\documentstyle[12pt]{article} %\pagestyle{empty} \topmargin=0in \textheight=240mm \oddsidemargin=0mm \textwidth=170mm %\baselineskip=7mm \renewcommand{\baselinestretch}{1.33} \renewcommand{\arraystretch}{1.33} \begin{document} %\load{\normalsize}{\rm} %\vspace*{\bigskipamount} %\vspace*{\bigskipamount} %\vspace*{\bigskipamount} \vspace*{\bigskipamount} \vspace*{\bigskipamount} \vspace*{\bigskipamount} \noindent \begin{center} {\bf NUMERICAL ANALYSIS OF THE INCOMPLETE BLOW-UP OF THE SOLUTIONS OF A CLASS OF NONLINEAR HEAT TRANSFER EQUATIONS\\ } \bigskip \bigskip S.N.~Dimova, T.P.~Tchernogorova\\ Faculty of Mathematics and Informatics, The University of Sofia,\\ James Bautchier 5, Sofia 1126, Bulgaria,\\ \end{center} \newcommand{\beq}{$$} \newcommand{\ds}{\displaystyle} \newcommand{\eeq}{$$} \bigskip \bigskip The possibility of nontrivial continuation of the blow-up solutions of the initial boundary value problem $$u_t = \nabla\left( u^{\sigma} \nabla u \right) + u^{\beta}, \ \ \ t>0, \ \ x\in \Omega\in R^N,$$ $$u(0,x) = u_o(x)\ge 0,\ \ x \in \Omega,$$ $$u(t,x) = 0, \ \ t \ge 0, \ \ \ x \in \partial \Omega,$$ \bigskip \noindent after the blow-up time is analyzed numerically. This phenomenon, called incomplete blow-up, takes place in the case $\sigma \ge 0$, $\beta > \beta^ \star$, where $\beta^ \star$ is one of the critical blow-up exponents: $\beta^ \star = (\sigma + 1)(N+2)/(N-2), \ N \ge 3$. The self-similar solutions, which describe the two stages of the process - up to and after the blow-up, are analyzed numerically. A special numerical technique for solving the problem for large spase dimensions $N$ is developed. The semilinear case $( \sigma = 0)$ and the quasilinear case $( \sigma > 0)$ are discussed. \end{document} 