Asymptotic behaviour in linear viscoelasticity
Author:
Jaime E. Muñoz Rivera
Journal:
Quart. Appl. Math. 52 (1994), 628-648
MSC:
Primary 73F15; Secondary 35B40, 35Q72, 45K05
DOI:
https://doi.org/10.1090/qam/1306041
MathSciNet review:
MR1306041
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Abstract: We study the asymptotic behaviour of the solution of the viscoelastic equation, and we prove for a bounded domain that the energy associated to this system approaches zero exponentially as time goes to infinity. Moreover, for the whole space ${\mathbb {R}^n}$ we will prove that the displacement vector field can be decomposed into two parts, solenoidal and irrotational, whose corresponding energies decay to zero uniformly as time goes to infinity with rates that depend on the regularity of the initial data.
G. Creus, Viscoelasticity basic theory and applications to concrete structures, (C. A. Brebbia and S. A. Orszag, eds.), Lecture Notes in Engineering, vol. 16, Springer-Verlag, New York, 1986
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1970)
C. M. Dafermos, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations (3), 178–219 (1979)
G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized 3-D viscoelasticity, Quart. Appl. Math. 48, 715–730 (1990)
J. L. Lions, Quelques méthodes de résolution de problèmes aux limites non lineares, Dunod Gauthier Villars, Paris, 1969
G. Creus, Viscoelasticity basic theory and applications to concrete structures, (C. A. Brebbia and S. A. Orszag, eds.), Lecture Notes in Engineering, vol. 16, Springer-Verlag, New York, 1986
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554–569 (1970)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1970)
C. M. Dafermos, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations (3), 178–219 (1979)
G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized 3-D viscoelasticity, Quart. Appl. Math. 48, 715–730 (1990)
J. L. Lions, Quelques méthodes de résolution de problèmes aux limites non lineares, Dunod Gauthier Villars, Paris, 1969
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Article copyright:
© Copyright 1994
American Mathematical Society