Maximum interval of definition of $\frac{dy}{dt} = f(t, y) $, where $f \in C^1$, $y(t_0) = y_0 $

Question

Given the equation $$\frac{dy}{dt} = f(t, y) $$where $f \in C^1$, $y(t_0) = y_0 $ and assuming that $|y - y_0| \leq L|t - t_0|$ (1) and $|L| \leq 1$

By using the Picard Iterations and by saying that $f(t, y)$ is locally Lipschitz, could I say that the maximum interval of definition of the solution of the IVP is $\mathbb{R}$ ?

I have some difficulties in understanding the meaning of the condition 1. Could someone explain me this too?

Thanks in advance.

Answer 1

This claim should have nothing to do with Picard iterations, only with the theorem about the local solution itself.

The corollaries about the maximal solution/maximal domain tell that the maximal domain is an open set and that the graph of the maximal solution leaves every bounded set inside the domain of the ODE (to both sides, "past" and "future"). So if you consider the box $[t_0-a,t_0+a]\times[y_0-La,y_0+La]$, then the solution can only leave this box to the sides, meaning that the maximal domain is larger than this interval. As this is true for any $a>0$, the maximal domain is indeed the whole of $\Bbb R$.

The condition $L\le1$ is not necessary.