G-convergence and homogenization of monotone damped hyperbolic equations

Abstract

Multiscale stochastic homogenization is studied for quasilinear hyperbolic problems. We consider the asymptotic behaviour of a sequence of realizations of the form ${\frac{\partial^2 u^\omega_{\varepsilon}}{\partial t^2}} - \mathrm{div}\left(a\left(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_{\varepsilon}\right)\right)-\Delta(\frac{\partial u^\omega_{\varepsilon}}{\partial t}) +G\left(T_3(\frac{x}{\varepsilon_3})\omega_3 ,t,\frac{\partial u^\omega_{\varepsilon}}{\partial t}\right)=f$. It is shown, under certain structure assumptions on the random maps $a\left(\omega_1,\omega_2,t,\xi\right)$ and $G\left(\omega_3,t,\eta\right)$, that the sequence $\{u^\omega_\epsilon\}$ of solutions converges weakly in $L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem ${\frac{\partial^2 u}{\partial t^2}} - \mathrm{div}\left( b \left( t,D u \right)\right)-\Delta(\frac{\partial u}{\partial t})+{\overline G}(t,\frac{\partial u}{\partial t}) = f$.