Complicated dynamics of scalar reaction diffusion equations with a nonlocal term

Bernold Fiedler, Peter Polácik

Research output: Contribution to journalArticlepeer-review

42 Scopus citations

Abstract

We consider the dynamics of scalar equations u, = uxx +f(x, u) + c(x)α(u), 0«x«l, where α denotes some weighted spatial average and Dirichlet boundary conditions are assumed. Prescribing f, c, α appropriately, it is shown that complicated dynamics can occur. Specifically, linearisations at equilibria can have any number of purely imaginary eigenvalues. Moreover, the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily, up to any finite order. These results are in marked contrast with the case α = 0, where bounded solutions are known to converge to equilibrium.

Original languageEnglish (US)
Pages (from-to)167-192
Number of pages26
JournalProceedings of the Royal Society of Edinburgh: Section A Mathematics
Volume115
Issue number1-2
DOIs
StatePublished - Jan 1 1990

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