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

We consider a family of semilinear parabolic problems with nonlinear boundary conditions \[ \left\{ \begin{aligned} u_t(x,t) &=Δu(x,t) -au(x,t) + f(u(x,t)),\ x \in Ω_ε\mbox{ and } t>0\,,\\ \displaystyle\frac{\partial u}{\partial N}(x,t) &=g(u(x,t)),\ x \in \partialΩ_ε\mbox{ and } t>0\,, \end{aligned} \right. \] where $Ω_0 \subset \mathbb{R}^n$ is a smooth (at least $\mathcal{C}^2$) domain , $Ω_ε = h_ε(Ω_0)$ and $h_ε$ 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 $ε>0$ sufficiently small in a suitable scale of fractional spaces, the associated semigroup has a global attractor $\mathcal{A}_ε$ and the family $\{\mathcal{A}_ε\}$ is continuous at $ε= 0$.

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