Existence and stability of bounded solutions for a system of parabolic equations

In this paper we study the existence and the stability of bounded solutions of the following nonlinear system of parabolic equations with homogeneous Dirichlet boundary conditions: u(t) = DDeltau + f (t, u), t greater than or equal to 0, u is an element of R-n, u = 0 on partial derivativeOmega, where f is an element of C-1 (R x R-n), D = diag(d(1), d(2),..., d(n)) is a diagonal matrix with d(i) > 0, i = 1, 2,..., n, and Omega is a sufficiently regular bounded domain in R-N (N = 1, 2,3). Roughly speaking we shall prove the following result: if f is globally Lipschitz with constant L, 3/4 < alpha < 1 and (lambda1d)1-alpha/Gamma(1-alpha) > 6ML, then the system has a bounded solution on R-n which is stable, where 2d = min{d(i): i = 1, 2,..., n}, (lambda(j)d(j)t)(alpha)e(-lambdaj (di/2)t) < M, lambdai are the eigenvalues of -Delta, and Gamma(.) is the well-known gamma function. Also, we prove that for a large class of functions f this bounded solution is almost periodic. (C) 2003 Published by Elsevier Science (USA).