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

Abstract

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: $\left(\begin{array}{cc}{\partial }_{t}u+{(-\Delta )}_p^s\left({\left|u\right|}^{m-1}u\right)=h(t,x,{\left|u\right|}^{m-1}u)& \text{in}(0,T)\times \Omega ,\\ u=0& \text{in}(0,T)\times R^d\setminus \Omega ,\\ u(0,\cdot )=u_0& \text{in}\Omega .\end{array}\right)$We also study further the the homogeneous case $h\left(u\right)={\left|u\right|}^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of weak solutions and stabilization.