Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 97, pp. 1-31.

Title: Continuity of attractors for C^1 perturbations of a smooth domain
       
Authors: Pricila S. Barbosa (Univ. Tecnologica Federal do Parana, Brazil)
         Antonio L. Pereira (Univ. de Sao Paulo, Sao Paulo, Brazil)

Abstract:
 We consider a family of semilinear parabolic problems with nonlinear boundary conditions
 $$\displaylines{ 
 u_t(x,t)=\Delta u(x,t) -au(x,t) + f(u(x,t)),\quad  x \in \Omega_\epsilon,\; t>0\,,\cr
 \frac{\partial u}{\partial N}(x,t)=g(u(x,t)), \quad  x \in \partial\Omega_\epsilon,\; t>0\,,
 }$$
 where $\Omega_0 \subset \mathbb{R}^n$ is a smooth (at least $\mathcal{C}^2$) domain,
 $\Omega_{\epsilon} = h_{\epsilon}(\Omega_0)$ and $h_{\epsilon}$ is a family of
 diffeomorphisms converging to the identity in the $\mathcal{C}^1$-norm.
 Assuming suitable regularity and dissipative conditions for the nonlinearites,
 we show that the problem is well posed for $\epsilon>0$ sufficiently small in a suitable
 scale of fractional spaces,  the associated semigroup has a global attractor
 $\mathcal{A}_{\epsilon}$ and the family $\{\mathcal{A}_{\epsilon}\}$  is continuous at
 $\epsilon = 0$.
 
Submitted December 31, 2019. Published September 21, 2020.
Math Subject Classifications: 35B41, 35K20, 58D25.
Key Words: Parabolic problem; perturbation of the domain; global attractor;
           continuity of attractors.