Motivated by this question, and my own related research interest, I'm thinking to see under what conditions/hypotheses, the initial point in the Cauchy-Picard-Lindelöf theorem lies on the boundary point of the open set where the solution is supposed to lie in for a short time. The precise details of what I'm seeking are:
Consider the initial value problem (IVP): $$\frac{dx}{dt} = f(x(t),t), \\ x(0) = x_0. $$
Let $U \subset \mathbb{R}^n$ be an open set. Let $x_0\in \partial{U}.$ ADD EXTRA CONDITIONS HERE. Let $ f: U \times [0,T] \to \mathbb{R}^n $ be a continuous function which satisfies the Lipschitz condition $$ |f(x_1,t) - f(x_2,t)| \leq M |x_1 - x_2|, \quad \forall (x_1,t), (x_2,t) \in U \times [0,T], $$ where $M$ is a given constant. Then for some positive $ \delta $, there is a unique solution $ x: [0,\delta] \to U $ of the above initial value problem.
The above theorem is wrong in full generality without extra conditions: just solve the geodesic flow equation $dx_i/dt=v_i; dv_i/dt=0, x_i(0)=0, v_i(0)=c_i \in \mathbb{R}, i=1,2. $ Take $U\subset \mathbb{R}^2$ to be the domain bounded by $y=x,y=x^2.$ Note that $0\in \partial{U}.$ The solutions are straight lines passing through the origin. But no straight line is contained inside $U$, since the boundary curves got the same tangents at $0.$ So this means that we may need more assumption on $U$ for the proposed theorem to hold. See the figure below.
To circumvent the previous issue, we can make $U$ to be locally convex at $x_0,$ i.e. $\exists \delta >0$ so that $U\cap B(x_0;\delta)$ is convex. But this seems to only prevent the straight lines emanating from $x_0$ going outside $U$ at a small enough time. But this doesn't seem to necessarily prevent the solutions to the above equation to leave $U$ at an arbitrary short time.
So my question is: is there a modified version of the above theorem where the initial point $x(0)=:x_0$ lies on the boundary of $U,$ yet the solution $x(t)$ stay within $U$ for $t\in [0,T].$
If yes, I'd love to see a reference(s). Thank you!
