Mathematica Bohemica, Vol. 148, No. 3, pp. 409-434, 2023
Stability result for a thermoelastic Bresse system with delay term in the internal feedback
Lamine Bouzettouta, Sabah Baibeche, Manel Abdelli, Amar Guesmia
Received October 4, 2021. Published online August 5, 2022.
Abstract: The studies considered here are concerend with a linear thermoelastic Bresse system with delay term in the feedback. The heat conduction is also given by Cattaneo's law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem using the semigroup method. Furthermore, based on the energy method, we establish an exponential stability result depending of a condition on the constants of the system that was first considered by A. Keddi, T. Apalara, S. A. Messaoudi in 2018.
References: [1] F. Alabau Boussouira, J. E. Muñoz Rivera, D. S. Almeida Júnior: Stability to weak dissipative Bresse system. J. Math. Anal. Appl. 374 (2011), 481-498. DOI 10.1016/j.jmaa.2010.07.046 | MR 2729236 | Zbl 1209.35018
[2] M. O. Alves, L. H. Fatori, M. A. J. Silva, R. N. Monteiro: Stability and optimality of decay rate for a weakly dissipative Bresse system. Math. Methods Appl. Sci. 38 (2015), 898-908. DOI 10.1002/mma.3115 | MR 3324487 | Zbl 1316.35178
[3] A. Benaissa, M. Miloudi, M. Mokhtari: Global existence and energy decay of solutions to a Bresse system with delay terms. Commentat. Math. Univ. Carol. 56 (2015), 169-186. DOI 10.14712/1213-7243.2015.116 | MR 3338731 | Zbl 1340.35198
[4] L. Bouzettouta, S. Zitouni, K. Zennir, A. Guesmia: Stability of Bresse system with internal distributed delay. J. Math. Comput. Sci. 7 (2017), 92-118.
[5] L. Bouzettouta, S. Zitouni, K. Zennir, H. Sissaoui: Well-posedness and decay of solutions to Bresse system with internal distributed delay. Int. J. Appl. Math. Stat. 56 (2017), 153-168. MR 3620629
[6] J. A. C. Bresse: Cours de méchanique appliquée. Mallet Bachelier, Paris (1859). (In French.)
[7] G. Chen: Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim. 17 (1979), 66-81. DOI 10.1137/0317007 | MR 0516857 | Zbl 0402.93016
[8] G. Chen: Control and stabilization for the wave equation in a bounded domain. II. SIAM J. Control Optim. 19 (1981), 114-122. DOI 10.1137/0319009 | MR 0603084 | Zbl 0461.93037
[9] 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. DOI 10.1137/0324007 | MR 0818942 | Zbl 0592.93047
[10] L. H. Fatori, J. E. Muñoz Rivera: Rates of decay to weak thermoelastic Bresse system. IMA J. Appl. Math. 75 (2010), 881-904. DOI 10.1093/imamat/hxq038 | MR 2740037 | Zbl 1209.80005
[11] H. D. Fernándes Sare, R. Racke: On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Arch. Ration. Mech. Anal. 194 (2009), 221-251. DOI 10.1007/s00205-009-0220-2 | MR 2533927 | Zbl 1251.74011
[12] F. A. Gallego, J. E. Muñoz Rivera: Decay rates for solutions to thermoelastic Bresse systems of types I and III. Electron. J. Differ. Equ. 2017 (2017), Article ID 73, 26 pages. MR 3651870 | Zbl 1370.35046
[13] A. Guesmia, M. Kafini: Bresse system with infinite memories. Math. Methods Appl. Sci. 38 (2015), 2389-2402. DOI 10.1002/mma.3228 | MR 3366806 | Zbl 1317.35007
[14] A. A. Keddi, T. A. Apalara, S. A. Messaoudi: Exponential and polynomial decay in a thermoelastic-Bresse system with second sound. Appl. Math. Optim. 77 (2018), 315-341. DOI 10.1007/s00245-016-9376-y | MR 3776342 | Zbl 1388.35188
[15] J. U. Kim, Y. Renardy: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25 (1987), 1417-1429. DOI 10.1137/0325078 | MR 0912448 | Zbl 0632.93057
[16] V. Komornik: Exact Controllability and Stabilization: The Multiplier Method. Research in Applied Mathematics 36. John Wiley & Sons, Chichester (1994). MR 1359765 | Zbl 0937.93003
