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2019, Volume 14, Issue 2: 341-369. Doi: 10.3934/nhm.2019014

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

Mogtaba Mohammed^1,2, , and
Mamadou Sango^2,

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
^* 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

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Abstract

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.

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Mathematics Subject Classification: Primary: 60H15, 80M35, 80M40; Secondary: 35L70.

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