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 $ΩT = Ω\times (0,T)$, where $Ω\subset \mathbb{R}^n$ ($n \geq 2$) is a bounded, convex $C^2$-domain and $T>0$. The inhomogeneity $f=(f^1,\dots,f^N)$ belongs to $L^{n+2+σ}(Ω_T,\mathbb{R}^N)$ for some $σ>0$. The coefficients $a\colon \mathbb{R}{>0} \to \mathbb{R}_{>0}$ 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.