A non-local inequality and global existence

TL;DR

This paper introduces new non-local integral inequalities and applies them to prove the global existence of solutions for a non-local heat equation with quadratic non-linearity, extending previous results to a broader parameter range.

Contribution

The paper develops novel non-local inequalities and demonstrates their use in establishing global solutions for a class of non-local heat equations, broadening the applicable parameter range.

Findings

Established new non-linear, non-local integral inequalities.

Proved global existence of solutions for a non-local heat equation.

Extended previous results to include all  in (0, 74/75).

Abstract

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 $\alpha\in (0, 74/75)$.