Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U \to \mathbb R$ with open intervals $I, U$ and $(x_0, y_0) \in I \times U$. Let $h : U \to \mathbb R$ be locally Lipschitz-continuous. If $b$ is not the right endpoint of the interval $I$, then there exists for every $\beta < b$ and every compact set $K \subseteq U$ a $\xi \in (\beta; b)$ with $\varphi(\xi) \notin K$. An analogous result holds for the left endpoint $a$ of $I$.
Said differently: If $(a,b) \ne I$, then $\varphi$ leaves every compactum, or if all values of $\varphi$ are contained in a compact set, then $\varphi$ is defined on the whole interval $I$.
I know that the IVP (1) has, in a sufficiently small open interval around $x_0$, a solution (existence result), and further if $h : U \to \mathbb R$ locally Lipschitz, then on every interval around $x_0$ the IVP (1) has at most one solution (uniqueness result). Also a solution $\varphi : (a,b) \to \mathbb R$ of the IVP (1) is called maximal, if for every other solution $\psi : (\alpha, \beta) \to \mathbb R$ we have: i) $(\alpha,\beta) \subseteq (a,b)$ and ii) $\psi(x) = \varphi(x)$ for $x \in (\alpha,\beta)$. For $h : U \to \mathbb R$ locally Lipschitz, then it has at most one maximal solution.
In my textbook the following picture is given:
And below it is written that all those solution curves go from "boundary to boundary", meaning the property mentioned in the above Theorem. But as I understand it, the lowermost curve does not fulfills the property, because it's "range" is contained in a compact set, or have I misunderstood something?
I am still a little bit intimidated by this Theorem; for what is it used, what is its essential meaning? Has anybody examples or something for illustration? In my textbook after this Theorem the chapter closes and nothing more is mentioned... Thanks!
If $h$ was globally Lipschitz, we would not worry about $K$; the solutions of the ODE would exist for all times. But the local Lipschitz condition leave the possibility of solutions escaping to infinity in finite time. A typical example is $y'=y^2$, for which $y(t)=1/(C-t)$.
As long as solution stays within a compact set, the local Lipschitz condition amounts to a global one, and the Picard theorem provides existence and uniqueness. Thus, for a solution to cease to exist, it has to exit every compact set.
The lowermost curve misses the shaded compact set entirely. But we can imagine a larger compact set which does contain it; in this way, the curve does illustrate the theorem.