Xu Haiy

Biodiesel production from used palm cooking oil in the presence of bifunctional γ-Al2O3-KOH supported catalysts

Biodiesel production from used palm cooking oil in the presence of bifunctional γ-Al 2 O 3 -KOH supported catalysts

Xu Ziming, Wenfu Tong, Yu Haiy, Xu Bin School of Food and Biological Engineering, Jiangsu University, Zhenjiang Jiangsu 212013 Biodiesel has been identified as a promising substitute for diesel derived from petroleum. Both edible and non-edible oils are used in the production of biodiesel. Used cooking oil can be a promising feedstock for low-cost biodiesel production. Moreover, the production of biodiesel from used cooking oil will not only avoid the competition of the same oil resources for food and fuel but will also overcome the waste cooking oil disposal problems. In this study, γ-Al 2 O 3 was modified with 5-20wt% of KOH to form γ-Al 2 O 3 -KOH bifunctional heterogeneous support. The supports were then impregnated with 5wt% Mn and tested for biodiesel production from used palm cooking oil. The catalyst which shows the highest biodiesel yield was then further modified with Mn, Sn and Zn to form bimetallic catalysts and the activity of these catalysts were tested for biodiesel production. Among different catalysts tested, Mo-Mn/γ-Al 2 O 3 -20 wt% KOH catalyst shows the highest biodiesel yield of 85.6% in 4 h reaction at 110 o C, methanol to oil molar ratio of 30:1, catalyst loading of 7 wt% and agitation speed of 600 rpm. Moreover, the catalyst shows substantial chemical stability and could be used for at least 6 times without any major loss in its catalytic activity. The physicochemical properties of the synthesized biodiesel were further studied and compared with ASTM and the European biodiesel specifications. The results show that the properties of the biodiesel produced comply with the international standard specifications. Keywords: γ-Al 2 O 3 -KOH, bifunctional catalyst, biodiesel, used palm cooking oil


Thierry Gallay

Romain Joly

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Keywords: travelling front, global stability, damped wave equation, Lyapunov function AMS classification codes (2000): 35B35, 35B40, 37L15, 37L70

 and relies on the fact that our system has a Lyapunov function in any Galilean frame.

Introduction

The aim of this paper is to describe the long-time behavior of a large class of solutions of the semilinear damped wave equation

αu tt + u t = u xx -V ′ (u) , (1.1)
where α > 0 is a parameter, V : R → R is a smooth bistable potential, and the unknown u = u(x, t) is a real-valued function of x ∈ R and t ≥ 0. Equations of this form appear in many different contexts, especially in physics and in biology. For instance, Eq. (1.1) describes the continuum limit of an infinite chain of coupled oscillators, the propagation of voltage along a nonlinear transmission line [START_REF] Dunbar | On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks[END_REF], and the evolution of an interacting population if the spatial spread of the individuals is modelled by a velocity jump process instead of the usual Brownian motion [START_REF] Goldstein | On diffusion by discontinuous movements, and on the telegraph equation[END_REF][START_REF] Hadeler | Reaction transport systems in biological modelling[END_REF][START_REF] Kac | A stochastic model related to the telegrapher's equation[END_REF]. As was already observed by several authors, the long-time asymptotics of the solutions of the damped wave equation (1.1) are quite similar to those of the corresponding reactiondiffusion equation u t = u xx -V ′ (u). In particular, if V ′ (u) vanishes rapidly enough as u → 0, the solutions of (1.1) originating from small and localized initial data converge as t → +∞ to the same self-similar profiles as in the parabolic case [START_REF] Th | Scaling variables and asymptotic expansions in damped wave equations[END_REF][START_REF] Ikehata | Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption[END_REF][START_REF] Karch | Selfsimilar profiles in large time asymptotics of solutions to damped wave equations[END_REF][START_REF] Nishihara | Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping[END_REF][START_REF] Nishihara | Global asymptotics for the damped wave equation with absorption in higher dimensional space[END_REF]. The analogy persists for solutions with nontrivial limits as x → ±∞, in which case the long-time asymptotics are often described by uniformly translating solutions of the form u(x, t) = h(x -st), which are usually called travelling fronts. Existence of such solutions for hyperbolic equations of the form (1.1) was first proved by Hadeler [START_REF] Hadeler | Hyperbolic travelling fronts[END_REF][START_REF] Hadeler | Travelling fronts for correlated random walks[END_REF], and a few stability results were subsequently obtained by Gallay & Raugel [START_REF] Th | Convergence to Travelling Waves in Damped Hyperbolic Equations[END_REF][START_REF] Th | Stability of travelling waves for a damped hyperbolic equation[END_REF][START_REF] Th | Stability of propagating fronts in damped hyperbolic equations[END_REF][START_REF] Th | Scaling variables and stability of hyperbolic fronts[END_REF].

While local stability is an important theoretical issue, in the applications one is often interested in global convergence results which ensure that, for a large class of initial data with a prescribed behavior at infinity, the solutions approach travelling fronts as t → +∞. For the scalar parabolic equation u t = u xx -V ′ (u), such results were obtained by Kolmogorov, Petrovski & Piskunov [START_REF] Kolmogorov | Etude de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF], by Kanel [START_REF] Kanel | Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory[END_REF][START_REF] Kanel | Stabilization of the solutions of the equations of combustion theory with finite initial functions[END_REF], and by Fife & McLeod [8,[START_REF] Fife | A phase plane discussion of convergence to travelling fronts for nonlinear diffusion[END_REF] under various assumptions on the potential. All the proofs use in an essential way comparison theorems based on the maximum principle. These techniques are very powerful to obtain global information on the solutions, and were also successfully applied to monotone parabolic systems [START_REF] Volpert | Traveling wave solutions of parabolic systems[END_REF][START_REF] Roquejoffre | Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems[END_REF] and to parabolic equations on infinite cylinders [START_REF] Roquejoffre | Convergence to travelling waves for solutions of a class of semilinear parabolic equations[END_REF][START_REF] Roquejoffre | Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders[END_REF].

However, unlike its parabolic counterpart, the damped wave equation (1.1) has no maximum principle in general. More precisely, solutions of (1.1) taking their values in some interval I ⊂ R obey a comparison principle only if

4α sup u∈I V ′′ (u) ≤ 1 , (1.2)
see [START_REF] Protter | Maximum Principles in Partial Differential Equations[END_REF] or [START_REF] Th | Stability of travelling waves for a damped hyperbolic equation[END_REF]Appendix A]. In physical terms, this condition means that the relaxation time α is small compared to the period of the nonlinear oscillations. In particular, if I is a neighborhood of a local minimum ū of V , inequality (1.2) implies that the linear oscillator αu tt + u t + V ′′ (ū)u = 0 is strongly damped, so that no oscillations occur. It was shown in [START_REF] Th | Stability of travelling waves for a damped hyperbolic equation[END_REF][START_REF] Th | Stability of propagating fronts in damped hyperbolic equations[END_REF] that the travelling fronts of (1.1) with a monostable nonlinearity are stable against large perturbations provided that the parameter α is sufficiently small so that the strong damping condition (1.2) holds for the solutions under consideration. In other words, the basin of attraction of the hyperbolic travelling fronts becomes arbitrarily large as α → 0, but if α is not assumed to be small there is no hope to use "parabolic" methods to obtain global stability results for the travelling fronts of the damped wave equation (1.1).

Recently, however, a different approach to the stability of travelling fronts has been developped by Risler [START_REF] Th | A variational proof of global stability for bistable travelling waves[END_REF][START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF]. The new method is purely variational and is therefore restricted to systems that possess a gradient structure, but its main interest lies in the fact that it does not rely on the maximum principle. The power of this approach is demonstrated in the pioneering work [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF] where global convergence results are obtained for the non-monotone reaction-diffusion system u t = u xx -∇V (u), with u ∈ R n and V : R n → R. The aim of the present article is to show that Risler's method can be adapted to the damped hyperbolic equation (1.1) and allows in this context to prove global convergence results without any smallness assumption on the parameter α. This will also be an opportunity to present the main arguments of [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF] in an alternative way, although some important ingredients of our proof have no counterpart in Risler's work.

Before stating our theorem, we need to specify the assumptions we make on the nonlinearity in (1.1). We suppose that V ∈ C 3 (R), and that there exist positive constants a, b such that uV ′ (u) ≥ au 2 -b , for all u ∈ R .

(1.3)

In particular, V (u) → +∞ as |u| → ∞. We also assume V (0) = 0 , V ′ (0) = 0 , V ′′ (0) > 0 , (1.4) V (1) < 0 , V ′ (1) = 0 , V ′′ (1) > 0 .

(1.5)

Finally we suppose that, except for V (0) and V (1), all critical values of V are positive:

u ∈ R V ′ (u) = 0 , V (u) ≤ 0 = {0 ; 1} . (1.6)
In other words V is a smooth, strictly coercive function which reaches its global minimum at u = 1 and has in addition a local minimum at u = 0. We call V a bistable potential because both u = 0 and u = 1 are stable equilibria of the one-dimensional dynamical system u = -V ′ (u). The simplest example of such a potential is represented in Fig. 1. Note however that V is allowed to have positive critical values, including local minima.

V

V (u) u Fig. 1: The simplest example of a potential V satisfying assumptions (1.3)- (1.6).

Under assumptions (1.4)- (1.6), it is well-known that the parabolic equation u t = u xx -V ′ (u) has a family of travelling fronts of the form u(x, t) = h(x-c * t-x 0 ) connecting the stable equilibria u = 1 and u = 0, see e.g. [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]. More precisely, there exists a unique speed c * > 0 such that the boundary value problem

h ′′ (y) + c * h ′ (y) -V ′ (h(y)) = 0 , y ∈ R , h(-∞) = 1 , h(+∞) = 0 , (1.7)
has a solution h : R → (0, 1), in which case the profile h itself is unique up to a translation. Moreover h ∈ C 4 (R), h ′ (y) < 0 for all y ∈ R, and h(y) converges exponentially toward its limits as y → ±∞. As was observed in [START_REF] Th | Stability of travelling waves for a damped hyperbolic equation[END_REF][START_REF] Hadeler | Hyperbolic travelling fronts[END_REF], for any α > 0 the damped hyperbolic equation (1.1) has a corresponding family of travelling fronts given by u(x, t) = h( 1 + αc 2 * x -c * t -x 0 ) , x 0 ∈ R .

(1.8)

Remark that the actual speed of these waves is not c * , but s * = c * / 1 + αc 2 * . In particular s * is smaller than 1/ √ α (the slope of the characteristics of Eq. (1.1)), which means that the travelling fronts (1.8) are always "subsonic". In what follows we shall refer to c * as the "parabolic speed" to distinguish it from the physical speed s * .

We are now in position to state our main result:

Theorem 1.1. Let α > 0 and let V ∈ C 3 (R) satisfy (1.3)-(1.6) above.
Then there exist positive constants δ and ν such that, for all initial data (u 0 , u

1 ) ∈ H 1 ul (R) × L 2 ul (R) satisfying lim sup ξ→-∞ ξ+1 ξ (u 0 (x) -1) 2 + u ′ 0 (x) 2 + u 1 (x) 2 dx ≤ δ , (1.9)
lim sup

ξ→+∞ ξ+1 ξ u 0 (x) 2 + u ′ 0 (x) 2 + u 1 (x) 2 dx ≤ δ , (1.10
)

equation (1.1) has a unique global solution (for positive times) such that u(•, 0) = u 0 , u t (•, 0) = u 1 . Moreover, there exists x 0 ∈ R such that sup x∈R u(x, t) -h( 1 + αc 2 * x -c * t -x 0 ) = O(e -νt ) , as t → +∞ . (1.11)
Remarks:

1. Loosely speaking Theorem 1.1 says that, if the initial data (u 0 , u 1 ) are close enough to the global equilibrium (1, 0) as x → -∞ and to the local equilibrium (0, 0) as x → +∞, the solution u(x, t) of (1.1) converges uniformly in space and exponentially fast in time toward a member of the family of travelling fronts (1.8). In particular, any solution which looks roughly like a travelling front at initial time will eventually approach a suitable translate of that front. It should be noted, however, that our result does not give any constructive estimate of the time needed to reach the asymptotic regime described by (1.11). Depending on the shape of the potential and of the initial data, very long transients can occur before the solution actually converges to a travelling front.

2. The definition of the uniformly local Lebesgue space L 2 ul (R) and the uniformly local Sobolev space H 1 ul (R) will be recalled at the beginning of Section 2. These spaces provide a very convenient framework to study infinite-energy solutions of the hyperbolic equation (1.1), but their knowledge is not necessary to understand the meaning of Theorem 1.1. In a first reading one can assume, for instance, that u ′ 0 and u 1 are bounded and uniformly continuous functions, in which case assumptions (1.9), (1.10) can be replaced by lim sup

x→-∞ (|u 0 (x) -1| + |u ′ 0 (x)| + |u 1 (x)|) ≤ δ , lim sup x→+∞ (|u 0 (x)| + |u ′ 0 (x)| + |u 1 (x)|) ≤ δ .
Also, to simplify the presentation, we have expressed our convergence result (1.11) in the uniform norm, but the proof will show that the solution u(x, t) of (1.1) converges to a travelling front in the uniformly local energy space H 1 ul (R) × L 2 ul (R), see (9.5) below. 3. The convergence rate ν in (1.11) is related to the spectral gap of the linearization of (1.1) at the travelling front. As is shown in Section 9, we can take ν = O(1) in the parabolic limit α → 0, whereas ν = O(1/α) as α → +∞. On the other hand, the parameter δ in (1.9), (1.10) must be chosen small enough so that the following two conditions are satisfied. First, the initial data (u 0 , u 1 ) should lie in the local basin of attraction of the steady state (0, 0) for large positive x, and in the basin of (1, 0) for large negative x. Next, the energy integral

0 -∞ e cx α 2 u 1 (x) 2 + 1 2 u ′ 0 (x) 2 + V (u 0 (x)) dx ,
which is well-defined for any c > 0, should diverge to -∞ as c → 0. The second condition is an essential ingredient of our variational proof, but we do not know if the conclusion of Theorem 1.1 still holds without such an assumption.

