Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions

Yuri Latushkin; Jan Prüss; Ronald Schnaubelt

doi:10.3934/dcdsb.2008.9.595

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2008, Volume 9, Issue 3&4: 595-633. Doi: 10.3934/dcdsb.2008.9.595

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Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions

1.
Department of Mathematics, University of Missouri, Columbia, MO 65211
2.
Fachbereich Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle
3.
Department of Mathematics, University of Karlsruhe, 76128 Karlsruhe

Received: March 2007

Revised: August 2007

Published: May 2008

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Abstract

We study quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains. Our main results concern the asymptotic behavior of the solutions in the vicinity of an equilibrium. The local center, center–stable, and center–unstable manifolds are constructed and their dynamical properties are established using nonautonomous cutoff functions. Under natural conditions, we show that each solution starting close to the center manifold converges to a solution on the center manifold.

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Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 35K50, 35K60.

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