A doubly nonlinear evolution problem involving the fractional p-Laplacian

TL;DR

This paper studies a complex nonlinear nonlocal evolution problem involving the fractional p-Laplacian, establishing well-posedness, regularity, and extinction properties of solutions in a rigorous mathematical framework.

Contribution

It introduces new well-posedness and regularity results for a doubly nonlinear fractional p-Laplacian evolution problem, extending understanding of solution behavior.

Findings

Established well-posedness in L^1 for mild solutions.

Derived regularity and decay estimates for solutions.

Proved finite-time extinction for homogeneous cases.

Abstract

In this article, we focus on a doubly nonlinear nonlocal parabolic initial boundary value problem driven by the fractional $p$-Laplacian equipped with homogeneous Dirichlet boundary conditions on a domain in $\mathbb{R}^{d}$ and composed with a continuous, strictly increasing function. We establish well-posedness in $L^1$ in the sense of mild solutions, a comparison principle, and for restricted initial data we obtain that mild solutions of the inhomogeneous evolution problem are strong. We obtain $L^{q}$-$L^{\infty}$ regularity estimates for mild solutions, implying decay estimates and extending the property of strong solutions for more initial data. Moreover, we prove local and global H\"older continuity results as well as a comparison principle that yields extinction in finite time of mild solutions to the homogeneous evolution equation.