Non-simultaneous Blow-up Research Articles

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

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 , on ∂Ω x (0,T ), where Ω is a bounded smooth domain in . 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 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.

Non-simultaneous blow-up in a semilinear parabolic system

In this paper we study the possibility of non-simultaneous blow-up for positive solutions of a fully coupled system of two semilinear heat equations, \(u_t=\Delta u + u^{p_{11}} v^{p_{12}}, v_t=\Delta v + u^{p_{21}} v^{p_{22}}\). Under adequate hypotheses, we prove that if u blows up then v can fail to blow up if and only if \(p_{11} >1\) and \( p_{21} < p_{11}-1 \).