Long Time Boundedness of Planar Jump Discontinuities for Homogeneous Hyperbolic Systems

Suppose that $L(\partial_t,\partial_x)$ is a homogeneous constant coefficient strongly hyperbolic partial differential operator on ${\mathbb R}^{1+d}$ and $H$ is a characteristic hyperplane. Suppose that in a conic neighborhood of the conormal variety of $H$, the characteristic variety of $L$ is the graph of a real analytic function $\tau(\xi)$ with ${\rm rank}\,\tau_{\xi\xi}$ identically equal to zero or the maximal possible value $d-1$. Suppose that the source function $f$ is compactly supported in $t\ge 0$ and piecewise smooth with singularities only on $H$. Then the solution of $Lu=f$ with $u=0$ for $t<0$ is uniformly bounded on ${\mathbb R}^{1+d}$. Typically when ${\rm rank}\,\tau_{\xi\xi}\ne 0$ on the conormal variety, the sup norm of the the jump in the gradient of $u$ across $H$ grows linearly with $t$.