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Multi-phase Stefan problems for a non-linear one-dimensional model of cell-to-cell adhesion and diffusion

Published online by Cambridge University Press:  18 January 2010

K. ANGUIGE
K. ANGUIGE*
Affiliation:
RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria email: keith.anguige@ricam.oeaw.ac.at
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Abstract

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We consider a family of multi-phase Stefan problems for a certain one-dimensional model of cell-to-cell adhesion and diffusion, which takes the form of a non-linear forward–backward parabolic equation. In each material phase the cell density stays either high or low, and phases are connected by jumps across an ‘unstable’ interval. We develop an existence theory for such problems which allows for the annihilation of phases and the subsequent continuation of solutions. Stability results for the long-time behaviour of solutions are also obtained, and, where necessary, the analysis is complemented by numerical simulations.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 21 , Issue 2 , April 2010 , pp. 109 - 136
Copyright
Copyright © Cambridge University Press 2010

References

[1]Adams, R. (1975) Sobolev Spaces, Academic Press.Google Scholar
[2]Amann, H. (1989) Dynamic theory of quasilinear parabolic systems. III. Global existence. Math. Z. 202, 219–250.Google Scholar
[3]Anguige, K. & Schmeiser, C. (2009) A one-dimensional model of cell diffusion and aggregation, incorporating volume-filling and cell-to-cell adhesion. J. Math. Biol. 58, 395–427.CrossRefGoogle ScholarPubMed
[4]Ladyženskaya, O., Solonnikov, V. & Ural'ceva, N. (1968) Linear and Quasilinear Equations of Parabolic Type. AMS Translations of Mathematical Monographs, Amer. Math. Soc, Providence, R. I. Vol. 23.Google Scholar
[5]Meirmanov, A. (1992) The Stefan Problem. de Gruyter Expositions in Mathematics. de Gruyter, Berlin.Google Scholar
[6]Sun, X. & Ward, M. (2000) The dynamics and coarsening of interfaces for the viscous Cahn–Hilliard equation in one spatial dimension. Stud. Appl. Math. 105, 203–234.CrossRefGoogle Scholar
[7]Taylor, M. (1996) Partial Differential Equations III. Springer.CrossRefGoogle Scholar
[8]Vazquez, J. L. (2007) The Porous-Medium Equation: Mathematical Theory. Oxford Science Publications.Google Scholar