American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and concentration of solutions for a Kirchhoff type problem with potentials
  • DCDS Home
  • This Issue
  • Next Article
    Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields
December  2016, 36(12): 7117-7136. doi: 10.3934/dcds.2016110

Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

Louis Tebou 1,

1. 

Department of Mathematics & Statistics, Florida International University, Miami, FL 33199

Received  November 2015 Revised  August 2016 Published  October 2016

We consider the dynamic elasticity equations with a locally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitable open subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multiplier techniques combined with the frequency domain method, we show that: i) the energy of this system decays polynomially when the damping coefficient is only bounded measurable, ii) the energy of this system decays exponentially when the damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition. These results generalize and improve, at the same time, on an earlier result of Liu and Rao involving the wave equation with Kelvin-Voigt damping; those authors proved the exponential decay of the energy provided that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient.
Keywords: Kelvin-Voigt damping., Stabilization, wave equation, localized dissipation, dynamic elasticity equations.
Mathematics Subject Classification: Primary: 93D15; Secondary: 74J05, 35L05, 74H4.
Citation: Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110
References:
[1]

F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems,, SIAM J. Control Optim., 37 (1999), 521.  doi: 10.1137/S0363012996313835.  Google Scholar

[2]

G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity,, Comm. Partial Differential Equations, 26 (2001), 1787.  doi: 10.1081/PDE-100107459.  Google Scholar

[3]

D. D. Ang, M. Ikehata, D. D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients,, Comm. Partial Differential Equations, 23 (1998), 371.  doi: 10.1080/03605309808821349.  Google Scholar

[4]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[5]

M. A. Astaburuaga and R. Coimbra Charão, Stabilization of the total energy for a system of elasticity with localized dissipation,, Differential Integral Equations, 15 (2002), 1357.   Google Scholar

[6]

H. T. Banks, R. C. Smith and Y. Wang, Modeling aspects for piezoelectric patch actuation of shells, plates and beams,, Quart. Appl. Math., 53 (1995), 353.   Google Scholar

[7]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary,, SIAM J. Control and Opt., 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar

[8]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425.  doi: 10.1002/mana.200410429.  Google Scholar

[9]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765.  doi: 10.1007/s00028-008-0424-1.  Google Scholar

[10]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[11]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Masson, (1983).   Google Scholar

[12]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers,, Asymptot. Anal., 14 (1997), 157.   Google Scholar

[13]

N. Burq and H. Christianson, Imperfect geometric control and overdamping for the damped wave equation,, Commun. Math. Phys., 336 (2015), 101.  doi: 10.1007/s00220-014-2247-y.  Google Scholar

[14]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310.  doi: 10.1137/S0363012902408010.  Google Scholar

[15]

G. Chen, Control and stabilization for the wave equation in a bounded domain,, SIAM J. Control Optim., 17 (1979), 66.  doi: 10.1137/0317007.  Google Scholar

[16]

G. Chen, S. A. Fulling, F. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping,, SIAM J. Appl. Math., 51 (1991), 266.  doi: 10.1137/0151015.  Google Scholar

[17]

C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations,, in Nonlinear evolution equations(M.G. Crandall ed.) Academic Press, 40 (1978), 103.   Google Scholar

[18]

B. Dehman and L. Robbiano, La propriété du prolongement unique pour un système elliptique. Le système de Lamé,, J. Math. Pures Appl., 72 (1993), 475.   Google Scholar

[19]

G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique,, Travaux et Recherches Mathématiques, (1972).   Google Scholar

[20]

A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations,, Asymptot. Anal., 22 (2000), 1.   Google Scholar

[21]

X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping,, SIAM J. Control Optim., 50 (2012), 1643.  doi: 10.1137/110833051.  Google Scholar

[22]

A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations,, J. Differential Equations, 59 (1985), 145.  doi: 10.1016/0022-0396(85)90151-2.  Google Scholar

[23]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Port. Math., 46 (1989), 245.   Google Scholar

[24]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, (1985), 43.   Google Scholar

[25]

V. Komornik, Rapid boundary stabilization of the wave equation,, SIAM J. Control Optim., 29 (1991), 197.  doi: 10.1137/0329011.  Google Scholar

[26]

V. Komornik, On the nonlinear boundary stabilization of the wave equation,, Chin. Ann. Math., 14 (1993), 153.   Google Scholar

[27]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM, (1994).   Google Scholar

[28]

V. Komornik, Rapid boundary stabilization of linear distributed systems,, SIAM J. Control and Optimization, 35 (1997), 1591.  doi: 10.1137/S0363012996301609.  Google Scholar

