Traveling fronts guided by the environment for reaction-diffusion equations

This paper deals with the existence of traveling fronts guided by the medium for a KPP reaction-diffusion equation coming from a model in population dynamics in which there is spatial spreading as well as genetic mutation of a quantitative genetic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of a threshold value on the existence of a nonzero asymptotic profile (a stationary limiting solution). When a nonzero asymptotic profile exists, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case. We also study here the bistable case. The equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if the area where theere is reaction is large enough and the non-existence if it is too small.


Introduction
This paper deals with the existence of bounded traveling fronts for the reactiondiffusion equation The function h will be of three different forms in this paper. The first two concern non-linear terms h(y, u) = f (u) − αg(y)u where f : R → R is C 1 , and either of KPP type, or of bistable type and g : R N −1 → R + is C 0 , g(0) = 0 and g |y|→+∞ −−−−−→ +∞. The existence of traveling front depends on the value of α > 0. The third case we consider here is when h(y, u) = f (u) for |y| ≤ L 1 and h(y, u) ≤ −mu for |y| ≥ L 2 where 0 < L 1 ≤ L 2 < ∞ are given parameters and f is of bistable form and h(y, u) + mu → 0 for |y| → +∞. We study the existence of traveling fronts depending on the value of L 1 and L 2 .
The problems we study in this paper bear some similarities with the question of traveling fronts in cylinders of [3]. However there are important differences that have to do with the fact that the cross section in [3] was bounded and only the Neumann condition was considered there. Whereas here, the problem is posed in the whole space and the solution vanish at infinity in directions orthogonal to the direction of propagation. We follow the same general scheme as in [3] and in particular the sliding method. But some new ideas are also required. In particular, first, we treat directly the KPP case without the approximation of the KPP non-linearity by a combustion non-linearity as in [3]. Then in the approach of Berestycki -Nirenberg [3] to traveling fronts in cylinders for the bistable case, a useful result of H. Matano [27] was involved in the proof. Here, we rely on stability ideas but also use energy minimization properties to bound the speed of the solution in the finite domain approximation. In particular, we do not use the precise exponential behavior that was used in [3]. Actually the developments of this method that we present in this paper can be used to somewhat simplify parts of [3]. They can also be applied to traveling fronts in cylinder with Robin or Dirichlet boundary conditions. 1 Equation (1) in the first case comes from a model in population dynamics [20] that we briefly describe now. Let u(t, x, v) represent the density of individuals at time t and position x that possess some given quantitative genetic trait represented by a continuous variable v ∈ R. The latter could be for example the size of wings or the height of an individual. We assume that individuals follow a brownian motion (i.e. they diffuse) in space with a constant diffusion coefficient ν, reproduce identically and disappear with a growth rate k(x, v) that depends on the position x and on the trait v. Furthermore, they also reproduce with mutation that is represented by a kernel K(x, v, w) and disappear due to competition with a constant L > 0. Thus, one is led to the following equation for u: We assume moreover that there exists a most adapted trait φ = φ(x) that may depends on the location x. The farther the trait of an individual is from the most adapted trait, the larger the probability of dying and not reproducing. Thus the growth rate can be written for example as k(x, v) = a − b |v − φ(x)| 2 with a and b > 0. Non-local reaction-diffusion equations of this type are quite difficult to handle from a mathematical standpoint as shown in [12] where behaviors very different from 1 The construction of traveling fronts for Neumann and Dirichlet conditions in cylinders given by [41] appears to be incomplete. Indeed, the continuity of the function φ on page 515 is not established before using Dini's Theorem to derive Lemma 3.2 there. those in local equation are brought to light. In a forthcoming numerical study [5], we show that depending on the value of α, the traveling fronts may not be monotonous anymore. This fact was also established in a different context for some related types of non-local equations in [12]. This leads us to introduce a simplified version of this model that emphasizes propagation guided by the environment. First, we assume that mutations are due to a diffusion process represented by a Brownian motion in the space of trait v. Furthermore, we assume that φ is linear. Then a rotation in the variables (x 1 , y) allows one to reduce the problem to the case where the most adapted trait is y = 0. Therefore we assume φ(x) = 0 and (2) can be rewritten as Lastly we assume that competition is only between individuals sharing the same trait which leads us to equation Equation (1) is a generalization of this equation. In [20], the authors observe numerically a generalized transition front spreading along the graph of φ for equation (2) (see [8,9,10,28] or [37] for the definition of generalized transition fronts). Here we want to prove theoretically (i) that there exists such a front for equation (1) at least for some values of the parameter α > 0 and (ii) that extinction occurs if α is too large. The latter condition can be interpreted as saying that the "area" of adapted traits is too thin compared to the diffusion. To remain consistent with the biological motivation, we only consider here non-negative and bounded solutions of (1).
Other types of models related to this one have been proposed in the literature. For example, the model developed by Kirkpatrick and Barton in 1997 [25] also studies the evolution of a population and of its mean trait. The main difference is that they have a system in u and v where u represents the population and v the mean trait is described by a specific equation. This model has been further explored many times since [22,24]. It is worth noting that these models use the same type of non-linearity for the adaptation to the environment and model the mutation with the Laplace operator as well rather than the integral operators.
This type of reaction-diffusion process in heterogeneous media also arises in many contexts in medicine. An important class of such models was treated in [15,33]. They deal with the propagation of a cortical spreading depression (CSD) in the human brain. These CSD's are transient depolarizations of the brain that slowly propagate in the cortex of several animal species after a stroke, a head injury, or seizures [38]. They also are suspected of being responsible for the aura in migraines with aura. CSD's are the subject of intensive research in biology since experiments blocking them during strokes in rodents have produced very promising results [19,30]. These observations however have not been confirmed in humans and the existence of CSD's in the human brain is still a matter of debate [29,23,1,39]. Since very few experiments and measurements on human brain are available be it for ethical or for technical reasons, mathematical models of a CSD is help in understanding their existence and conditions for propagation. In such a problem, the morphology of the brain and thus the geometry of the domain where CSD's propagate, is believed to play an important role. The brain is composed of gray matter where neuron's soma are and of the white matter where only axons are to be found. The rodent brain (on which most of the biological experiments are done) is rather smooth and composed almost entirely of gray matter. On the opposite, the human brain is very tortuous. The gray matter is a thin layer at the periphery of the brain with much thickness variations and convolutions, the rest of the brain being composed of white matter. According to mathematical models of CSDs [17,38,36,40], the depolarization amplitude follows a reaction-diffusion process of bistable type in the gray matter of the brain while it diffuses and is absorbed in the white matter of the brain. The modeling of CSD hence leads one to the study of equations of the following type: Here, f is of bistable type and |y| = L corresponds to the transition from gray matter to white matter. In [15], this equation was studied to prove that the thinness of the human gray matter (L small) may prevent the creation or the propagation of CSDs on large distances. It was proved by studying the energy in a traveling referential of the solution of (5) with a specific initial condition. The special case of (5) for N = 2 was described more completely in [18]. In [33], a numerical study shows that the convolutions of the brain have also a strong influence on the propagation of CSD. Finally, in [16], the effect of rapid variations of thickness of the gray matter was studied.
Finally, let us note that the same kind of equation arises in the modeling of tumor cords but with a slightly more complicate KPP non-linearity. We plan to investigate this model in our forthcoming work [6].
As already mentioned, the study of propagation of fronts and spreading properties in heterogeneous media is of intense current interest. For instance, the existence of fronts propagating in non-homogeneous geometries with obstacles has been established in Berestycki, Hamel and Matano [7]. Definitions of generalized waves have been given by Berestycki and Hamel in [9] and [10] where they are called generalized transition waves. Somewhat different approaches to generalizing the notions of traveling fronts have been proposed by H. Matano [28] and W. Shen [37]. The existence of fronts for non-homogeneous equations are established in [31] and [42].
Let us first introduce some notations before stating the main results.
Notations. We note x = (x 1 , y) ∈ R N where x 1 ∈ R and y ∈ R N −1 . Hence x is the space variable in R N , x 1 is its first coordinate and y is the vector of R N −1 composed of all the other coordinates of x. As usual B R = B(0, R) denotes a ball of radius R centered at 0, but here it will always mean the ball in R N −1 .
First we are interested in solutions of with α > 0. We will assume that f : R → R is C 1 and satisfies either one of the following conditions: The first case will be referred to as the KPP case and the second one will be called bistable case. Since we are only interested in solutions of (6) in [0, 1], we will further assume that f (s) ≤ 0 for s ≥ 1. Moreover we always assume and lim |y|→+∞ g(y) = +∞.
This paper is concerned with the long term behavior of (6) and with the existence of curved traveling fronts, i.e. solutions u(t, x) = U (x 1 −ct, y) with c ∈ R a constant and U : R N → R such that the limits lim s→±∞ U (s, .) exist uniformly and are not equal. Regarding these fronts, our main results are the following. Theorem 1.1. If f is of KPP type, there exists α 0 > 0 such that: • For α ≥ α 0 , there exists no traveling front solution of (6), • For α < α 0 there exists a threshold c * > 0 such that there exists a traveling front of speed c of equation (6) if and only if c ≥ c * .
This existence theorem gives us information on the behavior of the solution of the parabolic problem. In this paper we prove the following theorem: This means that there is a threshold value α 0 such that for α ≥ α 0 , there is extinction. On the contrary, when α ≤ α 0 , there is spreading and the state V (y) invades the whole space. The asymptotic speed of spreading is then c * . The property of asymptotic spreading is in the same spirit of the theorem of asymptotic speed of spreading in cylinders established by Mallordy and Roquejoffre in [26].
Regarding the case of bistable f we have the following result: • For α ≥ α * , there exists no traveling front solution of (6), • For α < α * , under condition 40 of Section 6, there exists a traveling front u of speed c > 0 solution of (6).
Lastly, the model for CSD's leads one to equations of the type where h(y, u) that verifies where 0 < L 1 ≤ L 2 < ∞ and m > 0 are given parameters and f is of bistable form.
In this paper we prove the following Theorem.
• For L 1 > L * (independently of L 2 ), assuming that there is a unique stable asymptotic profile of (50), there exists a traveling front of speed c > 0 solution of (9).
This result completes the study in [15] on the existence of CSD in the human brain. Indeed in [15] the transition from gray to white matter was instantaneous when biologically there is a smooth transition from gray to white matter. This Theorem confirms the intuition that CSD's can be found in part of the human brain where the gray matter is sufficiently thick but they can not propagate over large distances due to a thin gray matter in many parts of the human brain.
The paper is organized as follows. In section 2 we state some preliminary results that will be used in the sequel. Section 3 is dedicated to the study of the existence and uniqueness of non-zero asymptotic profiles for a traveling front solution of (6). In section 5 we study the large time behavior. There we prove extinction if α ≥ α 0 and convergence towards the front of minimal speed if α < α 0 . Then, section 6 is devoted to the study of the asymptotic profiles in the bistable case and section 7 to the existence of traveling front for α < α * in the bistable case. Lastly, in section 8 we describe the precise problem arising in the modeling of CSD's and state our main result in this framework.

