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

We want to analyze both regularizing effect and long, short time decay concerning a class of parabolic equations having first order superlinear terms. The model problem is the following: $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle u_t-\text {div }(A(t,x)|\nabla u|^{p-2}\nabla u)=\gamma |\nabla u|^q &{} \text {in}\,\,(0,T)\times \Omega ,\\ u=0 &{}\text {on}\,\,(0,T)\times \partial u(0,x)=u_0(x) {in}\,\, , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ an open bounded subset $${{\,\mathrm{{{\mathbb {R}}}}\,}}^N$$ $$N\ge 2$$ $$0<T\le \infty $$1<p<N$$ $$q<p$$ . assume that A(t, x) coercive, measurable matrix, growth rate q gradient term but still subnatural, $$\gamma positive constant, initial datum $$u_0$$ unbounded function belonging well precise Lebesgue space $$L^\sigma (\Omega )$$ for $$\sigma =\sigma (q,p,N)$$