Abstract
In this paper, we consider a system of two viscoelastic equations with Dirichlet boundary conditions. For certain class of relaxation functions and initial data, we establish general and optimal decay results. This result extends earlier one of Liu (Nonlinear Anal. TMA 71:2257–2267, 2010), in which only the usual exponential and polynomial decay rates are considered. The conditions of the relaxation functions \(g_{1}(t)\) and \(g_{2}(t)\) in our work appeared first in Messaoudi and Khulaifi (Appl. Math. Lett. 66:16–22, 2017).
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References
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Ferreira, J.: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043–1053 (2001)
Han, X.S., Wang, M.X.: Global existence and blow up of solutions for a system of nonlinear viscoelastic equation with damping and source. Nonlinear Anal. TMA 71, 5427–5450 (2009)
Houari, B.S., Messaoudi, S.A., Guesmia, A.: General decay of solutions of a nonlinear system of viscoelastic wave equations. Nonlinear Differ. Equ. Appl. 18, 659–684 (2011)
Lacroix-Sonrier, M.-T.: Distributions Espace de Sobolev Application (1998). Ellipses/Edition Marketing S.A.
Lions, J.L.: Quelques Metodes de Resolution des Problems aux Limites NonLineaires. Dunod, Paris (1969)
Liu, W.J.: Uniform decay of solutions for a quasilinear system of viscoelastic equations. Nonlinear Anal. TMA 71, 2257–2267 (2009)
Liu, W.J.: General decay and blow up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. TMA 73, 1890–1904 (2010)
Liu, W.J.: Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms. Front. Math. China 5(3), 555–574 (2010). https://doi.org/10.1007/s11464-010-0060-2
Messaoudi, S.A.: On the control of solution of a viscoelastic equation. J. Franklin Inst. 344(5), 765–776 (2007)
Messaoudi, S.A.: Existence and decay of solutions to a viscoelastic plate equation. Electron. J. Differ. Equ. 2016(22) (2016), 14 p.
Messaoudi, S.A., Khulaifi, W.A.: General and optimal decay for a quasilinear viscoelastic equation. Appl. Math. Lett. 66, 16–22 (2017)
Messaoudi, S.A., Tatar, N.E.: Exponential and polynomial decay for a quasilinear viscoelastic equation. Nonlinear Anal. TMA 68, 785–793 (2008)
Mustafa, M.: Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations. Nonlinear Anal. RWA 13, 452–463 (2012)
Wu, S.T.: General decay of solution for a viscoelastic equation with nonlinear damping and source term. Acta Math. Sci. 31B(4), 436–1448 (2011)
Xu, R.Z., Yang, Y.B., Liu, Y.C.: Global well-posedness for strongly damped viscoelastic wave equation. Appl. Anal. 92(1), 138–157 (2013)
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The authors are highly grateful for the referees valuable suggestions which improved this work a lot.
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He, L. On Decay of Solutions for a System of Coupled Viscoelastic Equations. Acta Appl Math 167, 171–198 (2020). https://doi.org/10.1007/s10440-019-00273-1
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DOI: https://doi.org/10.1007/s10440-019-00273-1