Well-posedness and $L^1-L^p$ Smoothing Effect of the Porous Media Equation under Poincar\'e Inequality

We investigate the well-posedness and uniqueness of the Cauchy problem for a class of porous media equations defined on $\mathbb{R}^d$, and demonstrate the $L^1-L^p$ smoothing effect. In particular, we establish that the logarithm of the ratio of the $L^p$ norm to the $L^1$ norm decreases super-exponentially fast during the initial phase, subsequently decaying to zero exponentially fast in the latter phase. This implies that if the initial data is solely in $L^1$, then for $t>0$, the solution will belong to $L^p$ for any $p\in [1,\infty)$. The results are obtained under the assumption of a Poincar\'e inequality.