Finite speed of propagation for linear symmetric hyperbolic systems: cone domination and independence of the extension

(Correct symbol domination.) For every $\mathrm{\mid }\xi \mathrm{\mid }\le 1$, $\mathrm{\parallel }{\sum }_i{\xi }_i{A}^i\mathrm{\parallel }\le {\sum }_i\mathrm{\mid }{\xi }_i\mathrm{\mid }\mathrm{\parallel }{A}^i\mathrm{\parallel }\le b\mathrm{\parallel }\xi {\mathrm{\parallel }}_1\le \sqrt{3}b$. Hence for every $z$ one has $z^{\mathsf{T}}(\lambda A^0+{\xi }_i{A}^i)z\ge (\lambda a-\sqrt{3}b)\mathrm{\mid }z{\mathrm{\mid }}^2=a\mathrm{\mid }z{\mathrm{\mid }}^2\ge 0$. This proves (a).

(Energy-current inequality on the compact cone.) Put $j^{\mu }=w^{\mathsf{T}}{A}^{\mu }w$. Symmetry and the equation give ${\mathrm{\partial }}_{\mu }{j}^{\mu }=w^{\mathsf{T}}({\mathrm{\partial }}_t{A}^0+{\sum }_i{\mathrm{\partial }}_i{A}^i-C-C^{\mathsf{T}})w$. On compact D let $M={\sup }_D\mathrm{\parallel }{\mathrm{\partial }}_t{A}^0+{\sum }_i{\mathrm{\partial }}_i{A}^i\mathrm{\parallel }+2K<\mathrm{\infty }$. Thus ${\mathrm{\partial }}_{\mu }{j}^{\mu }\le M\mathrm{\mid }w{\mathrm{\mid }}^2\le (M\mathrm{/}a)j^0$ throughout D.

(Truncated cone flux.) For $0<\tau <t_0$, apply the divergence theorem to $D_{\tau }=\{(s,x):0\le s\le \tau ,\mathrm{\mid }x-x_0\mathrm{\mid }\le \lambda (t_0-s)\}$. Base flux is $-E(0)$ and top flux is $E(\tau )$, where $E(s)={\int }_{\mathrm{\mid }x-x_0\mathrm{\mid }\le \lambda (t_0-s)}{j}^0(s,x)dx$. The lateral outward unit normal is $(\lambda ,\xi )\mathrm{/}\sqrt{1+{\lambda }^2}$, so its flux is nonnegative by step 1.1. Step 1.2 gives $E(\tau )\le E(0)+(M\mathrm{/}a){\int }_0^{\tau }E(s)ds$. The domains avoid the cone vertex, so their boundaries are piecewise C1 as required.

(Vanishing and the vertex.) $E$ is continuous and nonnegative; zero initial data give $E(0)=0$. Gronwall therefore gives $E(s)=0$ for $0\le s<t_0$. Positivity of $A^0$ and continuity of w imply w vanishes on each closed top ball, since a nonzero value would give positive integral on a neighborhood. Letting s increase to t_0 and using continuity also gives $w(t_0,x_0)=0$. This proves (b).

(Independence.) On the common neighborhood of D, subtract the two identical linear equations. The difference $w_1-w_2$ solves the homogeneous system with the common matrices $A^{\mu }$ and the common zeroth-order coefficient C; it vanishes on the base. The preceding argument uses only coefficients on D and gives equality throughout D. This proves (c).