Large Time Behavior of Solutions of Systems of Nonlinear Reaction-Diffusion Equations

Abstract

We discuss the asymptotic behavior of solutions of weakly coupled parabolic equations describing systems undergoing diffusion, convection and nonlinear interaction in a bounded spatial domain, $\Omega $. If the system admits a bounded invariant region $\Sigma $ of phase space then we isolate a parameter $\sigma $ which depends upon the size of $\Omega $, the lower bound for the diffusion matrix, the magnitude of convection and a measure of the strength or sensitivity of the reaction. For $\sigma > 0$ we show that every solution with initial values in $\Sigma $ and subject to homogeneous Neumann boundary conditions decays exponentially to a spatially homogeneous function of time. This limiting function is a solution of an ordinary differential equation whose $\omega $-limit sets are determined by the reaction mechanism alone. This result may be interpreted as giving a sufficient condition for the validity of the “lumped parameter” approximation of distributed systems by solutions of ordinary differential e...