[17] Z. Liu, B. Rao: Energy decay rate of the thermoelastic Bresse system. Z. Angew. Math. Phys. 60 (2009), 54-69. DOI 10.1007/s00033-008-6122-6 | MR 2469727 | Zbl 1161.74030
[18] Z. Liu, S. Zheng: Semigroups Associated with Dissipative Systems. Chapman & Hall/CRC Research Notes in Mathematics 398. Chapman & Hall/CRC, Boca Raton (1999). MR 1681343 | Zbl 0924.73003
[19] S. A. Messaoudi, M. I. Mustafa: On the internal and boundary stabilization of Timoshenko beams. NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008), 655-671. DOI 10.1007/s00030-008-7075-3 | MR 2465776 | Zbl 1163.74037
[20] S. A. Messaoudi, M. I. Mustafa: On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Math. Methods Appl. Sci. 32 (2009), 454-469. DOI 10.1002/mma.1047 | MR 2493590 | Zbl 1171.35014
[21] J. E. Munõz Rivera, R. Racke: Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst. 9 (2003), 1625-1639. DOI 10.3934/dcds.2003.9.1625 | MR 2017685 | Zbl 1047.35023
[22] M. I. Mustafa: A uniform stability result for thermoelasticity of type III with boundary distributed delay. J. Math. Anal. Appl. 415 (2014), 148-158. DOI 10.1016/j.jmaa.2014.01.080 | MR 3173160 | Zbl 1310.74006
[23] M. I. Mustafa, M. Kafini: Exponential decay in thermoelastic systems with internal distributed delay. Palest. J. Math. 2 (2013), 287-299. MR 3109903 | Zbl 1343.93066
[24] M. Nakao: Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl. 60 (1977), 542-549. DOI 10.1016/0022-247X(77)90040-3 | MR 0499564 | Zbl 0376.34051
[25] 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. DOI 10.1137/060648891 | MR 2272156 | Zbl 1180.35095
[26] S. Nicaise, C. Pignotti: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21 (2008), 935-958. MR 2483342 | Zbl 1224.35247
[27] D. Ouchnane: A stability result of a Timoshenko system in thermoelasticity of second sound with a delay term in the internal feedback. Georgian Math. J. 21 (2014), 475-489. DOI 10.1515/gmj-2014-0045 | MR 3284711 | Zbl 1304.35103
[28] J.-H. Park, J.-R. Kang: Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation. IMA J. Appl. Math. 76 (2011), 340-350. DOI 10.1093/imamat/hxq040 | MR 2781698 | Zbl 1219.35311
[29] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44. Springer, New York (1983). DOI 10.1007/978-1-4612-5561-1 | MR 0710486 | Zbl 0516.47023
[30] C. A. Raposo, J. Ferreira, M. L. Santos, N. N. O. Castro: Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 18 (2005), 535-541. DOI 10.1016/j.aml.2004.03.017 | MR 2127817 | Zbl 1072.74033
[31] M. L. Santos, A. Soufyane, D. S. Almeida Júnior: Asymptotic behavior to Bresse system with past history. Q. Appl. Math. 73 (2015), 23-54. DOI 10.1090/S0033-569X-2014-01382-4 | MR 3322725 | Zbl 1308.74066
[32] J. A. Soriano, J. E. Muñoz Rivera, L. H. Fatori: Bresse system with indefinite damping. J. Math. Anal. Appl. 387 (2012), 284-290. DOI 10.1016/j.jmaa.2011.08.072 | MR 2845750 | Zbl 1231.35113
[33] S. P. Timoshenko: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. (6) 41 (1921), 744-746. DOI 10.1080/14786442108636264
[34] A. Wehbe, W. Youssef: Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks. J. Math. Phys. 51 (2010), Article ID 103523, 17 pages. DOI 10.1063/1.3486094 | MR 2761337 | Zbl 1314.74035
[35] C. Q. Xu, S. P. Yung, L. K. Li: Stabilization of wave systems with input delay in the boundary control. ESAIM, Control Optim. Calc. Var. 12 (2006), 770-785. DOI 10.1051/cocv:2006021 | MR 2266817 | Zbl 1105.35016
[36] S. Zitouni, L. Bouzettouta, K. Zennir, D. Ouchenane: Exponential decay of thermo-elastic Bresse system with distributed delay term. Hacet. J. Math. Stat. 47 (2018), 1216-1230. DOI 10.15672/hjms.2017.498 | MR 3974506 | Zbl 07406314
