Note on the existence of classical solutions of derivative semilinear models for one dimensional wave equation

TL;DR

This paper extends the understanding of long-time existence of classical solutions for one-dimensional semilinear wave equations with derivative nonlinearities, providing new models and proof techniques.

Contribution

It introduces new models for derivative semilinear wave equations in one dimension and extends existing results to more general cases using classical iteration methods.

Findings

Established long-time existence of classical solutions

Extended models to cover more general derivative nonlinearities

Validated results with classical iteration techniques

Abstract

This note is a supplement with a new result to the review paper by Takamura [13] on nonlinear wave equations in one space dimension. We are focusing here to the long-time existence of classical solutions of semilinear wave equations in one space dimension, especially with derivative nonlinear terms of product-type. Our result is an extension of the single component case, but it is meaningful to provide models as possible as many to cover the optimality of the general theory. The proof is based on the classical iteration argument of the point-wise estimate of the solution.