Preliminary results
In our proofs, we will several times need the exponential decay of the asymptotic profile which can be easily proved from the following theorem established in [14].
loc (R N ) be a positive function. Assume that there exists γ > 0 and C > 0 such that Then, lim This result is established in [14], lemma 2.2. In the context of equation (1), we thus have the following corollary.
Then, for any γ > 0 there exists C > 0 such that Proof. The estimate on v comes directly from Theorem 2.1 and the estimate on |∇v| derives from standard global L p estimates.
3 The case of a KPP non-linearity. Asymptotic profiles.
In this section, we are interested in the asymptotic profiles of a traveling front solution of (6) as x 1 → ±∞. Hence, we are looking for solutions of the following equation We assume that f : R → R is C 1 , and f (s) < f ′ (0)s for s ∈ (0, 1) and s ∈ (0, 1] → f (s) s decreasing.
Since the constant function 0 is always solution, the problem is to know when there exist non-zero solutions. As we will see here, the existence of such a positive asymptotic profile is characterized by the sign of the principal eigenvalue of the linearized operator around 0. We now make precise this notion.

Principal eigenvalue of the linearized operator
To start with, let us define the natural weighted space The linearized operator about 0 is Lϕ = −∆ϕ + αg(y) − f ′ (0) ϕ for ϕ ∈ H. We are interested in the eigenvalues of L. Even though the problem is set on all of R N −1 , the term in αg(y) yields compactness of the injection H ֒→ L 2 (R N −1 ). Hence the existence of a principal eigenvalue is obtained as usual.
The operator L has a smallest eigenvalue Moreover there exists a unique positive eigenfunction associated to λ α of L 2 -norm equal to 1, called ϕ α in the following,. The eigenspace associated to λ α is spanned by ϕ α .
Proof. The proof is classical due to the compactness of H ֒→ L 2 (R N −1 ). We refer for example to [21].
Remark 1. If g(y) = |y| 2 , the problem can be rescaled and we obtain the harmonic oscillator for which principal eigenvalue and eigenfunction are well known [35]. In Since the existence of a positive solution of (10) will depend on the sign of the principal eigenvalue, the following proposition describes the behavior of λ α as a function of α. Proposition 1. The function α → λ α is continuous, increasing and concave for α ∈ (0, +∞). Moreover lim α→0 λ α = −f ′ (0) and for α large enough λ α > 0.
Concavity is classical. It suffices to observe that for each fixed ϕ, is an affine function of α and that λ α = inf ϕ∈H\{0} R α (ϕ).

