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

Abstract

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 ${\Omega }_0\subset {\mathbb{R}}^n$ 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 $\epsilon =0$.

arXiv admin note: text overlap with arXiv:1603.06104