Abstract
We consider the thermoelastic model following the type III theory for the Euler Bernoulli beam equation with tip. We prove that the corresponding semigroup is analytic. In particular, this implies: the smoothing effect over the initial data, the exponential stability of the semigroup and that the rate of decay of the semigroup is equal to the spectral bound of its generator (linear stability property).
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References
Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253–264 (1992)
Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)
Green, A.E., Naghdi, P.M.: A unified procedure for construction of theories of deformable media. I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua. Proc. R. Soc. Lond. A 448, 335–356 (1995). 357–377, 379–388
Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Liu, K., Liu, Z.: Boundary stabilization of a nonhomogeneous beam with rotatory inertia at the tip. J. Comput. Appl. Math. 114(1), 1–10 (2000). https://doi.org/10.1016/S0377-0427(99)00284-8
Liu, K., Liu, Z.: Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voight damping. SIAM J. Control Optim. 36(3), 1086–1098 (1998)
Liu, K., Chen, S., Liu, Z.: Spectrum and stability for elastic systems with global or local Kelvin-Voight damping. SIAM J. Appl. Math. 59(2), 651–668 (1998)
Quintanilla, R., Racke, R.: Stability in thermoelasticity of type III. Discrete Contin. Dyn. Syst., Ser. B 3, 383–400 (2003)
Reissig, M., Wang, Y.G.: Cauchy problems for linear thermoelastic systems of type III in one space variable. Math. Methods Appl. Sci. 28, 1359–1381 (2005)
Andrews, K.T., Shillor, M.: Vibrations of a beam with a damping tip body math. Math. Comput. Model. 35, 1033–1042 (2002)
Andrews, K.T., Fernández, J.R., Shillor, M.: A thermoviscoelastic beam with a tip body. Comput. Mech. 33, 225–234 (2004)
Yang, L., Wang, Y.G.: Well-posedness and decay estimates for cauchy problems of linear thermoelastic systems of type III in 3-D. Indiana Univ. Math. J. 55, 1333–1361 (2006)
Zhang, X., Zuazua, E.: Decay rates for the system of thermoelasticity of type III. Commun. Contemp. Math. 5, 25–83 (2003)
Acknowledgements
The authors would like to thank to CNPq project 310249/2018-0 for the financial support. To the Research project UBB 2020108 IF/I
Elena Ochoa was supported by CONICYT-PFCHA/doctorado nacional/2020-21200268.
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Ávalos, G.G., Muñoz Rivera, J. & Ochoa Ochoa, E. Analiticity of the Type III Thermoelastic Euler Bernoulli Model with Tip. Acta Appl Math 182, 9 (2022). https://doi.org/10.1007/s10440-022-00543-5
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DOI: https://doi.org/10.1007/s10440-022-00543-5