The proof of Theorem 1.1 relies on the fact that Eq. (1.1) has, at least formally, a whole family of Lyapunov functions. To see this, let u(x, t) be a solution of (1.1) whose initial data satisfy (1.9), (1.10). Given any c ≥ 0 we go to a uniformly translating frame by setting

u(x, t) = u c ( √ 1 + αc 2 x -ct , t) , or u c (y, t) = u x + ct √ 1 + αc 2 , t . (1.12)
The new function u c (y, t) is then a solution of the modified equation

αü c + uc -2αc u′ c = u ′′ c + cu ′ c -V ′ (u c ) , (1.13)
where uc (y, t) ≡ ∂ t u c (y, t) and u ′ c (y, t) ≡ ∂ y u c (y, t). If we now introduce the energy function

E c (t) = R e cy α 2 ( uc (y, t)) 2 + 1 2 (u ′ c (y, t)) 2 + V (u c (y, t)) dy , (1.14)
a direct calculation shows that

E ′ c (t) = -(1 + αc 2 ) R e cy ( uc (y, t)) 2 dy ≤ 0 . (1.15)
In other words, Eq. (1.1) possess (at least formally) a continuous family of non-equivalent Lyapunov functions, indexed by the parabolic speed c ≥ 0. In the parabolic case α = 0, it is shown in [START_REF] Th | A variational proof of global stability for bistable travelling waves[END_REF] that this rich Lyapunov structure is sufficient to prove the convergence (1.11) if we restrict ourselves to solutions which decay sufficiently rapidly to zero as x → +∞, and we believe that the approach of [START_REF] Th | A variational proof of global stability for bistable travelling waves[END_REF] works in the hyperbolic case too. However, it is important to realize that the solutions we consider in Theorem 1.1 are only supposed to be small for large positive x, and do not necessarily converge to zero as x → +∞. Under these assumptions the integral in (1.14), which contains the exponentially growing factor e cy , is usually divergent at +∞, so that the Lyapunov function E c is certainly not well-defined. This is a technical problem which seriously complicates the analysis. To overcome this difficulty, a possibility is to truncate the exponential factor e cy in (1.14) to make it integrable over R, see [START_REF] Fife | The approach of solutions of nonlinear diffusion equations to travelling front solutions[END_REF], [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF]. We choose here another solution which consists in decomposing the solution u(x, t) into a principal part v(x, t) which is compactly supported to the right, and a small remainder r(x, t) which decays exponentially to zero as t → +∞. The idea is then to study the approximate Lyapunov function defined by (1.14) with u c (x, t) replaced by v c (x, t), see Section 4 for more details.

As was already mentioned, the proof of Theorem 1.1 closely follows the previous work [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF] which deals with gradient reaction-diffusion systems of the form u t = u xx -∇V (u). There are, however, significant differences that we want to emphasize. First, the evolution defined by the damped hyperbolic equation (1.1) is not regularizing in finite time, but only asymptotically as t → +∞. As a consequence, the compactness arguments which play an essential role in the proof become slightly more delicate in the hyperbolic case. On the other hand, the solutions of (1.1) have a finite speed of propagation, a property which has no parabolic analog. Although this is not an essential ingredient of the proof, we shall take advantage of this fact here and there to get a priori estimates on the solutions of (1.1). Finally, an important property of the scalar equation (1.1) is that the associated elliptic problem (1.7) has a unique solution (h, c * ), and that the corresponding travelling front is a stable solution of (1.1). This is no longer true for the systems considered in [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF], in which several stable or unstable fronts with different speeds may connect the same pair of equilibria. In this more general situation, without additional assumptions one can only show that the solution u(x, t) approaches as t → ∞ the family of all travelling fronts with a given speed.

Besides these natural differences due to the properties of Eq. (1.1), we also made technical choices in our proof which substantially differ from [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF]. As was already mentioned, the most important one is that we give a meaning to the Lyapunov function E c by decomposing the solution u(x, t), and not by truncating the exponential weight e cy . The main avantage of this approach is that the behavior of the energy is then easier to control. However, new arguments are required which have no counterpart in [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF] or [START_REF] Th | A variational proof of global stability for bistable travelling waves[END_REF]. This is the case in particular of Section 6, which is the main technical step in our proof.

The rest of this paper is organized as follows. In Section 2, we briefly present the uniformly local spaces and we study the Cauchy problem for Eq. (1.1) in this framework. In Section 3, we prove the persistence of the boundary conditions (1.9), (1.10) and we decompose the solution of (1.1) as u(x, t) = v(x, t) + r(x, t), where v is compactly supported to the right and r decays exponentially as t → +∞. We also introduce the invasion point x(t) which tracks the position of the moving interface. The core of the proof starts in Section 4, where we control the behavior of the energy E c in a frame moving at constant speed s = c/ √ 1 + αc 2 . These estimates are used in Section 5 to prove that the average speed x(t)/t converges to a limit s ∞ ∈ (0, 1/ √ α) as t → +∞. The main technical step is Section 6, where we show that the energy stays uniformly bounded in a frame following the invasion point, see Proposition 6.1 for a precise statement. This allows us to prove in Section 7 that the solution u(x, t) converges as t → +∞ to a travelling front uniformly in any interval of the form (x(t) -L, +∞). The proof of Theorem 1.1 is then completed in two steps. In Section 8, we use an energy estimate in the laboratory frame to show that the solution u(x, t) converges uniformly on R to a travelling front, at least for a sequence of times. Finally, the local stability result established in Section 9 gives the convergence for all times and the exponential rate in (1.11).

Notations. The symbols K 0 , K 1 , . . . denote our main constants, which will be used throughout the paper. In contrast, the local constants C 0 , C 1 , . . . will change from a section to another. We also denote by C a positive constant which may change from place to place.

Acknowledgements. As is emphasized in the text our approach is essentially based on ideas and techniques introduced by Emmanuel Risler, to whom we are also indebted for many fruitful discussions. The work of Th.G was partially supported by the French Ministry of Research through grant ACI JC 1039.

Global existence and asymptotic compactness

In this section, we prove that the Cauchy problem for Eq. (1.1) is globally well-posed for positive times in the uniformly local energy space X = H 1 ul (R) × L 2 ul (R). We first recall the definitions of the uniformly local Sobolev spaces which provide a natural framework for the study of partial differential equations on unbounded domains, see e.g. [START_REF] Arrieta | Linear parabolic equations in locally uniform spaces[END_REF][START_REF] Efendiev | The attractor for a nonlinear reaction-diffusion system in an unbounded domain[END_REF][START_REF] Feireisl | Bounded, locally compact global attractors for semilinear damped wave equations on R N[END_REF][START_REF] Th | Energy flow in formally gradient partial differential equations on unbounded domains[END_REF][START_REF] Ginibre | The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, I. Compactness methods[END_REF][START_REF] Kato | The Cauchy problem for quasi-linear symmetric hyperbolic systems[END_REF][START_REF] Maekawa | The Navier-Stokes equations with initial data in uniformly local L p spaces[END_REF][START_REF] Matos | Universal blow-up rates for a semilinear heat equation and applications[END_REF][START_REF] Mielke | Attractors for modulation equations on unbounded domains -existence and comparison[END_REF].

For any u ∈ L 2 loc (R) we denote

u L 2 ul = sup ξ∈R ξ+1 ξ |u(x)| 2 dx 1/2 = sup ξ∈R u L 2 ([ξ,ξ+1]) ≤ ∞ . (2.1)
The uniformly local Lebesgue space is defined as

L 2 ul (R) = u ∈ L 2 loc (R) u L 2 ul < ∞ , lim ξ→0 T ξ u -u L 2 ul = 0 , (2.2)
where T ξ denotes the translation operator: (T ξ u)(x) = u(x -ξ). In a similar way, for any k ∈ N, we introduce the uniformly local Sobolev space

H k ul (R) = u ∈ H k loc (R) ∂ j u ∈ L 2 ul (R) for j = 0, 1, 2, . . . , k , (2.3)
which is equipped with the natural norm

u H k ul = k j=0 ∂ j u 2 L 2 ul 1/2 .
It is easy to verify that H k ul (R) is a Banach space, which is however neither reflexive nor separable. If C k bu (R) denotes the Banach space of all u ∈ C k (R) such that ∂ j u is bounded and uniformly continuous for j = 0, . . . , k, we have the continuous inclusions

C k bu (R) ֒→ H k ul (R) ֒→ C k-1 bu (R) .
In particular

H 1 ul (R) ֒→ C 0 bu (R) and u 2 L ∞ ≤ 2 u 2 H 1 ul for all u ∈ H 1 ul . Note also that H k ul (R) is an algebra for any k ≥ 1, i.e. uv H k ul ≤ C u H k ul v H k ul for all u, v ∈ H k ul (R). Finally the space C ∞ bu (R) is dense in H k ul (R)
for any k ∈ N. Remark: Some authors do not include in the definition of the uniformly local L 2 space the assumption that ξ → T ξ u is continuous for any u ∈ L 2 ul (R). The resulting uniformly local Sobolev spaces are of course larger, but also less convenient from a functional-analytic point of view. In particular, one looses the property that H k+1 ul (R) is dense in H k ul (R). As we shall see, the definitions (2.2), (2.3) guarantee that the damped wave equation (1.1) defines a continuous evolution in

H 1 ul (R) × L 2 ul (R). Let X = H 1 ul (R) × L 2 ul (R) and Y = H 2 ul (R) × H 1 ul (R).
The main result of this section is: Proposition 2.1. For all initial data (u 0 , u 1 ) ∈ X, Eq. (1.1) has a unique global solution

u ∈ C 0 ([0, +∞), H 1 ul (R)) ∩ C 1 ([0, +∞), L 2 ul (R)) satisfying u(0) = u 0 , u t (0) = u 1 .
This solution depends continuously on the initial data, uniformly in time on compact intervals. Moreover, there exists K ∞ > 0 (depending only on α and V ) such that

lim sup t→+∞ ( u(•, t) 2 H 1 ul + u t (•, t) 2 L 2 ul ) ≤ K ∞ .
(2.4)

Proof: Setting w = (u, u t ), we rewrite (1.1) as a first order evolution equation

w t = Aw + F (w) , (2.5)
where

A = 1 α 0 α ∂ 2 x -1 -1
, and

F (w) = 1 α 0 u -V ′ (u) . (2.6)
Using d'Alembert's formula for the solution of the wave equation αu tt = u xx , it is straightforward to verify that the linear operator A 0 on X defined by

D(A 0 ) = Y , A 0 = 1 α 0 α ∂ 2 x 0
, is the generator of a strongly continuous group of bounded linear operators in X. The same is true for the linear operator A, which is a bounded perturbation of A 0 , see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]Section 3.1]. On the other hand, as V ∈ C 3 (R) and H 1 ul (R) ֒→ C 0 bu (R), it is clear that the nonlinearity F maps X into Y , and that F is Lipschitz continuous on any bounded set B ⊂ X. Thus a classical argument shows that the Cauchy problem for (2.5) is locally well-posed in X, see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]Section 6.1] or [START_REF] Th | Energy flow in formally gradient partial differential equations on unbounded domains[END_REF]Section 7.2]. More precisely, for any r > 0, there exists T (r) > 0 such that, for all initial data w 0 ∈ X with w 0 X ≤ r, Eq. (2.5) has a unique (mild) solution w ∈ C 0 ([0, T ], X) satisfying w(0) = w 0 . This solution w(t) depends continuously on the initial data w 0 in X, uniformly for all t ∈ [0

, T ]. Moreover, if w 0 ∈ Y , then w ∈ C 1 ([0, T ], X) ∩ C 0 ([0, T ], Y
) is a classical solution of (2.5). To prove Proposition 2.1, it remains to show that all solutions of (2.5) stay bounded for positive times (hence can be extended to global solutions), and are eventually contained in an attracting ball whose radius is independent of the initial data.

Assume that w = (u, u t ) ∈ C 0 ([0, T ], X) is a solution of (2.5). Let ρ(x) = exp(-κ|x|), where κ > 0 is small enough so that 2 √ ακ ≤ 1 and κ 2 ≤ a, with a > 0 as in (1.3). For any ξ ∈ R and all t ∈ [0, T ], we define

E(ξ, t) = R (T ξ ρ)(x) α 2 u 2 t + αu 2 x + 2αV (u) + 1 2 u 2 + αuu t (x, t) dx , (2.7)
F (ξ, t) = R (T ξ ρ)(x) αu 2 t + u 2 x + au 2 (x, t) dx , (2.8)
where

V (u) = V (u) -V (1) ≥ 0 and (T ξ ρ)(x) = ρ(x -ξ).
We also denote

M(t) = sup ξ∈R E(ξ, t) , t ∈ [0, T ] . (2.9) Since u(•, t) ∈ H 1 ul (R) and u t (•, t) ∈ L 2 ul (R), it is clear that M(t) < ∞ for all t ∈ [0, T ]. Moreover, as V (u) ≥ 0 and |αuu t | ≤ 3α 2 4 u 2 t + 1 3 u 2 , we have E(ξ, t) ≥ R (T ξ ρ)(x) α 2 4 u 2 t + αu 2 x + 1 6 u 2 (x, t) dx .
Taking in both sides the supremum over ξ ∈ R and using the definitions (2.1)-( 2.3), we see that there exists

C 1 > 0 (depending only on α) such that w(•, t) 2 X ≡ u(•, t) 2 H 1 ul + u t (•, t) 2 L 2 ul ≤ C 1 M(t).
On the other hand, differentiating E(ξ, t) with respect to time, we find

∂ t E(ξ, t) = - R (T ξ ρ)(αu 2 t + u 2 x + uV ′ (u)) dx - R (T ξ ρ) ′ (uu x + 2αu x u t ) dx .
To estimate the last integral, we observe that

- R (T ξ ρ) ′ uu x dx = κ 2 2 R (T ξ ρ)u 2 dx -κu(ξ) 2 ≤ κ 2 2 R (T ξ ρ)u 2 dx , and
R (T ξ ρ) ′ 2αu x u t dx ≤ 2ακ R (T ξ ρ)|u x u t | dx ≤ √ ακ R (T ξ ρ)(αu 2 t + u 2 x ) dx .
Using (1.3) together with our assumptions on κ, we arrive at

