July  2015, 20(5): 1529-1553. doi: 10.3934/dcdsb.2015.20.1529

Convective nonlocal Cahn-Hilliard equations with reaction terms

1. 

Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

2. 

Dipartimento di Matematica, Politecnico di Milano, 20133 Milano

Received  May 2014 Revised  January 2015 Published  May 2015

We introduce and analyze the nonlocal variants of two Cahn-Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors.
Citation: Francesco Della Porta, Maurizio Grasselli. Convective nonlocal Cahn-Hilliard equations with reaction terms. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1529-1553. doi: 10.3934/dcdsb.2015.20.1529
References:
[1]

A. C. Aristotelous, O. Karakashian and S. M. Wise, A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2211-2238. doi: 10.3934/dcdsb.2013.18.2211.

[2]

M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A, 41 (1990), 6763-6771. doi: 10.1103/PhysRevA.41.6763.

[3]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equation, J. Nonlinear Sci., 7 (1997), 475-502 (Erratum, J. Nonlinear Sci. 8 (1998), p233). doi: 10.1007/s003329900037.

[4]

P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277. doi: 10.1016/j.jde.2004.07.003.

[5]

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.

[6]

A. L. Bertozzi, S. Esedo$\overlineg$lu 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.

[7]

A. L. Bertozzi, S. Esedo$\overlineg$lu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936. doi: 10.1137/060660631.

[8]

S. Bosia, M. Grasselli and A. Miranville, On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743. doi: 10.1002/mma.2832.

[9]

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.

[10]

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

[11]

R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM J. Appl. Math., 69 (2009), 1712-1738. doi: 10.1137/080728809.

[12]

R. Choksi, Scaling laws in microphase separation of diblock copolymers, J. Nonlinear Sci., 11 (2001), 223-236. doi: 10.1007/s00332-001-0456-y.

[13]

R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, J. Stat. Phys., 113 (2003), 151-176. doi: 10.1023/A:1025722804873.

[14]

R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM J. Appl. Dyn. Syst., 10 (2011), 1344-1362. doi: 10.1137/100784497.

[15]

L. Cherfils, H. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi-Esedo$\overlineg$lu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.

[16]

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.

[17]

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.

[18]

P. Colli, S. Frigeri and M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428-444. doi: 10.1016/j.jmaa.2011.08.008.

[19]

A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514. doi: 10.1016/0362-546X(94)00205-V.

[20]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.

[21]

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.

[22]

P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, 48 (2000), 26 pp.

[23]

S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint, WIAS Preprint, 1923 (2014), 34 pp.

[24]

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.

[25]

S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials, Dyn. Partial Differ. Equ., 9 (2012), 273-304. doi: 10.4310/DPDE.2012.v9.n4.a1.

[26]

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.

[27]

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.

[28]

G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett., 76 (1996), 1094-1097. doi: 10.1103/PhysRevLett.76.1094.

[29]

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.

[30]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729. doi: 10.1137/S0036139996313046.

[31]

S. C. Glotzer, E. A. Di Marzio and M. Muthukumar, Reaction-controlled morphology of phase separating mixtures, Phys. Rev. Lett., 74 (1995), 2034-2037. doi: 10.1103/PhysRevLett.74.2034.

[32]

Y. Huo, H. Zhang and Y. Yang, Effects of reversible chemical reaction on morphology and domain growth of phase separating binary mixtures with viscosity difference, Macromol. Theory Simul., 13 (2004), 280-289. doi: 10.1002/mats.200300021.

[33]

Y. Huo, X. Jiang, H. Zhang and Y. Yang, Hydrodynamic effects on phase separation of binary mixtures with reversible chemical reaction, J. Chem. Phys., 118 (2003), 9830-9837. doi: 10.1063/1.1571511.

[34]

T. P. Lodge, Block copolymers: past successes and future challenges, Macromol. Chem. Phys., 204 (2003), 265-273. doi: 10.1002/macp.200290073.

[35]

S.-O. 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.

[36]

S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.

[37]

P. Mansky, P. Chaikin and E. L. Thomas, Monolayer films of diblock copolymer microdomains for nanolithographic applications, J. Mater. Sci., 30 (1995), 1987-1992. doi: 10.1007/BF00353023.

