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

(0.1)${?}_{t}u?{\mathfrak{L}}^{\sigma ,\mu }[\phi (u)]=g(x,t)\text{in}{\mathbb{R}}^N\times (0,T),$

where φ 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., ${\mathfrak{L}}^{\sigma ,\mu }=\mathrm{\Delta }?{(?\mathrm{\Delta })}^{\frac{1}{2}}$. 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 ${\mathfrak{L}}^{\sigma ,\mu }$. Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained.