Higher integrability for singular doubly nonlinear systems

Abstract

We prove a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems whose prototype is $${\partial }_t\left(|u{|}^{q-1}u\right)-\mathrm{div}\left(|Du{|}^{p-2}Du\right)=\mathrm{div}\left(|F{|}^{p-2}F\right)\text{ in }{\Omega }_T:=\Omega \times (0,T)$$ with parameters $p>1$ and $q>0$ and $\Omega \subset {\mathbb{R}}^n$. In this paper, we are concerned with the ranges $q>1$ and $p>\frac{n(q+1)}{n+q+1}$. A key ingredient in the proof is an intrinsic geometry that takes both the solution $u$ and its spatial gradient $Du$ into account.