[29]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J.M.P.A., 69 (1990), 33.   Google Scholar

[30]

J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163.  doi: 10.1016/0022-0396(83)90073-6.  Google Scholar

[31]

J. Lagnese, Boundary stabilization of linear elastodynamic systems,, SIAM J. Control and Optim., 21 (1983), 968.  doi: 10.1137/0321059.  Google Scholar

[32]

J. Lagnese, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary,, Nonlinear Anal., 16 (1991), 35.  doi: 10.1016/0362-546X(91)90129-O.  Google Scholar

[33]

I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects,, Shape optimization and optimal design (Cambridge, 216 (2001), 333.   Google Scholar

[34]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507.   Google Scholar

[35]

I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,, Nonlinear Anal., 64 (2006), 1757.  doi: 10.1016/j.na.2005.07.024.  Google Scholar

[36]

I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13.  doi: 10.1006/jmaa.1999.6348.  Google Scholar

[37]

G. Lebeau, Equation des ondes amorties,, Algebraic and geometric methods in mathematical physics (Kaciveli, 19 (1996), 73.   Google Scholar

[38]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Arch. Ration. Mech. Anal., 148 (1999), 179.  doi: 10.1007/s002050050160.  Google Scholar

[39]

C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients,, Math. Ann., 331 (2005), 611.  doi: 10.1007/s00208-004-0597-z.  Google Scholar

[40]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués,, Vol. 1, (1988).   Google Scholar

[41]

K. Liu, Locally distributed control and damping for the conservative systems,, S.I.A.M J. Control and Opt., 35 (1997), 1574.  doi: 10.1137/S0363012995284928.  Google Scholar

[42]

K. Liu and Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,, S.I.A.M J. Control and Opt, 36 (1998), 1086.  doi: 10.1137/S0363012996310703.  Google Scholar

[43]

K. Liu and B. Rao, Stabilité exponentielle des équations des ondes avec amortissement local de Kelvin-Voigt,, C. R. Math. Acad. Sci. Paris, 339 (2004), 769.  doi: 10.1016/j.crma.2004.09.029.  Google Scholar

[44]

K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping,, Z. Angew. Math. Phys., 57 (2006), 419.  doi: 10.1007/s00033-005-0029-2.  Google Scholar

[45]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[46]

P. Martinez, PhD Thesis., University of Strasbourg, (1998).   Google Scholar

[47]

P. Martinez, Uniform boundary stabilization of elasticity systems of cubic cystals by nonlinear feedbacks,, Nonlinear Analysis 37 (1999), 37 (1999), 719.  doi: 10.1016/S0362-546X(98)00068-6.  Google Scholar

[48]

G. Nakamura, G. Uhlmann and J.-N. Wang, Unique continuation property for elliptic systems and crack determination in anisotropic elasticity,, Partial differential equations and inverse problems, (2004), 321.  doi: 10.1090/conm/362/06621.  Google Scholar

[49]

M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation,, Israel J. Math., 95 (1996), 25.  doi: 10.1007/BF02761033.  Google Scholar

[50]

M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation,, Math. Ann., 305 (1996), 403.  doi: 10.1007/BF01444231.  Google Scholar

[51]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[52]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar

[53]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains,, Indiana Univ. Math. J., 24 (1974), 79.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar

[54]

M. Renardy, On localized Kelvin-Voigt damping,, ZAMM Z. Angew. Math. Mech., 84 (2004), 280.  doi: 10.1002/zamm.200310100.  Google Scholar

[55]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations,, Recent progress and open problems. SIAM Rev., 20 (1978), 639.  doi: 10.1137/1020095.  Google Scholar

[56]

M. Slemrod, Weak asymptotic decay via a "Relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping,, Proc. Royal Soc. Edinburgh Sect. A, 113 (1989), 87.  doi: 10.1017/S0308210500023970.  Google Scholar

[57]

L. R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé,, C. R. Acad. Paris, 325 (1997), 1175.  doi: 10.1016/S0764-4442(97)83549-5.  Google Scholar

[58]

L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J.D.E., 145 (1998), 502.  doi: 10.1006/jdeq.1998.3416.  Google Scholar

[59]

L. R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with $L^r$ localizing coefficient,, Comm. in P.D.E., 23 (1998), 1839.  doi: 10.1080/03605309808821403.  Google Scholar

[60]

