Asymptotic speed of propagation for a viscous semilinear parabolic equation

We characterize the asymptotic speed of propagation of almost planar solutions to a semilinear viscous parabolic equation, with periodic nonlinearity.


equation (1) becomes
(2) v ε t − ∆v ε + ε 1−α g (v ε ) = 0. Therefore, the analysis of the limit as ε → 0 of solutions to (1) is related to the long time behavior of solutions of (2). When ε = 1 or equivalently α = 1, the long time behavior has been considered in the literature by several authors, also in the case the laplacian is substituted by a more general elliptic operator, and the existence of special solutions such as traveling or pulsating waves has been established (see [9,3,14,10,2,13,4,5,7] and references therein). When 1 0 g = 0, in [12,1] it has been studied the limit behavior of solutions of the following long time rescaling of (1) In this note we are interested in the case α ∈ [0, 1). The main result is Theorem 3.1 which provides the effective speed of propagation of solutions to (1) starting from almost planar initial datum. In particular we show that the solution to (1) where the average positive speed c(p) depends on g.
The proof of this result relies on maximum principle arguments and on a priori bounds on the gradient to solutions to (1) with planar initial data. These bounds are provided by Theorem 1.1 and are based on the so called Bernstein method.
The interesting feature we observe is that in the case α < 1, the asymptotic speed of propagation c(p) is just lower semicontinuous, but not continuous with respect to p, whereas when α = 1, the speed c(p) depends continuously on p, as it has been proved in [9,5]. In particular, we show in Proposition 2.1 that c(p) = − 1 0 g(s)ds p = 0 −( 1 0 g −1 (s)ds) −1 p = 0 so that the limit speed is discontinuous with respect to the slope p. Such phenomenon is unusual in homogenization problems, and indicates that the effective limit of (1) as ε → 0 is governed by a differential operator which is discontinuous in the gradient entry. An equation similar to (1), with n = 1 and α = 0, giving rise to a discontinuous effective limit problem, has been considered in [8].
This makes the analysis of this limit more challenging, and will be the topic of a paper in preparation [6], where we will consider the long time behaviour and the asymptotic limit of solutions to where g : R n × R → R is Lipschitz continuous and Z n+1 -periodic.

A priori estimates on the gradient
We consider in this section the more general case in which g : R n × R → R is a Lipschitz, Z n+1 periodic function.
We introduce, for any p ∈ R n , the initial value problem We recall that this problem admits a unique solution v ε ∈ C 2+γ,1+γ/2 for all γ ∈ (0, 1). Moreover we recall also that, due to Lipschitz regularity of g, a standard comparison principle among sub and supersolutions to (4) holds (see [15]).
We provide an apriori bound on the oscillation and on the gradient of the solutions to (4), which will be useful in the following. The approach is based on the so called Bernstein type method.
In the following we let w ε ( where v ε is the unique solution to (4). Notice that w ε solves Theorem 1.1. Given p ∈ R n , then there exists ε 0 = ε(|p|) such that for every ε ≤ ε 0 and every t ≥ 0 there holds where C n is a constant depending on the space dimension and g 1,∞ is the Lipschitz norm of g.
Proof. The proof of the theorem is based on similar arguments as in [9, Thm 2.4, Cor. 2.5]. We divide the proof in several steps.
Step 1: we prove that for every t ≥ 0 Observe that, due to periodicity of g, it satisfies the same equation as w ε : This implies immediately (6).
Step 2: properties of the function W ε (t) = sup x∈R n w ε (x, t).
For simplicity from now on we consider the case in which g ∈ C ∞ (R n+1 ). The case of g just Lipschitz is recovered by a standard approximation procedure.
Note that, by comparison principle, for all x, t. Moreover, again by comparison principle, This implies that the function satisfies, in viscosity sense, W ε t ≤ ε 1−α g ∞ . Let λ > 0 to be fixed later. We consider the function Step 3: we prove that choosing λ = Cε (1−α)/2 n 1/2 (1 + |p|)( Dg ∞ + g ∞ ), for C > 1, then for every ε sufficiently small Let t * such that Φ ε (t * ) ≥ Φ ε (t) for every t ∈ [0, T ]. If t * = 0, then Φ ε (t) ≤ 0 for every t and we are done. Assume that t * > 0. It is easy to show that there exists a sequence x k (t * ) such that We can choose the sequence such that and then we get the desired conclusion. Assume by contradiction that the claim is not true. Then, for every λ, there exists a subsequence such that lim k |Dw ε (x k (t * ), t * )| > 0. We compute So, using this equation, (8), (9) and (7), we get We multiply the first equation in (8) by w ε x i and compute at x k (t * ): we have that This implies, since |Dw ε (x k (t * ))| = 0, that We compute (10) at x k (t * ), recalling the definition of T , of the sequence x k (t * ) and (10) we get We claim it is possible to choose λ so that the right hand side of (11) is strictly negative getting then a contradiction. Indeed if we choose λ = Cε (1−α)/2 n 1/2 (1 + |p|)( Dg ∞ + g ∞ ), with C > 1, the claim is true for every ε sufficiently small.

Existence of almost-planar solutions
In this section, we show in fact that there exists for every p ∈ R n a positive constant c ε (p) such that solutions to (1) starting from hyperplanes z = p · x remain at a small distance from hyperplanes with the same normal and moving with (uniformly bounded in ε) speed c ε (p)ε α−1 . Moreover we study the limit as ε → 0 to c ε (p)ε α−1 : this will be the average speed of hyperplanes.
Theorem 2.1. Let v ε be the solution to (4). Then there exists a unique c ε (p) > 0 such that where K depends on the Lipschitz norm of g. Moreover The proof of this theorem is based on the apriori estimates provided in Theorem 1.1, and follows as in [9, Thm 3.1] (see also [6] for a proof of a more general result of this kind).

Remark 1.
Observe that if u ε is the solution to (1) with initial datum u ε (x, 0) = p · x, then v ε (x, t) = 1 ε u ε (εx, ε 2−α t) is the solution to (4). So Theorem 2.1 implies Remark 2. Note that, when α = 1, the speed c ε (p) does not depend on ε. When α = 1 and the laplacian in (1) is substituted by the mean curvature operator, the existence of traveling wave solutions has been established in [9], under suitable assumptions on the forcing term g.

Asymptotic speed of propagation
In this section we provide the asymptotic speed of propagation of solutions to (1) as ε → 0 starting from almost planar initial datum. When the initial datum is an hyperplane, we obtain the homogenization limit of solutions to (4).