If g vanishes on B r 0
The main part of the proof is still correct if g vanishes on B r0 but the result is slightly modified.
In this section, we assume that there exists r 0 > 0 such that (7) is substituted by the following assumption We define λ ∆ the principal eigenvalue of the Laplacian on B r0 with Dirichlet boundary conditions, i.e.
In this case, the principal eigenvalue of the linearized operator about 0 is well defined and Proposition 1 becomes Proposition 2. The function α → λ α is continuous, increasing and concave for α ∈ (0, +∞), and lim α→0 λ α = −f ′ (0). Now there are two cases: Proof. The proof of the first part of the proposition is exactly the same as in Proposition 1. We just have to prove i) and ii).
i) We assume that f ′ (0) < λ ∆ and argue by contradiction that λ α ≤ 0 for any α ∈ (0, +∞). As in the proof of proposition 1, we have for α → +∞ and any R > r 0 . As before ϕ α is bounded in H and up to extraction, we have λ α → λ ≤ 0 and the weak convergence in H and strong convergence in ii) By taking ϕ = φ 0 in the Rayleigh quotient (13), where φ 0 is the principal eigenvalue of the above problem in B r0 with Dirichlet boundary conditions, we see In the following, we will not state the results specifically for this case (14). However, the proofs and results developed here carry over to this case with the obvious modifications.
Proof. Let us fix α ≥ α 0 . Then λ α ≥ 0. Assume by contradiction that there exists a solution V of (10). Then the strong maximum principle shows that V > 0.
Since ϕ α is an eigenfunction of the linearized operator L and V is solution of (10), we have Now from corollary 1, V and ∇V are rapidly decreasing for |y| → ∞ and so we can apply Stokes formula ∆V We now turn to the case α < α 0 . For α < α 0 , the eigenvalue λ α is negative.
Hence V is a sub-solution of (10). The constant function 1 is a super-solution and V ≤ 1 if ε is small enough. Therefore by the suband super-solution method, there exists a solution V such that 0 < V ≤ V ≤ 1. Now consider V and W two non-zero solutions of (10). We argue by contradiction and assume that V ≡ W . Then for example Ω = {y ∈ R N −1 , V (y) < W (y)} is not empty. Introduce a cutoff function β ∈ C ∞ (R) with β = 0 on (−∞, 1/2], β = 1 on [1, +∞) and 0 < β ′ < 4 on (1/2, 1) and for all ε > 0, let us set β ε (s) = β s ε . Using equation (10), we have by Lebesgue's dominated convergence theorem. Owing to corollary 1, V , ∇V , W and ∇W have exponential decay and thus Stokes formula can be applied and we obtain In the term I 2 the integrand satisfies Therefore by Lebesgue's Theorem of dominated convergence, we infer that I 2 → 0. Next the term I 1 satisfies I 1 ≥ 0. Consequently, we may write: which is a contradiction in view of (12) as W > V in Ω. Hence V = W and the non-zero solution is unique.
The last point concerns the stability of the asymptotic profiles for α < α 0 . Let us start by studying the energy of V . For w ∈ H, we define the energy For α < α 0 , the unique positive solution of (10) V is stable in the energy sense, i.e. V is the global minimum of J α and, furthermore J α (V ) < 0 = J α (0).
Proof. Owing to the maximum principle, solutions of (10) are between 0 and 1.
Hence we can modify f on ] − ∞, 0[ such that it becomes odd and as a consequence, F can be considered as even. Since λ α the principal eigenvalue of the linearized operator about the zero solution is negative for α < α 0 , 0 cannot be the global minimum of J α . Now J α admits a global minimum that will be calledṼ for the argument. One can prove that |Ṽ | is also a global minimum of J α and hence |Ṽ | is a positive solution of (10). By uniqueness, |Ṽ | = V and thus V is a global minimum of J α . Since 0 is not a global minimum, necessarily J α (V ) < 0 = J α (0).
We now conclude with the linearized stability of V . Proof. Denote by ψ a positive eigenfunction associated with λ 1 [V ] and assume by contradiction that (10). From there, it would follow that there exists a solution of (10) between V + εψ and 1 but this contradicts the uniqueness of V .

