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

2021/02/28 by Juan Luis Vázquez, Vázquez, Juan Luis · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #35C06 #35K55 #35R11 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Class (philosophy) #Combinatorics #Diffusion #Diffusion equation #Extinction (optical mineralogy) #FOS: Mathematics #FOS: Physical sciences #Fractional Laplacian #Laplace operator #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Partial Differential Equations #Nonlinear system #Operator (biology) #Physics #Pure mathematics #Quantum mechanics #Range (aeronautics) #Space (punctuation) #Stability and Controllability of Differential Equations #Type (biology) #Uniqueness #math-ph #math.AP #math.MP #msc:35C06 #msc:35K55 #msc:35R11

paper · pdf · doi:10.48550/arxiv.2103.00552

published in arXiv (Cornell University) (Cornell University) · 47 pages, 3 figures New section on mass conservation added to this version

openalex publication_date 2021/02/28 · arxiv created 2021/05/21 · arxiv updated 2021/05/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/08/08

Abstract

We establish existence, uniqueness as well as quantitative estimates for\nsolutions to the fractional nonlinear diffusion equation, \∂_t u\n+ mathcal L_s,p (u)=0, where mathcal L_s,p=(-\Δ)_p^s is the\nstandard fractional p-Laplacian operator. We work in the range of exponents\n0<s<1 and 1<p<2, and in some sections sp<1. The equation is posed in the\nwhole space x\∈ mathbb R^N. We first obtain weighted global integral\nestimates that allow establishing the existence of solutions for a class of\nlarge data that is proved to be roughly optimal. We study the class of\nself-similar solutions of forward type, that we describe in detail when they\nexist. We also explain what happens when possible self-similar solutions do not\nexist. We establish the dichotomy positivity versus extinction for nonnegative\nsolutions at any given time. We analyze the conditions for extinction in finite\ntime.\n

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• The Cauchy problem of non-local space-time reaction-diffusion equation involving fractional p-Laplacian

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