A Cahn–Hilliard equation with a proliferation term for biological and chemical applications

Article type: Research Article

Authors: Fakih, Hussein

Affiliations: Laboratoire de Mathématiques et Applications, UMR CNRS 7348 – SP2MI, Université de Poitiers, Boulevard Marie et Pierre Curie – Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France. E-mail: Hussein.Fakih@math.univ-poitiers.fr

Abstract: In this paper, we are interested in the study of the solution to a generalization of the Cahn–Hilliard equation endowed with Neumann boundary conditions. This model has, in particular, applications in biology and in chemistry. We show that the solutions blow up in finite time or exist globally in time. We further prove that the relevant, from a biological and a chemical point of view, solutions converge to a constant as time goes to infinity. We finally give some numerical simulations which confirm the theoretical results.

Keywords: Cahn–Hilliard equation, proliferation term, blow up, convergence to a steady state, simulations

DOI: 10.3233/ASY-151306

Journal: Asymptotic Analysis, vol. 94, no. 1-2, pp. 71-104, 2015