Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain

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December 2014, 4(4): 451-463. doi: 10.3934/mcrf.2014.4.451

Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain

Ahmed Bchatnia ^1, and Aissa Guesmia ^2,

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

Received July 2013 Revised February 2014 Published September 2014

Abstract
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In this work, we consider the Lamé system in 3-dimension bounded domain with infinite memories. We prove, under some appropriate assumptions, that this system is well-posed and stable, and we get a general and precise estimate on the convergence of solutions to zero at infinity in terms of the growth of the infinite memories.

Keywords: infinite memory, asymptotic behavior, lamé system, general decay, Well-posedness, energy method., semigroup theory.

Mathematics Subject Classification: 35A01, 35B4.

Citation: Ahmed Bchatnia, Aissa Guesmia. Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain. Mathematical Control & Related Fields, 2014, 4 (4) : 451-463. doi: 10.3934/mcrf.2014.4.451

References:

[1]	M. Aassila, Strong asymptotic stability of isotropic elasticity systems with internal damping,, Acta Sci. Math. (Szeged), 64 (1998), 103. Google Scholar
[2]	F. Alabau-Boussouira and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems,, SIAM J. Control and Optim., 37 (1999), 521. doi: 10.1137/S0363012996313835. Google Scholar
[3]	F. Alabau-Boussouira, P. Cannarasa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations,, J. Evol. Equa., 2 (2002), 127. doi: 10.1007/s00028-002-8083-0. Google Scholar
[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. Google Scholar*
[5]	C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Ration. Mech. Anal., 37 (1970), 297. Google Scholar
[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. doi: 10.1006/jmaa.2001.7437. Google Scholar*
[7]	A. Guesmia, Energy decay for a damped nonlinear coupled system,, J. Math. Anal. Appl., 239 (1999), 38. doi: 10.1006/jmaa.1999.6534. Google Scholar
[8]	A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations,, Asymptotic Analysis, 22 (2000), 1. Google Scholar
[9]	A. Guesmia, Contributions à la Contrô Labilité Exacte et la Stabilisation Des SystèMes D'évolution,, Ph.D. Thesis, (2000). Google Scholar
[10]	A. Guesmia, Quelques résultats de stabilisation indirecte des systèmes couplés non dissipatifs,, Bull. Belg. Math. Soc., 15 (2008), 479. Google Scholar
[11]	A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory,, J. Math. Anal. Appl., 382* (2011), 748. doi: 10.1016/j.jmaa.2011.04.079. Google Scholar*
[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. doi: 10.1016/j.nonrwa.2011.08.004. Google Scholar
[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. Google Scholar*
[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. doi: 10.1006/jmaa.1998.5963. Google Scholar*
[15]	M. A. Horn, Stabilization of the dynamic system of elasticity by nonlinear boundary feedback,, Internal Ser. Numer. Math., 133* (1999), 201. Google Scholar*
[16]	B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems,, Comp. Appl. Math., 15 (1996), 199. Google Scholar
[17]	V. Komornik, Exact Controllability and Stabilization., The Multiplier Method, (1994). Google Scholar
[18]	V. Komornik, Boundary stabilization of linear elasticity systems,, Lecture Notes in Pure and Appl. Math., 174 (1996), 135. Google Scholar
[19]	J. E. Lagnese, Boundary stabilization of linear elastodynamic systems,, SIAM J. Control and Optim., 21 (1983), 968. doi: 10.1137/0321059. Google Scholar
[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. doi: 10.1016/0362-546X(91)90129-O. Google Scholar
[21]	P. Martinez, Stabilisation de Systèmes Distribués Semi Linéaires: Domaines Presques Étoilés et Inégalités Intégrales généralisées,, Ph. D. Thesis, (1998). Google Scholar
[22]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[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. doi: 10.3934/mcrf.2012.2.45. Google Scholar

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References:

[1]	M. Aassila, Strong asymptotic stability of isotropic elasticity systems with internal damping,, Acta Sci. Math. (Szeged), 64 (1998), 103. Google Scholar
[2]	F. Alabau-Boussouira and V. Komornik, Boundary observability, controllability and stabilization of linear elastodynamic systems,, SIAM J. Control and Optim., 37 (1999), 521. doi: 10.1137/S0363012996313835. Google Scholar
[3]	F. Alabau-Boussouira, P. Cannarasa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations,, J. Evol. Equa., 2 (2002), 127. doi: 10.1007/s00028-002-8083-0. Google Scholar
[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. Google Scholar*
[5]	C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Ration. Mech. Anal., 37 (1970), 297. Google Scholar
[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. doi: 10.1006/jmaa.2001.7437. Google Scholar*
[7]	A. Guesmia, Energy decay for a damped nonlinear coupled system,, J. Math. Anal. Appl., 239 (1999), 38. doi: 10.1006/jmaa.1999.6534. Google Scholar
[8]	A. Guesmia, On the decay estimates for elasticity systems with some localized dissipations,, Asymptotic Analysis, 22 (2000), 1. Google Scholar
[9]	A. Guesmia, Contributions à la Contrô Labilité Exacte et la Stabilisation Des SystèMes D'évolution,, Ph.D. Thesis, (2000). Google Scholar
[10]	A. Guesmia, Quelques résultats de stabilisation indirecte des systèmes couplés non dissipatifs,, Bull. Belg. Math. Soc., 15 (2008), 479. Google Scholar
[11]	A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory,, J. Math. Anal. Appl., 382* (2011), 748. doi: 10.1016/j.jmaa.2011.04.079. Google Scholar*
[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. doi: 10.1016/j.nonrwa.2011.08.004. Google Scholar
[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. Google Scholar*
[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. doi: 10.1006/jmaa.1998.5963. Google Scholar*
[15]	M. A. Horn, Stabilization of the dynamic system of elasticity by nonlinear boundary feedback,, Internal Ser. Numer. Math., 133* (1999), 201. Google Scholar*
[16]	B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems,, Comp. Appl. Math., 15 (1996), 199. Google Scholar
[17]	V. Komornik, Exact Controllability and Stabilization., The Multiplier Method, (1994). Google Scholar
[18]	V. Komornik, Boundary stabilization of linear elasticity systems,, Lecture Notes in Pure and Appl. Math., 174 (1996), 135. Google Scholar
[19]	J. E. Lagnese, Boundary stabilization of linear elastodynamic systems,, SIAM J. Control and Optim., 21 (1983), 968. doi: 10.1137/0321059. Google Scholar
[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. doi: 10.1016/0362-546X(91)90129-O. Google Scholar
[21]	P. Martinez, Stabilisation de Systèmes Distribués Semi Linéaires: Domaines Presques Étoilés et Inégalités Intégrales généralisées,, Ph. D. Thesis, (1998). Google Scholar
[22]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[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. doi: 10.3934/mcrf.2012.2.45. Google Scholar

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