Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping

Authors: Zhuangyi Liu and Songmu Zheng
Journal: Quart. Appl. Math. 55 (1997), 551-564
MSC: Primary 73H10; Secondary 35Q72, 73B30, 73F15, 73K10, 73K50
DOI: https://doi.org/10.1090/qam/1466148
MathSciNet review: MR1466148
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Abstract | References | Similar Articles | Additional Information

Abstract: The exponential stability of the semigroup associated with the Kirchhoff plate with thermal or viscoelastic damping and various boundary conditions is proved. This improves the corresponding results by Lagnese by showing that the semigroup is still exponentially stable even without feedback control on the boundary. The proof is essentially based on PDE techniques and the method is remarkable in the sense that it also throws light on applications to other higher-dimensional problems.

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  • [BLZ] J. A. Burns, Z. Liu, and S. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl. 179, 574-591 (1993) MR 1249839
  • [CFNS] G. Chen, S. A. Fulling, F. J. Narcowich, and S. Sun, Exponential decay of evolution equations with locally distributed damping, SIAM J. Appl. Math. 51, 266-301 (1991) MR 1089141
  • [D1] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal. 37, 297-308 (1970) MR 0281400
  • [D2] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7, 554-569 (1970) MR 0259670
  • [DM] W. Desch and R. K. Miller, Exponential stabilization of Volterra integrodifferential equations in Hilbert space, J. Differential Equations 70, 366-389 (1987) MR 915494
  • [FI] R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations arising in linear viscoelasticity, SIAM J. Math. Anal. 21, 374-393 (1990) MR 1038898
  • [GRT] J. S. Gibson, I. G. Rosen, and G. Tao, Approximation in control of thermoelastic systems, SIAM J. Control. Optim. 30, 1163-1189 (1992) MR 1178657
  • [H] F. L. Huang, Characteristic condition for exponential stability of linear dynamical system in Hilbert spaces, Ann. Differential Equations 1, 43-56 (1985) MR 834231
  • [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, 1200-1233 (1988) MR 957661
  • [K] J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23, 889-899 (1992) MR 1166563
  • [L] J. Lagnese, Boundary Stabilization of Thin Plates, Vol. 10 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1989 MR 1061153
  • [La] I. Lasiecka, Controllability of a viscoelastic Kirchhoff plate, Internat. Ser. Numer. Math. 91, 237-247 (1989) MR 1033062
  • [LM] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer, Heidelberg, 1972
  • [LZ1] Z. Liu and S. Zheng, Exponential stability of semigroup associated with thermoelastic system, Quart. Appl. Math. LI, 535-545 (1993) MR 1233528
  • [LZ2] Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math. LIV, 21-31 (1996) MR 1373836
  • [Pa] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983 MR 710486
  • [RR] J. E. M. Rivera and R. Racke, Smoothing properties, decay and global existence of solutions to nonlinear coupled systems of thermoelastic type, SFB series No. 287, Bonn University, 1993
  • [Re] K. Rektorys, Variational Methods in Mathematics, Science and Engineering, D. Reidel Publishing Company, Holland, 1977 MR 0487653
  • [S] M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 76, 97-133 (1981) MR 629700

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Additional Information

DOI: https://doi.org/10.1090/qam/1466148
Article copyright: © Copyright 1997 American Mathematical Society

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