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

TL;DR

This paper establishes existence, uniqueness, and long-term behavior of solutions for a nonlinear parabolic problem involving the fractional p-Laplacian in porous media, including special cases and stability analysis.

Contribution

It proves the existence and uniqueness of solutions for a fractional p-Laplacian parabolic problem and analyzes their global behavior and stabilization properties.

Findings

Existence and uniqueness of weak and mild solutions

Global time existence and behavior of solutions

Stabilization results for the solutions

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: \begin{equation*} \begin{cases} \partial_t u+(-\Delta)^s_p(|u|^{m-1}u)=h(t,x,|u|^{m-1}u) & \text{in} \; (0,T)\times \Omega,\\ u=0 & \text{in} \; (0,T) \times \mathbb{R}^d\backslash \Omega, \\ u(0,\cdot)=u_0 & \text{in} \; \Omega . \end{cases}\ \end{equation*} We also study further the the homogeneous case $h(u)=|u|^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of weak solutions and stabilization.