Global solutions to a non-local diffusion equation with quadratic non-linearity

Abstract

In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $α\in[0,2/3)$: $$ \partial_t u = {(-\triangle)^{-1}u} \triangle u + αu^2, \quad u(t=0) = u_0. $$ The initial condition $u_0$ is positive, radial, and non-increasing with $u_0\in L^1\cap L^{2+δ}(\threed)$ for some small $δ>0$. There is no size restriction on $u_0$. This model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: $u_t = \triangle u + αu^2$.