A method of energy estimates in L^∞ and its application to porous medium equations

Dedicated to Professor Kyuûya MASUDA on the occasion of his 60-th birthday

Abstract

The existence of time local C^∞-solutions is shown for Cauchy problem of the porous medium equations. Our arguments rely on the“ L^∞-energy method”developed in our previous paper [{16}] and a new method based on the theory of evolution equations in the L²-framework which enables us to handle with perturbations which can be decomposed into monotone parts and small parts in Sobolev spaces of higher order.