∂ t E(ξ, t) ≤ - 1 2 R (T ξ ρ)(αu 2 t + u 2 x + au 2 ) dx + 2b κ = - 1 2 F (ξ, t) + 2b κ . (2.10)
This differential inequality implies that the quantity M(t) defined in (2.9) is a decreasing function of time as long as it stays above a certain threshold. More precisely, we have:

Lemma 2.2. There exists C 2 > 0 (depending only on α, V ) such that, if M(t) ≥ C 2 for some t ∈ [0, T ] and E(ξ, t) ≥ M(t) -1 for some ξ ∈ R, then ∂ t E(ξ, t) ≤ -1.
Assuming this result to be true, we now conclude the proof of Proposition 2.1. It follows readily from Lemma 2.2 that M(t) ≤ max(C 2 , M(0) -t) for all t ∈ [0, T ], an estimate which holds for any solution w ∈ C 0 ([0, T ], X) of (2.5). This shows that any solution of (2.5) stays bounded in X for positive times (hence can be extended to a global solution), and that (2.4) holds with

K ∞ = C 1 C 2 .
Proof of Lemma 2.2. We use the same notations as in the proof of Proposition 2.1. Let C 3 = 2(1 + 2b/κ), and take L > 0 large enough so that e κL ≥ 3. Fix also ξ ∈ R

and t ∈ [0, T ]. If F (ξ, t) ≥ C 3 , then ∂ t E(ξ, t) ≤ -1 by (2.10). On the other hand, if F (ξ, t) ≤ C 3 , there exists C 4 > 0 (depending on α, V , L, and C 3 ) such that ξ+L ξ-L (T ξ ρ)(x) e(u, u x , u t )(x, t) dx ≤ C 4 , (2.11)
where e(u, u

x , u t ) = α 2 u 2 t + αu 2 x + 2αV (u) + 1 2 u 2 + αuu t ≥ 0. Inequality (2.11) holds because F (ξ, t) controls the norm of (u, u t ) in H 1 ([ξ -L, ξ + L]) × L 2 ([ξ -L, ξ + L]).
As a consequence of (2.7), (2.11) at least one of the following inequalities holds:

∞ ξ+L (T ξ ρ)(x) e(u, u x , u t )(x, t) dx ≥ 1 2 (E(ξ, t) -C 4 ) , or (2.12) ξ-L -∞ (T ξ ρ)(x) e(u, u x , u t )(x, t) dx ≥ 1 2 (E(ξ, t) -C 4 ) .
Suppose for instance that the first inequality in (2.12) holds. Then

E(ξ + L, t) ≥ ∞ ξ+L (T ξ+L ρ)(x) e(u, u x , u t )(x, t) dx ≥ 3 ∞ ξ+L (T ξ ρ)(x) e(u, u x , u t )(x, t) dx ≥ 3 2 (E(ξ, t) -C 4 ) , because (T ξ+L ρ)(x) ≥ 3(T ξ ρ)(x)
for all x ≥ ξ + L, by assumption on L. Using a similar argument in the other case we conclude that, if

F (ξ, t) ≤ C 3 , then max E(ξ + L, t), E(ξ -L, t) ≥ 3 2 (E(ξ, t) -C 4 ) . (2.13) Now, fix C 5 > 3(C 4 + 1). If M(t) ≥ C 5 and E(ξ, t) ≥ M(t) -1, we claim that F (ξ, t) > C 3 , so that ∂ t E(ξ, t) ≤ -1 by (2.10). Indeed, if F (ξ, t) ≤ C 3 , it follows from (2.13) that M(t) ≥ max E(ξ + L, t), E(ξ -L, t) ≥ 3 2 (M(t) -1 -C 4 ) ,
which contradicts the assumption that M(t) ≥ C 5 .

Remark: The proof of Proposition 2.1 can be simplified if we assume, in addition to

(1.3), that uV ′ (u) ≥ a ′ V (u) -b ′ for some positive constants a ′ , b ′ , but Lemma 2.

allows us to avoid this unnecessary assumption.

To conclude this section, we show that the solutions of (1.1) given by Proposition 2.1 are locally asymptotically compact, in the following sense:

Proposition 2.3. Let u ∈ C 0 ([0, +∞), H 1 ul (R)) ∩ C 1 ([0, +∞), L 2 ul (R)) be a solution of (1.1), and let {(x n , t n )} n∈N be a sequence in R × R + such that t n → +∞ as n → ∞.
Then there exists a subsequence, still denoted (x n , t n ), and a solution ū

∈ C 0 (R, H 1 ul (R)) ∩ C 1 (R, L 2 ul (R)) of (1.1) such that, for all L > 0 and all T > 0, sup t∈[-T,T ] u(x n +•, t n +t)-ū(•, t) H 1 ([-L,L]) + u t (x n +•, t n +t)-ūt (•, t) L 2 ([-L,L]) ---→ n→∞ 0 .
In other words, after extracting a subsequence, we can assume that the sequence {u(

x n + x, t n + t)} converges in C 0 ([-T, T ], H 1 loc (R)) ∩ C 1 ([-T, T ], L 2 loc (R)
) towards a solution ū(x, t) of (1.1). Proof: As in the proof of Proposition 2.1, we set w = (u, u t ) and we consider Eq. (2.5) instead of Eq. (1.1). If w 0 ∈ X, the solution of (2.5) with initial data w 0 has the following representation:

w(t) = e At w 0 + t 0 e A(t-s) F (w(s)) ds ≡ w 1 (t) + w 2 (t) .
As is easily verified, there exists C 6 > 0 and µ > 0 such that e At L(X) ≤ C 6 e -µt for all t ≥ 0 (this estimate will be established in a more general setting in Section 9, Lemma 9.2). Thus w 1 (t) = e At w 0 converges exponentially to zero as t → +∞, and can therefore be neglected. On the other hand, by Proposition 2.1, there exists C 7 > 0 such that w(t) X ≤ C 7 for all t ≥ 0. As F maps X into Y = D(A) and is Lipschitz on bounded sets, there exists

C 8 > 0 such that AF (w) X ≤ C 8 whenever w X ≤ C 7 . Since Aw 2 (t) = t 0 e A(t-s) AF (w(s)) ds, we deduce that Aw 2 (t) X ≤ C 6 t 0 e -µ(t-s) AF (w(s)) X ds ≤ C 6 C 8 µ , t ≥ 0 ,
hence there exists C 9 > 0 such that w 2 (t) Y ≤ C 9 for all t ≥ 0. In particular, given any

T > 0, the sequence {w 2 (x n + •, t n -T )} is bounded in H 2 ([-L, L]) × H 1 ([-L, L]) for any L > 0.
Extracting a subsequence and using a diagonal argument, we can assume that there exists w0 ∈ H 2 loc (R) × H 1 loc (R) such that, for any L > 0,

w 2 (x n + •, t n -T ) ---→ n→∞ w0 in H 1 ([-L, L]) × L 2 ([-L, L]) . (2.14)
By construction w0 Y ≤ C 9 , hence in particular w0 ∈ X. Note that (2.14) still holds if we replace w 2 by the full solution w, because w 1 (•, t) X converges to zero. Finally, let w(t) ∈ C 0 ([-T, +∞), X) be the solution of (2.5) with initial data w(•, -T ) = w0 . Since the evolution defined by (2.5) has a finite speed of propagation, it is clear that the solution w(t) depends continuously on the initial data w0 in the topology of H 1 loc (R) × L 2 loc (R), uniformly in time on compact intervals. Thus it follows from (2.14) that, for all L > 0,

sup t∈[-T,T ] w(x n + •, t n + t) -w(•, t) H 1 ([-L,L])×L 2 ([-L,L]) ---→ n→∞ 0 .
Repeating the argument for larger T and using a diagonal extraction, we conclude the proof of Proposition 2.3.

Pinching at infinity and splitting of the solution

In this section we prove that, if the initial data satisfy the boundary conditions (1.9),(1.10), the solution u(x, t) of Eq. (1.1) has the same properties for all positive times. As a consequence, we show that any such solution can be decomposed into a principal part v(x, t) which is compactly supported to the right, and a small remainder r(x, t) which decays exponentially to zero as t → +∞.

We first verify that, due to assumptions (1.4), (1.5), the homogeneous equilibria u = 0 and u = 1 are stable steady states of Eq. (1.1). Let (u 0 , u 1 ) ∈ X = H 1 ul (R) × L 2 ul (R), and let (u, u t ) be the solution of Eq. (1.1) with initial data (u 0 , u 1 ) given by Proposition 2.1.

Lemma 3.1. There exist positive constants

K i , δ i , µ i for i = 0, 1 such that a) If (u 0 , u 1 ) 2 X ≤ δ 0 , then (u(•, t), u t (•, t)) 2 X ≤ K 0 (u 0 , u 1 ) 2 X e -µ 0 t for all t ≥ 0. b) If (u 0 -1, u 1 ) 2 X ≤ δ 1 , then (u(•, t)-1, u t (•, t)) 2 X ≤ K 1 (u 0 -1, u 1 ) 2 X e -µ 1 t for all t ≥ 0.
Proof: It is sufficient to prove a), the other case being similar. Let β 0 = V ′′ (0) > 0, and choose ε 0 > 0 small enough so that

β 0 2 ≤ V ′′ (u) ≤ 2β 0 , for all u ∈ [-ε 0 , ε 0 ] . (3.1)
In particular, we have

β 0 u 2 2 ≤ uV ′ (u) ≤ 2β 0 u 2 , and β 0 u 2 4 ≤ V (u) ≤ β 0 u 2 , (3.2)
whenever |u| ≤ ε 0 . In analogy with (2.7), we introduce the functional

E 0 (ξ, t) = R (T ξ ρ)(x) α 2 u 2 t + αu 2 x + 2αV (u) + 1 2 u 2 + αuu t (x, t) dx ,
where ρ(x) = exp(-κ|x|) and κ > 0 is small enough so that 2 √ ακ ≤ 1 and 2κ 2 ≤ β 0 . If

u(•, t) 2 L ∞ ≤ 2 u(•, t) 2 H 1 ul ≤ ε 2 0 , it follows from (3.
2) and from the definitions (2.1)

-(2.3) that C -1 0 (u(•, t), u t (•, t)) 2 X ≤ sup ξ∈R E 0 (ξ, t) ≤ C 0 (u(•, t), u t (•, t)) 2 X ,
for some C 0 > 1. Under the same assumption, we find as in the proof of Proposition 2.1:

∂ t E 0 (ξ, t) = - R (T ξ ρ)(αu 2 t + u 2 x + uV ′ (u)) dx - R (T ξ ρ) ′ (uu x + 2αu x u t ) dx ≤ - R (T ξ ρ)( α 2 u 2 t + 1 2 u 2 x + β 0 4 u 2 ) dx ≤ -µ 0 E 0 (ξ, t) ,
for some µ 0 > 0. Now, let K 0 = C 2 0 and choose δ 0 > 0 small enough so that 2K 0 δ 0 < ε 2 0 . If (u 0 , u 1 ) 2 X ≤ δ 0 , the inequalities above imply that the solution (u,

u t ) satisfies (u(•, t), u t (•, t)) 2 X ≤ K 0 (u 0 , u 1 ) 2 X e -µ 0 t for all t ≥ 0. In particular, u(•, t) 2 L ∞ ≤ 2 u(•, t) 2 H 1 ul ≤ ε 2 0 e -µ 0 t for all t ≥ 0.
From now on, we assume that the initial data (u 0 , u 1 ) ∈ X satisfy the assumptions (1.9), (1.10) for some δ ≤ min(δ 0 , δ 1 )/2, and we let u

∈ C 0 ([0, +∞), H 1 ul (R)) ∩ C 1 ([0, +∞), L 2 ul (R)
) be the solution of (1.1) given by Theorem 1.1. Using Lemma 3.1 and the finite speed of propagation we show that, for all t ≥ 0, the solution u(x, t) stays close for large |x| to the homogenous equilibria u = 0 and u = 1. Proposition 3.2. If δ ≤ min(δ 0 , δ 1 )/2, the solution of (1.1) given by Theorem 1.1 satisfies, for all t ≥ 0, lim sup

ξ→+∞ ξ+1 ξ u(x, t) 2 + u x (x, t) 2 + u t (x, t) 2 dx ≤ K 0 δ 0 e -µ 0 t , (3.3) lim sup ξ→-∞ ξ+1 ξ (u(x, t) -1) 2 + u x (x, t) 2 + u t (x, t) 2 dx ≤ K 1 δ 1 e -µ 1 t . (3.4)
Proof: We only prove the first inequality, the second one being similar. Take ξ 0 ∈ R such that

ξ+1 ξ u 0 (x) 2 + u ′ 0 (x) 2 + u 1 (x) 2 dx ≤ 3δ 0 4
, for all ξ ≥ ξ 0 -4 .

We consider the modified initial data (r 0 , r 1 ) ∈ X defined by

r 0 (x) = θ(x -ξ 0 )u 0 (x) , r 1 (x) = θ(x -ξ 0 )u 1 (x) , x ∈ R , (3.5)
where θ(x) = min(1, (1 + x/4) + ) satisfies θ(x) = 1 for x ≥ 0, θ(x) = 0 for x ≤ -4, and |θ ′ (x)| ≤ 1/4 for all x. By construction (r 0 (x), r 1 (x)) = (u 0 (x), u 1 (x)) for all x ≥ ξ 0 , and

(r 0 , r 1 ) 2 X ≤ 4 3 sup ξ≥ξ 0 -4 ξ+1 ξ u 0 (x) 2 + u ′ 0 (x) 2 + u 1 (x) 2 dx ≤ δ 0 . If (r, r t ) ∈ C 0 ([0, +∞), X)
is the solution of (1.1) with initial data (r 0 , r 1 ), we know from Lemma 3.1 that

(r(•, t), r t (•, t)) 2 X ≤ K 0 δ 0 e -µ 0 t , for all t ≥ 0 . (3.6)
On the other hand, the finite speed of propagation implies that u(x, t) = r(x, t) for all t ≥ 0 and all x ≥ ξ 0 + t/ √ α. Both observations together imply (3.3).

