Boundary regularity for parabolic systems with nonstandard $(p,q)$-growth conditions in smooth convex domains

Abstract

We study the boundary regularity of local weak solutions to nonlinear parabolic systems of the form \begin{equation*} \partial_t u^i - \mathrm{div} \big( a(|Du|) Du^i \big)= f^i, \qquad i=1,\dots,N, \end{equation*} in a space-time cylinder ${\Omega }_T=\Omega \times (0,T)$, where $\Omega \subset {\mathbb{R}}^n$ ($n\ge 2$) is a bounded, convex $C^2$-domain and $T>0$. The inhomogeneity $f=(f^1,\dots ,f^N)$ belongs to $L^{n+2+\sigma }({\Omega }_T,{\mathbb{R}}^N)$ for some $\sigma >0$. The coefficients $a:{\mathbb{R}}_{>0}\to {\mathbb{R}}_{>0}$ are of Uhlenbeck-type and satisfy a nonstandard $(p,q)$-growth condition with \[ 2 \leq p \leq q < p + \frac{4}{n+2}. \] Our main result establishes a local Lipschitz estimate up to the lateral boundary for any local weak solution that vanishes on the lateral boundary of the cylinder.