Boundedness of solutions to quasilinear parabolic equations

We prove boundedness of the weak solutions to the Cauchy–Dirichlet problem for quasilinear parabolic equations whose prototype is the parabolic m-Laplacian. The nonlinear terms satisfy sub-controlled growth conditions with respect to the unknown function and its spatial gradient, while the behaviour in the independent variables is modelled in Lebesgue–Morrey spaces.