Global existence for the three-dimensional thermoelastic equations of Type II
Authors:
Yuming Qin, Shuxian Deng, Lan Huang, Zhiyong Ma and Xiaoke Su
Journal:
Quart. Appl. Math. 68 (2010), 333-348
MSC (2000):
Primary 35B35, 35M13, 35D30
DOI:
https://doi.org/10.1090/S0033-569X-10-01188-1
Published electronically:
February 19, 2010
MathSciNet review:
2663003
Full-text PDF Free Access
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Abstract: In this paper, we shall establish some global existence results for a 3D hyperbolic system arising from Green-Naghdi models of thermoelasticity of type II with a dissipative boundary condition for the displacement. The existence and exponential decay of energy for the linear problem has been solved by Lazzari and Nibbi, Journal of Mathematical Analysis and Applications, 338 (2008), 317–329. Furthermore, we shall establish the global existence of solutions to semilinear and nonlinear thermoelastic systems by using the semigroup approach.
References
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References
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- C. A. Bosello, B. Lazzari and R. Nibbi, A viscous boundary condition with memory in linear elasticity, Internat. J. Engrg. Sci. 45(2007), 94-110. MR 2314588 (2008e:74010)
- D. S. Chandrasekharaiah, A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation, J. Elasticity 43(1996), 279-283. MR 1415546 (97e:73009)
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- M. Fabrizio and A. Morro, A boundary condition with memory in electromagnetism, Arch. Rational Mech. Anal. 136(1996), 359-381. MR 1423012 (97k:78012)
- H. Gao and J. E. Muñoz Rivera, On the exponential stability of thermoelastic problem with memory, Appl. Anal. 78(2001), 379-403. MR 1883541 (2002j:74034)
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- B. Lazzari and R. Nibbi, On the energy decay of a linear hyperbolic thermoelastic system with dissipative boundary, J. Thermal Stresses 30(2007), 1-14.
- B. Lazzari and R. Nibbi, On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary, J. Math. Anal. Appl. 338(2008), 317-329. MR 2386418 (2009e:35281)
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- J. E. Muñoz Rivera and Y. Qin, Global existence and exponential stability in one-dimensional nonlinear thermoelasticity with thermal memory, Nonlinear Analysis 51(2002), 11-32. MR 1915739 (2003d:35254)
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- G. Propst and J. Prüss, On wave equations with boundary dissipation of memory type, J. Integral Equations Appl. 8(1996), 99-123. MR 1391147 (97d:35122)
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- Y. Qin and J. E. Muñoz Rivera, Global existence and exponential stability of solutions of thermoelastic equations of hyperbolic type, Journal of Elasticity 75(2004), 125-145. MR 2110169 (2005j:35216)
- Y. Qin, L. Xu and Z. Ma, Global existence and exponential stability for a thermoelastic equation of type II, J. Zhenzhou Univ. (Natural Science Edition) 40(2008), 1-11.
- R. Quintanilla, Existence in thermoelasticity without energy dissipation, J Thermal Stresses 25(2002), 195-202. MR 1883591 (2003b:74020)
- R. Quintanilla, Some remarks on growth and uniqueness in thermoelasticity, Int. J. Math. Sci. (2003), 617-623. MR 1969547 (2004b:74020)
- R. Quintanilla and R. Racke, Stability in thermoelasticity of type III, Discrete Contin. Dynamical Systems Ser. B 3(2003), 383-400. MR 1974153 (2004d:74019)
- R. Racke, Y. Shibata and S. Zheng, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math. 4(1993), 751-763. MR 1247439 (94j:35177)
- M. Reissing and Y. Wang, Cauchy problems for linear thermoelastic systems of type III in one space variable, Math. Methods Appl. Sci. 28(2005), 1359-1381. MR 2150160 (2006f:35280)
- L. Yang and Y. Wang, Well-posedness and decay estimates for Cauchy problems of linear thermoelastic systems of type III in 3-D, Indiana Univ. Math. J. 55(2006), 1333-1362. MR 2269415 (2007h:35335)
- X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Comm. Contemp. Math. 5(2003), 25-83. MR 1958019 (2004b:74023)
- S. Zheng, Nonlinear Evolution Equations, Monographs and Surveys in Pure and Applied Mathematics, 133, CRC Press, Boca Raton, FL, 2004. MR 2088362 (2006a:35001)
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Additional Information
Yuming Qin
Affiliation:
Department of Applied Mathematics, Donghua University, Shanghai 201620, People’s Republic of China
Email:
yuming_qin@hotmail.com
Shuxian Deng
Affiliation:
College of Information Science and Technology, Donghua University, Shanghai 201620, People’s Republic of China
Email:
dshuxian@163.com
Lan Huang
Affiliation:
College of Information Science and Technology, Donghua University, Shanghai 201620, People’s Republic of China
Email:
huanglan82@hotmail.com
Zhiyong Ma
Affiliation:
College of Information Science and Technology, Donghua University, Shanghai 201620, People’s Republic of China
Email:
mazhiyong1980@hotmail.com
Xiaoke Su
Affiliation:
College of Information Science and Technology, Donghua University, Shanghai 201620, People’s Republic of China
Email:
suxiaoke07@126.com
Keywords:
Thermoelastic equations of type II,
global existence,
semigroup approach.
Received by editor(s):
April 29, 2008
Published electronically:
February 19, 2010
Article copyright:
© Copyright 2010
Brown University