Hölder continuity of local weak solutions for parabolic equations exhibiting two degeneracies

Abstract

We consider equations of the form $${\partial}_{t} v - \mbox{div} ( \alpha (v) \nabla v) = 0 \ , $$ where $v \in [0,1]$ and $\alpha (v)$ degenerates for $v=0$ and $v=1$. We show that local weak solutions are locally Hölder continuous provided $\alpha$ behaves like a power near the two degeneracies. We adopt the technique of intrinsic rescaling developed by DiBenedetto.