American Institute of Mathematical Sciences

Advanced
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

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), 103-108.  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), 521-542. 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), 127-150. 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), 1-17.  Google Scholar

[5]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  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), 83-99. 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), 38-48. 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), 1-13.  Google Scholar

[9]

A. Guesmia, Contributions à la Contrô Labilité Exacte et la Stabilisation Des SystèMes D'évolution, Ph.D. Thesis, Louis Pasteur University, France, 2000. Google Scholar

[10]

A. Guesmia, Quelques résultats de stabilisation indirecte des systèmes couplés non dissipatifs, Bull. Belg. Math. Soc., 15 (2008), 479-497.  Google Scholar

[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.  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), 476-485. 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), 1-45.  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), 126-150. 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), 201-210.  Google Scholar

[16]

B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Comp. Appl. Math., 15 (1996), 199-212.  Google Scholar

[17]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.  Google Scholar

[18]

V. Komornik, Boundary stabilization of linear elasticity systems, Lecture Notes in Pure and Appl. Math., 174 (1996), 135-146.  Google Scholar

[19]

J. E. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 21 (1983), 968-984. 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), 35-54. 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, Louis Pasteur University, France, 1998. Google Scholar

[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.  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), 45-60. doi: 10.3934/mcrf.2012.2.45.  Google Scholar

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.  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), 521-542. 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), 127-150. 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), 1-17.  Google Scholar

[5]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  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), 83-99. 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), 38-48. 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), 1-13.  Google Scholar

[9]

A. Guesmia, Contributions à la Contrô Labilité Exacte et la Stabilisation Des SystèMes D'évolution, Ph.D. Thesis, Louis Pasteur University, France, 2000. Google Scholar

[10]

A. Guesmia, Quelques résultats de stabilisation indirecte des systèmes couplés non dissipatifs, Bull. Belg. Math. Soc., 15 (2008), 479-497.  Google Scholar

[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.  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), 476-485. 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), 1-45.  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), 126-150. 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), 201-210.  Google Scholar

[16]

B. V. Kapitonov, Uniform stabilization and exact controllability for a class of coupled hyperbolic systems, Comp. Appl. Math., 15 (1996), 199-212.  Google Scholar

[17]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.  Google Scholar

[18]

V. Komornik, Boundary stabilization of linear elasticity systems, Lecture Notes in Pure and Appl. Math., 174 (1996), 135-146.  Google Scholar

[19]

J. E. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control and Optim., 21 (1983), 968-984. 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), 35-54. 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, Louis Pasteur University, France, 1998. Google Scholar

[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.  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), 45-60. doi: 10.3934/mcrf.2012.2.45.  Google Scholar

[1]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[2]

Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations & Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035
[3]

Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure & Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457
[4]

George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151
[5]

Rainer Brunnhuber, Barbara Kaltenbacher. Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4515-4535. doi: 10.3934/dcds.2014.34.4515
[6]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297
[7]

Michele Colturato. Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics. Evolution Equations & Control Theory, 2018, 7 (2) : 217-245. doi: 10.3934/eect.2018011
[8]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2039-2064. doi: 10.3934/cpaa.2021057
[9]

Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259
[10]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[11]

Fuyi Xu, Meiling Chi, Lishan Liu, Yonghong Wu. On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2515-2559. doi: 10.3934/dcds.2020140
[12]

Tong Li. Well-posedness theory of an inhomogeneous traffic flow model. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 401-414. doi: 10.3934/dcdsb.2002.2.401
[13]

Carlos F. Daganzo. On the variational theory of traffic flow: well-posedness, duality and applications. Networks & Heterogeneous Media, 2006, 1 (4) : 601-619. doi: 10.3934/nhm.2006.1.601
[14]

Salim A. Messaoudi, Jamilu Hashim Hassan. New general decay results in a finite-memory bresse system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1637-1662. doi: 10.3934/cpaa.2019078
[15]

Jianghao Hao, Junna Zhang. General stability of abstract thermoelastic system with infinite memory and delay. Mathematical Control & Related Fields, 2021, 11 (2) : 353-371. doi: 10.3934/mcrf.2020040
[16]

Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407
[17]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
[18]

Gustavo Ponce, Jean-Claude Saut. Well-posedness for the Benney-Roskes/Zakharov- Rubenchik system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 811-825. doi: 10.3934/dcds.2005.13.811
[19]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126
[20]

Manas Bhatnagar, Hailiang Liu. Well-posedness and critical thresholds in a nonlocal Euler system with relaxation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5271-5289. doi: 10.3934/dcds.2021076

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (125)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences