Invariant manifolds for parabolic equations under perturbation of the domain

TL;DR

This paper investigates how small changes in the domain affect the stability structures of solutions to semilinear parabolic equations, establishing continuity properties of invariant manifolds under domain perturbations.

Contribution

It proves the upper and lower semicontinuity of invariant manifolds for parabolic equations under domain perturbations using Mosco convergence.

Findings

Invariant manifolds are semicontinuous under domain perturbations.

Continuity is established for both stable and unstable manifolds.

Results extend previous work to more general domain perturbations.

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].