A non-local inequality and global existence

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for u >= 0 and p is an element of (0, infinity) we obtain integral(R3) dx u(p+1)(x) <= (p+1/p)(2) integral(R3) dx {(-Delta)(-1) u(x)} vertical bar del u(p/2) (x)vertical bar(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 Krieger and Strain (in press) [4] to include all alpha is an element of (0, 74/75). (C) 2012 Elsevier Inc. All rights reserved.