The regularity for nonlinear parabolic systems of \(p\)-Laplacian type with critical growth (Q2441672)

The authors study the Hölder regularity of weak solutions to the evolutionary \(p\)-Laplacian system \[ \partial_tu^i -\sum_{\alpha,\beta=1}^m D_\alpha ((|Du|_g)^{p-2}g^{\alpha\beta}(z,u) D_\beta u^i )=F^i(z,u,Du),\quad i=1,\dots,n \] with critical growth on the gradient. It is established a natural criterion for proving that a small solution and its gradient are locally Hölder continuous almost everywhere. The result can be applied to study the regularity of the heat flow for \(m\)-dimensional \(H\)-systems as well as the \(m\)-harmonic flow.