On a generalized Cahn-Hilliard equation with biological applications | Zendy

Laurence Cherfils | Zendy; Alain Miranville | Zendy; Sergey Zelik | Zendy

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On a generalized Cahn-Hilliard equation with biological applications

Author(s) -

Laurence Cherfils,

Alain Miranville,

Sergey Zelik

Publication year - 2014

Publication title -

discrete and continuous dynamical systems - b

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.864

H-Index - 53

eISSN - 1553-524X

pISSN - 1531-3492

DOI - 10.3934/dcdsb.2014.19.2013

Subject(s) - generalization , infinity , bounded function , term (time) , von neumann architecture , cahn–hilliard equation , neumann boundary condition , mathematics , boundary (topology) , convergence (economics) , boundary value problem , mathematical analysis , partial differential equation , pure mathematics , physics , quantum mechanics , economics , economic growth

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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