A doubly nonlinear evolution problem involving the fractional p-Laplacian

Collier, Timthy; Hauer, Daniel

doi:10.48550/arxiv.2110.13401

Cited by 1 publication

(1 citation statement)

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“…α 2 [59] (see also [50,47] with 3 The estimate is purely nonlinear since it degenerates when m = 1. However, the stronger Aronson-Bénilan estimate [4] do hold for the linear case as well, but it relies on the operator itself having space-scaling.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities

Bonforte¹,

Endal²

2022

Preprint

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We establish boundedness estimates for solutions of generalized porous medium equations of the form ∂tu + (−L)[u m ] = 0 in R N × (0, T ), where m ≥ 1 and −L is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise L 1 -L ∞ -smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of −L and I − L.In the linear case m = 1, it is well-known that the L 1 -L ∞ -smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques.We establish a similar scenario in the nonlinear setting m > 1. First, we can show that operators for which ultracontractivity holds, also provide L 1 -L ∞ -smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like −L = I − J * . They do not regularize when m = 1, but we show that surprisingly enough they do so when m > 1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator.Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration.

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Section: Introduction and Main Resultsmentioning

confidence: 99%