Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 2, No 2 (2008)

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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.

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