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

TL;DR

This paper establishes the H"older continuity of sign-changing solutions to a class of doubly nonlinear parabolic equations in mixed degenerate and singular cases, using novel integral Harnack inequalities.

Contribution

It introduces new integral Harnack inequalities to prove regularity of solutions in challenging degenerate and singular regimes.

Findings

Proves H"older continuity for solutions with sign changes.

Develops new integral Harnack inequalities for these equations.

Handles both degenerate and singular parameter regimes.

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.