Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0025-5718-99-01063-7
MathSciNet review: 1627805
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. H. AMANN, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math J., 21 (1971), pp 125-146. MR 45:5558
  • 2. D. ARONSON, M. CRANDALL, L. PELETIER, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlin. Anal., Vol. 6 (1982), pp 1001-1002. MR 84j:35099
  • 3. J.G. BERRYMAN, C.J. HOLLAND, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal. 74 (1980), pp 379-388. MR 81m:35065
  • 4. A. FRIEDMAN, A.A. LACEY, Blowup of positive solutions of semilinear parabolic equations, Math. Anal. Appl., 132 (1998), pp 171-186.
  • 5. A. FRIEDMAN, B. MCLEOD, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34, (1985), pp 425-447.
  • 6. A. FRIEDMAN, B. MCLEOD, Blowup of nonlinear degenarate parabolic equations, Arch. Rational Mech. Anal. 96, (1986), pp 55-80.
  • 7. H.B. KELLER, Elliptic boundary value problems suggested by nonlinear diffusion process, Arch. Rational Mech. Anal., 96 (1986), pp 55-80.
  • 8. M.-N. LE ROUX, Semi-discretization in time of nonlinear parabolic equations with blowup of the solution, SIAM J. Numer. Anal. Vol. 31, no. 1, (1994), pp 170-195. MR 95a:65148
  • 9. M.-N. LE ROUX, Numerical solution of nonlinear reaction diffusion processes in plasmas, Proceedings of the Second Hellenic European Conference on Mathematics and Informatics, Sept. 94, Athenis, E.A. Lipitakis Editor. CMP 96:07
  • 10. M.-N. LE ROUX, Semidiscretization in time of a fast diffusion equation, J. Math. Anal. Appl. Vol. 137, no. 2, (1989), pp 354-370. MR 90k:65166
  • 11. M.-N. LE ROUX, Résolution numérique d'un problème de fast-diffusion, Publication CeReMaB, Bordeaux I, No. 9306.
  • 12. M.-N. LE ROUX, H. WILHELMSSON, External boundary effects on simultaneous diffusion and reaction processes, Physica Scripta, Vol. 40, (1989), pp 674-681.
  • 13. H.A. LEVINE, P.E. SACKS, Some existence and nonexistence theorems for solutions of degenerate parabolic equations, J. Diff. Equat., Vol. 52, (1984), pp 135-161. MR 85f:35120
  • 14. P.E. MAINGE, Résolution numérique d'équations de réaction-diffusion intervenant en physique des plasmas, Thèse (1996), Université Bordeaux 1.
  • 15. E.S. SABININA, A class of nonlinear degenerating parabolic equations, Soviet Math. Dokl. 143 (1962), pp 495-498.
  • 16. P.E. SACKS, Continuity of solutions of a singular parabolic equation, Nonlinear Anal., Vol. 7 (1983), pp 387-409. MR 84d:35081
  • 17. P.E. SACKS, The initial and boundary value problem for a class of degenerate parabolic equations, Part. Diff. Equat., Vol. 8 (1983), pp 693-733. MR 85h:35128
  • 18. P.E. SACKS, Global behavior for a class of nonlinear evolution equation, SIAM J. Math. Anal. Vol. 16, no. 2, (1985) pp 233-250. MR 85f:35031
  • 19. J. SIMON, Compact sets in the space $ L^{p}(0,T;B)$, Ann. Mat. Pura., 146 (1987), pp 65-96. MR 89c:46055

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 35K55, 35K57, 65M60

Retrieve articles in all journals with MSC (1991): 35K55, 35K57, 65M60


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

American Mathematical Society