This paper deals with the application of Stackelberg–Nash strategies for the controllability of parabolic partial differential equations, with nonlinear diffusion terms in the spatial variable with dimension N (here N is an any positive integer). We consider three cases: in the first one, the main control (the leader) acts in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in the second one, we consider a leader acting on the boundary while the followers are of the distributed kind; in the third one, we consider a coupled parabolic system with a leader and two followers acting in the interior of the domain. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we look for a leader that solves a controllability problem using trajectories techniques.