A General Fractional Porous Medium Equation

Source abstract

Abstract We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion: \input amssym $$\left\{ {\matrix{ {{{\partial u} \over {\partial t}} + \left( { ‐ \Delta } \right)^{\sigma /2} \left( {\left| u \right|^{m ‐ 1} u} \right) = 0,} \hfill &amp; {x \in {\Bbb R} ^N ,\,\,t &gt; 0,} \hfill \cr {u\left( {x,0} \right) = f\left( x \right),} \hfill &amp; {x \in {\Bbb R} ^N .} \hfill \cr } } \right.$$ We consider data \input amssym $f\in L^1(\Bbb{R}^N)$ and all exponents $0<\sigma<2\;and\;m>0$ . Existence and uniqueness of a strong solution is established for $m > {m_\ast}={(N-\sigma)_+}/N$ , giving rise to an L 1 ‐contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range ${0 < m} \le {m_\ast}$ existence and uniqueness happen under some restrictions, and the properties of the solutions are different from the ones for the case above m * . We also study the dependence of solutions on f , m , and σ. Moreover, we consider the above questions for the problem posed in a bounded domain. © 2012 Wiley Periodicals, Inc.