  We want to analyse both regularizing effect and long, short time decay
concerning parabolic Cauchy-Dirichlet problems of the type \begin{equation*}
\begin{cases} \begin{array}{ll} u_t-\text{div} (A(t,x)|\nabla u|^{p-2}\nabla
u)=\gamma |\nabla u|^q & \text{in}\,\,Q_T,\\ u=0
&\text{on}\,\,(0,T)\times\partial\Omega,\\ u(0,x)=u_0(x) &\text{in}\,\, \Omega.
\end{array} \end{cases} \end{equation*} We assume that $A(t,x)$ is a coercive,
bounded and measurable matrix, the growth rate $q$ of the gradient term is
superlinear but still subnatural, $\gamma>0$, the initial datum $u_0$ is an
unbounded function belonging to a well precise Lebesgue space
$L^\sigma(\Omega)$ for $\sigma=\sigma(q,p,N)$.
