Reaction Diffusion Equations with Nonlinear Boundary Conditions in Narrow Domains

Abstract

Second initial boundary problem in narrow domains of width $\epsilon \ll 1$ for linear second order differential equations with nonlinear boundary conditions is considered in this paper. Using probabilistic methods we show that the solution of such a problem converges as $\epsilon \downarrow 0$ to the solution of a standard reaction-diffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front.

27 Pages, 3 figures