Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

On the dynamics of a modified Cahn-Hilliard equation with biological applications
  • Home
  • /
  • On the dynamics of a modified Cahn-Hilliard equation with biological applications
  1. Home /
  2. Archives /
  3. Vol 50, No 1 (September 2017) /
  4. Articles

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

Authors

  • Xiaopeng Zhao

Keywords

Modified Cahn-Hilliard equation, attractors, asymptotic compactness

Abstract

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

References

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.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2017-08-19

How to Cite

1.
ZHAO, Xiaopeng. On the dynamics of a modified Cahn-Hilliard equation with biological applications. Topological Methods in Nonlinear Analysis. Online. 19 August 2017. Vol. 50, no. 1, pp. 169 - 183. [Accessed 8 July 2025].
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 50, No 1 (September 2017)

Section

Articles

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop