Stability of solutions of infinite systems of nonlinear differential-functional equations of parabolic type

Abstract

A parabolic initial boundary value problem and an associated elliptic Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations are considered. It is shown that the solutions of the parabolic problem is asymptotically stable and the limit of the solution of the parabolic problem as $t\to\infty$ is the solution of the associated elliptic problem. The result is based on the monotone methods.