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

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

2010-12-13

arxiv.org/abs/1012.2890v2

Mathematics

Analysis of PDEs

38 pages, made changes according to the referee reports, in press at Comm. PDE

Scientific paper

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

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