Reaction-diffusion systems on domains with thin channels

[摘要] The purpose of this work is to consider a weakly coupled system of semilinear parabolic equations; namely, a reaction-diffusion system, which is typically a model of chemical reaction, population biology, morphogenesis, etc. This system,considered in a bounded spatial domain, Omega subset of R(2), is written as u(t) = D Delta u + f(u), (t, x) is an element of (0, infinity) x Omega, with Newmann boundary conditions, u = (u(1),..., u(m)) and D is an m x m positive diagonal matrix. We suppose tha Omega(E) is smooth and that there exists a bounded and smooth set, Omega(0), such that R(epsilon) = Omega(epsilon)\Omega(0) approaches portions of a curve as epsilon --> 0. In fact we assume that R(epsilon) is a finite union of thin domains over a curve. We prove that the dynamic behavior of (1) is determined by a finite dimensional ODE dy/dt = h(epsilon)(y), and h(epsilon) --> h(0) in the C-1 sense, where h(0) is given. In fact we prove existence and convergence of inertial manifolds for (1). (C) 1995 Academic Press, Inc.

[发布日期] 1995-12-10