What mean a $0$ in the set of functions infinitely differentiable $C_0^\infty$
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It means "compactly supported", i.e. elements of that space identically vanish outside a compact subset.
Sometimes it means that the elements $u$ decay to zero "at infinity", in the sense that for any $\epsilon >0$ there exists a compact subset $K$ such that $|u|<\epsilon$ on $\complement K$.