A doubly nonlinear evolution problem involving the fractional p-Laplacian

2021/10/26 by Timthy Collier, Collier, Timthy, Daniel Hauer +1 · 1 citation
Computer Science · Engineering · Mathematics · #35B40 #35B65 #35K10 #35K55 #35R11 #47H06 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2110.13401

openalex publication_date 2021/10/26 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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 ℝ^d and composed with a continuous, strictly increasing function. We establish well-posedness in L¹ 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^∞ 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 Hölder continuity results as well as a comparison principle that yields extinction in finite time of mild solutions to the homogeneous evolution equation.

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