Global higher integrability for systems with $p$-growth structure in noncylindrical domains

We consider the Cauchy-Dirichlet problem to systems with $p$-growth structure with $1<p<\mathrm{\infty }$, whose prototype is \begin{equation*} \partial_t u- \operatorname{div} \big( |Du|^{p-2} Du \big) = \operatorname{div} \left( |F|^{p-2} F \right), \end{equation*} in a bounded noncylindrical domain $E\subset {\mathbb{R}}^{n+1}$. For $p>\frac{2(n+1)}{n+2}$ and domains $E$ that satisfy suitable regularity assumptions and do not grow or shrink too fast, we prove global higher integrability of $Du$. The result is already new in the case $p=2$.