Homogenization of semilinear parabolic equations with asymptotically degenerating coefficients

Анотація

An 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 $a^\varepsilon_{ij}(x)$ are of the order of $\varepsilon^{3+\gamma}$ ($0\le\gamma <1$ ) on a set of spherical annuluses $G^\alpha_\varepsilon$  thickness $d_\varepsilon = d \varepsilon^{2+\gamma}$. The annuluses are periodically with a period $\varepsilon$ distributed in $\Omega$. On the set $\Omega\setminus \bigcup_\alpha G^\alpha_\varepsilon$ these coefficients are constants. We study the asymptotical behaviour of the solutions $u^\varepsilon (x,t)$ of the problem as $\varepsilon\to 0$. It is shown that the asymptotic behaviour of the solutions is described by a system of a parabolic p.d.e. coupled with an o.d.e.