Asymptotic behavior of a nonisothermal Ginzburg-Landau model
Authors:
Maurizio Grasselli, Hao Wu and Songmu Zheng
Journal:
Quart. Appl. Math. 66 (2008), 743-770
MSC (2000):
Primary 35K55; Secondary 35B40, 35B41, 80A22, 82D55
DOI:
https://doi.org/10.1090/S0033-569X-08-01115-9
Published electronically:
October 3, 2008
MathSciNet review:
2465143
Full-text PDF Free Access
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Abstract: We analyze a system of nonlinear parabolic equations which describes the evolution of an order parameter $f$ and the relative temperature $v$ in a superconducting material occupying a bounded domain $\Omega \subset {\mathbb R}^n$, $n\leq 3$. Here the dependent variable $f$ is subject to the homogeneous Neumann boundary condition, while $v$ is equal to a time-dependent Dirichlet datum $\tilde u$ on the boundary. Therefore, the corresponding dynamical system is nonautonomous. Our main goal is to analyze the asymptotic behavior of its solutions. We first show that the system has a bounded absorbing set and a global attractor which are uniform with respect to a sufficiently general class of $\tilde u$. Then, we give sufficient conditions on $\tilde u$ which ensure the convergence of a given trajectory to a single stationary state and we estimate the convergence rate. Finally, we demonstrate the existence of an exponential attractor of finite fractal dimension for quasi-periodic or stabilizing boundary data $\tilde u$.
References
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References
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- P.W. Bates, S. Zheng, Inertial manifolds and inertial sets for the phase-field equations, J. Dynamics Differential Equations 4 (1992), 375-397. MR 1160925 (93h:35187)
- V. Berti, M. Fabrizio, A non-isothermal Ginzburg-Landau model in superconductivity: existence, uniqueness and asymptotic behavior, Nonlinear Anal. 66 (2007), 2565–2578. MR 2312606 (2008a:82098)
- D. Brochet, X. Chen, D. