On almost global existence and local well-posedness for some 3-D quasi-linear wave equations

TL;DR

This paper proves almost global existence and local well-posedness for certain 3-D quasi-linear wave equations with low-regularity data, utilizing space-time $L^2$ estimates under radial symmetry.

Contribution

It introduces a novel approach using space-time $L^2$ estimates to establish existence results for low-regularity data in 3-D quasi-linear wave equations.

Findings

Almost global existence for small initial data in $H^2 \times H^1$

Local well-posedness of the initial value problem

Effective use of space-time $L^2$ estimates under radial symmetry

Abstract

We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time $L^2$ estimate for the variable coefficient wave equation plays a central role for this purpose. Assuming radial symmetry, we establish the almost global existence of a strong solution for every small initial data in $H^2 \times H^1$. We also show that the initial value problem is locally well-posed.