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The semilinear heat equation with a Heaviside source term

Published online by Cambridge University Press:  16 July 2009

Roberto Gianni  and
Josephus Hulshof
Roberto Gianni
Affiliation:
Dipartimento di Matematica ‘Ulisse Dini’, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
Josephus Hulshof
Affiliation:
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
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Abstract

We consider the initial value problem for the equation ut = uxx + H(u), where H is the Heaviside graph, on a bounded interval with Dirichlet boundary conditions, and discuss existence, regularity and uniqueness of solutions and interfaces.

Information

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 4 , December 1992 , pp. 367 - 379
Copyright
Copyright © Cambridge University Press 1992

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References

[1]Norbury, J. & Stuart, A. M. 1989 A model for porous medium combustion. Quart. J. Mech. Appl. Math. 42, 159–178.CrossRefGoogle Scholar
[2]Norbury, J. & Stuart, A. M. 1987 Parabolic free boundary problems arising in porous medium combustion. IMA J. Appl. Math. 39, 241–257.CrossRefGoogle Scholar
[3]Lacey, A. A. 1981 The spatial dependence of supercritical reacting systems. IMA J. Appl. Math. 27, 71–84.CrossRefGoogle Scholar
[4]Rinzel, J. & Keller, J. B. 1973 Travelling wave solutions of a nerve conducting equation. Biophysics J. 13, 1313–1337.CrossRefGoogle ScholarPubMed
[5]Friedman, A. & Tzavaras, A. E. 1988 Combustion in a porous medium. Siam J. Math. Anal. 19 (3), 509–519.CrossRefGoogle Scholar
[6]Gianni, R. & Manucci, P. 1992 Existence theorems for a free boundary problem in combustion theory. (To appear in Quart. Appl. Math.)CrossRefGoogle Scholar
[7]Gianni, R. & Manucci, P. 1991 Some Existence Theorems for an N-dimensional Parabolic Equation with a Discontinuous Source Term. Preprint.Google Scholar
[8]Aguirre, J. & Escobedo, M. 1986sol;1987 A Cauchy problem for u - δu = u with 0 < p < 1. Ann. Sci Fac. Toulouse 8, 175–203.CrossRefGoogle Scholar
[9]Ladyzenskaja, O. A., Solonnikov, V. A. & Ural'ceva, N. N. 1968 Linear and quasi-linear equations of parabolic type. Trans. A.M.S., Rhode Island.Google Scholar
[10]Hulshof, J. 1987 An elliptic-parabolic free boundary problem: continuity of the interface. Proc. Roy. Soc. Edin. 106 A, 327–339.CrossRefGoogle Scholar
[11]Guillemin, V. & Pollack, A. 1974 Differential Topology. Prentice Hall.Google Scholar
[12]Bertsch, M. & Hulshof, J. 1986 Regularity results for an elliptic-parabolic free boundary problem. Trans. A.M.S. 297 (1), 337–350.CrossRefGoogle Scholar
[13]Hilhorst, D. & Hulshof, J. 1992 An elliptic-parabolic problem in combustion theory:convergence to travelling waves. (To appear in J. Nonl. Anal.)CrossRefGoogle Scholar