Upper semicontinuity of attractors of semilinear parabolic equations with asymptotically degenerating coefficients

I. D. Chueshov   ; L. S. Pankratov

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 6 (1999) no. 1, pp. 158-181

Résumé

The initial boundary value problem for semilinear parabolic equation $$ \frac{\partial u^\varepsilon}{\partial t}-\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\left(a^\varepsilon_{ij}(x)\frac{\partial u^\varepsilon}{\partial x_j}\right)+f(u^\varepsilon)=h^\varepsilon (x), \qquad x\in\Omega, \quad t\in(0,T), $$ with the coefficients $a^\varepsilon_{ij}(x)$ depending on a small parameter $\varepsilon$ is considered. We suppose that $a^\varepsilon_{ij}(x)$ have an order $\varepsilon^{3+\gamma}$ $(0 \le\gamma<1)$ on a set of spherical annuli $G^\alpha_\varepsilon$ having the thickness $d_\varepsilon=d\varepsilon^{2+\gamma}$. The annuli are periodically (with a period $\varepsilon$) distributed in $\Omega$. On the remaining part of the domain these coefficients are constants. The asymptotical behavior of the global attractor ${\mathcal A}_\varepsilon$ of the problem as $\varepsilon \rightarrow 0$ is studied. It is shown that the global attractors ${\mathcal A}_\varepsilon$ tend in a appropriate sense to a weak global attractor ${\mathcal A}$ of the homogenized model as $\varepsilon\to 0$. This model is a system of a parabolic p.d.e. coupled with an o.d.e.