Energy decay for hyperbolic thermoelastic systems of memory type
Authors:
Luci Harue Fatori and Jaime E. Muñoz Rivera
Journal:
Quart. Appl. Math. 59 (2001), 441-458
MSC:
Primary 74F05; Secondary 35B35, 35B40, 35L20, 35Q72, 74H40
DOI:
https://doi.org/10.1090/qam/1848527
MathSciNet review:
MR1848527
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Abstract: In this paper we study the hyperbolic thermoelastic system, which is obtained when, instead of Fourier’s law for the heat flux relation, we follow the linearized model proposed by Gurtin and Pipkin concerning the memory theory of heat conduction. In this case the thermoelastic model is fully hyperbolic. We show that the linear system is well posed and that the solution decays exponentially to zero as time goes to infinity.
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- Constantine M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7 (1970), 554–569. MR 259670, DOI https://doi.org/10.1016/0022-0396%2870%2990101-4
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J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969
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- Jaime E. Muñoz Rivera, Global smooth solutions for the Cauchy problem in nonlinear viscoelasticity, Differential Integral Equations 7 (1994), no. 1, 257–273. MR 1250950
J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math. 52, 629–648 (1994)
- Jaime E. Muñoz Rivera and Eugenio Cabanillas Lapa, Decay rates of solutions of an anisotropic inhomogeneous $n$-dimensional viscoelastic equation with polynomially decaying kernels, Comm. Math. Phys. 177 (1996), no. 3, 583–602. MR 1385077
- J. E. Muñoz Rivera, E. C. Lapa, and R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity 44 (1996), no. 1, 61–87. MR 1417809, DOI https://doi.org/10.1007/BF00042192
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B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18, 199–208 (1967)
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, Journal of Differential Equations 7, 554–569 (1970)
G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized 3-D viscoelasticity, Quart. Appl. Math. 48, 715–730 (1990)
M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rational Mech. Anal. 116, 139–152 (1991)
C. Giorgi and M. G. Naso, On the exponential stability of linear non-Fourier thermoviscoelastic bar, qquaderni del Seminario di Brescia 2/97 (1997)
M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 113–126 (1968)
J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, International Series of Numerical Math. 91, 211–235 (1989)
Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math. 54, 21–31 (1996)
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969
J. E. Muñoz Rivera, Asymptotic behaviour of energy in linear thermoviscoelasticity, Computational and Appl. Math. 11, 45–71 (1992)
J. E. Muñoz Rivera, Global smooth solutions for the Cauchy problem in nonlinear viscoelasticity, Differential Integral Equations 7, 257–273 (1994)
J. E. Muñoz Rivera, Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math. 52, 629–648 (1994)
J. E. Muñoz Rivera and E. Cabanillas, Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomial decaying kernels, Comm. Math. Physics 177, 583–602 (1996)
J. E. Muñoz Rivera, E. Cabanillas, and R. Barreto, Decay rates for viscoelastic plates with memory, Journal of Elasticity 44, 61–87 (1996)
O. J. Staffans, On a nonlinear hyperbolic Volterra Equation, Siam J. Math. Anal. 11, 793–812 (1980)
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© Copyright 2001
American Mathematical Society