Asymptotic behaviour of the energy in partially viscoelastic materials
Authors:
Jaime E. Muñoz Rivera and Alfonso Peres Salvatierra
Journal:
Quart. Appl. Math. 59 (2001), 557-578
MSC:
Primary 35B40; Secondary 35L70, 35Q72, 74D10, 74H40
DOI:
https://doi.org/10.1090/qam/1848535
MathSciNet review:
MR1848535
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Abstract: In this paper we study models of materials consisting of an elastic part (without memory) and a viscoelastic part, where the dissipation given by the memory is effective. We show that the solutions of the corresponding partial viscoelastic model decay exponentially to zero, provided the relaxation function also decays exponentially, no matter how small is the viscoelastic part of the material
- Claude Bardos, Gilles Lebeau, and Jeffrey Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065. MR 1178650, DOI https://doi.org/10.1137/0330055
- 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
- George Dassios and Filareti Zafiropoulos, Equipartition of energy in linearized $3$-D viscoelasticity, Quart. Appl. Math. 48 (1990), no. 4, 715–730. MR 1079915, DOI https://doi.org/10.1090/qam/1079915
- J. M. Greenberg and Ta Tsien Li, The effect of boundary damping for the quasilinear wave equation, J. Differential Equations 52 (1984), no. 1, 66–75. MR 737964, DOI https://doi.org/10.1016/0022-0396%2884%2990135-9
- 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
F. A. Khodja, A. Benabdallah, and D. Teniou, Stabilisation frontière et interne du système de la Thermoélasticité, Reprint de l’équipe de Mathématiques de Besançon 95/32
- Vilmos Komornik and Enrike Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. (9) 69 (1990), no. 1, 33–54. MR 1054123
- Vilmos Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim. 29 (1991), no. 1, 197–208. MR 1088227, DOI https://doi.org/10.1137/0329011
- 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
- I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Euler-Bernoulli equations with boundary control only in $\Delta w|_\Sigma $, Boll. Un. Mat. Ital. B (7) 5 (1991), no. 3, 665–702 (English, with Italian summary). MR 1127018
- Irena Lasiecka, Global uniform decay rates for the solutions to wave equation with nonlinear boundary conditions, Appl. Anal. 47 (1992), no. 2-3, 191–212. MR 1333954, DOI https://doi.org/10.1080/00036819208840140
- I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only, J. Differential Equations 95 (1992), no. 1, 169–182. MR 1142282, DOI https://doi.org/10.1016/0022-0396%2892%2990048-R
- J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contrôlabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. MR 953547
- Kangsheng Liu and Zhuangyi Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM J. Control Optim. 36 (1998), no. 3, 1086–1098. MR 1613917, DOI https://doi.org/10.1137/S0363012996310703
- Jaime E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math. 52 (1994), no. 4, 629–648. MR 1306041, DOI https://doi.org/10.1090/qam/1306041
- Jaime E. Muñoz Rivera, Global smooth solutions for the Cauchy problem in nonlinear viscoelasticity, Differential Integral Equations 7 (1994), no. 1, 257–273. MR 1250950
- Jaime E. Muñoz Rivera and Milton Lacerda Olivera, Stability in inhomogeneous and anisotropic thermoelasticity, Boll. Un. Mat. Ital. A (7) 11 (1997), no. 1, 115–127 (English, with Italian summary). MR 1438361
- Mitsuhiro Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann. 305 (1996), no. 3, 403–417. MR 1397430, DOI https://doi.org/10.1007/BF01444231
- Kosuke Ono, A stretched string equation with a boundary dissipation, Kyushu J. Math. 48 (1994), no. 2, 265–281. MR 1294530, DOI https://doi.org/10.2206/kyushujm.48.265
- 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
- J. P. Puel and M. Tucsnak, Global existence for the full von Kármán system, Appl. Math. Optim. 34 (1996), no. 2, 139–160. MR 1397777, DOI https://doi.org/10.1007/s002459900024
- James V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22 (1969), 807–823. MR 254433, DOI https://doi.org/10.1002/cpa.3160220605
- James Ralston, Gaussian beams and the propagation of singularities, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, DC, 1982, pp. 206–248. MR 716507
- Michael Renardy, On the type of certain $C_0$-semigroups, Comm. Partial Differential Equations 18 (1993), no. 7-8, 1299–1307. MR 1233196, DOI https://doi.org/10.1080/03605309308820975
