The existence of time local -solutions is shown for Cauchy problem of the porous medium equations. Our arguments rely on the "-energy method" developed in our previous paper  and a new method based on the theory of evolution equations in the -framework which enables us to handle with perturbations which can be decomposed into monotone parts and small parts in Sobolev spaces of higher order.
Yoshie SUGIYAMA. Mitsuharu ÔTANI. "A method of energy estimates in and its application to porous medium equations." J. Math. Soc. Japan 53 (4) 745 - 789, October, 2001. https://doi.org/10.2969/jmsj/05340745