Exponential and polynomial decay for a laminated beam with Fourier’s law of heat conduction and possible absence of structural damping. (English) Zbl 1476.35050
Summary: We study the well-posedness and decay properties of a one-dimensional thermoelastic laminated beam system either with or without structural damping, of which the heat conduction is given by Fourier’s law effective in the rotation angle displacements. We show that the system is well-posed by using the Lumer-Philips theorem, and prove that the system is exponentially stable if and only if the wave speeds are equal, by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem. Furthermore, we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal, by using the second-order energy method. When the speeds are not equal, whether the system without structural damping may has polynomial stability is left as an open problem.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L52 | Initial value problems for second-order hyperbolic systems |
| 74F05 | Thermal effects in solid mechanics |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 93D20 | Asymptotic stability in control theory |
References:
| [1] | Alharbi, A. M.; Scott, N. H., Wave stability in anisotropic generalized temperature-rate-dependent thermoelasticity, IMA J Appl Math, 81, 5, 750-778 (2016) · Zbl 1408.74004 · doi:10.1093/imamat/hxw022 |
| [2] | Almeida, D. S Jr; Santos, M. L.; Muñoz Rivera, J. E., Stability to 1-D thermoelastic Timoshenko beam acting on shear force, Z Angew Math Phys, 65, 6, 1233-1249 (2014) · Zbl 1316.35044 · doi:10.1007/s00033-013-0387-0 |
| [3] | Apalara, T. A., Uniform stability of a laminated beam with structural damping and second sound, Z Angew Math Phys, 68, 41, 1-16 (2017) · Zbl 1379.35182 |
| [4] | Apalara, T. A.; Messaoudi, S. A., 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) · Zbl 1326.35033 · doi:10.1007/s00245-014-9266-0 |
| [5] | Bajkowski, J.; Fernáandezw, J. R.; Kuttler, K. L.; Shillor, M., A thermoviscoelastic beam model for brakes, European J Appl Math, 15, 2, 181-202 (2004) · Zbl 1085.74033 · doi:10.1017/S0956792503005370 |
| [6] | Boulanouar, F.; Drabla, S., General boundary stabilization result of memory-type thermoelasticity with second sound, Electron J Differential Equations, 2014, 202, 1-18 (2014) · Zbl 1304.35084 |
| [7] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), New York: Springer, New York · Zbl 1220.46002 · doi:10.1007/978-0-387-70914-7 |
| [8] | Campelo, A. D S.; Almeida Juúnior, D. S.; Santos, M. L., Stability to the dissipative Reissner-Mindlin-Timoshenko acting on displacement equation, European J Appl Math, 27, 2, 157-193 (2016) · Zbl 1383.35227 · doi:10.1017/S0956792515000467 |
| [9] | Cao, X. G.; Liu, D. Y.; Xu, G. Q., Easy test for stability of laminated beams with structural damping and boundary feedback controls, J Dyn Control Syst, 13, 3, 313-336 (2007) · Zbl 1129.93010 · doi:10.1007/s10883-007-9022-8 |
| [10] | Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Falcão Nascimento, F. A.; Lasiecka, I.; Rodrigues, J. H., 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) · Zbl 1316.35034 · doi:10.1007/s00033-013-0380-7 |
| [11] | Chen, M. M.; Liu, W. J.; Zhou, W. C., 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) · Zbl 1404.35435 · doi:10.1515/anona-2016-0085 |
| [12] | Fareh, A.; Messaoudi, S. A., Stabilization of a type III thermoelastic Timoshenko system in the presence of a time-distributed delay, Math Nachr, 290, 7, 1017-1032 (2017) · Zbl 1434.35218 · doi:10.1002/mana.201500203 |
| [13] | Feng, B.; Ma, T. F.; Monteiro, R. N.; Raposo, C. A., Dynamics of laminated Timoshenko beams, J Dynam Differential Equations, 30, 4, 1489-1507 (2018) · Zbl 1403.35055 · doi:10.1007/s10884-017-9604-4 |
| [14] | Guesmia, A.; Messaoudi, S. A., On the stabilization of Timoshenko systems with memory and different speeds of wave propagation, Appl Math Comput, 219, 17, 9424-9437 (2013) · Zbl 1290.74019 |
| [15] | Hansen, S. W.; Spies, R., Structural damping in a laminated beams due to interfacial slip, J Sound Vibration, 204, 2, 183-202 (1997) · doi:10.1006/jsvi.1996.0913 |
