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

Smith, Hart F.; Sogge, Christopher D.

Commun. Partial Differ. Equations 25, No. 11-12, 2171-2183 (2000).

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.

Reviewer: Jean Ginibre (Orsay)

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

Keywords:

inhomogeneous wave equations with perturbed Laplacian

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