Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles

TL;DR

This paper establishes long-term existence results for high-dimensional quasilinear wave equations outside star-shaped obstacles, extending four-dimensional results to include nonlinearities depending on the solution itself.

Contribution

It introduces new techniques to handle nonlinearities depending on the solution in exterior domain energy methods, extending previous results.

Findings

Proves global existence for certain high-dimensional quasilinear wave equations

Extends four-dimensional results to higher dimensions

Handles nonlinearities depending on the solution itself

Abstract

We study long time existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. In particular, we obtain exterior domain analogs of the four dimensional results of H\"ormander where the nonlinearity is permitted to depend on the solution not just its first and second derivatives. Previous proofs in exterior domains omitted this dependence as it did not mesh well with the energy methods in use.