We consider the nonlinear heat equation u(t) - Delta u = vertical bar u vertical bar(p)+b
vertical bar del u vertical bar(q) in (0,infinity) x R-n, where n >= 1, p > 1, q >=
1, and b > 0. First, we focus our attention on positive solutions and obtain an optimal
Fujita-type result: any positive solution blows up in finite time if p <= 1+ 2/n or
q <= 1+ 1/n+1, while global classical positive solutions exist for suitably small
initial data when p > 1 + 2/n and q > 1+ 1/n+1. Although finite time blow-up cannot
be produced by the gradient term alone and should be considered as an effect of the
source term vertical bar u vertical bar(p), this result shows that the gradient term
induces an interesting phenomenon of discontinuity of the critical Fujita exponent,
jumping from p = 1+ 2/n to p = infinity as q reaches the value 1 + 1/n+1 from above.
Next, we investigate the case of sign-changing solutions and show that if p <= 1 +
2/n or 0 < (q - 1)(np - 1) <= 1, then the solution blows up in finite time for any
nontrivial initial data with nonnegative mean. Finally, a Fujita-type result, with
a different critical exponent, is obtained for sign-changing solutions to the inhomogeneous
version of this problem.