Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents

We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$\{\begin{array}{ll}{u}_t-div\mathcal{A}(x,t,\mathrm{\nabla }u)=0& \text{on }\mathrm{\Omega }\times (-T,T),\\ u=0& \text{on }\mathrm{\partial }\mathrm{\Omega }\times (-T,T),\end{array}$$ where the non-linear structure $$\mathcal{A}$(x, t,\nabla u)$ismodelledafterthevariableexponent$p(x,t)$-Laplace operator given by $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x,t)-2}\mathrm{\nabla }u$. To this end, we prove that the gradients satisfy a reverse H"older inequality near the boundary by constructing a suitable test function which is Lipschitz continuous and preserves the boundary values. In the interior case, such a result was proved in \cite{bogelein2014very} provided $p(x,t)\ge {\mathfrak{p}}^{-}\ge 2$ holds and was then extended to the singular case $\frac{2n}{n+2}< \mathfrak{p}^-\leq p(x,t)\leq \mathfrak{p}^+ \leq 2$ in \cite{li2017very}. This restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case ${\mathfrak{p}}^{+}\le 2$ and ${\mathfrak{p}}^{-}\ge 2$. In this paper, we develop a new unified intrinsic scaling, using which, we are able to extend the results of \cite{bogelein2014very,li2017very} to the full range $\frac{2n}{n+2} < \mathfrak{p}^- \leq p(x,t)\leq \mathfrak{p}^+<\infty$ and also obtain analogous results upto the boundary. \emph{The main novelty of this paper is that our methods are able to handle both the singular case and degenerate case simultaneously.} To simplify the exposition, we will only prove the higher integrability result near the boundary, provided the domain $\mathrm{\Omega }$ satisfies a uniform measure density condition. Our techniques are also applicable to higher order equations as well as systems.