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

**[1]**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) MR**1178650****[2]**C. M. Dafermos,*An abstract Volterra equation with application to linear viscoelasticity*, J. Differential Equations**7**, 554-589 (1970) MR**0259670****[3]**G. Dassios and F. Zafiropoulos,*Equipartition of energy in linearized 3-d viscoelasticity*, Quart. Appl. Math.**48**, 715-730 (1990) MR**1079915****[4]**J. M. Greenberg and Li Tatsien,*The effect of the boundary damping for the quasilinear wave equation*, J. Differential Equations**52**, 66-75 (1984) MR**737964****[5]**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) MR**1270110****[6]**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**[7]**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) MR**1054123****[8]**V. Komornik,*Rapid boundary stabilization of the wave equation*, SIAM J. Control and Optimization**29**, 197-208 (1991) MR**1088227****[9]**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 MR**1033061****[10]**I. Lasiecka and R. Triggiani,*Exact controllability and uniform stabilization of Euler-Bernoulli equations with, boundary control only in*, Bollettino Un. Mat. Ital. B(7)**5**, 665-702 (1991) MR**1127018****[11]**I. Lasiecka,*Global uniform decay rates for the solution to the wave equation with nonlinear boundary conditions*, Applicable Analysis**47**, 191-212 (1992) MR**1333954****[12]**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) MR**1142282****[13]**J. L. Lions,*Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués*, Tome 1, Masson, Paris, 1988 MR**953547****[14]**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) MR**1613917****[15]**J. Muñoz Rivera,*Asymptotic behaviour in linear viscoelasticity*, Quart. Appl. Math.**52**, 629-648 (1994) MR**1306041****[16]**J. Muñoz Rivera,*Global smooth solution for the Cauchy problem in nonlinear viscoelasticity*, Differential Integral Equations**7**, 257-273 (1994) MR**1250950****[17]**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) MR**1438361****[18]**M. Nakao,*Decay of solutions of the wave equation with a local nonlinear dissipation*, Mathematische Annalen**305**, 403-417 (1996) MR**1397430****[19]**K. Ono,*A stretched string equation with a boundary dissipation*, Kyushu J. of Math.**28**, 265-281 (1994) MR**1294530****[20]**J. P. Puel and M. Tucsnak,*Boundary stabilization for the von Kármán equation*, SIAM J. Control and Optimization**33**, 255-273 (1995) MR**1311669****[21]**J. P. Puel and M. Tucsnak,*Global existence for the full von Kármán system*, Applied Mathematics and Optimization**34**, 139-160 (1996) MR**1397777****[22]**J. Ralston,*Solution of the wave equation with localized energy*, Comm. Pure Appl. Math.**22**, 807-823 (1969) MR**0254433****[23]**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 MR**716507****[24]**M. Renardy,*On the type of certain -semigroups*, Communications in Partial Differential Equations**18**, 1299-1307 (1993) MR**1233196****[25]**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 MR**919738****[26]**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) MR**867754****[27]**E. Zuazua,*Exponential decay for the semilinear wave equation with locally distributed damping*, Communications in Partial Differential Equations**15**, 205-235 (1990) MR**1032629****[28]**E. Zuazua,*Uniform stabilization of the wave equation by nonlinear wave equation boundary feedback*, SIAM J. Control and Optimization**28**, 466-477 (1990) MR**1040470**

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DOI:
https://doi.org/10.1090/qam/1848535

Article copyright:
© Copyright 2001
American Mathematical Society