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

We develop, on general metric measure spaces, a well-posedness theory for the Cauchy problem of the signed porous medium equation and its fast diffusion counterpart \begin{equation*} \partial_t u=\mathcal{L}\left(|u|^{m-1}u\right), \qquad m>0, \end{equation*} where $\mathcal{L}$ is the generator of a symmetric Dirichlet form.

We prove that, for every initial datum $u_0\in L^{m+1}(M,\mu)$, there exists a unique function $u$ that weakly solves the equation in a suitable sense.

The proof is based on the Rothe method and the theory of monotone operators and only uses the definition of the Dirichlet form and its extended version; no extra properties of the form or geometry of the space are needed.

Consequently, the theory applies to a wide range of metric measure spaces, in particular, including non-smooth fractals.