Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Numerical solution of a fast diffusion equation
HTML articles powered by AMS MathViewer

by Marie-Noelle Le Roux and Paul-Emile Mainge PDF
Math. Comp. 68 (1999), 461-485 Request permission


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.
  • Herbert Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 125–146. MR 296498, DOI 10.1512/iumj.1971.21.21012
  • 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, DOI 10.1016/0362-546X(82)90072-4
  • 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, DOI 10.1007/BF00249681
  • A. FRIEDMAN, A.A. LACEY, Blowup of positive solutions of semilinear parabolic equations, Math. Anal. Appl., 132 (1998), pp 171-186.
  • A. FRIEDMAN, B. MCLEOD, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34, (1985), pp 425-447.
  • A. FRIEDMAN, B. MCLEOD, Blowup of nonlinear degenarate parabolic equations, Arch. Rational Mech. Anal. 96, (1986), pp 55-80.
  • H.B. KELLER, Elliptic boundary value problems suggested by nonlinear diffusion process, Arch. Rational Mech. Anal., 96 (1986), pp 55-80.
  • 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, DOI 10.1137/0731009
  • 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.
  • 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, DOI 10.1016/0022-247X(89)90251-5
  • M.-N. LE ROUX, Résolution numérique d’un problème de fast-diffusion, Publication CeReMaB, Bordeaux I, No. 9306.
  • M.-N. LE ROUX, H. WILHELMSSON, External boundary effects on simultaneous diffusion and reaction processes, Physica Scripta, Vol. 40, (1989), pp 674-681.
  • 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, DOI 10.1016/0022-0396(84)90174-8
  • P.E. MAINGE, Résolution numérique d’équations de réaction-diffusion intervenant en physique des plasmas, Thèse (1996), Université Bordeaux 1.
  • E.S. SABININA, A class of nonlinear degenerating parabolic equations, Soviet Math. Dokl. 143 (1962), pp 495-498.
  • Paul E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409. MR 696738, DOI 10.1016/0362-546X(83)90092-5
  • 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, DOI 10.1080/03605308308820283
  • 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
  • Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI 10.1007/BF01762360
Similar Articles
  • Retrieve articles in Mathematics of Computation 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:
  • Paul-Emile Mainge
  • Affiliation: GRAMM-Mathématiques, 351, cours de la Libération, F-33405 Talence Cedex, France
  • Received by editor(s): August 13, 1996
  • Received by editor(s) in revised form: May 5, 1997
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 461-485
  • MSC (1991): Primary 35K55, 35K57, 65M60
  • DOI:
  • MathSciNet review: 1627805