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)=u₀(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 $\sigma=\sigma(q,p,N)$.