Traveling fronts for a KPP non-linearity
This section is devoted to the definition of a speed c * for which a traveling front of equation (6) exists for α ∈ (0, α 0 ). The threshold of existence of the non-zero asymptotic profile is called α 0 as in the previous section. For 0 < α < α 0 , V denotes the unique non-zero asymptotic profile. As shown in the previous section, the energy of the non-zero profile J α (V ) is negative. A curved traveling front of speed c is a function u(x 1 − ct, y) solution of equation (6) and connecting the non-zero asymptotic state V to 0. Thus we are looking for a solution of where c ∈ R is also an unknown of the problem.
The construction of c * in Theorem 1.1 uses the sliding method following ideas of [3]. Note however that there are important differences with [3]. In that paper, the KPP case is derived by first solving the "combustion non-linearity" and then approach the KPP non-linearity as a limiting case of truncated functions. Contrary to [3] here, we derive directly the existence of a solution of the KPP case. Actually the method we present here can be applied to somewhat simplify the proof of [3] in the KPP case for cylinder with Neumann conditions. 4.1 Problem on a domain bounded in x 1 .
Let us fix a > 1 and c ∈ R for this subsection and consider the following problem: The aim of this subsection is to prove the following theorem: There exists a unique solution of (19), denoted u c a in the following. This solution decreases in the To prove this theorem, we require the following two propositions.
Proposition 3. Let u be a solution of (19).
Here we think of f as having been extended by 0 outside [0, 1]. Since f (s) ≤ 0 for all s ≥ 1, we observe that: • by the strong maximum principle, 0 < ψ R < M on B R .
• therefore ψ R tends to a function when R → +∞ and through local elliptic estimates, this function is a non-zero solution (≥ V ) of the asymptotic problem (10). By uniqueness, we obtain ψ R The solution u of (19) is a sub-solution of (21) and the constant function M is a super-solution. Using monotone iterations starting from the super-solution M , we build the same sequence as previously (for problem (20)) since by induction the solutions do not depend on x 1 ∈ (−a, a). Hence the sequence converges toward ψ R and we have u ≤ ψ R ≤ M . Now letting R → +∞ yields u ≤ V .
and u ≤ v on ∂Ω. Then u ≤ v on Ω.
Proof. By contradiction, suppose this is not true. Due to corollary 1 and proposition 3, u(x 1 , y) and v(x 1 , y) converge uniformly to 0 for |y| → +∞. Consequently, there exist (x 0 , y 0 ) ∈ Ω such that and subtracting the equation (22) with u from the one with v, we obtain Let us now turn to the proof of Theorem 4.1 using sliding method.
First u(x, y) = V (y) is a super-solution, 0 is a sub-solution and 0 ≤ u, so by monotone iterations, there exists a solution u of (19).
Proof of the lemma. By proposition 3, we have 0 ≤ u ≤ V (resp. 0 ≤ v ≤ V ) and using the strong maximum principle, we obtain 0 < u < V (resp. 0 Let us fix R > 0 such that g(y) > K α for y ∈ B R . By compactness and continuity of u and v, there exists This enables us to define Let us prove that h * = 0 and argue by contradiction that h * > 0. By continuity, Suppose that min with the definition of h * . Therefore min Writing in the usual way that u − v h * is solution of a linear elliptic equation in I h * × R N −1 and u − v h * ≥ 0 with u − v h * vanishing at the point (x * 1 , y * ), the strong maximum principle implies that u − v h * ≡ 0 which is impossible.
Applying the preceding lemma with h = 0 yields the uniqueness of the solution of (19). Taking u = v = u c a , one sees that u c a is monotone decreasing. Thus ∂ 1 u c a ≤ 0 and deriving equation (19) and applying once more the maximum principle gives ∂ 1 u c a < 0. It remains to study the behavior of u c a with respect to c. The continuity is deduced from the uniqueness of the solution and a priori estimates in the standard way. Now let c 1 < c 2 and denote by u 1 (resp. by u 2 ) the solution of (19) with c = c 1 (resp. c = c 2 ). Since ∂ 1 u 1 < 0, and u 1 > 0 is a super-solution of equation (19) with c = c 2 . By uniqueness of the solution, necessarily u 2 ≤ u 1 . Once more the strong maximum principle implies u 2 < u 1 .

Convergence to a solution on R N
Now that the equation is solved on a domain bounded in the x 1 -direction, the idea is to increase a up to infinity so that the domain tends to R N . However if c is chosen arbitrarily, the function u c a may converge toward the constant 0 or to V when a tends to infinity. Hence we adopt a normalization method as in [3]. The following theorem will define the value of the speed c depending on a to avoid those situations. We recall that since α < α 0 , λ α the principal eigenvalue of the linearized operator about the solution 0 is negative.
i) Assume c = 0. Let ϕ α be the positive eigenfunction of the linearized operator L associated with the first eigenvalue λ α < 0 and with the normalization Moreover by construction of V (cf section 3), v(−a, y) = ηϕ α (y) < V (y) if η is small enough. Then v(a, y) = 0 and v ≤ 1. Hence, v is a sub-solution of (19) for c = 0.
Proof of the lemma. We follow a similar proof to that of lemma 4: If the lemma does not stand, since kψ β − V tends to 0 at ∞, there exists y 0 ∈ By the choice of R, we get ∆(kψ β − V )(y 0 ) < 0 which yields a contradiction.
Let us now build the super-solution when c > 2 so it is indeed a super-solution of (19). Moreover Thus if a is large enough to have e − c 2 a < θ k , then for any c > 2 √ −λ α + ε, we get u c a (0, 0) < θ.
With the bounds on the speed c a it is now possible to pass to the limit as a tends to infinity. Proposition 5. There exists a sequence (a j ) j∈N such that a j → +∞, c aj → c * ∈ [0, 2 √ −λ α ] and u ca j aj → u in C 2 loc (R N ). The limit u is solution of Then u is necessarily a traveling front solution of (18) with c = c * .

Existence of traveling front for
In this section we still assume 0 < α < α 0 and we will prove the following theorem. Proof. We argue by contradiction and assume that there exists a traveling front u of speed c < 2 √ −λ α of (18). We are going to construct a small positive subsolution with compact support. To this end, we can find δ ∈ (0, f ′ (0)) such that c 2 + 4(λ α + 2δ) < 0 and η > 0 such that for all s ∈ [0, η], f (s) ≥ (f ′ (0) − δ)s.
We have already proved that for c < 2 √ −λ α , there exists no traveling front of speed c solution of (18) and that for c = c * = 2 √ −λ α there exists a traveling front of speed c. Let us prove that for any c > c * there exists at least a traveling front to conclude with Theorem 4.5. The proof goes as usual. We consider the following problem where u * is the traveling front of speed c * . The function u * (· + r, ·) is a strict supersolution of (26) (since c > c * ) when 0 is a strict sub-solution and 0 < u * (· + r, ·). Hence as in theorem 4.1, it can be proved that there exists a unique solution v r a of (26) and moreover ∂ x w r a < 0 and By uniqueness, w r a depends continuously on r ∈ R, so w r . Once again taking any sequence a n → +∞, up to an extraction u an → u in C 2 loc and u is a traveling front of speed c solution of (18).

The case of a KPP non-linearity. Asymptotic speed of spreading.
This section is concerned with the asymptotic behavior of the solutions of the parabolic problem where f is KPP and u 0 is an initial condition at least bounded.

Extinction for α ≥ α 0
Let us fix α ≥ α 0 . We recall that there is no positive asymptotic profile of (10). This section is devoted to the proof of this theorem.
Let us fix S = max(1, u 0 ∞ ). Then the constant functions 0 and S are respectively sub-and super-solutions of (27). Thus there exists u(t, x) a solution of (27) such that 0 ≤ u ≤ S. By the parabolic maximum principle, this solution is unique.
Let us define w the solution of (27) with the initial condition w(0, x) = S. Since the problem and the initial condition do not depend on x 1 , neither does w thus we will write w(t, y). By the maximum principle, 0 ≤ u ≤ w ≤ S and since S is a super-solution, ∂ t w ≤ 0. Thus w(t, y) and thus is a nonnegative asymptotic profile. Since α ≥ α 0 , W ≡ 0. So u(t, x) tends to 0 for t → +∞ uniformly in R N .

