Simultaneous vs. Non-simultaneous Blow-up in Numerical Approximations of a Parabolic System with Nonlinear Boundary Conditions

We study the asymptotic behavior of a semidiscrete numerical approximation for a pair of heat equations ut = ∆u, vt = ∆v in Ω × (0, T ); fully coupled by the boundary conditions ∂u ∂η = up11vp12 , ∂v ∂η = up21vp22 on ∂Ω× (0, T ), where Ω is a bounded smooth domain in Rd. We focus in the existence or not of non-simultaneous blow-up for a semidiscrete approximation (U, V ). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 − 1), which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover, we find that if the continuous problem has non-simultaneous blow-up then the same is true for the discrete one. We also prove some results about the convergence of the scheme and the convergence of the blow-up times.