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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.