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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: $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle u_t-\text {div }(A(t,x)|\nabla u|^{p-2}\nabla u)=\gamma |\nabla u|^q &{} \text {in}\,\,(0,T)\times \Omega ,\\ u=0 &{}\text {on}\,\,(0,T)\times \partial \Omega ,\\ u(0,x)=u_0(x) &{}\text {in}\,\, \Omega , \end{array}\right. } \end{aligned}$$ u t - div ( A ( t , x ) | ∇ u | p - 2 ∇ u ) = γ | ∇ u | q in ( 0 , T ) × Ω , u = 0 on ( 0 , T ) × ∂ Ω , u ( 0 , x ) = u 0 ( x ) in Ω , where $$\Omega $$ Ω is an open bounded subset of $${{\,\mathrm{{{\mathbb {R}}}}\,}}^N$$ R N , $$N\ge 2$$ N ≥ 2 , $$0<T\le \infty $$ 0 < T ≤ ∞ , $$1<p<N$$ 1 < p < N and $$q<p$$ q < p . 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, $$\gamma $$ γ is a positive constant, and the initial datum $$u_0$$ u 0 is an unbounded function belonging to a well precise Lebesgue space $$L^\sigma (\Omega )$$ L σ ( Ω ) for $$\sigma =\sigma (q,p,N)$$ σ = σ ( q , p , N ) .

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