On the Wiener test for degenerate parabolic equations with non-standard growth condition (Q955891)

In the present article, the boundary regularity for a weak solution of the equation \[ u_t - \sum_{i=1}^n\frac{\partial}{\partial x_i}\left(\left|\frac{\partial u}{\partial x_i}\right|^{p(x) - 2}\frac{\partial u}{\partial x_i}\right) = 0, \quad 2 < p_1 \leq p(x) \leq p_2, \] is studied near a nonsmooth boundary of a cylindrical domain \(\Omega_T \equiv \Omega\times (0,T)\). Equations of this type arise in the context of non-Newtonian fluid dynamics with viscosity depending on the temperature. The local boundedness and the interior Hölder continuity of weak solutions of the above-mentioned equation with \(p = p(x,y)\) were obtained by \textit{M. Xu} and \textit{Y. Chen} in [Acta Math. Sin., Engl. Ser. 22, No.~3, 793--806 (2006; Zbl 1107.35041)]. As a result, the author establishes the sufficient and necessary condition for regularity of a boundary point in terms of the \(p(x)\)-capacity and exposes new a priori pointwise estimates for some auxiliary solutions.