Examples of bounded solutions with nonstationary limit profiles for semilinear heat equations on $${\mathbb{R}}$$ R

Abstract

We consider bounded solutions of the semilinear heat equation \({u_t = u_{xx} + f(u) \,\,{\rm on}\,\, \mathbb{R}}\), where f is of a bistable type. We show that there always exist bounded solutions whose ω-limit set with respect to the locally uniform convergence contains functions which are not steady states. For balanced bistable nonlinearities, there are examples of such solutions with initial values u(x, 0) converging to 0 as \({|x| \to \infty}\). Our example with an unbalanced bistable nonlinearity shows that bounded solutions whose ω-limit set do not consist of steady states occur for a robust class of nonlinearities f.