Numerical Blow-Up Time for a Semilinear Parabolic Equation with Nonlinear Boundary Conditions

Abstract

We obtain some conditions under which the positive solution forsemidiscretizations of the semilinear equation $u_t=u_{xx}-a(x,t)f(u),0<x<1,t\in (0,T)$, with boundary conditions $u_x\left(0,t\right)=0$, $u_x\left(1,t\right)=b\left(t\right)g\left(u\left(1,t\right)\right)$, blows up in a finite time and estimate its semidiscrete blow-up time. We also establishthe convergence of the semidiscrete blow-up time and obtain some results aboutnumerical blow-up rate and set. Finally, we get an analogous result takinga discrete form of the above problem and give some computational results toillustrate some points of our analysis.