A fractional porous medium equation

We develop a theory of existence, uniqueness and regularity for the following porous medium equation with fractional diffusion,{∂u∂t+(−Δ)1/2(|u|m−1u)=0,x∈RN,t>0,u(x,0)=f(x),x∈RN, with m>m⁎=(N−1)/Nm>m⁎=(N−1)/N, N⩾1N⩾1 and f∈L1(RN)f∈L1(RN). An L1L1-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∈RNx∈RN, t>0t>0.