Quick version of the problem (detailed formal problem statement below) : In order to show that solutions can be extended on arbitrarily large time intervals I have tried showing that solutions evolve very slowly when close to the border of the domain, which allows to extend them to arbitrarily large times, but I was only able to do it on small examples. Are there simpler ways to extend solutions ?
Formal problem statement :
I need to extend the solutions of some ODEs of the form $d_t X = F(X)$ to arbitrarily large times in order to satisfy the hypotheses of a certain theorem (on stochastic approximations) which I want to use. Since I am working with an infinite family of functions $F$ I am looking for a general method.
The functions $F$'s coordinates are partial fractions whose denominators have strictly positive coefficients, so by the Cauchy-Lipschitz theorem, we have existence and uniqueness of solutions with initial condition in the interior of the simplex $\Delta = \{ \vec{x} \in \mathbb{R}_{\geqslant 0}^d \text{ | } \sum_i x_i = 1 \}$ (where $F$ is $C^1$-smooth) on certain open time intervals (but not necessarily defined for arbitrarily large times). My goal is to show that those solutions can be extend to arbitrarily large times and to understand what happens on/close to the simplex's boundary. Moreover, the scalar product $F \cdot (1, ..., 1)$ is null, so solutions stay in the hyperplane $\{ \vec{x} \in \mathbb{R}^d \text{ | } \sum_i x_i = 1 \}$.
A very important additional fact is that $F$ is also bounded on $\overset{\circ}{\Delta}$, but cannot be extended to all its border (one can imagine $\dfrac{a}{a+b}$ which cannot be extended to the origin $(0,0)$ but is bounded by $1$ on all $\overset{\circ}{\Delta}$).
I have found theorems on the extension of solutions of ODEs in dimension $d=1$, but in higher dimension I only know the following general results on ODEs where $F$ is defined on an open set (here $\overset{\circ}{\Delta}$ which is bounded) and is $C^1$ (or just locally Lipschitz) :
There is a set $V$ open in $\mathbb{R} \times \overset{\circ}{\Delta}$ where the integral flow $ \Phi : (t,x_0) \mapsto \phi_{x_0}(t) $ is defined and unique (where $\phi_{x_0}$ is the solution with initial condition $\phi(0) = x_0$).
If $\phi$ denotes a solution then the function $ t \mapsto (t, \phi(t)) $ goes out of any compact included in $V$
If for some $(t_0, x_0) \in \mathbb{R} \times \overset{\circ}{\Delta}$ and $r>0$ is such that the closed ball $B(x_0, r)$ is included in $\overset{\circ}{\Delta}$ then if we denote $M := \sup_{B(x_0, r)} \| F \| $, then the solution $\phi$ with initial condition $\phi(t_0) = x_0 $ exists on $] t_0 - \alpha; t_0 + \alpha [$ for any $\alpha < \tfrac{r}{M} $
Using result number 3 and the fact that $F$ is bounded on $\overset{\circ}{\Delta}$ (say by a positive constant $M$) I have shown that :
For any $x_0 \in \overset{\circ}{\Delta}$, if the maximal solution with initial condition $\phi(0) = x_0$ exists only on an interval of the form $]t_1; t_2[$ with $ t_2 < + \infty$, then there must be a point $x_{\infty} \in \partial \Delta$ towards which the solution converges, but converges "fast" in the sense that for $t<t_2$ close enough to $t_2$ we have $\| \phi(t) - x_{\infty} \| \leq M(t_2 -t)$.
I can show on some small examples, and using the specific expression of the corresponding function $F$, that close to the border, solutions either move away or converge towards the border but in the latter case, there distance to the border is bounded from below by a decreasing exponential, and thus cannot reach the border in finite time (which, by what was said above, implies that they can be extended to arbitrarily large times).
I am confident that in the general case the solutions do stay in $\overset{\circ}{\Delta}$ for arbitrarily large times but I don't know how to prove it. I wonder if there are easier ways to extend solutions to arbitrarily large times or methods that can work without much hypotheses on $F$'s precise expression.
Thank you in advance !