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Stable periodic solutions of a spatially homogeneous nonlocal reaction–diffusion equation

Published online by Cambridge University Press:  14 November 2011

Peter Poláčik  and
Vladimir Šošovička
Peter Poláčik
Affiliation:
Institute of Applied Mathematics, Comenius University, Mlynská Dolina, 84215 Bratislava, Slovakia
Vladimir Šošovička
Affiliation:
Institute of Applied Mathematics, Comenius University, Mlynská Dolina, 84215 Bratislava, Slovakia
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Abstract

Nonlocal reaction–diffusion equations of the form ut = uxx + F(u, α(u)), where are considered together with Neumann or Dirichlet boundary conditions. One of the main results deals with linearisation at equilibria. It states that, for any given set of complex numbers, one can arrange, choosing the equation properly, that this set is contained in the spectrum of the linearisation. The second main result shows that equations of the above form can undergo a supercritical Hopf bifurcation to an asymptotically stable periodic solution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 4 , 1996 , pp. 867 - 884
Copyright
Copyright © Royal Society of Edinburgh 1996

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