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Simultaneous vs. non-simultaneous blow-up in numerical approximations of aparabolic system with non-linear boundary conditions
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- Journal:
- ESAIM: Mathematical Modelling and Numerical Analysis / Volume 36 / Issue 1 / January 2002
- Published online by Cambridge University Press:
- 15 April 2002, pp. 55-68
- Print publication:
- January 2002
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- Article
- Export citation
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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 $\frac{\partial u}{\partial\eta} = u^{p_{11}}v^{p_{12}}$
, $\frac{\partial v}{\partial\eta} = u^{p_{21}}v^{p_{22}}$ 
on ∂Ω x (0,T), where Ω 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.
