Existence of reaction‐diffusion‐convection waves in unbounded strips

M. Belk,B. Kazmierczak,V. Volpert

doi:10.1155/ijmms.2005.169

Abstract

Existence of reaction‐diffusion‐convection waves in unbounded strips is proved in the case of small Rayleigh numbers. In the bistable case the wave is unique, in the monostable case they exist for all speeds greater than the minimal one. The proof uses the implicit function theorem. Its application is based on the Fredholm property, index, and solvability conditions for elliptic problems in unbounded domains.

Highlights

Propagation of reaction-diffusion waves, if it occurs in a liquid or in a gaseous medium, can be accompanied by natural convection
The first mathematical work devoted to reaction-diffusion fronts with convection is [19] where it is shown that in some cases the corresponding eigenvalue problem can be reduced to a monotone system and, a minimax representation for the principal eigenvalue can be obtained [14, 16, 20]
We show how to apply the implicit function theorem to such problems

Summary

Introduction

Propagation of reaction-diffusion waves, if it occurs in a liquid or in a gaseous medium, can be accompanied by natural convection. It will allow us in particular to prove existence of reaction-diffusion waves with convection This construction is based on the theory of elliptic problems in unbounded domains: Fredholm property, index, solvability conditions. We show how the Fredholm property of elliptic operators, their index, and solvability conditions allow the application of the implicit function theorem It is illustrated with some reaction-diffusion problems. There exists a unique monotone in x1 travelling wave, that is, a constant c and a classical solution w(x) of (2.14), (2.15) satisfying (2.16). The proof of this theorem can be found in [15].

Structural stability of families of solutions of operator equations

Existence of reaction-diffusion-convection waves