Conservation of the mass for solutions to a class of singular parabolic equations

Abstract

In this paper we deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with L^∞ coefficients, whose prototypes are the p-Laplacian $\left(\frac{2N}{N+1}<p<2\right)$ and the Porous medium equation $\left(\left(\frac{N-2}{N}\right)_+<m<1\right)$. In this range of the parameters p and m, we are in the so called fast diffusion case. We prove that the initial mass is preserved for all the times.