A well-posedness and exponential decay of solutions for a coupled Lamé system with viscoelastic term and logarithmic source terms

References

• Alabau-Boussouira F, Cannarsa P, Komornik V. Indirect internal stabilization of weakly coupled evolution equations. J Evol Equ. 2002;2:127–150. doi: 10.1007/s00028-002-8083-0
   Web of Science ®Google Scholar
• Bchatnia A, Guesmia A. Well-posedness and asymptotic stability for the Lame system with infinite memories in a bounded domain. Math Control Relat Fields. 2014;4(4):451–463. doi: 10.3934/mcrf.2014.4.451
   Web of Science ®Google Scholar
• Bchatnia A, Daoulatli M. Behavior of the energy for Lame systems in bounded domains with nonlinear damping and external force. Electron J Differ Equ. 2013;2013(1):1–17. doi: 10.1186/1687-1847-2013-1
   Web of Science ®Google Scholar
• Boumaza N, Boulaaras S. General decay for Kirchhoff type in viscoelasticity with not necessarily decreasing kernel. Math Method App Sci. 2018;41:6050–6069. doi: 10.1002/mma.5117
   Web of Science ®Google Scholar
• Cavalcanti M, Filho VND, Cavalcanti JSP, et al. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ Integral Equ. 2001;14:85–116.
   Google Scholar
• Cavalcanti M, Cavalcanti VND, Soriano JA. Exponential decay for the solution of semi linear viscoelastic wave equations with localized damping. Electron J Differ Equ. 2002;2002(44):1-14.
   Google Scholar
• Cavalcanti M, Oquendo HP. Frictional versus viscoelastic damping in a semi linear wave equation. SIAM J Control Optim. 2003;42:131–132. doi:10.1137/S0363012902408010
   Web of Science ®Google Scholar
• Cavalcanti M, Cavalcanti VND, Martinez P. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal. 2008;68:177–193. doi: 10.1016/j.na.2006.10.040
   Web of Science ®Google Scholar
• Cavalcanti MM, Oquendo HP. Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J Control Optim. 2003;42(4):1310–1324. doi: 10.1137/S0363012902408010
   Web of Science ®Google Scholar
• Komornik V, Rao B. Boundary stabilization of compactly coupled wave equations. Asymptot Anal. 1997;14:339–359.
   Web of Science ®Google Scholar
• Racke R. Instability of coupled systems with delay. Commun Pure Appl Anal. 2012;11(5):1753–1773. doi: 10.3934/cpaa.2012.11.1753
   Web of Science ®Google Scholar
• Lions JL. Quelques methodes de resolution des problemes aux limites non lineaires. Paris: Dunod; 1969. (in French)
   Google Scholar
• Bass RW, Zes D. Spillover nonlinearity, and flexible structures. In: Taylor LW, editor. The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems. Washington: NASA Conference Publication 10065, 1991. p. 1–14.
   Google Scholar
• Mu C, Ma J. On a system of nonlinear wave equatioins with Balakrishnan–Taylor damping. Z Angew Math Phys. 2014;65:91–113. doi: 10.1007/s00033-013-0324-2
   Web of Science ®Google Scholar
• Mesloub F, Boulaaras S. General decay for a viscoelastic problem with not necessarily decreasing kernel. J Appl Math Comput. 2017. doi:10.1007/S12190-017-1161-9.
   Google Scholar
• Ouchenane D, Boulaaras S, Mesloub F. General decay for a viscoelastic problem with not necessarily decreasing kernel. Appl Anal. doi:10.1080/00036811.2018.
   Google Scholar
• Medjden M, Tatar N-e. Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel. Appl Math Comput. 2005;167:1221–1235.
   Web of Science ®Google Scholar
• Nakao M. Decay of solutions of some nonlinear evolution equation. J Math Anal Appl. 1977;60:542–549. doi: 10.1016/0022-247X(77)90040-3
   Web of Science ®Google Scholar
• Ono K. Global existence, decay, and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings. J Differ Equ. 1997;137:273–301. doi: 10.1006/jdeq.1997.3263
   Web of Science ®Google Scholar
• Tatar N-e, Zarai A. Exponential stability and blow up for a problem with Balakrishnan-Taylor damping. Demonstr Math XLIV. 2011;1:67–90.
   Google Scholar
• Liu W. Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping. J Appl Math Comput. 2010;32:59–68. doi: 10.1007/s12190-009-0232-y
   Google Scholar
• Li MR, Tsai LY. Existence and nonexistence of global solutions of some systems of semilinear wave equations. Nonlinear Anal. 2003;54:1397–1415. doi: 10.1016/S0362-546X(03)00192-5
   Web of Science ®Google Scholar
• Bouizem Y, Boulaaras S, Djebbar B. Some existence results for an el-liptic equation of Kirchhoff-type with changing sign data and a logarithmic nonlinearity. Math Meth Appl Sci. 2019;42:2465–2474.
   Google Scholar
• Zarai A, Tatar N-e. Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping. Arch Math(BRNO). 2010;46:157–176.
   Google Scholar