A fractional porous medium equation

Details A fractional porous medium equation A fractional porous medium equation

2010-01-14

arxiv.org/abs/1001.2383v1

Mathematics

Analysis of PDEs

Scientific paper

We develop a theory of existence, uniqueness and regularity for a porous medium equation with fractional diffusion, $\frac{\partial u}{\partial t} + (-\Delta)^{1/2} (|u|^{m-1}u)=0$ in $\mathbb{R}^N$, with $m>m_*=(N-1)/N$, $N\ge1$ and $f\in L^1(\mathbb{R}^N)$. An $L^1$-contraction semigroup is constructed and the continuous dependence on data and exponent is established. Nonnegative solutions are proved to be continuous and strictly positive for all $x\in\mathbb{R}^N$, $t>0$.

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