Monotone continuous dependence of solutions of singular quenching parabolic problems

We prove the continuous dependence, with respect to the initial datum of solutions of the "quenching parabolic problem" partial derivative(t)u - Delta u + chi({u>0})u(-beta) = lambda u(p), with zero Dirichlet boundary conditions, when beta is an element of (0, 1), p is an element of (0, 1], lambda >= 0 and chi({u>0}) denotes the characteristic function of the set of points (x, t) where u(x, t) > 0. Notice that the absorption term chi({u>0})u(-beta) is singular and monotone decreasing which does not allow the application of standard monotonicity arguments.