Abstract

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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

$t\to\infty$ 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.