# American Institute of Mathematical Sciences

June  2019, 14(2): 341-369. doi: 10.3934/nhm.2019014

## Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

 1 Department of Mathematics, Faculty of Mathematical and Computer Sciences, University of Gezira, Wad Madani, P.O.Box 20, Sudan 2 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

* Corresponding author: Mogtaba Mohammed

Dedicated to the memory of Professor Salah-Eldin A. Mohammed (May 20, 1946 - Dec 21, 2016)

Received  June 2018 Revised  October 2018 Published  April 2019

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

Citation: Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
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