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) = \Delta u(x,t) - au(x,t) + f(u(x,t)), \,\,\, x \in \Omega_{\epsilon}, t > 0, \\ \frac{\partial u}{\partial N} (x,t) = g(u(x,t)), \,\,\, x \in \partial \Omega_{\epsilon}, 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, $m \geq 1$, which converge to the identity of $\Omega$ in $C^{\alpha}$ norm, if $\alpha <1$ but do not converge in the $C^{1}$ - norm and, $f,g: \mathbb{R} \rightarrow \mathbb{R}$ 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 $0 <\epsilon \leq \epsilon_{0}$, $\epsilon_{0} > 0$, in a suitable phase space, the associated semigroup has a global attractor $\mathcal{A}_{\epsilon}$ and the family $\{ \mathcal{A}_{\epsilon} \}_{0 \, < \, \epsilon \, \leq \, \epsilon_{0}}$ converges as $\epsilon \to 0$ to the attractor of the limiting problem: \begin{equation*}\ \left\{ \begin{array}{lll} u_{t}(x,t) = \Delta u(x,t) - au(x,t) + f(u(x,t)), \,\,\, x \in \Omega, t > 0, \\ \frac{\partial u}{\partial N} (x,t) = g(u(x,t))\mu, \,\,\, x \in \partial \Omega, t > 0, \end{array} \right. \end{equation*} where $\mu$ is essentially the limit of the Jacobian determinant of the diffeomorphism ${h_{\epsilon}}_{| \partial \Omega} : \partial \Omega \rightarrow \partial h_{\epsilon}(\Omega)$ (but does not depend on the particular family $h_{\epsilon})$.