Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T01:42:58.137Z Has data issue: false hasContentIssue false
  • English
  • Français

A free boundary problem arising in some reacting–diffusing system

Published online by Cambridge University Press:  14 November 2011

D. Hilhorst ,
Y. Nishiura  and
M. Mimura
D. Hilhorst
Affiliation:
Laboratoire d'Analyse Numérique Bâtiment 425, CNRS et Université de Paris-Sud, 91405 Orsay, France
Y. Nishiura
Affiliation:
Department of Mathematics, Faculty of Integrated Arts and Sciences, Hiroshima University, Hiroshima 730, Japan
M. Mimura
Affiliation:
Department of Mathematics, Faculty of Sciences, Hiroshima University, Hiroshima 730, Japan
Get access Rights & Permissions [Opens in a new window]

Synopsis

We prove the well-posedness for a one-dimensional free boundary problem arising from some reaction diffusion system. The interfacial point hits a boundary point in finite time or remains inside for all time. In the large diffusion limit, the system is reduced to ordinary differential equations of finite dimension.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 3-4 , 1991 , pp. 355 - 378
Copyright
Copyright © Royal Society of Edinburgh 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Chen, X. Y.. Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J. (in press).Google Scholar
2Chen, Y. G., Giga, Y. and Goto, S.. Uniquness and existence of viscosity solutions of generalized mean curvature equations, J. Differential Geometry (to appear).Google Scholar
3Conrad, F., Hilhorst, D. and Seidman, T. I.. Well-posedness of a moving boundary problem arising in a dissolution growth-process, Nonlinear Anal. TMA 15 (1990), 445–465.CrossRefGoogle Scholar
4Evans, L. C. and Spruck, J.. Motion of level sets by mean curvature I J. Differential Geometry (to appear).Google Scholar
5Fasano, A. and Primicerio, M.. General free boundary problems for the heat equation, I. J. Math. Anal. Appl. 57 (1977), 694–723.CrossRefGoogle Scholar
6Fasano, A. and Primicerio, M.. General free boundary problems for the heat equation, II. J. Math. Anal. Appl. 58 (1977), 202–231.CrossRefGoogle Scholar
7Fife, P. C.. Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics 53 (Philadelphia: SIAM, 1988).Google Scholar
8Hale, J.. Ordinary Differential Equations (New York: Krieger, 1980).Google Scholar
9Hilhorst, D. and Hulshof, J.. An ellipic-parabolic problem in cumbustion theory: Convergence to travelling waves, Nonlinear Anal. TMA (to appear).Google Scholar
10Ikeda, T. and Kobayashi, R.. Numerical approach to interfacial dynamics (preprint).Google Scholar
11Keener, J. R.. A geometrical theory for spiral waves in excitable media. SIAM J. Appl. Math. 46 (1986), 1039–1056.Google Scholar
12Ladyženskaha, O. A., Solonnikov, V. A. and Ural'ceva, N. N.. Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs 23 (Providence R. I.: American Mathematical Society, 1968).Google Scholar
13Nishiura, Y.. Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. 13 (1982), 555–593.Google Scholar
14Nishiura, Y. and Mimura, M.. Layer oscillations in reaction-diffusion systems. SIAM J. Appl. Math. 49 (1989), 481–514.Google Scholar
15Seidman, T. I.. The transient semi-conductor problem with generation terms, II, in Nonlinear Semi-groups, PDE's and Attractors (to appear).Google Scholar
16Temam, R.. Navier-Stokes Equations (Amsterdam: North Holland, 1977).Google Scholar