Solutions with a uniform time of existence of a class of characteristic semi-linear wave equations near S+

TL;DR

This paper proves existence and uniqueness of solutions for a class of semi-linear wave equations with null conditions, with initial data on a light-cone in Minkowski space, ensuring solutions persist uniformly near the cone.

Contribution

It establishes a new existence and uniqueness result for semi-linear wave equations with null conditions on characteristic initial data in Minkowski space.

Findings

Solutions exist and are unique near the light-cone

The neighborhood of the initial cone does not shrink at infinity

Application to wave maps in Minkowski space

Abstract

We prove existence and uniqueness of solution of a class of semi-linear wave equations with initial data prescribed on the light-cone with vertex the origin of a Minkowski space-time. The nonlinear term is assumed to satisfy a nullity condition which guarantee that the neighborhood of the initial cone on which we obtain our solution does not shrink to zero as one approaches infinity. This result is applied to wave maps on Minkowski space-time $\R^{n+1}$ with $n\ge 3$.