Decay rates of solutions to a von Kármán system for viscoelastic plates with memory
Authors:
Jaime E. Muñoz Rivera and Gustavo Perla Menzala
Journal:
Quart. Appl. Math. 57 (1999), 181-200
MSC:
Primary 74H45; Secondary 35Q72, 45K05, 74D10, 74K20
DOI:
https://doi.org/10.1090/qam/1672191
MathSciNet review:
MR1672191
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Abstract: We consider the dynamical von Kármán equations for viscoelastic plates under the presence of a long-range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially, then the first-order energy also decays exponentially, When the relaxation $g$ satisfies \[ - {c_1}{g^{1 + \frac {1}{p}}}\left ( t \right ) \le g’\left ( t \right ) \le - {c_0}g{\left ( t \right )^{1 + \frac {1}{p}}}, \qquad 0 \le g”\left ( t \right ) \le {c_2}{g^{1 + \frac {1}{p}}}\left ( t \right ) , \: \textrm {and} \] \[ g, {g^{1 + \frac {1}{p}}} \in {L^1}\left ( \mathbb {R} \right ) \textrm {with} \: p > 2 ,\] then the energy decays as $\frac {1}{\left ( 1 +t \right )^{p}}$. A new Liapunov functional is built for this problem.
E. Bisognin, V. Bisognin, G. Perla Menzala, and E. Zuazua, On exponential stability for von Kármán equations in the presence of thermal effects, Mathematical Methods in the Applied Sciences 21, 393–416 (1998)
- 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
- Constantine M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297–308. MR 281400, DOI https://doi.org/10.1007/BF00251609
- C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations 4 (1979), no. 3, 219–278. MR 522712, DOI https://doi.org/10.1080/03605307908820094
- I. D. Chueshov, Strong solutions and the attractor of a system of von Kármán equations, Mat. Sb. 181 (1990), no. 1, 25–36 (Russian); English transl., Math. USSR-Sb. 69 (1991), no. 1, 25–36. MR 1048828, DOI https://doi.org/10.1070/SM1991v069n01ABEH001230
- Mary Ann Horn and Irena Lasiecka, Uniform decay of weak solutions to a von Kármán plate with nonlinear boundary dissipation, Differential Integral Equations 7 (1994), no. 3-4, 885–908. MR 1270110
- Angelo Favini, Mary Ann Horn, Irena Lasiecka, and Daniel Tataru, Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation, Differential Integral Equations 9 (1996), no. 2, 267–294. MR 1364048
- W. J. Hrusa and J. A. Nohel, The Cauchy problem in one-dimensional nonlinear viscoelasticity, J. Differential Equations 59 (1985), no. 3, 388–412. MR 807854, DOI https://doi.org/10.1016/0022-0396%2885%2990147-0
- Herbert Koch and Andreas Stahel, Global existence of classical solutions to the dynamical von Kármán equations, Math. Methods Appl. Sci. 16 (1993), no. 8, 581–586. MR 1233041, DOI https://doi.org/10.1002/mma.1670160806
- John E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, Control and estimation of distributed parameter systems (Vorau, 1988) Internat. Ser. Numer. Math., vol. 91, Birkhäuser, Basel, 1989, pp. 211–236. MR 1033061
- J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 6, Masson, Paris, 1988. MR 953313
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quarterly of Applied Mathematics LII, No. 4, 629–648, December 1994
- 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
- Jean-Pierre Puel and Marius Tucsnak, Boundary stabilization for the von Kármán equations, SIAM J. Control Optim. 33 (1995), no. 1, 255–273. MR 1311669, DOI https://doi.org/10.1137/S0363012992228350
- Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738
E. Bisognin, V. Bisognin, G. Perla Menzala, and E. Zuazua, On exponential stability for von Kármán equations in the presence of thermal effects, Mathematical Methods in the Applied Sciences 21, 393–416 (1998)
C. M. Dafermos, An abstract Volterra equation with application to linear viscoelasticity, J. Differential Equations 7, 554–589 (1970)
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rat. Mech. Anal. 37, 297–308 (1970)
C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integro-differential equations, Comm. Partial Differential Equations 4, 219–278 (1979)
I. D. Chueshov, Strong solutions and the attractor of the von Kármán equations, Math. USSR Sbornik 69(1), 25–36 (1991)
M. A. Horn and I. Lasiecka, Uniform decay of weak solutions to a von Kármán plate with nonlinear boundary dissipation, Differential and Integral Equations 7(4), 885–908 (1994)
M. A. Horn, A. Favini, I. Lasiecka, and D. Tataru, Global existence, uniqueness and regularity to a von Kármán system with nonlinear boundary dissipation, Differential and Integral Equations 9(2), 267–294 (1996)
W. J. Hrusa and J. A. Nohel, The Cauchy problem in one dimensional nonlinear viscoelasticity, J. Diff. Eq. 58, 388–412 (1985)
H. Koch and A. Stahel, Global existence of classical solutions to the dynamical von Kármán equations, Math. Methods in the Applied Sciences 161, 581–586 (1993)
J. E. Lagnese, Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, International Series of Numerical Mathematics, Vol. 91, Birkhäuser-Verlag, Basel, 1989
J. E. Lagnese and J. L. Lions, Modelling analysis and control of thin plates, Collection Recherches en Mathématiques Appliquées 6, Masson, Paris, 1988
J. L. Lions, Quelques méthodes de resolution de problèmes aux limites non linéaires, Dunod Gauthiers Villars, Paris, 1969
J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, Springer-Verlag, New York, 1972
J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quarterly of Applied Mathematics LII, No. 4, 629–648, December 1994
J. E. Muñoz Rivera, E. C. Lapa, and R. Barreto, Decay rates for viscoelastic plates with memory, Journal of Elasticity 44, 61–87 (1996)
J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Kármán equations, SIAM J. Control and Optimization 33(1), 255–273 (1995)
M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical problems in viscoelasticity, Pitman monograph in Pure and Applied Mathematics 35 (1987)
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© Copyright 1999
American Mathematical Society