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 $${\partial }_{t}u-{\mathfrak{L}}^{\sigma ,\mu }[\phi (u)]=g(x,t)\text{in}{\mathbb{R}}^N\times (0,T),$$ where $\phi$ is merely continuous and nondecreasing and ${\mathfrak{L}}^{\sigma ,\mu }$ is the generator of a general symmetric Lévy process. This means that ${\mathfrak{L}}^{\sigma ,\mu }$ can have both local and nonlocal parts like e.g. ${\mathfrak{L}}^{\sigma ,\mu }=\Delta -(-\Delta {)}^{\frac{1}{2}}$. 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 ${\mathfrak{L}}^{\sigma ,\mu }$. Existence and a priori estimates are deduced from a numerical approximation, and energy type estimates are also obtained.

6 pages. Minor revision. Added details to Step 2 of the proof of Theorem 3.1