Higher integrability in the full subcritical range

Establish local higher integrability of the spatial gradient for weak solutions of the inhomogeneous doubly nonlinear parabolic system \(\partial_t(|u|^{q-1}u)-\operatorname{div}(|Du|^{p-2}Du)=\operatorname{div}(|F|^{p-2}F)\) throughout the full subcritical parameter range \(1<p\leq N(q+1)/(N+q+1)\) and \(q>0\), under the stated extra local integrability assumption on \(|u|\).

Background

The paper studies local boundedness and higher integrability for weak solutions of inhomogeneous doubly nonlinear parabolic systems in the subcritical regime. It proves local boundedness across the full subcritical range but establishes higher integrability of $|Du|$ only in the narrower range $p=N(q+1)/(N+q+1)$ and $(N+p)/(N-p)<q<(N+2)/(N-2)$.

The authors explain that the restriction to this narrower range is caused by the need to control the quantitative boundedness estimate in their higher-integrability argument. Thus, extending the higher-integrability result from the stated narrower range to the full subcritical range remains an explicitly unresolved problem.