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

Abstract

We investigate the continuity of solutions for general nonlinear parabolic equations of the form \begin{equation*} u_t-\sum_{i=1}^n\frac{\partial}{\partial x_i}\Bigl(\Bigl|\frac{\partial u}{\partial x_i}\Bigr|^{p(x)-2}\frac{\partial u}{\partial x_i}\Bigr)=0,\qquad 2 < p_1\leq p(x)\leq p_2 \end{equation*} near a nonsmooth boundary of a cylindrical domain. We prove the sufficient and necessary condition for regularity of a boundary point in terms of the $p(x)$-capacity.