Pulsating solutions for multidimensional bistable and multistable equations
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Mathematische Annalen
Pulsating solutions for multidimensional bistable and multistable equations Thomas Giletti1 · Luca Rossi2 Received: 11 June 2019 / Revised: 17 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We investigate the existence of pulsating front-like solutions for spatially periodic heterogeneous reaction–diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.
1 Introduction In this work we consider the reaction–diffusion equation ∂t u = div(A(x)∇u) + f (x, u), t ∈ R, x ∈ R N ,
(1.1)
where N ≥ 1 is the space dimension. The diffusion matrix field A = (Ai, j )1≤i, j≤N is always assumed to be smooth, symmetric and to satisfy the ellipticity condition ∃C1 , C2 ∈ (0, ∞), ∀x, ξ ∈ R N , C1 |ξ |2 ≤
Ai, j (x)ξi ξ j ≤ C2 |ξ |2 . (1.2)
i, j
Communicated by Y. Giga.
B
Luca Rossi [email protected]
1
Univ. Lorraine, IECL UMR 7502, Vandoeuvre-lès-Nancy, France
2
CNRS, EHESS, CAMS, Paris, France
123
T. Giletti, L. Rossi
As far as the regularity of the reaction term f (x, u) is concerned, we assume that it is at least globally Lipschitz continuous (a stronger hypothesis will be made in the general multistable case; see below). Equation (1.1) is spatially heterogeneous. As our goal is to construct travelling fronts, i.e., self-similar propagating solutions, we impose a spatial structure on the heterogeneity. More precisely, we assume that the terms in the equation are all periodic in space, with the same period. For simplicity and without loss of generality up to some change of variables, we choose the periodicity cell to be [0, 1] N , that is, ∀L ∈ Z N ,
A(· + L) ≡ A(·), f (· + L, ·) ≡ f (·, ·).
(1.3)
From now on, when we say that a function is periodic, we always understand that its period is (1, . . . , 1). In the spatially periodic case, one can consider the notion of pulsating travelling front, which we shall recall precisely below. Roughly, these are entire in time solutions which connect periodic steady states of the parabolic equation (1.1). The existence of such solutions is therefore deeply related to the underlying structure of (1.1) and its steady states. In this paper, we shall always assume that (1.1) admits at least two spatially periodic steady states: the constant 0 and a positive state p(x). ¯ Namely, we assume that f (·, 0) ≡ 0, as well as
div(A(x)∇ p) ¯ + f (x, p) ¯ = 0, N ¯ + L) ≡ p¯ > 0. ∀L ∈ Z , p(·
We shall restrict ourselves to solutions u(t, x) of (1.1) that satisfy the inequality
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