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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

2 u ε ω t 2 -divaT 1 x ε 1 ω 1 ,T 2 x ε 2 ω 2 ,t,Du ε ω -Δu ε ω t+GT 3 x ε 3 ω 3 ,t,u ε ω t=f·

It is shown, under certain structure assumptions on the random maps a(ω 1 ,ω 2 ,t,ξ) and G(ω 3 ,t,η)A, that the sequence {u ε ω } of solutions converges weakly in L p (0,T;W 0 1,p (Ω)) to the solution u of the homogenized problem 2 u t 2 -div(b(t,Du))-Δ(u t)+G ¯(t,u t)=f.

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