Decomposition of the solution:

The proof of Proposition 3.2 provides us with a useful decomposition of the solution of (1.1). Let

u(x, t) = v(x, t) + r(x, t) , x ∈ R , t ≥ 0 , (3.7)
where r(x, t) is the solution of (1.1) associated to the initial data (r 0 , r 1 ) defined in (3.5). By construction, the principal part v(x, t) vanishes identically for x ≥ ξ 0 + t/ √ α, and satisfies the modified equation

αv tt + v t = v xx -V ′ (v + r) + V ′ (r) , (3.8)
supplemented with the initial data (v 0 , v 1 ) = (u 0 -r 0 , u 1 -r 1 ). If we define

f (v, r) = V ′ (v) + V ′ (r) -V ′ (v + r) = -vr 1 0 1 0 V ′′′ (tv + sr) dt ds , (3.9)
we can rewrite (3.8) in the form

αv tt + v t = v xx -V ′ (v) + f (v, r) . (3.10)
The main advantage of working with (3.10) instead of (1.1) is that the energy functional (1.14) (with u replaced by v) is well-defined for all c > 0 since v(x, t) is compactly supported to the right. The price to pay is the additional term f (v, r) in (3.10), which we shall treat as a perturbation. Remark that, since v(x, t) and r(x, t) stay uniformly bounded for all t ≥ 0, the formula (3.9) shows that there exists K 2 > 0 such that

|f (v(x, t), r(x, t))| ≤ K 2 |v(x, t)| |r(x, t)| , x ∈ R , t ≥ 0 . (3.11) Moreover, we know that r(•, t) 2 L ∞ ≤ 2 r(•, t) 2 H 1 ul
≤ ε 2 0 e -µ 0 t for all t ≥ 0, hence (3.10) is really a small perturbation of (1.1) for large times. In particular, the asymptotic compactness property stated in Proposition 2.3 holds for the solution v(x, t) of (3.10), and by Proposition 2.1 there exists

M 0 > 0 such that v(•, t) 2 H 1 ul + v t (•, t) 2 L 2 ul ≤ M 2 0 , for all t ≥ 0 . (3.12)
The invasion point: As is explained in [START_REF] Th | A variational proof of global stability for bistable travelling waves[END_REF][START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF], to control the behavior of the solution u(x, t) of (1.1) using the energy functionals (1.14) it is necessary to track for all times the approximate position of the front interface. Since r(x, t) converges uniformly to zero as t → +∞, this can be done for the solution v(x, t) of (3.10) instead of u(x, t). We thus introduce the invasion point x(t) ∈ R defined for any t ≥ 0 by

x(t) = sup{x ∈ R | |v(x, t)| ≥ ε 0 } , (3.13)
where ε 0 is as in (3.1). It is clear that x(t) < ξ 0 + t/ √ α since v(x, t) vanishes identically for larger values of x. In the same way, using (3.4), one can prove that there exists ξ 1 ∈ R such that x(t) > ξ 1 -t/ √ α for all t ≥ 0. Note that x(t) is not necessarily a continuous function of t, although it follows from the definition (3.13) that x(t) is upper semi-continuous.

Energy estimates in a Galilean frame

As is explained in the introduction, the proof of Theorem 1.1 is based on the existence of Lyapunov functions for Eq. (1.1) in uniformly translating frames. The aim of this section is to define these functions rigorously and to study their basic properties. Let u(x, t) be a solution of (1.1) whose initial data satisfy the assumptions of Theorem 1.1. Given any c > 0, we go to a uniformly translating frame by setting, as in (1.12), u(x, t) = u c ( √ 1 + αc 2 x -ct , t). To avoid confusions, we always denote by y = √ 1 + αc 2 x -ct the space variable in the moving frame. Note that the physical speed s ∈ (0, 1/ √ α) of the frame is related to the parabolic speed c > 0 by the formulas By construction, both v c and r c belong to C 0 ([0, +∞),

s = c √ 1 + αc 2 , c = s √ 1 -αs 2 . ( 4
H 1 ul (R)) ∩ C 1 ([0, +∞), L 2 ul (R))
. Moreover, from (3.12) and Lemma 3.1, we know that

v c (t) 2 L ∞ ≤ 2 v(t) 2 H 1 ul ≤ M 2 0 , r c (t) 2 L ∞ ≤ 2 r(t) 2 H 1 ul ≤ ε 2 0 e -µ 0 t , (4.3)
for all t ≥ 0. In view of (1.13), (3.10), the evolution equations satisfied by v c , r c read

αr c + ṙc -2αc ṙ′ c = r ′′ c + cr ′ c -V ′ (r c ) , αv c + vc -2αc v′ c = v ′′ c + cv ′ c -V ′ (v c ) + f (v c , r c ) , (4.4)
where

f (v c , r c ) = -V ′ (v c + r c ) + V ′ (v c ) + V ′ (r c ).
Here and in the rest of the text, to simplify the notation and to avoid double subscripts, we denote vc (y, t)

= ∂ t v c (y, t), v ′ c (y, t) = ∂ y v c (y, t)
, and similarly for r c . In analogy with (3.13), we also define the invasion point in the moving frame by

y c (t) = √ 1 + αc 2 x(t) -ct = sup{y ∈ R | |v c (y, t)| ≥ ε 0 } . (4.5)

The energy functional

In a moving frame with parabolic speed c > 0, the energy functional involves an exponentially growing weight e cy , see (1.14). It is thus natural to introduce the following weighted spaces:

L 2 c (R) = {u ∈ L 2 loc (R) | e cy/2 u ∈ L 2 (R)} , (4.6)
H 1 c (R) = {u ∈ H 1 loc (R) | e cy/2 u ∈ L 2 (R) and e cy/2 u ′ ∈ L 2 (R)} .
Since v c (•, t) ∈ H 1 ul (R) and v c (y, t) vanishes for all sufficiently large y > 0, it is clear that v c (•, t) belongs to H 1 c (R) for any c > 0. Similarly, vc (•, t) belongs to L 2 c (R). The following quantity is thus well-defined for any y 0 ∈ R and all t ≥ 0:

E c (y 0 , t) = R e cy α 2 | vc | 2 + 1 2 |v ′ c | 2 + V (v c ) (y 0 + y, t) dy . (4.7)
We shall refer to E c (y 0 , t) as the energy of the solution v c (y, t) in the moving frame. The translation parameter y 0 is introduced here for later convenience. Changing y 0 results in a simple rescaling, as is clear from the identity

E c (y 0 , t) = e c(y 1 -y 0 ) E c (y 1 , t) . (4.8)
Due to the term f (v c , r c ) in (4.4), the energy E c (y 0 , t) is not necessarily a decreasing function of time. Indeed, a formal calculation gives

∂ t E c (y 0 , t) = -(1 + αc 2 ) R e cy | vc (y 0 + y, t)| 2 dy + R c (y 0 , t) , (4.9)
where R c (y 0 , t) = R e cy (f (v c , r c ) vc )(y 0 + y, t) dy . (4.10)

Using (3.11) and (4.3), it is easy to verify that R c (y 0 , t) is well-defined for all t ≥ 0 and depends continuously on time. A classical argument then shows that E c (y 0 , t) is indeed differentiable with respect to t and that (4.9) holds for all t ≥ 0. The purpose of this section is to show that, in appropriate situations, the remainder term R c in (4.9) is a negligible quantity which does not really affect the decay of the energy. Our first result in this direction is:

Lemma 4.1. There exists a positive constant K 3 , independent of c, such that

|R c (y 0 , t)| ≤ K 3 e -µt E c (y 0 , t) + K 3 c e c(y c (t)-y 0 ) , (4.11)
for all y 0 ∈ R and all t ≥ 0, where µ = µ 0 /2.

Proof: Using (3.11), (4.3), and (4.10), we obtain

|R c (y 0 , t)| ≤ K 2 ε 0 e -µt R e c(y-y 0 ) | vc v c |(y, t) dy ≤ K 2 ε 0 2 e -µt R e c(y-y 0 ) (| vc | 2 + |v c | 2 )(y, t) dy . If y ≥ y c (t), then |v c (y, t)| ≤ ε 0 by (4.5), hence |v c (y, t)| 2 ≤ (4/β 0 )V (v c (y, t)) by (3.2). Thus 1 2 (| vc (y, t)| 2 + |v c (y, t)| 2 ) ≤ C α 2 | vc (y, t)| 2 + 1 2 |v ′ c (y, t)| 2 + V (v c (y, t)) ,
where

C = max(α -1 , 2β -1 0 ). If y ≤ y c (t), we can bound 1 2 (| vc (y, t)| 2 + |v c (y, t)| 2 ) ≤ C α 2 | vc (y, t)| 2 + 1 2 |v ′ c (y, t)| 2 + V (v c (y, t)) + 1 2 v c (t) 2 L ∞ + C| min V | .
Combining these estimates and using (4.3), we thus obtain

|R c (y 0 , t)| ≤ K 2 ε 0 e -µt CE c (y 0 , t) + M 2 0 2 + C| min V | y c (t)
-∞ e c(y-y 0 ) dy

≤ K 3 e -µt E c (y 0 , t) + K 3 c e c(y c (t)-y 0 ) ,
which is the desired result.

The following corollary of Lemma 4.1 will turn out to be useful: Proposition 4.2. Assume that the invasion point satisfies, for some c + > 0, lim sup

t→+∞ y c + (t) t ≤ 0 .
Then, there exist η > 0 and K 4 > 0 such that, for all c ∈ [c + -η, c + + η], all y 0 ∈ R, and all t 1 ≥ t 0 ≥ 0, E c (y 0 , t 1 ) ≤ K 4 max(E c (y 0 , t 0 ) , e -cy 0 ) . (4.12)

Moreover

|R c (y 0 , t)| ≤ K 4 e -µt/2 max(E c (y 0 , t 0 ) , e -cy 0 ) , for all t ≥ t 0 . (4.13)

Proof: By assumption, for all ε > 0, there exists C ε > 0 such that y c + (t) ≤ εt + C ε for all t ≥ 0. In a frame moving at parabolic speed c, this bound becomes

y c (t) ≤ 1 + αc 2 1 + αc 2 + (c + + ε) -c t + 1 + αc 2 1 + αc 2 + C ε .
Thus, if we choose ε > 0 small enough, there exist η ∈ (0, c + ) and C 1 > 0 such that, for all c ∈ [c + -η, c + + η] and all t ≥ 0, we have cy c (t) ≤ (µt)/2 + C 1 . Using (4.9) and Lemma 4.1, we deduce that

∂ t E c (y 0 , t) ≤ |R c (y 0 , t)| ≤ K 3 e -µt (E c (y 0 , t) + C 2 e µt/2-cy 0 ) , (4.14)
for some C 2 > 0. Integrating this differential inequality between t 0 and t 1 , we obtain

E c (y 0 , t 1 ) ≤ e K 3
µ (e -µt 0 -e -µt 1 ) E c (y

0 , t 0 ) + K 3 C 2 t 1 t 0 e K 3
µ (e -µt -e -µt 1 ) e -µt/2-cy 0 dt ≤ K 4 max(E c (y 0 , t 0 ) , e -cy 0 ) , which proves (4.12). Estimate (4.13) is a direct consequence of (4.12) and (4.14).

Remark: Of course, if the initial data u 0 , u 1 decay rapidly enough as x → +∞, the decomposition (3.7) is not needed and we can use the energy (1.14) instead of (4.7). This is the point of view adopted in [START_REF] Th | A variational proof of global stability for bistable travelling waves[END_REF]. In a first reading of the paper, it is therefore possible to set r = 0 everywhere, in which case the remainder term R c (y 0 , t) disappears from (4.9) and the energy E c (y 0 , t) is a true Lyapunov function. However, once the the invasion point is under control, the results of this section show that R c (y 0 , t) becomes really negligible for large times. Thus the general case can be seen as a perturbation of the particular situation where r = 0, and the outline of the arguments is the same in both cases.

A Poincaré inequality

As was already noted in [START_REF] Muratov | A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type[END_REF] and [START_REF] Th | A variational proof of global stability for bistable travelling waves[END_REF], Poincaré inequalities hold in the weighted space The Poincaré inequality implies the following important lower bound on the energy. We recall that, for all y ≥ y c (t), one has |v c (y, t)| ≤ ε 0 by (4.5), so that V (v c (y, t)) ≥ 0 by (3.2). Thus

H 1 c (R) if c > 0.
E c (y 0 , t) = e -cy 0 y c (t) -∞ e cy α 2 | vc | 2 + 1 2 |v ′ c | 2 + V (v c ) (y, t) dy + e -cy 0 ∞ y c (t) e cy α 2 | vc | 2 + 1 2 |v ′ c | 2 + V (v c ) (y, t) dy ≥ e -cy 0 y c (t)
-∞ e cy (min V ) dy + e -cy 0 1 2

∞ y c (t)
e cy |v ′ c (y, t)| 2 dy .

Using now (4.16) and the fact that |v c (y c (t), t)| = ε 0 , we obtain

E c (y 0 , t) ≥ e c(y c (t)-y 0 ) min V c + cε 2 0 2 . (4.18)

Existence of the invasion speed

The purpose of this section is to show that the invasion point x(t) defined in (3.13) has a positive average speed as t → +∞:

Proposition 5.1. The limit lim t→+∞ x(t) t = s ∞ exists and lies in the interval (0, 1 √ α ).

We call s ∞ the invasion speed because this is the speed at which the front interface described by the solution u(x, t) "invades" the steady state u = 0. We prove that s ∞ < 1 √ α , which means that the invasion process is always "subsonic". This might not be the case if we drop the assumption that the equilibrium u = 0 is stable. For instance, if

h(x) = (1 + e x ) -1 , one can check that u(x, t) = h(x -st) is a solution of (1.1) provided that -V ′ (u) = u(1 -u)(s + γ(1 -2u))
, where γ = αs 2 -1 .

