## A non-local inequality and global existence    [PDF]

Philip T. Gressman, Joachim Krieger, Robert M. Strain
In this article we prove a collection of new non-linear and non-local
integral inequalities. As an example for $u\ge 0$ and $p\in (0,\infty)$ we
obtain $$\int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed} dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2.$$ We
use these inequalities to deduce global existence of solutions to a non-local
all $\alpha\in (0, 74/75)$.