Abstract
In previous work (Alves et al. in Z Angew Math Phys 69:106, 2018), by using the linear semigroup theory, Alves et al. investigated the existence and exponential stability results for a Moore–Gibson–Thompson model encompassing memory of type 1, 2 or 3 in a history space framework. In this paper, we continue to consider the similar problem with type 1 and establish explicit and general decay results of energy for system in both the subcritical and critical cases, by introducing suitable energy and perturbed Lyapunov functionals and following convex functions ideas presented in Guesmia (J Math Anal Appl 382:748–760, 2011). Our results allow a much larger class of the convolution kernels which improves the earlier related results.
Similar content being viewed by others
References
Alves, M.O., et al.: Moore–Gibson–Thompson equation with memory in a history framework: a semigroup approach. Z. Angew. Math. Phys. 69(4), 106 (2018)
Arnold, V.I.: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, 2nd edn. Springer, New York (1989)
Boulanouar, F., Drabla, S.: General boundary stabilization result of memory-type thermoelasticity with second sound. Electron. J. Differ. Equ. 2014(202), 18 (2014)
Caixeta, A.H., Lasiecka, I., Cavalcanti, V.N.D.: Global attractors for a third order in time nonlinear dynamics. J. Differ. Equ. 261(1), 113–147 (2016)
Caixeta, A.H., Lasiecka, I., Domingos Cavalcanti, V.N.: On long time behavior of Moore–Gibson–Thompson equation with molecular relaxation. Evol. Equ. Control Theory 5(4), 661–676 (2016)
Cavalcanti, M.M., Guesmia, A.: General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type. Differ. Integral Equ. 18(5), 583–600 (2005)
Chen, M., Liu, W., Zhou, W.: Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms. Adv. Nonlinear Anal. 7(4), 547–569 (2018)
Conejero, J.A., Lizama, C., Rodenas, F.: Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation. Appl. Math. Inf. Sci. 9(5), 2233–2238 (2015)
Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970)
Dell’Oro, F., Lasiecka, I., Pata, V.: The Moore–Gibson–Thompson equation with memory in the critical case. J. Differ. Equ. 261(7), 4188–4222 (2016)
Dell’Oro, F., Pata, V.: On the Moore–Gibson–Thompson equation and its relation to linear viscoelasticity. Appl. Math. Optim. 76(3), 641–655 (2017)
Dell’Oro, F., Pata, V.: On a fourth-order equation of Moore–Gibson–Thompson type. Milan J. Math. 85(2), 215–234 (2017)
Feng, B.: General decay for a viscoelastic wave equation with density and time delay term in \({\mathbb{R}}^n\). Taiwan. J. Math. 22(1), 205–223 (2018)
Feng, B.: General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history. Mediterr. J. Math. 15(3), 103 (2018)
Guesmia, A.: Asymptotic stability of abstract dissipative systems with infinite memory. J. Math. Anal. Appl. 382(2), 748–760 (2011)
Guesmia, A., Messaoudi, S.A., Soufyane, A.: Stabilization of a linear Timoshenko system with infinite history and applications to the Timoshenko-heat systems. Electron. J. Differ. Equ. 2012(193), 45 (2012)
Guesmia, A.: Asymptotic behavior for coupled abstract evolution equations with one infinite memory. Appl. Anal. 94(1), 184–217 (2015)
Kaltenbacher, B.: Mathematics of nonlinear acoustics. Evol. Equ. Control Theory 4(4), 447–491 (2015)
Kaltenbacher, B., Lasiecka, I.: Exponential decay for low and higher energies in the third order linear Moore–Gibson–Thompson equation with variable viscosity. Palest. J. Math. 1(1), 1–10 (2012)
Kaltenbacher, B., Lasiecka, I., Marchand, R.: Wellposedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound. Control Cybern. 40(4), 971–988 (2011)
Lasiecka, I., Wang, X.: Moore–Gibson–Thompson equation with memory, part I: exponential decay of energy. Z. Angew. Math. Phys. 67(2), 17 (2016)
Lasiecka, I., Wang, X.: Moore–Gibson–Thompson equation with memory, part II: general decay of energy. J. Differ. Equ. 259(12), 7610–7635 (2015)
Li, G., Kong, X.Y., Liu, W.J.: General decay for a laminated beam with structural damping and memory: the case of non-equal wave speeds. J. Integral Equ. Appl. 30(1), 95–116 (2018)
Liu, W.J., Chen, K.W., Yu, J.: Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term. IMA J. Math. Control Inf. 34(2), 521–542 (2017)
Liu, W.J., Chen, Z.J., Chen, D.Q.: New general decay results for a Moore–Gibson–Thompson equation with memory. Appl. Anal. (2019). https://doi.org/10.1080/00036811.2019.1577390
Liu, W.J., Wang, D.H., Chen, D.Q.: General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound. J. Therm. Stress. 41(6), 758–775 (2018)
Liu, W.J., Zhao, W.F.: Stabilization of a thermoelastic laminated beam with past history. Appl. Math. Optim. 80(1), 103–133 (2019)
Moore, F., Gibson, W.: Propagation of weak disturbances in a gas subject to relaxing effects. J. Aerosp. Sci. 27, 117–127 (1960)
Messaoudi, S.A., Al-Gharabli, M.M.: A general decay result of a viscoelastic equation with past history and boundary feedback. Z. Angew. Math. Phys. 66(4), 1519–1528 (2015)
Messaoudi, S.A., Apalara, T.A.: Asymptotic stability of thermoelasticity type III with delay term and infinite memory. IMA J. Math. Control Inf. 32(1), 75–95 (2015)
Mustafa, M.I.: Optimal decay rates for the viscoelastic wave equation. Math. Methods Appl. Sci. 41(1), 192–204 (2018)
Mustafa, M.I.: General decay result for nonlinear viscoelastic equations. J. Math. Anal. Appl. 457(1), 134–152 (2018)
Tahamtani, F., Peyravi, A.: Asymptotic behavior and blow-up of solution for a nonlinear viscoelastic wave equation with boundary dissipation. Taiwan. J. Math. 17(6), 1921–1943 (2013)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11771216), the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725], the Six Talent Peaks Project in Jiangsu Province (Grant No. 2015-XCL-020) and the Qing Lan Project of Jiangsu Province.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Liu, W., Chen, Z. General decay rate for a Moore–Gibson–Thompson equation with infinite history. Z. Angew. Math. Phys. 71, 43 (2020). https://doi.org/10.1007/s00033-020-1265-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-020-1265-1