March  2016, 5(1): 37-59. doi: 10.3934/eect.2016.5.37

Energy decay rates for solutions of the wave equation with linear damping in exterior domain

1. 

University of Dammam, King Faisal Road, 31952, Dammam, Saudi Arabia

Received  October 2014 Revised  January 2016 Published  March 2016

In this paper we study the behavior of the energy and the $L^{2}$ norm of solutions of the wave equation with localized linear damping in exterior domain. Let $u$ be a solution of the wave system with initial data $\left( u_{0},u_{1}\right) $. We assume that the damper is positive at infinity then under the Geometric Control Condition of Bardos et al [5] (1992), we prove that:
    1. If $(u_{0},u_{1}) $ belong to $H_{0}^{1}( \Omega) \times L^{2}( \Omega ) ,$ then the total energy $ E_{u}(t) \leq C_{0}(1+t) ^{-1}I_{0}$ and $\Vert u(t) \Vert _{L^{2}}^{2}\leq C_{0}I_{0},$ where \begin{equation*} I_{0}=\left\Vert u_{0}\right\Vert _{H^{1}}^{2}+\left\Vert u_{1}\right\Vert _{L^{2}}^{2}. \end{equation*}    2. If the initial data $\left( u_{0},u_{1}\right) $ belong to $ H_{0}^{1}\left( \Omega \right) \times L^{2}\left( \Omega \right) $ and verifies \begin{equation*} \left\Vert d\left( \cdot \right) \left( u_{1}+au_{0}\right) \right\Vert _{L^{2}}<+\infty , \end{equation*} then the total energy $E_{u}\left( t\right) \leq C_{2}\left( 1+t\right) ^{-2}I_{1}$ and $\left\Vert u\left( t\right) \right\Vert _{L^{2}}^{2} \leq C_{2} \left( 1+t\right) ^{-1}I_{1},$ where \begin{equation*} I_{1}=\left\Vert u_{0}\right\Vert _{H^{1}}^{2}+\left\Vert u_{1}\right\Vert _{L^{2}}^{2}+\left\Vert d\left( \cdot \right) \left( u_{1}+au_{0}\right) \right\Vert _{L^{2}}^{2} \end{equation*} and \begin{equation*} d\left( x\right) =\left\{ \begin{array}{lc} \left\vert x\right\vert & d\geq 3, \\ \left\vert x\right\vert \ln \left( B\left\vert x\right\vert \right) & d=2, \end{array} \right. . \end{equation*} with $B$ $\underset{x\in \Omega }{\inf } \left\vert x\right\vert \geq 2$.
Citation: Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations and Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
References:
[1]

L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Communications in Partial Differential Equations, 36 (2011), 797-818. doi: 10.1080/03605302.2010.534684.

[2]

L. Aloui and M. Khenissi, Stabilisation de l'équation des ondes dans un domaine extérieur, Rev. Mat. Iberoamerica, 18 (2002), 1-16. doi: 10.4171/RMI/309.

[3]

J. Bae and M. Nakao, Energy decay for the wave equation with boundary and localized dissipations in exterior domains, Math. Nachr., 278 (2005), 771-783. doi: 10.1002/mana.200310271.

[4]

A. Bchatnia and M. Daoulatli, Local energy decay for the wave equation with a nonlinear time dependent damping, Appl. Anal., 92 (2013), 2288-2308. doi: 10.1080/00036811.2012.734375.

[5]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optimization, 30 (1992), 1024-1065. doi: 10.1137/0330055.

[6]

N. Burq, Mesures semi-classiques et mesures de défaut, (French) [Semiclassical measures and defect measures] Séminaire Bourbaki, Astérisque, 1996/97 (1997), 167-195.

[7]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.

[8]

W. Dan and Y. Shibata, On a local energy decay of solutions of a dissipative wave equation, Funkcial. Ekvac., 38 (1995), 545-568.

[9]

M. Daoulatli, Local energy decay for the nonlinear dissipative wave equation in an exterior domain, Port. Math., 64 (2007), 39-65. doi: 10.4171/PM/1775.

[10]

M. Daoulatli, B. Dehman and M. Khenissi, Local energy decay for the elastic system with nonlinear damping in an exterior domain, SIAM J.Control Optim., 48 (2010), 5254-5275. doi: 10.1137/090757332.

