On the numerical computation of blowing‐up solutions for semilinear parabolic equations

Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end , with a finite unknown ‘blow‐up’ time T b have been studied in a previous work. Specifically, for ε a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ‘mass control’ property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve‐\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon‐u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end , \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow‐up time T b and the blow‐up solution u . For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end , with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end . Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end , we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end , defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end . In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end , the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end , verifies, $\|(u‐u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1‐\alpha}$\nopagenumbers\end , \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end . The two special cases of a single‐point blow‐up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p‐1})$\nopagenumbers\end . The estimate $|T_\eta‐T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd.