤ 1 in R (respectively w ≥ 0 in R), then w < 1 in R (respectively w > 0 in R). Therefore, since 0 < w(0) < 1, one gets that w < 1 in R if Z[w -1] = 0 (respectively w > 0 in R if Z[w] = 0)
. Now, to complete the proof, it remains to exclude the following three cases:

• Z[w -1] = 1, Z[w] = 0 and SGN [w -1] = [+, -]; • Z[w -1] = 2, Z[w] = 0 and SGN [w -1] = [+, -, +]; • Z[w -1] = 1, Z[w] = 0 and SGN [w -1] = [-, +].
In the first two cases, one can find a continuous function g : R → [0, 1] satisfying (1.6) such that g < w in R. Let u(t, x) be the solution of the Cauchy problem associated with (1.1) with initial condition u(0, x) = g(x). Then, the comparison principle immediately implies that u(t, x) < w(x) for all t ≥ 0 and x ∈ R.

(2.25)

On the other hand, it follows from Theorem 1.

1 (iii) that u(t, •) -U L (t + τ 0 , •) L ∞ (R) → 0 as t → +∞, for some τ 0 ∈ R.
Remember that the wave speed c L is positive. This implies in particular that u(t, 0) → 1 as t → +∞. Since w(0) ∈ (0, 1), we have u(t, 0) > w(0) for all large t, which is a contradiction with (2.25).

Finally, if the last case occurs, then one reaches a similar contradiction by noticing that equation (1.1) admits a pulsating front of the type U L (t, x) = φ L (x + c L t, x/L) satisfying the asymptotic conditions (1.13), and that such a front is also globally asymptotic stable, with positive speed c L (see Remark 1.4). The proof of Lemma 2.3 is thus complete. Lemma 2.4. Let L ≥ L * be fixed, let U L be a pulsating front of (1.1), and let

w ∈ C 2 (R) be a solution of (a L w ) + f L (x, w) = 0 in R. Then Z[U L (t, •) -w] ≤ 2 for all t ∈ R. Furthermore, if Z[U L (t, •) -w] = 2 for some t ∈ R, then there must hold SGN [U L (t, •) -w] = [+, -, +].
(2.26)

Proof. Let t * ∈ R be an arbitrary time such that U L (t * , x * ) = w(x * ) for some x * ∈ R. Denote α := w(x * ). Notice that α ∈ (0, 1) since 0 < U L < 1 in R 2 ,
; z) = H(z -x),
where H denotes the Heaviside function defined by H(y) = 0 if y < 0 and

H(y) = 1 if y ≥ 0. When z < x * , we define τ (z) := min t > 0 : u(t, x * ; z) = α . (2.27)
Let us first observe that τ (z) is well defined and 0 < τ (z) < +∞. Indeed, since c L > 0, it follows from Theorem 1.1 (iii) that lim t→+∞ u(t, x * ; z) = 1 > α. This, together with the continuity of u(•, x * ; z) in [0, +∞) and the fact that u(0, x * ; z) = 0 < α, immediately implies that the minimum in (2.27) is well defined and 0

< τ (z) < +∞. Moreover, since u(t, x; z) → 0 as z → -∞ locally uniformly in (t, x) ∈ [0, +∞) × R, we have lim z→-∞ τ (z) = +∞.
Furthermore, by the proof of [23, Lemma 3.1], there is an entire solution u ∞ : R 2 → (0, 1) of (1.1) that is steeper than any other entire solution between 0 and 1, 5 and such that the following limit exists in the topology of C 1;2 t;x;loc (R 2 ), up to extraction of a subsequence:

lim k→+∞ u(t + τ (-kL), x; -kL) = u ∞ (t, x). (2.28) Next, we claim that u ∞ (t, x) ≡ U L (t + t * , x). (2.29)
Indeed, since the pulsating front of (1.1) is unique up to shifts in time (see Theorem 1.1 (ii)) and since U L is steeper than any other entire solution between 0 and 1 by [31, Theorem 1.8], the function U L is then identically equal to u ∞ up to a shift in time. Since u ∞ (0, x * ) = U L (t * , x * ) = α, we obtain (2.29).

For clarity, we divide the remaining arguments into two parts, according to the behavior of w. First, if w is one of the types (a)-(c) listed in Lemma 2.3, then there exists k * ∈ N sufficiently large such that

Z[ u(0, •; -kL) -w] = 1 for all k ≥ k * .
It then follows from Lemma 2.2 that

Z[ u(t + τ (-kL), •; -kL) -w] ≤ 1 for all t ≥ -τ (-kL) and k ≥ k * .
Hence, by Lemma 2.1 and (2.28)-(2.29), we obtain that

Z[U L (t, •) -w] ≤ 1 for all t ∈ R. Notice actually that, since U L (t * , x * ) = w(x * ) = α and U L is not stationary, Lemma 2.2 implies that Z[U L (t * , •) -w] = 1.
Next, we consider the case where w is one of the types (d)-(e) in Lemma 2.3. In this situation, it is also easily seen that

Z[ u(0, •; -kL) -w] = 2, and SGN [ u(0, •; -kL) -w] = [+, -, +] for all large k ∈ N.
By using Lemmas 2.1 and 2.2 again, we have

Z[U L (t * , •) -w] ≤ 2, and SGN [U L (t * , •) -w] [+, -, +].
In particular, this implies that if

Z[U L (t * , •) -w] = 2, then SGN [U (t * , •) -w] = [+, -, +]. The proof of Lemma 2.4 is thus complete.
In the following lemma, we give a special solution of (2.23) that decays to 0 as x → +∞, with a decay rate that is controlled uniformly with respect to L ≥ L * . Lemma 2.5. For any L ≥ L * and any δ ∈ (0, δ 0 ], where δ 0 ∈ (0, 1/2) is the constant provided by (A2), there exists a solution w ∈ C 2 (R) of (2.23) such that

w(0) = δ, Z[w -1] = 0, w < 1 in R, Z[w] ≤ 1, (2.30)
and 0 < w(x) ≤ δe -µx for all x ≥ 0, (2.31)

for some constant µ > 0 independent of L.

Proof. For any L ≥ L * and any δ ∈ (0, δ 0 ], we show the existence of the desired solution by an approximation argument. For each n ∈ N, let us first consider the following problem in bounded interval :

(a L w ) + f L (x, w) = 0 in [0, n], w(0) = δ, w(n) = 0. (2.32)
Since f L (x, 0) ≡ 0 and since f L (x, δ) < 0 for all x ∈ R, it follows that w = 0 and w = δ are, respectively, a sub-solution and a super-solution of (2.32). Then, a classical iteration argument implies that there exists a solution

w n ∈ C 2 ([0, n]) such that 0 ≤ w n ≤ δ in [0, n]
. Furthermore, 0 < w n < δ in (0, n) from the strong maximum principle and, from the sliding method [START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF], each w n is unique and the sequence (w n ) n∈N is nondecreasing in n ∈ N, in the sense that

w n ≤ w m in [0, n] if n ≤ m. Define now w(x) := lim n→+∞ w n (x) for x ∈ [0, +∞).
It then follows from the standard elliptic estimates that w is a C 2 ([0, +∞)) solution of

(a L w ) + f L (x, w) = 0 in [0, +∞), w(0) = δ, 0 ≤ w ≤ δ in [0, +∞), hence 0 < w < δ in (0, +∞) from the strong maximum principle. Since f L (x, u) is continuous in R × R and globally Lipschitz-continuous in u ∈ R uniformly in x ∈ R, the function w can be extended to x ∈ R so that it is a C 2 (R)
(a L w n ) -γ 0 w n ≥ 0 for x ∈ [0, n]. Since a L (x) = a(x/L) is L-periodic and a is positive and at least of class C 1 (R), we can find a small constant µ > 0 (independent of L ≥ L * ) such that a L (x)µ 2 -a L (x)µ -γ 0 ≤ 0 for all x ∈ R. Letting w(x) := δe -µx for x ≥ 0, we compute that (a L w ) (x) -γ 0 w(x) = (a L (x)µ 2 -a L (x)µ -γ 0 ) w(x) ≤ 0 for all x ≥ 0.
It then follows from the elliptic weak maximum principle that w n (x) ≤ w(x) = δe -µx for all x ∈ [0, n] and n ∈ N, hence w(x) ≤ δe -µx for all x ≥ 0. This ends the proof of Lemma 2.5.

In the last lemma of this subsection, we consider the intersection of the pulsating front with the stationary solution of (2.23) obtained in Lemma 2.5.

Lemma 2.6. Let L ≥ L * be fixed, let U L be a pulsating front of (1.1), let δ ∈ (0, δ 0 ] and let w ∈ C 2 (R) be the solution of (2.23) provided by Lemma 2.5.

Then, Z[U L (t, •) -w] ≤ 1 for every t ∈ R. Furthermore, if Z[U L (t, •) -w] = 1 for some t ∈ R, then SGN [U L (t, •) -w] = [+, -].
Proof. The proof is similar to that of Lemma 2.4; therefore, we only give its outline. Let t * ∈ R be arbitrary. Without loss of generality, we suppose that there exists some x * ∈ R such that U L (t * , x * ) = w(x * ), hence w(x * ) ∈ (0, 1). To show the lemma, it suffices to prove that

Z[U L (t * , •) -w] ≤ 0, or Z[U L (t * , •) -w] = 1 and SGN [U L (t * , •) -w] = [+, -].
For any z < x * , let u(t, x; z) be the solution of (1.1) with Heaviside type initial condition H(z -•) and let τ (z) be defined as in (2.27) with α = w(x * ). By the proof of Lemma 2.4, we see that 0 < τ (z) < +∞, lim z→-∞ τ (z) = +∞, and that, up to extraction of a subsequence,

u(t + τ (-kL), x; -kL) → U L (t + t * , x) as k → +∞ in C 1;2 t;x;loc (R 2
). Since w satisfies (2.30) and lim x→+∞ w(x) = 0, it is straightforward to check that

Z[ u(0, •; -kL) -w] = 1 and SGN [ u(0, •; -kL) -w] = [+, -]
for all large k ∈ N. Passing to the limit as k → +∞, it follows from Lemmas 2.1 and 2.

2 that Z[U L (t * , •) -w] ≤ 1 and SGN [U L (t * , •) -w] [+, -],
yielding the desired results.

Proof of Theorem 1.7

For clarity, we divide the proof into three steps. In our arguments below, δ ∈ (0, 1/2] is given.

Step 1: (1.20) holds true with C > 0 independent of L ≥ L * . For any L ≥ L * and x ∈ R, since U L (t, x) is continuous and increasing in t with U L (-∞, x) = 0 and U L (+∞, x) = 1, the set

I L (x) := {t ∈ R : δ ≤ U L (t, x) ≤ 1 -δ} is a compact interval in R.
Denote by m(I L (x)) the length of this interval. It follows from Theorem 1.1 (ii) and (iv) that m(I L (x)) is continuous with respect to (L, x) ∈ [L * , +∞) × R.

Assume by contradiction that the desired result (1.20) is not true. Then, there exist sequences

(L n ) n∈N ⊂ [L * , +∞) and (x n ) n∈N ⊂ R such that m(I Ln (x n )) = m {t ∈ R : δ ≤ U Ln (t, x n ) ≤ 1 -δ} → +∞ as n → +∞.
(2.33)

From (1.5), one easily sees that m(

I n (x n )) = m(I n (x n +L n ))
. Therefore, without loss of generality, we may assume that (

x n ) n∈N ⊂ [0, L n ). Moreover, it follows from the continuity of m(I L (x)) in (L, x) that L n → +∞ as n → +∞. For each n ∈ N, since U Ln (•, x n ) is increasing and continuous, with U Ln (-∞, x n ) = 0 and U Ln (+∞, x n ) = 1, there exists a unique t n ∈ I Ln (x n ) such that U Ln (t n , x n ) = 1/2. Define U n (t, x) := U Ln (t + t n , x + x n ) for (t, x) ∈ R 2 .
Clearly, each function U n is an entire solution of

∂ t U n = ∂ x a x + x n L n ∂ x U n + f x + x n L n , U n in R 2 , (2.34)
with U n (0, 0) = 1/2. Up to extraction of some subsequence, one can assume that x n /L n → x ∞ ∈ [0, 1], and that, from standard parabolic estimates, there exists a function

U ∞ ∈ C 1;2 t;x (R 2 , [0, 1]) such that U n → U ∞ in C 1;2 t;x;loc (R 2 ) as n → +∞. (2.35)
Moreover, U ∞ satisfies U ∞ (0, 0) = 1/2 and it is a solution to

∂ t U ∞ = a(x ∞ )∂ xx U ∞ + f (x ∞ , U ∞ ) in R 2 . (2.36)
By the strong maximum principle, we have 0

< U ∞ < 1 in R 2 . Furthermore, since ∂ t U n > 0 in R 2 , we have ∂ t U ∞ ≥ 0 in R 2 . Finally, one infers from (2.33) that m {t ∈ R : δ ≤ U ∞ (t, 0) ≤ 1 -δ} = +∞. (2.37)
By the monotonicity of U ∞ in t and standard parabolic estimates applied to equation (2.36), it follows that there exist two steady states 0 ≤ p -≤ p + ≤ 1 of (2.36) such that

U ∞ (t, •) → p ± as t → ±∞ in C 2 loc (R). (2.38) It is clear that p ± satisfy a(x ∞ )p ± + f (x ∞ , p ± ) = 0 in R, (2.39)
and 0 ≤ p -(0) ≤ 1/2 ≤ p + (0) ≤ 1. Moreover, due to (2.37), one can conclude that either

δ ≤ p -(0) ≤ 1/2 or 1/2 ≤ p + (0) ≤ 1 -δ.
Then, by the strong maximum principle, we have

either 0 < p -< 1 in R, or 0 < p + < 1 in R.
Without of loss generality, we assume that the former case happens, since the latter case can be handled in a similar way. Then, thanks to the assumptions (A1) and (A3), according to the phase diagrams of equation (2.39), the solution p -can only be one of the following three types: either a constant function, or a non-constant periodic function, or a ground state solution such that p -(x) → 0 as x → ±∞. We will derive a contradiction in each of these three cases.

Case 1: p -is a constant solution, that is, p -≡ b(x ∞ ) in R.
In this case, for any non-constant periodic solution 0 < q < 1 of (2.39), one finds some points y 1 < z 1 < y 2 < z 2 and some constant

ε 0 > 0 such that q(y i ) -p -(y i ) ≥ ε 0 , and q(z i ) -p -(z i ) ≤ -ε 0 for i = 1, 2.
(2.40)

Remember that function u → f (x, u) is globally Lipschitz-continuous in u ∈ R uniformly in x ∈ R.
Then, for each n ∈ N, it follows from the classical ODE theory that the following problem

a x + x n L n w + f x + x n L n , w = 0 in R, (2.41)
with w(y 1 ) = q(y 1 ) and w (y 1 ) = q (y 1 ), admits a unique solution w n ∈ C 2 (R). Moreover, by standard elliptic estimates, we have w n → q as n → +∞ in C 2 loc (R). This implies in particular that there exists N ∈ N such that

|w n (x) -q(x)| ≤ ε 0 4 for all x ∈ [y 1 , z 2 ] and n ≥ N.
On the other hand, it follows from (2.35) and (2.38) that there exists s * < 0, with |s * | sufficiently large, such that, replacing N by a larger integer if necessary,

U n (s * , x) -p -(x) ≤ ε 0 4 for all x ∈ [y 1 , z 2 ] and n ≥ N.
Combining the above, we obtain that, for all large n,

w n (y i ) -U n (s * , y i ) ≥ ε 0 2 , and w n (z i ) -U n (s * , z i ) ≤ - ε 0 2 for i = 1, 2.
By continuity, the function x → U n (s * , x)-w n (x) has at least three sign changes in [y 1 , z 2 ], which is a contradiction with the fact that

Z[ U n (s * , •)-w n ] ≤ 2 (by Lemma 2.4 applied with U n (•, •-x n ) and w n (• -x n )).
Hence, Case 1 is ruled out. Case 2: p -is a non-constant periodic solution. In this case, letting q(x) ≡ b(x ∞ ), we see that q is a constant solution of (2.39). Then, we can find some points y 1 < z 1 < y 2 < z 2 and a constant ε 0 > 0 such that (2.40) holds true, and hence, the same reasoning as in Case 1 yields a contradiction. Therefore, Case 2 is ruled out too.

Case 3: p -is a ground state solution with p -(x) → 0 as x → ±∞. In the case, p -(x) is symmetrically decreasing and there exist some points y 1 < y 0 < y 2 and a constant

ε 0 > 0 such that b(x ∞ ) -p -( y i ) ≥ ε 0 for i = 1, 2, and b(x ∞ ) -p -( y 0 ) ≤ -ε 0 .
Similarly as in Case 1, for each n ∈ N, problem (2.41) with w( y 1 ) = b(x ∞ ) and w ( y 1 ) = 0 has a unique solution w n ∈ C 2 (R), and from standard elliptic estimates, w n → b(x ∞ ) as n → +∞ in C 2 loc (R). As a consequence, there exists a large integer N ∈ N such that

| w n (x) -b(x ∞ )| ≤ ε 0 4 for all x ∈ [ y 1 , y 2 ] and n ≥ N .
On the other hand, by using (2.35) and (2.38) again and by making some adjustment to N if necessary, one finds some negative time s * < 0, with | s * | large enough, such that

U n ( s * , x) -p -(x) ≤ ε 0 4 for all x ∈ [ y 1 , y 2 ] and n ≥ N .
Therefore, we obtain that, for each n ≥ N ,

w n ( y i ) -U n ( s * , y i ) ≥ ε 0 2 for i = 1, 2, and w n ( y 0 ) -U n ( s * , y 0 ) ≤ - ε 0 2 . (2.42)
Remember that by Lemma 2.4 (applied with

U n (•, • -x n ) and w n (• -x n )), we have Z[ U n ( s * , •) - w n ] ≤ 2. Then, (2.42) implies that Z[ U n ( s * , •) -w n ] = 2, and SGN [ U n ( s * , •) -w n ] = [-, +, -].
This last property contradicts property (2.26) of Lemma 2.4, hence Case 3 is ruled out too. Consequently, the case where 0 < p -< 1 cannot happen. Furthermore, with similar arguments, one can exclude the case where 0 < p + < 1. Therefore, our assumption (2.33) at the beginning of the proof is unreasonable. This ends the proof of Step 1.

Step 2: Up to extraction of some subsequence, we may assume that the sequence (x n / Ln ) n∈N ⊂ [0, 1] converges as n → +∞. By a slight abuse of notation, we still denote by x ∞ this limit. Then, standard parabolic estimates imply that, possibly up to extraction of a further subsequence, Ūn → Ū∞ in C 1;2 t;x;loc (R 2 ), where Ū∞ is an entire solution of (2.36). It is clear that ∂ t Ū∞ ≥ 0 in R 2 and ∂ t Ū∞ (0, 0) = 0. Furthermore, the strong maximum principle applied to the equation satisfied by ∂ t Ū∞ immediately gives that

∂ t Ū∞ ≡ 0 in R 2 . In other words, Ū∞ (t, x) = Ū∞ (x) is independent of t and it obeys a(x ∞ ) Ū ∞ + f (x ∞ , Ū∞ ) = 0 in R. Notice that 0 ≤ Ū∞ ≤ 1 in R and δ ≤ Ū∞ (0) ≤ 1 -δ.
It then follows from the strong maximum principle that 0 < Ū∞ < 1 in R. Furthermore, it follows from (A1) and (A3) that either Ū∞ is a constant solution (i.e., Ū∞ ≡ b(x ∞ )), or it is a non-constant periodic solution, or it is a ground state solution decaying to 0 as x → ±∞. Proceeding similarly as in Step 1, we can find a contradiction in each of these cases: namely, for each large n ∈ N, there exists then a solution wn of (2.41) (with Ln and xn instead of L n and x n ) such that either

Z[ Ūn (0, •) -wn ] ≥ 3, or Z[ Ūn (0, •) -wn ] = 2 and SGN [ Ūn (0, •) -wn ] = [-, +, -]
(the proof is even simpler than that in Step 1, as the limit function Ū∞ is stationary; therefore we do not repeat the details). Both situations are impossible, due to Lemma 2.4 (applied with Ūn (•, • -xn ) and wn (• -xn )). As a consequence, the proof of Step 2 is complete.

Step 3: (1.21) holds true with C > 0 independent of L ≥ L * . Assume by contradiction that there does not exist such a bound independent of L. Then, we can find sequences

(L n ) n∈N ⊂ [L * , +∞), (s n ) n∈N ⊂ R, (α n ) n∈N , (γ n ) n∈N ⊂ [δ, 1 -δ] and (x n ) n∈N , (y n ) n∈N ⊂ R such that U Ln (s n , x n ) = α n and U Ln (s n , y n ) = γ n for each n ∈ N,
and |x n -y n | → +∞ as n → +∞.
Without loss of generality, for each n ∈ N, we may assume that y n > x n , and thanks to (1.5), we may also assume that 0 ≤ x n < L n . Moreover, by Theorem 1.1 (ii) (iv) again, we have

L n → +∞ as n → +∞. Define now U n (t, x) := U Ln (t + s n , x + x n ) for (t, x) ∈ R 2 .
Clearly, each function U n (t, x) is an entire solution of (2.34) with U n (0, 0) = α n and U n (0, z n ) = γ n , where 0 < z n = y n -x n → +∞ as n → +∞. Up to extraction of some subsequence, we have

x n L n → x ∞ ∈ [0, 1], α n → α ∞ ∈ [δ, 1 -δ], γ n → γ ∞ ∈ [δ, 1 -δ], as n → +∞,
and by standard parabolic estimates, U n → U ∞ in C 1;2 t;x;loc (R 2 ) as n → +∞, where U ∞ is a solution of (2.36) with U ∞ (0, 0) = α ∞ .

Set δ * := min{δ 0 , δ}, where δ 0 is the constant provided by (A2). It follows from Lemma 2.5 that, for each n ∈ N, (2.23) with L = L n admits a solution w n ∈ C 2 (R) such that w n (0) = δ * , Z[w n -1] = 0, Z[w n ] ≤ 1 and 0 < w n (x) ≤ δ * e -µx for all x ≥ 0, where µ > 0 is independent of n ∈ N. Notice that for each n ∈ N, U n (t, x) is decreasing and continuous in t ∈ R, with U n (-∞, x) = 0 and U n (+∞, x) = 1 locally uniformly in x ∈ R. This implies in particular that there exists a unique time

τ n ∈ R such that U n (τ n , z n ) = w n (z n ). Since w n (z n ) < δ * ≤ δ ≤ γ n = U n (0, z n ) for each n ∈ N, it is clear that τ n < 0. Now, we claim that τ n → -∞ as n → +∞. (2.43)
Suppose to the contrary that (τ n ) n∈N converges, up to extraction of a subsequence, to τ ∞ ∈ (-∞, 0]. For each n ∈ N, let us write

z n := z n + z n , and V n (t, x) := U n (t, x + z n ) in R 2 ,
where z n ∈ L n N and z n ∈ [0, L n ). It is easily checked that each V n is an entire solution of

∂ t V n = ∂ x a x + x n + z n L n ∂ x V n + f x + x n + z n L n , V n in R 2 ,
with V n (0, 0) = γ n and V n (τ n , 0) = w n (z n ). Since z n → +∞ as n → +∞ and since w n (x) decays to 0 as x → +∞ uniformly in n ∈ N, it follows that lim n→+∞ V n (τ n , 0) = 0. Next, up to extraction of another subsequence, we may assume that z n /L n → z ∞ ∈ [0, 1] as n → +∞, and that, by standard parabolic estimates,

V n → V ∞ as n → +∞ in C 1;2 t;x;loc (R 2 )
, where V ∞ is an entire solution of

∂ t V ∞ = a(x ∞ + z ∞ )∂ xx V ∞ + f (x ∞ + z ∞ , V ∞ ) in R 2 . Since 0 ≤ V ∞ ≤ 1 and since V ∞ (τ ∞ , 0) = lim n→+∞ V n (τ n , 0) = 0, it follows from the strong maximum principle that V ∞ ≡ 0 in R 2 .
This contradicts the fact that V ∞ (0, 0) = lim n→+∞ γ n = γ ∞ ≥ δ. Therefore, our claim (2.43) is proved.

We are now ready to derive a contradiction. By the proof of Step 1, it follows that the sequence m({t ∈ R :

δ * ≤ U n (t, 0) ≤ 1 -δ * }) n∈N is bounded uniformly in n ∈ N. This together with U n (0, 0) = α n ∈ [δ, 1 -δ] ⊂ [δ * , 1 -δ * ] and ∂ t U n > 0 yields the existence of A > 0 such that {t ∈ R : δ * ≤ U n (t, 0) ≤ 1 -δ * } ⊂ [-A, A] for all n ∈ N. As a consequence, one finds t * < 0 such that U n (t * , 0) < δ * = w n (0) for all n ∈ N.
Furthermore, by (2.43) and the positivity of ∂ t U n , there exists n * ∈ N sufficiently large such that

U n * (t * , z n * ) > w n * (z n * ).
Combining the above, we immediately obtain that

Z[ U n * (t * , •) -w n * ] ≥ 1 and SGN [ U n * (t * , •) -w n * ] [-, +],
which is a contradiction with the conclusions of Lemma 2.6 (applied with U n * (•, • -x n * ) and

w n * (• -x n * )).
This ends the proof of Step 3, and the proof of Theorem 1.7 is thus complete.

Proof of Theorem 1.8

We first show statements (i). From Theorem 1.1 (ii), it suffices to prove that for any δ ∈ (0, 1/2], the interval

E L (y) := ξ ∈ R : δ ≤ φ L (ξ + ζ L (y), y) ≤ 1 -δ
is bounded uniformly with respect to y ∈ R and L ≥ L * . To do so, observe that the formula

U L (t, x) = φ L (x -c L t, x/L) implies that U L - ξ c L + Ly -ζ L (y) c L , Ly = φ L (ξ + ζ L (y), y) for all ξ ∈ R and y ∈ R. (2.44)
Clearly, U L ((Ly -ζ L (y))/c L , Ly) = 1/2 for all y ∈ R, by (1.15). Then, by Theorem 1.7, the following interval

I L (Ly) := t ∈ R : δ ≤ U L t + Ly -ζ L (y) c L , Ly ≤ 1 -δ is bounded uniformly in L ≥ L * and y ∈ R. Remember that c L is bounded in L ≥ L * ,
∂ ξ φ L (ξ + ζ L (y), y) = - 1 c L ∂ t U L - ξ c L + Ly -ζ L (y) c L , Ly .
Then by Theorem 1.7 and the fact that c L > 0 is bounded in L ≥ L * , we immediately obtain statement (ii) (with a constant β > 0 which is in general different from the one appearing in (1.22)).

Finally, we prove statement (iii

). Since ζ L (y) is continuous in (L, y) ∈ [L * , +∞) × R, it is clear that (L, y) → ζ L (y)/L is also continuous. It remains to show that this function is bounded. Since ζ L (y) is 1-periodic in y ∈ R, we only need to show that ζ L (y)/L is bounded in (L, y) ∈ [L * , +∞) × [0, 1]. It follows from (1.14) and Theorem 1.7 that the set {x ∈ R : U L (0, x) = 1/2} is bounded uniformly in L ≥ L * , that is, there exists L ≥ L * such that x ∈ R : U L (0, x) = 1/2 ⊂ [-L, L] for all L ≥ L * .
Remember that lim x→+∞ U L (0, x) = 0 for each L ≥ L * . This implies that U L (0, L(y + 1)) < 1/2 for all y ∈ [0, 1] and L > L. Thanks to (2.44) applied with ξ := L(y+1)-ζ L (y+1) and y+1 instead of y, it follows that φ L (L(y + 1) -ζ L (y + 1) + ζ L (y), y) = φ L (L(y + 1) -ζ L (y + 1) + ζ L (y), y + 1) = U L (0, L(y + 1)) < 1/2. Hence, the monotonicity of φ L in its first variable implies that L(y + 1) -ζ L (y + 1) > 0 for all y ∈ [0, 1] and L > L.

Consequently, by the periodicity of ζ L (y), we obtain ζ L (y)/L < y + 1 for all y ∈ [0, 1] and L > L.

In a similar way, by using the fact that lim x→-∞ U L (0, x) = 1 and the monotonicity of U L in its first variable, we can prove that L(y -2) -ζ L (y -2) < 0 for all y ∈ [0, 1] and L > L, hence ζ L (y)/L > y -2 for all y ∈ [0, 1] and L > L. Combining the above, we immediately obtain that -2 < ζ L (y)/L < 2 for all y ∈ [0, 1] and L > L. This together with the continuity of

ζ L (y)/L in [L * , +∞) × [0, 1] implies that ζ L (y)/L is bounded in [L * , +∞) × [0, 1]. The proof of Theorem 1.8 is thus complete. 3 Convergence of c L as L → +∞: proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2. When L is large, we consider the Cauchy problem associated with equation (1.1) under the large-time and large-space scaling, t → Lt, x → Lx. More precisely, letting v(t, x) = u(Lt, Lx) with u(t, x) being a solution of (1.1), the rescaled Cauchy problem reads

   v t = 1 L (a(x)v x ) x + Lf (x, v), t > 0, x ∈ R, v(0, x) = g(x), x ∈ R. (3.45)
We consider initial conditions g ∈ C(R, [0, 1]) which are independent of L and front-like, in the sense that lim inf

x→-∞ (g(x) -b(x)) > 0, lim sup x→+∞ (g(x) -b(x)) < 0, (3.46)
where x → b(x) ∈ (0, 1) is the zero of f (x, •) provided by (A1). For each L > 0, denote by v L (t, x) the solution of the Cauchy problem (3.45). By the strong maximum principle, we have 0 < v L (t, x) < 1 for all t > 0 and x ∈ R. Moreover, it is easily seen from (1.4) that for each L ≥ L * , with L * > 0 as in Theorem 1.1, φ L (L(x -c L t), x) is a solution of the first equation of (3.45), and it is defined for all (t, x) ∈ R 2 . As we mentioned in Section 1, we use Gärtner's result [START_REF] Gärtner | Bistable reaction-diffusion equations and excitable media[END_REF] to show the convergence of c L as L → +∞. Recall that the function c : R → R, y → c(y) denotes the unique front speeds of the family of homogeneous equations (1.8). The following lemma is a direct application of [START_REF] Gärtner | Bistable reaction-diffusion equations and excitable media[END_REF]Corollary,p. 140] to our one-dimensional spatially periodic equation (3.45) (see also Xin's review paper [START_REF] Xin | Front propagation in heterogeneous media[END_REF]Theorem 3.7] in the case where g is compactly supported):

