I'm interested in the speed of propagation of singularities of a solution to the following wave equation. \begin{equation} (\partial_{tt}+A)u = 0 \end{equation} where A is a second order elliptic operator. One could imagine either the purely highest order case $A=-c^2(x)\Delta$, or $A$ having the divergence form $A=-\nabla\cdot c^2\nabla$, or a mixed case. It seems to be common sense, that these singularities propagate with speed 1 (in the Riemannian metric induced by the principal symbol of $A$) along the characteristics of the PDE. Why is that?

So far I found out: From the theory for Hyperbolic PDE it is known that singularities occurring in the solution (for instance, in the initial condition) are propagated along the so-called characteristics of the PDE. The second common statement is the one of finite speed of propagation, which asserts that the solution $u$ at a fixed point $(t_0,x_0)$ only depends on the former values of $u$ given in the cone $C=\{(t,x), \operatorname{dist}(x,x_0)<t_0-t\}$, where $\operatorname{dist}(x,y)$ is the distance of $x$ to $y$ in the Riemannian metric induced by the second order coefficients of $A$.

Another important tool to describe the propagation of singularities is the concept of a wave front set. Roughly speaking, the wave front set $WF(u)$ of a distribution $u$ contains all points $(x,\xi)\in \mathbb R^n\times\mathbb R^n$, where $x$ is in the singular support of $u$ and $\xi$ gives the direction in which the product of $u$ and any $\phi\in C_0^\infty$ is not rapidly decreasing. Now for a wide class of operators (including hyperbolic operators), one can show that as soon as there is a point $(x_0,\xi_0)\notin WF(u)$, all points on any characteristic strip going through $(x_0,\xi_0)$ have to be regular as well. That is, the singularities propagate along the characteristics in the direction indicated by the wave front set, and they do not disappear.

The question is now: How can I glue this together with the finite speed of propagation to get a generalized version of Huygens principle for the singular support (rather than for the support of $u$ in the simple $(2n+1)D$ constant coefficient case)? In the case of constant coefficients, the boundary $\partial C$ of the cone $C$ seems to be exactly the characteristic surface going through $x_0$, and the singularities propagate along straight lines on the cone; that is, they propagate with the (constant) wave speed $c$. In the general case, I think that the characteristics cannot cross $\partial C$, as this would violate the computed propagation speed. But until now, I didn't see a proof that they actually lie on the boundary. That is, I wasn't able to exclude the case of a characteristic strip lying in the interior of $C$. As I mentioned, it seems to be common sense that such principles exist, but I didn't find an answer why this is correct.

[1]: Lars Hörmander. The Analysis of Linear Partial Differential Operators I-IV.

[2]: Lawrence C. Evans. Partial Differential Equations.


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