×

zbMATH — the first resource for mathematics

The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity. (English) Zbl 0922.35018
The author studies the exponential decay of solutions to the linear system of thermoviscoelasticity: \(u'' - \mu \Delta u -(\lambda+\mu)\nabla \text{div }u +\mu g*\Delta u +(\lambda+\mu)g*\nabla\text{div }u +\alpha\nabla\theta=0\), \(\theta' - \Delta \theta +\beta \text{div } u' =0\) subject to initial and Dirichlet boundary conditions. The assumptions made on the relaxation function \(g\) are standard and they imply that the static modulus is positive. The main result of this paper is to prove that the energy functional \(E(u,\theta,t)\) decays exponentially as \(t\to\infty\). It is known that in the higher-dimensional case the energy functional defined as \(E(u,\theta,t)={1\over 2} (\| u'(t)\|_2^2 +{\alpha\over \beta}\| \theta(t)\|_2^2 +\mu \| \nabla u(t)\|_2^2 +(\lambda+\mu) \| \text{div }u(t)\|_2^2)\), in general, does not tend to zero. In order to increase the loss of energy and subsequently to ensure the exponential decay of solutions the author introduces a boundary velocity feedback on a part of the boundary of a thermoviscoelastic body, which is clamped along the rest of its boundary.

