Exponential decay for a general class of nonautonomous abstract semilinear evolution equations with time-varying delay feedback

[1] W. Al-Khulaifi, T. Diagana, A. Guesmia, Well-posedness and stability results for some nonau-
tonomous abstract linear hyperbolic equations with memory, Semigroup Forum 105 (2022) 351–
373.
[2] S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, R. Schnaubelt, One-
Parameter Semigroups for Linear Evolution Equations, San Francisco State University, USA, 1991.
[3] A. Batkai, S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10. AK
Peters, Ltd., Wellesley, MA, 2005.
[4] A.N. Carvalho, T. Dlotko, M. J. Nascimento, Nonautonomous semilinear evolution equations with
almost sectorial operators, J. Evol. Equ. 8 (2008) 631–659.
[5] 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) 1–14.
[6] M.M. Cavalcanti, V.N. Domingos Cavalcanti, T.F. Ma, J. A. Soriano, Global existence and asymp-
totic stability for viscoelastic problems, Differ. Integral Equ. 15 (2002) 731–748.
[7] H. Chellaoua, Y. Boukhatem, Stability results for second-order abstract viscoelastic equation in
Hilbert spaces with time-varying delay, Z. fur Angew. Math. Phys. 72 (2021) 46.

[8] H. Chellaoua, Y. Boukhatem, B. Feng, Well-posedness and stability for an abstract evolution equa-
tion with history memory and time delay in Hilbert space, Adv. Differ. Equ. 28 (2023) 953–980.
[9] R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math.
Anal. 3.3 (1972) 428–445.
[10] R. Datko, J. Lagnese, M.P. Polis, An example on the effect of time delays in boundary feedback
stabilization of wave equations, SIAM J. Control Optim. 24 (1986) 152–156.
[11] 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.
[12] T. Diagana, Semilinear Evolution Equations and their Applications, Springer, Cham, 2018.
[13] M. Djemoui, H. Chellaoua, Y. Boukhatem, A nonautonomous delayed viscoelastic wave equation
with a linear damping: well-posedness and exponential stability, J. Math. Model 12 (2024) 319–
336.
[14] B. Feng, General decay for a viscoelastic wave equation with strong time-dependent delay, Bound.
Value Probl. 2017 (2017) 1–11.
[15] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite
memory and time delay, IMA J. Math. Control Inform. 30(4) (2013) 507–526.
[16] T. Jabeen, V. Lupulescu, Existence of mild solutions for a class of non-autonomous evolution equa-
tions with nonlocal initial conditions, J. Nonlinear Sci. Appl. 10(1) (2017).
[17] T. Kato, Linear and quasi-linear equations of evolution of hyperbolic type, In Hyperbolicity, of
C.I.M.E. Summer Sch. 72 (2011) 125–191.
[18] T. Kato, Linear evolution equations of ”hyperbolic” type. II, J. Math. Soc. Japan 25 (1973) 648–
666.
[19] M. Kirane, B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with
a delay, Z. Angew. Math. Phys. 62(6) (2011) 1065–1082.
[20] V. Komornik C. Pignotti, Energy decay for evolution equations with delay feedbacks, Math. Nachr.
295(2) (2022) 377–394.
[21] 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.
[22] 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.
[23] E. Fridman, S. Nicaise, and J. Valein, Stabilization of second order evolution equations with un-
bounded feedback with time-dependent delay, SIAM Journal on Control and Optimization 48(8)
(2010) 5028–5052.

[24] S. Nicaise, C. Pignotti, Stabilization of second-order evolution equations with time delay, Math.
Control Signals Syst. 26 (2014) 563–588.
[25] S. Nicaise, C. Pignotti, Exponential stability of abstract evolution equations with time delay, J.
Evol. Equ. 15 (2015) 107–129.
[26] S. Nicaise, Stability properties of dissipative evolution equations with nonautonomous and nonlin-
ear damping, arXiv: arXiv:2110.11122, 2021.
[27] V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun.
Pure Appl. Anal. 9 (2010) 721–730.
[28] A. Paolucci, C. Pignotti, Exponential decay for semilinear wave equations with viscoelastic damp-
ing and delay feedback, Math. Control Signals System33 (2021) 617–636.
[29] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,
Springer, New York, 1983.
[30] N.E. Tatar, Exponential decay for a viscoelastic problem with a singular kernel, Z. Angew. Math.
Phys. 60 (2009) 640–650.