[11]

M. Daoulatli, Behaviors of the energy of solutions of the wave equation with damping and external force, Journal of Mathematical Analysis and Applications, 389 (2012), 205-225. doi: 10.1016/j.jmaa.2011.11.051.

[12]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Ann. Scient. Éc. Norm. Sup., 36 (2003), 525-551. doi: 10.1016/S0012-9593(03)00021-1.

[13]

G. Francfort, An introduction to H-measures and their applications, Variational problems in materials science, 85-110, Progr. Nonlinear Differential Equations Appl., 68, Birkhuser, Basel, 2006. doi: 10.1007/3-7643-7565-5_7.

[14]

P. Gérard, Microlocal defect measures, Comm. Partial Diff. eq., 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.

[15]

P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Mathematical Journal, 71 (1993), 559-607. doi: 10.1215/S0012-7094-93-07122-0.

[16]

R. Ikehata, Energy decay of solutions for the semilinear dissipative wave equations in an exterior domain, Funkcial. Ekvac., 44 (2001), 487-499.

[17]

R. Ikehata, Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain, Journal of Differential Equations, 188 (2003), 390-405. doi: 10.1016/S0022-0396(02)00101-8.

[18]

R. Ikehata and T. Matsuyama, L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon., 55 (2002), 33-42.

[19]

R. Joly and L. Camille, Stabilization for the semilinear wave equation with geometric control condition, Anal. PDE, 6 (2013), 1089-1119. doi: 10.2140/apde.2013.6.1089.

[20]

S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Jpn., 47 (1995), 617-653. doi: 10.2969/jmsj/04740617.

[21]

M. Khenissi, Équation des ondes amorties dans un domaine extérieur, Bull. Soc. Math. France, 131 (2003), 211-228.

[22]

P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, 26. Academic Press, Inc., Boston, MA, 1989.

[23]

G. Lebeau, Equation des ondes amorties, In Algebraic and geometric methods in mathematical physics, proceedings of the Kaciveli Summer School, Crimea, Ukraine, Springer, 19 (1996), 73-109.

[24]

M. Nakao, Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation, J. Diff. Eq., 148 (1998), 388-406. doi: 10.1006/jdeq.1998.3468.

[25]

M. Nakao, Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations, Math. Z., 238 (2001), 781-797. doi: 10.1007/s002090100275.

[26]

M. Nakao, Decay of solutions to the Cauchy problem for the Klein-Gordon equation with a localized nonlinear dissipation, Hokkaido Math. J., 27 (1998), 245-271. doi: 10.14492/hokmj/1351001285.

[27]

K. Ono, $L^{1}$ estimates for dissipative wave equations in exterior domains, J. Math. Anal. Appl., 333 (2007), 1079-1092. doi: 10.1016/j.jmaa.2006.01.031.

[28]

R. Racke, Nonhomogeneous nonlinear damped wave equations in unbounded domains, Math. Methods Appl. Sci., 13 (1990), 481-491. doi: 10.1002/mma.1670130604.

[29]

D. Tataru, The $X^{s,\theta }$ spaces and unique continuation for solutions to the semilinear wave equations, Comm. P.D.E., 21 (1996), 841-887. doi: 10.1080/03605309608821210.

[30]

L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proceedings of the Royal Society of Edinburgh.Section A. Mathematics, 115 (1990), 193-230. doi: 10.1017/S0308210500020606.

[31]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. pures et appl., 70 (1992), 513-529.

show all references

References:
[1]

L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Communications in Partial Differential Equations, 36 (2011), 797-818. doi: 10.1080/03605302.2010.534684.

[2]

L. Aloui and M. Khenissi, Stabilisation de l'équation des ondes dans un domaine extérieur, Rev. Mat. Iberoamerica, 18 (2002), 1-16. doi: 10.4171/RMI/309.

[3]

J. Bae and M. Nakao, Energy decay for the wave equation with boundary and localized dissipations in exterior domains, Math. Nachr., 278 (2005), 771-783. doi: 10.1002/mana.200310271.

[4]

A. Bchatnia and M. Daoulatli, Local energy decay for the wave equation with a nonlinear time dependent damping, Appl. Anal., 92 (2013), 2288-2308. doi: 10.1080/00036811.2012.734375.

[5]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optimization, 30 (1992), 1024-1065. doi: 10.1137/0330055.

