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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.

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

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