  We consider the nonlinear heat equation $u_t-\Delta u =|u|^p+b |\nabla u|^q$
in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ 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 $p\leq 1+\frac{2}{n}$
or $q\leq 1+\frac{1}{n+1}$, while global classical positive solutions exist for
suitably small initial data when $p>1+\frac{2}{n}$ and $q> 1+\frac{1}{n+1}$.
Although finite time blow-up cannot be produced by the gradient term alone and
should be considered as an effect of the source term $|u|^p$, this result shows
that the gradient term induces an interesting phenomenon of discontinuity of
the critical Fujita exponent, jumping from $p=1+\frac{2}{n}$ to $p=\infty$ as
$q$ reaches the value $1+\frac{1}{n+1}$ from above. Next, we investigate the
case of sign-changing solutions and show that if $p\le 1+\frac{2}{n}$ or
$0<(q-1)(np-1)\le 1$, 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 % also obtained for sign-changing
solutions to the inhomogeneous version of this problem.
