Some existence and regularity results for porous media and fast diffusion equations with a gradient term

Abstract:

In this article we consider the problem \begin{equation}\tag {\textit {P}} \left \{\begin {array}{rclll} u_t-\Delta u^m&=&|\nabla u|^q + f(x,t),\quad u\ge 0 &\hbox { in } \Omega _T\equiv \Omega \times (0,T),\\ u(x,t)&=&0 &\quad \hbox { on } \partial \Omega \times (0,T),\\ u(x,0)&=&u_0(x)&\quad \hbox { in } \Omega , \end{array} \right . \end{equation} where $\Omega \subset \mathbb {R}^N$ is a bounded regular domain, $N\ge 1$, $1<q\le 2$, and $f\ge 0$, $u_0\ge 0$ are in a suitable class of measurable functions.

We obtain some results for the so-called elliptic-parabolic problems with measure data related to problem $(P)$ that we use to study the existence of solutions to problem $(P)$ according with the values of the parameters $q$ and $m$.