Continuity of attractors for $\mathcal{C}^1$ perturbations of a smooth domain

TL;DR

This paper studies how the global attractors of semilinear parabolic problems with nonlinear boundary conditions behave under small smooth domain perturbations, proving their continuity as the domain changes.

Contribution

It establishes the continuity of attractors for semilinear parabolic problems under $ ext{C}^1$ domain perturbations, extending the understanding of stability of attractors in variable domains.

Findings

Global attractors exist for small perturbations.

Attractors vary continuously with domain changes.

Results apply to smooth, nonlinear boundary conditions.

Abstract

We consider a family of semilinear parabolic problems with nonlinear boundary conditions \[ \left\{ \begin{aligned} u_t(x,t) &=\Delta u(x,t) -au(x,t) + f(u(x,t)),\ x \in \Omega_\epsilon \mbox{ and } t>0\,,\\ \displaystyle\frac{\partial u}{\partial N}(x,t) &=g(u(x,t)),\ x \in \partial\Omega_\epsilon \mbox{ and } t>0\,, \end{aligned} \right. \] 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 $\epsilon = 0$.