We study the evolution of solutions to the initial-boundary-value problem
\begin{alignat*}{3}
u_t&=(u^m)_{xx}+\lambda(u^q)_x, & \quad x&>0,&\quad t&\in(0,T), \\[2pt] -(u^m)_x(0,t)&=u^p(0,t), & &&t&\in(0,T), \\[2pt] u(x,0)&=u_0(x), & \quad x&>0,
\end{alignat*}
and give a rather complete characterization, in terms of the parameters $m\ge1$, $p,q>0$ and $\lambda>0$, of whether all solutions are global in time or, on the contrary, there exist blow-up solutions. We show that the presence of the convective term has a preventive effect on the blow-up (with respect to the case $\lambda=0$) and gives rise to a collapse of the region where all solutions blow up in this case. On the other hand, a new Fujita-type phenomenon takes place at the level $p=q$ and $0<\lambda<1$.