Aronson-Bénilan gradient estimates for porous medium equations under lower bounds of $N$-weighted Ricci curvature with $N<0$

The Aronson-Bénilan gradient estimate for the porous medium equation has been studied as a counterpart to the Li-Yau gradient estimate for the heat equation. In this paper, we give the Aronson-Bénilan gradient estimates for the porous medium equation on weighted Riemannian manifolds under lower bounds of $N$-weighted Ricci curvature with $\epsilon$-range for some $N<0$. This is a generalization of those estimates under constant lower $N$-weighted Ricci curvature bounds with $N\in [n,\mathrm{\infty })$.