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

Abstract

In this article we consider the problem (\textit {P}) { u t − Δ u m a m p ; = a m p ; | ∇ u | q + f ( x , t ) , u ≥ 0 a m p ;  in  Ω T ≡ Ω × ( 0 , T ) , u ( x , t ) a m p ; = a m p ; 0 a m p ;  on  ∂ Ω × ( 0 , T ) , u ( x , 0 ) a m p ; = a m p ; u 0 ( x ) a m p ;  in  Ω , \begin{equation}\tag {\textit {P}} \left \{\begin {array}{rclll} u_t-\Delta u^m&=&|abla 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 Ω ⊂ R N \Omega \subset \mathbb {R}^N is a bounded regular domain, N ≥ 1 N\ge 1 , 1 > q ≤ 2 1>q\le 2 , and f ≥ 0 f\ge 0 , u 0 ≥ 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 ) (P) that we use to study the existence of solutions to problem ( P ) (P) according with the values of the parameters q q and m m .