[38]

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

[39]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: evolutionary equations, IV (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[40]

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

[41]

A. Miranville, Asymptotic behaviour of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal., 92 (2013), 1308-1321. doi: 10.1080/00036811.2012.671301.

[42]

C. B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions, Comm. Math. Phys., 299 (2010), 45-87. doi: 10.1007/s00220-010-1094-8.

[43]

C. B. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type, Phys. Rev. E, 66 (2002), 066108, 25 pp. doi: 10.1103/PhysRevE.66.066108.

[44]

B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297. doi: 10.1080/03605308908820597.

[45]

Y. Nishiura and I. Ohnishi, Some mathematical aspects of the microphase separation in diblock copolymers, Phys. D, 84 (1995), 31-39. doi: 10.1016/0167-2789(95)00005-O.

[46]

A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985.

[47]

A. Novick-Cohen, The Cahn-Hilliard equation, Handbook of differential equations: Evolutionary equations, IV (2008), 201-228. doi: 10.1016/S1874-5717(08)00004-2.

[48]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. doi: 10.1021/ma00164a028.

[49]

Y. Oono and S. Puri, Computationally Efficient Modeling of Ordering of Quenched Phases, Phys. Rev. Lett., 58 (1987), 836-839. doi: 10.1103/PhysRevLett.58.836.

[50]

G. Schimperna, Global attractors for Cahn-Hilliard equations with nonconstant mobility, Nonlinearity, 20 (2007), 2365-2387. doi: 10.1088/0951-7715/20/10/006.

[51]

S. Villain Guillot, 1D Cahn-Hilliard equation for modulated phase systems, J. Phys. A, 43 (2010).

[52]

S. Walheim, E. Schaeffer, J. Mlynek and U. Steiner, Nanophase-separated polymer films as high-performance antireflection coatings, Science, 283 (1999), 520-522. doi: 10.1126/science.283.5401.520.

[53]

S. Zheng, Asymptotic behavior of solution to the Cahn-Hilliard equation, Appl. Anal., 23 (1986), 165-184. doi: 10.1080/00036818608839639.

show all references

References:
[1]

A. C. Aristotelous, O. Karakashian and S. M. Wise, A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2211-2238. doi: 10.3934/dcdsb.2013.18.2211.

[2]

M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A, 41 (1990), 6763-6771. doi: 10.1103/PhysRevA.41.6763.

[3]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equation, J. Nonlinear Sci., 7 (1997), 475-502 (Erratum, J. Nonlinear Sci. 8 (1998), p233). doi: 10.1007/s003329900037.

[4]

P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differential Equations, 212 (2005), 235-277. doi: 10.1016/j.jde.2004.07.003.

[5]

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.

[6]

A. L. Bertozzi, S. Esedo$\overlineg$lu 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.

[7]

A. L. Bertozzi, S. Esedo$\overlineg$lu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936. doi: 10.1137/060660631.

[8]

S. Bosia, M. Grasselli and A. Miranville, On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743. doi: 10.1002/mma.2832.

[9]

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.

[10]

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

[11]

R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM J. Appl. Math., 69 (2009), 1712-1738. doi: 10.1137/080728809.

[12]

R. Choksi, Scaling laws in microphase separation of diblock copolymers, J. Nonlinear Sci., 11 (2001), 223-236. doi: 10.1007/s00332-001-0456-y.

[13]

R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, J. Stat. Phys., 113 (2003), 151-176. doi: 10.1023/A:1025722804873.

[14]

R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM J. Appl. Dyn. Syst., 10 (2011), 1344-1362. doi: 10.1137/100784497.

[15]

L. Cherfils, H. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi-Esedo$\overlineg$lu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.

[16]

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.

[17]

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.

[18]

P. Colli, S. Frigeri and M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428-444. doi: 10.1016/j.jmaa.2011.08.008.

[19]

A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514. doi: 10.1016/0362-546X(94)00205-V.

[20]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.

[21]

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.

[22]

P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations, 48 (2000), 26 pp.

