Continuity of attractors for C^1 perturbations of a smooth domain

Abstract

We consider a family of semilinear parabolic problems with nonlinear boundary conditions $$\displaylines{ u_t(x,t)=\Delta u(x,t) -au(x,t) + f(u(x,t)),\quad x \in \Omega_\epsilon,\; t>0\,,\cr \frac{\partial u}{\partial N}(x,t)=g(u(x,t)), \quad x \in \partial\Omega_\epsilon,\; t>0\,, }$$ 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 \(\epsilon = 0\). For more information see https://ejde.math.txstate.edu/Volumes/2020/97/abstr.html