Let y0 > 0, z0 > 0 and α > 1.
(∗) \(\left\{\begin{array}{l}\frac{d y}{d t}=y^{\alpha} \quad \text { for } t>0, \\ y(0)=y_{0}\end{array}\right.\)
(∗∗) \(\left\{\begin{array}{l}\frac{d z}{d t}=-z^{\alpha} \quad \text { for } t>0, \\ z(0)=z_{0}\end{array}\right.\)
We say that the solution to a differential equation exists globally if it exists for all t > 0.
Which of the following statements is true?
1
Both (∗) and (∗∗) have global solutions
2
None of (∗) and (∗∗) have global solutions
3
There exists a global solution for (∗) and there exists a T < ∞ such that \(\displaystyle\lim _{t \rightarrow T}|z(t)|=+\infty\)
4
There exists a global solution for (∗∗) and there exists a T < ∞ such that \(\displaystyle \lim _{t \rightarrow T}|y(t)|=+\infty\)