## Topological Methods in Nonlinear Analysis

- Topol. Methods Nonlinear Anal.
- Volume 16, Number 1 (2000), 37-63.

### 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 $\varphi$.

#### Article information

**Source**

Topol. Methods Nonlinear Anal., Volume 16, Number 1 (2000), 37-63.

**Dates**

First available in Project Euclid: 22 August 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.tmna/1471875421

**Mathematical Reviews number (MathSciNet)**

MR1805038

**Zentralblatt MATH identifier**

0974.35050

#### Citation

Crooks, Elaine C. M. Stability of travelling-wave solutions for reaction-diffusion-convection systems. Topol. Methods Nonlinear Anal. 16 (2000), no. 1, 37--63. https://projecteuclid.org/euclid.tmna/1471875421