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.
Wave equation with compact supported Cauchy data
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wave-equation
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0$u_{tt}-\Delta u$ is not an equation maybe you mean it should be equal to 0 or some other constant function? – 2017-02-12
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0It is a straightforward consequence of the explicit formulae for the solutions. What kind of argument are you looking for? – 2017-02-12
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0I'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 – 2017-02-12
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0I guess you should add this to your question then. – 2017-02-12