- 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
- Wei Xi Shen and Song Mu Zheng, Global smooth solutions to the system of one-dimensional thermoelasticity with dissipation boundary conditions, Chinese Ann. Math. Ser. B 7 (1986), no. 3, 303–317. A Chinese summary appears in Chinese Ann. Math. Ser. A 7 (1986), no. 4, 502. MR 867754
- Enrike Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations 15 (1990), no. 2, 205–235. MR 1032629, DOI https://doi.org/10.1080/03605309908820684
- Enrike Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim. 28 (1990), no. 2, 466–477. MR 1040470, DOI https://doi.org/10.1137/0328025
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM Journal of Control and Optimization 30, 1024–1065 (1992)
C. M. Dafermos, An abstract Volterra equation with application to linear viscoelasticity, J. Differential Equations 7, 554–589 (1970)
G. Dassios and F. Zafiropoulos, Equipartition of energy in linearized 3-d viscoelasticity, Quart. Appl. Math. 48, 715–730 (1990)
J. M. Greenberg and Li Tatsien, The effect of the boundary damping for the quasilinear wave equation, J. Differential Equations 52, 66–75 (1984)
M. A. Horn and I. Lasiecka, Uniform decay of weak solutions to a von Kármán plate with nonlinear boundary dissipation, Differential Integral Equations 7, 885–908 (1994)
F. A. Khodja, A. Benabdallah, and D. Teniou, Stabilisation frontière et interne du système de la Thermoélasticité, Reprint de l’équipe de Mathématiques de Besançon 95/32
V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures et Appl. 69, 33–54 (1990)
V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control and Optimization 29, 197–208 (1991)
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
I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Euler-Bernoulli equations with, boundary control only in $\Delta w\left | \right . _{\Sigma }$, Bollettino Un. Mat. Ital. B(7) 5, 665–702 (1991)
I. Lasiecka, Global uniform decay rates for the solution to the wave equation with nonlinear boundary conditions, Applicable Analysis 47, 191–212 (1992)
I. Lasiecka, Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only, J. Differential Equations 95, 169–182 (1992)
J. L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1, Masson, Paris, 1988
K. Liu and Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping, SIAM Control and Optimization 36, 1086–1098 (1998)
J. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math. 52, 629–648 (1994)
J. Muñoz Rivera, Global smooth solution for the Cauchy problem in nonlinear viscoelasticity, Differential Integral Equations 7, 257–273 (1994)
J. E. Muñoz Rivera and M. de Lacerda Oliveira, Stability in inhomogeneous and anisotropic thermoelasticity, Bollettino Un. Mat. Ital. A(7) 11, 115–127 (1997)
M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Mathematische Annalen 305, 403–417 (1996)
K. Ono, A stretched string equation with a boundary dissipation, Kyushu J. of Math. 28, 265–281 (1994)
J. P. Puel and M. Tucsnak, Boundary stabilization for the von Kármán equation, SIAM J. Control and Optimization 33, 255–273 (1995)
J. P. Puel and M. Tucsnak, Global existence for the full von Kármán system, Applied Mathematics and Optimization 34, 139–160 (1996)
J. Ralston, Solution of the wave equation with localized energy, Comm. Pure Appl. Math. 22, 807–823 (1969)
J. Ralston Gaussian beams and propagation of singularities, Studies in Partial Differential Equations, MAA Studies in Math. 23, W. Littmann, ed., Math. Assoc. America, Washington, D.C., 1982, pp. 206–248
M. Renardy, On the type of certain $C_{0}$-semigroups, Communications in Partial Differential Equations 18, 1299–1307 (1993)
M. Renardy, W. J. Hrusa, and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Appl. Math. 35, Longman Sci. Tech., 1987
Wei Xi Shen and Song Mu Zheng, Global smooth solutions to the system of one-dimensional thermoelasticity with dissipation boundary conditions, Chinese Ann. of Math. Ser. B 7, 303–317 (1986)
E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Communications in Partial Differential Equations 15, 205–235 (1990)
E. Zuazua, Uniform stabilization of the wave equation by nonlinear wave equation boundary feedback, SIAM J. Control and Optimization 28, 466–477 (1990)
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