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Asymptotic behaviour of weak solutions to a boundary value problem for dynamic viscoelastic equations with memory

Published online by Cambridge University Press:  14 November 2011

Jin Liang  and
Qin Tiehu
Jin Liang
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14/4AS, Scotland, U.K.
Qin Tiehu
Affiliation:
Department of Mathematics, Fudan University, Shanghai 200433, P.R. of China
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Abstract

The paper discusses the asymptotic behaviour of weak solutions u(t, x), as t → ∞, to the boundary value problem for one-dimensional viscoelastic equations with singular memory. The changes of phase are admitted for the problem. One of our results is that ut(t, ·)⇀0 weakly in L2(0,1) as t → ∞.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 1 , 1995 , pp. 153 - 164
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Andrews, G. and Ball, J. M.. Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. J. Differential Equations 44 (1982), 306–341.CrossRefGoogle Scholar
2Ball, J. M. and James, R.. Fine phase mixture as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987), 13–52.CrossRefGoogle Scholar
3Dafermos, C. M. and Nohel, J. A.. A nonlinear hyperbolic Volterra equation in viscoelasticity. Amer. J. Math. Suppl. (1981), 87–116.Google Scholar
4Engler, H.. Weak solutions of a class of quasilinear hyperbolic integrodifferential equations describing viscoelastic materials. Arch. Rational Mech. Anal. 113 (1991), 1–38.CrossRefGoogle Scholar
5Ericken, J. L.. Equilibrium of bars. J. Elasticity 5 (1975), 191–201.CrossRefGoogle Scholar
6Gurtin, M. E. and Hrusa, W. J.. On energies for nonlinear viscoelastic materials of single-integral type. Quart. Appl. Math. 46 (1988), 381–392.CrossRefGoogle Scholar
7Hrusa, W.. A nonlinear functional differential equation in Banach space with applications to materials with fading memory. Arch. Rational Mech. Anal. 84 (1983), 99–137.CrossRefGoogle Scholar
8Hrusa, W. and Renardy, M.. On a class of quasilinear partial integrodifferential equations with singular kernels. J. Differential Equations 64 (1986), 195–220.CrossRefGoogle Scholar
9MacCamy, M. C.. A model for one-dimensional nonlinear viscoelasticity. Quart. Appl. Math. 35 (1977), 21–33.CrossRefGoogle Scholar
10Pego, R. L.. Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Rational Mech. Anal. 97 (1987), 353–394.CrossRefGoogle Scholar
11Tiehu, Qin. Global existence of weak solution to the boundary value problem for three-dimensional viscoelastic dynamic system (to appear).Google Scholar
12Renardy, M., Hrusa, W. J. and Nohel, J. A.. Mathematical problems in viscoelasticity (Harlow: Longman Scientific and Technical and New York: John Wiley, 1987).Google Scholar