Lemma 3.1. [30] Let g ∈ C(R, [0, 1]) satisfy (3.46) and let Γ 0 = {x ∈ R : g(x) = b(x)}.
For any L > 0, let v L (t, x) be the solution of (3.45) with initial condition g (g is independent of L). Then, as L → +∞,

v L (t, x) → 1 locally uniformly in {(t, x) ∈ (0, +∞) × R : ρ(x, Γ 0 ) < t}, v L (t, x) → 0 locally uniformly in {(t, x) ∈ (0, +∞) × R : ρ(x, Γ 0 ) > t},
where ρ(x, Γ 0 ) := inf{ρ(x, x 2 ) : x 2 ∈ Γ 0 } and ρ(•, •) denotes the signed distance function defined by

ρ(x 1 , x 2 ) := x 1 x 2 c -1 (y)dy for x 1 ∈ R, x 2 ∈ R.
Recall that c * is the constant defined in (1.11). Since the function y → c(y) is 1-periodic, the above result suggests that as L → +∞, the solution v L (t, x) propagates at a mean speed equal to c * . This however does not imply directly the convergence of the speeds c L to c * , since the profiles of the pulsating fronts, say at time 0, depend on L, unlike the initial condition g in Lemma 3.1. To circumvent this difficulty, we establish in Lemma 3.2 some further comparisons between the solutions v L (t, x) and the pulsating fronts φ L (L(x -c L t), x) for L ≥ L * , which easily lead to the proof of Theorem 1.2, on the basis of the uniform estimates proved in Section 2. 6To do so, let δ 0 and γ 0 be the positive constants provided by the assumption (A2). Since f (•, 0) = f (•, 1) = 0 in R, there is ε 0 ∈ (0, δ 0 /2] such that

∂ u f (x, u) ≤ - γ 0 2 for all u ∈ [-2ε 0 , 2ε 0 ] ∪ [1 -2ε 0 , 1 + 2ε 0 ] and x ∈ R. (3.47)
We now choose a special initial condition

g ∈ C(R, [0, 1]), independent of L ≥ L * , such that lim inf x→-∞ g(x) > 1 -ε 0 , lim sup x→+∞ g(x) < ε 0 , and Γ 0 = {0}. (3.48)
Notice that it is for instance possible to choose g in such a way that g = 1 in (-∞, -A] and g = 0 in [A, +∞), for some large A.

Lemma 3.2. For any L ≥ L * , with L * > 0 as in Theorem 1.1, let (φ L , c L ) be a pulsating front of (1.1) and let v L (t, x) be the solution of (3.45) with the initial condition g satisfying (3.48).

Then there exist constants C ± ≥ 0 and K 0 > 0 (all independent of L) such that

v L (t, x) ≥ φ L (L(x -c L t + C -) + K 0 ε 0 , x) -ε 0 e -γ 0 Lt/2 v L (t, x) ≤ φ L (L(x -c L t -C + ) -K 0 ε 0 , x) + ε 0 e -γ 0 Lt/2
for all t ≥ 0 and x ∈ R.

(3.49)

Proof. We use a Fife-McLeod [START_REF] Fife | The approach of solutions of non-linear diffusion equations to traveling front solutions[END_REF] type sub-and super-solutions method to show this lemma. We only give the construction of a super-solution, as the analysis for a sub-solution is analogous. First of all, since the initial function g satisfies (3.48), it follows from Theorem 1.8 (i) and (iii) that there exists a constant C + > 0 independent of L such that, for all L ≥ L * and x ∈ R,

φ L (L(x -C + ), x) + ε 0 = φ L L x -C + - ζ L (x) L + ζ L (x), x + ε 0 ≥ g(x). (3.50)
For any L ≥ L * , we then set

v L (t, x) := φ L L(x -c L t -C + ) + η L (t), x + q L (t) for t ≥ 0 and x ∈ R,
where t → q L (t) and t → η L (t) are some C 1 ([0, +∞)) functions satisfying

q L (0) = ε 0 , q L (t) < 0 < q L (t) for all t ≥ 0, η L (0) = 0, η L (t) < 0 for all t ≥ 0. (3.51)
By choosing suitable functions η L and q L later, we will show that v L is a super-solution of (3.45). Notice first that (3.50)-(3.51) imply that v L (0,

•) ≥ g = v L (0, •) in R for each L ≥ L * . Now, for (t, x) ∈ (0, +∞) × R, we define N L (t, x) := ∂ t v L (t, x) - 1 L ∂ x (a(x)∂ x v L (t, x)) -Lf (x, v L (t, x)). Since (t, x) → φ L (L(x -c L t -C + ), x
) is an entire solution of the first equation of (3.45), a straightforward calculation gives, for all (t, x) ∈ (0, +∞) × R,

N L (t, x) = η L (t)∂ ξ φ L (L(x -c L t -C + ), x) + q L (t) + Lf x, φ L (L(x -c L t -C + ) + η L (t), x) -Lf (x, v L (t, x)).
Since ∂ ξ φ L < 0 in R 2 , we get η L (t)∂ ξ φ L (L(x-c L t-C + ), x) > 0 and 0 < q L (t) ≤ ε 0 for all t ≥ 0 and x ∈ R, provided q L and η L fulfill (3.51). Then, on the one hand, for any pair (t, x)

∈ (0, +∞) × R such that φ L (L(x -c L t -C + ) + η L (t), x) ∈ (0, ε 0 ] × [1 -ε 0 , 1), it follows from (3.47) that N L (t, x) ≥ q L (t) + Lf (x, φ L (L(x -c L t -C + ) + η L (t), x)) -Lf (x, v L (t, x)) ≥ q L (t) + γ 0 L 2 q L (t).
On the other hand, by Theorem 1.8 (ii), there is β > 0 (independent of L ≥ L * ) such that ∂ ξ φ L (ξ, y) ≤ -β for all L ≥ L * and for all (ξ, y

) ∈ R 2 with ε 0 ≤ φ L (ξ, y) ≤ 1 -ε 0 . It follows that, if φ L (L(x -c L t -C + ) + η L (t), x) ∈ (ε 0 , 1 -ε 0 ), then N L (t, x) ≥ -η L (t)β + q L (t) + Lf (x, φ L (L(x -c L t -C + ) + η L (t), x)) -Lf (x, v L (t, x)) ≥ -η L (t)β + q L (t) -LC 1 q L (t),
where

C 1 := ∂ u f L ∞ (R×R) . Let us now choose q L (t) and η L (t) such that    q L (0) = ε 0 , q L (t) + γ 0 L 2 q L (t) = 0 for t ≥ 0, η L (0) = 0, -βη L (t) + q L (t) -LC 1 q L (t) = 0 for t ≥ 0.
Namely, we set q L (t) = ε 0 e -γ 0 Lt/2 and η L (t) = -ε 0 (γ 0 + 2C 1 ) βγ 0 (1 -e -γ 0 Lt/2 ) for t ≥ 0.

These functions q L and η L satisfy (3.51). Consequently, N L (t, x) ≥ 0 for all (t, x) ∈ (0, +∞) × R.

Finally, the comparison principle implies that, for any L ≥ L * , v L (t, x) ≥ v L (t, x) for all t ≥ 0 and x ∈ R. Taking K 0 = (γ 0 + 2C 1 )/(βγ 0 ) and using the fact that φ L (ξ, y) is decreasing in ξ, we obtain the second inequality of (3.49). As we have mentioned above, similar arguments imply the first one. The proof of Lemma 3.2 is thus compete.

We are now ready to complete the Without loss of generality, we assume that c ∞ < c * (as sketched below, the case where c ∞ > c * can be treated similarly). Then we have c Ln < (c ∞ + c * )/2 < c * for all large n ∈ N. Now, we can choose a large time t * > 0 and a positive integer x * (both independently of n) such that

c Ln t * + C + + 1 < x * < c * t * for all large n ∈ N, (3.52)
where C + ≥ 0 is the constant given by Lemma 3.2. Since each function φ Ln (ξ, y) is decreasing in ξ, it follows from the notations of Lemma 3.2 and the second inequality of (3.49) that

0 < v Ln (t * , x * ) ≤ φ Ln (L n (x * -c Ln t * -C + ) -K 0 ε 0 , x * ) + ε 0 e -γ 0 Lnt * /2 ≤ φ Ln (L n -K 0 ε 0 , x * ) + ε 0 e -γ 0 Lnt * /2 (3.53)
for all large n ∈ N. Moreover, since x * ∈ N and each function ζ Ln given by (1.15) is 1-periodic, we have ζ Ln (x * ) = ζ Ln (0) = 0. Therefore, passing to the limit as n → +∞ in (3.53), we see from Theorem 1.8 that φ Ln (L n -K 0 ε 0 , x * ) → 0 as n → +∞, hence v Ln (t * , x * ) → 0 as n → +∞.

On the other hand, we had chosen g such that Γ 0 = {x ∈ R : g(x) = b(x)} = {0}. Since x * is a positive integer and since the function y → c(y) is positive and 1-periodic, we have ρ(x * , Γ 0 ) = x * /c * , whence ρ(x * , Γ 0 ) < t * due to (3.52). It then follows from Lemma 3.1 that v Ln (t * , x * ) → 1 as n → +∞, yielding a contradiction. Therefore, the case where c ∞ < c * is ruled out.

In the case where c ∞ > c * , one can derive a similar contradiction by choosing a large time t * > 0 and a positive integer x * (both independently of n) such that

c Ln t * -C --1 > x * > c * t * for all large n ∈ N.
As a conclusion, c L → c * as L → +∞, and the proof of Theorem 1.2 is thus complete.

Convergence of the front profiles: proof of Theorem 1.6

This section is devoted to the proof of Theorem 1.6, that is, the convergence of the front profiles φ L as L → +∞, under the assumptions (A1)-(A4). The key step is to determine the asymptotic behavior as L → +∞ of the solutions of Cauchy problem of the following equation:

∂ t z L = a∂ xx z L + f y + x L , z L for t > 0, x ∈ R, (4.54)
where y ∈ R is arbitrary. We point out that a is here a positive constant, thanks to (A4). We recall that, for each y ∈ R, the couple (ψ(•, y), c(y)) denotes the front profile and front speed of the homogeneous equation (1.9)- (1.10). We first present some properties on (ψ(•, y), c(y)) in Section 4.1. In Section 4.2, we choose the initial condition of (4.54) sufficiently close to the homogeneous front ψ(•, y), and then construct a pair of sub-and super-solutions of (4.54). This will ensure that, at a certain time t, z L (t, •+L) is still close to ψ(•, y) provided that L is sufficiently large (see Proposition 4.7 in Section 4.3). We complete the proof of Theorem 1.6 in Section 4.4.

Preliminaries on homogeneous fronts

In this subsection, we present some properties regarding the homogeneous traveling front (ψ(•, y), c(y)) of (1.9)-(1.10) with respect to the parameter y. These properties will be used in our construction of sub-and super-solutions later and are also of interest in themselves. (ii) lim ξ→+∞ ψ(ξ, y) = 0 and lim ξ→-∞ ψ(ξ, y) = 1 uniformly in y ∈ R; furthermore, there exist positive constants µ 1 , µ 2 , M , C 1 and C 2 (all are independent of y) such that

0 < ψ(ξ, y) ≤ C 1 e -µ 1 ξ
for all ξ ≥ M and y ∈ R,

0 < 1 -ψ(ξ, y) ≤ C 2 e µ 2 ξ
for all ξ ≤ -M and y ∈ R; (4.55)

(iii) for any δ ∈ (0, 1/2], there exists γ = γ(δ) > 0 (independent of y) such that The proof of Proposition 4.1 is quite lengthy and is therefore postponed to the appendix in Section 5. Let us emphasize that this proposition holds without the extra assumption (A4).

∂ ξ ψ(ξ, y) ≤ -γ for all (ξ, y) ∈ R 2 such that δ ≤ ψ(ξ, y) ≤ 1 -δ; (

Sub-and super-solutions

In the remaining part of Section 4, we always assume that (A1)-(A4) hold. To present our suband super-solutions, we need some notations. Since the function z → c(z) is 1-periodic and of class C 1 (R) by Proposition 4.1 (iv), one infers that, for each y ∈ R and L > 0, the following ODE problem

X y,L (t) = c y + X y,L (t) L for t ≥ 0, X y,L (0) = 0, (4.57)
admits a unique solution X y,L : [0, +∞) → R, with X y,L = X y+1,L in [0, +∞) for each y ∈ R and L > 0. Furthermore, each function X y,L is increasing in [0, +∞) and X y,L (t) → +∞ as t → +∞, since min R c > 0. Let then T L > 0 be the unique time such that

X y,L (T L ) = L.
Since the function c is 1-periodic, it is then checked by integrating the function t → X y,L (t)/c(y + X y,L (t)/L) = 1 over [0, T L ] that T L does not depend on y and is equal to

T L = L c * (4.58)
where c * > 0 is the constant defined by (1.11). In particular, T L → +∞ as L → +∞ (uniformly in y ∈ R). Notice also that, again by 1-periodicity of the function c, one has

X y,L (kT L ) = kL for all k ∈ N. (4.59)
Our super-solution is stated in the following lemma.

Lemma 4.2. There exists ε 0 ∈ (0, δ 0 ), with δ 0 ∈ (0, 1/2) as in assumption (A2), such that, for every ε ∈ (0, ε 0 ], there exists L 1,ε > 0 such that, for every y ∈ R and

L ≥ L 1,ε , the function v + ε,y,L : [0, +∞) × R → R defined by v + ε,y,L (t, x) := ψ x -X y,L (t) + η ε,L (t), y + X y,L (t) L + q ε,L (t)
is a super-solution of (4.54) for t ≥ 0 and x ∈ R, where q ε,L and η ε,L are C 1 ([0, +∞)) functions satisfying q ε,L (0) = ε, q ε,L (t) < 0 < q ε,L (t) for all t ≥ 0,

η ε,L (0) = 0, η ε,L (t) < 0 for all t ≥ 0. (4.60)
Proof. First of all, by the assumptions (A1)-(A2) and the C 1 (R × [0, 1]) smoothness of f and its periodicity in x, together with (1.3), there exist δ 1 ∈ (0, 1/2) and γ 1 > 0 such that

∂ u f (x, u) ≤ -γ 1 for all x ∈ R and u ∈ [-δ 1 , δ 1 ] ∩ [1 -δ 1 , 1 + δ 1 ]. (4.61)
Without loss of generality, one can assume that δ 1 ≤ δ 0 , with δ 0 > 0 as in assumption (A4). Moreover, by Proposition 4.1 (iv) and the boundedness of the function z → c(z), there is a constant C 1 > 0 such that, for all y ∈ R and L > 0,

X y,L (t) L ∂ y ψ(ξ, z) ≤ C 1 L for all t ≥ 0 and (ξ, z) ∈ R × R. (4.62)
It further follows from Proposition 4.1 (i)-(ii) that there exist M 1 > 0 and

ε 0 ∈ (0, min{δ 1 /2, δ 0 }) ⊂ (0, 1/4), with δ 0 ∈ (0, 1/2) as in assumption (A2), such that, for all (ξ, z) ∈ R × R,            0 < ψ(ξ, z) ≤ δ 1 2 if ξ ≥ M 1 , 1 - δ 1 2 ≤ ψ(ξ, z) < 1 if ξ ≤ -M 1 , 2ε 0 ≤ ψ(ξ, z) ≤ 1 -2ε 0 if -M 1 < ξ < M 1 ,
where the last inequality is actually a consequence of the continuity of ψ : R 2 → (0, 1) and its periodicity in the second variable.

