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Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain
| 1. | Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Campus Universitaire 2092 - El Manar 2, Tunis, Tunisia |
| 2. | Department of Mathematics and Statistics, College of Sciences, King Fahd University of Petroleum and Minerals, P.O. Box 5005, Dhahran 31261, Saudi Arabia |
References:
| [1] |
M. Aassila, Strong asymptotic stability of isotropic elasticity systems with internal damping, Acta Sci. Math. (Szeged), 64 (1998), 103-108. |
| [2] |
F. Alabau-Boussouira and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 37 (1999), 521-542.
doi: 10.1137/S0363012996313835. |
| [3] |
F. Alabau-Boussouira, P. Cannarasa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equa., 2 (2002), 127-150.
doi: 10.1007/s00028-002-8083-0. |
| [4] |
A. Bchatnia and M. Daoulatli, Behavior of the energy for Lamé systems in bounded domains with nonlinear damping and external force, Electron. J. Diff. Equa., 2013 (2013), 1-17. |
| [5] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308. |
| [6] |
C. Giorgi, J. E. Muoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.
doi: 10.1006/jmaa.2001.7437. |
| [7] |
A. Guesmia, Energy decay for a damped nonlinear coupled system, J. Math. Anal. Appl., 239 (1999), 38-48.
doi: 10.1006/jmaa.1999.6534. |
| [8] |
A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations, Asymptotic Analysis, 22 (2000), 1-13. |
| [9] |
A. Guesmia, Contributions à la Contrô Labilité Exacte et la Stabilisation Des SystèMes D'évolution, Ph.D. Thesis, Louis Pasteur University, France, 2000. |
| [10] |
A. Guesmia, Quelques résultats de stabilisation indirecte des systèmes couplés non dissipatifs, Bull. Belg. Math. Soc., 15 (2008), 479-497. |
| [11] |
A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760.
doi: 10.1016/j.jmaa.2011.04.079. |
| [12] |
A. Guesmia and S. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and infinite history memories, Nonlinear Analysis, 13 (2012), 476-485.
doi: 10.1016/j.nonrwa.2011.08.004. |
| [13] |
A. Guesmia, S. Messaoudi and A. Soufyane, On the stabilization for a linear Timoshenko system with infinite history and applications to the coupled Timoshenko-heat systems, Electron. J. Diff. Equa., 2012 (2012), 1-45. |
| [14] |
M. A. Horn, Implications of sharp trace regularity results on boundary Stabilization of the system of linear elasticity, J. Math. Anal. Appl., 223 (1998), 126-150.
doi: 10.1006/jmaa.1998.5963. |
| [15] |
M. A. Horn, Stabilization of the dynamic system of elasticity by nonlinear boundary feedback, Internal Ser. Numer. Math., 133 (1999), 201-210. |
| [16] |
B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Comp. Appl. Math., 15 (1996), 199-212. |
| [17] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994. |
| [18] |
V. Komornik, Boundary stabilization of linear elasticity systems, Lecture Notes in Pure and Appl. Math., 174 (1996), 135-146. |
| [19] |
J. E. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 21 (1983), 968-984.
doi: 10.1137/0321059. |
| [20] |
J. E. Lagnese, Uniform asymptotic energy estimates for solution of the equation of dynamic plane elasticity with nonlinear dissipation at the boundary, Nonlinear Anal. T. M. A., 16 (1991), 35-54.
doi: 10.1016/0362-546X(91)90129-O. |
| [21] |
P. Martinez, Stabilisation de Systèmes Distribués Semi Linéaires: Domaines Presques Étoilés et Inégalités Intégrales généralisées, Ph. D. Thesis, Louis Pasteur University, France, 1998. |
| [22] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [23] |
L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms, Math. Control Relat. Fields, 2 (2012), 45-60.
doi: 10.3934/mcrf.2012.2.45. |
show all references
References:
| [1] |
M. Aassila, Strong asymptotic stability of isotropic elasticity systems with internal damping, Acta Sci. Math. (Szeged), 64 (1998), 103-108. |
| [2] |
F. Alabau-Boussouira and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 37 (1999), 521-542.
doi: 10.1137/S0363012996313835. |
| [3] |
F. Alabau-Boussouira, P. Cannarasa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equa., 2 (2002), 127-150.
doi: 10.1007/s00028-002-8083-0. |
| [4] |
A. Bchatnia and M. Daoulatli, Behavior of the energy for Lamé systems in bounded domains with nonlinear damping and external force, Electron. J. Diff. Equa., 2013 (2013), 1-17. |
| [5] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308. |
| [6] |
C. Giorgi, J. E. Muoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.
doi: 10.1006/jmaa.2001.7437. |
| [7] |
A. Guesmia, Energy decay for a damped nonlinear coupled system, J. Math. Anal. Appl., 239 (1999), 38-48.
doi: 10.1006/jmaa.1999.6534. |
| [8] |
A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations, Asymptotic Analysis, 22 (2000), 1-13. |
| [9] |
A. Guesmia, Contributions à la Contrô Labilité Exacte et la Stabilisation Des SystèMes D'évolution, Ph.D. Thesis, Louis Pasteur University, France, 2000. |
| [10] |
A. Guesmia, Quelques résultats de stabilisation indirecte des systèmes couplés non dissipatifs, Bull. Belg. Math. Soc., 15 (2008), 479-497. |
| [11] |
A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl., 382 (2011), 748-760.
doi: 10.1016/j.jmaa.2011.04.079. |
| [12] |
A. Guesmia and S. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and infinite history memories, Nonlinear Analysis, 13 (2012), 476-485.
doi: 10.1016/j.nonrwa.2011.08.004. |
| [13] |
A. Guesmia, S. Messaoudi and A. Soufyane, On the stabilization for a linear Timoshenko system with infinite history and applications to the coupled Timoshenko-heat systems, Electron. J. Diff. Equa., 2012 (2012), 1-45. |
| [14] |
M. A. Horn, Implications of sharp trace regularity results on boundary Stabilization of the system of linear elasticity, J. Math. Anal. Appl., 223 (1998), 126-150.
doi: 10.1006/jmaa.1998.5963. |
| [15] |
M. A. Horn, Stabilization of the dynamic system of elasticity by nonlinear boundary feedback, Internal Ser. Numer. Math., 133 (1999), 201-210. |
| [16] |
B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Comp. Appl. Math., 15 (1996), 199-212. |
| [17] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994. |
| [18] |
V. Komornik, Boundary stabilization of linear elasticity systems, Lecture Notes in Pure and Appl. Math., 174 (1996), 135-146. |
| [19] |
J. E. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 21 (1983), 968-984.
doi: 10.1137/0321059. |
| [20] |
J. E. Lagnese, Uniform asymptotic energy estimates for solution of the equation of dynamic plane elasticity with nonlinear dissipation at the boundary, Nonlinear Anal. T. M. A., 16 (1991), 35-54.
doi: 10.1016/0362-546X(91)90129-O. |
| [21] |
P. Martinez, Stabilisation de Systèmes Distribués Semi Linéaires: Domaines Presques Étoilés et Inégalités Intégrales généralisées, Ph. D. Thesis, Louis Pasteur University, France, 1998. |
| [22] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [23] |
L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms, Math. Control Relat. Fields, 2 (2012), 45-60.
doi: 10.3934/mcrf.2012.2.45. |
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