Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

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