If we choose s > 0 large enough so that γ > 0, the front h(x -st) is supersonic, but in that case we also have V ′′ (0) < 0, hence u = 0 is an unstable equilibrium of (1.1). Our proof of Proposition 5.1 follows closely the method introduced in [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF] and simplified in [START_REF] Th | A variational proof of global stability for bistable travelling waves[END_REF]. It is divided into three lemmas. Proof: Choose c > 0 large enough so that min V c + cε 2 0 2 > 0. By (4.18), there exists C 1 > 0 such that E c (0, t) ≥ C 1 e cy c (t) for all t ≥ 0. Inserting this bound into (4.11), we see that there exists C 2 > 0 such that

∂ t E c (0, t) ≤ |R c (0, t)| ≤ C 2 e -µt E c (0, t) , for all t ≥ 0 .
If we integrate this differential inequality as in the proof of Proposition 4.2, we find that E c (0, t) ≤ C 3 E c (0, 0) for all t ≥ 0. Going back to the lower bound E c (0, t) ≥ C 1 e cy c (t) , we conclude that y c (t) is bounded from above, hence lim sup

t→+∞ x(t) t = 1 √ 1 + αc 2 c + lim sup t→+∞ y c (t) t ≤ c √ 1 + αc 2 < 1 √ α ,
which is the desired result. Proof: We argue by contradiction and assume that lim sup(y c (t)/t) < 0 for all c > 0. Using (4.9), (4.10) together with the bound (3.11), we find

∂ t E c (0, t) = -(1 + αc 2 ) R e cy | vc | 2 (y, t) dy + R e cy (f (v c , r c ) vc )(y, t) dy ≤ 1 4 R e cy |f (v c , r c )| 2 (y, t) dy ≤ K 2 2 4 R e cy |v c (y, t)| 2 |r c (y, t)| 2 dy .
Our goal is to bound the right-hand side by a quantity which is integrable in time and independent of c if c is sufficiently small. To do that, we fix c ′ = 2 √ µ, where µ > 0 is as in Lemma 4.1, and we assume that c ∈ (0, c ′ ]. Denoting ρ = √ 1 + αc 2 / √ 1 + αc ′ 2 and using the definitions (4.2), we obtain the identity

R e cy |v c | 2 |r c | 2 (y, t) dy = R e cy |v c ′ | 2 |r c ′ | 2 (ρ -1 (y + ct) -c ′ t , t) dy = ρ e c(ρc ′ -c)t R e cρz |v c ′ | 2 |r c ′ | 2 (z, t) dz . Since ρ ≤ 1 and c(ρc ′ -c) ≤ µ by construction, we have R e cy |v c (y, t)| 2 |r c (y, t)| 2 dy ≤ e µt ∞ 0 e c ′ y |v c ′ | 2 |r c ′ | 2 (y, t) dy + 0 -∞ |v c ′ | 2 |r c ′ | 2 (y, t) dy .
Remark that the right-hand side is now independent of c. To bound the first integral, we proceed as in the proof of Lemma 4.1. Since y c ′ (t) is bounded from above by assumption, so is E c ′ (0, t) by Proposition 4.2 and we obtain

∞ 0 e c ′ y |v c ′ | 2 |r c ′ | 2 (y, t) dy ≤ ε 2 0 e -2µt R e c ′ y |v c ′ (y, t)| 2 dy ≤ Cε 2 0 e -2µt E c ′ (0, t) + (M 2 0 + | min V |) 1 c ′ e c ′ y c ′ (t) ≤ C 4 e -2µt ,
for some C 4 > 0. To estimate the second integral we observe that r(x, t) = 0 for x ≤ ξ 0 -4 -t/ √ α, because the initial data (r 0 , r 1 ) satisfy (3.5). Thus there exists

C 5 > 0 such that r c ′ (y, t) = 0 whenever y ≤ -C 5 (1 + t), hence 0 -∞ |v c ′ | 2 |r c ′ | 2 (y, t) dy ≤ C 5 (1 + t) v c ′ (t) 2 L ∞ r c ′ (t) 2 L ∞ ≤ C 5 (1 + t)M 2 0 ε 2 0 e -2µt .
Summarizing, we have shown the existence of a constant C 6 > 0 such that ∂ t E c (0, t) ≤ C 6 (1 + t) e -µt for all t ≥ 0 and all c ∈ (0, c ′ ]. In particular,

∞ 0 ∂ t E c (0, t) dt ≤ C 7 = C 6 (1 + µ)/µ 2 .
Now, if the initial data (u 0 , u 1 ), or equivalently (v 0 , v 1 ), satisfy the boundary condition (1.9) for some sufficiently small δ > 0, it is straightforward to verify that

E c (0, 0) √ 1 + αc 2 = R e c √ 1+αc 2 x α 2 |v 1 +sv ′ 0 | 2 + 1 2(1+αc 2 ) |v ′ 0 | 2 +V (v 0 ) (x) dx --→ c→0 -∞ , (5.1)
because V (1) < 0. In particular we can take c ∈ (0, c ′ ] small enough so that E c (0, 0) ≤ -2C 7 . Then E c (0, t) ≤ -C 7 for all t ≥ 0, and since E c (0, t) ≥ -e cy c (t) | min V |/c by (4.18), we conclude that y c (t) is bounded from below. Thus lim sup(y c (t)/t) ≥ 0, which is the desired contradiction. Proof: Again, we argue by contradiction and assume that

s -≡ lim inf t→+∞ x(t) t < lim sup t→+∞ x(t) t ≡ s + .
Then there exist two increasing sequences of times t n and t ′ n , both converging to +∞, such that

x(t n ) t n ---→ n→∞ s + , and x(t ′ n ) t ′ n ---→ n→∞ s -.
Given T > 0, we can assume in view of Proposition 2.3 that the sequence of functions (v, v)(x(t n ) + •, t n + •) converges in the space C 0 ([0, T ], H 1 loc (R) × L 2 loc (R)) to some limit (w, ẇ) which satisfies (1.1). Note that |w(0, 0)| = ε 0 , because |v(x(t n ), t n )| = ε 0 for all n by definition of the invasion point. Let c -, c + be the parabolic speeds associated to s -, s + according to (4.1) (if s -≤ 0, we simply take c -= 0). We choose any c ∈ (c -, c + ) such that c > c + -η, where η is the positive constant given by Proposition 4.2. By construction, y c (t n ) → +∞ and y c (t ′ n ) → -∞ as n → ∞. Applying (4.13) with y 0 = t 0 = 0, we see that there exists

C 8 > 0 such that |R c (0, t)| ≤ C 8 e -µt/2 for all t ≥ 0. Since ∂ t E c (0, t) ≤ R c (0, t), it follows that E c (0, t) ≥ E c (0, t ′ n ) -C 9 e -µt/2 for t ∈ [0, t ′ n ]
, where C 9 = 2C 8 /µ. Taking the limit n → ∞ and using (4.18), we conclude that E c (0, t) ≥ -C 9 e -µt/2 for all t ≥ 0.

On the other hand, using (4.7) and the estimate above on R c (0, t), we obtain

E c (0, t) -E c (0, 0) ≤ -(1 + αc 2 ) t 0 R e cy | vc (y, τ )| 2 dy dτ + C 9 (1 -e -µt/2 ) .
Recalling that E c (0, t) ≥ -C 9 e -µt/2 and setting t = t n + T , we find

E c (0, 0) ≥ -C 9 + (1 + αc 2 ) tn+T 0 R e cy | vc (y, t)| 2 dy dt ≥ -C 9 + (1 + αc 2 ) tn+T tn R e c(y c (tn)+y) | vc (y c (t n ) + y, t)| 2 dy dt , hence T 0 R e cy | vc (y c (t n ) + y, t n + t)| 2 dy dt ≤ e -cy c (tn) (1 + αc 2 ) (E c (0, 0) + C 9 ) ---→ n→∞ 0 . (5.2) Since vc (y, t) = v y + ct √ 1 + αc 2 , t + sv ′ y + ct √ 1 + αc 2 , t , where s = c √ 1 + αc 2 ,
it follows from (5.2) that

T 0 L -L | v + sv ′ | 2 (x(t n ) + x, t n + t)
dx dt converges to zero as n → ∞ for any L > 0. Passing to the limit, we conclude that ẇ(x, t) + sw ′ (x, t) = 0 for all t ∈ [0, T ] and (almost) all x ∈ R. The key point is that this identity must hold for all c ∈ (c -, c + ) such that c > c + -η; i.e., for all s in a nonempty open interval. Obviously, this implies that w ′ = ẇ = 0, hence, since |w(0, 0)| = ε 0 , w(x, t) is identically equal either to ε 0 or to -ε 0 . But this is impossible, because w must be a solution of (1.1) and we know from (3.2) that V ′ (±ε 0 ) = 0.

Control of the energy around the invasion point

Proposition 5.1 shows that the invasion point x(t) has an average speed s ∞ ∈ (0, 1/ √ α) as t → +∞. Our next objective is to prove that the solution v(x, t) of (3.8) converges in any neighborhood of the invasion point to the profile of a travelling front. To achieve this goal, a crucial step is to control the energy E c (y c (t), t) for c close to c ∞ , where

c ∞ = s ∞ 1 -αs 2 ∞ . (6.1)
The main result of this section is:

Using (4.9) and Proposition 4.2, we obtain for all n ∈ N:

(1 + αc 2 ) t ′ n tn-T R e cy | vc | 2 (y c (t n ) + y, t) dy dt ≤ E c (y c (t n ), t n -T ) -E c (y c (t n ), t ′ n ) + 2K 4 µ max E c (y c (t n ), t n -T ) , e -cy c (tn) .
In view of (6.2), we have max E c (y c (t n ), t n -T ) , e -cy c (tn) = e -cy c (tn) max(E c (0, t n -T ) , 1) ---→ n→∞ 0 , because y c (t n ) → +∞ and E c (0, t n -T ) is bounded from above due to (4.12). On the other hand, since y c (t ′ n ) -y c (t n ) → -∞ by assumption, it follows from (4.18) that

-E c (y c (t n ), t ′ n ) ≤ e c(y c (t ′ n )-y c (tn)) | min V | c ---→ n→∞ 0 .
Thus, we have shown:

(1 + αc 2 ) 0 -T R e cy | vc | 2 (y c (t n ) + y, t n + t) dy dt ---→ n→∞ 0 .
Proceeding as in the proof of Lemma 5.4, we conclude that ẇ(x, t)+sw ′ (x, t) = 0 for all t ∈ [-T, 0] and (almost) all x ∈ R, where s = c/ √ 1 + αc 2 . Now, the crucial observation is that, for any c ′ ∈ (c, c ∞ ), we still have

y c ′ (t n ) → +∞ and y c ′ (t ′ n ) -y c ′ (t n ) → -∞ as n → ∞.
The second claim follows immediately from the identity

y c 2 (t ′ ) -y c 2 (t) = 1 + αc 2 2 1 + αc 2 1 y c 1 (t ′ ) -y c 1 (t) + 1 + αc 2 2 1 + αc 2 1 c 1 -c 2 (t ′ -t) . (6.4)
Thus, repeating the same arguments, we conclude that ẇ + s ′ w ′ = 0 for all s ′ in a nonempty open interval. This implies that ẇ = w ′ = 0, and we obtain a contradiction as in Lemma 5.4.

Combining the bound (6.3) and the identity (6.4), we obtain the following useful estimate, which is valid in any frame whose speed is close enough to the invasion speed. Corollary 6.3. For all p > 0, there exist η > 0 and M > 0 such that, for all c ∈

[c ∞ -η, c ∞ + η], y c (t ′ ) -y c (t) ≥ -M -p(t ′ -t) , for all t ′ ≥ t ≥ 0 . (6.5)
The next proposition shows that, if the energy is sufficiently large at a given time, and if the invasion point stays bounded from above on a sufficiently long time interval, then a significant decay of energy must occur. Lemma 6.4. There exist positive constants η, t 0 , ω, and K 5 such that the following holds. For any c ∈ [c ∞ -η, c ∞ + η], any y 0 ∈ R, any t 1 ≥ t 0 , any T > 0, and any M > 0, if the invasion point satisfies y c (t) ≤ y 0 + M for all t ∈ [t 1 , t 1 + T ], then E c (y 0 , t 1 + T ) ≤ K 5 e -ωT E c (y 0 , t 1 ) + e cM . (6.6)

Proof: Let η be the positive number given by Proposition 4.

2 for c + = c ∞ and let c ∈ [c ∞ -η, c ∞ + η]. Given y 0 ∈ R, we define E c (y 0 , t) = R e cy α 2 | vc | 2 + 1 2 |v ′ c | 2 + V (v c ) + αγv c vc (y 0 + y, t) dy ,
where γ > 0 will be fixed later. Equation (4.4) satisfied by v c implies that

∂ t E c (y 0 , t) = R e cy -(1 + αc 2 )| vc | 2 + f (v c , r c ) vc + αγ| vc | 2 -γ(1 + 2αc 2 )v c vc -2αγcv ′ c vc -γ|v ′ c | 2 -γV ′ (v c )v c + γf (v c , r c )v c (y 0 + y, t) dy .
From (3.11) and ( 4.3) we know that

R e cy |f (v c , r c )|(| vc | + γ|v c |)(y 0 + y, t) dy ≤ K 2 ε 0 e -µt R e cy (|v c | 2 + | vc | 2 )(y 0 + y, t) dy ,
provided that γ ≤ 3/4. Thus, using the bound 2ab ≤ C -1 a 2 + Cb 2 and the Poincaré inequality (4.15), we obtain that, if γ is small enough and t 0 is large enough, the following estimate holds for all t ≥ t 0 :

∂ t E c (y 0 , t) ≤ - R e cy 1 2 | vc | 2 + γ 2 |v ′ c | 2 + γV ′ (v c )v c (y 0 + y, t) dy .
On the other hand, we know from (4.3) that |v c (y, t)| is uniformly bounded for all y ∈ R and all t ≥ 0, and from 2 for all y ≥ y c (t). Thus, there exist ω > 0 and C 0 > 0 such that

(3.2) that 2V ′ (v c (y, t))v c (y, t) ≥ V (v c (y, t)) ≥ (β 0 /4)v c (y, t)
∂ t E c (y 0 , t) ≤ -ωE c (y 0 , t) + C y c (t)-y 0 -∞ e cy |V ′ (v c )v c | + |V (v c )| (y 0 + y, t) dy ,
≤ -ωE c (y 0 , t) + C 0 e c(y c (t)-y 0 ) , (6.7) for all t ≥ t 0 . In a similar way, there exist C 1 > 1 and C 2 > 0 such that

C -1 1 E c (y 0 , t) -C 2 e c(y c (t)-y 0 ) ≤ E c (y 0 , t) ≤ C 1 E c (y 0 , t) + C 2 e c(y c (t)-y 0 ) , (6.8)
for all t ≥ t 0 . Remark that, in (6.7) and (6.8), all constants can be chosen to be independent of

y 0 ∈ R, of c ∈ [c ∞ -η, c ∞ + η],
and of t ≥ t 0 . Now, we fix t 1 ≥ t 0 and assume that y c (t) -y 0 ≤ M when t ∈ [t 1 , t 1 + T ], for some M > 0 and some T > 0. Integrating the differential inequality (6.7), we find

E c (y 0 , t 1 + T ) ≤ e -ωT E c (y 0 , t 1 ) + C 0 ω e cM .
Combining this result with (6.8), we arrive at

E c (y 0 , t 1 + T ) ≤ C 2 1 e -ωT E c (y 0 , t 1 ) + 2C 1 C 2 + C 0 C 1 ω e cM ,
which is the desired estimate.

Using the control on the invasion point given by Lemma 6.2 and the decay of energy described in Lemma 6.4, we are now able to prove the main result.

Proof of Proposition 6.1: Let t 0 , ω, K 5 be as in Lemma 6.4, and choose p > 0 such that 4c ∞ p ≤ min(ω, µ), where µ > 0 is as in Lemma 4.1. By Corollary 6.3, there exist η > 0 and M > 0 such that (6.5) holds, and without loss of generality we can assume that η < c ∞ and that M is large enough so that e -(c∞-η)M ≤ 1/2. In the rest of the proof, we fix some c ∈ [c ∞ -η, c ∞ + η] ⊂ (0, 2c ∞ ) (but all constants will be independent of c).

From (4.9) and Proposition 4.2 we know that, for all t ≥ 0 and all τ ≥ 0,

∂ τ E c (y c (t), t + τ ) ≤ K 4 e -µ(t+τ )/2 max E c (y c (t), t) , e -cy c (t)
≤ K 4 e -µ(t+τ )/2 e -cy c (t) max E c (0, t) , 1 .

Since, by (6.5), cy c (t) ≥ c(y c (0) -M -pt) ≥ -C -µt/2 for all t ≥ 0, and since E c (0, t) is uniformly bounded from above by Proposition 4.2, there exists C 3 > 0 such that, for all t ≥ 0 and all τ ≥ 0,

E c (y c (t), t + τ ) ≤ E c (y c (t), t) + C 3 . (6.9)
Now, we choose T ≥ 1 and C 4 ≥ C 3 such that the following inequalities hold:

4K 5 e cM e -ωT /2 ≤ 1 , and 4K 5 e 4cM e cp(1+T ) ≤ C 4 .

We claim that, if E c (y c (t), t) ≥ C 4 for some t ≥ t 0 , then there exists

t ′ ∈ [1, T ] such that E c (y c (t + t ′ ), t + t ′ ) ≤ 1 2 E c (y c (t), t).
To prove this claim, we distinguish two possible cases. If there exists τ ∈ [0, T ] such that y c (t + τ ) ≥ y c (t) + 3M + p, then t ′ = max(τ, 1) is a suitable choice. Indeed, by Corollary 6.3, we have y c (t + t ′ ) ≥ y c (t) + 2M and so, using (6.9), we find

E c (y c (t + t ′ ), t + t ′ ) = e c(y c (t)-y c (t+t ′ )) E c (y c (t), t + t ′ ) ≤ e -2cM (E c (y c (t), t) + C 3 ) ≤ 1 2 E c (y c (t), t)
, because e -2cM ≤ 1/4 and C 3 ≤ C 4 ≤ E c (y c (t), t) by assumption. On the other hand, if y c (t + τ ) ≤ y c (t) + 3M + p for all τ ∈ [0, T ], we can take t ′ = T because, due to Corollary 6.3, Lemma 6.4 and our choices of T , C 4 , and p, we have

E c (y c (t + T ), t + T ) = e c(y c (t)-y c (t+T )) E c (y c (t), t + T ) ≤ e c(M +pT ) K 5 e -ωT E c (y c (t), t) + e c(3M +p) ≤ 1 2 E c (y c (t), t) .
We now show that the claim above implies Proposition 6.1. To this purpose, we construct the following sequence of times. We take t 0 > 0 as in Lemma 6.4, and given t n we define t n+1 in the following way. If E c (y c (t n ), t n ) ≤ C 4 , we simply set t n+1 = t n + 1 and, using (6.9) and Corollary 6.3, we get

E c (y c (t n+1 ), t n+1 ) = e c(y c (tn)-y c (t n+1 )) E c (y c (t n ), t n+1 ) ≤ e c(M +p) (C 4 + C 3 ) . If E c (y c (t n ), t n ) ≥ C 4 we set t n+1 = t n + t ′ ,
where t ′ ≥ 1 is the time given by the claim above when t = t n . In this case, we know that E c (y c (t n+1 ), t n+1 ) ≤ 1 2 E c (y c (t n ), t n ). By contruction, the sequence {t n } goes to +∞ as n → ∞, and

lim sup n→∞ E c (y c (t n ), t n ) ≤ e c(M +p) (C 4 + C 3 ) .
To conclude the proof, we observe that (6.9) and Corollary 6.3 provides a control on the energy for the remaining times. Indeed, since t n+1 -t n ≤ T for all n, we have

y c (t) ≥ y c (t n ) -M -pT for all t ∈ [t n , t n+1 ], hence E c (y c (t), t) ≤ e c(M +pT ) E c (y c (t n ), t) ≤ e c(M +pT ) (E c (y c (t n ), t n ) + C 3 ) , t ∈ [t n , t n+1 ] .
A similar argument shows that E c (y c (t), t) is bounded from above for t ∈ [0, t 0 ].

Convergence to a travelling wave

The purpose of this section is to show that, for any L > 0, the solution v of (3.8) converges to a travelling front uniformly in the interval (x(t) -L, +∞). The key step is to prove that, in the frame moving at the invasion speed s ∞ , the energy dissipation around the invasion point converges to zero as t → +∞. Proposition 7.1. Let s ∞ be the invasion speed introduced in Proposition 5.1 and let c ∞ be the parabolic speed (6.1). For any T > 0, we have

t t-T R e c∞y | vc∞ (y c∞ (t) + y, τ )| 2 dy dτ ----→ t→+∞ 0 .
We start the proof with an auxiliary result showing that the energy E c (y c (t), t) is a continuous function of the parameter c. Lemma 7.2. Let η > 0 be as in Proposition 6.1. Given any T ≥ 0, there exists

K 6 > 0 such that, for all c 1 , c 2 ∈ [c ∞ -η/2, c ∞ + η/2]
, all t ≥ T and all τ ∈ [t -T, t], the following estimate holds:

|E c 1 (y c 1 (t), τ ) -E c 2 (y c 2 (t), τ )| ≤ K 6 |c 1 -c 2 | .
Proof: If we return to the original variables using the definitions (4.2), (4.5), we obtain the identity

E c (y c (t), τ ) = R e cy α 2 | vc | 2 + 1 2 |v ′ c | 2 + V (v c ) (y c (t) + y, τ ) dy (7.1) = √ 1+αc 2 e c(y c (τ )-y c (t)) R e c √ 1+αc 2 x α 2 | v+sv ′ | 2 + |v ′ | 2 2(1+αc 2 ) + V (v) (x(τ )+x, τ ) dx ,
where s = c/ √ 1 + αc 2 . Assume first that τ = t and c = c, where c = c ∞ + η. We know from Proposition 6.1 that E c (y c (t), t) is bounded (from above) for all times. On the other hand, we obtain a lower bound on the last member of (7.1) if we replace V (v(x(t) + x, t)) by zero if x ≥ 0 and by min V if x ≤ 0. Thus, using in addition the Poincaré inequality (4.15) to control |v| 2 in terms of |v ′ | 2 , we deduce from (7.1) that there exists a constant

C 0 > 0 such that R e c √ 1+αc 2 x | v| 2 + |v ′ | 2 + |v| 2 (x(t) + x, t) dx ≤ C 0 , for all t ≥ 0 . (7.2)
Using the uniform control (7.2), it is a straightforward exercise to verify that the last member of (7.

1) is indeed a Lipschitz function of c ∈ [c ∞ -η/2, c ∞ + η/2], uniformly in t ≥ T and τ ∈ [t -T, t].
The only potential difficulty comes from the exponential terms.

If we denote

Y c (t, τ ) = c(y c (τ ) -y c (t)) = c √ 1 + αc 2 (x(τ ) -x(t)) + c 2 (t -τ ) , we know from Corollary 6.3 that Y c (t, τ ) ≤ c(M + pT ) for all τ ∈ [t -T, t] and all c ∈ [c ∞ -η, c ∞ + η], hence e Yc 1 (t,τ ) -e Yc 2 (t,τ ) ≤ e max(Yc 1 (t,τ ),Yc 2 (t,τ )) |Y c 1 (t, τ ) -Y c 2 (t, τ )| ≤ C 1 |c 1 -c 2 | . On the other hand, if c 1 , c 2 ∈ [c ∞ -η/2, c ∞ + η/2], we can bound e c 1 √ 1+αc 2 1 x -e c 2 √ 1+αc 2 2 x ≤ C 2 |c 1 -c 2 | e c √ 1+αc 2 x if x ≥ 0 , e c √ 1+αc 2 x if x ≤ 0 ,
where c = c ∞ + η and c = c ∞ -η. Thus, using estimate (7.2) for x ≥ 0 and the uniform bound (3.12) for x ≤ 0, we obtain the desired conclusion.

Proof of Proposition 7.1: We argue by contradiction. Assume that there exist δ > 0, T > 0, and a sequence of times {t n } going to +∞ such that

tn tn-T R e c∞y | vc∞ (y c∞ (t n ) + y, t)| 2 dy dt ≥ δ , (7.3)
for all n ∈ N. Following an idea introduced in [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF], we shall arrive at a contradiction by considering the variation of the energy along the broken line connecting the points (t n , y c∞ (t n )) in the (t, y) plane, see Fig. 2. To this end we define, for all n ∈ N,

t
s n = 1 1 + αc 2 ∞ y c∞ (t n+1 ) -y c∞ (t n ) t n+1 -t n + s ∞ , and c n = s n 1 -αs 2 n .
As is easily verified, the speed s n is the slope of the line segment connecting (t n , x(t n )) and (t n+1 , x(t n+1 )); i.e., x(t n+1 ) -x(t n ) = s n (t n+1 -t n ). Since y c∞ (t)/t converges to zero as t → +∞ by Proposition 5.1, we can assume (up to extracting a subsequence) that s n ∈ (0, 1/ √ α) for all n ∈ N and that s n → s ∞ as n → ∞. Then the parabolic speed c n is well-defined for all n, and by construction y cn (t n+1 ) = y cn (t n ). Extracting another subsequence if needed, we can further assume that t n+1 ≥ t n + T and |c n -c ∞ | ≤ η/2 for all n ∈ N, where η > 0 is as in Proposition 6.1, and that the sum n≥0 |c n -c ∞ | is finite. Now we define, for each n ∈ N,

∆ n = E cn (y cn (t n ), t n ) -E c n+1 (y c n+1 (t n+1 ), t n+1 ) = E cn (y cn (t n ), t n ) -E cn (y cn (t n+1 ), t n+1 ) + E cn (y cn (t n+1 ), t n+1 ) -E c n+1 (y c n+1 (t n+1 ), t n+1 ) = ∆ 1 n + ∆ 2 n .
Since y cn (t n+1 ) = y cn (t n ), the quantity ∆

| ≤ K 6 |c n -c n+1 |, hence ∆ n = ∆ 1 n + ∆ 2 n ≥ δ -K 6 |c n -c n+1 | -C 4 T |c n -c ∞ | -C 3 e -µtn/4 , n ∈ N .
To conclude the proof, we observe that

E c 0 (y c 0 (t 0 ), t 0 ) -E c N (y c N (t N ), t N ) = N -1 n=0 ∆ n ---→ N →∞ +∞ , because t n ≥ nT and the sum n≥0 |c n -c ∞ | is finite. Thus E c N (y c N (t N ), t N ) → -∞
as N → ∞, in contradiction with the lower bound (4.18). Thus (7.3) cannot hold for all n ∈ N, and Proposition 7.1 is proved.

Using now, for the first time, the fact that the differential equation (1.7) has a frontlike solution for a single value of c * and that this solution is unique up to translations, we can establish the local convergence to a travelling front.

1+αc 2 * x | v(x(t) + x, t) + s * v ′ * (x)| 2 + |v ′ (x(t) + x, t) -v ′ * (x)| 2 + |v(x(t) + x, t) -v * (x)| 2 dx ----→ t→+∞ 0 ,
where v * (x) = h( 1 + αc 2 * x) and h is the solution of (1.7) normalized so that h(0) = ε 0 . In particular, v converges to a front uniformly in any interval of the type (x(t) -L, +∞).