Spreading for α < α 0
In this section we assume α < α 0 . So there exists a critical speed c * of existence of traveling front for (18). We assume that u 0 ∈ C 0 0 (R N ), i.e. u 0 is continuous and compactly supported, and that u 0 < V where V is the positive asymptotic profile solution of (10). We will prove the spreading of the solution of (27) but we first need the following theorem.
Proof. Let us consider the generalized sub-solution with compact support w(x 1 , y) defined in (25). This is possible since c < c * . Up to a decrease of ε > 0, we can assume that w ≤ z on R N . Now by applying the sliding method to w τ where w τ (x 1 , y) = w(x 1 + τ, y) and z, one can prove that w τ ≤ z for all τ ∈ R. We can thus define ∀y ∈ R N −1 z(y) = inf x1∈R z(x 1 , y) ≥ 0 and state that z ≡ 0. Now z is a super-solution of (10) since z = inf h∈R z(· + h, ·) and an infimum of solutions is a super-solution. Finally as in section 3, we can build a positive sub-solution of (10) smaller than z and thus by monotone iteration we have a solution of (10) between these suband super-solution. By uniqueness of the positive solution, we obtain V ≤ z. And due to condition in (28), we have z ≡ V .
Let us now turn to the precise study of the spreading of the solution of (27) for any c with 0 ≤ c < c * lim Proof. Fix c > c * . Let U denote a traveling front of speed c * . Since U (x 1 , ·) → V for x 1 → −∞ locally uniformly, there exists L ∈ R such that U (x 1 − L, y) > u 0 (x 1 , y) for all (x 1 , y) ∈ R N . Now considering v(t, x) = U (x 1 − L − c * t, y) and applying the comparison principle, we have u(t, x) ≤ v(t, x) for all t ≥ 0 and x ∈ R N . Thus since U is decreasing in Since c > c * and U (x 1 , y) x1→+∞ −−−−−→ 0 uniformly in y ∈ R N −1 . We see that sup x1≥ct u(t, x) → 0 as t → +∞. Since u(t, −x 1 , y) satisfies the same equation (27), this shows that sup x1≤−ct u(t, x) → 0 as well as t → +∞. Thus (29) is proved.
Assume now c < c * . Let us first prove the following weaken version of (30): Proof of lemma 5.4. Let us assume that c ≥ 0, the proof being similar for c ≤ 0.
Let v(t, x 1 , y) = u(t, x 1 − ct, y). Then v satisfies the equation with the initial datum v(0, x 1 , y) = u 0 (x, y) ≥ 0 and ≡ 0. Hence by the parabolic maximum principle, for all (x 1 , y) ∈ R N v(1, x 1 , y) > 0. Now since c < c * , in (25), we constructed w(x 1 , y) ≥ 0 a stationary non-zero sub-solution of (32) with compact support and w could be chosen arbitrary small. Hence we can assume w ≤ v(1, ·, ·). So ifw is the solution of then by comparison principle, ∀t ≥ 1 ∀(x 1 , y) ∈ R N v(t, x 1 , y) ≥w(t−1, x 1 , y). Now since w is a sub-solution,w is increasing with respect to t and 0 ≤w(t, x 1 , y) ≤ V (y). Therefore, by standard elliptic estimates,w(t, x 1 , y) t→+∞ −−−−→ z(x, y) and z is a solution of (28). By theorem 5.2, we have z ≡ V and this complete the proof of the lemma since by the comparison principlew(t − 1, which yields (31).
Let us now prove (30), that is the uniform convergence to V in the expanding slab {x 1 ≤ ct}. We will only prove it for 0 ≤ x 1 ≤ ct. Indeed using as before u(t, −x 1 , y), the general result follows from the convergence in the set {0 ≤ x 1 ≤ ct}.
Let c with 0 < c < c * be fixed and let ε > 0 be given (arbitrarily small). For R > 0 sufficiently large, we know that the principal eigenvalue λ R α of the problem (24) above is such that λ R α < 0. Denote by ψ R > 0 the corresponding eigenfunction of (24). Under these conditions we know that there exists a unique solution V R (y) > 0 of the profile equation in B R with Dirichlet condition: (Compare e.g. [4]). Moreover, it is straightforward to show that V R is increasing with R and that lim R→+∞ V R (y) = V (y). Let us choose R > 0 sufficiently large so that for all y ∈ B R V (y) < ε and for all y ∈ B R 0 < V (y) − V R (y) < ε. The proof of the uniform convergence to V for c < c * will rest on the following Proposition.

Proposition 7.
Let c be such that 0 < c < c * . Then, with R chosen as above, there exists a solution v c (x 1 , y) defined for satisfying the following properties: Postponing the proof of this proposition, let us complete the proof of Theorem 5.3. Extending v c by 0 for x 1 ≥ 0 turns v c into a (generalized) sub-solution of equation (34) in the cylinder R × B R (see [11]). Therefore v c (x 1 − c(t − t 0 ), y) is a sub-solution of the equation (27) in this cylinder for all t 0 ≥ 0 and all c ∈ (0, c * ).
By Lemma 5.4 (applied here in the case c = 0), we can fix t 0 > 0 sufficiently large such that for t ≥ t 0 we have We fixc ∈ (c, c * ) and we consider v(t, x 1 , y) = vc(x 1 −c(t − t 0 ), y). In the region D = (0, +∞) × B R , u is a solution and v a sub-solution of equation (27) and for any time Moreover, u(t 0 , x 1 , y) ≥ v(t 0 , x 1 , y) = 0 in D. The comparison principle then yields Outside of B R we already know that 0 < u < V < ε for any t ≥ 0, x 1 ∈ R and |y| ≥ R. Therefore lim sup Since this is true for all ε > 0 (and for −ct ≤ x 1 ≤ 0), we have thereby established (30). It now remains to prove Proposition 7 which we carry now. As in (25), we construct a sub-solution of the equation (34) with compact support, namely: In comparison with (25), there is a translation in x 1 such that the support of w now lies in R − × B R .
For any b < 0, let z b be the solution of Since V R is a super-solution and 0 a sub-solution, there exists a solution of this problem. By the sliding method of [3], we know that this solution is unique and Next, for b < −L, wince w is a sub-solution, we also know that This allows us to pass to the limit when b → −∞. Clearly y). By the lower bound, v c (x 1 , y) > w(x 1 , y) which shows that v c (x 1 , y) > 0 in R * − ×B R . Since ∂ 1 v c ≤ 0 and v c ≡ 0, we also know that ∂ 1 v c < 0 in R * − × B R . Now since lim x1→−∞ v c (x 1 , y) must be a positive solution of (33). Hence by uniqueness we get v c (−∞, y) = V R (y). This completes the proof of Proposition 7 and therefore of Theorem 5. 3 6 The case of a bistable non-linearity. Asymptotic profiles.
In this section we consider again equation (1) but in the bistable framework. That is, we assume that f is a C 1 function that satisfies the following assumptions for some θ ∈ (0, 1): We also assume that We are concerned here with the existence of traveling front solutions of (1), that is, (c, u) solution of (18). First we require some preliminary results on the equation (10) in the bistable case.

