A general fractional porous medium equation

الأصل · EN

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, {ll ∂ u/∂ t + (-Δ)σ/² (|u|ᵐ⁻¹u)=0, & xⁿ, t>0, [8pt] u(x,0) = f(x), & xⁿ.%. We consider data f∈ L¹(Rⁿ) and all exponents 0<σ<2 and m>0. Existence and uniqueness of a weak solution is established for m> m*=(N-σ)+ /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≤ 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 σ. Moreover, we consider the above questions for the problem posed in a bounded domain.