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

TL;DR

This paper establishes the global well-posedness of a non-local diffusion equation with quadratic non-linearity, which models plasma physics phenomena and differs significantly from classical heat equations.

Contribution

It proves the global existence and uniqueness of solutions for a non-local PDE with specific initial conditions, without size restrictions, highlighting its relevance to plasma physics.

Findings

Global solutions exist for all time under given conditions

The model shares structural similarities with Landau's equation

Behavior differs markedly from semi-linear heat equations

Abstract

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: $u_t = \triangle u + \alpha u^2$.