Continuity of attractors for a highly oscillatory family of perturbations of the square

Abstract

Consider the family of semilinear parabolic problems \begin{equation*} \left\{ \begin{array}{lll} u_{t}(x,t) = Δu(x,t) - au(x,t) + f(u(x,t)), \,\,\, x \in Ω_ε, t > 0, \\ \frac{\partial u}{\partial N} (x,t) = g(u(x,t)), \,\,\, x \in \partial Ω_ε, t > 0, \end{array} \right. \end{equation*} where $a>0$, $\Omega$ is the unit square, ${\Omega }_{\epsilon }=h_{\epsilon }(\Omega )$, $h_{\epsilon }$ is a family of $C^m$ - diffeomorphisms, $m\ge 1$, which converge to the identity of $\Omega$ in $C^{\alpha }$ norm, if $\alpha <1$ but do not converge in the $C^1$ - norm and, $f,g:\mathbb{R}\to \mathbb{R}$ are real functions. We show that a weak version of this problem, transported to the fixed domain $\Omega$ by a ``pull-back'' procedure, is well posed for $0<\epsilon \le {\epsilon }_0$, ${\epsilon }_0>0$, in a suitable phase space, the associated semigroup has a global attractor ${\mathcal{A}}_{\epsilon }$ and the family $\{{\mathcal{A}}_{\epsilon }{\}}_{0<\epsilon \le {\epsilon }_0}$ converges as $\epsilon \to 0$ to the attractor of the limiting problem: \begin{equation*}\ \left\{ \begin{array}{lll} u_{t}(x,t) = Δu(x,t) - au(x,t) + f(u(x,t)), \,\,\, x \in Ω, t > 0, \\ \frac{\partial u}{\partial N} (x,t) = g(u(x,t))μ, \,\,\, x \in \partial Ω, t > 0, \end{array} \right. \end{equation*} where $\mu$ is essentially the limit of the Jacobian determinant of the diffeomorphism ${h_{\epsilon }}_{|\partial \Omega }:\partial \Omega \to \partial h_{\epsilon }(\Omega )$ (but does not depend on the particular family $h_{\epsilon })$.