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

Abstract

We consider here the family of semilinear parabolic problems \begin{equation*} \begin{array}{rcl} \left\{ \begin{array}{rcl} u_t(x,t)&=&Δu(x,t) -au(x,t) + f(u(x,t)) ,\,\,\ x \in Ω_ε\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \\ \displaystyle\frac{\partial u}{\partial N}(x,t)&=&g(u(x,t)), \,\, x \in \partialΩ_ε\,\,\,\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 $\epsilon =0$.