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Global Strichartz estimates for nontrapping perturbations of the Laplacian. (English) Zbl 0972.35014

This paper is devoted to the proof of a class of Strichartz estimates globally in time for a class of inhomogeneous wave equations with perturbed Laplacian. The perturbed Laplacian is that associated with a Riemann metric \(g\) in the complement \(\Omega\) of a smooth bounded nontrapping obstacle, with Dirichlet condition on \(\partial\Omega\). Furthermore, the metric \(g\) is Euclidean outside of a bounded region. The proof uses (i) local in time Strichartz estimates for the same problem, previously derived by the same authors, (ii) the by now standard global Strichartz estimates in Minkowski space time, (iii) an exponential decay estimate for the local energy of the solution of the homogeneous equation, localized in a neighborhood of the region where the Laplacian is perturbed, and finally (iv) an abstract lemma of Christ and Kiselev showing that the homogeneous Strichartz estimates imply the inhomogeneous ones. The paper also includes a proof of that lemma.

MSC:

35B45 A priori estimates in context of PDEs
35L05 Wave equation
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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References:

[1] Beals M, Progress in Nonlinear Differential Equations and their Applications 21 pp 25– (1996)
[2] M. Christ, A. Kiselev: Maximal Inequality, preprint
[3] DOI: 10.1007/BF01390202 · Zbl 0449.53040
[4] Genibre J, J. Funct. Anal 133 pp 68– (1995)
[5] Grieser, D.Lpbounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries. UCLA
[6] Kapitanski L, Leningrad Math J 1 pp 693– (1990)
[7] DOI: 10.1353/ajm.1998.0039 · Zbl 0922.35028
[8] Lax, P.D and Philips, R.S. ”Scattering Theory Revised Edition”.
[9] DOI: 10.1006/jfan.1995.1075 · Zbl 0846.35085
[10] Mockenhaupt G, J. Amer. Math. Soc 6 pp 65– (1993)
[11] DOI: 10.1007/BF01181697 · Zbl 0538.35063
[12] Ralston J, J. Diff. Geom 12 pp 87– (1977)
[13] DOI: 10.1090/S0894-0347-1995-1308407-1
[14] DOI: 10.1016/0022-1236(70)90027-3 · Zbl 0189.40701
[15] DOI: 10.1215/S0012-7094-77-04430-1 · Zbl 0372.35001
[16] DOI: 10.1002/cpa.3160290102 · Zbl 0318.35009
[17] DOI: 10.1070/RM1975v030n02ABEH001406 · Zbl 0318.35006
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