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Matematicheskaya Fizika, Analiz, Geometriya [Mathematical Physics, Analysis, Geometry], 1998, Volume 5, Number 3/4, Pages 250–273
(Mi jmag440)
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This article is cited in 4 scientific papers (total 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
Citation:
L. Pankratov, “Homogenization of semilinear parabolic equations with asymptotically degenerating coefficients”, Mat. Fiz. Anal. Geom., 5:3/4 (1998), 250–273
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https://www.mathnet.ru/eng/jmag440 https://www.mathnet.ru/eng/jmag/v5/i3/p250
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