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

Elaine C. M. Crooks
2000 Topological Methods in Nonlinear Analysis  
We are concerned with the asymptotic behaviour of classical solutions of systems of the form 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 [5] .) We show that if φ is bounded, uniformly
more » ... uniformly continuously differentiable and such that w(x) − φ(x) is small when |x| is large, then there exists χ ∈ R such that Our approach extends an idea developed by Roquejoffre, Terman and Volpert in the convectionless case, where f is independent of u x . First φ is assumed to be increasing in x, and (2) proved via a homotopy argument. Then we deduce the result for arbitrary φ by showing that there is an increasing function in the ω−limit set of φ.
doi:10.12775/tmna.2000.029 fatcat:d7dshvu7m5g2lf6ir5uea2yrzu