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

We establish boundedness estimates for solutions of generalized porous medium equations the form $$ \partial_t u+(-\mathfrak{L})[u^m]=0\quad\quad\text{in $\mathbb{R}^N\times(0,T)$}, where $m\geq1$ and $-\mathfrak{L}$ is a linear, symmetric, nonnegative operator. The wide class operators we consider includes, but not limited to, L\'evy operators. Our quantitative take precise $L^1$--$L^\infty$-smoothing effects absolute bounds, their proofs are based on interplay between dual formulation problem Green function $I-\mathfrak{L}$. In linear case $m=1$, it well-known that effect, or ultracontractivity, equivalent to Nash inequalities. This also heat kernel estimates, which imply represent key ingredient in our techniques. similar scenario nonlinear setting $m>1$. First, can show ultracontractivity holds, provide case. converse implication true general. A counterexample given by $0$-order like $-\mathfrak{L}=I-J\ast$. They do regularize when surprisingly enough they so $m>1$, due convex nonlinearity. reveals striking property equations: nonlinearity allows better regularizing properties, almost independently Finally, smoothing effects, both nonlinear, families inequalities Gagliardo-Nirenberg-Sobolev type, explore equivalences settings through application Moser iteration.