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

Source 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 equations. In particular we show that all attractors of the ODE are stable as solutions of the PDE as well.