Continuity of attractors for a highly oscillatory family of perturbations of the square
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 $a > 0$, $Ω$ is the unit square, $Ω_ε = h_ε(Ω)$, $h_ε$ is a family of $C^{m}$ - diffeomorphisms, $m \geq 1$, which converge to the identity of $Ω$ in $C^α$ norm, if $α<1$ 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 $Ω$ by a ``pull-back'' procedure, is well posed for $0 <ε\leq ε_{0}$, $ε_{0} > 0$, in a suitable phase space, the associated semigroup has a global attractor $\mathcal{A}_ε$ and the family $\{ \mathcal{A}_ε \}_{0 \, < \, ε\, \leq \, ε_{0}}$ converges as $ε\to 0$ 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 $μ$ is essentially the limit of the Jacobian determinant of the diffeomorphism ${h_ε}_{| \partial Ω} : \partial Ω\rightarrow \partial h_ε(Ω)$ (but does not depend on the particular family $h_ε)$.