American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 10-18. doi: 10.3934/proc.2015.0010

Existence and multiplicity of stationary solutions for a Cahn--Hilliard-type equation in $\mathbb{R}^N$

Pablo Álvarez-Caudevilla 1,

1. 

Universidad Carlos III de Madrid, Av. Universidad 30, 28911-Leganés, Spain

Received  September 2014 Revised  January 2015 Published  November 2015

Solutions of the stationary semilinear Cahn--Hilliard-type equation $$ -\Delta^2 u - u -\Delta(|u|^{p-1}u)=0 \quad \mbox{in}     \mathbb{R}^N, \quad \mbox{with} \quad p>1, $$ which are exponentially decaying at infinity, are studied. Using the Mounting Pass Theorem allows us the determination of two different solutions. On the other hand, the application of Lusternik--Schnirel'man (L--S) Category Theory shows the existence of, at least, a countable family of solutions.
Keywords: variational setting, countable family of critical points., Stationary Cahn--Hilliard equation.
Mathematics Subject Classification: Primary: 35G20, 35K5.
Citation: Pablo Álvarez-Caudevilla. Existence and multiplicity of stationary solutions for a Cahn--Hilliard-type equation in $\mathbb{R}^N$. Conference Publications, 2015, 2015 (special) : 10-18. doi: 10.3934/proc.2015.0010
References:
[1]

P. Álvarez-Caudevilla, J.D. Evans and V.A. Galaktionov, Countable families of solutions of a limit stationary semilinear fourth-order Cahn-Hilliard-type equation I., Mountain Pass Theorem and Lusternik-Schnirel'man patterns in $\mathbbR^N$, ().   Google Scholar

[2]

P. Álvarez-Caudevilla and V.A. Galaktionov, Steady states, global existence and blow-up for fourth-order semilinear parabolic equations of Cahn-Hilliard type,, Advances Nonl. Stud., 12 (2012), 315.   Google Scholar

[3]

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.   Google Scholar

[4]

J.D. Evans, V.A. Galaktionov and J.F. Williams, Blow-up and global asymptotics of the limit unstable Cahn-Hilliard equation, SIAM, J. Math. Anal., 38 (2006), 64.   Google Scholar

[5]

V.A. Galaktionov, E. Mitidieri and S.I. Pohozaev, Variational approach to complicated similarity solutions of higher-order nonlinear PDEs. II,, Nonl. Anal.: RWA, 12 (2011), 2435.   Google Scholar

[6]

G.A. Klassen and E. Mitidieri, Standing wave solutions for a system derived from the FitzHugh-Nagumo equations for nerve conduction, SIAM, J. Math. Anal., 17 (1986), 74.   Google Scholar

[7]

A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965.   Google Scholar

[8]

L.A. Peletier and W.C. Troy, Spatial Patterns., Higher Order Models in Physics and Mechanics, (2001).   Google Scholar

show all references

References:
[1]

P. Álvarez-Caudevilla, J.D. Evans and V.A. Galaktionov, Countable families of solutions of a limit stationary semilinear fourth-order Cahn-Hilliard-type equation I., Mountain Pass Theorem and Lusternik-Schnirel'man patterns in $\mathbbR^N$, ().   Google Scholar

[2]

P. Álvarez-Caudevilla and V.A. Galaktionov, Steady states, global existence and blow-up for fourth-order semilinear parabolic equations of Cahn-Hilliard type,, Advances Nonl. Stud., 12 (2012), 315.   Google Scholar

[3]

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.   Google Scholar

[4]

J.D. Evans, V.A. Galaktionov and J.F. Williams, Blow-up and global asymptotics of the limit unstable Cahn-Hilliard equation, SIAM, J. Math. Anal., 38 (2006), 64.   Google Scholar

[5]

V.A. Galaktionov, E. Mitidieri and S.I. Pohozaev, Variational approach to complicated similarity solutions of higher-order nonlinear PDEs. II,, Nonl. Anal.: RWA, 12 (2011), 2435.   Google Scholar

[6]

G.A. Klassen and E. Mitidieri, Standing wave solutions for a system derived from the FitzHugh-Nagumo equations for nerve conduction, SIAM, J. Math. Anal., 17 (1986), 74.   Google Scholar

[7]

A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965.   Google Scholar

[8]

L.A. Peletier and W.C. Troy, Spatial Patterns., Higher Order Models in Physics and Mechanics, (2001).   Google Scholar

[1]

Alp Eden, Varga K. Kalantarov. 3D convective Cahn--Hilliard equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1075-1086. doi: 10.3934/cpaa.2007.6.1075
[2]

Jaemin Shin, Yongho Choi, Junseok Kim. An unconditionally stable numerical method for the viscous Cahn--Hilliard equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737
[3]

John W. Barrett, Harald Garcke, Robert Nürnberg. On sharp interface limits of Allen--Cahn/Cahn--Hilliard variational inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 1-14. doi: 10.3934/dcdss.2008.1.1
[4]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 609-629. doi: 10.3934/dcds.2007.19.609
[5]

Harald Garcke, Kei Fong Lam. Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4277-4308. doi: 10.3934/dcds.2017183
[6]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[7]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[8]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[9]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[10]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[11]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[12]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[13]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[14]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137
[15]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[16]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
[17]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881
[18]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020303
[19]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[20]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517

 Impact Factor: 

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences