@article{1192.35016,
author="Nguetseng, G. and Nnang, H. and Svanstedt, N.",
title="{$G$-convergence and homogenization of monotone damped hyperbolic
    equations.}",
language="English",
journal="Banach J. Math. Anal. ",
volume="4",
number="1",
pages="100-115",
year="2010",
abstract="{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 $\frac{\partial^2u}{\partial t^2}- \text{div}
    (b(t,Du))- \Delta(\frac{\partial u}{\partial t})+
    \overline{G}(t,\frac{\partial u}{\partial t})=f$.}",
keywords="{$G$-convergence; multiscale; stochastic homogenization; elliptic;
    parabolic; hyperbolic; homogenized problem}",
classmath="{*35B27 (Homogenization, etc.)
35B40 (Asymptotic behavior of solutions of PDE)
35R60 (PDE with randomness)
35L77 ( )
}",
}