A Positivity Criterion for the Wave Equation and Global Existence of Large Solutions

TL;DR

This paper establishes a positivity criterion for the wave equation in low dimensions and uses it to prove the existence of large global solutions for certain supercritical and other semilinear wave equations.

Contribution

It introduces a positivity criterion based on the fundamental solution and applies it to obtain new global existence results for various semilinear wave equations.

Findings

Existence of large global solutions in energy-supercritical cases

Application of positivity criterion to subcritical and critical cases

Extension of known results using positivity-based methods

Abstract

In dimensions one to three, the fundamental solution to the free wave equation is positive. Therefore, there exists a simple positivity criterion for solutions. We use this to obtain large global solutions to two well-studied energy-supercritical semilinear wave equations, as well as some new results in the subcritical and critical cases.