## Numerical blow-up solutions for some semilinear heat equations

Firmin K. N'Gohisse and Théodore K. Boni

### Abstract

This paper concerns the study of the numerical approximation for the following initial-boundary value problem, \begin{eqnarray*} &&u_{t}=u_{xx}+\frac{b}{x}u_{x}+ u^{p},\quad x\in(0,1),\quad t\in(0,T), \\ &&u_{x}(0,t)=0,\quad u(1,t)=0,\quad t\in(0,T), \\ &&u(x,0)=u_{0}(x),\quad x\in[0,1], \end{eqnarray*} where $b>0$ and $p>1$. We give some conditions under which the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. Under some assumptions, we also show that the semidiscrete blow-up time converges to the continuous blow-up time when the mesh size goes to zero. Finally, we give some numerical results to illustrate our analysis.

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### Key words

semidiscretizations, discretizations, semilinear heat equations, semidiscrete blow-up time

### AMS subject classifications

35B40, 35K65, 65M06

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