Global solution of 3D anisotropic wave equations with the same speed in one direction

Abstract

Motivated by the study of uniaxial crystal optics, we consider a coupled system of $3D$ anisotropic wave equations, in which they have the same speed only in one direction but distinct speeds in the other directions. Moreover, this system has a $1D$-type null structure along the common direction. Unlike the isotropic case, in which the celebrated Klainerman vector fields method is very successful, the main difficulty of the anisotropic case lies in the absence of enough joint commuting vector fields. After carefully analyzing the differences between group velocities of two waves, which are delicate near the same speed direction, the role of null structure, and a fine structure of the phase of oscillation in time, we prove global stability and scattering for small localized initial data.