The cutoff phenomenon, conceptualized at the origin for finite Markov chains, states that for a parametric family of evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt for certain choices of initial conditions, when the parameter tends to infinity. This threshold phenomenon can be seen as a critical competition between trend to equilibrium and worst initial condition. In this note, we investigate this phenomenon beyond stochastic processes, in the context of the analysis of nonlinear partial differential equations, by proving cutoff for the fast diffusion and porous medium Fokker-Planck equations on the Euclidean space, when the dimension tends to infinity. We formulate the phenomenon using quadratic Wasserstein distance, as well as using specific relative entropy and Fisher information. Our high dimensional asymptotic analysis uses the exact solvability of the model involving Barenblatt profiles. It includes the Ornstein-Uhlenbeck dynamics as a special linear case.