Proof: Fix T > 0 and let {t n } be a sequence of times going to +∞ as n → ∞. In view of Proposition 2.3, we can assume that the sequence of functions (v, v)(x(t n ) + •, t n + •) converges in the space C 0 ([-T, 0], H 1 loc (R) × L 2 loc (R)) to some limit (w, ẇ) which satisfies (1.1). By (4.2) and Proposition 7.1, we have

0 -T R e c∞ √ 1+αc 2 ∞ x | v + s ∞ v ′ | 2 (x(t n ) + x, t n + t) dx dt ---→ n→∞ 0 , hence ẇ(x, t) + s ∞ w ′ (x, t) = 0 for all t ∈ [-T, 0] and (almost) all x ∈ R. Setting w(x, t) = h( 1 + αc 2 ∞ x -c ∞ t), we see that h is a solution of the differential equation h ′′ + c ∞ h ′ -V ′ (h) = 0.
We also know that |h(x)| ≤ M 0 for all x ≤ 0, that |h(x)| ≤ ε 0 for all x ≥ 0, and that |h(0)| = |w(0, 0)| = ε 0 . Due to our assumptions (1.3)-(1.6) on the potential V , these properties together imply that c ∞ = c * , hence s ∞ = s * , and that h is the unique solution of (1.7) such that h(0) = ε 0 , see e.g. [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]. Since the limit is unique, we observe that the convergence above holds in fact for any sequence t n → +∞. In particular, if we denote v * (x) = h( 1 + αc 2 * x), we conclude that (v, v)(x(t) + •, t)

converges as t → +∞ to (v * , -s * v ′ * ) in H 1 ([-L, L]) × L 2 ([-L, L]
), for any L > 0. Using in addition the estimate (7.2) for x ≥ L, and the uniform bound (3.12) for x ≤ -L, we obtain the desired conclusion.

One can extract from the proof of Corollary 7.3 the following useful information on the invasion point: Lemma 7.4. For any T > 0 we have

sup |τ |≤T |x(t + τ ) -x(t) -s * τ | ----→ t→+∞ 0 .
Proof: Fix T > 0, and choose L > 0 large enough so that

h( 1 + αc 2 * L) ≤ ε 0 2
, and h(-

1 + αc 2 * L + c * T ) ≥ 1 + ε 0 2 , (7.4)
where h is as in Corollary 7.3. We claim that, for any δ > 0, there exists t 0 ≥ T such that, for all t ≥ t 0 ,

sup τ ∈[-T,0] sup x≥-L |v(x(t) + x, t + τ ) -h( 1 + αc 2 * x -c * τ )| ≤ δ . (7.5)
Indeed, if we restrict the values of x to a bounded interval I = [-L, L ′ ], where L ′ > 0, the analog of (7.5) follows immediately from the proof of Corollary 7.3 and the fact that H 1 (I) ֒→ L ∞ (I). On the other hand, by Lemma 6.2, there exists C > 0 such that x(t) ≥ x(t + τ ) -C for all τ ∈ [-T, 0]. In view of (7.2) we thus have

sup τ ∈[-T,0] sup x≥L ′ |v(x(t) + x, t + τ ) -h( 1 + αc 2 * x -c * τ )| ≤ sup τ ∈[-T,0] sup x≥L ′ -C |v(x(t + τ ) + x, t + τ )| + h( 1 + αc 2 * L ′ ) -----→ L ′ →+∞ 0 ,
uniformly in t. This proves (7.5).

We now assume that δ < min(ε 0 , 1 -ε 0 )/2. For any t ≥ t 0 and any τ ∈ [-T, 0], it follows from (7.4), (7.5)

), t + τ ) = h(0) = ε 0 , we obtain δ ≥ |ε 0 -h( 1 + αc 2 * (x(t + τ ) -x(t)) -c * τ )| ≥ m 1 + αc 2 * |x(t + τ ) -x(t) -s * τ | , where m = min |h ′ (y)| -1 + αc 2 * L ≤ y ≤ 1 + αc 2 * L + c * T > 0 . Thus |x(t + τ ) -x(t) -s * τ | ≤ δ/(m 1 + αc 2 *
) for all t ≥ t 0 and all τ ∈ [-T, 0]. Since δ > 0 was arbitrarily small, we obtain the desired conclusion.

Repair behind the front

As in the previous sections, we denote by u(x, t) a solution of (1.1) whose initial data fulfill the conditions (1.9), (1.10), where δ ≤ min(δ 0 , δ 1 )/2 is small enough so that (5.1) holds. We know from Corollary 7.3 that u(x, t) converges to a travelling front uniformly for x ∈ (x(t) -L, +∞), for any L > 0. To conclude the proof of Theorem 1.1, it remains to prove that u(x, t) converges uniformly to 1 far behind the invasion point. Following again the ideas introduced in [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF], we shall do this using a suitable energy estimate in the laboratory frame. Proposition 8.1. There exists a sequence t n → +∞ such that

u(x(t n ) + •, t n ) -v * H 1 ul + u(x(t n ) + •, t n ) + s * v ′ * L 2 ul ---→ n→∞ 0 , (8.1)
where v * is as in Corollary 7.3.

Remark: Using an additional argument as in [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF]Section 9.6], one can show that (8.1) holds in fact for all sequences t n → +∞. In our case, this follows from the local stability of the travelling front which will be established in the last section.

Proof: We recall that the solution of (1.1) has been decomposed as u(x, t) = v(x, t) + r(x, t), where the remainder (r, ṙ) converges exponentially to zero as t → +∞ in the uniformly local energy space X = H 1 ul (R) × L 2 ul (R). Using this remark and Corollary 7.3, we can construct a sequence of times {t n } satisfying t n+1 ≥ t n + n + 1 for all n ∈ N and such that, for all t ≥ t n , sup

z≥-2n z+1 z | u(x(t) + x, t) + s * v ′ * (x)| 2 + |u ′ (x(t) + x, t) -v ′ * (x)| 2 (8.2) + |u(x(t) + x, t) -v * (x)| 2 dx ≤ 1 n + 1 .
Without loss of generality, we also assume that t 0 ≥ 1. Let θ : R → [0, 1] be a smooth, nondecreasing function satisfying θ(x) = 0 for x ≤ -1, θ(x) = 1 for x ≥ 1, and 1 -1 θ(x) dx = 1. We define a smooth map x + : [t 0 , +∞) → R in the following way. For all n ∈ N, we set

x + (t) = ∞ 0 θ ′ (t -τ )x(τ ) dτ -n -θ 2t -t n -t n+1 t n+1 -t n , if t ∈ [t n , t n+1 ] .
We recall that t → x(t) is upper semi-continuous, hence measurable. Since R θ ′ (x) dx = 1 and R xθ ′ (x) dx = 0, we have for all t ≥ 1:

x(t) - ∞ 0 θ ′ (t -τ )x(τ ) dτ = ∞ 0 θ ′ (t -τ ) x(t) -x(τ ) -s * (t -τ ) dτ ,
and the right-hand side converges to zero as t → +∞ by Lemma 7.4. Thus, if n ∈ N is sufficiently large, we see that

x(t) -n -2 ≤ x + (t) ≤ x(t) -n + 1 , for t ∈ [t n , t n+1 ] . (8.3)
Similarly, since R θ ′′ (x) dx = 0 and R xθ ′′ (x) dx = -1, we have

s * - ∞ 0 θ ′′ (t -τ )x(τ ) dτ = ∞ 0 θ ′′ (t -τ ) x(t) -x(τ ) -s * (t -τ ) dτ ----→ t→+∞ 0 , hence |x ′ + (t) -s * | ≤ 1 if t ≥ 0
is sufficiently large. On the other hand, using the assumption (1.9) on the initial data and proceeding exactly as in the proof of Proposition 3.2, we see that there exists ξ

1 ∈ R such that sup z≤ξ 1 -t/ √ α z+1 z | u(x, t)| 2 + |u ′ (x, t)| 2 + |u(x, t) -1| 2 dx ≤ K 1 δ 1 e -µ 1 t . (8.4)
For all t ≥ 0, we set x -(t) = ξ 1 -2t/ √ α. Without loss of generality, we can assume that

x -(t) ≤ x + (t) for all t ≥ t 0 .

We next define, for all t ≥ t 0 ,

Φ(t) = R φ(x, t) α 2 | u(x, t)| 2 + 1 2 |u ′ (x, t)| 2 + V (u(x, t)) dx , where V (u) = V (u) -V (1) ≥ 0 and φ(x, t) =    e x-x -(t) if x ≤ x -(t) 1 if x -(t) ≤ x ≤ x + (t) e x + (t)-x if x ≥ x + (t) . A direct calculation shows that Φ ′ (t) = - R φ(x, t)| u(x, t)| 2 dx - x -(t) -∞ φ x ′ -(t) α 2 | u| 2 + 1 2 |u ′ | 2 + V (u) + uu ′ dx + ∞ x + (t) φ x ′ + (t) α 2 | u| 2 + 1 2 |u ′ | 2 + V (u) + uu ′ dx .
As is clear from (8.4), the second integral in the right-hand side converges to zero as t → +∞. Since x + (t) -x(t) → -∞ by (8.3), this is also the case for the last integral. Indeed, as v * (x) → 1 when x → -∞, it follows from (8.2), (8.3) that (u, u)(x

+ (t) + •, t) converges to (1, 0) in H 1 loc (R) × L 2 loc (R) as t → +∞.
Since Φ(t) ≥ 0 for all times, we conclude that, given any T > 0, there exists a sequence t ′ n → +∞ such that

t ′ n +T t ′ n -T R φ(x, t)| u(x, t)| 2 dx dt ---→ n→∞ 0 . (8.5)
Now, we claim that

sup z∈[x -(t ′ n ),x + (t ′ n )] z+1 z | u(x, t ′ n )| 2 + |u ′ (x, t ′ n )| 2 + |u(x, t ′ n ) -1| 2 dx ---→ n→∞ 0 . (8.6)
Assume on the contrary that, after extracting a subsequence, the left-hand side of (8.6) is bounded from below for all n ∈ N by some ε > 0. Then, given any ε ′ ∈ (0, ε), we can find a sequence {z n } such that z

n ∈ [x -(t ′ n ), x + (t ′ n )] for all n ∈ N and 1 0 | u(z n + x, t ′ n )| 2 + |u ′ (z n + x, t ′ n )| 2 + |u(z n + x, t ′ n ) -1| 2 dx = ε ′ .
Moreover, using (8.2), (8.4), and the definitions of x -(t), x + (t), we see that

x + (t ′ n ) -z n → +∞ and z n -x -(t ′ n ) → +∞ as n → ∞.
Without loss of generality, we can assume that ε ′ > 0 is sufficiently small so that the following property holds: if w : R → R is a bounded solution of the differential equation w ′′ -V ′ (w) = 0 such that |w(0) -1| 2 ≤ 2ε ′ , then w(x) = 1 for all x ∈ R. The existence of such an ε ′ follows from our assumptions (1.3)-(1.6) on the potential V .

Using once again Proposition 2.3, we can assume that, after extracting a subsequence, the sequence of functions (u, u)

(z n + •, t ′ n + •) converges in the space C 0 ([-T, T ], H 1 loc (R) ×
L 2 loc (R)) to some limit (w, ẇ) which satisfies (1.1). In view of (8.5), we have ẇ ≡ 0, so that w : R → R is a bounded solution of the differential equation w ′′ -V ′ (w) = 0. Moreover,

|w(0) -1| 2 ≤ 2 1 0 (|w ′ (x)| 2 + |w(x) -1| 2 ) dx = 2ε ′ ,
hence our assumption on ε ′ implies that w ≡ 1, which is clearly absurd. Thus (8.6) is established, and using in addition (8.2), (8.4) we see that (8.1) holds for the sequence {t ′ n }. This concludes the proof.

Local stability of the travelling front

The aim of this final section is to show that the family (1.8) of travelling fronts of (1.1) is asymptotically stable with shift in the uniformly local energy space X = H 1 ul (R) × L 2 ul (R). Together with Proposition 8.1, this will conclude the proof of Theorem 1.1. Whereas a lot is known about local stability of travelling fronts in parabolic systems (see e.g. [START_REF] Sattinger | On the Stability of Waves of Nonlinear Parabolic Systems[END_REF]), for the hyperbolic equation (1.1) with a bistable potential we are only aware of the note [START_REF] Th | Convergence to Travelling Waves in Damped Hyperbolic Equations[END_REF] where local stability in the usual energy space H 1 (R) × L 2 (R) is briefly discussed.

From now on, we fix c = c * and we denote by h : R → (0, 1) the unique solution of (1.7) such that h(0) = ε 0 . Linearizing (1.13) at the steady state u c = h, we obtain the evolution equation αu tt + u t -2αcu yt = u yy + cu y -g(y)u , (

where g(y) = V ′′ (h(y)). Proceeding as in Section 2, it is straightforward to verify that (9.1) defines a C 0 -group {S(t)} t∈R of bounded linear operators in X, the generator of which is the linear operator A given by

D(A) = Y , A = 1 α 0 α ∂ 2 y + c∂ y -g(y) -1 + 2αc∂ y , (9.2)
where Y = H 2 ul (R) × H 1 ul (R). By translation invariance, λ = 0 is an eigenvalue of A with eigenfunction (h ′ , 0). This eigenvalue is in fact simple, and the corresponding spectral projection reads

Π u v = N h ′ 0 R (ψ 1 u + ψ 2 v) dy , (9.3)
where ψ 2 (y) = e cy h ′ (y), ψ 1 = α -1 ψ 2 + 2cψ ′ 2 , and N > 0 is a normalization factor. One can check that ψ 1 , ψ 2 decay exponentially to zero as |y| → ∞. The main result of this section is: Proposition 9.1. There exist positive constants C 0 and ν such that S(t)(1 -Π) L(X) ≤ C 0 e -νt , for all t ≥ 0 .

