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

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of ∂tu − L σ,µ[ϕ(u)] = g(x, t) in R N (0.1) × (0, T ), where ϕ is merely continuous and nondecreasing and L σ,µ is the generator of a general symmetric L´evy process. This means that L σ,µ can have both local and nonlocal parts like e.g. L σ,µ = ∆ − (−∆) 1 2 . New uniqueness results for bounded distributional solutions of this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for L σ,µ. Existence and a priori estimates are deduced from a numerical approximation, and energy type estimates are also obtained.