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We prove boundary higher integrability for the (spatial) gradient of\emph{very weak} solutions of quasilinear parabolic equations of the form $$\left\{  \begin{array}{ll} u_t - div \mathcal{A}(x,t,\nabla u) = 0 &\quad \text{on} \\Omega \times (-T,T), \\ u = 0 &\quad \text{on} \ \partial \Omega \times(-T,T),  \end{array} \right. $$ where the non-linear structure $\mathcal{A}(x,t,\nabla u)$ is modelled after the variable exponent $p(x,t)$-Laplace operatorgiven by $|\nabla u|^{p(x,t)-2} \nabla u$. To this end, we prove that thegradients satisfy a reverse H\"older inequality near the boundary byconstructing a suitable test function which is Lipschitz continuous andpreserves the boundary values. In the interior case, such a result was provedin \cite{bogelein2014very} provided $p(x,t) \geq \mathfrak{p}^- \geq 2$ holdsand was then extended to the singular case $\frac{2n}{n+2}< \mathfrak{p}^-\leqp(x,t)\leq \mathfrak{p}^+ \leq 2$ in \cite{li2017very}. This restriction wasnecessary because the intrinsic scalings for quasilinear parabolic problems aredifferent in the case $\mathfrak{p}^+ \leq 2$ and $\mathfrak{p}^-\geq 2$.  In this paper, we develop a new unified intrinsic scaling, using which, weare able to extend the results of \cite{bogelein2014very,li2017very} to thefull 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 handleboth the singular case and degenerate case simultaneously.} To simplify theexposition, we will only prove the higher integrability result near theboundary, provided the domain $\Omega$ satisfies a uniform measure densitycondition. Our techniques are also applicable to higher order equations as wellas systems.

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