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

Abstract

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.

Upper heat kernel estimates for nonlocal operators via Aronson’s method

“…For a discussion on the equivalence of Nash -and Faber-Krahn inequalities, we refer the reader to [17]. Moreover, we would like to mention the recent article [7], where the relation between L 1 − L ∞ smoothing effects, on-diagonal upper heat kernel estimates, and functional inequalities are studied for fractional equations of porous medium type.…”