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Complicated dynamics of scalar reaction diffusion equations with a nonlocal term

Published online by Cambridge University Press:  14 November 2011

Bernold Fiedler  and
Peter Poláčik
Bernold Fiedler
Affiliation:
Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, D-6900 Heidelberg, West Germany
Peter Poláčik
Affiliation:
Institute of Applied Mathematics, Comenius University, Mlynská Dolina, 84 215 Bratislava, ČSSR
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Synopsis

We consider the dynamics of scalar equations ut, = uxx + f(x, u) + c(x)α(u), 0 < x < l, where α denotes some weighted spatial average and Dinchlet 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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 1-2 , 1990 , pp. 167 - 192
Copyright
Copyright © Royal Society of Edinburgh 1990

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