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Examples of bounded solutions with nonstationary limit profiles for semilinear heat equations on $${\mathbb{R}}$$. (English) Zbl 1319.35097
The author considers bounded solutions of the Cauchy problem for the semilinear parabolic equation $$u_t=u_{xx}+f(u)$$ on $$\mathbb R$$. It is showed that there always exist bounded solutions whose $$\omega$$-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 $$|x|$$ goes to infinity. A considered example with an unbalanced bistable nonlinearity shows that bounded solutions whose $$\omega$$-limit set does not consist of steady states occur for a robust class of nonlinearities $$f$$.

##### MSC:
 35K58 Semilinear parabolic equations 35K15 Initial value problems for second-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
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