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Existence of the free boundary in a multi-dimensional combustion problem*

Published online by Cambridge University Press:  14 November 2011

Roberto Gianni
Roberto Gianni
Affiliation:
Dipartimento di matematica ‘Ulisse Dini’, Viale Morgagni 67/a 50134, Firenze, Italy
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Extract

In this paper, we consider an n-dimensional semilinear equation of parabolic type with a discontinuous source term arising from combustion theory. We prove local existence for a classical solution having a ‘regular’ free boundary. In this regard, the free boundary is a surface through which the discontinuous source term exhibits a switch-like behaviour. We specify conditions under which this solution and its free boundary are global in time; moreover, we exhibit a special domain for which, for t tending to infinity, such a global-in-time solution converges, together with its free boundary, to the solution of the stationary problem and to its regular free boundary (which is proved to exist), respectively. We also prove uniqueness and continuous dependence theorems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 3 , 1995 , pp. 525 - 544
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Friedman, A.. Partial Differential Equations of Parabolic Type (Englewood Cliffs, N.J.: Prentice Hall, 1964).Google Scholar
2Gianni, R. and Hulshof, J.. A semilinear parabolic equation with a heaviside source term. European J. Appl. Math. 3 (1992), 367–379.CrossRefGoogle Scholar
3Gianni, R. and Mannucci, P.. Existence theorems for a free boundary problem in combustion theory. Quart. Appl. Math. LI (1993), 45–53.Google Scholar
4Gianni, R. and Mannucci, P.. Some existence theorems for an N-dimensional parabolic equation with a discontinuous source term. SIAM J. Math. Anal. 24 (1993), 618–633.CrossRefGoogle Scholar
5Kinderlehrer, D. and Stampacchia, G.. An Introduction to Variational Inequalities and Their Applications (New York: Academic Press, 1980).Google Scholar
6Kolmogorov, A. N. and Fomin, S. V.. Elementi di Teoria delle Funzioni e di Analisi Funzionale (Moscow: Edizioni MIR, 1980).Google Scholar
7Krasnov, M. L., Kiselev, A. I. and Makarenko, G. E.. Equazioni Integrali (Moscow: MIR, 1976).Google Scholar
8Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural'ceva, V. V.. Linear and quasilinear equations of parabolic type. Amer. Math. Soc. Transl. 23 (1968).Google Scholar
9Ladyzhenskaya, O. A. and Ural'ceva, N. N.. Equations aux Derivées Partielles de Type Elliptique (Paris: Dunod, 1968).Google Scholar
10Norbury, J. and Stuart, A. M.. A model for porous medium combustion. Quart. J. Mech. Appl. Math. 42 (1987), 159–178.CrossRefGoogle Scholar
11Norbury, J. and Stuart, A. M.. Parabolic free boundary problems arising in porous medium combustion. IMA J. Appl. Math. 39 (1987), 241–257.CrossRefGoogle Scholar
12Norbury, J. and Stuart, A. M.. Travelling combustion waves in a porous medium. Part 1: Existence. SIAM J. Appl. Math. 48 (1988), 155–169.CrossRefGoogle Scholar
13Norbury, J. and Stuart, A. M.. Travelling combustion waves in a porous medium. Part 2: Stability. SIAM J. Appl. Math. 48 (1988), 374–392.CrossRefGoogle Scholar