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 where and 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 –-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 and

.

In the linear case , it is well-known that the –

-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 . First, we can show that operators for which ultracontractivity holds, also provide –-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order Lévy operators like ⁎. They do not regularize when , but we show that surprisingly enough they do so when

, 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.