Asymptotic stability for a viscoelastic equation with the time-varying delay
Abstract
The goal of the present paper is to study the viscoelastic wave equation with the time-varying delay under initial-boundary value conditions. By using the multiplier method together with some properties of the convex functions, the explicit and general stability results of the total energy are proved under the general assumption on the relaxation function g.
Keyword : viscoelasticity, delay term, source term, energy decay
How to Cite
Liao, M., & Tan, Z. (2023). Asymptotic stability for a viscoelastic equation with the time-varying delay. Mathematical Modelling and Analysis, 28(1), 23–41. https://doi.org/10.3846/mma.2023.16160
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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W. Liu. General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback. Journal of Mathematical Physics, 54(4):043504, 2013. https://doi.org/10.1063/1.4799929
M.I. Mustafa. General decay result for nonlinear viscoelastic equations. Journal of Mathematical Analysis and Applications, 457(1):134–152, 2018. https://doi.org/10.1016/j.jmaa.2017.08.019
M.I. Mustafa. Optimal decay rates for the viscoelastic wave equation. Mathematical Methods in the Applied Sciences, 41(1):192–204, 2018. https://doi.org/10.1002/mma.4604
M.I. Mustafa. Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems. Zeitschrift fu¨r angewandte Mathematik und Physik, 72(67):1– 15, 2021. https://doi.org/10.1007/s00033-021-01498-7
M.I. Mustafa and M. Kafini. Energy decay for viscoelastic plates with distributed delay and source term. Zeitschrift fu¨r angewandte Mathematik und Physik, 67(36), 2016. https://doi.org/10.1007/s00033-016-0641-3
S. Nicaise and C. Pignotti. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM Journal on Control and Optimization, 45(5):1561–1585, 2006. https://doi.org/10.1137/060648891
S. Nicaise and C. Pignotti. Interior feedback stabilization of wave equations with time dependent delay. Electronic Journal of Differential Equations, 41:1– 20, 2011.
S.-T. Wu. Asymptotic behavior for a viscoelastic wave equation with a delay term. Taiwanese Journal of Mathematics, 17(3):765–784, 2013. https://doi.org/10.11650/tjm.17.2013.2517
S.-T. Wu. Blow-up of solution for a viscoelastic wave equation with delay. Acta Mathematica Scientia, 39(1):329–338, 2019. https://doi.org/10.1007/s10473-019-0124-7
S.T. Wu. Asymptotic behavior for a viscoelastic wave equation with a timevarying delay term. Journal of Partial Differential Equations, 29(1):22–35, 2016. https://doi.org/10.4208/jpde.v29.n1.3
N. Boumaza, M. Saker and B. Gheraibia. Asymptotic behavior for a viscoelastic Kirchhoff-type equation with delay and source terms. Acta Applicandae Mathematicae, 171(18):1–18, 2021. https://doi.org/10.1007/s10440-021-00387-5
H. Chellaoua and Y. Boukhatem. Optimal decay for second-order abstract viscoelastic equation in Hilbert spaces with infinite memory and time delay. Mathematical Methods in the Applied Sciences, 44(2):2071–2095, 2021. https://doi.org/10.1002/mma.6917
H. Chellaoua and Y. Boukhatem. Stability results for second-order abstract viscoelastic equation in hilbert spaces with time-varying delay. Zeitschrift fu¨r angewandte Mathematik und Physik, 72(46):1–18, 2021. https://doi.org/10.1007/s00033-021-01477-y
H. Chellaoua and Y. Boukhatem. Blow-up result for an abstract evolution problem with infinite memory and time-varying delay. Applicable Analysis, 101(13):4574–4597, 2022. https://doi.org/10.1080/00036811.2020.1863374
Q. Dai and Z. Yang. Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay. Zeitschrift fu¨r angewandte Mathematik und Physik, 65(5):885–903, 2014. https://doi.org/10.1007/s00033-013-0365-6
B. Feng. Well-posedness and exponential stability for a plate equation with time-varying delay and past history. Zeitschrift fu¨r angewandte Mathematik und Physik, 68(6):1–24, 2016. https://doi.org/10.1007/s00033-016-0753-9
B. Feng and H. Li. Decay rates for a coupled viscoelastic Lam´e system with strong damping. Mathematical Modelling and Analysis, 25(2):226–240, 2020. https://doi.org/10.3846/mma.2020.10383
J.H. Hassan and S.A. Messaoudi. General decay rate for a class of weakly dissipative second-order systems with memory. Mathematical Methods in the Applied Sciences, 42(8):2842–2853, 2019. https://doi.org/10.1002/mma.5554
M. Kafini and S.A. Messaoudi. A blow-up result in a nonlinear wave equation with delay. Mediterranean Journal of Mathematics, 13(1):237–247, 2016. https://doi.org/10.1007/s00009-014-0500-4
M. Kafini and S.A. Messaoudi. Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay. Applicable Analysis, 99(3):530– 547, 2020. https://doi.org/10.1080/00036811.2018.1504029
J.-R. Kang. Global nonexistence of solutions for viscoelastic wave equation with delay. Mathematical Methods in the Applied Sciences, 41(16):6834–6841, 2018. https://doi.org/10.1002/mma.5194
M. Kirane and B. Said-Houari. Existence and asymptotic stability of a viscoelastic wave equation with a delay. Zeitschrift fu¨r angewandte Mathematik und Physik, 62(6):1065–1082, 2011. https://doi.org/10.1007/s00033-011-0145-0
W. Liu. General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback. Journal of Mathematical Physics, 54(4):043504, 2013. https://doi.org/10.1063/1.4799929
M.I. Mustafa. General decay result for nonlinear viscoelastic equations. Journal of Mathematical Analysis and Applications, 457(1):134–152, 2018. https://doi.org/10.1016/j.jmaa.2017.08.019
M.I. Mustafa. Optimal decay rates for the viscoelastic wave equation. Mathematical Methods in the Applied Sciences, 41(1):192–204, 2018. https://doi.org/10.1002/mma.4604
M.I. Mustafa. Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems. Zeitschrift fu¨r angewandte Mathematik und Physik, 72(67):1– 15, 2021. https://doi.org/10.1007/s00033-021-01498-7
M.I. Mustafa and M. Kafini. Energy decay for viscoelastic plates with distributed delay and source term. Zeitschrift fu¨r angewandte Mathematik und Physik, 67(36), 2016. https://doi.org/10.1007/s00033-016-0641-3
S. Nicaise and C. Pignotti. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM Journal on Control and Optimization, 45(5):1561–1585, 2006. https://doi.org/10.1137/060648891
S. Nicaise and C. Pignotti. Interior feedback stabilization of wave equations with time dependent delay. Electronic Journal of Differential Equations, 41:1– 20, 2011.
S.-T. Wu. Asymptotic behavior for a viscoelastic wave equation with a delay term. Taiwanese Journal of Mathematics, 17(3):765–784, 2013. https://doi.org/10.11650/tjm.17.2013.2517
S.-T. Wu. Blow-up of solution for a viscoelastic wave equation with delay. Acta Mathematica Scientia, 39(1):329–338, 2019. https://doi.org/10.1007/s10473-019-0124-7
S.T. Wu. Asymptotic behavior for a viscoelastic wave equation with a timevarying delay term. Journal of Partial Differential Equations, 29(1):22–35, 2016. https://doi.org/10.4208/jpde.v29.n1.3