This article is cited in 4 papers

Homogenization of semilinear parabolic equations with asymptotically degenerating coefficients

L. Pankratov

Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., 310164, Kharkov, Ukraine

Abstract: 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 $\varepsilon^{3+\gamma}$ $(0\le \gamma<1)$ 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.

Received: 10.02.1997

Language: English

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