On distributional solutions of local and nonlocal problems of porous medium type

Abstract

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of(0.1)∂tu−Lσ,μ[φ(u)]=g(x,t)inRN×(0,T), where φ is merely continuous and nondecreasing, and Lσ,μ is the generator of a general symmetric Lévy process. This means that Lσ,μ can have both local and nonlocal parts like, e.g., Lσ,μ=Δ−(−Δ)12. New uniqueness results for bounded distributional solutions to this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for Lσ,μ. Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained.