On Abstract Strichartz Estimates and the Strauss Conjecture for Nontrapping Obstacles

TL;DR

This paper demonstrates how local energy decay estimates for linear wave equations with obstacles can lead to optimal global existence results for small nonlinear wave equations, extending to higher dimensions but limited in two dimensions.

Contribution

It establishes a method to derive global existence theorems for nonlinear wave equations using abstract Strichartz estimates and local energy decay, especially for dimensions three and four.

Findings

Linear decay estimates lead to global existence results in 3D and 4D.

Partial success in extending estimates to 2D cases.

Applications to nonlinear wave equations are limited in two dimensions.

Abstract

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.