Discontinuity, Nonlinearity, and Complexity
The Effect of Damping Terms on Decay Rate for System of Three Nonlinear Wave Equations with WeakMemories
Discontinuity, Nonlinearity, and Complexity 10(4) (2021) 635647  DOI:10.5890/DNC.2021.12.005
Derradji Guidad$^1$, Khaled Zennir$^{2,3}$ , Abdelhak Berkane$^4$, Mohamed Berbiche$^1$
$^1$ Department of Mathematics, College of Sciences, University Mohamed Khider Biskra, Algeria
$^2$ Department of Mathematics, College of Sciences
and Arts, Qassim University, ArRass, Saudi Arabia
$^3$ Laboratoire de Math'ematiques Appliqu'ees et de Mod'elisation, Universit'e 8 Mai 1945 Guelma. B.P. 401
Guelma 24000 Alg'erie
$^4$ Departement de Math'ematiques, Facult'e des sciences exact, Universit'e freres MentouriConstantine, Algeria
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Abstract
In this paper, we consider a very important problem from the point of view of application in sciences and engineering. A system of three wave equations having a different damping effects in an unbounded domain with strong external forces. Using the FaedoGalerkin method and some energy estimates, we will prove the existence of global solution in $\mathbb{R}^n$ owing to to the weighted function. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincar\'e's inequality, we obtain an unusual decay rate for the energy function.
Acknowledgments
The authors expresses sincerely thanks to the referees for their
constructive comments and suggestions that helped to improve this
paper.
\begin{thebibliography}{999}
References

[1] 
Papadopoulos, P.G. and Stavrakakis, N.M. (2001),
{Global existence and blowup results
for an equation
of Kirchhoff type
on $\mathbb R^N$},
Topol. Methods Nonlinear Anal.,
17(1), 91109.


[2] 
Lian, W. and Xu, R. (2020),
{Global wellposedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term},
Adv. Nonlinear Anal.,
9(1), 613632.


[3] 
Aliev, A.B. and Yusifova, G.I. (2017),
{Nonexistence of global solutions of Cauchy problems for systems of semilinear hyperbolic equations with positive initial energy},
Electronic Journal of Differential Equations,
2017(211), 110.


[4] 
Aliev, A.B. and Yusifova, G.I. (2017),
{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(1), 1119.


[5] 
Aliev, A.B. and Kazimov, A.A. (2013),
{Global solvability and behavior of solutions of the cauchy
problem for a system of two semilinear hyperbolic equations with dissipation},
Difer. Uravn., 49(4), 476486.


[6] 
Liu, W. (2010),
{Global existence, asymptotic behavior and blowup of solutions for coupled
KleinGordon equations with damping terms},
Nonlinear Anal., 73(1), 244255.


[7] 
Ye, Y. (2014),
{Global existence and nonexistence of solutions for coupled nonlinear wave
equations with damping and source terms},
Bull. Korean Math. Soc., 51(6), 16971710.


[8] 
Miyasita, T. and Zennir, Kh. (2019),
{A sharper decay rate for a viscoelastic wave equation with power nonlinearity},
Math. Meth. Appl. Sci., 17. DOI:10.1002/mma.5919.


[9] 
Liu, G. and Xia, S. (2015),
{Global existence and finite time blow up
for a class
of semilinear wave equations
on ${\mathbb R}^{N}$},
Comput. Math. Appl.,
70(6), 13451356.


[10] 
Zennir, Kh. (2015),
{General decay
of solutions
for damped wave equation
of Kirchhoff type
with density in ${\mathbb R}^{n}$},
Ann. Univ. Ferrara Sez. VII Sci. Mat.,
61(2), 381394.


[11] 
Zennir, Kh., Bayoud, M., and Georgiev, S. (2018),
{Decay
of solution
for degenerate wave equation
of Kirchhoff type in viscoelasticity},
Int. J. Appl. Comput. Math. Art.,
4(1), 118.


[12] 
Zitouni, S. and Zennir, Kh. (2017),
{On the existence
and
decay of solution
for viscoelastic wave equation
with nonlinear source
in weighted spaces},
Rend. Circ. Mat. Palermo,
66(3), 337353.


[13] 
Feng, B., Qin, Y., and Zhang, M. (2012),
{General decay for a system of nonlinear viscoelastic wave equations with weak damping},
Boundary Value Problems, 2012(164), 111.


[14] 
Wu, S.T. (2013),
{General decay of solutions for a nonlinear system of
viscoelastic wave equations with degenerate damping and
source terms},
J. Math. Anal. Appl., 406, 3448.


[15] 
Piskin, E. and Polat, N. (2013),
{Global existence, decay and blow up solutions for coupled nonlinear wave
equations with damping and source terms},
Turk. J. Math.,
37(4), 633651.


[16] 
Piskin, E. (2015),
{Blow up of positive initialenergy solutions
for coupled nonlinear wave equations with
degenerate damping and source terms},
Boundary Value Problems, 43, 111.


[17] 
Wu, J. and Li, S. (2011),
{Blowup for coupled nonlinear wave equations with damping and source},
Applied Mathematics Letters, 24, 10931098.


[18] 
Zennir, Kh. (2014),
{Growth of solutions with positive initial energy to system
of degeneratly Damed wave equations with memory},
Lobachevskii Journal of Mathematics, 35(2), 147156.


[19]  Karachalios, N.I. and Stavrakakis, N.M. (2001), {Global existence and blowup results for some nonlinear wave equations on $\mathbb R^N$},
Adv. Differential Equations, 6(2), 155174.
