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We consider here the family of semilinear parabolic problems\begin{equation*} \begin{array}{rcl} \left\{ \begin{array}{rcl}u_t(x,t)&=&\Delta u(x,t) -au(x,t) + f(u(x,t)) ,\,\,\ x \in \Omega_\epsilon\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \\ \displaystyle\frac{\partial u}{\partialN}(x,t)&=&g(u(x,t)), \,\, x \in \partial\Omega_\epsilon\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \end{array} \right. \end{array}\end{equation*} where $ {\Omega} $ is the unit square,$\Omega_{\epsilon}=h_{\epsilon}(\Omega)$ and $h_{\epsilon}$ is a family ofdiffeomorphisms converging to the identity in the $C^1$-norm. We show that theproblem is well posed for $\epsilon>0$ sufficiently small in a suitable phasespace, the associated semigroup has a global attractor $\mathcal{A}_{\epsilon}$and the family $\{\mathcal{A}_{\epsilon}\}$ is continuous at $\epsilon = 0$.

Continuity of attractors for a family of $C^1$ perturbations of the square