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

Abstract

In this paper, we prove a uniqueness theorem for a system of semilinear wave equations satisfying the null condition in R 1+3 . Suppose that two global solutions with C ∞ c 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. Contents 1. Introduction 1 2. Preliminaries 9 3. The Carleman estimates 12 4. Application of the Carleman estimates 24 5. Proof of the main theorems 28 References 37