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 ut−div(|∇u|p−2∇u+a(xt)|∇u|q−2∇u)=0,a(xt)≥0, under the generalized non-logarithmic Zhikov’s conditions |a(xt)−a(yτ)|⩽Aμ(r)rq−p,(xt),(yτ)∈Qr,r(x0t0),limr→0μ(r)rq−p=0,limr→0μ(r)=+∞,∫0μ−β(r)drr=+∞, with some β > 0.