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

Abstract

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:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle u_t-\text {div }(A(t,x)|\nabla u|^{p-2}\nabla u)=\gamma |\nabla u|^q &{} \text {in}\,\,(0,T)\times \Omega ,\\ u=0 &{}\text {on}\,\,(0,T)\times \partial \Omega ,\\ u(0,x)=u_0(x) &{}\text {in}\,\, \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega$ is an open bounded subset of ${{\,\mathrm{{{\mathbb {R}}}}\,}}^N$, $N\ge 2$, $0<T\le \infty$, $1<p<N$ and $q<p$. 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, $\gamma$ is a positive constant, and the initial datum $u_0$ is an unbounded function belonging to a well precise Lebesgue space $L^\sigma (\Omega )$ for $\sigma =\sigma (q,p,N)$.