Generalized travelling waves for reaction-diffusion equations

In this paper, we introduce a generalization of travelling waves for evolution equations. We are especially interested in reaction-diffusion equations and systems in heterogeneous media (with general operators and general geometry). Our goal is threefold. First we give several definitions, for transition waves, fronts, pulses, global mean speed of propagation, etc. Next, we discuss the meaning of these definitions in various contexts. Then, we report on several results of [4] (of which this is a companion paper) about these notions. We further establish here several new properties. For this definition to be meaningful we need to show two things. First, that the definition covers and unifies all classical cases (and does not introduce spurious objects). Second, that it allows one to understand propagation fronts in completely new situations. In particular we report here on a result about travelling fronts passing an obstacle. 1. Classical notions of travelling fronts 1.1. Planar fronts. Travelling fronts form a specially important class of timeglobal solutions of reaction-diffusion equations. They arise and play an important role in various fields such as biology, population dynamics, ecology, physics, combustion... In many situations, they describe the transition between two different states. Let us start with recalling the notion of classical travelling fronts in the homogeneous case, for the equation (1.1) ut = ∆u+ f(u) in R . For basic properties of the linear heat equations, which allow one to derive existence and uniqueness of the Cauchy problem associated with (1.1), we refer to the classical text of H. Brezis [14]. In the case of (1.1), a planar travelling front connecting the uniform steady states 0 and 1 (assuming f(0) = f(1) = 0) is a solution which propagates in a given unit direction e with a speed c, and which can then be written as u(t, x) = φ(x·e−ct) with φ(−∞) = 1 and φ(+∞) = 0. Two properties characterize such fronts: their 1991 Mathematics Subject Classification. Primary 35K55; Secondary 35B10, 35B40, 35K57.