Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation

In this article, we consider a Cahn-Hilliard model with boundary conditions of Wentzell type and mass conservation. We show that each solution of this problem converges to a steady state as time goes to infinity, provided that the potential function $f$ is real analytic and satisfies certain growth assumptions. Estimates of the rate of convergence to equilibrium are obtained as well. We also recall some results about the existence of global and exponential attractors and their properties.

[1]  S. Łojasiewicz.  Ensembles semi-analytiques , 1965 .

[2]  M. Grasselli,et al.  Asymptotic behavior of a nonisothermal Ginzburg-Landau model , 2008 .

[3]  L. Simon.  Asymptotics for a class of non-linear evolution equations, with applications to geometric problems , 1983 .

[4]  M. Grasselli,et al.  ON THE ASYMPTOTIC BEHAVIOR OF THE CAGINALP SYSTEM WITH DYNAMIC BOUNDARY CONDITIONS , 2008 .

[5]  Hao-qing Wu,et al.  Convergence to Equilibrium for the Cahn-Hilliard Equation with Wentzell Boundary Condition , 2004, 0705.3362.

[6]  Piotr Rybka,et al.  Convergence of solutions to cahn-hilliard equation , 1999 .

[7]  Leon Simon,et al.  Theorems on Regularity and Singularity of Energy Minimizing Maps , 1996 .

[8]  Jan Prüss,et al.  Convergence to steady states of solutions of the Cahn–Hilliard and Caginalp equations with dynamic boundary conditions , 2006 .

[9]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[10]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[11]  M. Grasselli,et al.  Convergence to equilibrium for a parabolic–hyperbolic phase-field system with dynamical boundary condition , 2007 .

[12]  Ciprian G. Gal,et al.  A Cahn–Hilliard model in bounded domains with permeable walls , 2006 .

[13]  Songmu Zheng,et al.  The Cahn-Hilliard equation with dynamic boundary conditions , 2003, Advances in Differential Equations.

[14]  S. Huang,et al.  Convergence in gradient-like systems which are asymptotically autonomous and analytic , 2001 .

[15]  Gisèle Ruiz Goldstein,et al.  Derivation and physical interpretation of general boundary conditions , 2006, Advances in Differential Equations.

[16]  Maurizio Grasselli,et al.  CONVERGENCE TO EQUILIBRIUM FOR A PARABOLIC–HYPERBOLIC PHASE-FIELD SYSTEM WITH NEUMANN BOUNDARY CONDITIONS , 2007 .

[17]  P. Polácik,et al.  Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains , 2002 .

[18]  L. Peletier,et al.  Steady states of the one-dimensional Cahn–Hilliard equation , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  A. Haraux,et al.  Systèmes dynamiques dissipatifs et applications , 1991 .

[20]  L. Segel,et al.  Nonlinear aspects of the Cahn-Hilliard equation , 1984 .

[21]  P. Polácik,et al.  Nonconvergent Bounded Trajectories in Semilinear Heat Equations , 1996 .

[22]  Philipp Maass,et al.  Novel Surface Modes in Spinodal Decomposition , 1997 .

[23]  Philipp Maass,et al.  Diverging time and length scales of spinodal decomposition modes in thin films , 1998 .

[24]  Sergey Zelik,et al.  Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions , 2005 .

[25]  M. Grinfeld,et al.  Counting stationary solutions of the Cahn–Hilliard equation by transversality arguments , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[26]  Jan Prüss,et al.  Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions , 2006 .

[27]  C. Gal.  Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions , 2008 .

[28]  Charles M. Elliott,et al.  On the Cahn-Hilliard equation , 1986 .

[29]  Sergey Zelik,et al.  Exponential attractors for a singularly perturbed Cahn‐Hilliard system , 2004 .

[30]  Alain Haraux,et al.  Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity , 2001 .

[31]  Hao-qing Wu,et al.  Convergence to equilibrium for the damped semilinear wave equation with critical exponent and diss , 2006 .

[32]  Hiroshi Matano,et al.  Convergence of solutions of one-dimensional semilinear parabolic equations , 1978 .

[33]  Bernd Rinn,et al.  Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions , 2001 .