Convergence and quasiconvergence properties of solutions of parabolic equations on the real line: an overview

We consider semilinear parabolic equations ut = uxx+f(u) on R. We give an overview of results on the large time behavior of bounded solutions, focusing in particular on their limit profiles as t→ ∞ with respect to the locally uniform convergence. The collection of such limit profiles, or, the ω-limit set of the solution, always contains a steady state. Questions of interest then are whether—or under what conditions—the ω-limit set consists of steady states, or even a single steady state. We give several theorems and examples pertinent to these questions.