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

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 ${\mathrm{\Omega }}_T=\mathrm{\Omega }\times (0,T)\subset {\mathbb{R}}^{N+1}$. 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 $\mathrm{\mid }u\mathrm{\mid }\in L_{\text{loc}}^{\text{rr}}({\mathrm{\Omega }}_T)$, we establish a quantitative bound for $\mathrm{\mid }u\mathrm{\mid }$. Here, the exponent $\text{rr}$ satisfies ${\lambda }_{\text{rr}}=N(p-q-1)+p\text{rr}>0$. In addition, we prove local higher integrability of $\mathrm{\mid }Du\mathrm{\mid }$ 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 $\mathrm{\mid }u\mathrm{\mid }\in L_{\text{loc}}^{\text{rr}}({\mathrm{\Omega }}_T)$ holds.