### Numerical blow-up and asymptotic behavior for a semilinear parabolic equation with a nonlinear boundary condition.

*Diabate Nabongo, Theodore K. Boni*

#### Abstract

This paper concerns the study of the numerical approximation for the following initial-boundary value problem:

%

\[(P)\\

\left\{%

\begin{array}{ll}

\hbox{$u_t(x,t)=u_{xx}(x,t)+au^{p}(x,t),\quad 00$,} \\

\hbox{$u(x,0)=u_{0}(x)>0,\quad 0\leq x \leq 1$,} \\

\end{array}%

\right.

\]

%

where $a>0$, $b>0$ and $p>q>1$. We show that under some conditions, the solution of a semidiscrete form of $(P)$ either decays uniformly to zero or blows up in a finite time. When the blow-up occurs, we estimate the semidiscrete blow-up time and prove that under some assumptions, the semidiscrete blow-up time converges to the real one when the mesh size goes to zero. When the semidiscrete solution goes to zero as $t$ goes to infinity, we give its asymptotic behavior. Finally, we give some numerical experiments to illustrate our analysis.

%

\[(P)\\

\left\{%

\begin{array}{ll}

\hbox{$u_t(x,t)=u_{xx}(x,t)+au^{p}(x,t),\quad 00$,} \\

\hbox{$u(x,0)=u_{0}(x)>0,\quad 0\leq x \leq 1$,} \\

\end{array}%

\right.

\]

%

where $a>0$, $b>0$ and $p>q>1$. We show that under some conditions, the solution of a semidiscrete form of $(P)$ either decays uniformly to zero or blows up in a finite time. When the blow-up occurs, we estimate the semidiscrete blow-up time and prove that under some assumptions, the semidiscrete blow-up time converges to the real one when the mesh size goes to zero. When the semidiscrete solution goes to zero as $t$ goes to infinity, we give its asymptotic behavior. Finally, we give some numerical experiments to illustrate our analysis.

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