L∞L∞ a priori bounds for gradients of solutions to quasilinear inhomogeneous fast-growing parabolic systems

We prove boundedness of gradients of solutions to quasilinear parabolic systems, the main part of which is a generalization to the pp-Laplacian and its right-hand side’s growth depending on the gradient is not slower (and generally strictly faster) than p−1p−1. This result may be seen as a generalization to the classical notion of a controllable growth of the right-hand side, introduced by Campanato, over gradients of pp-Laplacian-like systems. Energy estimates and a nonlinear iteration procedure of the Moser type are cornerstones of the used method.