Continuity of attractors for a family of $C^1$ perturbations of the square

TL;DR

This paper proves that for a family of perturbed square domains, the associated semilinear parabolic problems have well-defined global attractors that vary continuously with the domain perturbation.

Contribution

It establishes the continuity of attractors for a family of $C^1$ domain perturbations in semilinear parabolic problems on the square.

Findings

Existence of well-posed solutions for small perturbations.

Presence of a global attractor for each perturbed problem.

Continuity of the attractors as the domain perturbation vanishes.

Abstract

We consider here the family of semilinear parabolic problems \begin{equation*} \begin{array}{rcl} \left\{ \begin{array}{rcl} 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{array} \right. \end{array} \end{equation*} where $ {\Omega} $ is the unit square, $\Omega_{\epsilon}=h_{\epsilon}(\Omega)$ and $h_{\epsilon}$ is a family of diffeomorphisms converging to the identity in the $C^1$-norm. We show that the problem is well posed for $\epsilon>0$ sufficiently small in a suitable phase space, the associated semigroup has a global attractor $\mathcal{A}_{\epsilon}$ and the family $\{\mathcal{A}_{\epsilon}\}$ is continuous at $\epsilon = 0$.