Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary

We study a system of two porous medium type equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain conditions, one of its components becomes unbounded in finite time while the other remains bounded, a situation that is known in the literature as non-simultaneous blow-up. We characterize completely, in the case of nondecreasing in time solutions, the set of parameters appearing in the system for which non-simultaneous blow-up indeed occurs. Moreover, we obtain the blow-up rate and the blow-up set for the component which blows up. We also prove that in the range of exponents where each of the components may blow up on its own there are special initial data such that blow-up is simultaneous. Finally, we give conditions on the exponents which lead to non-simultaneous blow-up for every initial data.