A uniqueness theorem for 3D semilinear wave equations satisfying the null condition

TL;DR

This paper establishes a uniqueness theorem for 3D semilinear wave equations satisfying the null condition, showing that solutions with matching initial data and radiation fields are identical under certain geometric conditions.

Contribution

It introduces a new uniqueness result for solutions to semilinear wave equations under the null condition, considering initial data and radiation field constraints.

Findings

Solutions are unique outside a hyperboloid or everywhere, depending on initial data and radiation fields.

The theorem applies to solutions with smooth, compactly supported initial data.

It extends understanding of wave equation behavior under null condition constraints.

Abstract

In this paper, we prove a uniqueness theorem for a system of semilinear wave equations satisfying the null condition in $\mathbb{R}^{1+3}$. Suppose that two global solutions with $C_c^\infty$ initial data have equal initial data outside a ball and equal radiation fields outside a light cone. We show that these two solutions are equal either outside a hyperboloid or everywhere in the spacetime, depending on the sizes of the ball and the light cone.