A non-local inequality and global existence

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In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for u⩾0 and p∈(0,∞) we obtain∫R3dxup+1(x)⩽(p+1p)2∫R3dx{(−Δ)−1u(x)}|∇up2(x)|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 α∈(0,7475).