On fractional p-laplacian parabolic problem with general data

In this article the problem to be studied is the following (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } \O_{T}\equiv \Omega \times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u & \ge & 0 & \text{ in }\ren \times (0,T),\\ u(x,0) & = & u_0(x) & in \O, \end{array}% \right. where $\mathrm{\Omega }$ is a bounded domain, and $(-{\text{D}}_p^s)$ is the fractional p-Laplacian operator defined by $$(-{\text{D}}_p^s)u(x,t):=P\mathrm{.}V{\int }_{\text{ren}}{\displaystyle \frac{\mathrm{\mid }u(x,t)-u(y,t){\mathrm{\mid }}^{p-2}(u(x,t)-u(y,t))}{\mathrm{\mid }x-y{\mathrm{\mid }}^{N+ps}}}dy$$ with $1<p<N$, $s\in (0,1)$ and $f,u_0$ are measurable functions. The main goal of this work is to prove that if $(f,u_0)\in L^1(\O_T)\times L^1(\O)$,problem$(P)$ has a weak solution with suitable regularity. In addition, if $f_0,u_0$ are nonnegative, we show that the problem above has a nonnegative entropy solution. In the case of nonnegative data, we give also some quantitative and qualitative properties of the solution according the values of $p$.