The finite-time blow-up phenomenon for the semilinear heat equation was widely considered since 1960's both mathematically and numerically. In this paper, we would like to consider the computation of blow-up solutions for a coupled semilinear heat equations. To this end, we study the numerical solutions for an ODE system as a first step. We propose a scheme for the computation of the numerical blow-up time and prove the convergence for the numerical solution and the numerical blow-up time. Then we apply the idea to a semilinear parabolic system whose solution may blow up in a finite time. Several numerical examples are reported and discussed.

Abstract i
Contents ii
List of Figures iii
1 Introduction 1
2 A nonlinear ODE system 3
3 A finite difference scheme for (4) 10
4 Conclusion 17

[1] Abia, L., López-Marcos, J. C., Martínez, J.: The Euler method in the numerical integration of reaction-diffusion problems with blow-up. Appl. Numer. Math. 38, 287–313 (2001).
[2] Bandle, H., Brunner, C.: Blow-up in diffusion equations: a survey. J. Comp. Appl. Math. 97, 3–22 (1998).
[3] Chen, Y.-G.: Blow-up solutions to a finite difference analogue of u_t = ∆u + u^{1+α} in N-dimensional ball. Hokkaido Math. J. 21, 447–474 (1992).
[4] Cho, C.-H.: On the computation of numerical blow-up time. Jpn. J. Indust. Appl. Math. 30, 331–349 (2013).
[5] Cho, C.-H., Hamada, S., Okamoto, H.: On the finite difference approximation for a parabolic blow-up problem. Jpn J. Indust. Appl. Math. 24, 131–160 (2007).
[6] Cho, C.-H., Okamoto, H.: A finite difference scheme for an axisymmetirc nonlin-ear heat equation with blow-up. In preparation.
[7] Friedman, A., Giga, Y.: A single point blow-up for solutions of semilinear parabolic systems. J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 34, 65–79 (1987).
[8] Fujita, H.: On the blowing up of solutions to the Cauchy problem for u_t = ∆u + u^{1+α}. J. Faculty Science, U. of Tokyo 13, 109–124 (1966).
[9] Groisman, P.: Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions. Computing 76, 325–352 (2006).
[10] Levine, H. A.: The role of critical exponents in blow up theorems. SIAM Rev. 32, 262–288 (1990).
[11] Nakagawa, T.: Blowing up of a finite difference solution to u_t = u_{xx} + u^2. Appl. Math. Optim. 2, 337–350 (1976).
[12] Souplet, P.: Single-point blow-up for a semilinear parabolic system. J. Eur. Math. Soc. 11, 169–188 (2009).
[13] Werssler, F.: Single point blowup of similinear initial value problems. J. Differ. Equ. 55, 202–224 (1984).