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Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping
1. | Department of Mathematics & Statistics, Florida International University, Miami, FL 33199 |
References:
[1] |
F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM J. Control Optim., 37 (1999), 521-542.
doi: 10.1137/S0363012996313835. |
[2] |
G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. Partial Differential Equations, 26 (2001), 1787-1810.
doi: 10.1081/PDE-100107459. |
[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), 371-385.
doi: 10.1080/03605309808821349. |
[4] |
W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.
doi: 10.1090/S0002-9947-1988-0933321-3. |
[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), 1357-1376. |
[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), 353-381. |
[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), 1024-1065.
doi: 10.1137/0330055. |
[8] |
A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.
doi: 10.1002/mana.200410429. |
[9] |
C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.
doi: 10.1007/s00028-008-0424-1. |
[10] |
A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.
doi: 10.1007/s00208-009-0439-0. |
[11] |
H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983. |
[12] |
N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191. |
[13] |
N. Burq and H. Christianson, Imperfect geometric control and overdamping for the damped wave equation, Commun. Math. Phys., 336 (2015), 101-130.
doi: 10.1007/s00220-014-2247-y. |
[14] |
M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.
doi: 10.1137/S0363012902408010. |
[15] |
G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66-81.
doi: 10.1137/0317007. |
[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), 266-301.
doi: 10.1137/0151015. |
[17] |
C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations, in Nonlinear evolution equations(M.G. Crandall ed.) Academic Press, New-York, 40 (1978), 103-123. |
[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), 475-492. |
[19] |
G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, No. 21. Dunod, Paris, 1972. |
[20] |
A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations, Asymptot. Anal., 22 (2000), 1-13. |
[21] |
X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping, SIAM J. Control Optim., 50 (2012), 1643-1660.
doi: 10.1137/110833051. |
[22] |
A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59 (1985), 145-154.
doi: 10.1016/0022-0396(85)90151-2. |
[23] |
A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46 (1989), 245-258. |
[24] |
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. |
[25] |
V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.
doi: 10.1137/0329011. |
[26] |
V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chin. Ann. Math., 14 (1993), 153-164. |
[27] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM, Masson & John Wiley, Paris, 1994. |
[28] |
V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control and Optimization, 35 (1997), 1591-1613.
doi: 10.1137/S0363012996301609. |
[29] |
V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J.M.P.A.,, 69 (1990), 33-54. |
[30] |
J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.
doi: 10.1016/0022-0396(83)90073-6. |
[31] |
J. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 21 (1983), 968-984.
doi: 10.1137/0321059. |
[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), 35-54.
doi: 10.1016/0362-546X(91)90129-O. |
[33] |
I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects, Shape optimization and optimal design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math., Dekker, New York, 216 (2001), 333-354. |
[34] |
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. |
[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), 1757-1797.
doi: 10.1016/j.na.2005.07.024. |
[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), 13-57.
doi: 10.1006/jmaa.1999.6348. |
[37] |
G. Lebeau, Equation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., Kluwer Acad. Publ., Dordrecht, 19 (1996), 73-109. |
[38] |
G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231.
doi: 10.1007/s002050050160. |
[39] |
C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients, Math. Ann., 331 (2005), 611-629.
doi: 10.1007/s00208-004-0597-z. |
[40] |
J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués, Vol. 1, RMA 8, Masson, Paris, 1988. |
[41] |
K. Liu, Locally distributed control and damping for the conservative systems, S.I.A.M J. Control and Opt., 35 (1997), 1574-1590.
doi: 10.1137/S0363012995284928. |
[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), 1086-1098.
doi: 10.1137/S0363012996310703. |
[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), 769-774.
doi: 10.1016/j.crma.2004.09.029. |
[44] |
K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419-432.
doi: 10.1007/s00033-005-0029-2. |
[45] |
Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.
