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Numerical Treatment of Free Boundary Value Problems / Numerische Behandlung freier Randwertaufgaben pp 233–244Cite as

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The Occurrence of Pathologies in Some Stefan-Like Problems

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Abstract

A one-phase Stefan-like problem for the one-dimensional heat equation is considered and cases are discussed in which the problem does not possess global solution. They occur when the width of the domain tends to zero or when the speed of the free boundary becomes unbounded for finite time. Some examples of such pathologies are studied in detail and it is discussed how their occurrence is related to the data. Finally, the singularity of the solution is analyzed in the vicinity of the possible pathologies.

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References

  1. Cannon, J.R. — Hill, C.D.: Existence, uniqueness, stability and monotone dependence in a Stefan problem for the heat equation. J. Math. Mech. 17(1967), 1–20.

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Author information

Authors and Affiliations

  1. Istituto Matematico “U. Dini”, Viale Morgagni, 67/A, I 50134, Firenze, Italy

    Mario Primicerio

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  1. Mario Primicerio
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Editor information

Editors and Affiliations

  1. Institut für Mathematik, Technische Universität Clausthal, D-3392, Clausthal-Zellerfeld, Germany

    J. Albrecht

  2. Institut für Angewandte Mathematik, Universität Hamburg, Bundesstrasse 55, D-2 Hamburg 13, Germany

    L. Collatz

  3. Institut für Mathematik III, Freie Universität Berlin, Arnimallee 2-6, D-1 Berlin 33, Germany

    K.-H. Hoffmann

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© 1982 Springer Basel AG

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Primicerio, M. (1982). The Occurrence of Pathologies in Some Stefan-Like Problems. In: Albrecht, J., Collatz, L., Hoffmann, KH. (eds) Numerical Treatment of Free Boundary Value Problems / Numerische Behandlung freier Randwertaufgaben. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 58. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6563-0_18

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  • DOI: https://doi.org/10.1007/978-3-0348-6563-0_18

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-6565-4

  • Online ISBN: 978-3-0348-6563-0

  • eBook Packages: Springer Book Archive

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