The Dependence Of Blow-Up Time With Respect To Parameters For Small Reaction Diffusion Equation

In this paper we consider the following initial-boundary problem.

｛ut (x,t)-gu_xx(x,t)=f u(x,t)   x in(0,1),t in(0,T ),

     u(0,t)=0 u(1,t=0, t in [0,T],

     u( x,0 )= u₀(x), x in[0,T].

Where f s( ) is a positive, increasing, convex function for nonnegative value of s, f (0) >0,

, and ò ₀^¥    ds\f(s)<+¥, and gis a positive diffusion parameter. We find some conditions under which the solution of semi-discrete form of the above problem blows up in a finite time and estimate its semi-discrete blow up time. We also prove the convergence of the semidiscrete form blow-up time to the real one when the mesh size tends to zero. Finally, we give some numerical results to illustrate our analysis.