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 $\alpha\in (0, 74/75)$.
Categories: math.AP math-ph math.CA math.MP