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Numerical solution of a fast diffusion equation


Authors: Marie-Noelle Le Roux and Paul-Emile Mainge
Journal: Math. Comp. 68 (1999), 461-485
MSC (1991): Primary 35K55, 35K57, 65M60
MathSciNet review: 1627805
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Abstract: In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: $ u_{t}-\Delta u^{m}=\alpha u^{p_{1}}$ in $\Omega $, a smooth bounded domain in $ \mathbb{R}^{d} (d\geq 1)$ with the zero lateral boundary condition and with a positive initial condition, $ m\in \ ]0,1[$ (fast diffusion problem), $ \alpha \geq 0$ and $ p_{1}\geq m$. Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.


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Additional Information

Marie-Noelle Le Roux
Affiliation: GRAMM-Mathématiques, 351, cours de la Libération, F-33405 Talence Cedex, France
Email: m.n.leroux@math.u-bordeaux.fr

Paul-Emile Mainge
Affiliation: GRAMM-Mathématiques, 351, cours de la Libération, F-33405 Talence Cedex, France

DOI: https://doi.org/10.1090/S0025-5718-99-01063-7
Keywords: Reaction diffusion equations, parabolic problems
Received by editor(s): August 13, 1996
Received by editor(s) in revised form: May 5, 1997
Article copyright: © Copyright 1999 American Mathematical Society