In the arguments below, ε ∈ (0, ε 0 ] is arbitrary. To show Lemma 4.2, it suffices to find suitable C 1 ([0, +∞)) functions q ε,L and η ε,L satisfying (4.60) such that, if L is sufficiently large and independently of y, there holds

N (t, x) := ∂ t v + ε,y,L (t, x) -a∂ xx v + ε,y,L (t, x) -f y + x L , v + ε,y,L (t, x) ≥ 0 for all (t, x) ∈ [0, +∞) × R.
Since ψ is a solution of (1.9) and since here a is constant by assumption (A4), it is straightforward to check that, for any y ∈ R, L > 0 and any C 1 ([0, +∞)) functions q ε,L and η ε,L , one has

N (t, x) = X y,L (t) L ∂ y ψ + η ε,L (t)∂ ξ ψ + q ε,L (t) + f y + X y,L (t) L , ψ -f y + x L , v + ε,y,L (t, x)
for all (t, x) ∈ [0, +∞) × R, where ψ = ψ(x -X y,L (t) + η ε,L (t), y + X y,L (t)/L), and ∂ ξ ψ, ∂ y ψ stand for the partial derivatives of ψ with respect to the first variable and the second variable, respectively, evaluated at the same point (x -X y,L (t) + η ε,L (t), y + X y,L (t)/L). We complete the proof by considering three cases: (a) x -X y,L (t

) + η ε,L (t) ≥ M 1 , (b) x -X y,L (t) + η ε,L (t) ≤ -M 1 , (c) -M 1 < x -X y,L (t) + η ε,L (t) < M 1 .
In case (a), with q ε,L required to satisfy (4.60), we have

0 < ψ x -X y,L (t) + η ε,L (t), y + X y,L (t) L < v + ε,y,L (t, x) ≤ δ 1 2 + ε ≤ δ 1 2 + ε 0 ≤ δ 1 .
Since ψ is decreasing in its first variable and since η ε,L is required to satisfy (4.60), we have

η ε,L (t) ∂ ξ ψ x -X y,L (t) + η ε,L (t), y + X y,L (t) L ≥ 0.
Remember that f (x, u) is independent of x for u ∈ [0, δ 1 ] ⊂ [0, δ 0 ] by assumption (A4). It then follows from (4.61)-(4.62) that

N (t, x) ≥ - C 1 L + q ε,L (t) + f 0, ψ x -X y,L (t) + η ε,L (t), y + X y,L (t) L -f (0, v + ε,y,L (t, x)) ≥ - C 1 L + q ε,L (t) + γ 1 q ε,L (t). Let us choose L 1,ε = 2C 1 γ 1 ε > 0 (4.63) (notice that L 1,ε is independent of y ∈ R) and the function q ε,L such that q ε,L (0) = ε and - C 1 L + q ε,L (t) + γ 1 q ε,L (t) = 0 for t ≥ 0, (4.64) namely, q ε,L (t) = C 1 Lγ 1 + ε - C 1 Lγ 1 e -γ 1 t for t ≥ 0. (4.65)
It is clear that, for any L ≥ L 1,ε and for any y ∈ R, the function q ε,L satisfies (4.60) and

N (t, x) ≥ 0 for all (t, x) ∈ [0, +∞) × R such that x -X y,L (t) + η ε,L (t) ≥ M 1 , provided η ε,L satisfies (4.60) too.
Proceeding similarly as above, we can conclude that, for any L ≥ L 1,ε and y ∈ R, N (t, x) ≥ 0 for all (t, x) ∈ [0, +∞) × R such that x -X y,L (t) + η ε,L (t) ≤ -M 1 , as soon as η ε,L satisfies (4.60).

It remains to find a suitable function η ε,L such that N (t, x) ≥ 0 in case (c), i.e., -M 1 < x -X y,L (t) + η ε,L (t) < M 1 . In this case, we have

2ε 0 ≤ ψ x -X y,L (t) + η ε,L (t), y + X y,L (t) L ≤ 1 -2ε 0 and 2ε 0 ≤ v + ε,y,L (t, x) ≤ 1 -ε 0 (since 0 < q ε,L (t) ≤ ε ≤ ε 0 )
. It then follows from Proposition 4.1 (iii) that there exists β 1 > 0 (independent of ε, y, L, t and x) such that

∂ ξ ψ(x -X y,L (t) + η ε,L (t), y + X y,L (t)/L) ≤ -β 1 . Noticing that the function f is of class C 1 (R × [0, 1]
) and periodic in x, one finds some constants K 1 > 0 and K 2 > 0 (independent of ε, y, L, t and x) such that

|f (x 1 , u) -f (x 2 , u)| ≤ K 1 |x 1 -x 2 | for all x 1 , x 2 ∈ R and u ∈ [0, 1], |f (z, u 1 ) -f (z, u 2 )| ≤ K 2 |u 1 -u 2 | for all z ∈ R and u 1 , u 2 ∈ [0, 1].
Therefore, with η ε,L required to satisfy (4.60) (then being nonpositive in [0, +∞)), we have

N (t, x) ≥ - C 1 L -β 1 η ε,L (t) + q ε,L (t) + f y + X y,L (t) L , ψ x -X y,L (t) + η ε,L (t), y + X y,L (t) L -f y + x L , v + ε,y,L (t, x) ≥ - C 1 L -β 1 η ε,L (t) + q ε,L (t) -K 1 |X y,L (t) -x| L -K 2 q ε,L (t), ≥ - C 1 L -β 1 η ε,L (t) + q ε,L (t) -K 1 M 1 -η ε,L (t) L -K 2 q ε,L (t).
Then, with q ε,L (t) given by (4.64)-(4.65), by choosing η ε,L such that

η ε,L (0) = 0 and -β 1 η ε,L (t) + K 1 L η ε,L (t) -(γ 1 + K 2 )q ε,L (t) = K 1 M 1 L for t ≥ 0, we have N (t, x) ≥ 0 for (t, x) ∈ [0, +∞) × R such that -M 1 < x -X y,L (t) + η ε,L (t) < M 1 . It is straightforward to compute that η ε,L (t) = - γ 1 + K 2 β 1 (C 2 + C 3,ε,L )e K 1 t/(Lβ 1 ) -C 3,ε,L e -γ 1 t -C 2 for t ≥ 0, (4.66)
where

C 2 = C 1 β 1 γ 1 K 1 + M 1 β 1 K 2 + γ 1 and C 3,ε,L = ε -C 1 /(Lγ 1 ) γ 1 + K 1 /(Lβ 1 ) . It is clear that C 2 > 0 and C 3,ε,L > 0 if L ≥ L 1,ε
, whence the function η ε,L given in (4.66) satisfies (4.60).

Combining the above, we can conclude that for any ε ∈ (0, ε 0 ], y ∈ R and L ≥ L 1,ε , one has N (t, x) ≥ 0 for all (t, x) ∈ [0, +∞) × R. This ends the proof of Lemma 4.2.

The following lemma gives the sub-solution of problem (4.54).

Lemma 4.3. Let ε 0 ∈ (0, δ 0 ) be given by Lemma 4.2 and, for any ε ∈ (0, ε 0 ], let L 1,ε > 0 be given by (4.63) as in Lemma 4.2. Then, for every ε ∈ (0,

ε 0 ], y ∈ R and L ≥ L 1,ε , the function v - ε,y,L : [0, +∞) × R → R defined by v - ε,y,L (t, x) := ψ x -X y,L (t) -η ε,L (t), y + X y,L (t) L -q ε,L (t)
is a sub-solution of (4.54) for t ≥ 0 and x ∈ R, where q ε,L and η ε,L are C 1 ([0, +∞)) functions satisfying (4.60), given by (4.65) and (4.66), respectively.

Proof. The proof is analogous to that of Lemma 4.2; therefore, we omit the details.

Before going further on, we give two remarks about the functions q ε,L and η ε,L , which will be useful in the proof of our main result later.

Remark 4.4. For any ε ∈ (0, ε 0 ] and L ≥ L 1,ε = 2C 1 /(γ 1 ε), let T L = L/c * > 0 be the time provided by (4.58). For any given k ∈ N and τ > 0, the function η ε,L given by (4.66) is bounded in [0, kT L + τ ] uniformly with respect to ε ∈ (0, ε 0 ] and L ≥ L 1,ε . More precisely, we can find some constants 0 < A 1 < A 2 (independent of ε, L, k and τ ) such that

A 1 -A 2 e K 1 k/(c * β 1 )+K 1 τ γ 1 ε 0 /(2C 1 β 1 ) ≤ A 1 -A 2 e K 1 k/(c * β 1 )+K 1 τ /(Lβ 1 ) ≤ η ε,L (t) ≤ 0 for all 0 ≤ t ≤ kT L + τ .
Remark 4.5. We also point out that, for any ε ∈ (0, ε 0 ], the functions t → q ε,L (t) and t → η ε,L (t) converge locally uniformly in t ≥ 0 as L → +∞. More precisely, there holds

lim L→+∞ q ε,L (t) = ε e -γ 1 t and lim L→+∞ η ε,L (t) = -ε γ 1 + K 1 γ 1 β 1 (1 -e -γ 1 t ),
locally uniformly in t ∈ [0, +∞). Now, for any ε ∈ (0, ε 0 ], with ε 0 ∈ (0, δ 0 ) provided by Lemmas 4.2-4.3, we consider any family of continuous functions v 0 ε,y : R → [0, 1] such that

v 0 ε,y+1 ≡ v 0 ε,y in R, v 0 ε,y (0) = 1 2 and v 0 ε,y -ψ(•, y) L ∞ (R) ≤ ε for all y ∈ R. 7 (4.67)
Then, for any y ∈ R and L > 0, we denote by v ε,y,L : [0, +∞) × R → [0, 1] the solution of (4.54) with the initial condition v 0 ε,y . Since the function f is 1-periodic in its first variable, it follows that the functions v ε,y,L are 1-periodic with respect to the parameter y ∈ R. It is also easily seen

that max v - ε,y,L (0, x), 0 ≤ v 0 ε,y (x) ≤ min v + ε,y,L (0, x), 1 for all x ∈ R, (4.68)
with v ± ε,y,L are in Lemmas 4.2-4.3. Therefore, the next lemma is an immediate consequence of the comparison principle. Lemma 4.6. For any ε ∈ (0, ε 0 ], with ε 0 ∈ (0, δ 0 ) as in Lemmas 4.2-4.3, any y ∈ R, and any L ≥ L 1,ε with L 1,ε > 0 as in (4.63), there holds

max v - ε,y,L (t, x), 0 ≤ v ε,y,L (t, x) ≤ min v + ε,y,L (t, x)
, 1 for all t ≥ 0 and x ∈ R.

Asymptotic behavior of the Cauchy problem as L → +∞

In this subsection and the following one, we always let ε 0 ∈ (0, δ 0 ) be provided by Lemmas 4.2-4.3. Now, for any ε ∈ (0, ε 0 ], we consider the asymptotic behavior as L → +∞ of the solutions v ε,y,L of (4.54) with initial conditions v 0 ε,y satisfying (4.67), at a time T ε,y,L defined by

T ε,y,L := inf t > 0 : v ε,y,L (t, L) = 1 2 . ( 4.69)
Notice immediately that, since the functions v ε,y,L are 1-periodic with respect to y, so are the quantities T ε,y,L 's. Furthermore, for each y ∈ R and L ≥ L 1,ε , one has v ε,y,L (0, L) ≤ v + ε,y,L (0, L) = ψ(L, y) + ε by (4.60) and (4.68), hence lim sup

L→+∞ sup y∈R v ε,y,L (0, L) ≤ ε ≤ ε 0 < δ 0 < 1 2
since ψ(+∞, y) = 0 uniformly in y ∈ R by Proposition 4.1 (ii). Therefore, T ε,y,L > 0 for all L large enough, uniformly in y ∈ R. On the other hand, for each y ∈ R and

L ≥ L 1,ε , with T L as in (4.58), one has v ε,y,L (2T L , L) ≥ v - ε,y,L (2T L , L) ≥ ψ(-L -η ε,L (2T L ), y + 2) -ε
inf y∈R v ε,y,L (2T L , L) ≥ 1 -ε ≥ 1 -ε 0 > 1 -δ 0 > 1 2
by Remark 4.4 and since ψ(-∞, y) = 1 uniformly in y ∈ R by Proposition 4.1 (ii). As a consequence, there is

L 2,ε ≥ L 1,ε > 0 such that 0 < T ε,y,L ≤ 2T L < +∞ for all y ∈ R and L ≥ L 2,ε , (4.70)
and then v ε,y,L ( T ε,y,L , L) = 1 2 .

(4.71)

The following proposition shows that when L is sufficiently large, the profile v ε,y,L (t, • + L) at time t = T ε,y,L is close to that of the initial condition v ε,y,L (0, •) = v 0 ε,y . Proposition 4.7. For any ε ∈ (0, ε 0 ], with ε 0 ∈ (0, δ 0 ) as in Lemmas 4.2-4.3, there exists L 3,ε ≥ L 2,ε , with L 2,ε > 0 as in (4.70), such that, for any y ∈ R and L ≥ L 3,ε , there holds

v ε,y,L ( T ε,y,L , • + L) -ψ(•, y) L ∞ (R) ≤ ε 2 ,
where T ε,y,L is given by (4.69), and v ε,y,L solves (4.54) with initial conditions v 0 ε,y satisfying (4.67). For the proof of Proposition 4.7, let us first show two lemmas which are concerned with the comparison of T L and T ε,y,L . Lemma 4.8. For any fixed ε ∈ (0, ε 0 ], with ε 0 ∈ (0, δ 0 ) as in Lemmas 4.2-4.3, let T L and T ε,y,L be given by (4.58) and (4.69), respectively, for y ∈ R and L ≥ L 2,ε . Then, there is a constant

M ε ≥ 0 such that sup y∈R, L≥L 2,ε |T L -T ε,y,L | ≤ M ε ,
for all families of initial conditions v 0 ε,y ∈ C(R, [0, 1]) satisfying (4.67).

