CONVERGENCE TO EQUILIBRIUM FOR A PARABOLIC–HYPERBOLIC PHASE-FIELD SYSTEM WITH NEUMANN BOUNDARY CONDITIONS

This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature θ and the order parameter χ in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both ¸ and χ, under the assumption that the nonlinearities λ and ϕ are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz–Simon type inequality.

[1]  Danielle Hilhorst,et al.  Finite dimensional exponential attractor for the phase field model , 1993 .

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

[3]  G. F. Webb,et al.  Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces , 1979, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[4]  M. Jendoubi.  Convergence of Global and Bounded Solutions of the Wave Equation with Linear Dissipation and Analytic Nonlinearity , 1998 .

[5]  Songmu Zheng,et al.  On The Coupled Cahn-hilliard Equations , 1993 .

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

[7]  Eduard Feireisl,et al.  Long‐time convergence of solutions to a phase‐field system , 2001 .

[8]  Alain Haraux,et al.  Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity , 1999 .

[9]  C. M. Elliott,et al.  Global Existence and Stability of Solutions to the Phase Field Equations , 1990 .

[10]  Ángela Jiménez Casas,et al.  Asymptotic behaviour for a phase field model in higher order Sobolev spaces , 2002 .

[11]  Mohamed Ali Jendoubi,et al.  A Simple Unified Approach to Some Convergence Theorems of L. Simon , 1998 .

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

[13]  P. Laurençot.  Long-time behaviour for a model of phase-field type , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[14]  Sergey Zelik,et al.  Uniform exponential attractors for a singularly perturbed damped wave equation , 2003 .

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

[16]  Universal attractor and inertial sets for the phase field model , 1991 .

[17]  Peter Galenko,et al.  Diffuse-interface model for rapid phase transformations in nonequilibrium systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[19]  Eduard Feireisl,et al.  Convergence for Semilinear Degenerate Parabolic Equations in Several Space Dimensions , 2000 .

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

[21]  P. Bates,et al.  Inertial manifolds and inertial sets for the phase-field equations , 1992 .

[22]  Ralph Chill,et al.  On the Łojasiewicz–Simon gradient inequality , 2003 .

[23]  Vittorino Pata,et al.  Asymptotic behavior of a parabolic-hyperbolic system , 2004 .