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

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 $\mathrm{\Omega }$ is the unit square, ${\mathrm{\Omega }}_{ϵ}=h_{ϵ}(\mathrm{\Omega })$ and $h_{ϵ}$ is a family of diffeomorphisms converging to the identity in the $C^1$-norm. We show that the problem is well posed for $ϵ>0$ sufficiently small in a suitable phase space, the associated semigroup has a global attractor ${\mathcal{A}}_{ϵ}$ and the family $\{{\mathcal{A}}_{ϵ}\}$ is continuous at $ϵ=0$.