In this paper, we consider the positive solution to a Cauchy problem in $mathbb{B}^N$ of the fast diffusive equation: ${mid}x{mid}^mu_t={div}(mid{ abla}u{mid}^{p-2}{ abla}u)+{mid}x{mid}^nu^q$, with nontrivial, nonnegative initial data. Here $frac{2N+m}{N+m+1}$ < $p$ < 2, $q$ > 1 and 0 < $m{leq}n$ < $qm+N(q-1)$. We prove that $q_c=p-1{frac{p+n}{N+m}}$ is the critical Fujita exponent. That is, if 1 < $q{leq}q_c$, then every positive solution blows up in finite time, but for $q$ > $q_c$, there exist both global and non-global solutions to the problem.