MSC:
35B35 Stability in context of PDEs
74Hxx Dynamical problems in solid mechanics
74A15 Thermodynamics in solid mechanics
45K05 Integro-partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams, R., Sobolev spaces, (1975), Academic Press New York · Zbl 0314.46030
[2] \scF. Alabau and \scV. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems, SIAM J. Control Optim., to appear. · Zbl 0935.93037
[3] Burns, J.A.; Liu, Z.Y.; Zheng, S., On the energy decay of a linear thermoelastic bar, J. math. anal. appl., 179, 574-591, (1993) · Zbl 0803.35150
[4] Chen, G., Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. math. pures appl., 58, 249-273, (1979) · Zbl 0414.35044
[5] Dafermos, C.M., On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity, Arch. rational mech. anal., 29, 241-271, (1968) · Zbl 0183.37701
[6] Dafermos, C.M., Asymptotic stability in viscoelasticity, Arch. rational mech. anal., 37, 297-308, (1970) · Zbl 0214.24503
[7] Dafermos, C.M., An abstract Volterra equation with applications to linear viscoelasticity, J. differential equations, 7, 554-569, (1970) · Zbl 0212.45302
[8] Day, W.A., The decay of energy in a viscoelastic bady, Mathematika, 27, 268-286, (1980) · Zbl 0466.73034
[9] Dautray, R.; Lions, J.L., ()
[10] Desch, W.; Miller, R.K., Exponential stabilization of Volterra integrodifferential equations in Hilbert space, J. differential equations, 70, 366-389, (1987) · Zbl 0635.45029
[11] Grisvard, P., Contrôlabilité exacte des solutions de l’équation des ondes en présence de singularités, J. math. pures appl., 68, 215-259, (1989) · Zbl 0683.49012
[12] Hansen, S.W., Exponential energy decay in a linear thermoelastic rod, J. math. anal. appl., 167, 429-442, (1992) · Zbl 0755.73012
[13] Jiang, S.; Rivera, J.E.M., A global existence theorem for the Dirichlet problem in nonlinear n-dimensional viscoelasticity, Differential integral equations, 9, no. 4, 791-810, (1996) · Zbl 0862.35071
[14] Kim, J.U., On the energy decay of a linear thermoelastic bar and plate, SIAM J. math. anal., 23, 889-899, (1992) · Zbl 0755.73013
[15] Krstić, M.; Kanellakopoulos, I.; Kokotović, P., Nonlinear and adaptive control design, (1995), John Wiley and Sons, Inc New York · Zbl 0763.93043
[16] Komornik, V.; Zuazua, E., A direct method for the boundary stabilization of the wave equation, J. math. pures appl., 69, 33-54, (1990) · Zbl 0636.93064
[17] Lagnese, J., Decay of solutions of wave equations in a bounded region with boundary dissipation, J. differential equations, 50, 163-182, (1983) · Zbl 0536.35043
[18] Lagnese, J., Boundary stabilization of linear elastodynamic systems, SIAM J. control optim., 21, 968-984, (1983) · Zbl 0531.93044
[19] Lagnese, J., Boundary stabilization of thin plates, () · Zbl 0696.73034
[20] Leugering, G., On boundary feedback stabilization of a viscoelastic membrance, Dynamics and stability of systems, 4, 71-79, (1989) · Zbl 0681.93051
[21] Leugering, G., On boundary feedback stabilization of a viscoelastic beam, (), 57-69 · Zbl 0699.73037
[22] Leugering, G., A decomposition method for integro-partial differential equations and applications, J. math. pures appl., 71, 561-587, (1992) · Zbl 0830.45009
[23] Lebeau, G.; Zuazua, E., Sur la décroissance non uniforme de l’énérgie dans le système de la thermoélasticité linéaire, C. R. acad. sci. Paris, t. 324, 409-415, (1997), Série 1 · Zbl 0873.35011
[24] Lions, J.L., Contrôlabilité exacte perturbations et stabilisation de systèmes distribués, perturbations, (1988), Masson Paris Milan Barcelone Mexico, Tome 2 · Zbl 0653.93002
[25] \scW.J. Liu, Partial exact controllability and exponential stability in higher dimensional linear thermoelasticity, ESAIM: Control Optim. Calc. Var., to appear.
[26] Lagnese, J.; Lions, J.L., Modelling analysis and control of thin plates, (1989), Masson Paris · Zbl 0662.73039
[27] Lions, J.L.; Magenes, E., ()
[28] Liu, Z.; Renardy, M., A note on the equations of thermoelastic plates, Appl. math. letters, 8, 1-6, (1995) · Zbl 0826.35048
[29] Liu, Z.; Zheng, S., Exponential stability of the semigroup associated with a thermoelastic system, Quart. appl. math., LI, 535-545, (1993) · Zbl 0803.35014
[30] Liu, Z.; Zheng, S., On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. appl. math., LIV, 21-31, (1996) · Zbl 0868.35011
[31] Menzala, G.P.; Zuazua, E., Explicit exponential decay rates for solutions of von Kármán’s system of thermoelastic plates, C. R. acad. sci. Paris, t. 324, 49-54, (1997), Série I · Zbl 0873.73042
[32] [MZ_2]|\scG.P. Menzala and \scE. Zuazua, Energy decay rates for the Von Kármán system of thermoelastic plates, Adv. Differential Equations, to appear.
[33] Navarro, C.B., Asymptotic stability in linear thermoviscoelasticity, J. math. anal. appl., 65, 399-431, (1978) · Zbl 0408.73007
[34] Pazy, A., Semigroup of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023
[35] Rivera, J.E.M.; Barreto, R.K., Uniform rates of decay in nonlinear viscoelasticity for polynomial decaying kernels, Appl. anal., 60, 341-357, (1996) · Zbl 0871.35015
[36] Rivera, J.E.M., Global smooth solutions for the Cauchy problem in nonlinear viscoelasticity, Differential integral equations, 7, no. 1, 257-273, (1994) · Zbl 0786.45011
[37] Rivera, J.E.M.; Lapa, E.C., Decay rates of solutions of an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomially decaying kernels, Comm. math. phys., 177, no. 3, 583-602, (1996) · Zbl 0852.73026
[38] Walker, J.A., Dynamical systems and evolution equations, theory and applications, (1980), Plenum Press New York · Zbl 0421.34050
[39] Zuazua, E., Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. control optim., 28, no. 2, 466-477, (1990) · Zbl 0695.93090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.