Harnack's inequality for degenerate double phase parabolic equations under the non-logarithmic Zhikov's condition

TL;DR

This paper establishes Harnack's inequalities for solutions of degenerate parabolic equations with non-standard growth conditions, under a generalized non-logarithmic Zhikov's condition, extending classical results to more complex variable coefficient scenarios.

Contribution

It introduces a new framework for Harnack's inequalities applicable to degenerate parabolic equations with $(p,q)$ growth under non-logarithmic Zhikov's conditions, broadening the scope of regularity theory.

Findings

Proved Harnack's inequalities for solutions with $(p,q)$ growth.

Extended classical inequalities to equations with variable coefficients.

Established conditions under which inequalities hold for degenerate parabolic equations.

Abstract

We prove Harnack's type inequalities for bounded non-negative solutions of degenerate parabolic equations with $(p,q)$ growth $$ u_{t}-{\rm div}\left(\mid \nabla u \mid^{p-2}\nabla u + a(x,t) \mid \nabla u \mid^{q-2}\nabla u \right)=0,\quad a(x,t) \geq 0 , $$ under the generalized non-logarithmic Zhikovs conditions $$ \mid a(x,t)-a(y,\tau)\mid \leqslant A\mu(r) r^{q-p},\quad (x,t),(y,\tau)\in Q_{r,r}(x_{0},t_{0}),$$ $$\lim\limits_{r\rightarrow 0}\mu(r) r^{q-p}=0,\quad \lim\limits_{r\rightarrow 0}\mu(r)=+\infty,\quad \int\limits_{0} \mu^{-\beta}(r)\frac{dr}{r} =+\infty,$$ with some $\beta >0$.