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Free boundaries in fractional filtration equations

Quirós, F (U. Autónoma de Madrid)
Thursday 26 June 2014, 16:00-16:30

Seminar Room 1, Newton Institute


Co-authors: Arturo de Pablo (U. Carlos III de Madrid), Ana Rodríguez (U. Politécnica de Madrid), Juan Luis Vázquez (U. Autónoma de Madrid)

In this talk we will review some recent results on the Cauchy problem for the fractional filtration equation $\partial_t u+(-\Delta)^{\sigma/2}\varphi(u)$, $\sigma\in(0,2)$, where the nonlinearity $\varphi$ satisfies some not very restrictive conditions.

Solutions to these problems become instantaneously positive if the initial data are nonnegative. However, a free boundary emerges in certain cases in which there is a distinguished value, as in the fractional Stefan problem, $\varphi(u)=(u-1)_+$, or in the ‘mesa’ limit, $m\to\infty$, for the fractional porous media equation, $\varphi(u)=u^m$.


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