# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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, Hyperbolic equations, (1989), 11-31. doi: 10.2307/479055. [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), 1024-1065. doi: 10.1137/0330055. [3] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroup, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [4] L. Bunimovich, On the ergodic properties of nowhere dispersing billiards, Comm. Math. Phys., 65 (1979), 295-312. doi: 10.1007/BF01197884. [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), 1-29. doi: 10.1007/BF02392877. [6] N. Burq, A. Hassell and J. Wunsch, Spreading of quasimode in the Bunimovich stadium, Proc. Amer. Math. Soc., 135 (2007), 1029-1037. doi: 10.1090/S0002-9939-06-08597-2. [7] N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Let., 14 (2007), 35-47. [8] F. Cardoso and G. Vodev, On the stabilization of the wave equation by the boundary, Serdica Math. J., 28 (2002), 233-240. doi: 10.1016/S0924-0136(02)00391-6. [9] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59 (1985), 145-154. doi: 10.1016/0022-0396(85)90151-2. [10] G. Lebeau, Equation des ondes amorties, in "Algebraic and Geometric Methods in Mathematical Physics" (eds. A. Boutet de Monvel and V. Marchenko), Kluwer Academic Publishers, (1996), 73-109. [11] G. Lebeau and L.Robbiano, Stabilizaion de léquation des ondes par le bord, Duke Math. J., 86 (1997), 465-490. doi: 10.1215/S0012-7094-97-08614-2. [12] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4. [13] J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001. [14] H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains, Math. Res. Letters, 16 (2009), 881-894. [15] K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Diff. Eq., 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016. [16] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain, Discrete and Continuous Dynamical Systems, 20 (2008), 1057-1093. doi: 10.3934/dcds.2008.20.1057. [17] J. Rauch and M. Taylor, Exponential decay of solutions to symmetric hyperbolic equations in bounded domeins, Indiana J. Math., 24 (1974), 79-86. doi: 10.1512/iumj.1974.24.24004. [18] J. Ralston, Gaussian beams and propagation of singularities, in "Studies in Partial Differential Equations" (eds. W. Littman), MAA studies on Mathematics, 23 Math. Assoc. Amer, (1982), 206-248.

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##### 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, Hyperbolic equations, (1989), 11-31. doi: 10.2307/479055. [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), 1024-1065. doi: 10.1137/0330055. [3] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroup, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [4] L. Bunimovich, On the ergodic properties of nowhere dispersing billiards, Comm. Math. Phys., 65 (1979), 295-312. doi: 10.1007/BF01197884. [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), 1-29. doi: 10.1007/BF02392877. [6] N. Burq, A. Hassell and J. Wunsch, Spreading of quasimode in the Bunimovich stadium, Proc. Amer. Math. Soc., 135 (2007), 1029-1037. doi: 10.1090/S0002-9939-06-08597-2. [7] N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Let., 14 (2007), 35-47. [8] F. Cardoso and G. Vodev, On the stabilization of the wave equation by the boundary, Serdica Math. J., 28 (2002), 233-240. doi: 10.1016/S0924-0136(02)00391-6. [9] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations, 59 (1985), 145-154. doi: 10.1016/0022-0396(85)90151-2. [10] G. Lebeau, Equation des ondes amorties, in "Algebraic and Geometric Methods in Mathematical Physics" (eds. A. Boutet de Monvel and V. Marchenko), Kluwer Academic Publishers, (1996), 73-109. [11] G. Lebeau and L.Robbiano, Stabilizaion de léquation des ondes par le bord, Duke Math. J., 86 (1997), 465-490. doi: 10.1215/S0012-7094-97-08614-2. [12] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4. [13] J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001. [14] H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains, Math. Res. Letters, 16 (2009), 881-894. [15] K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Diff. Eq., 240 (2007), 92-124. doi: 10.1016/j.jde.2007.05.016. [16] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain, Discrete and Continuous Dynamical Systems, 20 (2008), 1057-1093. doi: 10.3934/dcds.2008.20.1057. [17] J. Rauch and M. Taylor, Exponential decay of solutions to symmetric hyperbolic equations in bounded domeins, Indiana J. Math., 24 (1974), 79-86. doi: 10.1512/iumj.1974.24.24004. [18] J. Ralston, Gaussian beams and propagation of singularities, in "Studies in Partial Differential Equations" (eds. W. Littman), MAA studies on Mathematics, 23 Math. Assoc. Amer, (1982), 206-248.
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