Hostname: page-component-7bb8b95d7b-pwrkn Total loading time: 0 Render date: 2024-09-25T21:32:56.794Z Has data issue: false hasContentIssue false
  • English
  • Français

Generation of Interface for an Allen-Cahn Equation with Nonlinear Diffusion

Published online by Cambridge University Press:  27 July 2010

M. Alfaro  and
D. Hilhorst
M. Alfaro
Affiliation:
I3M, Université de Montpellier 2, CC051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France.
D. Hilhorst*
Affiliation:
CNRS et Laboratoire de Mathématiques, Université de Paris-Sud 11, 91405 Orsay Cedex, France.
*
* Corresponding author. E-mail: danielle.hilhorst@math.u-psud.fr
Get access

Abstract

In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property.

Keywords

degenerate diffusionsingular perturbationmotion by mean curvaturepopulation dynamics
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 5: Reaction-diffusion waves , 2010 , pp. 1 - 12
Copyright
© EDP Sciences, 2010

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

Alfaro, M.. The singular limit of a chemotaxis-growth system with general initial data. Adv. Differential Equations, 11 (2006), no. 11, 1227–1260. Google Scholar
Alfaro, M., Hilhorst, D., Matano, H.. The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system. J. Differential Equations, 245 (2008), 505–565.CrossRefGoogle Scholar
Aronson, D., Crandall, M. G., Peletier, L. A.. Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal., 6 (1982), 1001–1022.CrossRefGoogle Scholar
Carl, E. A.. Population control in arctic ground squirrels. Ecology, 52 (1971), 395–413.CrossRefGoogle Scholar
Chen, X.. Generation and propagation of interfaces for reaction-diffusion equations. J. Differential Equations, 96 (1992), 116–141.CrossRefGoogle Scholar
Chen, X.. Generation and propagation of interfaces for reaction-diffusion systems. Trans. Amer. Math. Soc., 334 (1992), 877–913.CrossRefGoogle Scholar
DiBenedetto, E.. Continuity of weak solutions to a general porous medium equation. Indiana University Mathematics J., 32 (1983), 83–118.CrossRefGoogle Scholar
Feireisl, E.. Front propagation for degenerate parabolic equations. Nonlinear Anal., 35 (1999), 735–746.CrossRefGoogle Scholar
Gurney, W. S. C., Nisbet, R. M.. The regulation of inhomogeneous populations. J. Theoret. Biol., 52 (1975), 441–457.CrossRefGoogle ScholarPubMed
Gurtin, M. E., MacCamy, R. C.. On the diffusion of biological populations. Math. Biosci., 33 (1979), 35–49.CrossRefGoogle Scholar
Hilhorst, D., Kersner, R., Logak, E., Mimura, M.. Interface dynamics of the Fisher equation with degenerate diffusion. J. Differential Equations, 244 (2008), 2872–2889.CrossRefGoogle Scholar
J. L. Vásquez. The porous medium equation. Mathematical theory. Oxford University Press, Oxford, 2007.