The asymptotic behavior of classical solutions to the mixed initial-boundary value problem in finite thermo-viscoelasticity
Authors:
C. E. Beevers and M. Šilhavý
Journal:
Quart. Appl. Math. 46 (1988), 319-329
MSC:
Primary 73F15; Secondary 73G15
DOI:
https://doi.org/10.1090/qam/950605
MathSciNet review:
950605
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Abstract: In this paper we consider the asymptotic stability of a class of solutions to the mixed initial-boundary value problem in nonlinear thermo-viscoelasticity. The continuum model is a viscoelastic material of rate type with the thermal conduction obeying Fourier’s law. The work in this article generalizes in two ways the results obtained by the present authors in a previous paper [1], The results in this present paper are valid for nonisothermal conditions and for a genuinely nonlinear viscous stress.
- C. E. Beevers and M. Šilhavý, Asymptotic stability in nonlinear viscoelasticity, Quart. Appl. Math. 42 (1984), no. 3, 281–294. MR 757166, DOI https://doi.org/10.1090/S0033-569X-1984-0757166-9
- Constantine M. Dafermos, Can dissipation prevent the breaking of waves?, Transactions of the Twenty-Sixth Conference of Army Mathematicians (Hanover, N.H., 1980) ARO Rep. 81, vol. 1, U. S. Army Res. Office, Research Triangle Park, N.C., 1981, pp. 187–198. MR 605324
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- C. E. Beevers and R. E. Craine, Asymptotic stability in nonlinear elastic materials with dissipation, J. Elasticity 19 (1988), no. 2, 101–110. MR 937625, DOI https://doi.org/10.1007/BF00040889
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- M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 76 (1981), no. 2, 97–133. MR 629700, DOI https://doi.org/10.1007/BF00251248
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C. E. Beevers and M.Šilhavý, Asymptotic stability in nonlinear viscoelasticity, Quart. Appl. Math. 42, 281–294 (1984)
C. M. Dafermos, Can dissipation prevent the breaking of waves?, Trans. of the 26th Conf. of U. S. Army Math., 187–198 (1981)
F. Bloom, On the damped nonlinear evolution equation ${w_{tt}} = \sigma {\left ( w \right )_{xx}} - y{w_t}$, J. Math. Anal. and Applics. 96, 551–583 (1983)
M. Slemrod, Damped conservation laws in continuum mechanics, Heriot-Watt Symposium Vol. III, ed. R. J. Knops, Research Notes in Mathematics, Vol. 3, Pitman (1979)
C. E. Beevers and R. E. Craine, Asymptotic stability in nonlinear elastic materials with dissipation, J. Elasticity (to appear)
C. M. Dafermos, The mixed initial-boundary value problem for the equation of nonlinear one-dimensional viscoelasticity, J. Diff. Eq. 6, 71–86 (1969)
H. Hattori, Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations, Quart. Appl. Math. 40, 113–127 (1982)
M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional thermoelasticity, Arch. Rat. Mech. Anal. 76, 97–133 (1981)
J. L. Ericksen, Internat. J. Solids Struct. 2, 573 (1966)
M. E. Gurtin, Proc. IUTAM Symp. at Lehigh University USA, edited by D. F. Carlson and R. T. Shield, Martinus Nijhoff, The Hague-Boston-London (1982)
M. Arons and R. E. Craine, The continuous dependence of solutions on data in finite thermoelastostatics, Internat. J. Engrg. Sci. (1985), to appear
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© Copyright 1988
American Mathematical Society