Asymptotic spreading in heterogeneous diffusive excitable media

Abstract

We establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction-diffusion equations of the type: (t, x, u) with compactly supported initial conditions at t = 0. Here, A, q, f have a general dependence in t ∈ R + and x ∈ R N . We establish properties of families of propagation sets which are defined as families of subsets (S t ) t 0 of R N such that lim inf t→+∞ {inf x∈S t u(t, x)} > 0. The aim is to characterize such families as sharply as possible. In particular, we give some conditions under which: (1) a given pathforms a family of propagation sets, or (2) one can find such a family with S t ⊃ {x ∈ R N , |x| r(t)} and lim t→+∞ r(t) = +∞. This second property is called here complete spreading. Furthermore, in the case q ≡ 0 and inf (t,x)∈R + ×R N f u (t, x, 0) > 0, as well as under some more general assumptions, we show that there is a positive spreading speed, that is, r(t) can be chosen so that lim inf t→+∞ r(t)/t > 0. In the general case, we also show the existence of an explicit upper bound C > 0 such that lim sup t→+∞ r(t)/t < C. On the other hand, we provide explicit examples of reactiondiffusion equations such that for an arbitrary ε > 0, any family of propagation sets (S t ) t 0 has to satisfy S t ⊂ {x ∈ R N , |x| εt} for large t. In connection with spreading properties, we derive some new unique-* Corresponding author. 2147 ness results for the entire solutions of this type of equations. Lastly, in the case of space-time periodic media, we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero.

Propagation in Fisher–KPP type equations with fractional diffusion in periodic media

“…Their proof uses probabilistic tools; proofs using dynamical systems or PDE arguments are given in [12], [8], [2]. For α ∈ (0, 1) and µ ≡ 1 in (2), propagation is exponential in time.…”

“…µ est non constante et périodique), si u 0 est à support compact, Freidlin et Gärtner [10] prouvent, avec des outils probabilistes, qu'il existe une vitesse notée w * (e) dans chaque direction de propagation e ∈ S d−1 (la définition et l'expression de w * (e) sont données en (4)). D'autres démonstrations, utilisant des arguments de systèmes dynamiques ou d'EDP, sont proposées dans [12], [8], [2].…”

Persistence and propagation in periodic reaction-diffusion models

“…These minimal speeds are of great importance from the point of view of the applications. Indeed, under the assumptions of Theorem 3.1, it is known [3,17,47] Furthermore, the more aggregated the medium, the larger the speed, in the sense of the following result. We point out that the same statement does not hold in general in higher dimensions N ≥ 2, see [35].…”

A multi-dimensional bistable nonlinear diffusion equation in a periodic medium

“…The regularized profile equation is supplemented with the limit behaviour (6). Such a regularisation procedure has been widely used to study pulsating front and we refer for instance to [4].…”