Electron. J. Differential Equations, Vol. 2020 (2020), No. 97, pp. 1-31.

Continuity of attractors for C^1 perturbations of a smooth domain

Pricila S. Barbosa, Antonio L. Pereira

We consider a family of semilinear parabolic problems with nonlinear boundary conditions

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.

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Pricila S. Barbosa
Universidade Tecnológica Federal do Paraná,
Paraná, Brazil
email: pricilabarbosa@utfpr.edu.br
Antônio L. Pereira
Instituto de Matemática e Estatística
Universidade de São Paulo
São Paulo, Brazil
email: alpereir@ime.usp.br

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