Reaction–diffusion equations with nonlinear boundary conditions in narrow domains

Abstract

Second initial boundary problem in narrow domains of width ǫ ≪ 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 ǫ ↓ 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.