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

TL;DR

This paper establishes boundary higher integrability for very weak solutions of quasilinear parabolic equations with variable exponents, extending previous interior results to the boundary and handling both singular and degenerate cases simultaneously.

Contribution

It develops a new unified intrinsic scaling method that extends boundary higher integrability results to the full range of variable exponents, including singular and degenerate cases.

Findings

Proves boundary reverse Hölder inequality for gradients of solutions.

Extends interior higher integrability results to boundary cases.

Handles full range of variable exponents, including singular and degenerate cases.

Abstract

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.