Gradient higher integrability for degenerate/singular parabolic double phase problems

This article proves an interior gradient higher integrability result for weak solutions to the parabolic double phase problems. The prototype equation for the parabolic double phase problem of p-Laplace type reads as uₜ - div (|∇ u|ᵖ⁻²∇ u+ a(z)|∇ u|q-2∇ u)=0 where 2n/n+2<p≤ q < ∞ and the coefficient $a(z)$ is a non-negative H\"{o}lder continuous function on ΩT=Ω × (0, T), Ω ⊂ Rⁿ. Recently, this problem has been studied by Kim, Kinnunen and Moring [19] for the degenerate case and Kim and S\"{a}rki\"{o} [20] for the singular case. We introduce a new intrinsic scaling that can handle the degenerate and the singular case simultaneously. This scaling can also be used to obtain similar results in the variable exponent case.