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Numerical Blow-up Solutions for Nonlinear Parabolic Equations

Volume 7, Number 2 (2008)

Numerical Blow-up Solutions for Nonlinear Parabolic Equations

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Abstract

This paper concerns the study of the numerical approximation for the following initial-boundary value problem:
\begin{equation}
u_t(x,t)=(u^{m}(x,t))_{xx}+\alpha u^{p}(x,t),\quad x\in(0,1),\quad
t\in(0,T),
\end{equation}
\begin{equation}
u(0,t)=0,\quad u(1,t)=0,\quad t\in(0,T),
\end{equation}
\begin{equation}
u(x,0)=u_{0}(x)\geq 0,\quad x\in[0,1],
\end{equation}
where $p\geq m>1$ and $\alpha>0$. When $p=m$, we show that there exists a positive number $\alpha^{*}$ such that if $\alpha>\alpha^{*}$ then any solution of a semidiscrete form of (0.1)--(0.3) blows up in a finite time whereas if $\alpha<\alpha^{*}$ then any solution exists globally and decays to zero. We have obtained the same result using a discrete form of (0.1)--(0.3). When $p>m$, we prove that any solution of a semidiscrete form of (0.1)--(0.3) blows up in a finite time and estimate its semidiscrete blow-up time. In this last case, under some assumptions, we show that the semidiscrete blow-up time converges to the real one when the mesh size goes to zero. Finally, we give some numerical experiments to illustrate our analysis.