A General Fractional Porous Medium Equation

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion: amssym $$\{ {{ {{{∂ u} {∂ t}} + ( { ‐ Δ } )σ /2 ( {| u |m ‐ 1 u} ) = 0,} & {x ∈ { R} ^N ,\,\,t > 0,} {u( {x,0} ) = f( x ),} & {x ∈ { R} ^N .} } } .We consider data amssymf∈ L^1({R}^N)$ and all exponents $0<σ<2\;and\;m>0$ . Existence and uniqueness of a strong solution is established for $ m > {m_}={(N-σ)_+}/N$ , giving rise to an L 1 ‐contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range ${0 < m} ≤ {m_}$ 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.