The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation

We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $\phi(s)=s|s|^{m-1}$, assuming a $\nu$-Sobolev inequality with $\nu>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0<m<2/\nu$ and $\ell^1$-$\ell^q$ smoothing in the range $m>2/\nu$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.