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 operator given by $|\nabla u|^{ p(x,t)-2} \nabla u$. 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 $p(x,t) \geq \mathfrak{ p}^- \geq 2$ 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 $\mathfrak{ p}^+ \leq 2$ and $\mathfrak{ p}^-\geq 2$. 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 $\Omega$ satisfies a uniform measure density condition. Our techniques are also applicable to higher order equations as well as systems.
Categories: math.AP
