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
MathSciNet review: 1627805
Full-text PDF Free Access

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. Herbert Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 125–146. MR 0296498
  • 2. Donald Aronson, Michael G. Crandall, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), no. 10, 1001–1022. MR 678053, 10.1016/0362-546X(82)90072-4
  • 3. James G. Berryman and Charles J. Holland, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal. 74 (1980), no. 4, 379–388. MR 588035, 10.1007/BF00249681
  • 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. Marie-Noëlle Le Roux, Semidiscretization in time of nonlinear parabolic equations with blowup of the solution, SIAM J. Numer. Anal. 31 (1994), no. 1, 170–195. MR 1259971, 10.1137/0731009
  • 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, Semi-discretization in time of a fast diffusion equation, J. Math. Anal. Appl. 137 (1989), no. 2, 354–370. MR 984965, 10.1016/0022-247X(89)90251-5
  • 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. Howard A. Levine and Paul E. Sacks, Some existence and nonexistence theorems for solutions of degenerate parabolic equations, J. Differential Equations 52 (1984), no. 2, 135–161. MR 741265, 10.1016/0022-0396(84)90174-8
  • 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. Paul E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409. MR 696738, 10.1016/0362-546X(83)90092-5
  • 17. Paul E. Sacks, The initial and boundary value problem for a class of degenerate parabolic equations, Comm. Partial Differential Equations 8 (1983), no. 7, 693–733. MR 700733, 10.1080/03605308308820283
  • 18. G. I. Kresin and V. G. Maz′ya, The maximum principle for second-order elliptic and parabolic systems, Dokl. Akad. Nauk SSSR 273 (1983), no. 1, 38–41 (Russian). MR 722884
  • 19. Jacques Simon, Compact sets in the space 𝐿^{𝑝}(0,𝑇;𝐵), Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, 10.1007/BF01762360

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

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

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