A General Fractional Porous Medium Equation

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion,    @u @t + (−�) �/2 (|u| m−1 u) = 0, x ∈ R N , t > 0, u(x,0) = f(x), x ∈ R N . We consider data f ∈ L 1 (R N ) and all exponents 0 0. Existence and uniqueness of a weak solution is established for m > m∗ = (N − �)+/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 of solutions with good properties happen under some restrictions, and the properties are different from the case above m∗. We also study the dependence of solutions on f,m and �. Moreover, we consider the above questions for the problem posed in a bounded domain.

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