I want to prove that the problem below is not Lipschitz at $t = 0$.
$ \left\{ \begin{array}{ll} \frac{d}{dt}y(t) = 3(y(t))^\frac{2}{3}, t \geqslant 0\\ y(0) = 0 \end{array} \right. $
I managed to develop the part below, but I don't know how to continue.
$||f(t,y_1) - f(t,y_2)|| = ||3(y_1(t))^\frac{2}{3} - 3(y_2(t))^\frac{2}{3}|| = 3||(y_1(t))^\frac{2}{3} - (y_2(t))^\frac{2}{3}||$.
In my lay knowledge, I would have to do the condition $\leq$ but I don't know how to elaborate it. Incidentally, in that case, would I be able to prove that it is not Lipschitz for any $t > 0$?
Thank you!