[23]

S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint, WIAS Preprint, 1923 (2014), 34 pp.

[24]

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.

[25]

S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials, Dyn. Partial Differ. Equ., 9 (2012), 273-304. doi: 10.4310/DPDE.2012.v9.n4.a1.

[26]

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.

[27]

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.

[28]

G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett., 76 (1996), 1094-1097. doi: 10.1103/PhysRevLett.76.1094.

[29]

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.

[30]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions. II. Interface motion, SIAM J. Appl. Math., 58 (1998), 1707-1729. doi: 10.1137/S0036139996313046.

[31]

S. C. Glotzer, E. A. Di Marzio and M. Muthukumar, Reaction-controlled morphology of phase separating mixtures, Phys. Rev. Lett., 74 (1995), 2034-2037. doi: 10.1103/PhysRevLett.74.2034.

[32]

Y. Huo, H. Zhang and Y. Yang, Effects of reversible chemical reaction on morphology and domain growth of phase separating binary mixtures with viscosity difference, Macromol. Theory Simul., 13 (2004), 280-289. doi: 10.1002/mats.200300021.

[33]

Y. Huo, X. Jiang, H. Zhang and Y. Yang, Hydrodynamic effects on phase separation of binary mixtures with reversible chemical reaction, J. Chem. Phys., 118 (2003), 9830-9837. doi: 10.1063/1.1571511.

[34]

T. P. Lodge, Block copolymers: past successes and future challenges, Macromol. Chem. Phys., 204 (2003), 265-273. doi: 10.1002/macp.200290073.

[35]

S.-O. 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.

[36]

S.-O. Londen and H. Petzeltová, Convergence of solutions of a non-local phase-field system, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 653-670.

[37]

P. Mansky, P. Chaikin and E. L. Thomas, Monolayer films of diblock copolymer microdomains for nanolithographic applications, J. Mater. Sci., 30 (1995), 1987-1992. doi: 10.1007/BF00353023.

[38]

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

[39]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of differential equations: evolutionary equations, IV (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[40]

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

[41]

A. Miranville, Asymptotic behaviour of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal., 92 (2013), 1308-1321. doi: 10.1080/00036811.2012.671301.

[42]

C. B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions, Comm. Math. Phys., 299 (2010), 45-87. doi: 10.1007/s00220-010-1094-8.

[43]

C. B. Muratov, Theory of domain patterns in systems with long-range interactions of Coulomb type, Phys. Rev. E, 66 (2002), 066108, 25 pp. doi: 10.1103/PhysRevE.66.066108.

[44]

B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297. doi: 10.1080/03605308908820597.

[45]

Y. Nishiura and I. Ohnishi, Some mathematical aspects of the microphase separation in diblock copolymers, Phys. D, 84 (1995), 31-39. doi: 10.1016/0167-2789(95)00005-O.

[46]

A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985.

[47]

A. Novick-Cohen, The Cahn-Hilliard equation, Handbook of differential equations: Evolutionary equations, IV (2008), 201-228. doi: 10.1016/S1874-5717(08)00004-2.

[48]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. doi: 10.1021/ma00164a028.

[49]

Y. Oono and S. Puri, Computationally Efficient Modeling of Ordering of Quenched Phases, Phys. Rev. Lett., 58 (1987), 836-839. doi: 10.1103/PhysRevLett.58.836.

[50]

G. Schimperna, Global attractors for Cahn-Hilliard equations with nonconstant mobility, Nonlinearity, 20 (2007), 2365-2387. doi: 10.1088/0951-7715/20/10/006.

[51]

S. Villain Guillot, 1D Cahn-Hilliard equation for modulated phase systems, J. Phys. A, 43 (2010).

[52]

S. Walheim, E. Schaeffer, J. Mlynek and U. Steiner, Nanophase-separated polymer films as high-performance antireflection coatings, Science, 283 (1999), 520-522. doi: 10.1126/science.283.5401.520.

[53]

S. Zheng, Asymptotic behavior of solution to the Cahn-Hilliard equation, Appl. Anal., 23 (1986), 165-184. doi: 10.1080/00036818608839639.

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