In this paper, we consider the initial boundary value problem for the 2-D hyperbolic viscous Cahn-Hilliard equation. Firstly, we prove the existence and uniqueness of the local solution by the Galerkin method and contraction mapping principle. Then, using the potential well theory, we study the global well-posedness of the solution with initial data at different levels of initial energy, i.e., subcritical initial energy, critical initial energy and arbitrary positive initial energy. For subcritical initial energy, we prove the global existence, asymptotic behavior and finite time blowup of the solution. Moreover, we extend these results to the critical initial energy using the scaling technique. For arbitrary positive initial energy, including the sup-critical initial energy, we obtain the sufficient conditions for finite time blow-up of the solution. As a further study for estimating the blowup time, we give a unified expression of the lower bound of blowup time for all three initial energy levels and estimate the upper bound of blowup time for subcritical and critical initial energy.
[1] Cahn J W, Hilliard J E. Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys, 1958, 28(2): 258-267
[2] Buff F P, Lovett R, Stillinger J F. Interfacial density profile for fluids in the critical region. Phys Rev Lett, 1965, 15(15): 621-623
[3] Caginalp G. An analysis of a phase field model of a free boundary. Arch Rational Mech Anal, 1986, 92: 205-245
[4] Evans L C, Soner H M, Souganidis P E. Phase transitions and generalized motion by mean curvature. Comm Pure Appl Math, 1992, 45(9): 1097-1123
[5] Elliott C M, Garcke H. On the Cahn-Hilliard equation with degenerate mobility. SIAM J Math Anal, 1996, 27(2): 404-423
[6] Pacard F, Ritor$\rm{\acute{e}}$ M. From constant mean curvature hypersurfaces to the gradient theory of phase transitions. J Differential Geom, 2003, 64(3): 359-423
[7] Caffarelli L A, Muler N E. An $L^{\infty}$ bound for solutions of the Cahn-Hilliard equation. Arch Rational Mech Anal, 1995, 133: 129-144
[8] Liu S Q, Wang F, Zhao H J. Global existence and asymptotics of solutions of the Cahn-Hilliard equation. J Differential Equations, 2007, 238(2): 426-469
[9] Elliott C M, Zheng S M. On the Cahn-Hilliard equation. Arch Rational Mech Anal, 1986, 96(4): 339-357
[10] Li D. A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials. Discrete Contin Dyn Syst, 2022, 42(5): 2453-2460
[11] Rybka P, Hoffmann K H. Convergence of solutions to Cahn-Hilliard equation. Comm Partial Differential Equations, 1999, 24(4/5): 1055-1077
[12] Miranville A. The Cahn-Hilliard equation with a nonlinear source term. J Differential Equations, 2021, 294: 88-117
[13] Duan N, Wang J, Zhao X P. Well-posedness and large time behavior for Cahn-Hilliard-Oono equation. Z Angew Math Phys, 2023, 74(6): Art 226
[14] Galenko P, Jou D. Diffuse-interface model for rapid phase transformations in nonequilibrium systems. Phys Rev E, 2005, 71(4): Art 046125
[15] Galenko P, Lebedev V. Nonequilibrium effects in spinodal decomposition of a binary system. Phys Lett A, 2008, 372(7): 985-989
[16] Grasselli M, Schimperna G, Zelik S. Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term. Nonlinearity, 2010, 23(3): 707-737
[17] Segatti A. On the hyperbolic relaxation of the Cahn-Hilliard equation in 3D: approximation and long time behaviour. Math Models Methods Appl Sci, 2007, 17(3): 411-437
[18] Khanmamedov A, Yayla S. Global attractors for the 2D hyperbolic Cahn-Hilliard equations. Z Angew Math Phys, 2018, 69(14): Art 17
[19] Grasselli M, Schimperna G, Zelik S. On the 2D Cahn-Hilliard equation with inertial term. Comm Partial Differential Equations, 2009, 34(2): 137-170
[20] Gurtin M E. Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Phys D, 1996, 92(3/4): 178-192
[21] Zheng S M, Milani A. Global attractors for singular perturbations of the Cahn-Hilliard equations. J Differential Equations, 2005, 209(1): 101-139
[22] Zheng S M, Milani A. Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations. Nonlinear Anal, 2004, 57(5/6): 843-877
[23] Gatti S, Grasselli M, Pata V, Miranville A. Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D. Math Models Methods Appl Sci, 2005, 15(2): 165-198
[24] Kania M B. Global attractor for the perturbed viscous Cahn-Hilliard equation. Colloq Math, 2007, 109(2): 217-229
[25] Wang W K, Wu Z G. Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions. J Math Anal Appl, 2012, 387(1): 349-358
[26] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math, 1975, 22(3/4): 273-303
[27] Xu R Z, Su J. Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J Funct Anal, 2013, 264(12): 2732-2763
[28] Pang Y, R$\rm{\breve{a}}$dulescu V D, Xu R Z. Global existence and finite time blow-up for the $m$-Laplacian parabolic problem. Acta Math Sin Engl Ser, 2023, 39(8): 1497-1524
[29] Chen H, Xu H Y. Global existence and blow-up in finite time for a class of finitely degenerate semilinear pseudo-parabolic equations. Acta Math Sin Engl Ser, 2019, 35(7): 1143-1162
[30] Chen Y X.Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity. Commun Anal Mech
1)2023, 15(4): 658-694
[31] Xu R Z, Lian W, Niu Y. Global well-posedness of coupled parabolic systems. Sci China Math, 2020, 63(2): 321-356
[32] Chen H, Liu G W.Global existence, uniform decay and exponential growth for a class of semi-linear wave equation with strong damping. Acta Math Sci
2)2013, 33B(1): 41-58
[33] Lian W, Xu R Z. Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv Nonlinear Anal, 2020, 9(1): 613-632
[34] Luo Y B, Xu R Z, Yang C. Global well-posedness for a class of semilinear hyperbolic equations with singular potentials on manifolds with conical singularities. Calc Var Partial Differential Equations, 2022, 61(6): Art 210
[35] Temam R.Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer-Verlag, 1997
[36] Yang C, R$\rm{\breve{a}}$dulescu V D, Xu R Z, Zhang M Y. Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models. Adv Nonlinear Stud, 2022, 22(1): 436-468
[37] Walter W L. Ordinary Differential Equations.New York: Springer-Verlag, 1998
[38] Evans L C. Partial Differential Equations. Providence: American Mathematical Society, 1998
[39] Xu R Z. Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Quart Appl Math, 2010, 68(3): 459-468
[40] Han J B, Wang K Y, Xu R Z, Yang C. Global quantitative stability of wave equations with strong and weak dampings. J Differential Equations, 2024, 390: 228-344
[41] Han J B, Xu R Z, Yang C. Continuous dependence on initial data and high energy blowup time estimate for porous elastic system. Commun Anal Mech, 2023, 15(2): 214-244
[42] Lax P D. Functional Analysis. New York: Wiley, 2002
[43] Kalantarov V K, Ladyzhenskaya O A. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. J Soviet Math, 1978, 10: 53-70
[44] Gazzola F, Squassina M. Global solutions and finite time blow up for damped semilinear wave equations. Ann Inst H $\rm{Poincar\acute{e}}$ C Anal Non $\rm{Lin\acute{e}aire}$, 2006, 23(2): 185-207
[45] Yang Y B, Xu R Z. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Commun Pure Appl Anal, 2019, 18(3): 1351-1358