[6]

N. Burq, Mesures semi-classiques et mesures de défaut, (French) [Semiclassical measures and defect measures] Séminaire Bourbaki, Astérisque, 1996/97 (1997), 167-195.

[7]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.

[8]

W. Dan and Y. Shibata, On a local energy decay of solutions of a dissipative wave equation, Funkcial. Ekvac., 38 (1995), 545-568.

[9]

M. Daoulatli, Local energy decay for the nonlinear dissipative wave equation in an exterior domain, Port. Math., 64 (2007), 39-65. doi: 10.4171/PM/1775.

[10]

M. Daoulatli, B. Dehman and M. Khenissi, Local energy decay for the elastic system with nonlinear damping in an exterior domain, SIAM J.Control Optim., 48 (2010), 5254-5275. doi: 10.1137/090757332.

[11]

M. Daoulatli, Behaviors of the energy of solutions of the wave equation with damping and external force, Journal of Mathematical Analysis and Applications, 389 (2012), 205-225. doi: 10.1016/j.jmaa.2011.11.051.

[12]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Ann. Scient. Éc. Norm. Sup., 36 (2003), 525-551. doi: 10.1016/S0012-9593(03)00021-1.

[13]

G. Francfort, An introduction to H-measures and their applications, Variational problems in materials science, 85-110, Progr. Nonlinear Differential Equations Appl., 68, Birkhuser, Basel, 2006. doi: 10.1007/3-7643-7565-5_7.

[14]

P. Gérard, Microlocal defect measures, Comm. Partial Diff. eq., 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.

[15]

P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Mathematical Journal, 71 (1993), 559-607. doi: 10.1215/S0012-7094-93-07122-0.

[16]

R. Ikehata, Energy decay of solutions for the semilinear dissipative wave equations in an exterior domain, Funkcial. Ekvac., 44 (2001), 487-499.

[17]

R. Ikehata, Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain, Journal of Differential Equations, 188 (2003), 390-405. doi: 10.1016/S0022-0396(02)00101-8.

[18]

R. Ikehata and T. Matsuyama, L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon., 55 (2002), 33-42.

[19]

R. Joly and L. Camille, Stabilization for the semilinear wave equation with geometric control condition, Anal. PDE, 6 (2013), 1089-1119. doi: 10.2140/apde.2013.6.1089.

[20]

S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Jpn., 47 (1995), 617-653. doi: 10.2969/jmsj/04740617.

[21]

M. Khenissi, Équation des ondes amorties dans un domaine extérieur, Bull. Soc. Math. France, 131 (2003), 211-228.

[22]

P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, 26. Academic Press, Inc., Boston, MA, 1989.

[23]

G. Lebeau, Equation des ondes amorties, In Algebraic and geometric methods in mathematical physics, proceedings of the Kaciveli Summer School, Crimea, Ukraine, Springer, 19 (1996), 73-109.

[24]

M. Nakao, Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation, J. Diff. Eq., 148 (1998), 388-406. doi: 10.1006/jdeq.1998.3468.

[25]

M. Nakao, Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations, Math. Z., 238 (2001), 781-797. doi: 10.1007/s002090100275.

[26]

M. Nakao, Decay of solutions to the Cauchy problem for the Klein-Gordon equation with a localized nonlinear dissipation, Hokkaido Math. J., 27 (1998), 245-271. doi: 10.14492/hokmj/1351001285.

[27]

K. Ono, $L^{1}$ estimates for dissipative wave equations in exterior domains, J. Math. Anal. Appl., 333 (2007), 1079-1092. doi: 10.1016/j.jmaa.2006.01.031.

[28]

R. Racke, Nonhomogeneous nonlinear damped wave equations in unbounded domains, Math. Methods Appl. Sci., 13 (1990), 481-491. doi: 10.1002/mma.1670130604.

[29]

D. Tataru, The $X^{s,\theta }$ spaces and unique continuation for solutions to the semilinear wave equations, Comm. P.D.E., 21 (1996), 841-887. doi: 10.1080/03605309608821210.

[30]

L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proceedings of the Royal Society of Edinburgh.Section A. Mathematics, 115 (1990), 193-230. doi: 10.1017/S0308210500020606.

[31]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. pures et appl., 70 (1992), 513-529.

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