Quick question about Picard–Lindelöf theorem.
We know the existence and uniqueness of a local solution around $[t_0-\epsilon,t_0+\epsilon ]$. But what if we have Lipschitz continuity everywhere, does this mean that a unique solution exists say on the whole of $\mathbb{R}$.
For example $\frac{d}{dx}(x(t))=x(t)^3$ and $x(0)=1$ can we conclude without solving this that the solution will exist everywhere and be unique? I don't know how to apply the Picard–Lindelöf theorem here.