Hölder-Zygmund estimates for degenerate parabolic systems (Q2438108)

The author considers energi solutions of the inhomogeneous parabolic \(p\)-Laplacian system \(\partial_t u-\text{div}(|Du|^{p-2}\times Du)=-\text{div}g.\) It is shown that in the case \(p\geq 2\) if the right hand side \(g\) is locally in \(L^\infty(BMO),\) then \(u\) is locally in \(L^\infty(C^1),\) where \(C^1\) is the 1-Hölder-Zygmund space. This is the borderline case of the Calderón-Zygmund theory. There are obtained local quantitative estimates and it is shown that finer properties of \(g\) are conserved by \(Du.\) It is proved a new decay for gradients of \(p\)-caloric solutions for all \(\frac{2n}{n+2}<p<\infty.\)