L. R. Tcheugoué Tébou, A direct method for the stabilization of some locally damped semilinear wave equations,, C. R. Acad. Sci. Paris, 342 (2006), 859.  doi: 10.1016/j.crma.2006.04.010.  Google Scholar

[61]

L. Tebou, Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation,, Systems and Control Letters, 56 (2007), 538.  doi: 10.1016/j.sysconle.2007.03.003.  Google Scholar

[62]

L. Tebou, On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation,, Differential Integral Equations, 19 (2006), 785.   Google Scholar

[63]

L. Tebou, A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 561.  doi: 10.1051/cocv:2007066.  Google Scholar

[64]

L. Tebou, A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping,, C. R. Acad. Sci. Paris, 350 (2012), 603.  doi: 10.1016/j.crma.2012.06.005.  Google Scholar

[65]

L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations,, C. R. Acad. Sci. Paris, 350 (2012), 57.  doi: 10.1016/j.crma.2011.12.001.  Google Scholar

[66]

D. Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions,, Nonlinear Anal., {67 (2007), 512.  doi: 10.1016/j.na.2006.06.007.  Google Scholar

[67]

R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation,, Proc. Amer. Math. Soc., 105 (1989), 375.  doi: 10.1090/S0002-9939-1989-0953013-0.  Google Scholar

[68]

P. F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients,, SIAM J. Control Optim., 37 (1999), 1568.  doi: 10.1137/S0363012997331482.  Google Scholar

[69]

P. F. Yao, Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach,, Chapman and Hall/CRC Press, (2011).  doi: 10.1201/b11042.  Google Scholar

[70]

H. Yu, Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions,, ESAIM Control Optim. Calc. Var., 17 (2011), 761.  doi: 10.1051/cocv/2010021.  Google Scholar

[71]

Q. Zhang, Exponential stability of an elastic string with local Kelvin-Voigt damping,, ZAMP, 61 (2010), 1009.  doi: 10.1007/s00033-010-0064-5.  Google Scholar

[72]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Commun. P.D.E., 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar

[73]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures. Appl., 70 (1991), 513.   Google Scholar

show all references

References:
[1]

F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems,, SIAM J. Control Optim., 37 (1999), 521.  doi: 10.1137/S0363012996313835.  Google Scholar

[2]

G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity,, Comm. Partial Differential Equations, 26 (2001), 1787.  doi: 10.1081/PDE-100107459.  Google Scholar

[3]

D. D. Ang, M. Ikehata, D. D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients,, Comm. Partial Differential Equations, 23 (1998), 371.  doi: 10.1080/03605309808821349.  Google Scholar

[4]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[5]

M. A. Astaburuaga and R. Coimbra Charão, Stabilization of the total energy for a system of elasticity with localized dissipation,, Differential Integral Equations, 15 (2002), 1357.   Google Scholar

[6]

H. T. Banks, R. C. Smith and Y. Wang, Modeling aspects for piezoelectric patch actuation of shells, plates and beams,, Quart. Appl. Math., 53 (1995), 353.   Google Scholar

[7]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary,, SIAM J. Control and Opt., 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar

[8]

A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups,, Math. Nachr., 279 (2006), 1425.  doi: 10.1002/mana.200410429.  Google Scholar

[9]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765.  doi: 10.1007/s00028-008-0424-1.  Google Scholar

[10]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[11]

H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Masson, (1983).   Google Scholar

[12]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers,, Asymptot. Anal., 14 (1997), 157.   Google Scholar

[13]

N. Burq and H. Christianson, Imperfect geometric control and overdamping for the damped wave equation,, Commun. Math. Phys., 336 (2015), 101.  doi: 10.1007/s00220-014-2247-y.  Google Scholar

[14]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation,, SIAM J. Control Optim., 42 (2003), 1310.  doi: 10.1137/S0363012902408010.  Google Scholar

[15]

G. Chen, Control and stabilization for the wave equation in a bounded domain,, SIAM J. Control Optim., 17 (1979), 66.  doi: 10.1137/0317007.  Google Scholar

[16]

G. Chen, S. A. Fulling, F. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping,, SIAM J. Appl. Math., 51 (1991), 266.  doi: 10.1137/0151015.  Google Scholar

[17]

C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations,, in Nonlinear evolution equations(M.G. Crandall ed.) Academic Press, 40 (1978), 103.   Google Scholar

[18]

B. Dehman and L. Robbiano, La propriété du prolongement unique pour un système elliptique. Le système de Lamé,, J. Math. Pures Appl., 72 (1993), 475.   Google Scholar

[19]

