On the well-posedness of porous medium equations on general metric measure spaces

Abstract

On general metric measure spaces, we develop a new well-posedness theory for the signed porous medium equation and its fast diffusion counterpart \[ \partial_t u = \mathcal{L}\left(|u|^{m-1}u\right), \qquad m>0, \] where $\mathcal{L}$ is the associated non-positive self-adjoint operator of a symmetric Dirichlet form. The theory does not rely on a Gelfand triple or compact embeddings; instead, it is built upon the extended Dirichlet space ${\mathcal{F}}_e$ and auxiliary spaces $V^q:=L^q\cap {\mathcal{F}}_e$, whose uniform convexity plays a key role in the proof. The proof uses only the existence of the Dirichlet form and its extension; no additional regularity of the form or geometric assumptions on the underlying space are needed. Consequently, the results apply to a wide range of metric measure spaces, including non-smooth fractals.

39 pages, 2 figures. Have corrected nearly all errors in old versions. Comments are welcome