The ODE (12) can be smoothly extended as long as $w$ remains bounded positively below and $(1-t)w^{\prime}+w$ remains bounded (which imply boundedness of $w^{\prime\prime}$, and in particular the boundedness of $w,w^{\prime}$). Notice

$$\frac{d}{dt}((1-t)w^{\prime}+w)=(1-t)w^{\prime\prime}>0,$$

so $(1-t)w^{\prime}+w$ is monotone increasing, and in particular positive. By

$$\frac{d}{dt}((1-t)w^{\prime}+w)^{n-1}=\frac{1-t}{w^{3}},$$

we see $(1-t)w^{\prime}+w$ will be bounded as long as $w$ is bounded positively below.

The convexity of $w$ is ensured whenever the solution is smooth. Thus for some small $\epsilon>0$,

$$w(t)\geq w(\epsilon)-w^{\prime}(\epsilon)(t-\epsilon),\quad t\geq\epsilon.$$

For small $\epsilon$, we have $w_{0}-w_{0}\epsilon<w(\epsilon)<w_{0}$ and $-w_{0}<w^{\prime}(\epsilon)<0$, so $w(\epsilon)/w^{\prime}(\epsilon)>1-\epsilon$, whence $w(t)$ has an a priori lower bound slightly beyond $t=1$. ∎