Decay rate for systems of $ m $-nonlinear wave equations with new viscoelastic structures

Mohamed Biomy; Mohamed Biomy

doi:10.3934/math.2021326

AIMS Mathematics

2021, Volume 6, Issue 6: 5502-5517. doi: 10.3934/math.2021326

Previous Article Next Article

Research article

Decay rate for systems of $ m $-nonlinear wave equations with new viscoelastic structures

Mohamed Biomy ^{1,2
,
,}

1.
Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass, Saudi Arabia
2.
Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said, 42511, Egypt

Received: 07 December 2020 Accepted: 15 March 2021 Published: 17 March 2021
MSC : 35L70, 35L05, 35B35

The article discusses the effect of weak and strong damping terms on decay rate for systems of nonlinear $ m $- wave equations with new viscoelastic structures. The factors that allowed system (1.1) to coexist for a long time are the strong nonlinearities in the sources. We showed, under a novel condition on the kernel function in (2.4), a new scenario for energy decay in (3.7) by using an appropriate energy estimates. This result extend the results in ^[18,27] for system of $ m $-equations inspired from the paper ^[1].
- wave equation,
- strong nonlinear system,
- global solution,
- exponential decay rate
Citation: Mohamed Biomy. Decay rate for systems of $ m $-nonlinear wave equations with new viscoelastic structures[J]. AIMS Mathematics, 2021, 6(6): 5502-5517. doi: 10.3934/math.2021326

Related Papers:

Abstract

The article discusses the effect of weak and strong damping terms on decay rate for systems of nonlinear $ m $- wave equations with new viscoelastic structures. The factors that allowed system (1.1) to coexist for a long time are the strong nonlinearities in the sources. We showed, under a novel condition on the kernel function in (2.4), a new scenario for energy decay in (3.7) by using an appropriate energy estimates. This result extend the results in ^[18,27] for system of $ m $-equations inspired from the paper ^[1].

