Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous medium

In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian: \begin{equation*} \begin{cases} \partial_t u+(-Δ)^s_p(|u|^{m-1}u)=h(t,x,|u|^{m-1}u) & $\text{in}$ ; (0,T)\times Ω,\ u=0 & $\text{in}$ ; (0,T) \times $\mathbb{R}$^d\backslash Ω, \ u(0,\cdot)=u_0 & $\text{in}$ ; Ω. \end{cases}\ \end{equation*} We also study further the the homogeneous case $h(u)=\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of weak solutions and stabilization.