Partial regularity for degenerate parabolic systems with non-standard growth and discontinuous coefficients

Abstract

This article studies the partial H\"older continuity of weak solutions to certain degenerate parabolic systems whose model is the differentiable parabolic $p(x,t)$-Laplacian system, \begin{equation*}\partial_t u-\operatorname{div}[\mu(z)(1+|Du|^2)^{\frac{p(z)-2}{2}}Du]=0,\qquad p(z)\geq2.\end{equation*} Here, the exponential function $p(z)$ satisfies a logarithmic continuity condition. We show that if $\mu(z)$ satisfies a certain VMO-type condition, then $u$ is locally H\"older continuous except for a measure zero set.