  In this paper we prove the global in time well-posedness of the following
non-local diffusion equation with $\alpha \in[0,2/3)$: $$ \partial_t u =
{(-\triangle)^{-1}u} \triangle u + \alpha u^2, \quad u(t=0) = u_0. $$ The
initial condition $u_0$ is positive, radial, and non-increasing with $u_0\in
L^1\cap L^{2+\delta}(\threed)$ for some small $\delta >0$. 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: $u_t =
\triangle u + \alpha u^2$.
