Stability of travelling-wave solutions for reaction-diffusion-convection systems

Abstract

We are concerned with the asymptotic behaviour of classical solutions of systems of the form $$ \begin{cases} u_{t} = A u_{xx} + f(u, u_{x}) &\text{for } x \in {\mathbb R},\ t> 0,\ u(x,t) \in {\mathbb R}^N,\\ u(x,0) = \varphi (x),& \end{cases}\tag{1} $$ where $A$ is a positive-definite diagonal matrix and $f$ is a "bistable" nonlinearity satisfying conditions which guarantee the existence of a comparison principle for (1). Suppose that (1) has a travelling-front solution $w$ with velocity $c$, that connects two stable equilibria of $f$. (There are hypotheses on $f$ under which such a front is known to exist [E. C. M. Crooks and J. F. Toland, Travelling waves for reaction-diffusion-convection systems, Topol. Methods Nonlinear Anal. 11 (1998), 19–43].) We show that if $\varphi$ is bounded, uniformly continuously differentiable and such that $\Vert w(x) - \varphi (x) \Vert $ is small when $|x|$ is large, then there exists $\chi \in {\mathbb R}$ such that $$ \Vert u(\cdot, t) - w(\cdot + \chi - ct) \Vert _{BUC^{1}} \rightarrow 0 \quad\text{as } t \rightarrow \infty.\tag{2} $$ Our approach extends an idea developed by Roquejoffre, Terman and Volpert in the convectionless case, where $f$ is independent of $u_{x}$. First $\varphi$ is assumed to be increasing in $x$, and (2) proved via a homotopy argument. Then we deduce the result for arbitrary $\varphi$ by showing that there is an increasing function in the $\omega$-limit set of $\varphi$.