Description

• Title:
  Stability of travelling-wave solutions for reaction-diffusion-convection systems
• Creator:
  Crooks, Elaine C. M.
• Description:
  We are concerned with the asymptotic behaviour of classical solutions of systems of the form $$ \cases u_{t} = A u_{xx} + f(u, u_{x}) &\text{for } x \in {\mathbb R},\ t> 0,\ u(x,t) \in {\mathbb R}^N,\\ u(x,0) = \varphi (x),& \endcases\tag{1} $$ 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, < i> Travelling waves for reaction-diffusion-convection systems< /i> , Topol. Methods Nonlinear Anal. < b> 11< /b> (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 $\varphi$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2000.09.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 16, No 1 (September 2000); 37 - 63;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2000.029/9974
• Subject:
  Travelling waves;
  fronts;
  stability;
  bistable nonlinearity;
  comparison principles;
  reaction-diffusion-convection systems;
  35K45;
  35K40;
  35K55;
  35B35
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2000.029
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other