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We study the effect of domain perturbation on invariant manifolds forsemilinear parabolic equations subject to Dirichlet boundary condition. UnderMosco convergence assumption on the domains, we prove the upper and lowersemicontinuity of both the local unstable invariant manifold and the localstable invariant manifold near a hyperbolic equilibrium. The continuity resultsare 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].

Invariant manifolds for parabolic equations under perturbation of the domain