Reaction Diffusion Equations with Nonlinear Boundary Conditions in Narrow Domains

TL;DR

This paper studies reaction-diffusion equations with nonlinear boundary conditions in very narrow domains, showing that solutions converge to lower-dimensional equations and analyzing wave front propagation and jumps.

Contribution

It introduces a probabilistic approach to analyze the limit behavior of reaction-diffusion equations in narrow domains with nonlinear boundaries, revealing wave front dynamics.

Findings

Solutions converge to lower-dimensional reaction-diffusion equations as domain width shrinks.

Conditions for wave front jumps are identified.

Probabilistic methods effectively analyze boundary effects 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.