Revisit to Fritz John's paper on the blow-up of nonlinear wave equations

TL;DR

This paper revisits Fritz John's 1979 result on the blow-up of solutions to certain nonlinear wave equations, providing a simplified proof using a Gronwall's inequality.

Contribution

It offers a simplified proof of John's blow-up result for the wave equation with power nonlinearity, enhancing understanding of the original proof.

Findings

Confirmed blow-up for 1<p<1+√2 with compact support

Provided a more accessible proof technique

Clarified conditions for solution nonexistence

Abstract

In Fritz John's famous paper (1979), he discovered that for the wave equation $\Box u=|u|^p$, where $1<p<1+\sqrt{2}$ and $\Box$ denoting the d'Alembertian, there is no global solution for any nontrivial and compactly supported initial data. This paper is intended to simplify his proof by applying a Gronwall's type inequality.