(9.4) Using Proposition 9.1 and classical arguments which can be found in [START_REF] Sattinger | On the Stability of Waves of Nonlinear Parabolic Systems[END_REF] or [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF], it is easy to show that the family of all translates of the steady state (h, 0) is normally hyperbolic and asymptotically stable for the evolution defined by (1.13) on X. In other words, any solution of (1.13) whose initial data lie in a small tubular neighborhood of this family of equilibria (in the topology of X) converges exponentially fast as t → +∞ to some element of the family. Now, if u(x, t) is a solution of (1.1) satisfying the assumptions of Theorem 1.1, Proposition 8.1 shows that the corresponding solution of (1.13) eventually enters such a tubular neighborhood. Thus there exists x 0 ∈ R such that Obviously g ∞ (y) ≥ m for all y ∈ R, where m = min(V ′′ (0), V ′′ (1)) > 0. Let A ∞ be the linear operator on X obtained by replacing g with g ∞ in the definition (9.2) of A, and let S ∞ (t) be the C 0 -group generated by A ∞ . For any u ∈ H 1 ul (R), the map y → (g(y)-g ∞ (y))u(y) belongs to H 1 (R) and converges exponentially to zero as y → ±∞. In particular, the linear operator A -A ∞ : (u, v) → α -1 (0, (g ∞ -g)u) is compact in X. Moreover, the group S ∞ (t) is bounded in H 2 (R) × H 1 (R) and, due to the finite speed of propagation, it preserves the exponential decay in space. Therefore, the Duhamel perturbation formula

S(t)w = S ∞ (t)w + t 0 S ∞ (t -τ )(A -A ∞ )S(τ )w dτ , w ∈ X ,
shows that S(t) -S ∞ (t) is compact in X for any t ∈ R. In particular, S(t) and S ∞ (t) have the same essential spectrum (in what follows, we use the notion of essential spectrum adopted in [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF]). The first step in the proof of Proposition 9.1 is: Lemma 9.2. There exist positive constants C 1 and ν ∞ such that S ∞ (t) L(X) ≤ C 1 e -ν∞t , for all t ≥ 0 .

Proof: We proceed as in the proof of Proposition 2.1. Let ρ(y) = exp(-κ|y|), where κ > 0 is small enough so that 8καc ≤ 1, 3κ ≤ c, and 4κc ≤ m. If u(x, t) is a solution of the evolution equation αu tt + u t -2αcu yt = u yy + cu y -g ∞ (y)u associated with S ∞ (t), we define for all ξ ∈ R and all t ≥ 0: Taking the imaginary part of both sides, we find z 2 = βc, because 1 + 2αz 1 > 0 by assumption. Using this information and taking the real part of (9.8), we arrive at

Ê(ξ, t) = R ρ(y -ξ)
α z 1 + 1 2α 2 + α + 1 c 2 2 z 2 2 = 1 4α -m . (9.9)
This is clearly impossible if 4αm > 1. In the converse case, equation (9.9) defines an ellipse in C which is entirely contained in the half-plane {Re(z) ≤ µ α }, thus contradicting our assumption on z.

Proof of Proposition 9.3: Assume that λ ∈ σ(A) satisfies Re(λ) > µ α . Then λ is an eigenvalue of A, and there exists a nonzero u ∈ H 2 ul (R) satisfying (9.7). Since g(y) converges exponentially to V ′′ (0) as y → +∞, we know that u(y) = Aφ 1 (y) + Bφ 2 (y) for some A, B ∈ C, where φ 1 , φ 2 are particular solutions of (9. In particular, we have µ ∈ R, hence λ ∈ R because Re(λ) > -1/(2α). Furthermore, since µ = 0 is an eigenvalue of L with eigenfunction U(y) = e γy h ′ (y) < 0, we know from Sturm-Liouville theory that all the other eigenvalues of L are strictly negative. Finally, it follows from Bargmann's bound and the min-max principle that L has only a finite number of eigenvalues, see e.g. [START_REF] Simon | Schrödinger operators in the twentieth century[END_REF]. We conclude that the spectrum of A in the halfplane {z ∈ C | Re(z) > µ α } consists of the eigenvalue λ = 0 and, possibly, of a finite number of negative eigenvalues.

It is now easy to conclude the proof of Proposition 9.1. We know from Proposition 9.3 that λ = 0 is a simple, isolated eigenvalue of A, and that the rest of the spectrum lies in the half-plane {Re(λ) ≤ -ν} for some ν > 0. Going back to the semigroup S(t) generated by A, we infer that for any t > 0 the spectrum of S(t) is entirely contained in the disk {z ∈ C | |z| ≤ e -νt }, where ν = min(ν ∞ , ν) and ν ∞ is as in Lemma 9.2, except for the simple eigenvalue z = 1 which is due to the translation invariance. If we remove that eigenvalue by restricting S(t) to the invariant subspace ker Π, where Π is the spectral projection (9.3), we obtain estimate (9.4), possibly with a slightly smaller ν. The proof of Theorem 1.1 is now complete.

Proposition 4 . 3 . 4 ∞ y 1 e 1 e 1 e 2 y 2 y 1 e 2 y 1 e 2 ∞ y 1 e 1 e 1 e 1 e

 4341112212121111 Let c > 0 and let v c ∈ H 1 c (R). Then e cy |v c (y)| 2 → 0 as y → +∞. Moreover, for any y 1 ∈ R, c 2 cy |v c (y)| 2 dy ≤ ∞ y cy |v ′ c (y)| 2 dy , (4.15) and ce cy 1 |v c (y 1 )| 2 ≤ ∞ y cy |v ′ c (y)| 2 dy . (4.16)Proof: A simple integration shows that, for all y 1 ≤ y 2 ,e cy 2 |v c (y 2 )| 2 -e cy 1 |v c (y 1 )| 2 = cy v ′ c (y)v c (y) dy + c y cy |v c (y)| 2 dy .When y 2 goes to +∞, both integrals in the right-hand side have a finite limit since v c ∈ H 1 c (R). Thus the first term in the left-hand side also has a limit, which is necessarily zero since y → e cy |v c (y)| 2 ∈ L 1 (R). It follows thate cy 1 |v c (y 1 )| 2 ≤ cy |v ′ c (y)v c (y)| dy -c ∞ y cy |v c (y)| 2 dy .(4.17)Now, for any d > -c, we have|2v c v ′ c | ≤ (c + d)|v c | 2 + 1 c+d |v ′ c | 2 .Inserting this bound in (4.17) we finde cy 1 |v c (y 1 )| 2 ≤ 1 c + d ∞ y cy |v ′ c(y)| 2 dy + d ∞ y cy |v c (y)| 2 dy , from which (4.15) follows by taking d = -c/2 and (4.16) by choosing d = 0.

Lemma 5 . 2 .

 52 One has lim sup t→+∞

Lemma 5 . 3 .

 53 One has lim sup t→+∞

Lemma 5 . 4 .

 54 One has lim inf t→+∞

Fig. 2 :

 2 Fig. 2: The behavior of the energy along the broken line.

Corollary 7 . 3 .

 73 The invasion speed satisfies s ∞ = s * ≡ c * / 1 + αc 2 * , and we have R e c * √

  u c (•, t) -h(• -x 0 ) H 1 ul + uc (•, t) L 2 ul = O(e -νt ) , as t → +∞ ,(9.5) which implies (1.11) and concludes the proof of Theorem 1.1. It remains to prove Proposition 9.1. Let g ∞ (y) = V ′′ (1

4 U

 4 7) satisfying lim y→+∞ φ 1 (y) e γy e √ γ 2 +δ + y = 1 , lim y→+∞ φ 2 (y) e γy e - √ γ 2 +δ + y = 1 , where γ = c 2 (1 + 2αλ) and δ + = V ′′ (0) + λ(1 + αλ), see [3, Section 3.8]. But Lemma 9.4 implies that Re γ 2 + δ + > Re(γ) > 0, hence we must have B = 0 because φ 2 (y) is unbounded as y → +∞. Thus u(y) = Aφ 1 (y) ≈ A e -γy e - √ γ 2 +δ + y , as y → +∞ , and a similar argument shows that u(y) ≈ C e -γy e √ γ 2 +δ -y , as y → -∞ ,for some C ∈ C, where δ -= V ′′ (1) + λ(1 + αλ). These obervations reveal in particular that the bounded solutions of (9.7) form a one-dimensional family, hence λ is a simple eigenvalue of A. Moreover, if we set U(y) = e γy u(y), then U(y) decays exponentially to zero as y → ±∞, and a direct calculation shows that U solves the differential equationU ′′ -g(y) + c 2 = λ(1 + αλ)(1 + αc 2 )U , y ∈ R . (9.10) Thus µ = λ(1+αλ)(1+αc 2 ) is an eigenvalue of the selfadjoint operator L = ∂ 2 y -(g+c 2 /4).

  1 n is the variation of the energy E cn (y 0 , t) at a fixed point y 0 ∈ R on the time interval [t n , t n+1 ]. By (4.9) and Proposition 4.2, we have cn (y cn (t n ), t) dt , and |R cn (y cn (t n ), t)| ≤ K 4 e -µt/2 e -cny cn (tn) max(E cn (0, t n ) , 1) . But E cn (0, t n ) is bounded by from above uniformly in n by Proposition 4.2, and since y cn (t n )/t n converges to zero as n → ∞ we can assume without loss of generality that c n y cn (t n ) ≥ -µt n /4 for all n ∈ N. Thus cny | vcn (y cn (t n+1 ) + y, t)| 2 dy dt -C 3 e -µtn/4 , for some C 3 > 0. Moreover, since |c n -c ∞ | ≤ η/2, the proof of Lemma 7.2 shows that, for all t ∈ [t n+1 -T, t n+1 ], c∞y | vc∞ (y c∞ (t n+1 )+y, t)| 2 dy dt ≤ C 4 |c n -c ∞ | , for some C 4 > 0. Combining both estimates and using the assumption (7.3), we thus obtain ∆ 1 n ≥ δ -C 4 T |c n -c ∞ | -C 3 e -µtn/4 , n ∈ N . On the other hand, the quantity ∆ 2 n represents the change in the energy E c (y c (t n+1 ), t n+1 ) when c varies from c n to c n+1 . By Lemma 7.2, we have |∆ 2 n

	∆ 1 n = (1 + αc 2 n )	t n+1 tn	R	e cny | vcn (y cn (t n+1 ) + y, t)| 2 dy dt -	t n+1
	∆ 1 n ≥	t n+1 t n+1 -T R

tn R e R e cny | vcn (y cn (t n+1 )+y, t)| 2 dy dt -R e

  that |v(x(t) -L, t + τ )| > ε 0 , and sup x≥L |v(x(t) + x, t + τ )| < ε 0 .By the definition (3.13) of the invasion point, this means that x(t+τ ) ∈ [x(t)-L, x(t)+L]. Using (7.5) with x = x(t + τ ) -x(t) and recalling that v(x(t + τ

  α 2 u 2 t + αu 2 y + αg ∞ u 2 + 1 2 u 2 + αuu t (y, t) dy . Since |αuu t | ≤ 1 2 (u 2 + α 2 u 2 t ), there exists a constant C 1 ≥ 1 such that

	C -1 1	(u(•, t), u(•, t)) 2 X ≤ sup ξ∈R	Ê(ξ, t) ≤ C 1 (u(•, t), u(•, t)) 2 X .	(9.6)

Proposition 6.1. There exists a positive constant η such that, for all c ∈ [c ∞ -η, c ∞ + η], the energy E c (y c (t), t) is a bounded function of t ≥ 0.

Remarks:

1. It is important to realize that, unlike in the previous sections, we do not consider in Proposition 6.1 the energy E c (y 0 , t) located at some fixed point y 0 in the moving frame, but the energy E c (y c (t), t) located at the invasion point. From (4.18) we know that E c (y c (t), t) ≥ (min V )/c for all t ≥ 0, hence the only problem is to find an upper bound. If c is close to c ∞ , it follows from Proposition 4.2 (with c + = c ∞ ) that E c (y c (t 0 ), t) is bounded from above for all t ≥ t 0 ≥ 0. Thus, using the relation

In agreement with Proposition 6.1, we thus expect that the energy E c (y c (t), t) stays bounded for all times if c is close to c ∞ = c * , and blows up if c > c h . 3. The fact that the conclusion of Proposition 6.1 holds not only for c = c ∞ but for all c in a neighborhood of the invasion speed is one of the key points of our convergence proof. It will allow in Section 7 to control the variation of the energy E c (y c (t), t) when the parameter c is increased, a difficult task due to the exponential weight e cy in (4.7). This problem is completely avoided in the alternative approach of [START_REF] Risler | Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure[END_REF] where only bounded weights are used.

The first step in the proof of Proposition 6.1 consists in showing that the invasion point x(t) cannot make arbitrarily large jumps to the left. Lemma 6.2. There exists η > 0 such that, for all c ∈ (c ∞ -η, c ∞ ), there exists a positive constant M c such that,

Proof: Let η be the positive constant given by Proposition 4.2 for c + = c ∞ . To prove (6.3), we argue by contradiction. Assume that there exist a speed c ∈ (c ∞ -η, c ∞ ) and two sequences of times {t n } and {t ′ n } such that t ′ n > t n ≥ 0 for all n ∈ N and

we know that y c (t) → +∞ as t → +∞, and therefore we must have t n → +∞ as n → ∞. Thus, if we fix any T > 0, we can apply Proposition 2.3 and assume without loss of generality that the sequence of functions (v, v)(x(t n ) + •, t n + •) converges in the space C 0 ([-T, 0], H 1 loc (R) × L 2 loc (R)) toward some limit (w, ẇ) which satisfies (1.1).

On the other hand, differentiating Ê(ξ, t) with respect to t and using our assumptions on κ, we obtain

for some ν ∞ > 0. Thus Ê(ξ, t) ≤ Ê(ξ, 0) e -2ν∞t for all t ≥ 0, and using in addition (9.6) we obtain the desired estimate.

Since S(t) is a compact perturbation of S ∞ (t) for any t ≥ 0, it follows from Lemma 9.2 that the spectrum of S(t) outside the disk {z ∈ C | |z| ≤ e -ν∞t } consists of isolated eigenvalues with finite multiplicities. By the spectral mapping theorem, any such eigenvalue has the form e λt , where λ ∈ C is an eigenvalue of the generator A. If w = (u, v) ∈ Y satisfies Aw = λw, it follows from (9.2) that v = λu and

It remains to study the nonlinear eigenvalue problem (9.7). Let

where m = min(V ′′ (0), V ′′ (1)) > 0. One can check that µ α = sup{Re(λ) | λ ∈ σ ess (A)}.

The key observation is:

where k ∈ N depends on α and V .

The proof of Proposition 9.3 relies on the following elementary result: Lemma 9.4. Assume that z ∈ C satisfies Re(z) > µ α , and let

Then Re γ 2 + δ > Re(γ) > 0.

Proof: Since Re(z) > µ α ≥ -1/(2α), it is clear that Re(γ) > 0. On the other hand, if z > 0, we have γ > 0 and δ > 0, hence γ 2 + δ > γ. Thus, to prove Lemma 9.