Journal article

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

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