Invariant manifolds for parabolic equations under perturbation of the domain |
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| Institution: | 1. Angewandte Mathematik Münster, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstraße 62, D-48149 Münster, Germany;2. Politecnico di Bari, Dipartimento di Meccanica, Matematica e Management, Via Amendola, 126/B, I-70126 Bari, Italy;1. Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle, Germany;2. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA;1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China;2. School of Mathematics and Statistics Science, Ludong University, Yantai 264025, PR China |
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| Abstract: | We study the effect of domain perturbation on invariant manifolds for semilinear parabolic equations subject to the Dirichlet boundary condition. Under the Mosco convergence assumption on the domains, we prove the upper and lower semicontinuity of both the local unstable invariant manifold and the local stable invariant manifold near a hyperbolic equilibrium. The continuity results are obtained by keeping track of the construction of invariant manifolds in P.W. Bates, C.K.R.T. Jones, Invariant manifolds for semilinear partial differential equations, in: Dynamics Reported, Vol. 2, in: Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38]. |
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