Higher integrability for singular doubly nonlinear systems

Abstract

AbstractWe prove a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems whose prototype is $$\begin{aligned} \partial _t \left( |u|^{q-1}u \right) -{{\,\textrm{div}\,}}\left( |Du|^{p-2} Du \right) = {{\,\textrm{div}\,}}\left( |F|^{p-2} F \right) \quad \text { in } \Omega _T:= \Omega \times (0,T) \end{aligned}$$ ∂ t | u | q - 1 u - div | D u | p - 2 D u = div | F | p - 2 F in Ω T : = Ω × ( 0 , T ) with parameters $$p>1$$ p > 1 and $$q>0$$ q > 0 and $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ 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 > 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.