Well-posedness and exponential stability of a thermoelastic-Bresse system with second sound and delay

Abstract

In this paper, we consider a one-dimensional thermoelastic-Bresse system with a delay term, where the heat conduction is given by Cattaneo’s law effective in the shear angle displacement. We prove that the system is well-posed by using the semigroup method, and show, using the multiplier method, that the dissipation induced by the heat is strong enough to exponentially stabilize the system in the presence of a “small" delay when the stable number is zero.

Keywords

• thermoelastic-Bresse system,second sound,exponential decay,time delay

References

1. [1] T.A. Apalara, Well-posedness and exponential stability for a linear damped Timo- shenko system with second sound and internal distributed delay, Electron. J. Differ- ential Equations 2014 (254), 1–15, 2014.
2. [2] T.A. Apalara and S.A. Messaoudi, An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay, Appl. Math. Optim. 71 (3), 449–472, 2015.
3. [3] M. Bresse, Cours de Mecanique Appliquee par M. Bresse rsistance des matriaux et stabilit des constructions, Mallet-Bachelier, Paris, 1859.
4. [4] A.C. Casal and J.I. Díaz, On the complex Ginzburg-Landau equation with a delayed feedback, Math. Models Methods Appl. Sci. 16 (1), 1–17, 2006.
5. [5] M.M. Cavalcanti et al., Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys. 65 (6), 1189–1206, 2014.
6. [6] M.M. Chen, W.J. Liu and W.C. Zhou, 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.
7. [7] Z.J.Chen, W.J. Liu and D.Q. Chen, General decay rates for a laminated beam with memory, Taiwanese J. Math. 23 (5), 1227–1252, 2019.
8. [8] L.H. Fatori and J.E.M. Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math. 75 (6), 881–904, 2010.

9. [9] M. Kafini et al., Well-posedness and stability results in a Timoshenko-type system of thermoelasticity of type III with delay, Z. Angew. Math. Phys. 66 (4), 1499–1517, 2015.
10. [10] A.A. Keddi, T.A. Apalara and S.A. Messaoudi, Exponential and polynomial decay in a thermoelastic-Bresse system with second sound, Appl. Math. Optim. 77 (2), 315–341, 2018.
11. [11] V. Komornik, Exact controllability and stabilization, The multiplier method. Masson- John Wiley, Paris, 1994.
12. [12] J.E. Lagnese, G. Leugering and E.J.P.G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Methods Appl. Sci. 16 (5),327–358, 1993.
13. [13] J.E. Lagnese, G. Leugering and E.J.P.G. Schmidt, Modeling, analysis and control of dynamic elastic Multi-Link structures, Systems & Control: Foundations & Applica- tions, Boston, MA, 1994.
14. [14] G. Li, X.Y. Kong and W.J. Liu, General decay for a laminated beam with structural damping and memory: the case of non-equal wave speeds, J. Integral Equations Appl. 30 (1), 95–116, 2018.
15. [15] G. Li, Y. Luan, J.Y. Yu and F.D. Jiang, Well-posedness and exponential stabil- ity of a flexible structure with second sound and time delay, Appl. Anal. DOI: 10.1080/00036811.2018.1478081.
16. [16] G. Li, D.H. Wang and B.Q. Zhu, Well-posedness and decay of solutions for a trans- mission problem with history and delay, Electron. J. Differential Equations, 2016 (23), 1–21, 2016.
17. [17] G.W. Liu, Well-posedness and exponential decay of solutions for a transmission prob- lem with distributed delay, Electron. J. Differential Equations, 2017 (174), 1-13, 2017.
18. [18] W.J. Liu and M.M. Chen, Well-posedness and exponential decay for a porous ther- moelastic system with second sound and a time-varying delay term in the internal feedback, Contin. Mech. Thermodyn. 29 (3), 731–746, 2017.
19. [19] W.J. Liu, K.W. Chen and J. Yu, Asymptotic stability for a non-autonomous full von Kármán beam with thermo-viscoelastic damping, Appl. Anal. 97 (3), 400–414, 2018.
20. [20] W.J. Liu, D.H. Wang and D.Q. Chen, General decay of solution for a transmis- sion problem in infinite memory-type thermoelasticity with second sound, J. Therm. Stresses 41 (6), 758–775, 2018.
21. [21] W.J. Liu, J.Y. Yu and G. Li, Exponential stability of a flexible structure with second sound, Ann. Polon. Math. DOI: 10.4064/ap171116-31-8.
22. [22] W.J. Liu and W.F. Zhao, Stabilization of a thermoelastic laminated beam with past history, Appl. Math. Optim. 80, 103-133, 2019.
23. [23] S. Nicaise and 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 (5), 1561–1585, 2006.
24. [24] S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (3), 693–722, 2011.
25. [25] S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (3), 559–581, 2009.
26. [26] A. Pazy, Semigroups of linear operators and applications to partial differential equa- tions. Springer, New York, 1983.
27. [27] Y. Qin, J. Ren and T. Wei, Global existence, asymptotic stability, and uniform at- tractors for non-autonomous thermoelastic systems with constant time delay, J. Math. Phys. 53 (6), 063701, 1–20, 2012.
28. [28] M.L. Santos, D.S. Almeida Júnior and J.E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Differential Equations, 253 (9), 2715–2733, 2012.
29. [29] D.H. Wang, G. Li and B.Q. Zhu, Exponential energy decay of solution for a trans- mission problem with viscoelastic term and delay, Mathematics, 4 (42), 1–13, 2016.
30. [30] D.H. Wang, G. Li and B.Q. Zhu, Well-posedness and general decay of solution for a transmission problem with viscoelastic term and delay, J. Nonlinear Sci. Appl. 9 (3), 1202–1215, 2016.
31. [31] B. Wu, S.Y. Wu, J. Yu and Z.W. Wang, Determining the memory kernel from a fixed point measurement data for a parabolic equation with memory effect, Comput. Appl. Math. 37 (2), 1877–1893, 2018.
32. [32] S.T. Wu, Asymptotic behavior for a viscoelastic wave equation with a delay term, Taiwanese J. Math. 17 (3), 765–784, 2013.
33. [33] X.B. Zhang and H.L. Zhu, Hopf bifurcation and chaos of a delayed finance system, Complexity, Article ID: 6715036, 1–18, 2019.
34. [34] X.B. Zhang, H.Y. Zhao and Z.S. Feng, Spatio-temporal complexity of a delayed diffu- sive model for plant invasion, Comput. Math. Appl. 76 (11-12), 2575–2612, 2018.