Growing solutions of the fractional p-Laplacian equation in the Fast Diffusion range
Author
Vázquez, Juan LuisEntity
UAM. Departamento de MatemáticasPublisher
ElsevierDate
2022-01-01Citation
10.1016/j.na.2021.112575
Nonlinear Analysis: Theory, Methods and Applications 214 (2021): 112575
ISSN
0362-546X (print)DOI
10.1016/j.na.2021.112575Funded by
Author partially funded by Project PGC2018-098440-B-I00 (Spain)Project
Gobierno de España. PGC2018-098440-B-I00Editor's Version
https://doi.org/10.1016/j.na.2021.112575Subjects
Extinction; Fractional operators; Nonlinear parabolic equations; p-Laplacian operator; Self-similar solutions; Solutions with growing data; MatemáticasRights
© 2021 The Authors
Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
Abstract
We establish existence, uniqueness as well as quantitative estimates for solutions u(t,x) to the fractional nonlinear diffusion equation, ∂tu+Ls,p(u) = 0, where Ls,p= (−Δ)sp 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 we need sp <1. The equation is posed in the whole space x ∈ RN. 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 use the estimates to 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 analyse the conditions for extinction in finite time
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