Abstract
We consider the singularly perturbed semilinear parabolic problem u_t-d^2\Delta u+u=f(u) with homogeneous Neumann boundary conditions on a smoothly bounded domain \Omega\subseteq{\mathbb{R}}^N. Here f is superlinear at 0, and infinity and has subcritical growth.
For small d>0 we construct a compact connected invariant set X_d in the boundary of the domain of attraction of the asymptotically stable equilibrium $0$. The main features of $X_d$ are that it consists of positive functions that are pairwise non-comparable, and its topology is at least as rich as the topology of $\partial\Omega$ in a certain sense. If the number of equilibria in $X_d$ is finite, then this implies the existence of connecting orbits within $X_d$ that are not a consequence of a well known result by Matano.