---

Discontinuity, Nonlinearity, and Complexity 10(4) (2021) 635--647 | 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, Ar-Rass, 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 Mentouri-Constantine, Algeria

Download Full Text PDF

 

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 Faedo-Galerkin 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. [1]	Papadopoulos, P.G. and Stavrakakis, N.M. (2001), {Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$}, Topol. Methods Nonlinear Anal., 17(1), 91-109.

2. [2]	Lian, W. and Xu, R. (2020), {Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term}, Adv. Nonlinear Anal., 9(1), 613-632.

3. [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), 1-10.

4. [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), 11-19.

5. [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), 476-486.

6. [6]	Liu, W. (2010), {Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms}, Nonlinear Anal., 73(1), 244-255.

7. [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), 1697-1710.

8. [8]	Miyasita, T. and Zennir, Kh. (2019), {A sharper decay rate for a viscoelastic wave equation with power nonlinearity}, Math. Meth. Appl. Sci., 1-7. DOI:10.1002/mma.5919.

9. [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), 1345-1356.

10. [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), 381-394.

11. [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), 1-18.

12. [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), 337-353.

13. [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), 1-11.

14. [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, 34-48.

15. [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), 633-651.

16. [16]	Piskin, E. (2015), {Blow up of positive initial-energy solutions for coupled nonlinear wave equations with degenerate damping and source terms}, Boundary Value Problems, 43, 1-11.

17. [17]	Wu, J. and Li, S. (2011), {Blow-up for coupled nonlinear wave equations with damping and source}, Applied Mathematics Letters, 24, 1093-1098.

18. [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), 147-156.

19. [19]	Karachalios, N.I. and Stavrakakis, N.M. (2001), {Global existence and blow-up results for some nonlinear wave equations on $\mathbb R^N$}, Adv. Differential Equations, 6(2), 155-174.