We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ${u}_{t}=\Delta u$, ${v}_{t}=\Delta v$ in $\Omega \times (0,T)$; fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta}={u}^{{p}_{11}}{v}^{{p}_{12}}$, $\frac{\partial v}{\partial \eta}={u}^{{p}_{21}}{v}^{{p}_{22}}$ on $\partial \Omega \times (0,T)$, where $\Omega $ is a bounded smooth domain in ${\mathbb{R}}^{d}$. 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 ${p}_{11}>1$ and ${p}_{21}<2({p}_{11}-1)$, 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.

Classification: 65M60, 65M20, 35K60, 35B40

Keywords: blow-up, parabolic equations, semi-discretization in space, asymptotic behavior, non-linear boundary conditions

@article{M2AN_2002__36_1_55_0, author = {Acosta, Gabriel and Bonder, Juli\'an Fern\'andez and Groisman, Pablo and Rossi, Julio Daniel}, title = {Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {36}, number = {1}, year = {2002}, pages = {55-68}, doi = {10.1051/m2an:2002003}, zbl = {1003.65097}, mrnumber = {1916292}, language = {en}, url = {http://www.numdam.org/item/M2AN_2002__36_1_55_0} }

Acosta, Gabriel; Bonder, Julián Fernández; Groisman, Pablo; Rossi, Julio Daniel. Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 1, pp. 55-68. doi : 10.1051/m2an:2002003. http://www.numdam.org/item/M2AN_2002__36_1_55_0/

[1] Blow-up for semidiscretizations of reaction diffusion equations. Appl. Numer. Math. 20 (1996) 145-156. | Zbl 0857.65096

, and ,[2] On the blow-up time convergence of semidiscretizations of reaction diffusion equations. Appl. Numer. Math.26 (1998) 399-414. | Zbl 0929.65070

, and ,[3] Numerical approximation of a parabolic problem with nonlinear boundary condition in several space dimensions. Preprint. | Zbl 0997.35025

, , and .[4] Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations. 72 (1988) 201-269. | Zbl 0658.34011

,[5] Blow-up in diffusion equations: a survey. J. Comput. Appl. Math. 97 (1998) 3-22. | Zbl 0932.65098

and ,[6] A rescaling algorithm for the numerical calculation of blowing up solution. Comm. Pure Appl. Math. 41 (1988) 841-863. | Zbl 0652.65070

and ,[7] Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17 (1996) 305-327. | Zbl 0860.35050

, and ,[8] Asymptotic behaviours of blowing up solutions for finite difference analogue of ${u}_{t}={u}_{xx}+{u}^{1+\alpha}$. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986) 541-574. | Zbl 0616.65098

,[9] The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058

,[10] Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete Contin. Dyn. Syst. 4 (1998) 497-506. | Zbl 0951.65088

, and ,[11] Global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30 (1993) 1622-1663. | Zbl 0792.65066

and ,[12] Blow-up vs. spurious steady solutions. Proc. Amer. Math. Soc. 129 (2001) 139-144. | Zbl 0970.35003

and ,[13] Approximation of dissipative partial differential equations over long time intervals, in D.F. Griffiths et al., Eds., Numerical Analysis 1993. Proc. 15th Dundee Biennal Conf. on Numerical Analysis, June 29-July 2nd, 1993, University of Dundee, UK, in Pitman Res. Notes Math. Ser. 303, Longman Scientific & Technical, Harlow (1994) 180-207. | Zbl 0795.65031

, and ,[14] Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992). | MR 1212084 | Zbl 0777.35001

,[15] Simultaneousvs. non-simultaneous blow-up. N. Z. J. Math. 29 (2000) 55-59. | Zbl 0951.35019

and ,[16] On existence and nonexistence in the large for an N-dimensional system of heat equations with nontrivial coupling at the boundary. N. Z. J. Math. 26 (1997) 275-285. | Zbl 0891.35053

,[17] Blow-up in QuasiLinear Parabolic Equations, in Walter de Gruyter, Ed., de Gruyter Expositions in Mathematics 19, Berlin (1995). | Zbl 1020.35001

, , and ,[18] Dynamical systems and numerical analysis, in Cambridge Monographs on Applied and Computational Mathematics 2, Cambridge University Press, Cambridge (1998). | MR 1402909 | Zbl 0913.65068

and ,