Well-Posedness and L1−Lp Smoothing Effect of the Porous Media Equation Under Poincaré Inequality
MCML Authors
Abstract
Abstract
We investigate the well-posedness and uniqueness of the Cauchy problem for a class of porous media equations defined on ℝd, and demonstrate the L1−Lp smoothing effect. In particular, we establish that the logarithm of the ratio of the Lp norm to the L1 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 L1, then for t>0, the solution will belong to Lp for any p∈[1,∞). The results are obtained under the assumption of a Poincaré inequality.
BibTeXKey: Sun25