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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.