Propagation of singularities for a system of semilinear wave equations with null condition

Abstract

We study the propagation of singularities for a system of semilinear wave equations satisfying the null condition in one space dimension. We show that if a solution $(u,v)$ to the system is in $H_{loc}^s(\mathrm{\Omega })\cap H_{ml}^r(0,x_0,{\tau }_0,{\xi }_0)$, then $(u,v)\in H_{ml}^r(\mathrm{\Gamma })$ as long as , where $\mathrm{\Omega }\subset {ℝ}^2$ is an open set and $\mathrm{\Gamma }$ is a null bicharacteristic of $\square$ passing through $(0,x_0,{\tau }_0,{\xi }_0)$.