Existence of asymptotic profiles in the bistable case
Consider equation under the same assumption (7) and (8) as above for the function g.
The existence of solutions depends on α and is obtained in the following theorem.
The rest of this section is devoted to the proof of this Theorem. This Theorem follows from the observation that for α > 0 any positive solution u(y) of (38) satisfies u(y) → 0 as |y| → ∞. This is obtained from Corollary 1.
Next, by the maximum principle, any solution of (38) satisfies 0 ≤ u ≤ 1 (we think of f (s) as having been extended by 0 outside [0, 1]). Now u ≡ 1 is a super-solution of problem (38). Any solution of (38) for α is a sub-solution of (38) for any parameter β ≤ α. Therefore, if there exists a positive bounded solution of (38) for α, there also exists a positive solution for any 0 < β ≤ α.
Next, we claim that for small enough α > 0, (38) admits a positive solution. Indeed, consider the functional defined on H: Recall that f is extended by 0 outside [0, 1], thus F is bounded. Since g(r) → ∞ as r → ∞, it is straightforward to show that there exists a minimizer v of J(w): J(v) = min{J(w), w ∈ H}. Furthermore, we know that v ≥ 0 and v is a solution of (38) (see Theorem 3.3 for details).
Let us show that for α > 0 small enough J(v) < 0. To this end, let ζ R be defined by Then ζ R ∈ H and where |A| denotes the volume of A and C is a constant. Since −F (1) < 0 by (37), we see that by choosing R large enough, J 0 (ζ R ) < 0. Then for such an R fixed, we see that J α (ζ R ) < 0 provided α > 0 is small enough. This guarantees that J α (v) < 0. It follows that v ≡ 0. By the maximum principle, we then have 0 < v < 1. This shows that for small α > 0, (38) admits a positive solution.
The next step is to prove that the set of α > 0 such that (38) has a solution is a closed set. Let α j → α * be a sequence such that (38) admits a solution u j such that 0 < u j < 1 for all j. Note that by the maximum principle, θ < max u j < 1. The sequence (u j ) is bounded by 1 and by standard elliptic estimates is locally compact. Therefore, one can extract a subsequence u j such that u j → u * uniformly on compact sets in the C 2 -norm. Therefore u * is a solution of (38) for the value α = α * . We know that u * ≥ 0, but since max u j > θ, we see that max u * ≥ θ. Indeed by section 2, u j (y) → 0 as |y| → ∞ uniformly with respect to j. Therefore u * > 0 and (38) also has a positive solution for α * . This shows that the set of α such that (38) has a positive solution is an interval (0, α * ] with 0 < α * < ∞.
Considering the evolution equation we see that t → z(t, y) ≥ 0 is decreasing and therefore has a limit. This limit is necessarily the maximal positive solution V = V α for the α for which (38) has a positive solution, that is α ∈ (0, α * ], or is 0 in the opposite case, that is when α > α * . The existence of a second solution when 0 < α < α * is inspired from a work of P. Rabinowitz [34]. In a slightly different formulation, the existence of pairs of solutions is established in [34] by a topological degree argument for bistable type nonlinearities and another type of parameter dependance. The use of the topological degree involves compact operators and the results of [34] are set in the framework of bounded domains. A similar construction can be carried here owing to the condition (8) g(r) → +∞ as r → +∞. Indeed, under this condition, the injection H ֒→ L 2 (R N −1 ) is compact.This allows one to construct a compact operator and to carry the argument of [34] to the present framework.
Since we will not use the second solution, we will leave out the details of the proof of the existence of a second solution.

Stable asymptotic profiles
As we have seen, a solution of (38) is obtained by the minimization of J = J α defined above. The proof of the existence of the previous solution for α > 0 small yields the following result.
Proposition 8. There exists 0 < α * ≤ α * such that for all α ∈ (0, α * ) there exists a minimum v α > 0 of J α and such that In the following, we require the notion of stable solution. Definition 6.3. Let v be a solution of (38). Eigenvalues of the linearized problem about v are defined as the eigenvalues λ of The principle eigenvalue is uniquely determined by the existence of a corresponding eigenfunction ϕ with ϕ > 0. We say that v is (weakly) stable if the principal eigenvalue λ = λ 1 [v] of the linearized problem satisfies λ 1 [v] ≥ 0.
It is well known that the maximal solution V (y) given by Theorem 6.1 when 0 < α ≤ α * is weakly stable. Likewise, the minimum solution of the energy of the Proposition 8 above, when 0 < α < α * , is a weakly stable solution.
In the following we consider the case 0 < α < α * and we make the following assumption.
There exists a unique positive stable solution of (10). (40) This condition implies that the minimizer solution v α : J α (v α ) = min{J α (v), v ∈ H} coincides with the maximum solution V . We leave it as an open problem to give sufficient conditions for the uniqueness of the stable solution. Uniqueness results have been given for analogous problems but with α = 0, which would rather correspond to the minimal solution in our framework [32]. Likewise it would be interesting to give sufficient conditions that ensure that α * = α * . Condition (40) has several implications that we can state.
Proof. The proof follows the observation in [3]. However, it requires new elements in view of the unbounded domain. If v 1 < v 2 , let ϕ 2 be a principal eigenfunction of the linearized problem corresponding to λ 1 [v 2 ]. Since 0 and V are the only stable solutions, λ 1 [v 2 ] < 0. We claim that for ε > 0 sufficiently small, v = v 1 − εϕ 2 is a super-solution of (38). Indeed The right hand side is positive if ε > 0 is sufficiently small. Next, given R > 0, we can choose ε > 0 small enough so that v 1 < v 2 − εϕ 2 in B R . We choose R so that v 1 (y) ≤ δ for all |y| ≥ R and f is decreasing on [0, δ]. We claim that then v 1 ≤ v 2 − εϕ 2 in R N −1 \ B R . Argue by contradiction. In this were not the case, then, since v 1 , v 2 and ϕ 2 converge to 0 at infinity, there exists y, |y| > R such that . Therefore, we have reached a contradiction. This shows that v 1 ≤ v 2 − εϕ 2 . Now we have a super-solution v above a sub-solution v 1 . This implies that there exists a stable solution v such that v 1 ≤ v ≤ v 2 − εϕ 2 < V . This however is in contradiction with condition (40).
From this property, we derive the following useful consequence.
Proposition 10. Let α ∈ (0, α * ) and let W be the maximal solution of equation (38) with the value α * of the parameter. Then, any other solution v of (38) with parameter α that is not the maximal solution cannot be above W .
This immediately follows from the previous proposition as W is a sub-solution of the equation for the value α < α * and W < V .
A consequence of this proposition is Proposition 11. For α ∈ (0, α * ] and under condition (40), the maximal solution V is isolated in L ∞ topology. Therefore, there exists θ 1 > θ such that if v is a solution of (38) with v(0) ≥ θ 1 then v ≡ V .
As we have done before,we can prove that if v is a solution such that Now, if there exist a sequence v n of solutions of (38) such that v n (0) → V (0) then by elliptic estimates v n → W a positive solution of (38) and W (0) = V (0) so W ≡ V by the maximum principle which contradicts the fact that V is isolated.

