Suppose $u(x,t)$ where $x \in \mathbb{R}^3$ and $t \in \mathbb{R}$ satisfies the speed varying wave equation:
$$ \sum_{i=1}^3\frac{\partial^2}{\partial x_i ^2}u - \frac{1}{c(x)^2} \frac{\partial^2}{\partial t ^2} u =0 $$
where $0 < c(x) < c$ for some $c > 0$. Then if the initial wave $u(x,0)$ vanishes outside a ball $B(x_0, R)$ of radius $R$ centred at $x_0$ then does $u(x,t)$ necessarily vanish outside the ball $B(x_0, R+ct)$ for $t>0$ ?
It sounds physically intuitive but is there a formal proof? Is there a formulation for weak solutions to the wave equation (e.g., in Sobolev space) ? I know that if $c(x)$ is constant then it follows from Kirchoff integral theorem and this answer states that it can be generalized but seems quite complex, so I was wondering if we can avoid that machinery in this more mild question.
See https://users.math.msu.edu/users/yanb/847-full-note.pdf Theorem 4.19 where you can use inequality 4.29 instead of 4.30 in the hypothesis. I suspect it gives this exact bound.