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

$$\begin{aligned} \partial _t \left( |u|^{q-1}u \right) -{{\,\textrm{div}\,}}\left( |Du|^{p-2} Du \right) = {{\,\textrm{div}\,}}\left( |F|^{p-2} F \right) \quad \text { in } \Omega _T:= \Omega \times (0,T) \end{aligned}$$

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.