Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains

AbstractWe consider the Dirichlet problem for the semilinear heat equation (0.1)ut=Δu+g(x,u),x∈Ω,where Ω is an arbitrary bounded domain in RN, 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 R2 and g is of finite differentiability class. We also show that (0.1) can have an infinite-dimensional manifold of nonconvergent bounded trajectories