American Institute of Mathematical Sciences

Advanced
August  2018, 38(8): 3765-3788. doi: 10.3934/dcds.2018163

Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction

Annalisa Iuorio 1,, and  Stefano Melchionna 2,

1. 

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040, Vienna, Austria
2. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

* Corresponding author: Annalisa Iuorio

Received  July 2017 Revised  January 2018 Published  May 2018

Fund Project: The authors would like to acknowledge the Austrian Science Fund (FWF) for financial support (AI through project W1245, SM through project P27052-N25)

In this paper we study the long-time behavior of a nonlocal Cahn-Hilliard system with singular potential, degenerate mobility, and a reaction term. In particular, we prove the existence of a global attractor with finite fractal dimension, the existence of an exponential attractor, and convergence to equilibria for two physically relevant classes of reaction terms.

Keywords: Nonlocal Cahn-Hilliard equation, reaction term, global attractor, exponential attractor, convergence to equilibria.
Mathematics Subject Classification: Primary: 37L30; Secondary: 45K05.
Citation: 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
References:
[1]

H. Abels and M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193.  doi: 10.1016/j.na.2006.10.002.  Google Scholar

[2]

G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284.  doi: 10.1017/S0956792598003453.  Google Scholar

[3]

M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A, 41 (1990), p6763. Google Scholar

[4]

P. W. Bates and J. Han, The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation, J. Math. Anal. Appl., 311 (2005), 289-312.  doi: 10.1016/j.jmaa.2005.02.041.  Google Scholar

[5]

A. L. Bertozzi, S. Esedoḡlu and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291.  doi: 10.1109/TIP.2006.887728.  Google Scholar

[6]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.  doi: 10.1002/9781118788295.ch11.  Google Scholar

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[8]

L. Cherfils, H. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.  doi: 10.3934/ipi.2015.9.105.  Google Scholar

[9]

L. Cherfils, H. Fakih and A. Miranville, On the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with logarithmic nonlinear terms, SIAM J. Imaging Sci., 8 (2015), 1123-1140.  doi: 10.1137/140985627.  Google Scholar

[10]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[11]

L. Cherfils, A. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[12]

F. Della Porta and M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1529–1553, arXiv: math/0605406. doi: 10.3934/dcdsb.2015.20.1529.  Google Scholar

[13]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Math. Acad. Sci., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[14]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.  doi: 10.1137/S0036141094267662.  Google Scholar

[15]

L. C. Evans, Partial Differential Equations, American Mathematical Society, New York, 1998.  Google Scholar

[16]

H. Fakih, A Cahn-Hilliard equation with a proliferation term for biological and chemical applications, Asymptot. Anal., 94 (2015), 71-104.  doi: 10.3233/ASY-151306.  Google Scholar

[17]

E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21.  doi: 10.1016/j.jde.2003.10.026.  Google Scholar

[18]

S. Frigeri and M. Grasselli, Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Dynam. Differential Equations, 24 (2012), 827-856.  doi: 10.1007/s10884-012-9272-3.  Google Scholar

[19]

S. Frigeri, M. Grasselli and E. Rocca, A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity, 28 (2015), 1257-1293.  doi: 10.1088/0951-7715/28/5/1257.  Google Scholar

[20]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31.  doi: 10.1016/S0022-247X(02)00425-0.  Google Scholar

[21]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar

[22]

C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179.  doi: 10.3934/dcds.2014.34.145.  Google Scholar

[23]

H. Garcke, B. Nestler and B. Stoth, On anisotropic order parameter models for multi-phase systems and their sharp interface limits, Phys. D, 115 (1998), 87-108.  doi: 10.1016/S0167-2789(97)00227-3.  Google Scholar

[24]

H. Garcke, M. Rumpf and U. Weikard, The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies, Interfaces Free Bound., 3 (2001), 101-118.  doi: 10.4171/IFB/34.  Google Scholar

[25]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, J. Stat. Phys, 87 (1997), 37-61.  doi: 10.1007/BF02181479.  Google Scholar

[26]

Z. Guan, J. S. Lowengrub, C. Wang and S. M. Wise, Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71.  doi: 10.1016/j.jcp.2014.08.001.  Google Scholar

[27]

E. Khain and L. M. Sander, Generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 051129. doi: 10.1103/PhysRevE.77.051129.  Google Scholar

[28]

N. Q. Le, A Gamma-convergence approach to the Cahn-Hilliard equation, Calc. Var. Partial Differential Equations, 32 (2008), 499-522.  doi: 10.1007/s00526-007-0150-5.  Google Scholar

[29]

M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary Γ-convergence, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 6, 31 pp. doi: 10.1007/s00030-018-0495-9.  Google Scholar

[30]

S. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.  doi: 10.3934/dcdss.2011.4.653.  Google Scholar

[31]

S. Londen and H. Petzeltová, Regularity and separation from potential barriers for a non-local phase-field system, J. Math. Anal. Appl., 379 (2011), 724-735.  doi: 10.1016/j.jmaa.2011.02.003.  Google Scholar

[32]

S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497.   Google Scholar

[33]

A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.   Google Scholar

[34]

L. Modica and S. Mortola, Un esempio di Γ-convergenza, Boll. Unione Mat. Ital. B (5), 14 (1977), 285-299.   Google Scholar

[35]

L. Nirenberg, On elliptic partial differential equations, in Il Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali), Springer, (2011), 1–48. doi: 10.1007/978-3-642-10926-3_1.  Google Scholar

[36]

J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Vol. 28, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

show all references

References:
[1]

