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, p. 55-68

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 $\Omega ×\left(0,T\right)$; 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 $\partial \Omega ×\left(0,T\right)$, where $\Omega$ is a bounded smooth domain in ${ℝ}^{d}$. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation $\left(U,V\right)$. 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 ${p}_{21}<2\left({p}_{11}-1\right)$, 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.

DOI : https://doi.org/10.1051/m2an:2002003
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/

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