G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique,, Travaux et Recherches Mathématiques, (1972).   Google Scholar

[20]

A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations,, Asymptot. Anal., 22 (2000), 1.   Google Scholar

[21]

X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping,, SIAM J. Control Optim., 50 (2012), 1643.  doi: 10.1137/110833051.  Google Scholar

[22]

A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations,, J. Differential Equations, 59 (1985), 145.  doi: 10.1016/0022-0396(85)90151-2.  Google Scholar

[23]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Port. Math., 46 (1989), 245.   Google Scholar

[24]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, (1985), 43.   Google Scholar

[25]

V. Komornik, Rapid boundary stabilization of the wave equation,, SIAM J. Control Optim., 29 (1991), 197.  doi: 10.1137/0329011.  Google Scholar

[26]

V. Komornik, On the nonlinear boundary stabilization of the wave equation,, Chin. Ann. Math., 14 (1993), 153.   Google Scholar

[27]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM, (1994).   Google Scholar

[28]

V. Komornik, Rapid boundary stabilization of linear distributed systems,, SIAM J. Control and Optimization, 35 (1997), 1591.  doi: 10.1137/S0363012996301609.  Google Scholar

[29]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J.M.P.A., 69 (1990), 33.   Google Scholar

[30]

J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163.  doi: 10.1016/0022-0396(83)90073-6.  Google Scholar

[31]

J. Lagnese, Boundary stabilization of linear elastodynamic systems,, SIAM J. Control and Optim., 21 (1983), 968.  doi: 10.1137/0321059.  Google Scholar

[32]

J. Lagnese, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary,, Nonlinear Anal., 16 (1991), 35.  doi: 10.1016/0362-546X(91)90129-O.  Google Scholar

[33]

I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects,, Shape optimization and optimal design (Cambridge, 216 (2001), 333.   Google Scholar

[34]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507.   Google Scholar

[35]

I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,, Nonlinear Anal., 64 (2006), 1757.  doi: 10.1016/j.na.2005.07.024.  Google Scholar

[36]

I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13.  doi: 10.1006/jmaa.1999.6348.  Google Scholar

[37]

G. Lebeau, Equation des ondes amorties,, Algebraic and geometric methods in mathematical physics (Kaciveli, 19 (1996), 73.   Google Scholar

[38]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Arch. Ration. Mech. Anal., 148 (1999), 179.  doi: 10.1007/s002050050160.  Google Scholar

[39]

C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients,, Math. Ann., 331 (2005), 611.  doi: 10.1007/s00208-004-0597-z.  Google Scholar

[40]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués,, Vol. 1, (1988).   Google Scholar

[41]

K. Liu, Locally distributed control and damping for the conservative systems,, S.I.A.M J. Control and Opt., 35 (1997), 1574.  doi: 10.1137/S0363012995284928.  Google Scholar

[42]

K. Liu and Z. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,, S.I.A.M J. Control and Opt, 36 (1998), 1086.  doi: 10.1137/S0363012996310703.  Google Scholar

[43]

K. Liu and B. Rao, Stabilité exponentielle des équations des ondes avec amortissement local de Kelvin-Voigt,, C. R. Math. Acad. Sci. Paris, 339 (2004), 769.  doi: 10.1016/j.crma.2004.09.029.  Google Scholar

[44]

K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping,, Z. Angew. Math. Phys., 57 (2006), 419.  doi: 10.1007/s00033-005-0029-2.  Google Scholar

[45]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[46]

P. Martinez, PhD Thesis., University of Strasbourg, (1998).   Google Scholar

[47]

P. Martinez, Uniform boundary stabilization of elasticity systems of cubic cystals by nonlinear feedbacks,, Nonlinear Analysis 37 (1999), 37 (1999), 719.  doi: 10.1016/S0362-546X(98)00068-6.  Google Scholar

[48]

G. Nakamura, G. Uhlmann and J.-N. Wang, Unique continuation property for elliptic systems and crack determination in anisotropic elasticity,, Partial differential equations and inverse problems, (2004), 321.  doi: 10.1090/conm/362/06621.  Google Scholar

[49]

M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation,, Israel J. Math., 95 (1996), 25.  doi: 10.1007/BF02761033.  Google Scholar

[50]

M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation,, Math. Ann., 305 (1996), 403.  doi: 10.1007/BF01444231.  Google Scholar

[51]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[52]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar

[53]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains,, Indiana Univ. Math. J., 24 (1974), 79.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar

[54]