Traveling fronts for a bistable non-linearity
In this section we assume that f if of bistable type and satisfies (35)- (37). In addition, we assume that 0 < α < α * and that condition (40) is fulfilled. Therefore, there exists a unique non-zero stable solution V (y) = V α (y) of the profile equation (38). Therefore V > 0, J α (V ) = min{J(w), w ∈ H}, λ 1 [V ] ≥ 0 and V is isolated in the L ∞ topology. Furthermore V is the maximal solution. Any other non-zero solution w satisfies 0 < w < V in R N −1 and λ 1 [w] < 0 where λ 1 [w] is the principal eigenvalue of the linearized problem defined in definition 6.3.
In this section, we prove the existence of a traveling front solution of (1) representing an invasion of 0 by the state V at positive speed. Such a solution is given as a pair (c, u) of −∆u − c∂ 1 u + αg(y)u = f (u) in R N u(−∞, y) = V (y), u(+∞, y) = 0 (41) with c < 0 and u : R N → (0, 1). We follow the construction of a solution given above. Namely, let a ≥ 1 and in the slab Σ a = (−a, a) × R N −1 , consider the problem −∆u − c∂ 1 u + αg(y)u = f (u) in Σ a , u(−a, y) = V (y), u(+a, y) = 0.
We recall that for any c ∈ R, for a fixed, there exists a unique solution u = u c of (42). Furthermore, 0 < u < V and ∂ 1 u < 0 in Σ a .The mapping c → u c is decreasing. Up to here, the procedure is the same as before. From this point on however, we need to modify the above argument since we used the fact that f was positive.
Our first task is to prove the following Proposition 12. There exists a unique (c a , u a ) such that u a is a solution of (42) for speed c a and u a satisfies the normalization condition Proof. The bound from above is obtained simply by comparison with the one dimensional problem. Indeed, consider the ODE problem for z = z(x 1 ): It is known that there exists a unique value γ a for which (44) has a (unique) solution z. Furthermore, lim a→+∞ γ a = γ * where γ * is the unique speed of traveling fronts for the 1D equation Comparing (44) with (42), we see that for each c = γ a , the solution z of (44) is a super-solution of (42), thus z > u γ a and for all y ∈ R N −1 , u γ a (0, y) < z(0) = θ.
Since c → u c is decreasing, we see that Assume now that max y∈R N −1 u c (0, y) < θ for all c ∈ R. Passing to the limit for c → −∞, u c converges toward a positive solution v of (38) with max v < θ. By the maximum principle, it is impossible thus there exists a unique c a ∈ (−∞, γ a ) such that (43) is fulfilled.
Since γ a → γ * < ∞ as a → +∞ and a → γ a is a continuous function, this shows that sup a≥1 c a < ∞.
Since a → c a is continuous, in order to complete the proof of the Proposition, it suffices to show that lim inf a→∞ c a ≥ 0. For this, we argue by contradiction and assume that for a sequence a j → +∞ there holds c aj < 0. For the sake of simplicity, we write a instead of a j . Since c → u c is decreasing, from this we infer that along this subsequence, the solution v = v a of Due to Proposition 11, there exist θ 1 > θ such that if an asymptotic profile v solution of (38) There is a point b = b j , −a < b < 0 such that v a (b, 0) = θ 1 . We now translate the solution to center it on x 1 = b. That is, we letṽ a (x 1 , y) = v a (x 1 + b, y) defined for x 1 ∈ (−a − b, a − b) and y ∈ R N −1 . The interval (−a − b, a − b) either converges (along a subsequence) to (−∞, +∞) or to some (−d, +∞) with 0 ≤ d < ∞. In both cases, by standard elliptic estimates, one can strike out a subsequence ofṽ a , denoted againṽ a , such thatṽ a converges locally to some function w where w satisfies: In case the interval is converging to (−d, +∞), in addition we know that w(−d, y) = V (y). If the interval converges to R, then lim x→−∞ w(x 1 , y) exists and is some function W (y) which is then a solution of the profile equation (38). But since w(0, 0) = θ 1 , we know that W (0) ≥ θ 1 . By the definition of θ 1 , this implies that where now 0 ≤ d ≤ +∞. We also know that w(+∞, y) = ψ(y) exists with 0 ≤ ψ < V . Multiply (45) by ∂ 1 w and integrate over (−d, +∞) × R N −1 to get In all cases, we get Since V minimize J α , we obtain V ≡ ψ and w(x 1 , y) = V (y) for all x 1 ∈ (−d, +∞) but this contradicts the renormalization w(0, 0) = θ 1 .
We have thus reached a contradiction. This shows that for large a, c a ≥ 0, which completes the proof of the Proposition.
Let us now turn to the proof of the existence of traveling front solutions of (41). Since c a and u a are bounded, by standard elliptic estimates, we can strike out a sequence a = a j → ∞ (we continue to denote subsequences by a) such that c a → c ≥ 0 and u a → u. We know that (c, u) satisfies the equation −∆u + c∂ 1 u + αg(y)u = f (u) in R N with ∂ 1 u ≤ 0 and max R N −1 u(0, ·) = θ. It remains to identify the limits as x 1 → ±∞. These lim x1→±∞ u(x 1 , y) = u ± (y) exist and are solutions of the asymptotic profile equation (38). Now since 0 ≤ u + (y) = lim x1→+∞ u(x 1 , y) ≤ θ and all positive solutions w of (38) satisfy max w > θ, we have u + ≡ 0.
We claim that u − (y) = lim x1→−∞ u(x 1 , y) coincides with V (y). Clearly, 0 < u − ≤ V . Argue by contradiction that u − ≡ V , implying u − < V . By assumption, u − is an unstable solution of (38) in the sense that λ 1 [u − ] < 0. Let us construct a super-solution of the stationary equation, that is a w with such that w is a compact perturbation of u − and as close as we wish to u − .
Consider the linearized equation about u − : We know that λ 1 [u − ] is the limit of the Dirichlet principal eigenvalue in a ball when the radius goes to infinity (This follows from the Rayleigh quotient minimization). Therefore, R > 0 can be chosen sufficiently large so that the principal eigenvalue µ and associated eigenfunction ψ of Consider the function ζ(x 1 , y) = cos(ωx 1 )ψ(y) defined for x 1 ∈ (−L, L) with L = π 2ω and |y| < R. We note D = (−L, L) × B R . This function is positive and satisfies: Choose L large enough so that µ + ω 2 < 0. Then let w(x 1 , y) = u − (y) − εζ(x 1 , y) with ε > 0 and (x 1 , y) ∈ D. This function satisfies Since µ + ω 2 < 0, we can choose ε sufficiently small so that −∆w + αg(y)w − f (w) ≥ 0 in D and w > 0.
Furthermore, because εζ = 0 on ∂D and εζ > 0 in D, that is w < u − in D, if we extend w by choosing w(x 1 , y) = u − (y) for all (x 1 , y) ∈ D, we have constructed a (generalized) super-solution of the problem (see e.g. [11]). Let us now derive a contradiction. We consider two cases.
• Cases (i): Suppose c > 0. Then U (t, x 1 , y) = u(x 1 − ct, y) is a solution of the evolution equation y). Furthermore, for all times U (t, x 1 , y) ≤ u − (y). Since w is a compact perturbation of u − for a time t 0 sufficiently negative, we get ∀(x 1 , y) ∈ R N U (t 0 , x 1 , y) ≤ w(x 1 , t).
• Case (ii): The case that remains to be studied is c = 0 (since we already have c ≥ 0). Then u(x 1 , y) is a stationary solution of the same equation that w is a supersolution of. Since u(−∞, y) = u − (y) and u(+∞, y) = 0, and since w = u − outside a compact set, after a translation, we can assume that u h = u(x 1 +h, y) ≤ w(x 1 , y)(for large enough h). Define h * = inf{h ∈ R, u(x 1 + h, y) ≤ w(x 1 , y) in R N }.
Clearly, h * > −∞ (for w < u − at some points). Then w(x 1 , y) ≥ u(x 1 +h * , y) = u h * and min(w − u h * ) = 0 is necessarily achieved at a point of D. Since w(x 1 , y) = u − (y) > u(x 1 + h, y) for all h if (x 1 , y) ∈ D, we see that the maximum is achieved at an interior point of D. Writing w − u h * ≥ 0 as a super-solution of a linear elliptic equation in D, we derive a contradiction with the strong maximum principle.
Therefore (c, u) is a solution of the traveling front equation (41).

