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

Abstract

We study the Cauchy problem for a weighted porous medium equation on $\R$ associated with a Gibbs probability measure $\pi=e^{-V}$. Under a Poincar\'e inequality for $\pi$ and the convexity assumption on $V$, we prove well-posedness and uniqueness of non-negative weak solutions with initial data in $L^1(\R,\pi)$. We also establish an $L^1$--$L^p$ smoothing effect at every positive time. More precisely, for every admissible $p>1$, we show that the logarithm of the ratio between the $L^p(\R,\pi)$ norm of the solution and its conserved $L^1(\R,\pi)$ mass first decays at a super-exponential rate and then decays exponentially to zero. In particular, even if the initial datum belongs only to $L^1(\R,\pi)$, the solution belongs to $L^p(\R,\pi)$ for every finite $p>1$ and every $t>0$.