A nonautonomous delayed viscoelastic wave equation with a linear damping: well-posedness and exponential stability

[1] E.M. Ait Benhassi, K. Ammari, S. Boulite, L. Maniar, Feedback stabilization of a class of evolution
equations with delay, J. Evol. Equ. 9 (2009) 103–121.
[2] W. Al-Khulaifi, T. Diagana, A. Guesmia, Well-posedness and stability results for some nonau-
tonomous abstract linear hyperbolic equations with memory, Semigr. Forum. 105 (2022) 351–373.
[3] F. Alabau-Boussouira, P. Cannarsa, D. Sforza, Decay estimates for second order evolution equa-
tions with memory, J. Funct. Anal. 254 (2008) 1342–1372.
[4] K. Ammari, S. Nicaise, C. Pignotti, Feedback boundary stabilization of wave equations with inte-
rior delay, Syst. Control Lett. 59 (2010) 623–628.
[5] A. Benaissa, A. Benguessoum, S.A. Messaoudi, Global existence and energy decay of solutions
to a viscoelastic wave equation with a delay term in the non-linear internal feedback, Int. J. Dyn.
Syst. Differ. Equ. 5 (2014) 1–26.
[6] S. Berrimi, S.A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a
nonlinear source, Nonlinear Anal. Theory Methods Appl. 64 (2006) 2314–2331.
[7] M.M. Cavalcanti, V.D. Cavalcanti, J. Ferreira, Existence and uniform decay for a non-linear vis-
coelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001) 1043–1053.
[8] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.A. Soriano, Exponential decay for the solution
of semilinear viscoelastic wave equations with localized damping, Electron. J. Differ. Equ. 2002
(2002) 44.
[9] M.M. Cavalcanti, V.N. Domingos Cavalcanti, T.F. Ma, J.A. Soriano, Global existence and asymp-
totic stability for viscoelastic problems, Nonlinear Anal. Differ. Equ. 15 (2002) 731–748.
[10] H. Chellaoua, Stability result for an abstract time delayed evolution equation with arbitrary decay
of viscoelasticity, Proc. Indian Acad. Sci. Math. Sci. 2 (2020) 7–25.
[11] H. Chellaoua, Y. Boukhatem, Optimal decay for second order abstract viscoelastic equation in
Hilbert spaces with infinite memory and time delay, Math. Methods Appl. Sci. 44 (2021) 2071–
2095.
[12] H. Chellaoua, Y. Boukhatem, Stability results for second order abstract viscoelastic equation in
Hilbert spaces with time-varying delay, Z. Angew. Math. Phys. 72 (2021) 46.
[13] H. Chellaoua, Y. Boukhatem, General Decay for Semilinear Abstract Second order Viscoelastic
Equation in Hilbert Spaces with Time Delay, Bol. Soc. Paran. Mat. 41 (2023) 1–18.

[14] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37 (1970) 297–
308.
[15] Q. Dai, Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave
equation with a delay, Z. Angew. Math. Phys. 65 (2014) 885–903.
[16] T. Diagana, Semilinear evolution equations and their applications, Springer International Publish-
ing, 2018.
[17] B. Feng, Global well-posedness and stability for a viscoelastic plate equation with a time delay,
Math. Probl. Eng. 201 (2015) 585021.
[18] B. Feng, General decay for a viscoelastic wave equation with strong time-dependent delay, Bound.
Value Probl. 2017 (2017) 57.
[19] B. Feng, General decay for a viscoelastic wave equation with density and time delay term in Rn,
Taiwan. J. Math. 22 (2018) 205–223.
[20] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite
memory and time delay, IMA J. Math. Control Inform. 30 (2013) 507–526.
[21] A. Guesmia, New general decay rates of solutions for two viscoelastic wave equations with infinite
memory, Math. Model. Anal. 25 (2020) 351–373.
[22] A. Guesmia, N.E. Tatar, Some well-posedness and stability results for abstract hyperbolic equations
with infinite memory and distributed time delay, Commun. Pure Appl. Anal. 14 (2015) 457–491.
[23] M. Kirane, B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with
a delay, Z. Angew. Math. Phys. 62 (2011) 1065–1082.
[24] Y. Latushkin, T. Randolph, R. Schnaubelt, Exponential dichotomy and mild solutions of nonau-
tonomous equations in Banach spaces, J. Dynam. Differ. Equ. 10 (1998) 489–510.
[25] S.A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl. 341
(2008) 1457–1467.
[26] S.A. Messaoudi, N.E. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equa-
tion, Nonlinear Anal. Theory Methods Appl. 68 (2008) 785–793.
[27] J.E. Muoz Rivera, A.P. Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic
materials, Quart. Appl. Math. 59 (2001) 557–578.
[28] M.I. Mustafa, M. Kafini, Energy decay for viscoelastic plates with distributed delay and source
term, Z. Angew. Math. Phys. 67 (2016) 36.
[29] M.I. Mustafa, S.A. Messaoudi, General stability result for viscoelastic wave equations, J. Math.
Phys. 53 (2012) 053702.
[30] S. Nicaise, C. Pignotti, Stability and instability results of the wave equation with a delay term in
the boundary or internal feedbacks, SIAM J. Control Optim. 45 (2006) 1561–1585.

[31] S. Nicaise, C. Pignotti, Stabilization of the wave equation with boundary or internal distributed
delay, Differ. Integral Equ. 21 (2008) 935–958.
[32] S. Nicaise, C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay,
Electron. J. Differ. Equ. 2011 (2011) 41.
[33] S. Nicaise, C. Pignotti, Asymptotic stability of second order evolution equations with intermittent
delay, Adv. Differ. Equ. 17 (2012) 879–902.
[34] S. Nicaise, C. Pignotti, Stabilization of second order evolution equations with time delay, Math.
Control. Signals, Syst. 26 (2014) 563–588.
[35] S. Nicaise, C. Pignotti, Exponential stability of abstract evolution equations with time delay, J.
Evol. Equ. 15 (2015) 107–129.
[36] S. Nicaise, J. Valein, Stabilization of second order evolution equations with unbounded feedback
with delay, ESAIM Control Optim. Calc. Var. 16 (2010) 420–456.
[37] S. Nicaise, J. Valein, E. Fridman, Stability of the heat and of the wave equations with boundary
time varying delays, Discrete Contin. Dyn. Syst. S. 2 (2009) 559–581.
[38] S.H. Park, Global existence, energy decay and blow up of solutions for wave equations with time
delay and logarithmic source, Adv. Differ. Equ. 1 (2020) 631.
[39] V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math. 64 (2006) 499–513.
[40] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New
York, Springer, 1983.
[41] C. Pignotti, A note on stabilization of locally damped wave equations with time delay, Syst. Control
Lett. 61 (2012) 92–97.
[42] C. Pignotti, Stability results for second order evolution equations with memory and switching time-
delay, J. Dyn. Differ. Equ. 29 (2017) 1309–1324.
[43] M. Remil, A. Hakem, Global existence and asymptotic behavior of solutions to the viscoelastic
wave equation with a constant delay term, Facta Univ. Math. Inform. 32 (2017) 485–502.
[44] B. Said-Houari, S.A. Messaoudi, A. Guesmia, General decay of solutions of a nonlinear system of
viscoelastic wave equations, Nonlinear Differ. Equ. Appl. 18 (2011) 659–684.
[45] N.E. Tatar, Arbitrary decays in linear viscoelasticity, J. Math. Phys. 52 (2011) 013502 .
[46] S. Yu, Polynomial stability of solutions for a system of non linear viscoelastic equations, Appl.
Anal. 88 (2009) 1039–1051.