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Zbl 1192.35016
Nguetseng, G.; Nnang, H.; Svanstedt, N.
$G$-convergence and homogenization of monotone damped hyperbolic equations. (English)
[J] Banach J. Math. Anal. 4, No. 1, 100-115, electronic only (2010). ISSN 1735-8787/e

Summary: Multiscale stochastic homogenization is studied for quasilinear hyperbolic problems. We consider the asymptotic behaviour of a sequence of realizations of the form $$\frac{\partial^2u_\varepsilon^\omega}{\partial t^2}- \text{div} \left(a\left(T_1 \bigg(\frac{x}{\varepsilon_1}\bigg) \omega_1, T_2\bigg(\frac{x}{\varepsilon_2}\bigg)\omega_2,t, Du_\varepsilon^\omega\right)\right)- \Delta\bigg( \frac{\partial u_\varepsilon^\omega}{\partial t}\bigg)+ G\left(T_3 \bigg(\frac{x}{\varepsilon_3}\bigg) \omega_3,t, \frac{\partial u_\varepsilon^\omega}{\partial t}\right)=f.$$ It is shown, under certain structure assumptions on the random maps $a(\omega_1, \omega_2,t,\xi)$ and $G(\omega_3,t,\eta)A$, that the sequence $\{u_\varepsilon^\omega\}$ of solutions converges weakly in $L^p(0,T;W_0^{1,p}(\Omega))$ to the solution $u$ of the homogenized problem $\frac{\partial^2u}{\partial t^2}- \text{div} (b(t,Du))- \Delta(\frac{\partial u}{\partial t})+ \overline{G}(t,\frac{\partial u}{\partial t})=f$.

MSC 2000:: *35B27 Homogenization, etc.
35B40 Asymptotic behavior of solutions of PDE
35R60 PDE with randomness
35L77

Keywords: $G$-convergence; multiscale; stochastic homogenization; elliptic; parabolic; hyperbolic; homogenized problem

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