Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions

¹ Instituto de Ciencias, Univ. Nac. Gral. Sarmiento, J.M. Gutierrez entre Verdi y J.L. Suarez (1613), Los Polvorines, Buenos Aires, Argentina. gacosta@ungs.edu.ar.
² Departamento de Matemática, FCEyN, UBA (1428), Buenos Aires, Argentina. jfbonder@dm.uba.ar. andjrossi@dm.uba.ar.
³ Universidad de San Andrés, Vito Dumas 284 (1644), Victoria, Buenos Aires, Argentina. pgroisman@udesa.edu.ar.

Received: 23 April 2001
Revised: 12 October 2001

Abstract

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations u_t = Δu, v_t = Δv in Ω x (0,T); fully coupled by the boundary conditions , on ∂Ω x (0,T), where Ω is a bounded smooth domain in . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete 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.