# American Institute of Mathematical Sciences

November  2014, 34(11): 4371-4388. doi: 10.3934/dcds.2014.34.4371

## Polynomial stabilization of some dissipative hyperbolic systems

 1 UR Analyse et Contrôle des Edp (05/UR/15-01), Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, 5019 Monastir, Tunisia 2 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1 3 Université de Valenciennes et du Hainaut Cambrésis, LAMAV and FR CNRS 2956, Le Mont Houy, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9

Received  September 2013 Revised  January 2014 Published  May 2014

We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
Citation: Kais Ammari, Eduard Feireisl, Serge Nicaise. Polynomial stabilization of some dissipative hyperbolic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4371-4388. doi: 10.3934/dcds.2014.34.4371
##### References:
 [1] G. Allaire, Homogenization of the Navier-Stokes equations and derivation of Brinkman's law, In Mathématiques appliquées aux sciences de l'ingénieur (Santiago, 1989), pages 7-20. Cépaduès, Toulouse, 1991. [2] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386. doi: 10.1051/cocv:2001114. [3] P. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math., 81 (1999), 497-520. doi: 10.1007/s002110050401. [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] C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Rend. Sem. Mat. Univ. Politec. Torino 1988, (Special Issue), (1989), 11-31. Nonlinear hyperbolic equations in applied sciences. [6] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055. [7] 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. [8] 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. [9] 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. [10] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math., 46 (1989), 245-258. [11] J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978. [12] M. J. Lighthill, On sound generated aerodynamically. I. General theory, Proc. Roy. Soc. London. Ser. A., 211 (1952), 564-587. doi: 10.1098/rspa.1952.0060. [13] M. J. Lighthill, On sound generated aerodynamically. II. Turbulence as a source of sound, Proc. Roy. Soc. London. Ser. A., 222 (1954), 1-32. doi: 10.1098/rspa.1954.0049. [14] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. [15] K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928. [16] 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. [17] K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Differential Equations, 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016. [18] L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 57-62. doi: 10.1016/j.crma.2011.12.001. [19] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asymptotic Anal., 1 (1988), 161-185. [20] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl. (9), 70 (1991), 513-529.

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##### References:
 [1] G. Allaire, Homogenization of the Navier-Stokes equations and derivation of Brinkman's law, In Mathématiques appliquées aux sciences de l'ingénieur (Santiago, 1989), pages 7-20. Cépaduès, Toulouse, 1991. [2] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386. doi: 10.1051/cocv:2001114. [3] P. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math., 81 (1999), 497-520. doi: 10.1007/s002110050401. [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] C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Rend. Sem. Mat. Univ. Politec. Torino 1988, (Special Issue), (1989), 11-31. Nonlinear hyperbolic equations in applied sciences. [6] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055. [7] 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. [8] 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. [9] 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. [10] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math., 46 (1989), 245-258. [11] J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge, 1978. [12] M. J. Lighthill, On sound generated aerodynamically. I. General theory, Proc. Roy. Soc. London. Ser. A., 211 (1952), 564-587. doi: 10.1098/rspa.1952.0060. [13] M. J. Lighthill, On sound generated aerodynamically. II. Turbulence as a source of sound, Proc. Roy. Soc. London. Ser. A., 222 (1954), 1-32. doi: 10.1098/rspa.1954.0049. [14] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. [15] K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928. [16] 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. [17] K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Differential Equations, 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016. [18] L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations, C. R. Math. Acad. Sci. Paris, 350 (2012), 57-62. doi: 10.1016/j.crma.2011.12.001. [19] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems, Asymptotic Anal., 1 (1988), 161-185. [20] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl. (9), 70 (1991), 513-529.
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