The model of cortical spreading depression
We consider here more general versions of the model (5) described in the Introduction. The problems studied in this paper have the following general form In the modeling context N = 2 and 3 are the cases of interest. As indicated in the Introduction, this equation also describes cortical spreading depressions (CSD). There the wave propagates in a medium composed of two different components, the gray and white matters of the brain, with a narrow transition area separating them. Thus we consider in this section functions h(y, u) of the following type: where 0 < L 1 ≤ L 2 < ∞ and K ≥ m > 0 are given parameters and f is of bistable form. That is we assume that f verifies conditions (35)-(37)of section 6. Note that in particular, we assume 1 0 f (s)ds > 0.

The asymptotic profile equation
We start as usual with the profile equation We recall that λ 1 [V ] is the principal eigenvalue of the linearized equation about V . This eigenvalue can be defined as Associated with (50) is the energy functional: where H(y, z) = z 0 h(y, s)ds. Note that owing to condition (48), J(w) is well defined for all w ∈ H 1 (R N −1 ). Theorem 8.1. There exist critical radii 0 < L * ≤ L * < ∞ with the following properties: i) For L 2 < L * , there is no solution other than 0 to the asymptotic profile equation (50).
ii) For L 1 > L * (independently of L 2 ), there exists a maximum solution V of (50) and this solution is stable in the sense that λ 1 [V ] ≥ 0.
iii) For all L 1 > L * , the minimum of the energy functional is achieved at some non-zero function V J (y), i.e.
J(V J ) = min Proof. i) Since the equation implies that −∆u + mu ≤ 0 for all |y| ≥ L 2 in R N −1 , and u > 0 is bounded, by Theorem 2.1 we know that u and |∇u| have exponential decay as |y| → +∞. Then we get We know, by Sobolev embedding and Hölder inequality, that where η(L 2 ) → 0 as L 2 → 0. Therefore for L 2 small enough, these inequalities yield u ≡ 0. ii) Next, since 1 is a super-solution of the equation in R N −1 , there exists a maximum solution of equation (50) that we denote V . By what we have just seen, V ≡ 0 for L 2 sufficiently small. Let us now show that V > 0 for L 1 sufficiently large.
Let us consider the energy restricted to the ball of radius R ≤ L 1 where F (z) = 1 0 f (s)ds. We know (see the proof of Theorem 6.1) that for R sufficiently large there exists a minimum w R of Then w R > 0 in B R and w R is solution of −∆w R = f (w R ) in B R and w R = 0 on ∂B R . Extending w R by 0 outside the ball B R , we get a global (generalized) subsolution. The solution V is such that V ≥ w R (since V is the maximum solution). This implies that V ≡ 0 and therefore V > 0 in R N −1 for L 1 ≥ R. iii) Now for L 1 ≥ R, clearly inf w∈H 1 (R N −1 ) J(w) ≤ J R (w R ) < 0.
Let us now show that the infimum is achieved.

Traveling fronts for the CSD model
In this section, we prove Theorem 1.4. The proof is similar as in Section 7. There we used that h(y, u) = f (u) − αg(y)u with g → +∞. But actually, the same properties that were entailed one can derived for h(y, u) ≤ −mu for large |y|. We start by constructing a solution of ∆u a − c a ∂ 1 u a = h(y, u a ) in (−a, a) × R N −1 u a (−a, y) = V (y), u a (+a, y) = 0, that verifies sup for a ≥ 1 and where θ is the unstable 0 of f (u), that is f (θ) = 0 and 0 < θ < 1. We recall that c a is uniquely determined by the renormalization condition (52).
The function z ca a is a supersolution of (51) thus u a ≤ z ca a . In view of (52) this implies that z ca a (0) ≥ θ and this implies that c a ≤ γ a where γ a is the unique value of c such that the solution of (53) verifies z γa a (0) = θ. This as before yields the upper bound for c a .
The lower bound is achieved in the same manner as in the section 7 and the convergence for a → +∞ also.