Invariant manifolds for parabolic equations under perturbation of the domain

Parinya Sa Ngiamsunthorn

Abstract

We study the effect of domain perturbation on invariant manifolds for semilinear parabolic equations subject to Dirichlet boundary condition. Under 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 and C. K. R. T. Jones [Dynam. Report. Ser. Dynam. Systems Appl. Vol. 2, 1–38, 1989].

Keywords: domain perturbation, invariant manifolds, upper and lower semicontinuity, semilinear parabolic equations, Mosco convergence.

AMS Subject Classification: Primary 37L05; secondary 35K58, 35B20.