References

[1]	A. B. Aliev, G. I. Yusifova, Nonexistence of global solutions of Cauchy problems for systems of semilinear hyperbolic equations with positive initial energy, Electron. J. Differ. Eq., 211 (2017), 1–10.
[2]	A. B. Aliev, A. A. Kazimov, Global Solvability and Behavior of Solutions of the Cauchy Problem for a System of two Semilinear Hyperbolic Equations with Dissipation, Differ. Equations, 49 (2013), 457–467. doi: 10.1134/S001226611304006X
[3]	A. B. Aliev, G. I. Yusifova, Nonexistence of global solutions of the Cauchy problem for the systems of three semilinear hyperbolic equations with positive initial energy, Transactions Issue Mathematics, Azerbaijan National Academy of Sciences, 37 (2017), 11–19.
[4]	A. Braik, A. Beniani, Kh. Zennir, Well-posedness and general decay for Moore–Gibson–Thompson equation in viscoelasticity with delay term, Ric. Mat., (2021), 1–22.
[5]	S. Boulaaras, A. Draifia, Kh. Zennir, General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity, Math. Meth. Appl. Sci., 42 (2019), 4795–4814. doi: 10.1002/mma.5693
[6]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24 (2001), 1043–1053. doi: 10.1002/mma.250
[7]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, I. Lasiecka, W. M. Claudete, Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density, Adv. Nonlinear Anal., 6 (2017), 121–145. doi: 10.1515/anona-2016-0027
[8]	E. F. Doungmo Goufo, I. Tchangou Toudjeu, Analysis of recent fractional evolution equations and applications, Chaos, Solitons and Fractals, 126 (2019), 337–350. doi: 10.1016/j.chaos.2019.07.016
[9]	E. F. Doungmo Goufo, Yazir Khan, Stella Mugisha, Control parameter & solutions to generalized evolution equations of stationarity, relaxation and diffusion, Results Phys., 9 (2018), 1502–1507. doi: 10.1016/j.rinp.2018.04.051
[10]	H. Dridi, Kh. Zennir, Well-posedness and energy decay for some thermoelastic systems of Timoshenko type with Kelvin-Voigt damping, SeMA Journal, (2021), 1–16.
[11]	H. Dridi, B. Feng, Kh. Zennir, Stability of Timoshenko system coupled with thermal law of Gurtin-Pipkin affecting on shear force, Appl. Anal., (2021), 1–22.
[12]	B. Feng, Y. Qin, M. Zhang, General decay for a system of nonlinear viscoelastic wave equations with weak damping, Bound. Value Probl., 2012 (2012), 1–11. doi: 10.1186/1687-2770-2012-1
[13]	N. I. Karachalios, N. M. Stavrakakis, Global existence and blow-up results for some nonlinear wave equations on $\mathbb R^N$, Adv. Differ. Eq., 6 (2001), 155–174.
[14]	W. Lian, R. 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.
[15]	G. Liu, S. Xia, Global existence and finite time blow up for a class of semilinear wave equations on ${\mathbb R}^{N}$, Comput. Math. Appl., 70 (2015), 1345–1356. doi: 10.1016/j.camwa.2015.07.021
[16]	W. Liu, Global existence, asymptotic behavior and blow-up of solutions for coupled Klein–Gordon equations with damping terms, Nonlinear Anal-Theor, 73 (2010), 244–255. doi: 10.1016/j.na.2010.03.017
[17]	Q. Li, L. He, General decay and blow-up of solutions for a nonlinear viscoelastic wave equation with strong damping, Bound. Value Probl., 2018 (2018), 1–22. doi: 10.1186/s13661-017-0918-2
[18]	T. Miyasita, Kh. Zennir, A sharper decay rate for a viscoelastic wave equation with power nonlinearity, Math. Meth. Appl. Sci., 43 (2020), 1138–1144. doi: 10.1002/mma.5919
[19]	S. C. Oukouomi Noutchie, E. F. Doungmo Goufo, Exact Solutions of Fragmentation Equations with General Fragmentation Rates and Separable Particles Distribution Kernels, Math. Probl. Eng., 2014 (2014), 1–5.
[20]	P. G. Papadopoulos, N. M. Stavrakakis, Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$, Topol. Meth. Nonl. Anal., 17 (2001), 91–109. doi: 10.12775/TMNA.2001.006
[21]	S. T. Wu, General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms, J. Math. Anal. Appl., 406 (2013), 34–48. doi: 10.1016/j.jmaa.2013.04.029
[22]	Y. Ye, Global existence and nonexistence of solutions for coupled nonlinear wave equations with damping and source terms, B. Korean Math. Soc., 51 (2014), 1697–1710. doi: 10.4134/BKMS.2014.51.6.1697
[23]	Kh. Zennir, General decay of solutions for damped wave equation of Kirchhoff type with density in ${\mathbb R}^{n}$, Annali dell'Universita'di Ferrara, 61 (2015), 381–394. doi: 10.1007/s11565-015-0223-x
[24]	Kh. Zennir, Stabilization for solutions of plate equation with time-varying delay and weak-viscoelasticity in ${\bf{R}}^n$, Russian Math., 64 (2020), 21–33.
[25]	Kh. Zennir, M. Bayoud, S. Georgiev, Decay of solution for degenerate wave equation of Kirchhoff type in viscoelasticity, Int. J. Appl. Comput. Math., 4 (2018), 1–18.
[26]	Kh. Zennir, T. Miyasita, Dynamics of a coupled system for nonlinear damped wave equations with variable exponents, ZAMM Journal of applied mathematics and mechanics: Zeitschrift fur angewandte Mathematik und Mechanik, (2020), e202000094.
[27]	Kh. Zennir, S. S. Alodhaibi, A Novel Decay Rate for a Coupled System of Nonlinear Viscoelastic Wave Equations, Mathematics, 8 (2020), 203. doi: 10.3390/math8020203
[28]	S. Zitouni, Kh. Zennir, On the existence and decay of solution for viscoelastic wave equation with nonlinear source in weighted spaces, Rend. Circ. Mat. Palermo, 66 (2017), 337–353.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)