M. Renardy, On localized Kelvin-Voigt damping,, ZAMM Z. Angew. Math. Mech., 84 (2004), 280.  doi: 10.1002/zamm.200310100.  Google Scholar

[55]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations,, Recent progress and open problems. SIAM Rev., 20 (1978), 639.  doi: 10.1137/1020095.  Google Scholar

[56]

M. Slemrod, Weak asymptotic decay via a "Relaxed invariance principle" for a wave equation with nonlinear, nonmonotone damping,, Proc. Royal Soc. Edinburgh Sect. A, 113 (1989), 87.  doi: 10.1017/S0308210500023970.  Google Scholar

[57]

L. R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé,, C. R. Acad. Paris, 325 (1997), 1175.  doi: 10.1016/S0764-4442(97)83549-5.  Google Scholar

[58]

L. R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping,, J.D.E., 145 (1998), 502.  doi: 10.1006/jdeq.1998.3416.  Google Scholar

[59]

L. R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with $L^r$ localizing coefficient,, Comm. in P.D.E., 23 (1998), 1839.  doi: 10.1080/03605309808821403.  Google Scholar

[60]

L. R. Tcheugoué Tébou, A direct method for the stabilization of some locally damped semilinear wave equations,, C. R. Acad. Sci. Paris, 342 (2006), 859.  doi: 10.1016/j.crma.2006.04.010.  Google Scholar

[61]

L. Tebou, Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation,, Systems and Control Letters, 56 (2007), 538.  doi: 10.1016/j.sysconle.2007.03.003.  Google Scholar

[62]

L. Tebou, On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation,, Differential Integral Equations, 19 (2006), 785.   Google Scholar

[63]

L. Tebou, A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations,, ESAIM Control Optim. Calc. Var., 14 (2008), 561.  doi: 10.1051/cocv:2007066.  Google Scholar

[64]

L. Tebou, A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping,, C. R. Acad. Sci. Paris, 350 (2012), 603.  doi: 10.1016/j.crma.2012.06.005.  Google Scholar

[65]

L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations,, C. R. Acad. Sci. Paris, 350 (2012), 57.  doi: 10.1016/j.crma.2011.12.001.  Google Scholar

[66]

D. Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions,, Nonlinear Anal., {67 (2007), 512.  doi: 10.1016/j.na.2006.06.007.  Google Scholar

[67]

R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation,, Proc. Amer. Math. Soc., 105 (1989), 375.  doi: 10.1090/S0002-9939-1989-0953013-0.  Google Scholar

[68]

P. F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients,, SIAM J. Control Optim., 37 (1999), 1568.  doi: 10.1137/S0363012997331482.  Google Scholar

[69]

P. F. Yao, Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach,, Chapman and Hall/CRC Press, (2011).  doi: 10.1201/b11042.  Google Scholar

[70]

H. Yu, Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions,, ESAIM Control Optim. Calc. Var., 17 (2011), 761.  doi: 10.1051/cocv/2010021.  Google Scholar

[71]

Q. Zhang, Exponential stability of an elastic string with local Kelvin-Voigt damping,, ZAMP, 61 (2010), 1009.  doi: 10.1007/s00033-010-0064-5.  Google Scholar

[72]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Commun. P.D.E., 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar

[73]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains,, J. Math. Pures. Appl., 70 (1991), 513.   Google Scholar

[1]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
[2]

Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029
[3]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[4]

Miroslav Bulíček, Josef Málek, K. R. Rajagopal. On Kelvin-Voigt model and its generalizations. Evolution Equations & Control Theory, 2012, 1 (1) : 17-42. doi: 10.3934/eect.2012.1.17
[5]

Xiaobin Yao, Qiaozhen Ma, Tingting Liu. Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1889-1917. doi: 10.3934/dcdsb.2018247
[6]

Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075
[7]

Manil T. Mohan. On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations. Evolution Equations & Control Theory, 2020, 9 (2) : 301-339. doi: 10.3934/eect.2020007
[8]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251
[9]

Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303
[10]

To Fu Ma, Paulo Nicanor Seminario-Huertas. Attractors for semilinear wave equations with localized damping and external forces. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2219-2233. doi: 10.3934/cpaa.2020097
[11]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315
[12]

Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731
[13]

Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[14]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
[15]

Mohammad Akil, Ali Wehbe. Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control & Related Fields, 2019, 9 (1) : 97-116. doi: 10.3934/mcrf.2019005
[16]

Manil T. Mohan. Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3393-3436. doi: 10.3934/dcdsb.2020067
[17]

Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control & Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17
[18]

Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084
[19]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547
[20]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences