I have learned that for an ordinary differential equation of the form:
\begin{align}
\dot{x}(t)&=f(x,t) \\
x(t_{0})&=x_{0}
\end{align}
If $\;\;f:\mathbb{R}^{n}\rightarrow{}\mathbb{R}^{n}$ is globally Lipschitz continuous on $\mathbb{R}^{n}$, then there exists a unique solution to the ODE.
My question is: since only global Lipschitz continuity is sufficient for this ODE to have a unique solution, does this mean that local Lipschitz continuity is not sufficient? If this is true, can someone please provide an example where $f$ is locally, but not globally, Lipschitz continuous, and there does NOT exist a unique solution to the ODE?
Best Answer
Local Lipschitz continuity suffices to have uniqueness. What it does not suffice to is global existence. That is, if $f$ is local but not global Lipschitz, then the (unique) solution to the Cauchy problem might cease to exist (blow up) in finite time. The prototypical example is the Cauchy problem \begin{equation*} \begin{cases} \dot x = x^2 \\ x(0)=x_0 \end{cases} \end{equation*} which has the unique solution \begin{equation} x(t)=\frac1{x_0^{-1}-t}, \end{equation} that ceases to exist at $t=x_0^{-1}$.