Proof. We fix ε ∈ (0, ε 0 ] throughout the proof. We first prove the existence of some real numbers

L + 2,ε ∈ [L 2,ε , +∞) and M + ε ≥ 0 such that sup y∈R, L≥L + 2,ε ( T ε,y,L -T L ) ≤ M + ε (4.72)
for all families of initial conditions v 0 ε,y ∈ C(R, [0, 1]) satisfying (4.67). Assume by contradiction, there exist sequences (y n ) n∈N ⊂ R, (L n ) n∈N ⊂ [L 2,ε , +∞), and initial conditions (v 0 ε,yn ) n∈N ⊂ C(R, [0, 1]) satisfying (4.67), such that L n → +∞ and T ε,yn,Ln -T Ln → +∞ as n → +∞. Then for each n ∈ N, by Lemma 4.6, we have

       v ε,yn,Ln (t, x) ≥ ψ x -X yn,Ln (t) -η ε,Ln (t), y n + X yn,Ln (t) L n -q ε,Ln (t) v ε,yn,Ln (t, x) ≤ ψ x -X yn,Ln (t) + η ε,Ln (t), y n + X yn,Ln (t) L n + q ε,Ln (t) (4.73)
for all t ≥ 0 and x ∈ R. We already know from (4.70) that T ε,yn,Ln ≤ 2T Ln for all n ∈ N. Next, for any n ∈ N, choosing t = T ε,yn,Ln and x = L n in the first inequality of (4.73) yields

1 2 = v ε,yn,Ln ( T ε,yn,Ln , L n ) ≥ ψ L n -X yn,Ln ( T ε,yn,Ln ) -η ε,Ln ( T ε,yn,Ln ), y n + X yn,Ln ( T ε,yn,Ln ) L n -q ε,Ln ( T ε,yn,Ln ).
Since we have assumed T ε,yn,Ln -T Ln → +∞ as n → +∞, we may assume without loss of generality that T ε,yn,Ln > T Ln = L n /c * for all n ∈ N (hence, T ε,yn,Ln → +∞ as n → +∞). Then, we have

X yn,Ln ( T ε,yn,Ln ) = X yn,Ln (T Ln ) =Ln + T ε,yn,Ln T Ln c y n + X yn,Ln (t) L n dt ≥ L n + c -( T ε,yn,Ln -T Ln ),
where c -= min z∈R c(z) > 0. Remember that the function (ξ, z)

→ ψ(ξ, z) is decreasing in ξ ∈ R. It follows that 1 2 ≥ ψ -c -( T ε,yn,Ln -T Ln ) -η ε,Ln ( T ε,yn,Ln ), y n + X yn,Ln ( T ε,yn,Ln ) L n -q ε,Ln ( T ε,yn,Ln ). (4.74)
Since T ε,yn,Ln → +∞ as n → +∞, we see from (4.65) that q ε,Ln ( T ε,yn,Ln ) → 0 as n → +∞. Furthermore, since 0 < T ε,yn,Ln ≤ 2T Ln by (4.70), one infers from Remark 4.4 that the sequence (η ε,Ln ( T ε,yn,Ln )) n∈N is bounded. Passing to the limit as n → +∞ in (4.74) and using Proposition 4.1 (ii), we obtain 1/2 ≥ 1, which is impossible. Thus, (4.72) is proved.

Similarly as above, one can prove the existence of some real numbers

L - 2,ε ∈ [L 2,ε , +∞) and M - ε ≥ 0 such that sup y∈R, L≥L - 2,ε (T L -T ε,y,L ) ≤ M - ε for all families of initial conditions v 0 ε,y ∈ C(R, [0, 1]
) satisfying (4.67), by using this time in the previous paragraph the second inequality of (4.73) instead of the first one, together with q ε,Ln ≤ ε < 1/2 in [0, +∞) by (4.60).

Finally, since 0 < T ε,y,L ≤ 2T L = 2L/c * for all y ∈ R and L ≥ L 2,ε by (4.58) and (4.70), the desired conclusion of Lemma 4.8 follows, with

M ε := max{M + ε , M - ε , 2L + 2,ε /c * , 2L - 2,ε /c * }.
Lemma 4.9. For any fixed ε ∈ (0, ε 0 ], with ε 0 ∈ (0, δ 0 ) as in Lemmas 4.2-4.3, let T L and T ε,y,L be given by (4.58) and (4.69), respectively, for y ∈ R and L ≥ L 2,ε . Then there holds

lim L→+∞ X y,L (t + T ε,y,L ) X y,L (T L ) = 1 (4.75)
and lim

L→+∞ X y,L (t + T ε,y,L ) -X y,L (T L ) -c(y)(t + T ε,y,L -T L ) = 0 (4.76)
locally uniformly with respect to t ∈ R, and uniformly with respect to y ∈ R and with respect to the families of initial conditions v 0 ε,y ∈ C(R, [0, 1]) satisfying (4.67).

Proof. For any y ∈ R, L ≥ L 2,ε , and t ∈ R, one has X y,L (T L ) = L and

X y,L (t + T ε,y,L ) X y,L (T L ) -1 = t+ T ε,y,L T L c y + X y,L (s) L ds X y,L (T L ) ≤ c + | T ε,y,L -T L + t| L ,
where c + = max z∈R c(z). Lemma 4.8 then immediately gives (4.75). It remains to show (4.76). Let y ∈ R, L ≥ L 2,ε and t ∈ R be arbitrary. Without loss of generality, we may assume that t + T ε,y,L ≥ T L (the case t + T ε,y,L ≤ T L can be treated identically). Since

X y,L (t + T ε,y,L ) -X y,L (T L ) -c(y)(t + T ε,y,L -T L ) = t+ T ε,y,L T L c y + X y,L (s)
X y,L (t + T ε,y,L ) -X y,L (T L ) -c(y)(t + T ε,y,L -T L ) ≤ c L ∞ (R) × |t + T ε,y,L -T L | × max s∈[T L , t+ T ε,y,L ] X y,L (s) X y,L (T L ) -1 .
Lemma 4.8 and (4.75) then yield (4.76).

Now, we are ready to give the

Proof of Proposition 4.7. Fix any ε ∈ (0, ε 0 ], and assume by contradiction that the conclusion is not true. Then, there exist sequences (y n We first claim that the sequence (x n -L n ) n∈N is bounded. Suppose by way of contradiction that x n -L n → +∞ as n → +∞ (as we will sketch below, the case where x n -L n → -∞ as n → +∞ can be treated analogously). Choosing t = T ε,yn,Ln (notice that T ε,yn,Ln → +∞ as n → +∞ by (4.58) and Lemma 4.8) and x = x n in the second inequality of (4.73), we have 0 ≤ v ε,yn,Ln ( T ε,yn,Ln , x n ) ≤ ψ x n -L n + X yn,Ln (T Ln ) -X yn,Ln ( T ε,yn,Ln ) + η ε,Ln ( T ε,yn,Ln ), y n + X yn,Ln ( T ε,yn,Ln ) L n +q ε,Ln ( T ε,yn,Ln ).

) n∈N ⊂ R, (L n ) n∈N ⊂ [L 2,ε , +∞), (x n ) n∈N ⊂ R,
Due to Lemma 4.8 and the definition of X yn,Ln (t), the sequence (X yn,Ln (T Ln )-X n ( T ε,yn,Ln )) n∈N is bounded. Moreover, by Remark 4.4 and (4.70), the sequence (η ε,Ln ( T ε,yn,Ln )) n∈N is also bounded. Thus, since ψ(ξ, y) → 0 as ξ → +∞ uniformly in y ∈ R by Proposition 4.1 (ii), we have

ψ x n -L n + X yn,Ln (T Ln ) -X yn,Ln ( T ε,yn,Ln ) + η ε,Ln ( T ε,yn,Ln ), y n + X yn,Ln ( T ε,yn,Ln ) L n → 0
as n → +∞. This together with q ε,Ln ( T ε,yn,Ln ) → 0 as n → +∞ (from (4.65) and Lemma 4.8) yields v ε,yn,Ln ( T ε,yn,Ln , x n ) → 0 as n → +∞, hence v ε,yn,Ln ( T ε,yn,Ln , x n ) -ψ(x n -L n , y n ) → 0 as n → +∞, which is a contradiction with (4.77).

Similarly, if x n -L n → -∞ as n → +∞, by using the first inequality of (4.73), one can conclude that v ε,yn,Ln ( T ε,yn,Ln , x n ) → 1 and ψ(x n -L n , y n ) → 1 as n → +∞, which is also a contradiction with (4.77). Therefore, the sequence (x n -L n ) n∈N is bounded.

Next, for each n ∈ N, we set w n (t, x) = v ε,yn,Ln (t + T ε,yn,Ln , x + L n ) for (t, x) ∈ [-T ε,yn,Ln , +∞) × R. Clearly, w n (0, 0) = 1/2 and w n solves

∂ t w n = a∂ xx w n + f y n + x L n , w n for t > -T ε,yn,Ln and x ∈ R.
It is also easily seen from (4.73) that, for any t ∈ R, x ∈ R and any n ∈ N such that t+ T ε,yn,Ln ≥ 0, w n (t, x) satisfies w n (t, x) ≥ ψ x+X yn,Ln (T Ln )-X yn,Ln (t+ T ε,yn,Ln )-η ε,Ln (t+ T ε,yn,Ln ), y n + X yn,Ln (t+ T ε,yn,Ln ) L n -q ε,Ln (t + T ε,yn,Ln ) and w n (t, x) ≤ ψ x+X yn,Ln (T Ln )-X yn,Ln (t+ T ε,yn,Ln )+η ε,Ln (t+ T ε,yn,Ln ), y n + X yn,Ln (t+ T ε,yn,Ln ) L n + q ε,Ln (t + T ε,yn,Ln ).

Thanks to Lemma 4.8 and the boundedness of the sequence (x n -L n ) n∈N , one can find some τ ∞ ∈ R, ξ ∞ ∈ R and y ∞ ∈ [0, 1], such that, up to extraction of some subsequence,

T ε,yn,Ln -T Ln → τ ∞ , x n -L n → ξ ∞ and y n → y ∞ as n → +∞.
Furthermore, by standard parabolic estimates and possibly up to extraction of a further subsequence, there is a function w ∞ ∈ C 1;2 t;x (R 2 ) such that w n → w ∞ in C 1;2 t;x;loc (R 2 ) as n → +∞. Clearly, w ∞ is an entire solution of then (4.79) follows immediately from Remark 4.4). Passing to the limit as n → +∞ in the above inequalities on w n (t, x), it follows from Lemmas 4.8 and 4.9, together with the continuity of the map z → c(z) and the 1-periodicity of ψ(ξ, y) with respect to y, that

∂ t w ∞ = a∂ xx w ∞ + f (y ∞ , w ∞ ) for t ∈ R and x ∈ R. ( 4
ψ (x -c(y ∞ )(t + τ ∞ ) + C, y ∞ ) ≤ w ∞ (t, x) ≤ ψ (x -c(y ∞ )(t + τ ∞ ) -C, y ∞ )
for all (t, x) ∈ R 2 . Namely, w ∞ is an entire solution of (4.78) which is trapped between two shifts of the corresponding traveling front ψ(x -c(y )t, y ∞ ). Remember that the reaction f (y ∞ , •) is of the bistable type from (A1). It then follows from [9, Theorem 3.1] that there exists x 0 ∈ R such that w ∞ (t, x) ≡ ψ(x-c(y ∞ )t+x 0 , y ∞ ) for all (t, x) ∈ R 2 . Furthermore, notice that w n (0, 0) = 1/2 for each n ∈ N, hence w ∞ (0, 0) = 1/2. This together with the normalization condition (1.10) implies that x 0 has to be 0. Therefore, we have

w ∞ (t, x) ≡ ψ(x -c(y ∞ )t, y ∞ ) for all (t, x) ∈ R 2 .
On the other hand, it follows from (4.77) that |w n (0,

x n -L n ) -ψ(x n -L n , y n )| > ε/2 for all n ∈ N. Taking the limit as n → +∞ yields |w ∞ (0, ξ ∞ ) -ψ(ξ ∞ , y ∞ )| ≥ ε/2,
which is a contradiction. The proof of Proposition 4.7 is thus complete.

In addition to the previous observations, the last step before doing the proof of Theorem 1.6 in Section 4.4 is the global stability of the pulsating fronts φ L , as stated below.

Proposition 4.10. For any ε ∈ (0, ε 0 ] with ε 0 ∈ (0, δ 0 ) as in Lemmas 4.2-4.3, and for any L ≥ L * with L * > 0 as in Theorem 1.1, let (v ε,y,L ) y∈R be the family the solutions of (4.54) with initial conditions (v 0 ε,y ) y∈R satisfying (4.67). Then, there exists a 1-periodic real-valued function y → ξ ε,y,L such that

lim t→+∞ v ε,y,L (t, •) -φ L • -c L t + ξ ε,y,L , • L + y L ∞ (R)
= 0 for every y ∈ R. (4.80)

Proof. Let ε ∈ (0, ε 0 ] ⊂ (0, δ 0 ) and L ≥ L * be fixed. For each y ∈ R, the function

u ε,y,L : (t, x) → u ε,y,L (t, x) := v ε,y,L (t, x -Ly)
is a solution of (4.67). It then follows from Theorem 1.1 (iii) that there exists a unique ξ ε,y,L ∈ R such that lim

∂ t u ε,y,L = a∂ xx u ε,y,L + f (x/L, u ε,y,L ) for t > 0 and x ∈ R, with initial condition u ε,y,L (0, •) = v 0 ε,y (•-Ly) ∈ C(R, [0, 1]) such that u ε,y,L (0, •)-ψ(•-Ly, y) ≤ ε ≤ ε 0 < δ 0 , by
t→+∞ u ε,y,L (t, •) -φ L • -c L t + ξ ε,y,L , • L L ∞ (R) = 0.
Choosing ξ ε,y,L = ξ ε,y,L + Ly, we see from the above that (4.80) holds. Lastly, since v ε,y,L and φ L (•, y) are 1-periodic in y, and since φ L (ξ, y) is decreasing with respect to ξ, it follows that ξ ε,y,L in (4.80) is unique and necessarily 1-periodic in y. The proof of Proposition 4.10 is thus complete.

Proof of Theorem 1.6

For clarity, we proceed with several steps.

Step 1: approximation of the front profiles φ L by ψ(•, y). In this step, we fix any ε ∈ (0, ε 0 ], with ε 0 ∈ (0, δ 0 ) as in Lemmas 4.2-4.3 and δ 0 ∈ (0, 1/2) as in assumption (A2). Let L 4,ε := max{L * , L 3,ε }, with L * > 0 and L 3,ε > 0 provided by Theorem 1.1 and Proposition 4.7, respectively.

We shall prove in this step that there exists a real number L 5,ε ∈ [L 4,ε , +∞) such that, for every L ≥ L 5,ε , there is a 1-periodic real-valued function y → ξ * ε,y,L such that

φ L • + ξ * ε,y,L , • L + y -ψ(•, y) L ∞ (R)
≤ ε for every y ∈ R and L ≥ L 5,ε .

