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Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity. (English) Zbl 0786.73009
The paper deals with the existence and uniqueness of global solutions of the initial-boundary value problem (Neumann problem with stress-free, insulated boundaries) of one-dimensional coupled thermoelasticity. The constitutive relations are somewhat simplified by assuming that heat flux depends only on the temperature gradient. The Neumann problem is considered with stress free, insulated boundaries. The initial conditions on displacement gradient, velocity and temperature are supposed to be given. Based on some assumptions concerning constitutive relations and initial data, a global existence and uniqueness theorem is proved by carefully and deftly evaluating certain norm estimates in appropriate Sobolev spaces. Decay rate of solutions to the linearized problem is also calculated.

74A15 Thermodynamics in solid mechanics
74B99 Elastic materials
35Q72 Other PDE from mechanics (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
[1] Adams, R. A.: Sobolev spaces. (1975) · Zbl 0314.46030
[2] Carlson, D. E.: Linear thermoelasticity. Handbuch der physik vla/2, 297-346 (1972)
[3] Chrze\ogonek szczyk, A.: Some existence results in dynamical thermoelasticity. Part I. Nonlinear case. Arch. mech. 39, 605-617 (1987) · Zbl 0682.73004
[4] Dafermos, C. M.; Hsiao, L.: Development of singularities in solutions of the equations of nonlinear thermoelasticity. Q. appl. Math. 44, 463-474 (1986) · Zbl 0661.35009
[5] Hrusa, W. J.; Messaoudi, S. A.: On formulation of singularities in one-dimensional nonlinear thermoelasticity. Archs ration. Mech. analysis 111, 135-151 (1990) · Zbl 0712.73023
[6] Hrusa, W. J.; Tarabek, M. A.: On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelesticity. Q. appl. Math. 47, 631-644 (1989) · Zbl 0692.73005
[7] Jiang, S.: Global existence of smooth solutions in one-dimensional nonlinear thermoelasticity. Proc. R. Soc. edinb. 115A, 257-274 (1990) · Zbl 0723.35044
[8] Jiang, S.: Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity. (1990) · Zbl 0723.35044
[9] Jiang, S.; Racke, R.: On some quasilinear hyperbolic-parabolic initial boundary value problems. Math. meth. Appl. sci. 12, 315-339 (1990) · Zbl 0706.35098
[10] Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magneto-hydrodynamics. Thesis (1983)
[11] Racke, R.: On the Cauchy problem in nonlinear 3-d-thermoelasticity. Math. Z. 203, 649-682 (1990) · Zbl 0701.73002
[12] Racke, R.: Blow-up in nonlinear three-dimensional thermoelasticity. Math. meth. Appl. sci. 12, 267-273 (1990) · Zbl 0705.35081
[13] Racke, R.; Shibata, Y.: Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity. Archs ration. Mech. analysis 116, 1-34 (1991) · Zbl 0756.73012
[14] Rivera^J.E.M., Energy decay rates in linear thermoelasticity, Funkcialaj Ekvacioj (to appear). · Zbl 0838.73006
[15] Shibata, Y.: On a local existence theorem for some quasilinear hyperbolic-parabolic coupled system with Neumann type boundary condition. (1988)
[16] Shibata, Y.: Neumann problem for one-dimensional nonlinear thermoelasticity. (1990)
[17] Slemrod, M.: Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity. Archs ration. Mech. analysis 76, 93-134 (1981) · Zbl 0481.73009
[18] Zheng, S.; Shen, W.: Global solutions to the Cauchy problem of a class of quasilinear hyperbolic parabolic coupled systems. Sci. sin (Ser.A) 30, 1133-1149 (1987) · Zbl 0649.35013
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