L^{\infty} a priori bounds for gradients of solutions to quasilinear inhomogenous fast-growing parabolic systems

TL;DR

This paper establishes L^{} bounds for the gradients of solutions to a class of quasilinear parabolic systems with fast-growing inhomogeneous terms, extending classical growth control concepts.

Contribution

It generalizes the boundedness results for gradients to systems involving p-Laplacian operators with right-hand sides that grow at or faster than p - 1.

Findings

Proves boundedness of gradients for solutions to the specified systems.

Extends classical growth control to more general inhomogeneous terms.

Uses energy estimates and nonlinear Moser iteration techniques.

Abstract

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