Non-simultaneous quenching in a semilinear parabolic system with weak singularities of logarithmic type

In this paper, we are interested in the possibility of non-simultaneous quenching for positive solutions of a coupled system of two semilinear parabolic equations with weak singularities of logarithmic type, ut = uxx + log(αv), vt = vxx + log(βu), 0 < α, β < 1, with homogeneous Neumann boundary conditions and positive initial data. Under some assumptions on the initial data and parameters α, β, we prove that the quenching is always non-simultaneous. We also give the quenching rate when the quenching is non-simultaneous. Finally, we show that our results can be used to a blow-up problem.