1202.4088 (Philip T. Gressman et al.)

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
heat equation with a quadratic non-linearity for large radial monotonic
positive initial conditions. Specifically, we improve \cite{ksLM} to include
all $\alpha\in (0, 74/75)$.

View original: http://arxiv.org/abs/1202.4088