Reaction-diffusion waves with nonlinear boundary conditions

A reaction-diffusion equation with nonlinear boundary condition is considered in a two-dimensional infinite strip. Existence of waves in the bistable case is proved by the Leray-Schauder method. 1. Formulation of the problem. Reaction-diffusion problems with nonlinear boundary conditions arise in various applications. In physiology, such problems describe in particular development of atherosclerosis and other inflammatory diseases [3]. In this context, nonlinear boundary conditions show the influx of white blood cells from blood flow into the tissue where the inflammation occurs. Among other possible applications, let us indicate molecular transport through biological membrane where some molecules can amplify their own transport opening membrane channels, as it is the case, for example, with calcium induced calcium release [1]. In this work we consider the reaction-diffusion equation ∂u ∂t = ∆u+ f(u), (1) with nonlinear boundary conditions: y = 0 : ∂u ∂y = 0, y = 1 : ∂u ∂y = g(u). (2) Here f and g are sufficiently smooth functions, −∞ < x < ∞, 0 < y < 1. We will study the existence of travelling wave solutions of this problem, that is of solutions of the equation ∆u+ c ∂u ∂x + f(u) = 0 (3) with the same boundary conditions. Here c is an unknown constant, the wave speed, and the variable x in equation (3) is identified with the variable x− ct in equation (1). We assume that f(u±) = 0, g(u±) = 0 for some u+ and u−, and f (u±) < 0, g (u±) < 0. (4) 2010 Mathematics Subject Classification. Primary: 35K57; Secondary: 35J60.