doi: 10.1007/s00033-004-3073-4. |
[46] |
P. Martinez, PhD Thesis. University of Strasbourg, France, 1998. |
[47] |
P. Martinez, Uniform boundary stabilization of elasticity systems of cubic cystals by nonlinear feedbacks, Nonlinear Analysis 37 (1999), 719-733.
doi: 10.1016/S0362-546X(98)00068-6. |
[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, 321-338, Contemp. Math., 362, Amer. Math. Soc., Providence, RI, 2004.
doi: 10.1090/conm/362/06621. |
[49] |
M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation, Israel J. Math., 95(1996), 25-42.
doi: 10.1007/BF02761033. |
[50] |
M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417.
doi: 10.1007/BF01444231. |
[51] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[52] |
J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[53] |
J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.
doi: 10.1512/iumj.1975.24.24004. |
[54] |
M. Renardy, On localized Kelvin-Voigt damping, ZAMM Z. Angew. Math. Mech., 84 (2004), 280-283.
doi: 10.1002/zamm.200310100. |
[55] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations, Recent progress and open problems. SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[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), 87-97.
doi: 10.1017/S0308210500023970. |
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L. R. Tcheugoué Tébou, Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé, C. R. Acad. Paris, Série I, 325 (1997), 1175-1179.
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L. R. Tcheugoué Tébou, A direct method for the stabilization of some locally damped semilinear wave equations, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 859-864.
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L. Tebou, Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation, Systems and Control Letters, 56 (2007), 538-545.
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L. Tebou, On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation, Differential Integral Equations, 19 (2006), 785-798. |
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L. Tebou, A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations, ESAIM Control Optim. Calc. Var., 14 (2008), 561-574.
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L. Tebou, A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping, C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 603-608.
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L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations, C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 57-62.
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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), 512-544.
doi: 10.1016/j.na.2006.06.007. |
[67] |
R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383.
doi: 10.1090/S0002-9939-1989-0953013-0. |
[68] |
P. F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim., 37 (1999), 1568-1599.
doi: 10.1137/S0363012997331482. |
[69] |
P. F. Yao, Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach, Chapman and Hall/CRC Press, Boca Raton, Florida, 2011.
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H. Yu, Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions, ESAIM Control Optim. Calc. Var., 17 (2011), 761-770.
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[71] |
Q. Zhang, Exponential stability of an elastic string with local Kelvin-Voigt damping, ZAMP, 61 (2010), 1009-1015.
doi: 10.1007/s00033-010-0064-5. |
[72] |
E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Commun. P.D.E., 15 (1990), 205-235.
doi: 10.1080/03605309908820684. |
[73] |
E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures. Appl., 70 (1991), 513-529. |
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), 521-542.
doi: 10.1137/S0363012996313835. |
[2] |
G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. Partial Differential Equations, 26 (2001), 1787-1810.
doi: 10.1081/PDE-100107459. |
[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), 371-385.
doi: 10.1080/03605309808821349. |
[4] |
W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.
doi: 10.1090/S0002-9947-1988-0933321-3. |
[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), 1357-1376. |
[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), 353-381. |
[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), 1024-1065.
doi: 10.1137/0330055. |
[8] |
A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.
doi: 10.1002/mana.200410429. |
[9] |
C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.
doi: 10.1007/s00028-008-0424-1. |
[10] |
A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.
doi: 10.1007/s00208-009-0439-0. |
[11] |
H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983. |
[12] |
N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191. |
[13] |
N. Burq and H. Christianson, Imperfect geometric control and overdamping for the damped wave equation, Commun. Math. Phys., 336 (2015), 101-130.
doi: 10.1007/s00220-014-2247-y. |
[14] |
M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.
doi: 10.1137/S0363012902408010. |
[15] |
G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66-81.
doi: 10.1137/0317007. |
[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), 266-301.
