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arXiv · 1205.2223

Classical solutions for a logarithmic fractional diffusion equation

Arturo de Pablo·Fernando Quirós·Ana Rodríguez·Juan Luis Vázquez
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Abstract

We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $$ \partial_tu+(-\Delta)^{1/2}\log(1+u)=0, $$ posed for $x\in \mathbb{R}$, with nonnegative initial data in some function space of $L \logL$ type. The solutions are shown to become bounded and $C^\infty$ smooth in $(x,t)$ for all positive times. We also reformulate this equation as a transport equation with nonlocal velocity and critical viscosity, a topic of current relevance. Interesting functional inequalities are involved.

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Arturo de Pablo, Fernando Quirós, Ana Rodríguez, Juan Luis Vázquez. 2012-05-10. Classical solutions for a logarithmic fractional diffusion equation. https://arxiv.org/abs/1205.2223

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