Stability of thermoelastic Timoshenko system with variable delay in the internal feedback

Xinfeng Ge; Keqin Su; Xinfeng Ge; Keqin Su

doi:10.3934/era.2024160

Electronic Research Archive

2024, Volume 32, Issue 5: 3457-3476. doi: 10.3934/era.2024160

Previous Article Next Article

Research article Special Issues

Stability of thermoelastic Timoshenko system with variable delay in the internal feedback

Xinfeng Ge ¹,
Keqin Su ^{2
,
,}

1.
School of Electrical and Mechanical Engineering (Engineering Training Centre), Xuchang University, Xuchang 461000, China
2.
College of Information and Management Science, Henan Agricultural University, Zhengzhou 450046, China

Received: 21 February 2024 Revised: 13 May 2024 Accepted: 20 May 2024 Published: 28 May 2024

Based on the Fourier law of heat conduction, this paper was concerned with the thermoelastic Timoshenko system with memory and variable delay in the internal feedback, which describes the transverse vibration of a beam. By the Lummer-Phillips theorem and the variable norm technique suitable for the nonautonomous operator, the stability of the coupled system has been derived in space $ \mathscr{H} $.
- Timoshenko system,
- exponential stability,
- variable delay,
- memory
Citation: Xinfeng Ge, Keqin Su. Stability of thermoelastic Timoshenko system with variable delay in the internal feedback[J]. Electronic Research Archive, 2024, 32(5): 3457-3476. doi: 10.3934/era.2024160

Related Papers:

Abstract

Based on the Fourier law of heat conduction, this paper was concerned with the thermoelastic Timoshenko system with memory and variable delay in the internal feedback, which describes the transverse vibration of a beam. By the Lummer-Phillips theorem and the variable norm technique suitable for the nonautonomous operator, the stability of the coupled system has been derived in space $ \mathscr{H} $.

