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Steady state solutions of a bi-stable quasi-linear equation with saturating flux

Published online by Cambridge University Press:  17 February 2011

M. BURNS  and
M. GRINFELD
M. BURNS
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, UK email: martin.f.burns@strath.ac.uk
M. GRINFELD
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, UK email: martin.f.burns@strath.ac.uk
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Abstract

In this paper, we consider the bi-stable equation proposed by Rosenau to replace the Allen–Cahn equation in the case of large gradients. We discuss the bifurcation problem for stationary solutions of this equation on an interval as the diffusion coefficient and the length of the interval are varied, concentrating on classical solutions.

Keywords

Prescribed mean curvature equationbi-stable non-linearitystationary solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 4 , August 2011 , pp. 317 - 331
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Bonheure, D., Habets, P., Obersnel, F. & Omari, P. (2007) Classical and non-classical solutions of a prescribed curvature equation. J. Differ. Equ. 243, 208–237.CrossRefGoogle Scholar
[2]Burns, M. (2011) Reaction-Diffusion Equations with Saturating Flux, Ph.D. thesis. University of Strathclyde, UK.Google Scholar
[3]Dascal, L., Kamin, S. & Sochen, N. (2005) A variational inequality for discontinuous solutions of degenerate parabolic equations. Rev. R. Acad. Cien. Ser. A Math. 99, 243–256.Google Scholar
[4]Demengel, F. & Temam, R. (1984) Convex functions of a measure and applications. Indiana Univ. Math. J. 33, 673–709.CrossRefGoogle Scholar
[5]Duncan, D. B., Grinfeld, M. & Stoleriu, I. (2000) Coarsening in an integro-differential model of phase transitions. Eur. J. Appl. Math. 11, 561–572.CrossRefGoogle Scholar
[6]Evans, L. C. & Gariepy, R. F. (1992) Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton.Google Scholar
[7]Fife, P. C. (1996) An integro-differential analog of semilinear parabolic PDEs. In: Partial Differential Equations and Applications, Marcel Dekker, New York.Google Scholar
[8]Golubitsky, M. & Schaeffer, D. G. (1985) Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York.CrossRefGoogle Scholar
[9]Habets, P. & Omari, P. (2007) Multiple positive solutions of a one-dimensional prescribed mean curvature problem. Commun. Contemp. Math. 9, 701–730.CrossRefGoogle Scholar
[10]Obersnel, F. (2007) Classical and non-classical sign changing solutions of a one-dimensional autonomous prescribed curvature equation. Adv. Nonlinear Stud. 7, 1–13.CrossRefGoogle Scholar
[11]Pan, H. (2009) One-dimensional prescribed mean curvature equation with exponential nonlinearity. Nonlinear Anal. 70, 999–1010.CrossRefGoogle Scholar
[12]Prüss, J., Simonett, G. & Zacher, R. (2009) On convergence of solutions to equilibria for quasilinear parabolic problems. J. Differ. Equ. 246, 3902–3931.CrossRefGoogle Scholar
[13]Rosenau, P. (1989) Extension of Landau-Ginzburg free-energy functionals to high-gradient domains. Phys. Rev. A 39, 6614–6617.CrossRefGoogle ScholarPubMed
[14]Rosenau, P. (1990) Free-energy functionals at the high-gradient limit. Phys. Rev. A 41, 2227–2230.CrossRefGoogle ScholarPubMed
[15]Schaaf, R. (1991) Global Solution Branches of Two Point Boundary Value Problems, Springer-Verlag, Berlin.Google Scholar
[16]Smoller, J. & Wasserman, A. (1981) Global bifurcation of steady state solutions. J. Differ. Equ. 39, 269–290.CrossRefGoogle Scholar