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

Анотація

The initial boundary value problem for semilinear parabolic equation $$\frac{\partial u^\varepsilon}{\partial t}-\displaystyle\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\left(a^\varepsilon_{ij}(x)\frac{\partial u^\varepsilon}{\partial x_j}\right)+f(u^\varepsilon)=h^\varepsilon(x), x\in\Omega,t\in (0,T);$$ 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 $d\varepsilon^{2+\gamma}$. 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 $\varepsilon\to 0$ 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.