Abstract
We consider bounded solutions of the semilinear heat equation \(u_t=u_{xx}+f(u)\) on \(R\), where \(f\) is of the unbalanced bistable type. We examine the \(\omega \)-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at \(x=\pm \infty \), the \(\omega \)-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of \(f\).
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Supported in part by NSF Grant DMS–1161923.
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Dedicated to John Mallet-Paret on the occasion of his 60th birthday.
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Poláčik, P. Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations. J Dyn Diff Equat 28, 605–625 (2016). https://doi.org/10.1007/s10884-014-9421-y
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DOI: https://doi.org/10.1007/s10884-014-9421-y
Keywords
- Bistable reaction–diffusion equation
- Localized initial data
- Asymptotic behavior
- Nonconvergent solutions
- Threshold solutions