Asymptotic behavior of the numerical solutions for a system of nonlinear integrodifferential reaction-diffusion equations

This paper is concerned with the asymptotic behavior of the finite difference solutions of a coupled system of nonlinear integrodifferential reaction-diffusion equations. The existence of the finite difference solution and the monotone iteration process for solving the finite difference system are given. This includes an existence-uniqueness-comparison theorem. From the monotone iteration process, an attractor of the numerical time-dependent solution is obtained. This attractor is a sector between the pair of coupled quasisolutions of the corresponding numerical steady-state problem, which are obtained from a monotone iteration process. A sufficient condition, ensuring that the two coupled quasisolutions coincide, is given. Also given is the application to a reaction-diffusion problem with three different types of reaction functions, including some numerical results which validate the theory analysis.