I have a question regarding the exercise below. Sorry if I am not writing it in LaTex but it would be a mess to rewrite all this. I do not understand why, at point b) the author writes: We surely have global existence for all initial $k_0 ∈ [0,k_2]$ while this is not guaranteed for $k_0 > k_2$. Why is this true?
In order not to have global existence, the solution $k(t)$ would have to blow up in finite time. But if the solution starts in the interval $[0,k_2]$, it can't get out of that interval (since it can't cross the equilibrium solutions), so that can't happen.
On the other hand, if you start to the right of $k_2$ it may perhaps happen that $k(t) \nearrow +\infty$ as $k \nearrow T$, for some finite time $T$.