Stabilization of the moving front solution to the reaction-diffusion-advection problem

Abstract

INTRODUCTION Propagating fronts are characteristic for many physical phenomena. In case of reaction-diffusion-advection processes, these can be combustion or strain fronts. Solutions having large gradients to problems of this type also arise in nonlinear acoustics. Such problems include, for example, the Burgers equation, as well as equations with modular nonlinearity. The stationary reaction-diffusion-advection equations can be used for modelling of wind field distribution in the presence of plant heterogeneity. The domain where the solution has a large gradient is called the internal transition layer. The numerical implementation of solutions to problems with internal transition layers requires a preliminary analysis of the existence conditions and stability. In particular, for the numerical solution of some applied problems, the calculation method for establishing is often used, when the solution of the boundary value problem for the elliptic equation is found numerically as the solution of the corresponding initial-boundary value problem for the parabolic equation over a sufficiently long period of time. To implement this method, information on the asymptotic stability and the domain of attraction of the stationary solution is needed. In this paper, we consider the initial-boundary-value problem for reaction-diffusion-advection equation and the question of its moving front type solution stabilizing over an infinitely large time interval to the solution of the corresponding stationary problem. The existence of moving front solution is investigated in. The existence conditions of an asymptotically stable solution to the stationary problem are known from. To prove the stabilization theorem, in this paper we use the method of upper and lower solutions, which for this class of problems is justified in. The main idea of the proof is to show that the upper and lower solutions of the initial-boundary-value problem on an asymptotically large time interval fall into the attraction domain of the stationary solution. The upper and lower solutions with large gradients in the region of the internal transition layer are constructed according to the asymptotic method of differential inequalities as modifications of asymptotic approximations of the solutions to these problems in a small parameter. A small parameter here is the width of the inner transition layer with respect to the width of the front propagation region. The study conducted in this work gives an answer about non-local domain of attraction of the stationary solution. In addition, an estimate of the time interval is obtained in which the solution of the front type falls into the local domain of attraction of the stationary solution, that is, in fact, the criterion for the numerical solution stationing.