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$G$-convergence and homogenization of monotone damped hyperbolic equations. (English) Zbl 1192.35016
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 $\frac{\partial^2u}{\partial t^2}- \text{div} (b(t,Du))- \Delta(\frac{\partial u}{\partial t})+ \overline{G}(t,\frac{\partial u}{\partial t})=f$.

35B27Homogenization; equations in media with periodic structure (PDE)
35B40Asymptotic behavior of solutions of PDE
35R60PDEs with randomness, stochastic PDE
35L77Quasilinear higher-order hyperbolic equations
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