Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the exponential stability of linear viscoelasticity and thermoviscoelasticity

Authors: Zhuangyi Liu and Songmu Zheng
Journal: Quart. Appl. Math. 54 (1996), 21-31
MSC: Primary 73F15; Secondary 35B35, 35Q72, 73B30
DOI: https://doi.org/10.1090/qam/1373836
MathSciNet review: MR1373836
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The exponential stability of the semigroup associated with one-dimensional linear viscoelastic and thermoviscoelastic equations with several types of boundary conditions is proved for a class of kernel functions, including the weakly singular kernels.

References [Enhancements On Off] (What's this?)

  • [BF] J. A. Burns and R. H. Fabiano, Feedback control of a hyperbolic partial differential equation with viscoelastic damping, Control Theory and Advanced Technology, Vol. 5, No. 2, 1989, pp. 157-188
  • [BLZ] J. A. Burns, Z. Liu, and S. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl. (to appear)
  • [D] W. A. Day, The decay of energy in a viscoelastic body, Mathematika, Vol. 27, 1980, pp. 268-286
  • [Da1] C. M. Dafermos, On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity, Arch. Rat. Mech. Anal. 29, 241-271 (1968)
  • [Da2] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal. 37, 297-308 (1970)
  • [Da3] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554-569 (1970)
  • [DM1] W. Desch and R. K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert space, J. Differential Equations 70, 366-389 (1987)
  • [DM2] W. Desch and R. K. Miller, Exponential stabilization of Volterra integral equations with singular kernels, J. Integral Equations Appl. 1, 397-433 (1988)
  • [FI1] R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations in linear viscoelasticity, SIAM J. Math. Anal. 21 (2), 374-393 (1990)
  • [FI2] R. H. Fabiano and K. Ito, An approximation framework for equations in linear viscoelasticity with strongly singular kernels, Quart. Appl. Math. 52, 65-81 (1994)
  • [FL] M. Fabrizio and B. Lazzari, On the existence and asymptotic stability of solutions for linearly viscoelastic solids, Arch. Rat. Mech. Anal. 116, 139-152 (1991)
  • [GRT] J. S. Gibson, I. G. Rosen, and G. Tao, Approximation in control of thermoelastic systems, SIAM J. Control and Optimization 30 5, 1163-1189 (1992)
  • [H] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl. 167, 429-442 (1992)
  • [Hu] F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations 1 (1), 43-56 (1985)
  • [HRW] K. B. Hannsgen, Y. Renardy, and R. L. Wheeler, Effectiveness and robustness with respect to time delays of boundary feedback stabilization in one-dimensional viscoelasticity, SIAM J. Control Optim. 26 5, 1200-1233 (1988)
  • [HW1] K. B. Hannsgen and R. L. Wheeler, Viscoelastic and boundary feedback damping: Precise energy decay rates when creep modes are dominant, J. Integral Equations Appl. 2, 495-527 (1990)
  • [HW2] K. B. Hannsgen and R. L. Wheeler, Moment conditions for a Volterra integral equation in a Banach space, Proceedings of Delay Differential Equations and Dynamical Systems, Claremont, Vol. 1475 of Springer Lecture Notes in Mathematics, Springer-Verlag, 1991, pp. 204-209
  • [J] S. Jiang, Global solution of the Neumann problem in one-dimensional nonlinear thermoelasticity, preprint, 1991
  • [K] J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (4), 889-899 (1992)
  • [L] Z. Liu, Approximation and control of a thermoviscoelastic system, Ph.D. dissertation, Virginia Polytechnic Institute and State University, Department of Mathematics, August, 1989
  • [La] J. Lagnese, Boundary Stabilization of Thin Plates, Vol. 10 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1989
  • [Le1] G. Leugering, On boundary feedback stabilization of a viscoelastic membrane, Dynamics and Stability of Systems 4 (1), 71-79 (1989)
  • [Le2] G. Leugering, On boundary feedback stabilizability of a viscoelastic beam, Proc. Roy. Soc. Edinburgh Sect. A 114 (1), 57-69 (1990)
  • [LZ] Z. Liu and S. Zheng, Exponential stability of the semigroup associated with a thermoelastic system, Quart. Appl. Math. 51, 535-545 (1993)
  • [Na] C. B. Navaro, Asymptotic stability in linear thermoviscoelasticity, J. Math. Appl. 65, 399-431 (1978)
  • [Pa] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential equations, Springer, New York, 1983
  • [R] J. E. M. Rivera, Energy decay rate in linear thermoelasticity, Funkcial Ekvac. 35, 19-30 (1992)
  • [S] M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch Rat. Mech. Anal. 76, 97-133 (1981)
  • [Sh] Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelasticity, preprint, 1991

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73F15, 35B35, 35Q72, 73B30

Retrieve articles in all journals with MSC: 73F15, 35B35, 35Q72, 73B30

Additional Information

DOI: https://doi.org/10.1090/qam/1373836
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society