Global boundedness of the weak solutions to componentwise coercive parabolic systems

We prove essential boundedness of the weak solutions to the Cauchy--Dirichlet problem for the quasilinear parabolic system $$ \bu_t- \mathrm{div\,}\big(\bA(x,t,\bu,D\bu)\big)= \bb(x,t,\bu,D\bu) $$ which is modeled on the $p$-Laplacian vectorial operator. The nonlinear terms are given by Carath\'eodory functions and support controlled growth with respect to $\bu$ and $D\bu,$ while their dependence on $(x,t)$ is expressed in terms of suitable Lebesgue scales. Our result is proved by assuming additionally componentwise coercivity of the system and appropriate componentwise control of the lower-order terms.