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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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