Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source

Abstract

We consider in this article a generalized Cahn–Hilliard equation with mass source (nonlinear reaction term) which has applications in biology. We are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Dirichlet boundary conditions and then Neumann boundary conditions. The latter require additional assumptions on the mass source term to obtain the dissipativity. Indeed, otherwise, the order parameter u can blow up in finite time. We also give numerical simulations which confirm the theoretical results.

Keywords:

• Cahn–Hilliard equation
• mass source
• tumor growth
• well-posedness
• global attractor
• exponential attractor
• simulations

AMS Subject Classifications:

• 35K55
• 35B40
• 35J60

Acknowledgements

The author would like to thank L. Cherfils and A. Miranville, his supervisors, for many stimulating discussions and useful comments on the subject of the paper. He also wishes to thank the anonymous referee for her/his careful reading of the manuscript and useful suggestions.

Notes

No potential conflict of interest was reported by the author.