After reading a differential equations text and a bunch of sites in a google search, I'm still not perfectly clear on the difference between local and global Lipschitz and what these imply for the solutions of differential equations. I'll state my current understanding, but if I'm wrong on any point I'd appreciate a correction.
- This is the Lipschitz condition:
$$|f(t,u)-f(t,v)|<M|u-v|$$
A function is locally Lipschitz if, for any compact subset of the domain, there is an $M$ for which it satisfies the condition on that subset.
A function is globally Lipschitz, which we also just call Lipschitz, if it satisfies this condition for one choice of $M$ on all of $\mathbb{R}$.
If a differential equation $y'=f(t,y)$ is locally Lipschitz and has an initial condition, then its solution is unique but existence is not guaranteed. If we further ensure that $\frac{\partial f}{\partial y}$ is continuous then existence is guaranteed.
If $y'=f(t,y)$ is globally Lipschitz and has an initial condition then existence and uniqueness are guaranteed.