Homogenization of semilinear parabolic equations with asymptotically degenerating coefficients

L. Pankratov

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 5 (1998), pp. 250-273.

Résumé

An 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 $a^\varepsilon_{ij}(x)$ are of the order of $\varepsilon^{3+\gamma}$ $(0\le \gamma1)$ on a set of spherical annuluses $G^\alpha_\varepsilon$ of a thickness $d_\varepsilon = d\varepsilon^{2+\gamma}$. The annuluses are periodically with a period $\varepsilon$ distributed in $\Omega$. On the set $\Omega\setminus U_\alpha G^\alpha_\varepsilon$ these coefficients are constants. We study the asymptotical behaviour of the solutions $u^\varepsilon(x,t)$ of the problem as $\varepsilon \rightarrow 0$. 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.