April  2013, 33(4): 1583-1601. doi: 10.3934/dcds.2013.33.1583

Boundary stabilization of the waves in partially rectangular domains

1. 

Faculty of Education, Wakayama University, 930 Sakaedani, Wakayama-shi, Wakayama-Ken 640-8510, Japan

Received  April 2011 Revised  September 2012 Published  October 2012

We study the energy decay to the wave equation with a dissipative boundary condition on partially rectangular domains. We give a polynomial order energy decay under the assumption that the damping term may vanish on the rectangular part. A resolvent estimate for the correspondent stationary problem is proved.
Citation: Hisashi Nishiyama. Boundary stabilization of the waves in partially rectangular domains. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1583-1601. doi: 10.3934/dcds.2013.33.1583
References:
[1]

C. Bardos, G. Lebeau and J. Rauch, Un example d'urilization des notions de propagation pour le controle et la stabilisation de problems hyperboliques,, Rend. sem. Mat. Univ. Pol. Torino Fascicolo speciale, 1988 (1989), 11.  doi: 10.2307/479055.  Google Scholar

[2]

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

[3]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroup,, Math. Ann., 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[4]

L. Bunimovich, On the ergodic properties of nowhere dispersing billiards,, Comm. Math. Phys., 65 (1979), 295.  doi: 10.1007/BF01197884.  Google Scholar

[5]

N. Burq, Décroissace de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel,, Acta Math., 1980 (1998), 1.  doi: 10.1007/BF02392877.  Google Scholar

[6]

N. Burq, A. Hassell and J. Wunsch, Spreading of quasimode in the Bunimovich stadium,, Proc. Amer. Math. Soc., 135 (2007), 1029.  doi: 10.1090/S0002-9939-06-08597-2.  Google Scholar

[7]

N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Let., 14 (2007), 35.   Google Scholar

[8]

F. Cardoso and G. Vodev, On the stabilization of the wave equation by the boundary,, Serdica Math. J., 28 (2002), 233.  doi: 10.1016/S0924-0136(02)00391-6.  Google Scholar

[9]

A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations,, J. Differential Equations, 59 (1985), 145.  doi: 10.1016/0022-0396(85)90151-2.  Google Scholar

[10]

G. Lebeau, Equation des ondes amorties,, in, (1996), 73.   Google Scholar

[11]

G. Lebeau and L.Robbiano, Stabilizaion de léquation des ondes par le bord,, Duke Math. J., 86 (1997), 465.  doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar

[12]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[13]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Review, 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar

[14]

H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains,, Math. Res. Letters, 16 (2009), 881.   Google Scholar

[15]

K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Diff. Eq., 240 (2007), 92.  doi: 10.1016/j.jde.2007.05.016.  Google Scholar

[16]

K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain,, Discrete and Continuous Dynamical Systems, 20 (2008), 1057.  doi: 10.3934/dcds.2008.20.1057.  Google Scholar

[17]

J. Rauch and M. Taylor, Exponential decay of solutions to symmetric hyperbolic equations in bounded domeins,, Indiana J. Math., 24 (1974), 79.  doi: 10.1512/iumj.1974.24.24004.  Google Scholar

[18]

J. Ralston, Gaussian beams and propagation of singularities,, in, 23 (1982), 206.   Google Scholar

show all references

References:
[1]

C. Bardos, G. Lebeau and J. Rauch, Un example d'urilization des notions de propagation pour le controle et la stabilisation de problems hyperboliques,, Rend. sem. Mat. Univ. Pol. Torino Fascicolo speciale, 1988 (1989), 11.  doi: 10.2307/479055.  Google Scholar

[2]

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

[3]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroup,, Math. Ann., 347 (2010), 455.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[4]

L. Bunimovich, On the ergodic properties of nowhere dispersing billiards,, Comm. Math. Phys., 65 (1979), 295.  doi: 10.1007/BF01197884.  Google Scholar

[5]

N. Burq, Décroissace de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel,, Acta Math., 1980 (1998), 1.  doi: 10.1007/BF02392877.  Google Scholar

[6]

N. Burq, A. Hassell and J. Wunsch, Spreading of quasimode in the Bunimovich stadium,, Proc. Amer. Math. Soc., 135 (2007), 1029.  doi: 10.1090/S0002-9939-06-08597-2.  Google Scholar

[7]

N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Let., 14 (2007), 35.   Google Scholar

[8]

F. Cardoso and G. Vodev, On the stabilization of the wave equation by the boundary,, Serdica Math. J., 28 (2002), 233.  doi: 10.1016/S0924-0136(02)00391-6.  Google Scholar

[9]

A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations,, J. Differential Equations, 59 (1985), 145.  doi: 10.1016/0022-0396(85)90151-2.  Google Scholar

[10]

G. Lebeau, Equation des ondes amorties,, in, (1996), 73.   Google Scholar

[11]

G. Lebeau and L.Robbiano, Stabilizaion de léquation des ondes par le bord,, Duke Math. J., 86 (1997), 465.  doi: 10.1215/S0012-7094-97-08614-2.  Google Scholar

[12]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630.  doi: 10.1007/s00033-004-3073-4.  Google Scholar

[13]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Review, 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar

[14]

H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains,, Math. Res. Letters, 16 (2009), 881.   Google Scholar

[15]

K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Diff. Eq., 240 (2007), 92.  doi: 10.1016/j.jde.2007.05.016.  Google Scholar

[16]

K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain,, Discrete and Continuous Dynamical Systems, 20 (2008), 1057.  doi: 10.3934/dcds.2008.20.1057.  Google Scholar

[17]

J. Rauch and M. Taylor, Exponential decay of solutions to symmetric hyperbolic equations in bounded domeins,, Indiana J. Math., 24 (1974), 79.  doi: 10.1512/iumj.1974.24.24004.  Google Scholar

[18]

J. Ralston, Gaussian beams and propagation of singularities,, in, 23 (1982), 206.   Google Scholar

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