Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains

P. Poláčik, F. Simondon

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Abstract

We consider the Dirichlet problem for the semilinear heat equation ut = Δu + g(x, u), xεΩ, where Ω is an arbitrary bounded domain in ℝN, N≥2, with C2 boundary. We find a C∞-function g(x, u) such that (0.1) has a bounded solution whose ω-limit set is a continuum of equilibria. This extends and improves an earlier result of the first author with Rybakowski, in which Ω is a disk in ℝ2 and g is of finite differentiability class. We also show that (0.1) can have an infinite-dimensional manifold of nonconvergent bounded trajectories.

Original languageEnglish (US)
Pages (from-to)586-610
Number of pages25
JournalJournal of Differential Equations
Volume186
Issue number2
DOIs
StatePublished - Dec 10 2002

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