References

[1]	S. P. Timoshenke, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Mag., 41 (1921), 744–746. https://doi.org/10.1080/14786442108636264 doi: 10.1080/14786442108636264
[2]	J. E. M. Rivera, R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst., 9 (2003), 1625–1639. https://doi.org/10.3934/dcds.2003.9.1625 doi: 10.3934/dcds.2003.9.1625
[3]	J. E. M. Rivera, R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068–1083. https://doi.org/10.1016/j.jmaa.2007.11.012 doi: 10.1016/j.jmaa.2007.11.012
[4]	M. I. Mustafa, S. A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyn. Control Syst., 16 (2010), 211–226. https://doi.org/10.1007/s10883-010-9090-z doi: 10.1007/s10883-010-9090-z
[5]	D. S. A. Jnior, M. L. Santos, J. E. M. Rivera, Stability to 1-D thermoelastic Timoshenko beam acting on shear force, Z. Angew. Math. Phys., 65 (2014), 1233–1249. https://doi.org/10.1007/s00033-013-0387-0 doi: 10.1007/s00033-013-0387-0
[6]	M. O. Alves, A. H. Caixeta, M. A. J. Silva, J. H. Rodrigues, D. S. A. Junior, On a Timoshenko system with thermal coupling on both the bending moment and the shear force, J. Evol. Equations, 20 (2020), 295–320. https://doi.org/10.1007/s00028-019-00522-8 doi: 10.1007/s00028-019-00522-8
[7]	M. O. Alves, E. H. G. Tavares, M. A. J. Silva, J. H. Rodrigues, On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system, SIAM J. Math. Anal., 51 (2019), 4520–4543. https://doi.org/10.1137/18M1191774 doi: 10.1137/18M1191774
[8]	M. S. Alves, M. A. J. Silva, T. F. Ma, J. E. M. Rivera, Invariance of decay rate with respect to boundary conditions in thermoelastic Timoshenko systems, Z. Angew. Math. Phys., 67 (2016), 70. https://doi.org/10.1007/s00033-016-0662-y doi: 10.1007/s00033-016-0662-y
[9]	M. S. Alves, M. A. J. Silva, T. F. Ma, J. E. M. Rivera, Non-homogeneous thermoelastic Timoshenko systems, Bull. Braz. Math. Soc., 48 (2017), 461–484. https://doi.org/10.1007/s00574-017-0030-3 doi: 10.1007/s00574-017-0030-3
[10]	L. H. Fatori, R. N. Monteiro, H. D. F. Sare, The Timoshenko system with history and Cattaneo law, Appl. Math. Comput., 228 (2014), 128–140. https://doi.org/10.1016/j.amc.2013.11.054 doi: 10.1016/j.amc.2013.11.054
[11]	J. E. M. Rivera, R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248–278. https://doi.org/10.1016/S0022-247X(02)00436-5 doi: 10.1016/S0022-247X(02)00436-5
[12]	M. L. Santos, D. S. A. Jnior, J. E. M. Rivera, The stability number of the Timoshenko system with second sound, J. Differ. Equations, 253 (2012), 2715–2733. https://doi.org/10.1016/j.jde.2012.07.012 doi: 10.1016/j.jde.2012.07.012
[13]	H. D. F. Sare, R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221–251. https://doi.org/10.1007/s00205-009-0220-2 doi: 10.1007/s00205-009-0220-2
[14]	M. A. J. Silva, R. Racke, Effect of history and heat models on the staility of thermoelastic Timoshenko systems, J. Differ. Equations, 275 (2021), 167–203. https://doi.org/10.1016/j.jde.2020.11.041 doi: 10.1016/j.jde.2020.11.041
[15]	F. Ammar-Khodja, S. Kerbal, A. Soufyane, Stabilization of the nonuniform Timoshenko beam, J. Math. Anal. Appl., 327 (2007), 525–538. https://doi.org/10.1016/j.jmaa.2006.04.016 doi: 10.1016/j.jmaa.2006.04.016
[16]	H. D. F. Sare, J. E. M. Rivera, Exponential decay of Timoshenko system with indefinite memory dissipation, Adv. Differ. Equations, 13 (2008), 733–752. https://doi.org/10.57262/ade/1355867334 doi: 10.57262/ade/1355867334
[17]	Z. Ma, L. Zhang, X. Yang, Exponential stability for a Timoshenko-type system with history, J. Math. Anal. Appl., 380 (2011), 299–312. https://doi.org/10.1016/j.jmaa.2011.02.078 doi: 10.1016/j.jmaa.2011.02.078
[18]	J. E. M. Rivera, H. D. F. Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482–502. https://doi.org/10.1016/j.jmaa.2007.07.012 doi: 10.1016/j.jmaa.2007.07.012
[19]	T. Caraballo, J. Real, Attractors for 2D Navier-Stokes models with delays, J. Differ. Equations, 205 (2004), 271–297. https://doi.org/10.1016/j.jde.2004.04.012 doi: 10.1016/j.jde.2004.04.012
[20]	J. M. G. Luengo, P. M. Rubio, G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model, Discrete Contin. Dyn. Syst. - Ser. A, 34 (2014), 4085–4105. Available from: http://hdl.handle.net/11441/25923.
[21]	X. Yang, L. Li, X. Yan, L. Ding, The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay, Electron. Res. Arch., 28 (2020), 1395–1418. https://doi.org/10.3934/era.2020074 doi: 10.3934/era.2020074
[22]	B. Feng, M. L. Pelicer, Global existence and exponential stability for a nonlinear Timoshenko system with delay, Boundary Value Probl., 2015 (2015), 206. https://doi.org/10.1186/s13661-015-0468-4 doi: 10.1186/s13661-015-0468-4
[23]	B. Feng, X. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Appl. Anal., 96 (2017), 606–625. https://doi.org/10.1080/00036811.2016.1148139 doi: 10.1080/00036811.2016.1148139
[24]	X. Yang, J. Zhang, Y. Lu, Dynamics of the nonlinear Timoshenko system with variable delay, Appl. Math. Optim., 83 (2021), 297–326. https://doi.org/10.1007/s00245-018-9539-0 doi: 10.1007/s00245-018-9539-0
[25]	Z. Liu, S. Zheng, Semigroups Associated with Dissipative Systems, Chapman and Hall/CRC, 1999.
[26]	T. Kato, Linear and quasi-linear equations of evolution of hyperbolic type, in Hyperbolicity, (2011), 125–191. https://doi.org/10.1007/978-3-642-11105-1_4
[27]	M. Hu, X. Yang, J. Yuan, Stability and dynamics for Lam$\acute{e}$ system with degenerate memory and time-varying delay, Appl. Math. Optim., 89 (2024), 14. https://doi.org/10.1007/s00245-023-10080-8 doi: 10.1007/s00245-023-10080-8

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)