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

Gabriel Acosta; Julián Fernández Bonder; Pablo Groisman; Julio Daniel Rossi

- Volume: 36, Issue: 1, page 55-68
- ISSN: 0764-583X

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topAcosta, Gabriel, et al. "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 36.1 (2002): 55-68. <http://eudml.org/doc/245881>.

@article{Acosta2002,

abstract = {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.},

author = {Acosta, Gabriel, Bonder, Julián Fernández, Groisman, Pablo, Rossi, Julio Daniel},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {blow-up; parabolic equations; semi-discretization in space; asymptotic behavior; non-linear boundary conditions; nonlinear boundary conditions; convergence},

language = {eng},

number = {1},

pages = {55-68},

publisher = {EDP-Sciences},

title = {Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions},

url = {http://eudml.org/doc/245881},

volume = {36},

year = {2002},

}

TY - JOUR

AU - Acosta, Gabriel

AU - Bonder, Julián Fernández

AU - Groisman, Pablo

AU - Rossi, Julio Daniel

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 1

SP - 55

EP - 68

AB - 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.

LA - eng

KW - blow-up; parabolic equations; semi-discretization in space; asymptotic behavior; non-linear boundary conditions; nonlinear boundary conditions; convergence

UR - http://eudml.org/doc/245881

ER -

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