On Abstract Strichartz Estimates and the Strauss Conjecture for Nontrapping Obstacles

Details On Abstract Strichartz Estimates and the Strauss Conjecture for Nontrapping Obstacles On Abstract Strichartz Estimates and the Strauss Conjecture for Nontrapping Obstacles

2008-05-12

arxiv.org/abs/0805.1673v2

Mathematics

Analysis of PDEs

21 pages, simplified existence proof and corrected typos

Scientific paper

The purpose of this paper is to show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small amplitude nonlinear wave equations with power nonlinearities. To achieve this goal, at least for spatial dimensions $n=3$ and 4, we shall show how the aforementioned linear decay estimates can be combined with "abstract Strichartz" estimates for the free wave equation to prove corresponding estimates for the perturbed wave equation when $n\ge3$. As we shall see, we are only partially successful in the latter endeavor when the dimension is equal to two, and therefore, at present, our applications to nonlinear wave equations in this case are limited.

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