(4.81)

To do so, for any y ∈ R and L ≥ L 4,ε (≥ L 3,ε ≥ L 2,ε ), let v ε,y,L be the solution of (4.54) with initial condition v 0 ε,y satisfying (4.67) (for instance, say, take here v 0 ε,y = ψ(•, y)). By Proposition 4.7 and (4.71), we have

v ε,y,L ( T ε,y,L , L) = 1 2 and v ε,y,L T ε,y,L , • + L -ψ(•, y) L ∞ (R) ≤ ε 2 ≤ ε,
where T ε,y,L is defined by (4.69)-(4.70). Since f is 1-periodic with respect to first variable, the function (t, x) → v ε,y,L (t + T ε,y,L , x + L) is still a solution of (4.54), now with the initial condition v ε,y,L ( T ε,y,L , • + L) ∈ C(R, [0, 1]). Furthermore, since both v ε,y,L and T ε,y,L are 1-periodic with respect to y, one has v ε,y+1,L ( T ε,y+1,L , • + L) ≡ v ε,y,L ( T ε,y,L , • + L) in R for each y ∈ R. By using Proposition 4.7 again, we infer that

v ε,y,L T 2 ε,y,L , • + 2L -ψ(•, y) L ∞ (R) ≤ ε 2 ≤ ε,
where T 2 ε,y,L := min t > T ε,y,L : v ε,y,L (t, 2L) = 1/2 (the quantities T 2 ε,y,L are well defined real numbers, as are the T ε,y,L 's, from the same arguments as in (4.69)-(4.71)). Then, a simple induction argument implies that, for any k ∈ N,

v ε,y,L T k ε,y,L , • + kL, y -ψ(•, y) L ∞ (R) ≤ ε 2 , (4.82)
where

T k ε,y,L = min t > T k-1 ε,y,L : v ε,y,L (t, kL) = 1/2 .
T k+1 ε,y,L -T k ε,y,L -T L < +∞,
where T L is defined by (4.58). Since T L → +∞ as L → +∞ (independently of y ∈ R), there exists L 5,ε ∈ [L 4,ε , +∞) such that T k ε,y,L → +∞ as k → +∞ uniformly in y ∈ R and L ≥ L 5,ε . On the other hand, for each L ≥ L 5,ε (≥ L 4,ε ≥ L * ), it follows from Proposition 4.10 that there exists a 1-periodic real-valued function y → ξ ε,y,L such that

lim t→+∞ v ε,y,L (t, •) -φ L • -c L t + ξ ε,y,L , • L + y L ∞ (R) = 0
for every y ∈ R. Therefore, for each L ≥ L 5,ε and y ∈ [0, 1), there is k ε,y,L ∈ N such that

v ε,y,L T k ε,y,L ε,y,L , • + k ε,y,L L -φ L • + ξ * ε,y,L , • L + y L ∞ (R) ≤ ε 2 ,
where ξ * ε,y,L := k ε,y,L L-c L T k ε,y,L ε,y,L +ξ ε,y,L (we here use the 1-periodicity of φ in its second variable). Combining this with (4.82) at k = k ε,y,L , we obtain that

φ L • +ξ * ε,y,L , • L + y -ψ(•, y) L ∞ (R) ≤ ε. (4.83)
We can then extend the function y → ξ * ε,y,L as a 1-periodic function, so that (4.83) holds for all y ∈ R, since φ L (ξ, y) and ψ(ξ, y) are 1-periodic with respect to y ∈ R. The proof of Step 1 is thus complete.

Step 2: proof of lim

ε→0 sup y∈R, L≥L 5,ε ξ * ε,y,L -ζ L (y) = 0, ( 4
φ L ξ * ε,y,L , y -φ L (ζ L (y), y) =1/2 ≤ ε ≤ ε 0 < 1 4 . ( 4

.85)

Then, by Theorem 1.8 (i)-(ii), it follows that there exists a constant B 1 > 0 such that

|ξ * ε,y,L -ζ L (y)| ≤ B 1 for all ε ∈ (0, ε 0 ], y ∈ R and L ≥ L 5,ε , (4.86)
and also that (4.84) holds.

Step 3: lim sup L→+∞ |L -c L T L | < +∞, with T L given by (4.58). 8 Notice that for any y ∈ R and

L ≥ L 5,ε 0 ≥ L 4,ε 0 = max{L * , L 3,ε 0 }, the function (t, x) → φ L x -c L t + ξ * ε 0 ,y,L , x L + y = U L t - ξ * ε 0 ,y,L c L + Ly c L , x + Ly
is the solution of (4.54) with initial condition φ L (• + ξ * ε 0 ,y,L , •/L + y). Since L ≥ L 5,ε 0 ≥ L 1,ε 0 , it follows from (4.81) and Lemmas 4.2-4.3 (with their notations) that

v - ε 0 ,y,L (t, x) ≤ φ L x -c L t + ξ * ε 0 ,y,L , x L + y ≤ v + ε 0 ,y,L (t, x) for all t ≥ 0 and x ∈ R. (4.87)
In particular, choosing t = T L and x = L yields

ψ(-η ε 0 ,L (T L ), y) -q ε 0 ,L (T L ) ≤ φ L (L -c L T L + ξ * ε 0 ,y,L , y) ≤ ψ(η ε 0 ,L (T L ), y) + q ε 0 ,L (T L ).
Since lim L→+∞ q ε 0 ,L (T L ) = 0 by (4.65) and since sup L≥L 5,ε 0 |η ε 0 ,L (T L )| < +∞ by Remark 4.4, it then follows from Proposition 4.1 (namely, the continuity of ψ : R 2 → (0, 1) and its 1-periodicity in its second variable) that there exist κ 0 ∈ (0, 1/2) and L ≥ L 5,ε 0 such that

κ 0 ≤ φ L L -c L T L + ξ * ε 0 ,y,L , y ≤ 1 -κ 0 for all y ∈ R and L ≥ L.
Hence, by Theorem 1.8 (i), there exists B 2 > 0 such that

L -c L T L + ξ * ε 0 ,y,L -ζ L (y) ≤ B 2 for all y ∈ R and L ≥ L.
This together with (4.86) implies that |L -c L T L | ≤ B 1 + B 2 for all L ≥ L, which yields the desired result.

Step 4: proof the convergence (1.16). We fix any σ > 0. We have to show the existence of L * σ > 0 such that for any

L ≥ L * σ , sup y∈R φ L • + ζ L (y), • L + y -ψ(•, y) L ∞ (R) ≤ σ. ( 4
:= sup L≥L * ∂ ξ φ L L ∞ (R 2 ) = sup L≥L * c -1 L ∂ t U L L ∞ (R 2 ) ∈ [0, +∞). Next, by Step 2, there is ε σ ∈ (0, min{ε 0 , σ/2}] such that M × |ξ * εσ,y,L -ζ L (y)| ≤ σ 2 for all y ∈ R and L ≥ L 5,εσ .
Hence,

φ L • + ζ L (y), • L + y -φ L • + ξ * εσ,y,L , • L + y L ∞ (R) ≤ σ 2 for all y ∈ R and L ≥ L 5,εσ .
Then, (4.81) (applied with ε σ ) and the inequality ε σ ≤ σ/2 give (4.88) for all L ≥ L * σ := L 5,εσ > 0. This ends the proof of Step 4.

Step 5: the convergence (1.17) holds locally uniformly in x ∈ R. Let us first show that, for any given A > 0, sup Clearly, t y,L,x ≥ 0. It is easily seen from the definition of X y,L (t) in (4.57) that X y,L (t y,L,x ) = x. Choosing x and t = t y,L,x in (4.87) (these inequalities actually hold as in Step 3 with ε 0 replaced by any ε ∈ (0, ε 0 ], and for any L ≥ L 5,ε ) gives

y∈R, L≥L 5,ε , x∈[0,A] ξ * ε,y,L + L y+x/L y 1 - c L c(s) ds -ζ L y + x L → 0 as ε → 0. ( 4
ψ -η ε,L (t y,L,x ), y + X y,L (t y,L,x ) L -q ε,L (t y,L,x ) ≤ φ L L y+x/L y 1 - c L c(s) ds + ξ * ε,y,L , x L + y ≤ ψ η ε,L (t y,L,x ), y + X y,L (t y,L,x ) L + q ε,L (t y,L,x ).
0, y + X y,L (t y,L,x ) L = φ L ζ L x L + y , x L + y = 1 2 .
It then follows that Step 6: proof of the estimate (1.18). By (4.84) and the continuity of the map (L, y) → ζ L (y) in [L * , +∞)×R and its 1-periodicity in y, in order to show (1.18), it suffices to prove the existence of ε

φ L L y+x/L y 1 - c L c(s) ds + ξ * ε,y,L , x L + y -φ L ζ L x L + y , x L + y ≤ max ψ η ε,L (t y,L,x ), y + X y,L (t y,L,x ) L -ψ 0, y + X y,L (t y,L,x ) L , ψ -η ε,L (t y,L,x ), y + X y,L (t y,L,x ) L -ψ 0, y + X y,L (t y,L,x ) L +q ε,L (t y,L,x
ζ L y + x + A L -ζ L y + A L -L y+(x+A)/L y+A/L 1 - c L c (
1 ∈ (0, ε 0 ] such that sup ε∈(0,ε 1 ], y∈R, L≥L 5,ε ξ * ε,y,L + L y+•/L y 1 - c L c(s) ds -ζ L y + • L L ∞ ([-L,L]) < +∞. ( 4

.93)

Consider any ε ∈ (0, ε 0 ], y ∈ R, L ≥ L 5,ε , and x ∈ [0, L]. Let t y,L,x be as in (4.90). There holds 0 ≤ t y,L,x ≤ T L and X y,L (t y,L,x ) = x, since X y,L (T L ) = L ≥ x and X y,L is increasing. It is also easily seen that the inequalities (4.91) remain valid. Moreover, by (4.60) and Remark 4.4, we have 0 < q ε,L (t y,L,x ) ≤ ε and -B 3 ≤ η ε,L (T L ) ≤ η ε,L (t y,L,x ) ≤ 0, where B 3 is a positive constant independent of ε, y, L and x. Then, by the continuity of the map ψ : R 2 → (0, 1) and its 1-periodicity with respect to its second variable, there exist ε 1 ∈ (0, ε 0 ] and κ 1 ∈ (0, 1/2) such that same token the continuity of the functions ψ(•, y) in C 2 loc (R) with respect to y. More precisely, let us first show that the family (c(y)) y∈R is bounded. Since the function y → 1 0 f (y, u)du is positive, continuous and 1-periodic, and since f is bounded in R × [0, 1], there exists s 0 ∈ (0, 1) such that s 0 0 f (y, u)du > 0 for all y ∈ R. For each y ∈ R, let ξ 0 (y) be the unique real number such that ψ(ξ 0 (y), y) = s 0 . Then, integrating the first equation of (1.9) against ∂ ξ ψ(•, y) over [ξ 0 (y), +∞), we obtain

κ 1 ≤ φ L L y+x/L y 1 - c L c(s) ds + ξ * ε,y,L , x L + y ≤ 1 -κ 1 for all ε ∈ (0, ε 1 ], y ∈ R, L ≥ L 5,
L = L/c * that ξ * ε,y,L -ζ L y + x L + L y+x/L y 1 - c L c(s) ds = ξ * ε,y,L -ζ L y + x + L L + L y+(x+L)/L y 1 - c L c(s) ds -(L -c L T L ) ≤ B 4 + |L -c L T L |, for all ε ∈ (0, ε 1 ], y ∈ R, L ≥ L 5,
(∂ ξ ψ(ξ 0 (y), y)) 2 ≥ 2 a(y) s 0 0 f (y, u)du > 0.
Letting ξ 1 (y) ∈ R be such that ∂ ξ ψ(ξ 1 (y), y) = min ξ∈R ∂ ξ ψ(ξ, y) (such a ξ 1 (y) exists since ∂ ξ ψ(±∞, y) = 0 and the function ∂ ξ ψ(•, y) is negative and continuous), we get ∂ ξξ ψ(ξ 1 (y), y) = 0. Since ψ(ξ, y) is decreasing in ξ ∈ R and since c(y) > 0, it follows that

0 < c(y) = f (y, ψ(ξ 1 (y), y)) ∂ ξ ψ(ξ 1 (y), y) ≤ max x∈R, u∈[0,1] |f (x, u)| -∂ ξ ψ(ξ 0 (y), y) ≤ max x∈R, u∈[0,1] |f (x, u)| min x∈R 2a -1 (x) s 0 0 f (x, u)du .
Since the last quantity is a real number independent of y ∈ R, this completes the proof of the boundedness of (c(y)) y∈R .

Next, we show that c(y n ) → c(y 0 ) as n → +∞. Due to the periodicity, we may assume, without loss of generality, that (y n ) n∈N ⊂ [0, 1] and y 0 ∈ [0, 1]. Assume by contradiction that c(y n ) does not converge to c(y 0 ) as n → +∞. Then, we can find some c ∞ ∈ [0, c + ] such that c ∞ = c(y 0 ) and, up to extraction of some sequence, c(y n ) → c ∞ as n → +∞.

Remember that x → b(x) ∈ (0, 1) is the function given in (A1) such that f (x, b(x)) = 0 for all x ∈ R. Pick any ε 0 ∈ (0, b(y 0 )). For each n ∈ N, let z n be the unique real number such that ψ(z n , y n ) = ε 0 . Writing ψ n (ξ) = ψ(ξ + z n , y n ), we obtain that ψ n (ξ) is decreasing in ξ ∈ R and

a(y n )ψ n + c(y n )ψ n + f (y n , ψ n ) = 0 in R, 0 < ψ n < 1 in R and ψ n (0) = ε 0 .
By the standard elliptic estimates, we find a C 2 (R) nonincreasing function 0 ≤ ψ ∞ ≤ 1 such that, up to extraction of some subsequence, ψ n → ψ ∞ in C 2 loc (R) as n → +∞, and the function ψ ∞ solves a(y 0 )ψ ∞ + c ∞ ψ ∞ + f (y 0 , ψ ∞ ) = 0 in R, and ψ ∞ (0) = ε 0 .

By the strong maximum principle, we have 0 < ψ ∞ < 1 in R. Since ψ ∞ is nonincreasing, the limits ψ ∞ (±∞) := lim ξ→±∞ ψ ∞ (ξ) exist and they are zeros of f (y 0 , •). Since 0 < ψ ∞ (0) = ε 0 < b(y 0 ), and since 0, 1, b(y 0 ) are the only three zeros of f (y 0 , •), it is easily checked that ψ ∞ (+∞) = 0 and that either ψ ∞ (-∞) = 1 or ψ ∞ (-∞) = b(y 0 ). In other words, either ψ ∞ (x -c ∞ t) is a traveling front of (1.8) with y = y 0 connecting 0 and 1, or it is a traveling front connecting 0 and b(y 0 ). The former case is impossible, since the uniqueness of front speed of bistable traveling front [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF] would give c ∞ = c(y 0 ), a contradiction with our assumption c ∞ = c(y 0 ). The latter case leads to a contradiction as well, since equation (1.8) (with y = y 0 ) restricted to u ∈ [0, b(y 0 )] is of the monostable type and it is well known [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF] that there exists a constant c * > 0 such that a (nonincreasing) traveling front with speed c connecting b(y 0 ) and 0 exists if and only if c ≤ -c * . This contradicts the fact that c ∞ ≥ 0. Therefore, c(y n ) → c(y 0 ) as n → +∞. Finally, we also notice that, from the strong maximum principle applied to the nonpositive function ψ ∞ , one has ψ ∞ < 0 in R. Let then z ∞ ∈ R be the unique point such that ψ(z ∞ , y 0 ) = ε 0 . Then, by the uniqueness of traveling wave of (1.8) connecting 0 and 1, it is easily seen from the above proof that the whole sequence (ψ n ) n∈N converges to ψ(• + z ∞ , y 0 ) in C 2 loc (R) as n → +∞. This implies in particular that z n → z ∞ as n → +∞. Therefore, by the normalization condition (1.10), we obtain ψ(•, y n ) → ψ(•, y 0 ) in C 2 loc (R) as n → +∞. This ends the proof of Lemma 5.1.

We are now ready to give the Proof of Proposition 4.1 (ii)-(iii). We first prove that ψ(ξ, y) tends to its limits as ξ → ±∞ uniformly in y ∈ R. We only consider the convergence as ξ → +∞, since the proof of the other one is identical. Assume by contradiction that there exist ε 1 ∈ (0, 1) and a sequence (ξ n , y n ) n∈N ⊂ R 2 such that ξ n → +∞ as n → +∞ and ψ(ξ n , y n ) ≥ ε 1 for each n ∈ N. Due to the periodicity, without loss of generality, we may assume that (y n ) n∈N ⊂ [0, 1] and that y n → y ∞ as n → +∞ for some y ∞ ∈ [0, 1]. Let ξ ∞ ∈ R be the unique point such that ψ(ξ ∞ , y ∞ ) = ε 1 /2. Since ξ n → +∞ as n → +∞, we have ξ n > ξ ∞ for all large n, whence ψ(ξ ∞ , y n ) > ψ(ξ n , y n ) ≥ ε 1 , since ψ(•, y n ) is decreasing in R. By Lemma 5.1, passing to the limit as n → +∞, we obtain ψ(ξ ∞ , y ∞ ) ≥ ε 1 , which is impossible. Therefore, ψ(ξ, y) → 0 as ξ → +∞ uniformly in y ∈ R.

