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Title: A general decay estimate for a finite memory thermoelastic Bresse system (English)
Author: Enyi, Cyril Dennis
Author: Mukiawa, Soh Edwin
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 67
Issue: 5
Year: 2022
Pages: 633-656
Summary lang: English
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Category: math
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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. (English)
Keyword: general decay
Keyword: Bresse system
Keyword: nonequal speed
Keyword: viscoelastic
Keyword: thermoelastic
MSC: 35B35
MSC: 35B40
MSC: 35L05
MSC: 35L20
MSC: 74D10
idZBL: Zbl 07613016
idMR: MR4484890
DOI: 10.21136/AM.2022.0224-20
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Date available: 2022-09-15T09:16:40Z
Last updated: 2024-11-04
Stable URL: http://hdl.handle.net/10338.dmlcz/151029
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Reference: [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. Zbl 1316.35178, MR 3324487, 10.1002/mma.3115
Reference: [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. Zbl 1437.35055, MR 4029814, 10.1137/18M1191774
Reference: [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. Zbl 1131.74303, MR 2001030, 10.1016/S0022-0396(03)00185-2
Reference: [4] Bresse, J. A. C.: Cours de mécanique appliquée.Mallet Bachelier, Paris (1859), French.
Reference: [5] Dell'Oro, F.: Asymptotic stability of thermoelastic systems of Bresse type.J. Differ. Equations 258 (2015), 3902-3927. Zbl 1311.35024, MR 3322987, 10.1016/j.jde.2015.01.025
Reference: [6] Arwadi, T. El, Youssef, W.: On the stabilization of the Bresse beam with Kelvin-Voigt damping.Appl. Math. Optim. 83 (2021), 1831-1857. Zbl 7371826, MR 4261274, 10.1007/s00245-019-09611-z
Reference: [7] Enyi, C. D.: Dynamics of a new thermoelastic Timoshenko system with second sound.Results Appl. Math. 12 (2021), Article ID 100204, 20 pages. Zbl 1481.35053, MR 4330084, 10.1016/j.rinam.2021.100204
Reference: [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. Zbl 1470.35057, MR 4270141, 10.1007/s40840-020-01035-1
Reference: [9] Fatori, L. H., Monteiro, R. N.: The optimal decay rate for a weak dissipative Bresse system.Appl. Math. Lett. 25 (2012), 600-604. Zbl 1325.74065, MR 2856041, 10.1016/j.aml.2011.09.067
Reference: [10] Guesmia, A., Kafini, M.: Bresse system with infinite memories.Math. Methods Appl. Sci. 38 (2015), 2389-2402. Zbl 1317.35007, MR 3366806, 10.1002/mma.3228
Reference: [11] Guesmia, A., Messaoudi, S. A.: On the control of a viscoelastic damped Timoshenko-type system.Appl. Math. Comput. 206 (2008), 589-597. Zbl 1154.74030, MR 2483034, 10.1016/j.amc.2008.05.122
Reference: [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. Zbl 1183.35036, MR 2571867, 10.1002/mma.1125
Reference: [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. Zbl 0773.73060, MR 1217432, 10.1002/mma.1670160503
Reference: [14] Lions, J. L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Vol. III.Die Grundlehren der mathematischen Wissenschaften 183. Springer, Berlin (1973). Zbl 0251.35001, MR 0350179, 10.1007/978-3-642-65393-3
Reference: [15] Liu, Z., Rao, B.: Energy decay rate of the thermoelastic Bresse system.Z. Angew. Math. Phys. 60 (2009), 54-69. Zbl 1161.74030, MR 2469727, 10.1007/s00033-008-6122-6
Reference: [16] Ma, T. F., Monteiro, R. N.: Singular limit and long-time dynamics of Bresse systems.SIAM J. Math. Anal. 49 (2017), 2468-2495. Zbl 1391.35071, MR 3668597, 10.1137/15M1039894
Reference: [17] Messaoudi, S. A., Hassan, J. H.: New general decay results in finite-memory Bresse system.Commun. Pure Appl. Anal. 18 (2019), 1637-1662. Zbl 1421.35028, MR 3927415, 10.3934/cpaa.2019078
Reference: [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. Zbl 1106.35333, MR 1944350, 10.1016/S0022-247X(02)00436-5
Reference: [19] Soriano, J. A., Rivera, J. E. Muñoz, Fatori, L. H.: Bresse system with indefinite damping.J. Math. Anal. Appl. 387 (2012), 284-290. Zbl 1231.35113, MR 2845750, 10.1016/j.jmaa.2011.08.072
Reference: [20] Soufyane, A.: Stabilisation de la poutre de Timoshenko.C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999), 731-734 French. Zbl 0943.74042, MR 1680836, 10.1016/S0764-4442(99)80244-4
Reference: [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. Zbl 1304.35105, MR 3274656, 10.3934/eect.2014.3.713
Reference: [22] Tatar, N.-E.: Stabilization of a viscoelastic Timoshenko beam.Appl. Anal. 92 (2013), 27-43. Zbl 1261.35087, MR 3007921, 10.1080/00036811.2011.587810
Reference: [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. Zbl 1314.74035, MR 2761337, 10.1063/1.3486094
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