Improvement of the general theory for one dimensional nonlinear wave equations related to the combined effect

TL;DR

This paper advances the theoretical understanding of one-dimensional nonlinear wave equations by improving lifespan estimates for solutions with small initial data, addressing a problem anticipated over three decades ago.

Contribution

It provides an improved lower bound estimate of the lifespan for solutions to nonlinear wave equations considering combined effects, filling a long-standing gap in the theory.

Findings

Enhanced lifespan estimates for solutions

Addresses combined effect scenarios

Fills a 30-year-old theoretical gap

Abstract

We focus on the general theory to the Cauchy problem for one dimensional nonlinear wave equations with small initial data. In the general theory, we aim to obtain the lower bound estimate of the lifespan of classical solution. In this paper, we improve it in some case related to the combined effect, which was expected complete more than 30 years ago.