Regularity for the weak solutions to certain parabolic systems under certain growth condition

In this paper, we consider the regularity of the weak solutions to a quasilinear parabolic systems which is a generalization of p-Laplacian of the type u(t)(i) - (A(alpha)(i)(del u))(x alpha) = f(i) (x, t, u, del u), i = 1, ... , N where the main part satisfies some ellipticity and f(i) satisfies certain growth conditions. We prove boundedness of the solutions and the gradients of solutions to the systems by the means of the energy estimates and a nonlinear iteration procedure of the Moser type in this paper. (C) 2018 Elsevier Inc. All rights reserved.