New scenario of decay rate for system of three nonlinear wave equations with visco-elasticities

Yousif Altayeb; Yousif Altayeb

doi:10.3934/math.2021425

AIMS Mathematics

2021, Volume 6, Issue 7: 7251-7265. doi: 10.3934/math.2021425

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New scenario of decay rate for system of three nonlinear wave equations with visco-elasticities

Yousif Altayeb ^,

Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass, Saudi Arabia

Received: 26 January 2021 Accepted: 20 April 2021 Published: 29 April 2021
MSC : 35B35, 35L05, 35L70

A system of three semilinear wave equations with strong external forces in $ \mathbb{R}^n $ is considered. We use weighted phase spaces, where the problem is well defined, to compensate the lack of Poincare's inequality. Using the Faedo-Galerkin method and some energy estimates, we prove the existence of global solution. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincaré's inequality, we obtain an unusual decay rate for the energy function. It is a generalization of similar results in ^[16,29,31]. The work is relevant in the sense that the problem is more complex than what can be found in the literature.
- damped wave equation,
- power nonlinearity,
- global solution,
- Faedo-Galerkin approximation,
- decay rate
Citation: Yousif Altayeb. New scenario of decay rate for system of three nonlinear wave equations with visco-elasticities[J]. AIMS Mathematics, 2021, 6(7): 7251-7265. doi: 10.3934/math.2021425

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Abstract

A system of three semilinear wave equations with strong external forces in $ \mathbb{R}^n $ is considered. We use weighted phase spaces, where the problem is well defined, to compensate the lack of Poincare's inequality. Using the Faedo-Galerkin method and some energy estimates, we prove the existence of global solution. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincaré's inequality, we obtain an unusual decay rate for the energy function. It is a generalization of similar results in ^[16,29,31]. The work is relevant in the sense that the problem is more complex than what can be found in the literature.

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. Diff. Equ., 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, Diff. Equ., 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 of NAS of Azerbaijan, Issue Mathematics, 37 (2017), 11–19.
[4]	A. Benaissa, D. Ouchenane, K. Zennir, Blow up of positive initial-energy solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Stud., 19, (2012), 523–535.
[5]	A. Beniani, A. Benaissa, K. Zennir, Polynomial decay of solutions to the cauchy problem for a Petrowsky-Petrowsky system in $R^{n}$, Acta Appl. Math., 146 (2016), 67–79. doi: 10.1007/s10440-016-0058-1
[6]	S. Boulaaras, A. Draifia, K. Zennir, General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity, Math. Method. Appl. Sci., 42 (2019), 4795–4814. doi: 10.1002/mma.5693
[7]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Method. Appl. Sci., 24 (2001), 1043–1053. doi: 10.1002/mma.250
[8]	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
[9]	B. Feng, K. zennir, K. L. Lakhdar, General decay of solutions to an extensible viscoelastic plate equation with a nonlinear time-varying delay feedback, B. Malays Math. Sci. So., 42 (2019), 2265–2285. doi: 10.1007/s40840-018-0602-4
[10]	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
[11]	F. Hebhoub, K. Zennir, T. Miyasita, M. Biomy, Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources, AIMS MAthematics, 6 (2020), 442–455.
[12]	N. I. Karachalios, N. M. Stavrakakis, Global existence and blow-up results for some nonlinear wave equations on $\mathbb R^n$, Adv. Differential Equ., 6 (2001), 155–174.
[13]	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.
[14]	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
[15]	S. A. Messaoudi, B. Said-Houari, Global nonexistence of positive initial-energy solutions of a systemof nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277–287. doi: 10.1016/j.jmaa.2009.10.050
[16]	T. Miyasita, K. Zennir, A sharper decay rate for a viscoelastic wave equation with power nonlinearity, Math. Method. Appl. Sci., 43 (2019), 1138–1144.
[17]	D. Ouchenane, K. Zennir, M. Bayoud, Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms, Ukr. Math. J., 65 (2013), 723–739. doi: 10.1007/s11253-013-0809-3
[18]	P. G. Papadopoulos, N. M. Stavrakakis, Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$, Topol. Method. Nonl. An., 17 (2001), 91–109.
[19]	E. Piskin, F. Ekinci, K. Zennir, Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms, Theor. Appl. Mech., 47 (2020), 123–154. doi: 10.2298/TAM200428008P
[20]	K. Bouhali, F. Ellaggoune, Existence and decay of solution to coupled system of viscoelastic wave equations with strong damping in $\mathbb R^n$, Bol. Soc. Paran. Mat, 39 (2021), 31–52. doi: 10.5269/bspm.41175
[21]	K. Bouhali, F. Ellaggoune, Viscoelastic wave equation with logarithmic non-linearities in $\mathbb R^n$, JPDE, 30 (2017), 47–63. doi: 10.4208/jpde.v30.n1.4
[22]	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
[23]	J. Wu, S. Li, Blow-up for coupled nonlinear wave equations with damping and source, Appl. Math. Lett., 24 (2011), 1093–1098. doi: 10.1016/j.aml.2011.01.030
[24]	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
[25]	K. Zennir, General decay of solutions for damped wave equation of Kirchhoff type with density in ${\mathbb R}^{n}$, Ann. Univ. Ferrara, 61 (2015), 381–394. doi: 10.1007/s11565-015-0223-x
[26]	K. 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.
[27]	K. Zennir, Growth of solutions with positive initial energy to system of degeneratly Damed wave equations with memory, Lobachevskii J. Math., 35 (2014), 147–156. doi: 10.1134/S1995080214020139
[28]	K. Zennir, T. Miyasita, Dynamics of a coupled system for nonlinear damped wave equations with variable exponents, Z. Angew. Math. Mech., 2020. https://doi.org/10.1002/zamm.202000094.
[29]	K. Zennir, S. S. Alodhaibi, A Novel Decay Rate for a Coupled System of Nonlinear Viscoelastic Wave Equations, Mathematics, 8 (2020), 1–12.
[30]	K. Zennir, D. Ouchenane, A. Choucha, M. Biomy, Well-posedness and stability for Bresse-Timoshenko type systems with thermodiffusion effects and nonlinear damping, AIMS Math., 6 (2021), 2704–2721. doi: 10.3934/math.2021164
[31]	S. Zitouni, K. Zennir, On the existence and decay of solution for viscoelastic wave equation with nonlinear source in weighted spaces, Rend. Circ. Mat. Palermo, Ⅱ. Ser., 66 (2017), 337–353.

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