Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel
Abstract
:1. Introduction
2. Preliminaries and Assumptions
3. General Decay and Polynomial Decay
4. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
- Cavalcanti, M.; Cavalcanti, V.N.D.; Soriano, J.A. Exponential decay for the solution of semi linear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 2002, 2002, 1–14. [Google Scholar]
- Cavalcanti, M.; Cavalcanti, V.N.D.; Martinez, P. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 2008, 68, 177–193. [Google Scholar] [CrossRef]
- Enqvist, K.; McDonald, J. Q-balls and baryogenesis in the MSSM. Phys. Lett. B 1998, 425, 309–321. [Google Scholar] [CrossRef] [Green Version]
- Kirchhoff, G. Vorlesungen uber Mechanik; Tauber: Leipzig, Germany, 1883. [Google Scholar]
- Gorka, P. Logarithmic quantum mechanics: Existence of the ground state. Found. Phys. Lett. 2006, 19, 591–601. [Google Scholar] [CrossRef]
- Mu, C.; Ma, J. On a system of nonlinear wave equatioins with Balakrishnan-Taylor damping. Z. Angew. Math. Phys. 2014, 65, 91–113. [Google Scholar] [CrossRef]
- Mesloub, F.; Boulaaras, S. General decay for a viscoelastic problem with not necessarily decreasing kernel. J. Appl. Math. Comput. 2018, 58, 647–665. [Google Scholar] [CrossRef]
- Ouchenane, D.; Boulaaras, S.; Mesloub, F. General decay for a class of viscoelastic problem with not necessarily decreasing kernel. Appl. Anal. 2018. [Google Scholar] [CrossRef]
- Medjden, M.; Tatar, N.E. Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel. Appl. Math. Comput. 2005, 167, 1221–1235. [Google Scholar] [CrossRef]
- Boumaza, N.; Boulaaras, S. General Decay for Kirchhoff Type in Viscoelasticity with Not Necessarily Decreasing Kernel; John Wiley & Sons, Ltd.: New York, NY, USA, 2018. [Google Scholar] [CrossRef]
- Zara, A.; Tatar, N.E. Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping. Arch. Math. 2010, 46, 157–176. [Google Scholar]
- Alabau-Boussouira, F.; Cannarsa, P. A general method for proving sharp energy decay rates for memory-dissipative evolution equations. C. R. Math. Acad. Sci. Paris Ser. I 2009, 347, 867–872. [Google Scholar] [CrossRef] [Green Version]
- Bass, R.W.; Zes, D. Spillover nonlinearity, and flexible structures. In Proceedings of the Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, Williamsburg, VA, USA, 11–13 July 1991; pp. 1–14. [Google Scholar]
- Bartkowski, K.; Gorka, P. One-dimensional Klein-Gordon equation with logarithmic nonlinearities. J. Phys. A 2008, 41, 355201. [Google Scholar] [CrossRef]
- Barrow, J.; Parsons, P. Inflationary models with logarithmic potentials. Phys. Rev. D 1995, 52, 5576–5587. [Google Scholar] [CrossRef] [Green Version]
- Cavalcanti, M.; Filho, V.N.D.; Cavalcanti, J.S.P.; Soriano, J.A. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integr. Equ. 2001, 14, 85–116. [Google Scholar]
- Cavalcanti, M.; Oquendo, H.P. Frictional versus viscoelastic damping in a semi linear wave equation. SIAM J. Control Optim. 2003, 42, 1310–1324. [Google Scholar] [CrossRef]
- Nakao, M. Decay of solutions of some nonlinear evolution Equation. J. Math. Anal. Appl. 1977, 60, 542–549. [Google Scholar] [CrossRef]
- Ono, K. Global existence, decay, and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Differ. Equ. 1997, 137, 273–301. [Google Scholar] [CrossRef]
- Berrimi, S.; Messaoudi, S.A. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 2004, 2002, 1–10. [Google Scholar]
- Tatar, N.E.; Zarai, A. Exponential stability and blow up for a problem with Balakrishnan-Taylor damping. Demonstr. Math. 2011, 44, 67–90. [Google Scholar] [CrossRef]
- Liu, W. Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping. J. Appl. Math. Comput. 2010, 32, 59–68. [Google Scholar] [CrossRef]
- Li, M.R.; Tsai, L.Y. Existence and nonexistence of global solutions of some systems of semilinear wave equations. Nonlinear Anal. 2013, 54, 1397–1415. [Google Scholar] [CrossRef]
- Zara, A.; Draifia, A.; Boulaaras, S. Blow up of solutions for a system of nonlocal singular viscoelatic equations. Appl. Anal. 2017. [Google Scholar] [CrossRef]
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Boulaaras, S.; Draifia, A.; Alnegga, M. Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel. Symmetry 2019, 11, 226. https://doi.org/10.3390/sym11020226
Boulaaras S, Draifia A, Alnegga M. Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel. Symmetry. 2019; 11(2):226. https://doi.org/10.3390/sym11020226
Chicago/Turabian StyleBoulaaras, Salah, Alaeddin Draifia, and Mohammad Alnegga. 2019. "Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel" Symmetry 11, no. 2: 226. https://doi.org/10.3390/sym11020226