Growing Solutions of the fractional $p$-Laplacian equation in the Fast Diffusion Range

TL;DR

This paper investigates the existence, uniqueness, and behavior of solutions to a fractional $p$-Laplacian diffusion equation in the fast diffusion range, including self-similar solutions and extinction phenomena.

Contribution

It provides new existence and uniqueness results, detailed analysis of self-similar solutions, and conditions for extinction in finite time for the fractional $p$-Laplacian equation.

Findings

Existence of solutions for large data with optimal estimates

Characterization of self-similar solutions and their existence conditions

Dichotomy between positivity and extinction in solutions

Abstract

We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-\Delta)_p^s$ is the standard fractional $p$-Laplacian operator. We work in the range of exponents $0<s<1$ and $1<p<2$, and in some sections $sp<1$. The equation is posed in the whole space $x\in {\mathbb R}^N$. We first obtain weighted global integral estimates that allow establishing the existence of solutions for a class of large data that is proved to be roughly optimal. We study the class of self-similar solutions of forward type, that we describe in detail when they exist. We also explain what happens when possible self-similar solutions do not exist. We establish the dichotomy positivity versus extinction for nonnegative solutions at any given time. We analyze the conditions for extinction in finite time.