Long Time Boundedness of Planar Jump Discontinuities for Homogeneous Hyperbolic Systems

Abstract

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 $\mathrm{rank}{\tau }_{\xi \xi }$ 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 $\mathrm{rank}{\tau }_{\xi \xi }\ne 0$ on the conormal variety, the sup norm of the the jump in the gradient of $u$ across $H$ grows linearly with $t$.

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