Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains

We consider the Dirichlet problem for the semilinear heat equation ut=Δu+g(x,u), x Ω,where Ω is an arbitrary bounded domain in , N 2, with C2 boundary. We find a C∞-function g(x,u) such that (0.1) has a bounded solution whose ω-limit set is a continuum of equilibria. This extends and improves an earlier result of the first author with Rybakowski, in which Ω is a disk in and g is of finite differentiability class. We also show that (0.1) can have an infinite-dimensional manifold of nonconvergent bounded trajectories.