Asymptotic analysis of a double porosity model with thin fissures

L. S. Pankratov ; V. A. Rybalko

Sbornik. Mathematics, Tome 194 (2003) no. 1, pp. 123-150

Résumé

An initial-boundary-value problem is considered for the parabolic equation $$ \Phi^\varepsilon(x)u^\varepsilon_t-\operatorname{div}(A^\varepsilon(x) \nabla u^\varepsilon)=f^\varepsilon(x), \qquad x\in\Omega, \quad t>0, $$ with discontinuous diffusion tensor $A^\varepsilon(x)$. This tensor is assumed to degenerate as $\varepsilon\to0$ in the whole of the domain $\Omega$ except on a set ${\mathscr F}^{(\varepsilon)}$ of asymptotically small measure. It is shown that the behaviour of the solutions $u^\varepsilon$ as $\varepsilon\to0$ is described by a homogenized model with memory.

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