Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
general decay; Bresse system; nonequal speed; viscoelastic; thermoelastic
general decay; Bresse system; nonequal speed; viscoelastic; thermoelastic
Summary:
This work considers a Bresse system with viscoelastic damping on the vertical displacement and heat conduction effect on the shear angle displacement. A general stability result with minimal condition on the relaxation function is obtained. The system under investigation, to the best of our knowledge, is new and has not been studied before in the literature. What is more interesting is the fact that our result holds without the imposition of the equal speed of wave propagation condition, and differentiation of the equations of the system, as against the usual practice in the literature.
References:
[1] Alves, M. O., Fatori, L. H., Silva, M. A. Jorge, Monteiro, R. N.: 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
[2] Alves, M. O., Tavares, E. H. Gomes, Silva, M. A. Jorge, Rodrigues, J. H.: On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J. Math. Anal. 51 (2019), 4520-4543. DOI 10.1137/18M1191774 | MR 4029814 | Zbl 1437.35055
[3] Ammar-Khodja, F., Benabdallah, A., Rivera, J. E. Muñoz, Racke, R.: Energy decay for Timoshenko systems of memory type. J. Differ. Equations 194 (2003), 82-115. DOI 10.1016/S0022-0396(03)00185-2 | MR 2001030 | Zbl 1131.74303
[4] Bresse, J. A. C.: Cours de mécanique appliquée. Mallet Bachelier, Paris (1859), French.
[5] Dell'Oro, F.: Asymptotic stability of thermoelastic systems of Bresse type. J. Differ. Equations 258 (2015), 3902-3927. DOI 10.1016/j.jde.2015.01.025 | MR 3322987 | Zbl 1311.35024
[6] Arwadi, T. El, Youssef, W.: On the stabilization of the Bresse beam with Kelvin-Voigt damping. Appl. Math. Optim. 83 (2021), 1831-1857. DOI 10.1007/s00245-019-09611-z | MR 4261274 | Zbl 7371826
[7] Enyi, C. D.: Dynamics of a new thermoelastic Timoshenko system with second sound. Results Appl. Math. 12 (2021), Article ID 100204, 20 pages. DOI 10.1016/j.rinam.2021.100204 | MR 4330084 | Zbl 1481.35053
[8] Enyi, C. D., Feng, B.: Stability result for a new viscoelastic-thermoelastic Timoshenko system. Bull. Malays. Math. Sci. Soc. (2) 44 (2021), 1837-1866. DOI 10.1007/s40840-020-01035-1 | MR 4270141 | Zbl 1470.35057
[9] Fatori, L. H., Monteiro, R. N.: The optimal decay rate for a weak dissipative Bresse system. Appl. Math. Lett. 25 (2012), 600-604. DOI 10.1016/j.aml.2011.09.067 | MR 2856041 | Zbl 1325.74065
[10] Guesmia, A., Kafini, M.: Bresse system with infinite memories. Math. Methods Appl. Sci. 38 (2015), 2389-2402. DOI 10.1002/mma.3228 | MR 3366806 | Zbl 1317.35007
[11] Guesmia, A., Messaoudi, S. A.: On the control of a viscoelastic damped Timoshenko-type system. Appl. Math. Comput. 206 (2008), 589-597. DOI 10.1016/j.amc.2008.05.122 | MR 2483034 | Zbl 1154.74030
[12] Guesmia, A., Messaoudi, S. A.: General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math. Methods Appl. Sci. 32 (2009), 2102-2122. DOI 10.1002/mma.1125 | MR 2571867 | Zbl 1183.35036
[13] Lagnese, J. E., Leugering, G., Schmidt, E. J. P. G.: Modelling of dynamic networks of thin thermoelastic beams. Math. Methods Appl. Sci. 16 (1993), 327-358. DOI 10.1002/mma.1670160503 | MR 1217432 | Zbl 0773.73060
[14] Lions, J. L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Vol. III. Die Grundlehren der mathematischen Wissenschaften 183. Springer, Berlin (1973). DOI 10.1007/978-3-642-65393-3 | MR 0350179 | Zbl 0251.35001
[15] Liu, Z., Rao, B.: 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
[16] Ma, T. F., Monteiro, R. N.: Singular limit and long-time dynamics of Bresse systems. SIAM J. Math. Anal. 49 (2017), 2468-2495. DOI 10.1137/15M1039894 | MR 3668597 | Zbl 1391.35071
[17] Messaoudi, S. A., Hassan, J. H.: New general decay results in finite-memory Bresse system. Commun. Pure Appl. Anal. 18 (2019), 1637-1662. DOI 10.3934/cpaa.2019078 | MR 3927415 | Zbl 1421.35028
[18] Rivera, J. E. Muñoz, Racke, R.: Mildly dissipative nonlinear Timoshenko systems -- global existence and exponential stability. J. Math. Anal. Appl. 276 (2002), 248-278. DOI 10.1016/S0022-247X(02)00436-5 | MR 1944350 | Zbl 1106.35333
[19] Soriano, J. A., Rivera, J. E. Muñoz, Fatori, L. H.: 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
[20] Soufyane, A.: Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999), 731-734 French. DOI 10.1016/S0764-4442(99)80244-4 | MR 1680836 | Zbl 0943.74042
[21] Soufyane, A., Said-Houari, B.: The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evol. Equ. Control Theory 3 (2014), 713-738. DOI 10.3934/eect.2014.3.713 | MR 3274656 | Zbl 1304.35105
[22] Tatar, N.-E.: Stabilization of a viscoelastic Timoshenko beam. Appl. Anal. 92 (2013), 27-43. DOI 10.1080/00036811.2011.587810 | MR 3007921 | Zbl 1261.35087
[23] 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 10.1063/1.3486094 | MR 2761337 | Zbl 1314.74035
Search
Partner of
