G-Convergence and Homogenization of Monotone Damped Hyperbolic Equations

Paper in proceeding, 2010

Multiscale stochastic homogenization is studied for quasilinear hyperbolic problems. We consider the asymptotic behaviour of a sequence of realizations of the form partial derivative(2)u(epsilon)(omega)/partial derivative t(2) - div (a(T(1)(x/epsilon(1))omega(1), T(2)(x/epsilon(2))omega(2), t, Du(epsilon)(omega))) - Delta(partial derivative u(epsilon)(omega)/partial derivative t) + G(T(3)(x/epsilon(3))omega(3,) t, partial derivative u(epsilon)(omega)/partial derivative t) = 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), that the sequence {u(epsilon)(omega)} of solutions converges weakly in L(p)(0, T; W(0)(1,p)(Omega)) to the solution u of the homogenized problem partial derivative(2)u/partial derivative t(2) - div (b(t, (Du)) - Delta(partial derivative u/partial derivative t) + (G) over bar (t, partial derivative u/partial derivative t) = f.