Next, we show that ψ(ξ, y) decays to its limiting states as ξ → ±∞ exponentially fast uniformly in y ∈ R. We only give the proof of the first inequality of (4.55), since that of the second one is analogous. Since ψ(ξ, y) → 0 as ξ → +∞ uniformly in y, there exists some M > 0 (independent of y) such that 0 < ψ(ξ, y) ≤ δ 0 for all ξ ≥ M and y ∈ R, where δ 0 ∈ (0, 1/2) is the positive constant provided by (A2). Then, by the first line of (1.2), for each y ∈ R, we have a(y)∂ ξξ ψ(ξ, y) + c(y)∂ ξ ψ(ξ, y) -γ 0 ψ(ξ, y) ≥ 0 for all ξ ≥ M.

Since the functions y → a(y) and y → c(y) are continuous (see Lemma 5.1), positive and 1periodic, we can choose a constant µ 1 such that 0 < µ 1 ≤ min y∈R c(y) + c 2 (y) + 4γ 0 a(y) 2a(y) .

It is then easily checked that a(y)(-µ 1 ) 2 + c(y) × (-µ 1 ) -γ 0 ≤ 0 for all y ∈ R. Define v(ξ) = C 1 e -µ 1 ξ , where C 1 = δ 0 e µ 1 M . Clearly, for each y ∈ R, one has a(y)v (ξ) + c(y)v (ξ) -γ 0 v(ξ) ≤ 0 for all ξ ≥ M . Then, by the elliptic weak maximum principle, we obtain the first inequality of (4.55). The second one can be proved in a similar way by using the second line of (1.2). Finally, statement (iii) is an easy consequence of statements (i)-(ii) and the fact that ψ : R 2 → (0, 1) is continuous in R 2 (due to Lemma 5.1). This ends the proof. Remark 5.2. Before proceeding with the proof of statement (iv), we collect some easy corollaries of the estimate (4.55), which will be used frequently later. First of all, by (4.55) and standard elliptic estimates applied to (1.9) and its derivative, we can find a constant C 1 > 0 (independent of y ∈ R) such that √ c 2 (y)+4γ 0 a(y))/(2a(y))) ξ , 1} for all (ξ, y) ∈ R 2 .

(5.97)

Therefore, w * (ξ, y) decays no slower than e -µ 1 ξ as ξ → +∞, where µ 1 is given by µ 1 = min y∈R -c(y) + c 2 (y) + 4γ 0 a(y) 2a(y) > 0, and, as above, there is a constant C 3 > 0 such that w * (•, y) H 2 (R) ≤ C 3 for all y ∈ R. Now, we turn to the proof of statement (iv) of Proposition 4.1. The proof shares some similarities with the proof of [18, Theorem 1.2] which established the existence of pulsating fronts of (1.1) when L is small and the convergence of those fronts as L → 0 by using the implicit function theorem (see also [START_REF] Heinze | Wave solutions to reaction-diffusion systems in perforated domains[END_REF]Theorem 1.1] for the study of pulsating fronts in perforated domains). Here, we will apply the implicit function theorem under a similar setting to show the C 1 -smoothness of the homogeneous fronts (ψ(•, y), c(y)) with respect to y ∈ R.

Let us first introduce a family of auxiliary operators. In the sequel, we fix a real number β > 0. For any c ∈ R and y ∈ R, we define M c,y (v) = a(y)v + cv -βv for v ∈ H 2 (R).

Clearly, each M c,y maps H 2 (R) into L 2 (R). In the following lemma, we present some basic properties of this operator. (5.98) (ii) For any g ∈ L 2 (R), M -1 cn,yn (g n ) → M -1 c,y (g) in H 2 (R) (5.99)

as n → +∞ for all sequences (g n ) n∈N ⊂ L 2 (R), (c n ) n∈N ⊂ R, (y n ) n∈R ⊂ R such that g n -g L 2 (R) → 0, c n → c and y n → y as n → +∞. Furthermore, the above convergence is uniform in (g, c, y) ∈ B A × R for any A > 0, where B A is the ball given by B A = {(g, c) ∈ L 2 (R) × R : g L 2 (R) + |c| ≤ A}.

Proof. The proof of the invertibility of M c,y follows from similar arguments used in the proof of [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]Lemma 3.1]; therefore, we omit the details. To obtain the estimate (5.98), let v = M -1 c,y (g). Integrating M c,y (v) = g against v over R gives R a(y)(v

) 2 + βv 2 = - R gv, whence R a(y)(v ) 2 + β 2 v 2 ≤ 1 2β R g 2 .
This together with M c,y (v) = g implies that

v L 2 (R) ≤ 1 β g L 2 (R) , v L 2 (R) ≤ 1 2β min x∈R a(x) g L 2 (R)
for all ( v, c, y) ∈ H 2 (R) × R × R, where We will apply the implicit function theorem for the function G : H 2 (R) × R × R → H 2 (R) × R and show the C 1 -smoothness of the functions y → c(y) and y → ψ(•, y) at y = y 0 . To do so, we need the following lemma.

         ∂ v G 1 (v,
Lemma 5.4. The operator Q y 0 = ∂ (v,c) G(0, c(y 0 ), y 0 ) : H 2 (R) × R → H 2 (R) × R is invertible. Furthermore, there exists a constant C > 0 independent of y 0 such that for any ( g, d) ∈ H 2 (R)×R,

Q -1 y 0 ( g, d) H 2 (R)×R ≤ C ( g, d) H 2 (R)×R , (5.100)
where the space H 2 (R) × R is endowed with norm ( g, d) H 2 (R)×R = g H 2 (R) + | d|.

Proof. Let H be the linearization of (1.9) with respect to ψ at (ψ(•, y), c(y)) with y = y 0 . Namely, we define H(v) = a(y 0 )v + c(y 0 )v + ∂ u f (y 0 , ψ(•, y 0 ))v for v ∈ H 2 (R).

By [START_REF] Heinze | Wave solutions to reaction-diffusion systems in perforated domains[END_REF], the operator H and the adjoint operator H * , given by H * (v) = a(y 0 )v -c(y 0 )v + ∂ u f (y 0 , ψ(•, y 0 ))v for v ∈ H 2 (R), have algebraically simple eigenvalue 0. Moreover, the kernel ker(H) is equal to R∂ ξ ψ(•, y 0 ) and ker(H * ) is equal to Rw * , with w * := w * (•, y 0 ) given by (5.96). Moreover, it is also known from [START_REF] Heinze | Wave solutions to reaction-diffusion systems in perforated domains[END_REF] that the range of H is closed in L 2 (R). Based on these properties, one can conclude that the operator Q y 0 : H 2 (R) × R → H 2 (R) × R is invertible. The proof follows the same lines as those used in [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]Lemma 3.4] and [START_REF] Heinze | Wave solutions to reaction-diffusion systems in perforated domains[END_REF]Lemma 2.4]; therefore, we do not repeat the details here. Next, we show the estimate (5.100) (with a constant C independent of y 0 ). To do so, by linearity, it is sufficient to show that sup

Corollary 1 . 3 .

 13 Let (A1)-(A2) hold. Then, we have (i) if (A3) holds, then c L > 0 for all large L, and c L → c * as L → +∞; (ii) if 1 0 f (x, u)du < 0 and ∂ u f (x, b(x)) > 0 for all x ∈ R, then c L < 0 for all large L, and c L → c * as L → +∞; (iii) if the functions a, ∂ u f (•, 0) and ∂ u f (•, 1) are constants and if

  Therefore, the Cauchy-Schwarz inequality yields c 0 ≥ c * and even c 0 > c * as soon as b is not constant. This is in sharp contrast with the inequality lim L→0 c * L ≤ lim L→+∞ c * L under the Fisher-KPP assumption (1.7). On the other hand, if b ≡ b ∈ (0, 1/2) is constant, it follows from [18, Theorems 1.2 and 1.4] that equation (1.1) admits a pulsating front with speed c L for every small L > 0, 4 and that c L → c 0 as L → 0, where c 0 = √ 2a H (1/2 -b) is the unique speed of a traveling front connecting 0 and 1 for the homogeneous equation u t = a H u xx + u(1 -u)(u -b), with a H = ( 1 0 a(y) -1 dy) -1 being the harmonic mean of the diffusion coefficient a. With the notations (1.9), one has c(y) = 2a(y) (1/2 -b) for each y ∈ R, and Theorem 1.2 implies that

Theorem 1 . 8 .

 18 Let (A1)-(A3) hold. For each L ≥ L * , with L * as in Theorem 1.1, let φ L be a front profile of (1.1) and let ζ L be defined by (1.15) and ζ L (0) = 0. Then, (i) the following two convergences hold uniformly in L ≥ L * and y ∈ R: lim ξ→-∞ φ L (ξ + ζ L (y), y) = 1 and lim ξ→+∞ φ L (ξ + ζ L (y), y) = 0;

  we set SGN [w] = [sgn(w)] (where sgn(w) ∈ {-, +} is the classical sign function), and we set SGN [w] = [ ] (the empty word) if w ≡ 0. By definition, the length of the word SGN [w] is equal to Z[w] + 1.

  (1.22) holds true with β > 0 independent of L ≥ L * . Assume by contradiction that there exist sequences ( Ln ) n∈N ⊂ [L * , +∞) and ( tn , xn ) n∈N ⊂ R 2 such that UL n ( tn , xn ) ∈ [δ, 1 -δ] for each n ∈ N, and ∂ t UL n ( tn , xn ) → 0 as n → +∞.Due to (1.5), we may assume without loss of generality that 0 ≤ xn < Ln . By Theorem 1.1 (ii) (iv) and standard parabolic estimates, it follows that Ln → +∞ as n → +∞. Define now Ūn (t, x) := UL n (t + tn , x + xn ) for (t, x) ∈ R 2 .

Proof of Theorem 1 . 2 .

 12 Assume by contradiction that c L dose not converge to c * as L → +∞. Since the family (c L ) L≥L * ⊂ (0, +∞) is bounded by Theorem 1.1 (i), one finds a sequence (L n ) n∈N ⊂ [L * , +∞) with L n → +∞ as n → +∞ and a real number c ∞ ≥ 0 such that c Ln → c ∞ as n → +∞, and c ∞ = c * .

Proposition 4 . 1 .

 41 Let (A1)-(A3) hold. Then, the following statements hold true:(i) the function y → c(y) is positive, 1-periodic, and the function (ξ, y) → ψ(ξ, y) ∈ (0, 1) is 1-periodic in y ∈ R and decreasing in ξ ∈ R;

  iv) the function y → c(y) is of class C 1 (R), and the function (ξ, y) → ψ(ξ, y) is of class C 2;1 ξ;y (R 2 ), and satisfies sup ξ∈R, y∈R |∂ y ψ(ξ, y)| < +∞. (4.56)

  L -c(y) ds, and since the function z → c(z) is 1-periodic and of class C 1 by Proposition 4.1 (iv), we have c(y) = c(y + 1) and

1 L<

 1 .88)To do so, we first recall that the map L → c L is continuous and positive from Theorem 1.1 (i) and (iv), and that c L → c * > 0 by Theorem 1.2 (or by Step 3 of the present proof), hence sup L≥L * c -+∞. It then follows from standard parabolic estimates that M

  .89) For any ε ∈ (0, ε 0 ], y ∈ R, L ≥ L 5,ε and x ∈ [0, A], define t y,L,x := L

( 4 .

 4 91)Moreover, by the normalization conditions (1.10) and (1.15), we have ψ

  s) ds → 0 as ε → 0. Replacing y by y + A/L, we obtain (1.17) in the case where x ∈ [-A, 0]. The proof of Step 5 is thus compete.

  ε , and x ∈ [-L, 0]. By the conclusion of Step 3 and the continuity of c L and T L with respect to L, the last quantity is then bounded uniformly with respect to ε ∈ (0, ε 1 ], y ∈ R, L ≥ L 5,ε , and x ∈ [-L, 0]. Combining the above, we obtain (4.93), hence (1.18) is proved. This ends the proof of Theorem 1.6.

1

  -ψ(•, y) L 2 ((-∞,0)) + ψ(•, y) L 2 ((0,+∞)) + ∂ ξ ψ(•, y) H 2 (R) ≤ C 1 (5.95) for all y ∈ R. Furthermore, for ξ ∈ R and y ∈ R, letting w * (ξ, y) := exp c(y) a(y) ξ ∂ ξ ψ(ξ, y), (5.96) one observes from the proof of (4.55) and standard elliptic estimates that there is a constant C 2 > 0 such that |∂ ξ ψ(ξ, y)| ≤ C 2 min{e -((c(y)+

Lemma 5 . 3 .

 53 Fix β > 0. We have (i) For any c ∈ R and y ∈ R, the operator M c,y :H 2 (R) → L 2 (R) is invertible. Furthermore,for every A > 0, there exists a constant C > 0 (depending on β and A, but independent of y) such that for any c ∈ [-A, A], y ∈ R and g ∈ L 2 (R),M -1 c,y (g) H 2 (R) ≤ C g L 2 (R) .

  c, y)( v) = v + M -1 c,y ∂ u f (y, v + ψ(•, y 0 )) v + β v , ∂ c G 1 (v, c, y)( c) = -c M -1 c,y ∂ ξ M -1 c,y (K(v, c, y)) -ψ(•, y 0 ) , ∂ y G 1 (v, c, y)( y) = -y M -1 c,y a (y)∂ ξξ M -1 c,y (K(v, c, y)) -ψ(•, y 0 ) -∂ x f (y, v + ψ(•, y 0 )) ,and∂ v G 2 (v, c, y)( v) = v(0), ∂ c G 2 (v, c, y)( c) = 0, ∂ y G 2 (v, c, y)( y) = 0.

y 0

 0 ∈R, ( g, d)∈SQ -1 y 0 ( g, d) H 2 (R)×R < +∞,whereS := {( g, d) ∈ H 2 (R) × R : ( g, d) H 2 (R)×R = 1}. For any y 0 ∈ R and ( g, d) ∈ S, set ( v, c) = Q -1 y 0 ( g, d) and w = v -g. It then follows from the definition of Q y 0 that w = -M -1 c(y 0 ),y 0 ∂ u f (y 0 , ψ(•, y 0 )) v + β v -cM -1 c(y 0 ),y 0 ∂ ξ ψ(•, y 0 ) ,whence, owing to the definition of H,H( w) = -∂ u f (y 0 , ψ(•, y 0 )) g -β g -c∂ ξ ψ(•, y 0 ).Testing this equation with w * (•, y 0 ) ∈ ker(H * ) (which is given by (5.96)), we obtain c R ∂ ξ ψ(ξ, y 0 ) w * (ξ, y 0 ) dξ = -R ∂ u f (y 0 , ψ(ξ, y 0 )) g(ξ) + β g(ξ) w * (ξ, y 0 ) dξ.

  solution of (2.23) with w(0) = δ. It further follows from Lemma 2.3 that this solution must satisfy Z[w -1] = 0, w < 1 in R, and Z[w] ≤ 1 (more precisely, only cases (b) or (c) are possible).Now, to complete the proof, it remains to show the estimate (2.31). Since 0 ≤ w n ≤ δ in [0, n], by the first line of (1.2), we have, for each n ∈ N,

  by Theorem 1.1 (i). This together with(2.44) immediately implies that the interval E Next, we show statement (ii). It follows from (2.44) that for any (ξ, y) ∈ R×R with ξ ∈ E L (y), we have -ξ/c L ∈ I L (Ly) and

L (y) is bounded uniformly in y ∈ R and L ≥ L * . The proof of statements (i) of Theorem 1.8 is thus complete.

  and initial conditions (v 0 ε,yn ) n∈N ⊂ C(R, [0, 1]) satisfying (4.67) such that L n → +∞ as n → +∞, andv ε,yn,Ln ( T ε,yn,Ln , x n ) -ψ(x n -L n , y n ) >

	ε 2	for all n ∈ N.	(4.77)
	Remember that, for each n ∈ N, the inequalities stated in (4.73) hold true. By 1-periodicity
	of the functions v ε,y,L and ψ(•, y) with respect to y, we may assume without loss of generality
	that (y

n ) n∈N ⊂ [0, 1].

