Asymptotic analysis of a double porosity model with thin fissures

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.