How can I proof that the maximal solution to
\begin{align} x'(t) = x(t)^{x(t)}, \quad x(0) = x_0, \end{align}
where $x_0 > 0$ and $t \geq 0$, is not (globally) defined on $\mathbb{R}_{0}^{+}$?
I am given the hint that it might help to first look at $x_0 > 1$. Unfortunately, that does not help. I am close to giving up on this one so any help is appreciated.
Perhaps a little more direct, we know that $x^x=e^{x\ln x}$ has a minimum of $e^{-e^{-1}}>0.6922>\frac12$ at $x=e^{-1}$. So we know that the solution is growing at least with the rate $\frac12$. So we can be sure that $x(t)\ge 2$ for $t\ge 4$, assuming the domain of the maximal solution $x$ extends far enough to contain $t=4$.
After that, $x'\ge x^2$ so that $$ x(t)\ge\frac{x(4)}{1-x(4)(t-4)} $$ forces a divergence to infinity shortly after that point (as $x(4)\ge 2$).
