Higher integrability for singular doubly nonlinear systems

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