### AIMS Mathematics

2016, Issue 1: 64-76. doi: 10.3934/Math.2016.1.64
Research article Special Issues

# On the viscous Cahn-Hilliard equation with singular potential and inertial term

• Received: 20 April 2016 Accepted: 28 April 2016 Published: 05 May 2016
• We consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term utt. The equation also contains a semilinear term f(u) of “singular” type. Namely, the function f is defined only on a bounded interval of R corresponding to the physically admissible values of the unknown u, and diverges as u approaches the extrema of that interval. In view of its interaction with the inertial term utt, the term f(u) is diffcult to be treated mathematically. Based on an approach originally devised for the strongly damped wave equation, we propose a suitable concept of weak solution based on duality methods and prove an existence result.

Citation: Scala Riccardo, Schimperna Giulio. On the viscous Cahn-Hilliard equation with singular potential and inertial term[J]. AIMS Mathematics, 2016, 1(1): 64-76. doi: 10.3934/Math.2016.1.64

### Related Papers:

• We consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term utt. The equation also contains a semilinear term f(u) of “singular” type. Namely, the function f is defined only on a bounded interval of R corresponding to the physically admissible values of the unknown u, and diverges as u approaches the extrema of that interval. In view of its interaction with the inertial term utt, the term f(u) is diffcult to be treated mathematically. Based on an approach originally devised for the strongly damped wave equation, we propose a suitable concept of weak solution based on duality methods and prove an existence result.

 [1] H. Attouch (1984) Variational Convergence for Functions and Operators.Pitman, London . [2] V. Barbu (1976) Nonlinear Semigroups and Differential Equations in Banach Spaces.Noordhoff, Leyden . [3] V. Barbu, P. Colli, G. Gilardi, M. Grasselli (2000) Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation.Differential Integral Equations 13: 1233-1262. [4] V. Barbu, Y. Guo, M.A. Rammaha, D. Toundykov (2012) Convex integrals on Sobolev spaces.J. Convex Anal. 19: 837-852. [5] E. Bonetti, E. Rocca, R. Scala, G. Schimperna (2015) On the strongly damped wave equation with constraint.arXiv:1503.01911, submitted . [6] A. Bonfoh (2005) Existence and continuity of uniform exponential attractors for a singular perturbation of a generalized Cahn-Hilliard equation.Asymptotic Anal. 43: 233-247. [7] A. Bonfoh, M. Grasselli, A. Miranville (2008) Large time behavior of a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation, Math.Methods Appl. Sci. 31: 695-734. [8] H. Br′ezis (1972) Int′egrales convexes dans les espaces de Sobolev. (French).Israel J. Math. 13: 9-23. [9] J.W. Cahn, J.E. Hilliard (1958) Free energy of a nonuniform system. I. Interfacial free energy.J. Chem. Phys. 28: 258-267. [10] G. Dal Maso, A. De Simone, M.G. Mora (2006) Quasistatic evolution problems for linearly elastic-pefectly plastic materials, Arch.Rational Mech. Anal. 180: 237-291. [11] P. Galenko (2001) Phase-field models with relaxation of the diffusion flux in nonequilibrium solidification of a binary system.Phys. Lett. A 287: 190-197. [12] P. Galenko, D. Jou (2005) Diffuse-interface model for rapid phase transformations in nonequilibrium systems.Phys. Rev. E 71: 046125. [13] S. Gatti, M. Grasselli, A. Miranville, V. Pata (2005) On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation.J. Math. Anal. Appl. 312: 230-247. [14] S. Gatti, M. Grasselli, A. Miranville, V. Pata (2005) Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D, Math.Models Methods Appl. Sci. 15: 165-198. [15] M. Grasselli, H. Petzeltov′a, G. Schimperna (2007) Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term.J. Differential Equations 239: 38-60. [16] M. Grasselli, G. Schimperna, S. Zelik (2009) On the 2D Cahn-Hilliard equation with inertial term, Comm.Partial Differential Equations 34: 137-170. [17] M. Grasselli, G. Schimperna, A. Segatti, S. Zelik (2009) On the 3D Cahn-Hilliard equation with inertial term.J. Evol. Equ. 9: 371-404. [18] M. Grasselli, G. Schimperna, S. Zelik (2010) Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term.Nonlinearity 23: 707-737. [19] M.B. Kania (2007) Global attractor for the perturbed viscous Cahn-Hilliard equation.Colloq. Math 109: 217-229. [20] A. Miranville, S. Zelik (2004) Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math.Methods Appl. Sci. 27: 545-582. [21] A. Novick-Cohen (1998) The Cahn-Hilliard equation: mathematical and modeling perspectives.Adv. Math. Sci. Appl. 8: 965-985. [22] A. Novick-CohenOn the viscous Cahn-Hilliard equation. In: J.M. Ball ed., Material Instabilities in Continuum Mechanics.Oxford University Press, New York 329-342. [23] G. Schimperna, I. Pawłow (2013) On a class of Cahn-Hilliard models with nonlinear diffusion.SIAM J. Math. Anal. 45: 31-63. [24] R. Scala (2015) A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint, submitted. [25] J. Simon (1987) Compact sets in the space Lp(0; T; B).Ann. Mat. Pura Appl. 146: 65-96. [26] S. Zheng, A.J. Milani (2004) Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations.Nonlinear Anal. 57: 843-877. [27] S. Zheng, A.J. Milani (2005) Global attractors for singular perturbations of the Cahn-Hilliard equations.J. Differential Equations 209: 101-139.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.427 1.6