Geodesic
Stability result for a thermoelastic Bresse system with delay term in the internal feedback
Mathematica Bohemica, Tome 148 (2023) no. 3, pp. 409-434
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/MB.2022.0154-21
Classification : 35B40, 74H40, 74H55, 93D15, 93D20
Keywords: Bresse system; delay; decay rate; energy method; semigroup method; thermoelastic 
@article{10_21136_MB_2022_0154_21,
     author = {Bouzettouta, Lamine and Baibeche, Sabah and Abdelli, Manel and Guesmia, Amar},
     title = {Stability result for a thermoelastic {Bresse} system with delay term in the internal feedback},
     journal = {Mathematica Bohemica},
     pages = {409--434},
     year = {2023},
     volume = {148},
     number = {3},
     doi = {10.21136/MB.2022.0154-21},
     mrnumber = {4628618},
     zbl = {07729582},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2022.0154-21/}
}
TY  - JOUR
AU  - Bouzettouta, Lamine
AU  - Baibeche, Sabah
AU  - Abdelli, Manel
AU  - Guesmia, Amar
TI  - Stability result for a thermoelastic Bresse system with delay term in the internal feedback
JO  - Mathematica Bohemica
PY  - 2023
SP  - 409
EP  - 434
VL  - 148
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2022.0154-21/
DO  - 10.21136/MB.2022.0154-21
LA  - en
ID  - 10_21136_MB_2022_0154_21
ER  - 
%0 Journal Article
%A Bouzettouta, Lamine
%A Baibeche, Sabah
%A Abdelli, Manel
%A Guesmia, Amar
%T Stability result for a thermoelastic Bresse system with delay term in the internal feedback
%J Mathematica Bohemica
%D 2023
%P 409-434
%V 148
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2022.0154-21/
%R 10.21136/MB.2022.0154-21
%G en
%F 10_21136_MB_2022_0154_21
Bouzettouta, Lamine; Baibeche, Sabah; Abdelli, Manel; Guesmia, Amar. Stability result for a thermoelastic Bresse system with delay term in the internal feedback. Mathematica Bohemica, Tome 148 (2023) no. 3, pp. 409-434. doi: 10.21136/MB.2022.0154-21

[1] Boussouira, F. Alabau, Rivera, J. E. Muñoz, Júnior, D. S. Almeida: Stability to weak dissipative Bresse system. J. Math. Anal. Appl. 374 (2011), 481-498. | DOI | MR | JFM

[2] Alves, M. O., Fatori, L. H., Silva, M. A. J., Monteiro, R. N.: Stability and optimality of decay rate for a weakly dissipative Bresse system. Math. Methods Appl. Sci. 38 (2015), 898-908. | DOI | MR | JFM

[3] Benaissa, A., Miloudi, M., Mokhtari, M.: Global existence and energy decay of solutions to a Bresse system with delay terms. Commentat. Math. Univ. Carol. 56 (2015), 169-186. | DOI | MR | JFM

[4] Bouzettouta, L., Zitouni, S., Zennir, K., Guesmia, A.: Stability of Bresse system with internal distributed delay. J. Math. Comput. Sci. 7 (2017), 92-118.

[5] Bouzettouta, L., Zitouni, S., Zennir, K., Sissaoui, H.: Well-posedness and decay of solutions to Bresse system with internal distributed delay. Int. J. Appl. Math. Stat. 56 (2017), 153-168. | MR

[6] Bresse, J. A. C.: Cours de méchanique appliquée. Mallet Bachelier, Paris (1859), French.

