Konstantinos Spiliopoulos (University of Maryland )

Title: Reaction-Diffusion Equations with Nonlinear Boundary Conditions in Narrow Domains

Abstract: We will consider the second initial boundary problem in narrow domains of width $\epsilon\ll 1$ for linear second order differential equations with nonlinear boundary conditions. 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. In addition, an important and interesting problem, which is related to the previous one and will be presented here, is the Wiener process with instantaneous reflection in a narrow tube which, in contrast to before, is assumed to be non-smooth asymptotically.