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

Abstract

We prove Harnack's type inequalities for bounded non-negative solutions of degenerate parabolic equations with $(p,q)$ growth $$u_t-\mathrm{div}\left(\mid \nabla u{\mid }^{p-2}\nabla u+a(x,t)\mid \nabla u{\mid }^{q-2}\nabla u\right)=0,a(x,t)\ge 0,$$ under the generalized non-logarithmic Zhikovs conditions $$\mid a(x,t)-a(y,\tau )\mid ⩽A\mu (r)r^{q-p},(x,t),(y,\tau )\in Q_{r,r}(x_0,t_0),$$ $$\underset{r\to 0}{\lim }\mu (r)r^{q-p}=0,\underset{r\to 0}{\lim }\mu (r)=+\infty ,\underset{0}{\int }{\mu }^{-\beta }(r)\frac{dr}{r}=+\infty ,$$ with some $\beta >0$.