We consider the following initial-boundary value problem

where Ω is a bounded domain in  with smooth boundary  L is an elliptic operator,    are positive constants,  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  the solution  extents in a finite time. Some numerical results are given to illustrate our analysis.

nonlinear parabolic, systems, blow-up, parabolic equations, extinction, numerical blow-up time, numerical quenching time, existence, finite difference method.