A new strict decay rate for systems of longitudinal $ m $-nonlinear viscoelastic wave equations

Keltoum Bouhali; Sulima Ahmed Zubair; Wiem Abedelmonem Salah Ben Khalifa; Najla ELzein AbuKaswi Osman; Khaled Zennir; Keltoum Bouhali; Sulima Ahmed Zubair; Wiem Abedelmonem Salah Ben Khalifa; Najla ELzein AbuKaswi Osman; Khaled Zennir

doi:10.3934/math.2023046

AIMS Mathematics

2023, Volume 8, Issue 1: 962-976. doi: 10.3934/math.2023046

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A new strict decay rate for systems of longitudinal $ m $-nonlinear viscoelastic wave equations

1.
Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass, Saudi Arabia
2.
Departement de Mathématiques, Faculté des sciences, Université 20 Aôut 1955, Skikda, Algérie
3.
Department of mathematics, college of Education, Bahry University, Sudan
4.
College of Arts, Imam Abdulrahman Bin Faisal University, Saudi Arabia
5.
Laboratoire de Mathématiques Appliquées et de Modélisation, Université 8 Mai 1945 Guelma, Guelma 24000, Algeria

Received: 14 August 2022 Revised: 28 September 2022 Accepted: 30 September 2022 Published: 13 October 2022
MSC : 35B35, 35L05, 35L70

Recent years have been marked by a significant increase in interest in solving nonlinear equations that arise in various fields of natural science. This trend is associated with the creation of a new method of mathematical physics. The present study is devoted to the analysis of the propagation of $ m $-nonlinear viscoelastic waves equations in an unbounded domain. The physical properties are determined by the equations of the linear theory of viscoelasticity. This article shows the main effect and interaction between the different weak and strong damping terms on the behavior of solutions. We found, under a novel condition on the kernel functions, an energy decay rate by using an appropriate energy estimates.
- damped wave equation,
- strong nonlinear system,
- global solution,
- exponential decay rate
Citation: Keltoum Bouhali, Sulima Ahmed Zubair, Wiem Abedelmonem Salah Ben Khalifa, Najla ELzein AbuKaswi Osman, Khaled Zennir. A new strict decay rate for systems of longitudinal $ m $-nonlinear viscoelastic wave equations[J]. AIMS Mathematics, 2023, 8(1): 962-976. doi: 10.3934/math.2023046

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Abstract

Recent years have been marked by a significant increase in interest in solving nonlinear equations that arise in various fields of natural science. This trend is associated with the creation of a new method of mathematical physics. The present study is devoted to the analysis of the propagation of $ m $-nonlinear viscoelastic waves equations in an unbounded domain. The physical properties are determined by the equations of the linear theory of viscoelasticity. This article shows the main effect and interaction between the different weak and strong damping terms on the behavior of solutions. We found, under a novel condition on the kernel functions, an energy decay rate by using an appropriate energy estimates.

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