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Stabilization of the wave equation in an exterior domain. (Stabilisation pour l’équation des ondes dans un domaine extérieur.) (French) Zbl 1016.35044

The Dirichlet initial boundary value problem for the damped wave equation \[ (\partial^2_0- \Delta)u+ a\partial_0 u= 0\quad\text{in }\mathbb{R}_{>0}\times \Omega,\quad u= 0\quad\text{on }\mathbb{R}_{>0}\times\partial\Omega, \] with prescribed initial data \(u(0+)= f_1\), \(\partial_0 u(0+)= f_2\) in \(\Omega\) is considered in an exterior domain \(\Omega\) of \(\mathbb{R}^n\), \(n\) odd, with smooth boundary. The multiplicative damping coefficient \(a\) is assumed to be smooth and nonnegative. Under a geometric assumption (exterior geometric control) the authors are – by using semigroup methods and Lax-Phillips scattering theory – able to show exponential energy decay for compactly supported initial data.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35P25 Scattering theory for PDEs
93D15 Stabilization of systems by feedback
93C20 Control/observation systems governed by partial differential equations
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References:

[1] Bardos, C., Lebeau, G. et Rauch, J., Sharp sufficient conditions for the observation, control and stabilisation of waves from the boundary. SIAM J. Control Optim. 305 (1992), 1024–1065. · Zbl 0786.93009 · doi:10.1137/0330055
[2] Burq, N., Décroissance de l’energie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180 (1998), 16–29. · Zbl 0918.35081 · doi:10.1007/BF02392877
[3] Gérard, P., Microlocal defect measures. Comm. Partial Differential Equations 16 (1991), 1761–1794. · Zbl 0770.35001 · doi:10.1080/03605309108820822
[4] Gérard, P., Oscillations and concentrations effets in semi-linear dispersive wave equation. J. Funct. Anal. 41 (1996), no. 1, 60–98. · Zbl 0868.35075 · doi:10.1006/jfan.1996.0122
[5] Lax, D., Morawetz, C. S., Phillips, R. S., Exponential decay of solution of the wave equation in the exterior of a star shaped obstacle. Comm. Pure Appl. Math. 16 (1963), 477–486. · Zbl 0161.08001 · doi:10.1002/cpa.3160160407
[6] Lax, P. D., Phillips, R. S., Scattering theory decay . Academic Press, New York, 1967. · Zbl 0186.16301
[7] Lebeau, G., Control for hyperbolic equations. Actes du colloque de Saint Jean de Monts (1991). · Zbl 0753.35053
[8] Lebeau, G., Equations des ondes amorties, en Algebraic and Geometric Methods in Math.Physic , A. Boutet de Monvel and V. Marchenko (eds), 73–109. Kluwer Academic, The Netherlands, 1996.
[9] Lions, J. L., Contrôlabilité exacte. Perturbation et stabilisation des systèmes distribués , R.M.A., Masson, 1988. · Zbl 0653.93002
[10] Melrose, R., Sjöstrand, J., Singularities of boundary value problems I. Comm. Pure. Appl. Math. 31 (1978), 593–617. · Zbl 0368.35020 · doi:10.1002/cpa.3160310504
[11] Melrose, R., Sjöstrand, J., Singularities of boundary value problems II, Comm. Pure. Appl. Math. 35 (1982), 129–168. · Zbl 0546.35083 · doi:10.1002/cpa.3160350202
[12] Melrose, R., Singularities and energy decay in acoustical scattering. Duke Math. J. 46 (1979), 43–59. · Zbl 0415.35050 · doi:10.1215/S0012-7094-79-04604-0
[13] Morawetz, C. S., Decay for solutions of the exterior problem for the wave equation. Comm. Pure. Appl. Math. 28 (1975), 229–264. · Zbl 0304.35064 · doi:10.1002/cpa.3160280204
[14] Morawetz, C. S., Ralston, J., Strauss, W., Decay of solutions of the wave equation outside non-trapping obstacles. Comm. Pure. Appl. Math. 30 (1977), 447–508. · Zbl 0372.35008 · doi:10.1002/cpa.3160300405
[15] Ralston, J., Solution of the wave equation with localized energy. Comm. Pure. Appl. Math. 22 (1969), 807–823. · Zbl 0209.40402 · doi:10.1002/cpa.3160220605
[16] Rauch, J., Taylor, M., Exponential decay of solutions for the hyperbolic equation in bouded domain. Indiana Univ. Math. J. 24 (1972), 74–86. · Zbl 0281.35012 · doi:10.1512/iumj.1974.24.24004
[17] Strauss, W. A., Dispersal of waves vanishing on the boundary of a exterior domain. Comm. Pure. Appl. Math. 28 (1975), 265–278. · Zbl 0297.35047 · doi:10.1002/cpa.3160280205
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