*preprint*

**Inserted:** 19 jan 2024

**Year:** 2017

**Abstract:**

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 $\sigma=\sigma(q,p,N)$.