doi: 10.1137/0151015. |
[17] |
C. M. Dafermos, Asymptotic behaviour of solutions of evolution equations, in Nonlinear evolution equations(M.G. Crandall ed.) Academic Press, New-York, 40 (1978), 103-123. |
[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), 475-492. |
[19] |
G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, No. 21. Dunod, Paris, 1972. |
[20] |
A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations, Asymptot. Anal., 22 (2000), 1-13. |
[21] |
X. Fu, Sharp decay rates for the weakly coupled hyperbolic system with one internal damping, SIAM J. Control Optim., 50 (2012), 1643-1660.
doi: 10.1137/110833051. |
[22] |
A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59 (1985), 145-154.
doi: 10.1016/0022-0396(85)90151-2. |
[23] |
A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Port. Math., 46 (1989), 245-258. |
[24] |
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. |
[25] |
V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.
doi: 10.1137/0329011. |
[26] |
V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chin. Ann. Math., 14 (1993), 153-164. |
[27] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM, Masson & John Wiley, Paris, 1994. |
[28] |
V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control and Optimization, 35 (1997), 1591-1613.
doi: 10.1137/S0363012996301609. |
[29] |
V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J.M.P.A.,, 69 (1990), 33-54. |
[30] |
J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.
doi: 10.1016/0022-0396(83)90073-6. |
[31] |
J. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 21 (1983), 968-984.
doi: 10.1137/0321059. |
[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), 35-54.
doi: 10.1016/0362-546X(91)90129-O. |
[33] |
I. Lasiecka, Nonlinear boundary feedback stabilization of dynamic elasticity with thermal effects, Shape optimization and optimal design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math., Dekker, New York, 216 (2001), 333-354. |
[34] |
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. |
[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), 1757-1797.
doi: 10.1016/j.na.2005.07.024. |
[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), 13-57.
doi: 10.1006/jmaa.1999.6348. |
[37] |
G. Lebeau, Equation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., Kluwer Acad. Publ., Dordrecht, 19 (1996), 73-109. |
[38] |
G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal., 148 (1999), 179-231.
doi: 10.1007/s002050050160. |
[39] |
C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients, Math. Ann., 331 (2005), 611-629.
doi: 10.1007/s00208-004-0597-z. |
[40] |
J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués, Vol. 1, RMA 8, Masson, Paris, 1988. |
[41] |
K. Liu, Locally distributed control and damping for the conservative systems, S.I.A.M J. Control and Opt., 35 (1997), 1574-1590.
doi: 10.1137/S0363012995284928. |
[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), 1086-1098.
doi: 10.1137/S0363012996310703. |
[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), 769-774.
doi: 10.1016/j.crma.2004.09.029. |
[44] |
K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419-432.
doi: 10.1007/s00033-005-0029-2. |
[45] |
Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.
doi: 10.1007/s00033-004-3073-4. |
[46] |
P. Martinez, PhD Thesis. University of Strasbourg, France, 1998. |
[47] |
P. Martinez, Uniform boundary stabilization of elasticity systems of cubic cystals by nonlinear feedbacks, Nonlinear Analysis 37 (1999), 719-733.
doi: 10.1016/S0362-546X(98)00068-6. |
[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, 321-338, Contemp. Math., 362, Amer. Math. Soc., Providence, RI, 2004.
doi: 10.1090/conm/362/06621. |
[49] |
M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation, Israel J. Math., 95(1996), 25-42.
doi: 10.1007/BF02761033. |
[50] |
M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305 (1996), 403-417.
doi: 10.1007/BF01444231. |
[51] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[52] |
J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[53] |
J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.
doi: 10.1512/iumj.1975.24.24004. |
[54] |
M. Renardy, On localized Kelvin-Voigt damping, ZAMM Z. Angew. Math. Mech., 84 (2004), 280-283.
doi: 10.1002/zamm.200310100. |
[55] |
D. L. Russell, Controllability and stabilizability theory for linear partial differential equations, Recent progress and open problems. SIAM Rev., 20 (1978), 639-739.
doi: 10.1137/1020095. |
[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), 87-97.
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