The author studies the global behaviour of positive solutions to \[ u_ t=\Delta\varphi(u)\text{ in }B_ R\times(0,T),\;\partial\varphi(u)/\partial\nu=f(u)\text{ on }S_ R\times(0,T),\;u(x,0)=u_ 0(x)\text{ in }B_ R, \] where \(B_ R=\bigl\{| x|<R\bigr\}\), \(S_ R=\bigl\{| x|=R\bigr\}\) and \(u_ 0\) is a smooth nonnegative function such that \(\partial\varphi(u_ 0)/\partial\nu=f(u_ 0)\) on \(S_ R\). Here \(f\) and \(\varphi\) are increasing functions of \(u\), positive for \(u\) positive, which go to infinity as \(u\to\infty\), and \(0<\varphi'(u)<\infty\) in \(R\).
It is proved that when \(\varphi'(u)\geq C>0\) in \(\mathbb{R}\) there exists a unique global solution when \(f\) is sublinear and finite time blow-up occurs when \(\int^ \infty(ds/f(s))<\infty\). On the other hand, if one allows \(\varphi'(u)\) to go to zero as \(u\to\infty\), finite time blow-up may occur with \(f\) being sublinear. Precise relations between \(\varphi\) and \(f\) guarantee global existence or finite time blow-up.