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{\bf
NUMERICAL ANALYSIS OF THE INCOMPLETE BLOW-UP OF THE SOLUTIONS
OF A CLASS OF NONLINEAR HEAT TRANSFER EQUATIONS\\
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S.N.~Dimova, T.P.~Tchernogorova\\
Faculty of Mathematics and Informatics, The University of Sofia,\\
James Bautchier 5, Sofia 1126, Bulgaria,\\
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The possibility of nontrivial continuation of the blow-up solutions of the
initial boundary value problem

$$u_t = \nabla\left( u^{\sigma} \nabla u \right) + u^{\beta}, \ \ \
t>0, \ \ x\in \Omega\in R^N,$$
$$u(0,x) = u_o(x)\ge 0,\ \ x \in \Omega,$$
$$u(t,x) = 0, \ \ t \ge 0, \ \ \ x \in \partial \Omega,$$

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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.

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