Stability result for an abstract time delayed evolution equation with arbitrary decay of viscoelasticity

Abstract

In this paper, we consider a second-order abstract semilinear evolution equation with past
history and time delay. Under suitable conditions on initial data and the kernel memory function, we prove the well-posedness by using the semigroup arguments. The stability result is
also established defining a suitable Lyapunov functional. This work extends previous works
with time delay for a much wider class of kernels. Some applications are also given to illustrate
the result.

Keywords

• Abstract evolution equation,Past history,Time delay

References

1. [1] F. Alabau-Boussouira, S. Nicaise and C. Pignotti, Exponential stability of the wave equation with memory and time delay, New Prospects in Direct, Inverse and Control Problems for Evolution Equations. Springer, Cham, (2014), 1-22.
2. [2] M. Aassila, M. M. Cavalcanti and V. N. Domingos Cavalcanti, Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Dif., 15 (2002), 155-180.
3. [3] A. B`atkai, S. Piazzera, Semigroups for delay equations, Research Notes in Mathematics, 10. AK Peters, Ltd., Wellesley, MA, (2005).
4. [4] A. Benaissa, A. K. Benaissa and S. A. Messaoudi, Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks, J. Math. Phys., 53 (2012), 1-19.
5. [5] Y. Boukhatem, B. Benabderrahmane, General decay for a viscoelastic equation of variable coefficients with a time-varying delay in the boundary feedback and acoustic boundary conditions, Acta Math. Sci., 37(5) (2017), 1453-1471.
6. [6] Y. Boukhatem, B. Benabderrahmane, General Decay for a Viscoelastic Equation of Variable Coefficients in the Presence of Past History with Delay Term in the Boundary Feedback and Acoustic Boundary Conditions, Acta Appl. Math., 154(1) (2018), 131-152.
7. [7] M. M. Cavalcanti, V. N. Domingos and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Elect. J. Diff. Equa., 44 (2002), 1-14.
8. [8] Q. Dai and Y. Zhifeng, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay.” Z. Angew. Math. Phys., 65(5) (2014), 885-903.

9. [9] CM. Dafermos, Asymptotic stability in viscoelasticity, Arch. ration. mech. anal., 37 (1970), 297-308.
10. [10] R. Datko, Two questions concerning the boundary control of certain elastic systems, J. Diff. Equa., 1 (1991), 27-44.
11. [11] A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control I., 30 (2013), 507-526.
12. [12] A. Guesmia, N. E.Tatar, Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay, Commun. Pur. Appl. Anal., 14 (2015), 457-491.
13. [13] M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.
14. [14] S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.
15. [15] 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.
16. [16] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim., 2 (2010), 420-456.
17. [17] S. Nicaise, C. Pignotti, Stabilization of second-order evolution equations with time delay, Math. Control Signal., 26(4) (2014), 563-588.
18. [18] S. Nicaise, C. Pignotti, Exponential stability of abstract evolution equations with time delay, J. Evol. Equ., 15 (2015), 107-129.
19. [19] S. Nicaise, C. Pignotti, Well-posedness and stability results for nonlinear abstract evolution equations with time delays, J. Evol. Equ., 18 (2018), 947-971.
20. [20] V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 3 (2006), 499-513
21. [21] V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure. Appl. Anal., 9 (2010), 721-730.
22. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 277. Springer, New York (1983).
23. [23] N.E. Tatar, A new class of kernels leading to an arbitrary decay in viscoelasticity, Mediterr. J. Math., 10(1) (2013), 213-226.
24. [24] N. E. Tatar, On a large class of kernels yielding exponential stability in viscoelasticity, Appl. Math. Comp., 215 (2009), 2298-2306.
25. [25] N. E. Tatar, Uniform decay in viscoelasticity for kernels with small non-decreasingness zones, Appl. Math. Comp., 218 (2012), 7939-7946.