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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

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

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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).
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