On the regularity of the solutions of the Neumann problem for quasilinear parabolic systems

Partial regularity is proved of the generalized solution $u\colon\mathbf\Omega\times(0,T)\to\mathbf R^N$, $\mathbf\Omega\subset\mathbf R^n$, $n>2$, $N>1$, of a quasilinear parabolic system with nonsmooth conormal derivative. It is assumed that the functions forming the system and the boundary condition have controlled orders of nonlinearities, and their singularities are anisotropic with respect to the spatial variables and time. $L_p$-estimates of the gradient of $u$ in a neighborhood of $\partial\mathbf\Omega\times(0,T)$ are preliminarily deduced.