A numerical simulation for the blow-up of semi-linear diffusion equations

Abstract

Many mathematical models have the property of developing singularities at a finite time; in particular, the solution u(x, t) of the semi-linear parabolic Equation (1) may blow up at a finite time T. In this paper, we consider the numerical solution with blow-up. We discretize the space variables with a spectral method and the discrete method used to advance in time is an exponential time differencing scheme. This numerical simulation confirms the theoretical results of Herrero and Velzquez [M.A. Herrero and J.J.L. Velzquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincare 10 (1993), pp. 131–189.] in the one-dimensional problem. Later, we use this method as an experimental approach to describe the various possible asymptotic behaviours with two-space variables.

Keywords:

• semi-linear diffusion equations
• blow-up

1991 Mathematics Subject Classifications :

• Primary: 65M70
• Secondary: 65M20
• 35K05
• 35K55
• 35K57

Acknowledgements

The authors would like to thank Miguel Escobedo for fruitful conversations concerning this paper. The comments and extensive remarks of the anonymous referees are appreciated. This work was supported by the Basque Country University with the project 9/UPV 00127.310-15969/2004.