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June  2011, 4(3): 653-670. doi: 10.3934/dcdss.2011.4.653

Convergence of solutions of a non-local phase-field system

Stig-Olof Londen 1, and  Hana Petzeltová 2,

1. 

Aalto University School of Science and Technology, PB 1000, 02015 TKK, Finland
2. 

Mathematical Institute AV ČR, Žitná 25, 115 67 Praha 1

Received  January 2009 Revised  August 2009 Published  November 2010

We show that solutions of a two-phase model involving a non-local interactive term separate from the pure phases from a certain time on, even if this is not the case initially. This result allows us to apply a generalized Lojasiewicz-Simon theorem and to establish the convergence of solutions to a single stationary state as time goes to infinity.
Keywords: Non-local phase-field systems; separation property; convergence to equilibria..
Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35K65; 45K0.
Citation: Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653
References:
[1]

N. D. Alikakos, $L^p$-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.

[2]

C. K. Chen and P. C. Fife, Nonlocal models of phase transitions in solids, Adv. Math. Sci. Appl., 10 (2000), 821-849.

[3]

L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potential, J. Math. Anal. Appl., 343 (2008), 557-566.

[4]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.

[5]

E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Łojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21.

[6]

E. Feireisl and H. Petzeltová, Non-standard applications of the Łojasiewicz-Simon theory, stabilization to equilibria of solutions to phase-field models, Banach Center Publications, 81 (2008), 175-184.

[7]

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynamics Differential Equations, 12 (2000), 647-673.

[8]

H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems, Disc. Cont. Dyn. Syst, 15 (2006), 505-528.

[9]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31.

[10]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions I. Macroscopic limits, J. Statist. Phys., 87 (1997), 37-61.

[11]

M. Grasselli, H. Petzeltová and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations, 239 (2007), 38-60.

[12]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.

[13]

M. Grasselli, H. Petzeltová and G. Schimperna, A nonlocal phase-field system with inertial term, Quart. Appl. Math., 65 (2007), 451-469.

[14]

E. Rocca and R. Rossi, Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials, J. Differential Equations, 345 (2008), 3327-3375.

[15]

W. P. Ziemer, "Weakly Differentiable Functions," Springer-Verlag, New York, 1989.

show all references

References:
[1]

N. D. Alikakos, $L^p$-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.

[2]

C. K. Chen and P. C. Fife, Nonlocal models of phase transitions in solids, Adv. Math. Sci. Appl., 10 (2000), 821-849.

[3]

L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potential, J. Math. Anal. Appl., 343 (2008), 557-566.

[4]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.

[5]

E. Feireisl, F. Issard-Roch and H. Petzeltová, A non-smooth version of the Łojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21.

[6]

E. Feireisl and H. Petzeltová, Non-standard applications of the Łojasiewicz-Simon theory, stabilization to equilibria of solutions to phase-field models, Banach Center Publications, 81 (2008), 175-184.

[7]

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynamics Differential Equations, 12 (2000), 647-673.

[8]

H. Gajewski and J. A. Griepentrog, A descent method for the free energy of multicomponent systems, Disc. Cont. Dyn. Syst, 15 (2006), 505-528.

[9]

H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), 11-31.

[10]

G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions I. Macroscopic limits, J. Statist. Phys., 87 (1997), 37-61.

[11]

M. Grasselli, H. Petzeltová and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations, 239 (2007), 38-60.

[12]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.

[13]

M. Grasselli, H. Petzeltová and G. Schimperna, A nonlocal phase-field system with inertial term, Quart. Appl. Math., 65 (2007), 451-469.

[14]

E. Rocca and R. Rossi, Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials, J. Differential Equations, 345 (2008), 3327-3375.

[15]

W. P. Ziemer, "Weakly Differentiable Functions," Springer-Verlag, New York, 1989.

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