H. Abels and M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193.  doi: 10.1016/j.na.2006.10.002.  Google Scholar

[2]

G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284.  doi: 10.1017/S0956792598003453.  Google Scholar

[3]

M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A, 41 (1990), p6763. Google Scholar

[4]

P. W. Bates and J. Han, The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation, J. Math. Anal. Appl., 311 (2005), 289-312.  doi: 10.1016/j.jmaa.2005.02.041.  Google Scholar

[5]

A. L. Bertozzi, S. Esedoḡlu and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291.  doi: 10.1109/TIP.2006.887728.  Google Scholar

[6]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.  doi: 10.1002/9781118788295.ch11.  Google Scholar

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[8]

L. Cherfils, H. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.  doi: 10.3934/ipi.2015.9.105.  Google Scholar

[9]

L. Cherfils, H. Fakih and A. Miranville, On the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with logarithmic nonlinear terms, SIAM J. Imaging Sci., 8 (2015), 1123-1140.  doi: 10.1137/140985627.  Google Scholar

[10]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.  Google Scholar

[11]

L. Cherfils, A. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[12]

F. Della Porta and M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1529–1553, arXiv: math/0605406. doi: 10.3934/dcdsb.2015.20.1529.  Google Scholar

[13]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Math. Acad. Sci., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[14]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.  doi: 10.1137/S0036141094267662.  Google Scholar

[15]

L. C. Evans, Partial Differential Equations, American Mathematical Society, New York, 1998.  Google Scholar

[16]

H. Fakih, A Cahn-Hilliard equation with a proliferation term for biological and chemical applications, Asymptot. Anal., 94 (2015), 71-104.  doi: 10.3233/ASY-151306.  Google Scholar

[17]

E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21.  doi: 10.1016/j.jde.2003.10.026.  Google Scholar

[18]

S. Frigeri and M. Grasselli, Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Dynam. Differential Equations, 24 (2012), 827-856.  doi: 10.1007/s10884-012-9272-3.  Google Scholar

[19]

S. Frigeri, M. Grasselli and E. Rocca, A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity, 28 (2015), 1257-1293.  doi: 10.1088/0951-7715/28/5/1257.  Google Scholar

[20]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31.  doi: 10.1016/S0022-247X(02)00425-0.  Google Scholar

[21]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar

[22]

C. G. Gal and M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34 (2014), 145-179.  doi: 10.3934/dcds.2014.34.145.  Google Scholar

[23]

H. Garcke, B. Nestler and B. Stoth, On anisotropic order parameter models for multi-phase systems and their sharp interface limits, Phys. D, 115 (1998), 87-108.  doi: 10.1016/S0167-2789(97)00227-3.  Google Scholar

[24]

H. Garcke, M. Rumpf and U. Weikard, The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies, Interfaces Free Bound., 3 (2001), 101-118.  doi: 10.4171/IFB/34.  Google Scholar

[25]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. Ⅰ. Macroscopic limits, J. Stat. Phys, 87 (1997), 37-61.  doi: 10.1007/BF02181479.  Google Scholar

[26]

Z. Guan, J. S. Lowengrub, C. Wang and S. M. Wise, Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71.  doi: 10.1016/j.jcp.2014.08.001.  Google Scholar

[27]

E. Khain and L. M. Sander, Generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 051129. doi: 10.1103/PhysRevE.77.051129.  Google Scholar

[28]

N. Q. Le, A Gamma-convergence approach to the Cahn-Hilliard equation, Calc. Var. Partial Differential Equations, 32 (2008), 499-522.  doi: 10.1007/s00526-007-0150-5.  Google Scholar

[29]

M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary Γ-convergence, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 6, 31 pp. doi: 10.1007/s00030-018-0495-9.  Google Scholar

[30]

S. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.  doi: 10.3934/dcdss.2011.4.653.  Google Scholar

[31]

S. Londen and H. Petzeltová, Regularity and separation from potential barriers for a non-local phase-field system, J. Math. Anal. Appl., 379 (2011), 724-735.  doi: 10.1016/j.jmaa.2011.02.003.  Google Scholar

[32]

S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497.   Google Scholar

[33]

A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.   Google Scholar

[34]

L. Modica and S. Mortola, Un esempio di Γ-convergenza, Boll. Unione Mat. Ital. B (5), 14 (1977), 285-299.   Google Scholar

[35]

L. Nirenberg, On elliptic partial differential equations, in Il Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali), Springer, (2011), 1–48. doi: 10.1007/978-3-642-10926-3_1.  Google Scholar

[36]

J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Vol. 28, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

Figure 1.  Illustration of $r_\kappa$ for a reaction term $g$ satisfying (A1), (A2).
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113
[2]

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
[3]

Francesco Della Porta, Maurizio Grasselli. Convective nonlocal Cahn-Hilliard equations with reaction terms. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1529-1553. doi: 10.3934/dcdsb.2015.20.1529
[4]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
[5]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145
[6]

Aibo Liu, Changchun Liu. Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term. Evolution Equations & Control Theory, 2015, 4 (3) : 315-324. doi: 10.3934/eect.2015.4.315
[7]

Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855
[8]

Maurizio Grasselli, Nicolas Lecoq, Morgan Pierre. A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term. Conference Publications, 2011, 2011 (Special) : 543-552. doi: 10.3934/proc.2011.2011.543
[9]

Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103
[10]

D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
[11]

Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151
[12]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[13]

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
[14]

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
[15]

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
[16]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
[17]

Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859
[18]

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
[19]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[20]

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

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (171)
  • HTML views (163)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences