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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.

Information

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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References

1Calsina, A. and Perelló, C.. Equations for biological evolution (Preprint).Google Scholar
2Chafee, N.. The electric balast resistor: homogeneous and nonhomogeneous equilibria. In Nonlinear Differential Equations: Invariance, Stability and Bifurcation, eds Mottoni, P. de and Salvadori, L., pp. 97–127 (New York: Academic Press, 1981).CrossRefGoogle Scholar
3Fiedler, B. and Polàčik, P.. Complicated dynamics of scalar reaction–diffusion equations with a nonlocal term. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 167–92.CrossRefGoogle Scholar
4Freitas, P.. A nonlocal Sturm–Liouville eigenvalue problem. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 169–88.CrossRefGoogle Scholar
5Freitas, P.. Bifurcation and stability of stationary solutions of nonlocal scalar reaction–diffusion equations. J. Dynam. Differential Equations 4 (1994), 613–29.CrossRefGoogle Scholar
6Freitas, P.. Stability of stationary solutions for a scalar nonlocal reaction–diffusion equation (Preprint).Google Scholar
7Freitas, P. and Grinfeld, M.. Stationary solutions of an equation modelling ohmic heating. Appl. Math. Lett. 7 (1994), 1–6.CrossRefGoogle Scholar
8Furter, J. and Grinfeld, M.. Local vs. non-local interactions in population dynamics. J. Math. Biol. 27(1988), 65–80.CrossRefGoogle Scholar
9Hale, J. K.. Asymptotic behavior of dissipative systems (Providence, RI: AMS Publications, 1988).Google Scholar
10Henry, D.. Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840 (New York: Springer, 1981).CrossRefGoogle Scholar
11Lacey, A.. Thermal runaway in a nonlocal problem modelling ohmic heating (Preprint).Google Scholar
12Matano, H.. Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ. 18 (1978), 221–7.Google Scholar
13Medved, M.. Fundamentals of dynamical systems and bifurcation theory (New York: Adam Hilger, 1992).Google Scholar
14Vanderbauwhede, A.. Center manifolds, normal forms and elementary bifurcations. Dynamics Reported, eds Kirchgraber, U. and Walther, H.-O., 2 (1989), 89–169.CrossRefGoogle Scholar
15Vanderbauwhede, A. and Iooss, G.. Center manifold theory in infinite dimensions. Dynamics Reported (New Series), eds Jones, C. K. R. T., Kirchgraber, U. and Walther, H. O., 1 (1992), 125–63.CrossRefGoogle Scholar
16Zelenyak, T. I.. Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable. Differential Equations 4 (1968), 17–22.Google Scholar