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

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

ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 1, pp. 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 $\Omega \times (0,T)$; 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 ${ℝ}^d$. 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
Mots clés : blow-up, parabolic equations, semi-discretization in space, asymptotic behavior, non-linear boundary conditions

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