Well posedness and stability result for a thermoelastic laminated beam with structural damping

Ricerche di Matematica - Trang 1-25 - 2022
Djellali Fayssal1
1Mathematical Modeling and Numerical Simulation Laboratory, Badji-Mokhtar University, Annaba, Algeria

Tóm tắt

The main object of the present paper is the study of thermoelastic laminated beam with structural damping, where the heat conduction is given by the classical Fourier’s law and acting on both the rotation angle and the transverse displacements. We establish an exponential stability result for the considered problem in case of equal wave speeds. In the opposite case, we show the lack of exponential stability. Furthermore, a polynomial stability result is established.

Tài liệu tham khảo

Almeida Júnior, D.S., 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) Alves, M.S., Monteiro, R.N.: Exponential stability of laminated Timoshenko beams with boundary/internal controls. J. Math. Anal. Appl. 482(1), 1–16 (2020) Apalara, T.A.: Uniform stability of a laminated beam with structural damping and second sound. Z. Angew. Math. Phys. 68(2), 1–16 (2017) Apalara, T.A.: On the stability of a thermoelastic laminated beam. Acta Math. Scientia. 39(6), 1517–1524 (2019) Apalara, T.A., Nass, A.M., Al Sulaimani, H.: On a laminated timoshenko beam with nonlinear structural damping. Math. Comput. Appl. 25(2), 35 (2020) Apalara, T.A., Raposo, C.A., Nonato, C.A.S.: Exponential stability for laminated beams with a frictional damping. Archiv der Mathematik. 114(4), 471–480 (2020) Apalara, T.A.: Exponential stability of laminated beams with interfacial slip. Mechanics of Solids. 56(1), 131–137 (2021) Enyi, C.D., Mukiawa, S.E.: Dynamics of a thermoelastic-laminated beam problem. AIMS Mathematics. 5(5), 5261–5286 (2020) Feng, B.: Well-posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks. Math. Meth. Appl. Sci. 41(3), 1162–1174 (2018) Feng, B., Ma, T.F., Monteiro, R.N., Raposo, C.A.: Dynamics of laminated timoshenko beams. J. Dyn. Diff. Equat. 30(4), 1489–1507 (2018) Feng, B., Soufyane, A.: Memory-type boundary control of a laminated timoshenko beam. Math. Mech. Solids. 25(8), 1568–1588 (2020) Feng, B., Almeida Júnior, D..S., Ramos, A.J.A.: Exponential stabilization of laminated beams with history memories. Mathematische Nachrichten. 294(3), 559–579 (2021) Feng, B.: On a thermoelastic laminated timoshenko beam: well posedness and stability. Complexity. 1–13 (2021). https://doi.org/10.1155/2020/5139419 Gearhart, L.: Spectral theory for contraction semigroups on Hilbert space. Trans. Am. Math. Soc. 236, 385–394 (1978) Hansen, S.W., Spies, R.D.: Structural damping in laminated beams due to interfacial slip. J. sound vibration. 204(2), 183–202 (1997) Huang, F.: Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Diff. Eqns. 1(1), 43–56 (1985) Liu, W., Zhao, W.: Stabilization of a thermoelastic laminated beam with past history. Appl. Math. Optim. 80(1), 103–133 (2019) Liu, W., Zhao, W.: Exponential and polynomial decay for a laminated beam with Fourier’s law of heat conduction and possible absence of structural damping. Front. Math. China. 16(4), 997–1021 (2021) Liu, W., Zhao, W.: On the stability of a laminated beam with structural damping and Gurtin-Pipkin thermal law. Nonlinear Analysis: Modelling and Control. 26(3), 396–418 (2021) Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Chapman Hall/CRC, Boca, Raton (1999) Lo, A., Tatar, N.E.: Stabilization of laminated beams with interfacial slip. Electron. J. Differ. Eqns. 2015(129), 1–14 (2015) Lo, A., Tatar, N.E.: Uniform stability of a laminated beam with structural memory. Qual. Theory Dyn. Syst. 15(2), 517–540 (2016) Lo, A., Tatar, N.E.: Exponential stabilization of a structure with interfacial slip. Discrete Contin. Dyn. Syst. 36(11), 6285–6306 (2016) Mukiawa, S.E., Apalara, T.A., Messaoudi, S.A.: A stability result for a memory-type laminated-thermoelastic system with Maxwell-Cattaneo heat conduction. J. Thermal Stresses. 43(11), 1437–1466 (2020) Mukiawa, S.E., Apalara, T.A., Messaoudi, S.A.: Stability rate of a thermoelastic laminated beam: Case of equal-wave speed and nonequal-wave speed of propagation. AIMS Mathematics. 6(1), 333–361 (2021) 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) Mustafa, M.I.: Laminated timoshenko beams with viscoelastic damping. J. Math. Anal. Appl. 466(1), 619–641 (2018) Mustafa, M.I.: On the stabilization of viscoelastic laminated beams with interfacial slip. Z. Angew. Math. Phys. 69(2), 1–14 (2018) Mustafa, M.I.: Boundary control of laminated beams with interfacial slip. J. Math. Phys. 59(5), 1–9 (2018) Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983) Prüss, J.: On the spectrum of \(C_{0}\)-semigroups. Trans. Am. Math. Soc. 284(2), 847–857 (1984) Raposo, C.A.: Exponential stability for a structure with interfacial slip and frictional damping. Appl. Math. Lett. 53, 85–91 (2016) Tatar, N.E.: Stabilization of a laminated beam with interfacial slip by boundary controls. Bound. Value Probl. 2015, 1–11 (2015) 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)