  In this paper we deal with parabolic problems whose simplest model is
  $$
  \begin{cases}
  u'- \Delta_{p} u + B\frac{|\nabla u|^p}{u} = 0 & \text{in} (0,T) \times
\Omega,\newline u(0,x)= u_0 (x) &\text{in}\ \Omega, \newline u(t,x)=0
&\text{on}\ (0,T) \times \partial\Omega, \end{cases} $$
  where $T>0$, $N\geq 2$, $p>1$, $B > 0$, and $u_{0}$ is a positive function in
$L^{\infty}(\Omega)$ bounded away from zero.
