Regularity of solutions to quasilinear parabolic systems with time-nonsmooth principal matrix and the Neumann boundary condition (Q1661604)

The authors consider smoothness properties of weak solution \(u\) to non diagonal quasilinear parabolic system of equations \[ u_t - \mathrm{div}(a(z,u)\nabla u) = g(z),\quad z =(x,t)\in Q_1^+(0) \] on the space-time half cylinder \(Q_1^+(0)\) satisfying on the flat part of the boundary \(\Gamma_1(0)\) Neumann boundary condition \[ \frac{\partial u(z')}{\partial n_a}:= a(z',u)\nabla u\cdot n(x')),\quad z'=(x',0,t) \in \Gamma_1(0) , \] where \(Q_1^+(0)=B_1^+(0)\times (-1,0)\), \(B_1(0)\) is an open unit ball in \(\mathbb{R}^n\) and \(B_1^+(0)= B_1(0)\cap\{x_n >0\}\), \(\gamma_1(0)= B_1(0)\cap\{x_n=0\}\), \(\Gamma_1(0)= \gamma_1(0)\times(-1,0)\).