Regularity results for a class of doubly nonlinear very singular parabolic equations

The aim of this paper is to present several properties of the nonnegative weak solutions to a class of very singular equations whose prototype is ut=div(um−1|Du|p−2Du),p>1and3−p<2.Namely, we prove Llocr and Llocr−Lloc∞ estimates and Harnack estimates. Note that 3−p=m+p is a critical value: under this threshold the energy estimates hold with a reverse sign.