A fractional porous medium equation

Abstract

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 − 1 u ) = 0 , x ∈ R N , t > 0 , u ( x , 0 ) = f ( x ) , x ∈ R N , with m > m ⁎ = ( N − 1 ) / N , N ⩾ 1 and f ∈ L 1 ( 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 ∈ R N , t > 0 .