Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
Instituto de Ciencias, Univ. Nac. Gral. Sarmiento, J.M. Gutierrez entre Verdi y J.L. Suarez (1613), Los Polvorines, Buenos Aires, Argentina. firstname.lastname@example.org.
2 Departamento de Matemática, FCEyN, UBA (1428), Buenos Aires, Argentina. email@example.com. firstname.lastname@example.org.
3 Universidad de San Andrés, Vito Dumas 284 (1644), Victoria, Buenos Aires, Argentina. email@example.com.
Revised: 12 October 2001
We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = 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.
Mathematics Subject Classification: 65M60 / 65M20 / 35K60 / 35B40
Key words: Blow-up / parabolic equations / semi-discretization in space / asymptotic behavior / non-linear boundary conditions.
© EDP Sciences, SMAI, 2002