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Symmetry and convergence properties for non-negative solutions of nonautonomous reaction–diffusion problems

Published online by Cambridge University Press:  14 November 2011

Peter Hess  and
P. Poláčik
Peter Hess
Affiliation:
Mathematisches Institut, Universität Zurich, Rämistrasse 74, 8001 Züurich, Switzerland
P. Poláčik
Affiliation:
Institute of Applied Mathematics, Comenius University, Mlynská Dolina, 824 15 Bratislava, Slovakia
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Abstract

Nonautonomous parabolic equations of the form ut − Δu = f(u, t) on a symmetric domain are considered. Using the moving-hyperplane method, it is proved that any bounded nonnegative solution symmetrises as t → ∞. This is then used to show that for nonlinearities periodic in t, any non-negative bounded solution approaches a periodic solution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 3 , 1994 , pp. 573 - 587
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Angenent, S. B.. Solutions of the 1 – d porous medium equation are determined by their free boundary J. London Math. Soc. 42 (1990), 339–353.CrossRefGoogle Scholar
2Berestycki, H. and Nirenberg, L.. On the method of moving planes and the sliding method (preprint, École Normale Supérieure, 1991).CrossRefGoogle Scholar
3Brunovsky, P. and Poláčik, P.. Generic hyperbolicity for reaction–diffusion equations on symmetric domains. J. Appl. Math. Phys. (ZAMP) 38 (1987), 172–183.CrossRefGoogle Scholar
4Brunovsky, and Poláčik, P. and Sandstede, B.. Convergence in general periodic parabolic equations in one space dimension. Nonlinear Anal. 18 (1992), 209–215.CrossRefGoogle Scholar
5Castro, A. and Shivaji, R.. Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric. Comm. Partial Differential Equations 14 (1989), 1091–1100.CrossRefGoogle Scholar
6Chen, M., Chen, X.-Y. and Hale, J. K.. Structural stability for time-periodic one-dimensional parabolic equations. J. Differential Equations 96 (1992), 355–418.CrossRefGoogle Scholar
7Chen, X.-Y. and Matano, H.. Convergence, asymptotic periodicity, and finite-point blow up in onedimensional semilinear heat equations. J. Differential Equations 78 (1989), 160—190.CrossRefGoogle Scholar
8Chow, S. N. and Hale, J. K.. Methods of Bifurcation Theory (New York: Springer, 1982).CrossRefGoogle Scholar
9Dancer, E. N. and Hess, P.. Stable subharmonic solutions in periodic reaction-diffusion equations. J. Differential Equations (to appear).Google Scholar
10Dancer, E. N. and Hess, P.. The symmetry of positive solutions of periodic-parabolic problems (preprint).Google Scholar
11Gidas, B., Ni, W. and Nirenberg, L.. Symmetry and related properties by the maximum principle. Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
12Henry, D.. Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations. J. Differential Equations 59 (1985), 165–205.CrossRefGoogle Scholar
13Hess, P.. Asymptotics in semilinear periodic diffusion equations with Dirichlet or Robin boundary conditions. Arch. Rational Mech. Anal. 116 (1991), 91–99.CrossRefGoogle Scholar
14Hess, P.. Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics 247 (Harlow: Longman, 1991).Google Scholar
15Hess, P. and Poláčik, P.. Boundedness of prime periods of stable cycles and convergence to fixed points in discrete monotone dynamical systems. SIAM J. Math. Anal, (to appear).Google Scholar
16Hess, P. and Weinberger, H.. Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses. J. Math. Biol. 28 (1990), 83–98.CrossRefGoogle ScholarPubMed
17Poláčik, P. and Tereščak, I.. Convergence to cycles as a typical asymptotic behaviour in smooth strongly monotone discrete-time dynamical systems. Arch. Rational Mech. Anal. 116 (1991), 339–360.CrossRefGoogle Scholar
18Poláčik, P. and Tereščak, I.. Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations. J. Dynamical Differential Equations 5 (1993)279–303.CrossRefGoogle Scholar
19Serrin, J.. A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971), 304–318.CrossRefGoogle Scholar
20Shub, M.. Global Stability of Dynamic Systems (New York: Springer, 1987).CrossRefGoogle Scholar
21Takáč, P.. Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems. Proc. Amer. Math. Soc. 115 (1992), 691–698CrossRefGoogle Scholar
22Takáč, P.. A construction of stable subharmonic orbits in monotonic time-periodic dynamical systems Mh. Math. 115 (1993), 215–244.CrossRefGoogle Scholar
23Watson, G. N., A Treatise on the Theory of Bessel Functions (Cambridge: University Press 1944).Google Scholar