On the dynamics of a modified Cahn-Hilliard equation with biological applications

Xiaopeng Zhao

DOI: http://dx.doi.org/10.12775/TMNA.2017.025


We study the global solvability and dynamical behaviour of the modified Cahn-Hilliard equation with biological applications in the Sobolev space $H^1(\mathbb{R}^N)$.


Modified Cahn-Hilliard equation, attractors, asymptotic compactness

Full Text:



H. Amann, Linear and Quasilinear Parabolic Problems, Birkhaüser, Basel, 1995.

J.M. Arrieta, J.W. Cholewa, T. Dlotko and A. Rodrı́guez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal. 56 (2004), 515–554.

A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258–267.

L. Cherfils, A. Miranville and S. Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math. 79 (2011), 561–596.

J.W. Cholewa and A. Rodrı́guez-Bernal, On the Cahn–Hilliard equation in H 1 (RN ), J. Differential Equations 253 (2012), 3678–3726.

D.S. Cohen and J.D. Murray, A generalized diffusion model for growth and dispersal in a population, J. Math. Biology 12 (1981), 237–249.

P. Colli, G. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal. 4 (2015), 311–325.

A. Debussche and L. Dettori, On the Cahn–Hilliard equation with a logarithmic free energy, Nonlinear Anal. 24 (1995), 1491–1514.

T. Dlotko, Global attractor for the Cahn–Hilliard equation in H 2 and H 3 , J. Differential Equations 113 (1994), 381–393.

T. Dlotko, M.B. Kania and C. Sun, Analysis of the viscous Cahn–Hilliard equation in RN , J. Differential Equations 252 (2012), 2771–2791.

T. Dlotko and C. Sun, Dynamics of the modified viscous Cahn–Hilliard equation in RN , Topol. Methods Nonlinear Anal. 35 (2010), 277–294.

A. Eden and V.K. Kalantarov, The convective Cahn–Hilliard equation, Appl. Math. Lett. 20 (2007), 455–461.

A. Eden and V.K. Kalantarov, 3D convective Cahn–Hilliard equation, Comm. Pure Appl. Anal. 6 (2007), 1075–1086.

C.M. Elliott and S. Zheng, On the Cahn–Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), 339–357.

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn–Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal. 8 (2009), 881–912.

M. Gurtin, Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance, Phys. D 92 (1996), 178–192.

J.K. Hale, Asymptotic Behaviour of Dissipative Systems, American Mathematical Society, Providence, 1988.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.

E. Khain and L.M. Sander, Generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E 77 (2008), 051129.

B. Liu and C.V. Pao, Integral representation of a generalized diffusion model in population problems, J. Integral Equations 6 (1984), 175–185.

A. Miranville, Asymptotic behavior of a generalized Cahn–Hilliard equation with a proliferation term, Appl. Anal. 92 (2013), 1308–1321.

A. Miranville, A. Pitrus and J.M. Rakotoson, Dynamical aspect of a generalized Cahn–Hilliard equation based on a microforce balance, Asymptot. Anal. 16 (1998), 315–345.

A. Novick-Cohen and L.A. Segel, Nonlinear aspects of the Cahn–Hilliard equation, Phys. D 10 (1984), 277–298.

A. Novick-Cohen, Energy methods for the Cahn–Hilliard equation, Quart. Appl. Math. 46 (1988), 681–690.

M. Polat, A.O. Celebi and N. Caliskan, Global attractors for the 3D viscous Cahn–Hilliard equations in an unbounded domain, Appl. Anal. 88 (2009), 1157–1171.

A. Rodrı́guez-Bernal and B. Wang, Attractors for partly dissipative reaction diffusion systems in Rn , J. Math. Anal. Appl. 252 (2000), 790–803.

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, Springer, New York, 1988.

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D 128 (1999), 41–52.

H. Wu and S. Zheng, Convergence to equilibrium for the Cahn–Hilliard equation with dynamic boundary conditions, J. Differential Equations 204 (2004), 511–531.

X. Zhao, N. Duan and B. Liu, Optimal control problem of a generalized Ginzburg–Landau model equation in population problems, Math. Meth. Appl. Sci. 37 (2014), 435–446.

X. Zhao and C. Liu, Optimal control of the convective Cahn–Hilliard equation in 2D case, Appl. Math. Optim. 70 (2014), 61–82.


  • There are currently no refbacks.

Partnerzy platformy czasopism