Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions

TL;DR

This paper proves the obstacle version of the Strauss conjecture in 4D and extends almost global existence results for nonlinear wave equations in obstacle settings using weighted energy and Hardy inequalities.

Contribution

It establishes the Strauss conjecture for obstacle problems in four dimensions and extends almost global existence results to obstacle scenarios in (4+1)-dimensional Minkowski space.

Findings

Obstacle version of Strauss conjecture holds in 4D.

Almost global existence for nonlinear wave equations in obstacle setting.

Uses weighted energy and Hardy inequalities effectively.

Abstract

We show the obstacle version of the Strauss conjecture holds when the spatial dimension is equal to 4. We also show that an almost global existence theorem of H\"ormander for (4+1)-dimensional Minkowski space holds in the obstacle setting. We use weighed space-time variants of the energy inequality and a variant of the classical Hardy inequality.