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 $$ \partial_t \left(|u|^{q-1}u \right) -\operatorname{div} \left( |Du|^{p-2} Du \right) = \operatorname{div} \left( |F|^{p-2} F \right) \quad \text{ in } \Omega_T := \Omega \times (0,T) $$ 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.