[7] Chen, G.: Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim. 17 (1979), 66-81. | DOI | MR | JFM

[8] Chen, G.: Control and stabilization for the wave equation in a bounded domain. II. SIAM J. Control Optim. 19 (1981), 114-122. | DOI | MR | JFM

[9] Datko, R., Lagnese, J., Polis, M. P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986), 152-156. | DOI | MR | JFM

[10] Fatori, L. H., Rivera, J. E. Muñoz: Rates of decay to weak thermoelastic Bresse system. IMA J. Appl. Math. 75 (2010), 881-904. | DOI | MR | JFM

[11] Sare, H. D. Fernándes, Racke, R.: On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Arch. Ration. Mech. Anal. 194 (2009), 221-251. | DOI | MR | JFM

[12] Gallego, F. A., Rivera, J. E. Muñoz: 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 | JFM

[13] Guesmia, A., Kafini, M.: Bresse system with infinite memories. Math. Methods Appl. Sci. 38 (2015), 2389-2402. | DOI | MR | JFM

[14] Keddi, A. A., Apalara, T. A., Messaoudi, S. A.: Exponential and polynomial decay in a thermoelastic-Bresse system with second sound. Appl. Math. Optim. 77 (2018), 315-341. | DOI | MR | JFM

[15] Kim, J. U., Renardy, Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25 (1987), 1417-1429. | DOI | MR | JFM

[16] Komornik, V.: Exact Controllability and Stabilization: The Multiplier Method. Research in Applied Mathematics 36. John Wiley & Sons, Chichester (1994). | MR | JFM

[17] Liu, Z., Rao, B.: Energy decay rate of the thermoelastic Bresse system. Z. Angew. Math. Phys. 60 (2009), 54-69. | DOI | MR | JFM

[18] Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Chapman & Hall/CRC Research Notes in Mathematics 398. Chapman & Hall/CRC, Boca Raton (1999). | MR | JFM

[19] Messaoudi, S. A., Mustafa, M. I.: On the internal and boundary stabilization of Timoshenko beams. NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008), 655-671. | DOI | MR | JFM

[20] Messaoudi, S. A., Mustafa, M. I.: On the stabilization of the Timoshenko system by a weak nonlinear dissipation. Math. Methods Appl. Sci. 32 (2009), 454-469. | DOI | MR | JFM

[21] Rivera, J. E. Munõz, Racke, R.: Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst. 9 (2003), 1625-1639. | DOI | MR | JFM

[22] Mustafa, M. I.: A uniform stability result for thermoelasticity of type III with boundary distributed delay. J. Math. Anal. Appl. 415 (2014), 148-158. | DOI | MR | JFM

[23] Mustafa, M. I., Kafini, M.: Exponential decay in thermoelastic systems with internal distributed delay. Palest. J. Math. 2 (2013), 287-299. | MR | JFM

[24] Nakao, M.: Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl. 60 (1977), 542-549. | DOI | MR | JFM

[25] Nicaise, S., Pignotti, C.: 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 | MR | JFM

[26] Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21 (2008), 935-958. | MR | JFM

[27] Ouchnane, D.: 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 | MR | JFM

[28] Park, J.-H., Kang, J.-R.: Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation. IMA J. Appl. Math. 76 (2011), 340-350. | DOI | MR | JFM

[29] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44. Springer, New York (1983). | DOI | MR | JFM

[30] Raposo, C. A., Ferreira, J., Santos, M. L., Castro, N. N. O.: Exponential stability for the Timoshenko system with two weak dampings. Appl. Math. Lett. 18 (2005), 535-541. | DOI | MR | JFM

[31] Santos, M. L., Soufyane, A., Júnior, D. S. Almeida: Asymptotic behavior to Bresse system with past history. Q. Appl. Math. 73 (2015), 23-54. | DOI | MR | JFM

[32] Soriano, J. A., Rivera, J. E. Muñoz, Fatori, L. H.: Bresse system with indefinite damping. J. Math. Anal. Appl. 387 (2012), 284-290. | DOI | MR | JFM

[33] Timoshenko, S. P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. (6) 41 (1921), 744-746. | DOI

[34] Wehbe, A., Youssef, W.: 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 | MR | JFM

[35] Xu, C. Q., Yung, S. P., Li, L. K.: Stabilization of wave systems with input delay in the boundary control. ESAIM, Control Optim. Calc. Var. 12 (2006), 770-785. | DOI | MR | JFM

[36] Zitouni, S., Bouzettouta, L., Zennir, K., Ouchenane, D.: Exponential decay of thermo-elastic Bresse system with distributed delay term. Hacet. J. Math. Stat. 47 (2018), 1216-1230. | DOI | MR | JFM

Cité par Sources :