Convergence and Quasiconvergence Properties of Solutions of Parabolic Equations on the Real Line: An Overview

Abstract

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