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arXiv · 1603.06104

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

Pricila S. Barbosa·Antônio L. Pereira·Marcone C. Pereira
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Abstract

We consider here the family of semilinear parabolic problems \begin{equation*} \begin{array}{rcl} \left\{ \begin{array}{rcl} u_t(x,t)&=&Δu(x,t) -au(x,t) + f(u(x,t)) ,\,\,\ x \in Ω_ε\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \\ \displaystyle\frac{\partial u}{\partial N}(x,t)&=&g(u(x,t)), \,\, x \in \partialΩ_ε\,\,\,\mbox{and}\,\,\,\,\,\,t>0\,, \end{array} \right. \end{array} \end{equation*} where $ Ω $ is the unit square, $Ω_ε=h_ε(Ω)$ and $h_ε$ is a family of diffeomorphisms converging to the identity in the $C^1$-norm. We show that the problem is well posed for $ε>0$ sufficiently small in a suitable phase space, the associated semigroup has a global attractor $\mathcal{A}_ε$ and the family $\{\mathcal{A}_ε\}$ is continuous at $ε= 0$.

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Pricila S. Barbosa, Antônio L. Pereira, Marcone C. Pereira. 2016-03-31. Continuity of attractors for a family of $C^1$ perturbations of the square. https://arxiv.org/abs/1603.06104

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