Gevrey class for locally thermoelastic beam equations
Tóm tắt
In this article, we use the Euler–Bernoulli model to study the vibrations of a beam composed of two components, one consisting of a thermoelastic material and the other of a simply elastic material that does not produce dissipation. Our main result is that the semigroup associated with this model is differentiable. In particular, our proof implies the following properties of the semigroup (1) It is of Gevrey class 12. (2) It is exponentially stable. (3) It possesses the property of linear stability and has a regularizing effect on the initial data.
Tài liệu tham khảo
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Belinskiy, B., Lasiecka, I.: Gevrey’s and trace regularity of a semigroup associated with beam equation and non-monotone boundary conditions. J. Math. Anal. Appl. (2007). https://doi.org/10.1016/j.jmaa.2006.10.025
Crandall, M.G., Pazy, A.: On the differentiability of weak solutions of a differential equation in Banach space. J. Math. Mech. 18, 1007–1016 (1969)
Gao, H.J., Zhao, Y.J.: Asymptotic behaviour and exponential stability for thermoelastic problem with localized damping. Appl. Math. Mech. 27, 1557–1568 (2006). https://doi.org/10.1007/s10483-006-1114-1
Huang, F.: On the mathematical model for linear elastic systems with analytic damping. SIAM J. Control Optim. 26(3), 714–724 (1988). https://doi.org/10.1137/0326041
Lasiecka, I., Triggiani, R.: Analyticity of thermo-elastic semigroups with free boundary conditions. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze Série 4 Tome 27(3–4), 457–482 (1998)
Liu, K., Liu, Z.: Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping. SIAM J. Control Optim. 36(3), 1086–1098 (1998). https://doi.org/10.1137/S0363012996310703
Liu, Z.-Y., Renardy, M.: A note on the equations of a thermoelastic plate. Appl. Math. Lett. 8, 1–6 (1995). https://doi.org/10.1016/0893-9659(95)00020-Q
Liu, K., Liu, Z.: Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces. J. Differ. Equ. 141, 340–355 (1997). https://doi.org/10.1006/jdeq.1997.3331
Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Rivera, J.E.M., Oquendo, H.P.: The transmission problem for thermoelastic beams. J. Therm. Stresses 24, 1137–1158 (2001)
Rivera, J.E.M., Oquendo, H.P.: A transmission problem for thermoelastic plates. Q. Appl. Math. LXII, 273–293 (2004). https://doi.org/10.1090/qam/2054600
Sare, H.F., Rivera, J.: Analyticity of transmission problem to thermoelastic plates. Q. Appl. Math. 69, 1–13 (2011). https://doi.org/10.1090/S0033-569X-2010-01187-6
Taylor, S.: Ph.D. thesis, Chapter “Gevrey Semigroups”. School of Mathematics, University of Minnesota (1989)
Vila Bravo, J.C., Rivera, J.E.M.: The transmission problem to thermoelastic plate of hyperbolic type. IMA J. Appl. Math. 74, 950–962 (2009). https://doi.org/10.1093/imamat/hxp022