A General Fractional Porous Medium Equation

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion: input amssym $$left{ {matrix{ {{{partial u} over {partial t}} + left( { ‐ Delta } right)^{sigma /2} left( {left| u right|^{m ‐ 1} u} right) = 0,} hfill & {x in {Bbb R} ^N ,,,t > 0,} hfill cr {uleft( {x,0} right) = fleft( x right),} hfill & {x in {Bbb R} ^N .} hfill cr } } right.$$ We consider data input amssym $fin L^1(Bbb{R}^N)$ and all exponents $00$ . Existence and uniqueness of a strong solution is established for $ m > {m_ast}={(N-sigma)_+}/N$ , giving rise to an L¹‐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.