On the H\"{o}lder continuity of signed solutions to doubly nonlinear parabolic equations in the mixed degenerate/singular cases

Abstract

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