Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source. Issue 2 (25th January 2017)

• Record Type: Journal Article
• Authors: Fakih, Hussein
• Is Part Of: Applicable analysis. Volume 96:Issue 2(2017)
• Page Start: 324
• Page End: 348
• Publication Date: 2017-01-25
• Subjects: Cahn–Hilliard equation -- mass source -- tumor growth -- well-posedness -- global attractor -- exponential attractor -- simulations; 35K55 -- 35B40 -- 35J60; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2015.1135241
• Languages: English
• ISSNs: 0003-6811

Abstract

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.