The evolution fractional p-Laplacian equation in $\mathbb{R}^N$. Fundamental solution and asymptotic behaviour

TL;DR

This paper studies the fractional p-Laplacian evolution equation in Euclidean space, establishing well-posedness, constructing fundamental solutions, and analyzing their asymptotic convergence for solutions with finite mass.

Contribution

It introduces the first construction of self-similar fundamental solutions for the fractional p-Laplacian and proves convergence of general solutions to these fundamental solutions.

Findings

Solutions form non-expansive semigroups with regularity properties

Existence of self-similar fundamental solutions for all masses

Finite-mass solutions converge to fundamental solutions over time

Abstract

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$ (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue $L^q$ spaces is well posed, and show that the solutions form a family of non-expansive semigroups with regularity and other interesting properties. As main results, we construct the self-similar fundamental solution for every mass value $M,$ and prove that general finite-mass solutions converge towards that fundamental solution having the same mass in all $L^q$ spaces.A number of additional properties and estimates complete the picture.