Spreading speeds for some reaction-diffusion equations with general initial conditions

TL;DR

This paper investigates the asymptotic spreading speeds of solutions to reaction-diffusion equations in heterogeneous media with general initial conditions, extending previous results and revealing complex dynamics.

Contribution

It introduces new concepts of spreading speed, characterizes them for oscillatory initial conditions, and constructs initial states leading to intricate propagation behaviors.

Findings

Complete characterization of spreading speeds for oscillatory initial conditions

Introduction of generalized spreading speed concepts

Construction of initial conditions with complex dynamics

Abstract

This paper is devoted to the study of some qualitative and quantitative aspects of nonlinear propagation phenomena in diffusive media. More precisely, we consider the case a reaction-diffusion equation in a periodic medium with ignition-type nonlinearity, the heterogeneity being on the nonlinearity, the operator and the domain. Contrary to previous works, we study the asymptotic spreading properties of the solutions of the Cauchy problem with general initial conditions which satisfy very mild assumptions at infinity. We introduce several concepts generalizing the notion of spreading speed and we give a complete characterization of it when the initial condition is asymptotically oscillatory at infinity. Furthermore we construct, even in the homogeneous one-dimensional case, a class of initial conditions for which highly nontrivial dynamics can be exhibited.