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

Abstract

We prove Harnack-type inequalities for bounded non-negative solutions of the degenerate parabolic equations with $(p,q)$ growth \begin{equation*} u_{t}-\mathrm{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 , \end{equation*} under the generalized non-logarithmic Zhikov's conditions \begin{equation*} \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}), \end{equation*} \begin{equation*} \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, \end{equation*} \noindent with some ~$\beta >0$.