Reaction–diffusion equations with nonlinear boundary conditions in narrow domains

Abstract

Second initial boundary problem in narrow domains of width ε�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 ε↓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.