Global Solutions to a Non-Local Diffusion Equation with Quadratic Non-Linearity

In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $\alpha\, \varepsilon [0, 2/3)$: $\partial_tu=\{(-\Delta)^{-1}u\}\Delta u+\alpha u^2, \,\, u(t=0)=u_0$. The initial condition $u_0$ is positive, radial, and non-increasing with u_0 \varepsilon L^1 \cap L^{2+(\mathbb{R^3) 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: $u_t = \Delta u + \alpha u^2$.