Optimal regularity and fine asymptotics for the porous medium equation in bounded domains

We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time $T^{\ast }$. More precisely, we show that solutions are $C^{2,\alpha }(\bar{\mathrm{\Omega }})$ in space, with $\alpha =\frac{1}{m}$, and $C^{\mathrm{\infty }}$ in time (uniformly in $x\in \bar{\mathrm{\Omega }}$), for $t>T^{\ast }$. Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence $O(t^{-1-\gamma })$, and we prove that the convergence holds in the $C^{1,\alpha }(\bar{\mathrm{\Omega }})$ topology.