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Local energy decay of the wave equation in an exterior problem and without resonance in the neighborhood of the real line. (Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel.) (French) Zbl 0918.35081
The aim of this paper is to estimate the rate of decay of the local energy of the wave equation for the exterior problem and without resonance in the neighbourhood of the real line. The author shows that the decay is logarithmic in the case when the initial data have compact support. The proof is based on the existence of a region of the form \(\{\lambda\in\mathbb{C}: \text{Im }\lambda< Ce^{-\varepsilon| \lambda|}\}\) which does not contain poles of the scattering matrix. This last fast follows by using Carleman’s inequalities for Helmholtz operator.
Reviewer: C.Popa (Iaşi)

MSC:
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L05 Wave equation
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[1] [FL]Fernandez, C. &Lavine, R., Lower bounds for resonance width in potential and obstacle scattering.Comm. Math. Phys., 128 (1990), 263–284. · Zbl 0712.35072
[2] [G]Gérard, C., Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes.Mém. Soc. Math. France (N.S.), 31 (1988), 1–146. · Zbl 0654.35081
[3] [I1]Ikawa M., Decay of solutions of the wave equation in the exterior of several convex bodies.Ann. Inst. Fourier (Grenoble) 38 (1988), 113–146. · Zbl 0636.35045
[4] [I2] – Decay of solutions of the wave equation in the exterior of two convex bodies.Osaka J. Math. 19 (1982), 459–509. · Zbl 0498.35008
[5] [I3] –, Trapping obstacles, with a sequence of poles converging to the real axis.Osaka J. Math., 22 (1985), 657–689. · Zbl 0617.35102
[6] [I4] –, On the poles of the scattering matrix for two strictly convex obstacles.J. Math. Kyoto Univ., 23 (1983), 127–194. · Zbl 0561.35060
[7] [Le]Lebeau, G., Equation des ondes amorties, dansAlgebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), pp. 73–109. Kluwer Acad. Publ., Dordrecht, 1996.
[8] [LP1]Lax, P. D., &Phillips, R. S., The acoustic equation with an indefinite energy form and the Schrödinger equation.J. Funct. Anal. 1 (1967), 37–83. · Zbl 0186.16401
[9] [LP2] –Scattering Theory, 2e édition. Pure Appl. Math., 26. Academic Press, Boston, MA, 1989.
[10] [LR1]Lebeau, G. &Robbiano, L., Contrôle exact de l’équation de la chaleur.Comm. Partial Differential Equations, 20 (1995), 335–356. · Zbl 0819.35071
[11] [LR2]Lebeau, G. & Robbiano, L., Stabilisation de l’équation des ondes, par le bord. Prépublication de l’université de Paris-Sud 95-40, 1995.
[12] [Mi]Milnor, J.,Lectures on the h-Cobordism Theorem, Princeton Univ. Press, Princeton, NJ, 1965. · Zbl 0161.20302
[13] [Mo1]Morawetz, C. S., The decay of solutions of the exterior initial-boundary value problem for the wave equation.Comm. Pure Appl. Math., 14 (1961), 561–568. · Zbl 0101.07701
[14] [Mo2] –, Decay of solutions of the exterior problem for the wave equation.Comm. Pure Appl. Math., 28 (1975), 229–264. · Zbl 0304.35064
[15] [MS]Melrose, R. B. &Sjöstrand, J., Singularities of boundary value problems I.Comm. Pure Appl. Math., 31 (1978), 593–617. · Zbl 0378.35014
[16] [N]Nédélec, J. C., Quelques propriétés, logarithmiques des fonctions, de Hankel.C. R. Acad. Sci. Paris Sér. I. Math., 314 (1992), 507–510. · Zbl 0747.34020
[17] [Ra1]Ralston, J. V., Solutions of the wave equation with localized energy.Comm. Pure Appl. Math. 22 (1969), 807–823. · Zbl 0209.40402
[18] [Ra2] –, Trapped rays in spherically symmetric media and poles of the scattering matrix.Comm. Pure Appl. Math., 24 (1971), 571–582. · Zbl 0216.13001
[19] [RS]Reed, M. &Simon, B.,Methods of Modern Mathematical Physics, Volumes I–IV. Academic Press, New York-London, 1972–1979.
[20] [Va]Vainberg, B. R.,Asymptotic Methods in Equations of Mathematical Physics, Gordon & Breach, New York, 1989. · Zbl 0743.35001
[21] [Vo]Vodev, G., Sharp bounds on the number of scattering poles in even-dimensional spaces.Duke Math. J., 74 (1994), 1–17. · Zbl 0813.35075
[22] [Wal]Walker, H. F., Some remarks on the local energy decay of solutions of the initialboundary value problem for the wave equation in unbounded domains.J. Differential Equations, 23 (1977), 459–471. · Zbl 0337.35046
[23] [Wat]Watson, G. N.,A Treatise on the Theory of Bessel Functions, 2e édition. Cambridge Univ. Press, Cambridge, 1944. · Zbl 0063.08184
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