Global existence for globally Lipschitz vector fields

Statement

Let $J$ be an interval, let $K \ge 0$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be continuous. Assume that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in \mathbb{R}^n$. Then for each $t_0 \in J$ and $x_0 \in \mathbb{R}^n$, the initial value problem $x' = f(t, x)$, $x(t_0) = x_0$ has exactly one solution defined on all of $J$.

Remarks

The point is that the solution does not blow up: the Lipschitz bound gives $\|f(t, x)\| \le \|f(t, 0)\| + K \|x\|$, so $\|x(t)\|$ grows at most exponentially. A linear system $x' = A(t) x + b(t)$ with continuous $A$ and $b$ on $J$ satisfies the hypothesis on each compact subinterval. So its solutions exist on all of $J$. A field that is only locally Lipschitz can give blow-up in finite time.

Explanations

L1 human

If the field grows at most linearly in the state, solutions cannot run away to infinity in finite time. So the solution exists as long as the field is defined.