Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations

Gui-Qiang Chen; Benoît Perthame

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Reiterated homogenization is studied for divergence structure parabolic problems of the form $\partial u_{ε} / \partial t - div (a (x, x / ε, x / ε^{2}, t, t / ε^{k}) \nabla u_{ε}) = f$ . It is shown that under standard assumptions on the function $a (x, y_{1}, y_{2}, t, τ)$ the sequence ${u_{ϵ}}$ of solutions converges weakly in $L^{2} (0, T; H_{0}^{1} (Ω))$ to the solution $u$ of the homogenized problem $\partial u / \partial t - div (b (x, t) \nabla u) = f$ .

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We study the homogenization of parabolic or hyperbolic equations like $ρ_{ε} \frac{\partial^{n} u_{ε}}{\partial t^{n}} - div (a_{ε} \nabla u_{ε}) = f in Ω \times (0, T) + boundary conditions, n \in {1, 2},$ when the coefficients $ρ_{ε}$ , $a_{ε}$ (defined in $Ø$ ) take possibly high values on a $ε$ -periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.