Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations

TL;DR

This paper presents an improved version of Kato's lemma for ordinary differential inequalities, providing a simpler proof that enhances the analysis of lifespan bounds for solutions to semilinear wave equations in high dimensions.

Contribution

It introduces a refined Kato's lemma with a straightforward proof, eliminating the need for rescaling arguments in lifespan analysis of semilinear wave equations.

Findings

Provides an improved Kato's lemma with a simpler proof

Enhances the analysis of solution lifespan bounds in high-dimensional wave equations

Eliminates the need for rescaling arguments in the analysis

Abstract

We are interested in the upper bound of the lifespan of solutions of semilinear wave equations from above. For the sub-critical case in high dimensions, it has been believed that the basic tools of its analysis are Kato's lemma on ordinary differential inequalities and the rescaling argument in the functional method. But there is a small lack of delicate analysis and no published paper about this. Here we give a simple alternative proof by means of improved Kato's lemma without any rescaling argument.