Complicated dynamics of scalar reaction diffusion equations with a nonlocal term

Bernold Fiedler, Peter Polácik

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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 - 1990

Bibliographical note

Funding Information:
We happily remember the initiating suggestions by Jack Hale, who always so very generously shares his deep insights. We are also indebted to N. Chafee and J. Mallet-Paret for supportive discussions, and to H. Fattorini and A. Wyler for bringing the pole assignment theorem to our attention. This work was financially supported by the Deutsche Forschungsgemeinschaft.

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