Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains

We consider the Dirichlet problem for the semilinear heat equation u(t) = Deltau + g(x, u), x is an element of Omega, where Omega is an arbitrary bounded domain in R-N, N greater than or equal to 2, with C-2 boundary. We find a C-infinity-function g(x,u) such that (0.1) has it bounded Solution whose omega-limit set is a continuum of equilibria, This extends and improves an earlier result of the first author with Rybakowski, in which Omega is a disk in R-2 and g is of finite differentiability class. We also show that (0.1) can hake an infinite-dirnensional manifold of nonconvergent bounded trajectories, (C) 2002 Elsevier Science (USA). All rights reserved.