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Simulation of the blow-up and the quenching time for a positive solutions of
singular boundary value problems for a nonlinear parabolic systems

Halima Nachid(1), Benjamin Yekre (2) and, Yoro Gozo (3)
(1) International University of Grand-Bassam, BP 564 Grand-Bassam, CÔTE d'IVOIRE
et Labo. de Modélisation Mathématique et de Calcul Économique LM2CE settat, MAROC
(1) (2) (3) Laboratoire de Mathématiques et Informatique, UFR-SFA,
Université Nangui Abrogoua, 02 BP 801 Abidjan 02, CÔTE d'IVOIRE

Mathematics Subject Classification: (MSC 2010) 35K55, 35B40, 65M06
Key words : Nonlinear parabolic systems, blow-up, parabolic equations, extinction, numerical blow-up time, numerical quenching time, existence, finite difference method. 


We consider the following initial-boundary value problem.

(| ut(x,t) = aLu - αupf(v) in Ω ×(0,∞ ),
||{ vt(x,t) = bLu + cLv + Qupf(v)  in  Ω × (0,∞ ),
   ∂u-      -∂v
|||(  ∂N = 0,  ∂N  = 0  on  ∂Ω× (0,∞ ),
  u(x,0) = u0(x) > 0, u(x,0) = u0(x) > 0 in Ω,

where Ω is a bounded domain in N with smooth boundary Ω, L is an elliptic operator, p > 0, α0, a,b,c,Q are positive constants, f(s) is a positive and increasing function for the positive values of s.

We find some conditions under which solutions of the above system either exists globally or blow up in a finite time. We also prove that if 0 < p < 1, the solutions u,v) extincts in a finite time. Some numerical results are given to illustrate our analysis.

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