On the Hölder continuity of signed solutions to doubly nonlinear parabolic equations in the mixed degenerate/singular cases

Abstract

We prove the Hölder continuity of sign-changing solutions to the equation of the type $$\frac{\partial }{\partial t}(|u{|}^{q-1}u)-div(|Du{|}^{p-2}Du)=0,$$ where numbers $p$, $q$ satisfy the conditions $$0<q<p-1\text{and}p<2,$$ or $$q>p-1\text{and}p>2.$$ Our proof uses new versions of the integral Harnack type inequalities for sign-changing solutions.