On the Local boundedness and higher integrability for the subcritical doubly nonlinear parabolic systems

Abstract

We consider the inhomogeneous doubly nonlinear parabolic systems of the form \begin{equation*}\partial_t (|u|^{q-1}u)-\operatorname{div}(|Du|^{p-2}Du)=\operatorname{div}(|F|^{p-2}F)\end{equation*} in a bounded space-time cylinder ${\Omega }_T=\Omega \times (0,T)\subset {\mathbb{R}}^{N+1}$. We study the local regularity properties for weak solutions in the subcritical range $p\le \frac{N(q+1)}{N+q+1}$ and $0<q<\frac{N+2}{N-2}$. Under an extra integrability assumption $|u|\in L_{\loc}^{\rr}(Ω_T)$, we establish a quantitative bound for $|u|$. Here, the exponent $\rr$ satisfies $\mathrmλ_{\rr}=N(p-q-1)+p\rr>0$. In addition, we prove local higher integrability of $|Du|$ in the range $p=\frac{N(q+1)}{N+q+1}$ and $\frac{N+p}{N-p}<q<\frac{N+2}{N-2}$, provided that $|u|\in L_{\loc}^{\rr}(Ω_T)$ holds.

arXiv commentThe proof of Lemma 4.7 is wrong and the range (1.5) is an empty set