| [16] | Huang, F. L., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann Differential Equations, 1, 1, 43-56 (1985) · Zbl 0593.34048 |
| [17] | Keddi, A. A.; Apalaras, T. A.; Messaoudi, S. A., Exponential and polynomial decay in a thermoelastic-Bresse system with second sound, Appl Math Optim, 77, 2, 315-341 (2018) · Zbl 1388.35188 · doi:10.1007/s00245-016-9376-y |
| [18] | Komornik, V., Exact Controllability and Stabilization: The Multiplier Method, Research in Applied Mathematics (1994), Chichester/Paris: Wiley/Masson, Chichester/Paris · Zbl 0937.93003 |
| [19] | Liu, W. J.; Chen, K. W.; Yu, J., Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, IMA J Math Control Inform, 34, 2, 521-542 (2017) · Zbl 1458.93082 |
| [20] | Liu, W. J.; Chen, K. W.; Yu, J., Asymptotic stability for a non-autonomous full von Kármán beam with thermo-viscoelastic damping, Appl Anal, 97, 3, 400-414 (2018) · Zbl 1466.35041 · doi:10.1080/00036811.2016.1268688 |
| [21] | Liu, W. J.; Kong, X. Y.; Li, G., Asymptotic stability for a laminated beam with structural damping and infinite memory, Math Mech Solids, 25, 10, 1979-2004 (2020) · Zbl 07259267 · doi:10.1177/1081286520917440 |
| [22] | Liu, W. J.; Kong, X. Y.; Li, G., Lack of exponential decay for a laminated beam with structural damping and second sound, Ann Polon Math, 124, 3, 281-290 (2020) · Zbl 1445.35065 · doi:10.4064/ap181224-17-9 |
| [23] | Liu, W. J.; Luan, Y.; Liu, Y. D.; Li, G., Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III, Math Methods Appl Sci, 43, 6, 3148-3166 (2020) · Zbl 1446.35023 · doi:10.1002/mma.6108 |
| [24] | Liu, W. J.; Zhao, W. F., Stabilization of a thermoelastic laminated beam with past history, Appl Math Optim, 80, 1, 103-133 (2019) · Zbl 1418.35343 · doi:10.1007/s00245-017-9460-y |
| [25] | Liu, W. J.; Zhao, W. F., On the stability of a laminated beam with structural damping and Gurtin-Pipkin thermal law, Nonlinear Anal Model Control, 26, 3, 396-418 (2021) · Zbl 1466.35091 · doi:10.15388/namc.2021.26.23051 |
| [26] | Messaoudi, S. A.; Fareh, A., Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds, Arab J Math (Springer), 2, 2, 199-207 (2013) · Zbl 1272.35038 · doi:10.1007/s40065-012-0061-y |
| [27] | Muñoz Rivera, J. E.; Racke, R., Mildly dissipative nonlinear Timoshenko systems—global existence and exponential stability, J Math Anal Appl, 276, 1, 248-278 (2002) · Zbl 1106.35333 · doi:10.1016/S0022-247X(02)00436-5 |
| [28] | Muñoz Rivera, J. E.; Racke, R., Global stability for damped Timoshenko systems, Discrete Contin Dyn Syst, 9, 6, 1625-1639 (2003) · Zbl 1047.35023 · doi:10.3934/dcds.2003.9.1625 |
| [29] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), New York: Springer, New York · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 |
| [30] | Pruss, J., On the spectrum of Co-semigroups, Trans Amer Math Soc, 284, 2, 847-857 (1984) · Zbl 0572.47030 · doi:10.2307/1999112 |
| [31] | Qin, Y.; Yang, X. G.; Ma, Z., Global existence of solutions for the thermoelastic Bresse system, Commun Pure Appl Anal, 13, 4, 1395-1406 (2014) · Zbl 1292.76010 · doi:10.3934/cpaa.2014.13.1395 |
| [32] | Raposo, C. A., Exponential stability for a structure with interfacial slip and frictional damping, Appl Math Lett, 53, 85-91 (2016) · Zbl 1332.35042 · doi:10.1016/j.aml.2015.10.005 |
| [33] | Tahamtani, F.; Peyravi, A., Asymptotic behavior and blow-up of solution for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J Math, 17, 6, 1921-1943 (2013) · Zbl 1286.35152 · doi:10.11650/tjm.17.2013.3034 |
| [34] | Wang D H, Liu W J. Lack of exponential decay for a thermoelastic laminated beam under Cattaneo’s law of heat conduction. Ric Mat, 2020, doi:10.1007/s11587-020-00527-3 |
| [35] | Wang, J. M.; Xu, G. Q.; Yung, S. P., Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J Control Optim, 44, 5, 1575-1597 (2005) · Zbl 1132.93021 · doi:10.1137/040610003 |
| [36] | Xu, L. P.; Luo, J. W., Global attractiveness and exponential decay of neutral stochastic functional differential equations driven by fBm with Hurst parameter less than 1/2, Front Math China, 13, 6, 1469-1487 (2018) · Zbl 1409.60100 · doi:10.1007/s11464-018-0728-6 |
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