Geodesic
Continuous dependence of attractors on the shape of domain
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 26, Tome 221 (1995), pp. 58-66 
A. V. Babin; S. Yu. Pilyugin. Continuous dependence of attractors on the shape of domain. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 26, Tome 221 (1995), pp. 58-66. http://geodesic.mathdoc.fr/item/ZNSL_1995_221_a3/
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     author = {A. V. Babin and S. Yu. Pilyugin},
     title = {Continuous dependence of attractors on the shape of domain},
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     pages = {58--66},
     year = {1995},
     volume = {221},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1995_221_a3/}
}
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Let $\Omega_0$ be a bounded domain in $\mathbb R^n$, let $\mathcal G$ be a family of diffeomorphisms, and let $\Omega_G=G(\Omega_0)$ for $G\in\mathcal G$. Denote by $\Sigma_t(G)$ the semigroup generated by a fixed parabolic PDE with Dirichlet boundary conditions on the boundary of $\Omega_G$. Let $A_G$ be the global attractor $\Sigma_t(G)$. Conditions are given under which a generic diffeomorphism $G\in\mathcal G$ is a continuity point of the map $G\mapsto A_G$. Bibliography: 12 titles.