  .78) Notice that for any t ∈ R, lim n→+∞ q ε,Ln (t + T ε,yn,Ln ) = 0 and there exists a constant C > 0 independent of t such that-C ≤ lim inf n→+∞ η ε,yn (t + T ε,yn,Ln ) ≤ lim sup n→+∞ η ε,yn (t + T ε,yn,Ln ) ≤ 0 (4.79) (indeed, since the sequence (T Ln -T ε,yn,Ln ) n∈N is bounded by Lemma 4.8 and since T Ln → +∞ as n → +∞, it follows that, for any t ∈ R, one has 0 ≤ t + T ε,yn,Ln ≤ 2T Ln for all large n;

  ).By definition of t y,L,x , we have t y,L,x /L → 0 as L → +∞ uniformly in y ∈ R and x ∈ [0, A]. Furthermore, thanks to (4.60) and (4.66), and since L 5,ε ≥ L 1,ε → +∞ as ε → 0 by (4.63), one infers that sup

												(4.92)
	y	y+x/L	1 -	c L c(s)	ds + ξ * ε,y,L ,	x L	+ y -φ L ζ L	x L	+ y ,	x L	+ y	→ 0
									=1/2
	as ε → 0. This together with Theorem 1.8 (i)-(ii) implies (4.89) (the proof is actually similar
	to that of Step 2; therefore, we omit the details). Combining (4.84) and (4.89), we immediately
	obtain that (1.17) holds in the case where x ∈ [0, A].
	It remains to consider the case where x ∈ [-A, 0]. Clearly, the convergence (4.89) holds with x
	replaced by A and by x + A ∈ [0, A] as well. Since ξ * ε,y,L is independent of the variable x, it then
	follows that
	sup
	y∈R, L≥L 5,ε , x∈[-A,0]

y∈R, L≥L 5,ε , x∈[0,A] q ε,L (t y,L,x ) ≥0 +|η ε,L (t y,L,x )| → 0 as ε → 0.

Passing to the limit as ε → 0 in (4.92) and using the boundedness of ∂ ξ ψ in R 2 (which itself follows from standard elliptic estimates), one infers that sup y∈R, L≥L 5,ε , x∈[0,A] φ L L

  ε , and x ∈ [0, L]. It further follows from Theorem 1.8 that there exists B 4 ∈ [0, +∞) such that

	sup ε∈(0,ε 1 ], y∈R, L≥L 5,ε , x∈[0,L]	ξ * ε,y,L + L	y	y+x/L	1 -	c L c(s)	ds -ζ L y +	x L	≤ B 4 .	(4.94)

This means that (4.93) holds true with [-L, L] replaced by [0, L]. Next, for any x ∈ [-L, 0], we have x+L ∈ [0, L]. Since z → ζ L (z) and z → c(z) are 1-periodic, it then follows from (4.94) and the formula T

Together with (A1) and the C 1 smoothness of f , condition (A3) implies that b ∈ C 1 (R).

This case is named after the seminal works[START_REF] Fisher | The advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] on homogeneous equations.

In multidimensional periodic media, even if the minimal speed c * L (e) of pulsating fronts in a given direction e is not monotone in general with respect to L for equations with periodic drifts, the existence of a finite limit of c * L (e) as L → +∞ is still known, see[35,

[START_REF] Nadin | The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator[END_REF]. We also refer to[START_REF] Alfaro | Varying the direction of propagation in reaction-dffusion equations in periodic media[END_REF][START_REF] Ding | Admissible speeds in spatially periodic bistable reaction-diffusion equations[END_REF][START_REF] Giletti | Pulsating solutions for multidimensional bistable and multistable equations[END_REF][START_REF] Guo | Propagating speeds of bistable transition fronts in spatially periodic media[END_REF] for further results on the dependence of the minimal or unique speeds of multidimensional pulsating monostable or bistable fronts with respect to the direction e.

Notice that, in this case, pulsating fronts are even known to exist for all L > 0 if the function a is uniformly close to a constant[27,

[START_REF] Xin | Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity[END_REF].

The steepness is understood in the following sense: for any two entire solutions u1 : R × R → (0, 1) and u2 : R × R → (0, 1) of (1.1), we say that u1 is steeper than u2 provided that SGN [u1(t + t , •) -u2(t, •)] [+, -] for any t and t in R.

An alternate approach could be to use[START_REF] Gärtner | Bistable reaction-diffusion equations and excitable media[END_REF] Theorem 4.1] with L-dependent initial conditions φL(L•, •) combined with Theorem 1.8. We here use Lemma 3.1 and the comparisons established in Lemma 3.2, since the method of proof based on sub-and super-solutions will serve as an archetype to get more involved comparisons in Section 4.2.

A typical example is given by: v 0 ε,y = ψ(•, y) for y ∈ R, but other functions v 0 ε,y will be used in the proof of Theorem 1.6 in Section 4.4.

With this property, we here recover that cL → c * as L → +∞.

* This work has received funding from Excellence Initiative of Aix-Marseille Université -A*MIDEX, a French "Investissements d'Avenir" programme, and from the French ANR RESISTE (ANR-18-CE45-0019) project. The first author is partly supported by NSFC (12001206). The second author acknowledges support of the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne Université), and LabEx CARMIN (ANR-10-LABX-59-01). The third author is partially supported by the NSFC (11971454).

Appendix

The appendix is devoted to the proof of Proposition 4.1. The periodicity of the functions y → c(y) and (ξ, y) → ψ(ξ, y) with respect to y follows easily from the assumption that a(y) and f (y, u) are 1-periodic in y and the fact that (ψ(•, y), c(y)) is the unique solution to (1.9)-(1.10). Since c(y) and 1 0 f (y, u)du have the same sign, the assumption (A3) immediately implies that c(y) > 0 for all y ∈ R. Moreover, it is well known (see e.g., [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF]) that ψ(ξ, y) is decreasing in ξ ∈ R. Thus, statement (i) of Proposition 4.1 is proved. Now, we show statements (ii)-(iv) of Proposition 4.1. Let us first establish the continuity of c(y) and ψ(•, y) with respect to y, as stated in the following lemma.

Lemma 5.1. For any sequence (y n ) n∈N ⊂ R such that y n → y 0 as n → +∞ for some y 0 ∈ R, we have c(y n ) → c(y 0 ) as n → +∞, and ψ(•, y n ) → ψ(•, y 0 ) in C 2 loc (R) as n → +∞.

Proof. The continuity of the speeds c(y) with respect to y could alternatively be derived from [START_REF] Gärtner | Bistable reaction-diffusion equations and excitable media[END_REF]Proposition 2.3]. We here provide a short proof based on explicit estimates and we show with the and

Thus, part (i) of Lemma 5.3 follows.

Next, we prove the convergence (5.99). For each n ∈ N, let u n = M -1 cn,yn (g n ) and u = M -1 c,y (g). Let A > 0 be fixed and (g, c, y) ∈ B A × R. Without loss of generality, we may assume that (g n , c n , y n ) ∈ B 2A × R for all n ∈ N. By the proof of (5.98), we have u

where C 1 is a positive constant depending only on β and A. Notice that

for all n ∈ N. By the proof of (5.98) again, we find a positive constant C 2 depending only on β and A such that

for all n ∈ N. This implies that (5.99) holds uniformly in (g, c, y) ∈ B A × R. The proof of Lemma 5.3 is thus complete.

In addition to β > 0, we also consider in the sequel an arbitrary real number y 0 . Now, for any (v, c, y)

Clearly, K(0, c(y 0 ), y 0 )(ξ) = 0 for all ξ ∈ R, by (1.9). By Remark 5.2, we know that ψ(•, y 0 ) ∈ L 2 ((0, +∞)), 1 -ψ(•, y 0 ) ∈ L 2 ((-∞, 0)), and ∂ ξ ψ(•, y 0 ) ∈ H 2 (R). Moreover, since the function f (x, u) satisfies (A1) and is globally Lipschitz continuous with respect to u uniformly in x ∈ R, it follows that the function ξ → f (y, v(ξ) + ψ(ξ, y 0 )) belongs to L 2 (R). Therefore, for any (v, c, y)

It is easily seen from Lemma 5.

Remember that (ψ(•, y), c(y)) is the unique solution of (1.9)-(1.10). It is straightforward to check that for any y ∈ R, we have ψ(•, y) -ψ(•, y 0 ) ∈ H 2 (R) by (5.95), and

On the other hand, thanks to Lemma 5.3 (ii) and the assumption that the function

, and f (x, u), ∂ u f (x, u) are globally Lipschitz continuous in u uniformly in x ∈ R, by similar arguments to those used in the proof of [START_REF] Ding | Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat[END_REF]Lemma 3.4], one can check that the function G :

is simpler in our case, since no singularity occurs) and that the derivatives are given by

From Remark 5.2, the map y → R ∂ ξ ψ(ξ, y)w * (ξ, y)dξ is positive, periodic, continuous in R, and w * (•, y) L 2 (R) is bounded uniformly in y ∈ R. Therefore, there is a constant C 1 > 0 (independent of y 0 ∈ R and ( g, d) ∈ S) such that

(5.101)

We first claim that w 1 L 2 (R) is bounded uniformly with respect to y 0 ∈ R and ( g, d) ∈ S. Assume by contradiction that this is not true. Then, thanks to (5.101) and the 1-periodicity with respect to y 0 , there exist a sequence (y n

For each n ∈ N, writing

and

we have g n L 2 (R) → 0 as n → +∞ by (5.95), and M c(yn),yn

Up to extraction of a sequence, we can assume that y n → y * ∈ [0, 1] as n → +∞. It follows from Lemma 5.3 and the boundedness of the sequence (c(y n )) n∈N that the sequence (q n ) n∈N is bounded in H 2 (R). Hence, up to extraction of another subsequence, the sequence (q n ) n∈N converges in H 2 (R) weakly and in C 1 loc (R) to some q * ∈ H 2 (R). This implies that a(y * )q * + c(y * )q * + ∂ u f (y * , ψ(•, y * ))q * = 0. Furthermore, q * is orthogonal to ∂ ξ ψ(•, y * ) in L 2 (R), by (5.97) and Lemma 5.1. As a consequence, q * = 0.

Therefore, q n → 0 in C 1 loc (R), hence in L 2 loc (R), as n → +∞,. In order to get a contradiction, we further show that this convergence holds in L 2 (R). Indeed, by Proposition 4.1 (ii) and the assumption (A2), there exists a constant M > 0 (independent of n ∈ N) such that ∂ u f (y n , ψ(ξ, y n )) ≤ -γ 0 /2 for all ξ ∈ (-∞, -M ] ∪ [M , +∞) and n ∈ N. Integrating the equation H n (q n ) = g n against q n over (M , +∞), we get

This implies in particular that q n L 2 ((M ,+∞)) → 0 as n → +∞. The same analysis over (-∞, -M ) gives q n L 2 ((-∞,-M )) → 0 as n → +∞. Finally, the sequence (q n ) n∈N tends to 0 in L 2 (R), which is a contradiction with the fact that q n L 2 (R) = 1 for each n ∈ N. We can thus conclude that w 1 L 2 (R) is bounded uniformly with respect to y 0 ∈ R and ( g, d) ∈ S. Furthermore, since M c(y 0 ),y 0 ( w 1 ) = -∂ u f (y 0 , ψ(•, y 0 )) ( w 1 + g) -β( w 1 + g) -c∂ ξ ψ(•, y 0 ), it follows from (5.95), (5.101) and Lemma 5.3, together with the bondedness of c(y 0 ) with respect to y 0 ∈ R, that there exists a positive constant C 2 (independent of y 0 and ( g, d) ∈ S) such that

(5.102)

we have w 2 (0) = d -g(0) -w 1 (0). This together with Proposition 4.1 (iii), (1.10), (5.95) and the fact that w 2 ∈ R∂ ξ ψ(•, y 0 ) implies that

for some constant C 4 > 0 independent of y 0 and ( g, d) ∈ S. Combining this with (5.101)-(5.102) and the fact that v = w + g, we obtain that Q -1 y 0 ( g, d) H 2 (R)×R ≤ C 5 for some constant C 5 > 0 independent of y 0 and ( g, d) ∈ S. This completes the proof of Lemma 5.4.

Based on the above preparations, we are now ready to complete the Proof of Proposition 4.1 (iv). For any y 0 ∈ R, we define

Clearly, T maps R into H 2 (R) × R, and G(T (y), y) = (0, 0) for all y ∈ R, by (1.9) and the definitions of T and G. Applying the implicit function theorem to the function G :

Denote the derivative operator at y ∈ (y 0 -δ, y 0 + δ) by A y : R → H 2 (R) × R. Thus, the function y → A y is continuous from (y 0 -δ, y 0 + δ) to L(R, H 2 (R) × R). At y = y 0 , we have

Since the function f is of class C 1 in R 2 and satisfies (A1), it follows from (5.95) that there exists a constant C 1 > 0 independent of y 0 such that a (y 0 )∂ ξξ ψ(•, y 0

Remember that the periodic function y 0 → c(y 0 ) is bounded by Lemma 5.1. Then, by Lemmas 5.3 and 5.4, we find a constant C 2 > 0, independent of y 0 ∈ R, such that A y 0 ( y)

where we identify Writing A y 0 = (A 1 y 0 , A 2 y 0 ) ∈ H 2 (R) × R for each y 0 ∈ R, we see from (5.104) and the previous paragraph that the functions y 0 → c(y 0 ) and y 0 → ψ(•, y 0 ) are of class C 1 in R with derivatives ∂ y ψ(•, y 0 ) = A 1 y 0 and c (y 0 ) = A 2 y 0 at each y 0 ∈ R. It further follows from (5.103) that ∂ y ψ(•, y 0 ) H 2 (R) ≤ C 2 for all y 0 ∈ R, with C 2 independent of y 0 . Using the Sobolev inequality, we immediately obtain (4.56). Finally, by standard elliptic estimates and Lemma 5.1, the function (ξ, y) → ψ(ξ, y) is of class C 2;1 ξ;y (R 2 ). This completes the proof of Proposition 4.1.