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.
A.C.Aristotelous, O.A.Karakashian and S.M.Wise, Adaptive, second order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source, IMA J. Numer. Anal. (2014), 1–32.
2.
J.W.Cahn, On spinodal decomposition, Acta Metall.9 (1961), 795–801.
3.
J.W.Cahn and J.E.Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys.28 (1958), 258–267.
4.
V.Chalupecki, Numerical studies of Cahn–Hilliard equations and applications in image processing, in: Proceedings of Czech–Japanese Seminar in Applied Mathematics, Czech Technical University, Prague, 4–7 August, 2004.
5.
L.Cherfils, H.Fakih and A.Miranville, On the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation with logarithmic nonlinear terms, SIAM J. Imag. Sci.8 (2015), 1123–1140.
6.
L.Cherfils, A.Miranville and S.Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math.79 (2011), 561–596.
7.
L.Cherfils, A.Miranville and S.Zelik, On a generalized Cahn–Hilliard equation with biological applications, Discrete Cont. Dyn. Systems B19 (2014), 2013–2026.
8.
L.Cherfils, M.Petcu and M.Pierre, A numerical analysis of the Cahn–Hilliard equation with dynamic boundary conditions, Discrete Cont. Dyn. Systems27 (2010), 1511–1533.
9.
D.Cohen and J.M.Murray, A generalized diffusion model for growth and dispersion in a population, J. Math. Biol.12 (1981), 237–248.
10.
F.Della Porta and M.Grasselli, Convective nonlocal Cahn–Hilliard equations with reaction terms, Discrete Cont. Dyn. Systems B20 (2015), 1529–1553.
11.
I.C.Dolcetta, S.F.Vita and R.March, Area-preserving curve-shortening flows: From phase separation to image processing, Interfaces Free Bound.4 (2002), 325–343.
12.
C.M.Elliott, The Cahn–Hilliard model for the kinetics of phase separation, in: Mathematical Models for Phase Change Problems, J.F.Rodrigues, ed., International Series of Numerical Mathematics, Vol. 88, Birkhäuser, Basel, 1989.
13.
C.M.Elliott, D.A.French and F.A.Milner, A second order splitting method for the Cahn–Hilliard equation, Numer. Math.54 (1989), 575–590.
14.
H.Fakih, Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source, submitted.
E.Khain and L.M.Sander, A generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E77 (2008), 051129.
17.
I.Klapper and J.Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E74 (2006), 0319021.
18.
A.Miranville, Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comp.1 (2011), 523–536.
19.
A.Miranville, Asymptotic behavior of a generalized Cahn–Hilliard equation with a proliferation term, Appl. Anal.92 (2013), 1308–1321.
20.
B.Nicolaenko, B.Scheurer and R.Temam, Some global dynamical properties of a class of pattern formation equations, Commun. Diff. Eqns14 (1989), 245–297.
21.
A.Novick-Cohen, The Cahn–Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl.8 (1998), 965–985.
22.
Y.Oono and S.Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett.58 (1987), 836–839.
23.
A.Oron, S.H.Davis and S.G.Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys.69 (1997), 931–980.
24.
R.Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn, Springer-Verlag, New York, 1997.
25.
U.Thiele and E.Knobloch, Thin liquid films on a slightly inclined heated plate, Phys. D190 (2004), 213–248.
26.
S.Tremaine, On the origin of irregular structure in Saturn’s rings, Astron. J.125 (2003), 894–901.
27.
J.Verdasca, P.Borckmans and G.Dewel, Chemically frozen phase separation in an adsorbed layer, Phys. Rev. E52 (1995), 4616–4619.
28.
S.Villain-Guillot, Phases modulées et dynamique de Cahn–Hilliard, Habilitation thesis, Université Bordeaux 1, 2010.
