Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

Abstract

This paper investigates time-global wave-like solutions of heterogeneous reaction–diffusion equations: ${\partial }_{t}u-a(x){\partial }_{xx}u-b(x){\partial }_{x}u=f(x,u)$ in $\mathbb{R}\times \mathbb{R}$, where the coefficients $a$, $a_x$, $a_{xx}$, $1/a$, $b$, $b_x$ and $f$ are only assumed to be measurable and bounded in $x\in \mathbb{R}$ and the nonlinearity $f$ is Lipschitz-continuous in $u\in [0,1]$, with $f(x,0)=f(x,1)=0$ for all $x\in \mathbb{R}$. In this general framework, the notion of spatial transition wave has been introduced by Berestycki and Hamel [4]. Such waves always exist for one-dimensional ignition-type equations [22,27], but not for monostable ones [26]. We introduce in the present paper a new notion of wave-like solutions, called critical travelling waves since their definition relies on a geometrical comparison in the class of time-global solutions trapped between $0$ and $1$. Critical travelling waves always exist, whatever the nonlinearity of the equation is, are monotonic in time and unique up to normalization. They are spatial transition waves if such waves exist. Moreover, if the equation is of monostable type, for example if $b\equiv 0$ and $f(x,u)=c(x)u(1-u)$, with ${\inf }_{\mathbb{R}}c>0$, then critical travelling waves have minimum least mean speed. If the coefficients are homogeneous/periodic, then we recover the classical notion of planar/pulsating travelling wave. If the heterogeneity of the coefficients is compactly supported, then critical transition waves are either a spatial transition wave with minimal global mean speed or bump-like solutions if spatial transition does not exist. In the bistable framework, the nature of the critical travelling waves depends on the existence of non-trivial steady states. Hence, the notion of critical travelling wave provides a unifying framework to earlier scattered existence results for wave-like solutions. We conclude by proving that in the monostable framework, critical travelling waves attract, in a sense and under additional assumptions, the solution of the Cauchy problem associated with a Heaviside initial datum.