General and optimal decay rates for a system of wave equations with damping and a coupled source term

Qian Li; Yanyuan Xing; Qian Li; Yanyuan Xing

doi:10.3934/math.20241425

AIMS Mathematics

2024, Volume 9, Issue 10: 29404-29424. doi: 10.3934/math.20241425

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Research article

General and optimal decay rates for a system of wave equations with damping and a coupled source term

Qian Li ^,,
Yanyuan Xing

Department of Mathematics, Changzhi University, Changzhi 046011, China

Received: 18 July 2024 Revised: 02 October 2024 Accepted: 12 October 2024 Published: 17 October 2024
MSC : 35L05, 35L20

In this article, we aim to investigate the decay characteristics of a system consisting of two viscoelastic wave equations with Dirichlet boundary conditions, where the dispersion term and nonlinear weak damping term are taken into account. Under appropriate conditions, we establish both general and optimal decay results. This work generalizes and improves earlier results in the literature.
- general decay,
- coupled equations,
- Lyapunov function,
- relaxation function
Citation: Qian Li, Yanyuan Xing. General and optimal decay rates for a system of wave equations with damping and a coupled source term[J]. AIMS Mathematics, 2024, 9(10): 29404-29424. doi: 10.3934/math.20241425

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Abstract

In this article, we aim to investigate the decay characteristics of a system consisting of two viscoelastic wave equations with Dirichlet boundary conditions, where the dispersion term and nonlinear weak damping term are taken into account. Under appropriate conditions, we establish both general and optimal decay results. This work generalizes and improves earlier results in the literature.

References

[1]	W. Lian, R. Z. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613–632. https://doi.org/10.1515/anona-2020-0016 doi: 10.1515/anona-2020-0016
[2]	Y. X. Chen, R. Z. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 1–39. https://doi.org/10.1016/j.na.2019.111664 doi: 10.1016/j.na.2019.111664
[3]	R. Z. Xu, W. Lian, Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., 63 (2020), 321–356. https://doi.org/10.1007/s11425-017-9280-x doi: 10.1007/s11425-017-9280-x
[4]	R. Z. Xu, Y. B. Yang, Y. C. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138–157. https://doi.org/10.1080/00036811.2011.601456 doi: 10.1080/00036811.2011.601456
[5]	S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457–1467. https://doi.org/10.1016/j.jmaa.2007.11.048 doi: 10.1016/j.jmaa.2007.11.048
[6]	S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589–2598. https://doi.org/10.1016/j.na.2007.08.035 doi: 10.1016/j.na.2007.08.035
[7]	W. J. Liu, General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source, Nonlinear Anal. Theory Methods Appl., 73 (2010), 1890–1904. https://doi.org/10.1016/j.na.2010.05.023 doi: 10.1016/j.na.2010.05.023
[8]	S. A. Messaoudi, W. Al-khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Appl. Math. Lett., 66 (2017), 16–22. https://doi.org/10.1016/j.aml.2016.11.002 doi: 10.1016/j.aml.2016.11.002
[9]	M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134–152. https://doi.org/10.1016/j.jmaa.2017.08.019 doi: 10.1016/j.jmaa.2017.08.019
[10]	Q. Li, A blow-up result for a system of coupled viscoelastic equations with arbitrary positive initial energy, Bound. Value Probl., 2021 (2021), 61. https://doi.org/10.1186/s13661-021-01538-1 doi: 10.1186/s13661-021-01538-1
[11]	Q. Li, The general decay of solution for a system of wave equations with damping and coupled source term, Acta Math. Appl. Sin., 45 (2021), 380–400.
[12]	X. P. Liu, P. H. Shi, General decay of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Acta Math. Appl. Sin., 35 (2012), 283–296.
[13]	B. Said-Houari, S. A. Messaoudi, A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wave equation, Nonlinear Differ. Equ. Appl., 18 (2011), 659–684. https://doi.org/10.1007/s00030-011-0112-7 doi: 10.1007/s00030-011-0112-7
[14]	W. J. Liu, Uniform decay of solutions for a quasilinear system of viscoelastic equation, Nonlinear Anal., 71 (2009), 2257–2267. https://doi.org/10.1016/j.na.2009.01.060 doi: 10.1016/j.na.2009.01.060
[15]	X. S. Han, M. X. Wang, Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal., 71 (2009), 5427–5450. https://doi.org/10.1016/j.na.2009.04.031 doi: 10.1016/j.na.2009.04.031
[16]	M. M. Cavalcanti, H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310–1324. https://doi.org/10.1137/S0363012902408010 doi: 10.1137/S0363012902408010
[17]	S. A. Messaoudi, On the control of solutions of a viscoelastic equation, J. Franklin Inst., 344 (2007), 765–778. https://doi.org/10.1016/j.jfranklin.2006.02.029 doi: 10.1016/j.jfranklin.2006.02.029
[18]	L. F. He, On decay and blow-up of solutions for a system of viscoelastic equations with weak damping and source terms, J. Inequal. Appl., 2019 (2019), 200. https://doi.org/10.1186/s13660-019-2155-y doi: 10.1186/s13660-019-2155-y

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