On a generalized Cahn-Hilliardequation with biological applications

Laurence Cherfils; Alain Miranville; Sergey Zelik

doi:10.3934/dcdsb.2014.19.2013

Article Contents

2014, Volume 19, Issue 7: 2013-2026. Doi: 10.3934/dcdsb.2014.19.2013

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On a generalized Cahn-Hilliard equation with biological applications

1.
Université de La Rochelle, Laboratoire Mathématiques, Image et Applications, Avenue Michel Crépeau, F-17042 La Rochelle Cedex
2.
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex
3.
Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received: March 2013

Revised: May 2013

Published: September 2014

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we are interested in the study of the asymptotic behavior of a generalization of the Cahn-Hilliard equation with a proliferation term and endowed with Neumann boundary conditions. Such a model has, in particular, applications in biology. We show that either the average of the local density of cells is bounded, in which case we have a global in time solution, or the solution blows up in finite time. We further prove that the relevant, from a biological point of view, solutions converge to $1$ as time goes to infinity. We finally give some numerical simulations which confirm the theoretical results.

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Mathematics Subject Classification: 35B40, 35B44, 35K55.

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