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

TL;DR

This paper investigates existence and regularity of solutions for porous media and fast diffusion equations with a gradient term, considering measure data and parameter-dependent conditions.

Contribution

It provides new existence and regularity results for nonlinear PDEs with measure data, extending understanding of porous media and fast diffusion equations with gradient terms.

Findings

Existence results depend on parameters q and m.

Regularity results for solutions are established.

Connections between elliptic-parabolic problems and measure data are explored.

Abstract

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 $\O\subset \ren$, $N\ge 2$, is a bounded regular domain, $1<q\le 2$, 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 $m$.