Harnack's Inequality for Degenerate Double Phase Parabolic Equations under the Non-Logarithmic Zhikov's Condition

We prove Harnack-type inequalities for bounded non-negative solutions of the degenerate parabolic equations with (p, q) growth u_t-div(|∇ u|^p-2∇ u+a(x,t)|∇ u|^q-2∇ u)=0,0.5em a(x,t)≥ 0, under the generalized non-logarithmic Zhikov’s conditions [ [ |a(x,t)-a(y,τ)|⩽ Aμ (r)r^q-p, (x,t),(y,τ)∈Q_r,r(x_0,t_0), ]; [ r→ 0limμ (r)r^q-p=0, r→ 0limμ (r)=+∞, 0∫μ^-β(r)dr/r ]=+∞, ] with some β > 0.