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

Mogtaba Mohammed; Mamadou Sango; Mogtaba Mohammed; Mamadou Sango

doi:10.3934/nhm.2019014

Networks and Heterogeneous Media

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
,
,},
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

Received: 01 June 2018 Revised: 01 October 2018
Primary: 60H15, 80M35, 80M40; Secondary: 35L70

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.
- Homogenization,
- hyperbolic SPDEs,
- two-scale convergence,
- nonlinear damping and forcing,
- probabilistic compactness results
Citation: Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing[J]. Networks and Heterogeneous Media, 2019, 14(2): 341-369. doi: 10.3934/nhm.2019014

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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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