I am interested in uniqueness results for autonomous nonlinear ODEs which do not satisfy a local Lipschitz condition. To show my though process I give some preliminary results and successively state my questions in paragraphs 3), 4) and 5).
Consider the following IVP \begin{align} \dot{x}(t) & = f(x(t)),\\ x(0) & = x^0 \in \mathbb{R}^n, \end{align} with $f: \mathbb{R}^n \to \mathbb{R}^n$, continuous and locally Lipschitz continuous almost everywhere. Continuity gives the existence of solutions by Peano's Theorem. When do we get unique solutions?
1) The setting above is obviously to broad to guarantee uniqueness. See the classical counter example for the one-dimensional case ($n=1$): \begin{align} \dot{x}(t) & = f(x(t)) = 2\sqrt{|x(t)|},\\ x(0) & = 0, \end{align} with solutions $x(t) = 0$ and $x(t) = t^2$.
2) I will change the ODE above to get a related system. There are nonunique solutions because the point $x = 0$ where $f(x)$ is not locally Lipschitz has the property $f(0) = 0$. A solution can stay in $x = 0$ or slowly move away from it. Therefore, I want to consider the system \begin{align} \dot{x}(t) & = f(x(t)) = 2\sqrt{|x(t)|} + \varepsilon,\\ x(0) & = 0, \end{align} with $\varepsilon \neq 0$. I think in this case the initial value problem possesses unique (global) solutions. For $\varepsilon > 0$ we have $f(x) \ge \varepsilon > 0$ and in this case we get unique solutions. For the case $\varepsilon < 0$ there exists a neighborhood of $x = 0$ where $f(x) < \delta < 0$. So locally there is a unique solution which we then can extend in a unique way using the Lipschitz continuity of $\sqrt{x}$ away from zero. This might not be completely rigorous so please point out any flaws.
3) This observation brings me to my first conjecture:
Let $f:\mathbb{R} \to \mathbb{R}$ be continuous and $\Omega \subset \mathbb{R}$. Assume the following assumptions are met.
- $f$ is locally Lipschitz continuous on $\mathbb{R} \setminus \Omega$.
- $\Omega$ consist out of isolated points.
- $f(x) \neq 0$ for all $x \in \Omega$.
Then the IVP
\begin{align} \dot{x}(t) & = f(x(t)),\\ x(0) & = x^0 \in \mathbb{R}, \end{align}
has a uniqe local solution for all $x^0 \in \mathbb{R}$. Is this true? If it is false, can someone give a counter example? If this is true, can it be shown the way I discussed the system in 2)?
In addition, I want to know if this holds in the higher dimensions:
4) Does 3) hold in $\mathbb{R}^n$?
5) (Bonus) I would also like to know if one can extend the assumptions in 3) to get unique global solutions. Maybe this can be attained if one can guarantee that the elements in $\Omega$ are uniformly isolated ($\lVert \omega_1 - \omega_2\rVert > 4\delta > 0$ for $\omega_1 \neq \omega_2 \in \Omega$) while imposing a global Lipschitz condition on $\mathbb{R}^n \setminus \bigcup_{\omega \in \Omega} B(\omega, \delta)$ with an additional bound of $f$ on $\bigcup_{\omega \in \Omega} B(\omega, \delta)$.
Thank you for taking the time to read everything. I would be really glad if some of you could answer my questions or hint me at some helpful sources!
Kind regards, AverageJoe