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

In this paper we consider the problem (P)\qquad \{{array}{rclll} u_t-\D u^m&=&|\n 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 x\in \Omega {array}. where $\text{O}\subset \text{ren}$, $N\ge 2$,isaboundedregulardomain,$1<q\le 2$,and$f\ge 0$,$u_0\ge 0$ are in a suitable class of functions. We obtain some results for 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 $m$.