Qualitative results for a relativistic wave equation with multiplicative noise and damping terms

Hatice Taskesen; Hatice Taskesen

doi:10.3934/math.2023778

AIMS Mathematics

2023, Volume 8, Issue 7: 15232-15254. doi: 10.3934/math.2023778

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Qualitative results for a relativistic wave equation with multiplicative noise and damping terms

Hatice Taskesen ^,

Department of Mathematics, Van Yuzuncu Yil University, 65080, Van, Turkey

Received: 27 December 2022 Revised: 17 March 2023 Accepted: 03 April 2023 Published: 25 April 2023
MSC : 35A01, 35B44, 35Q40, 60H15

Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time.
- relativistic wave equation,
- local existence,
- blow up,
- energy inequality,
- noise
Citation: Hatice Taskesen. Qualitative results for a relativistic wave equation with multiplicative noise and damping terms[J]. AIMS Mathematics, 2023, 8(7): 15232-15254. doi: 10.3934/math.2023778

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Abstract

Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time.

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