# Wave equation with compact supported Cauchy data

Consider the wave equation $u_{tt}-\Delta u =0$ in $\mathbb{R}^n \times [0,\infty)$ and let $u(x,0)=g(x) \in C^{\infty}_C(\mathbb{R}^n)$, $u_t(x,0)=0$. What is a rigurous argument that the solution will be compact supported too for every $t\geq 0$? Using the fact that for a compact set where the Cauchy data vanishes there is a cone where the solution is identically zero.

• $u_{tt}-\Delta u$ is not an equation maybe you mean it should be equal to 0 or some other constant function? – mathreadler Feb 12 '17 at 11:32
• It is a straightforward consequence of the explicit formulae for the solutions. What kind of argument are you looking for? – Michał Miśkiewicz Feb 12 '17 at 12:29
• I'm looking for an argument using the fact that the solution is equal zero in a cone over a compact set where the Cauchy data vanishes – Hamilcar Feb 12 '17 at 13:04
• I guess you should add this to your question then. – Michał Miśkiewicz Feb 12 '17 at 13:39