Boundary stabilization of the waves in partially rectangular domains

Hisashi Nishiyama

doi:10.3934/dcds.2013.33.1583

Article Contents

2013, Volume 33, Issue 4: 1583-1601. Doi: 10.3934/dcds.2013.33.1583

This issue Previous Article Periodic solutions of Liénard equations with resonant isochronous potentials Next Article Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations

Boundary stabilization of the waves in partially rectangular domains

Hisashi Nishiyama^1,

1.
Faculty of Education, Wakayama University, 930 Sakaedani, Wakayama-shi, Wakayama-Ken 640-8510

Received: April 2011

Revised: September 2012

Published: April 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L05, 47A10, 47B44, 47D06, 49J20.

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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.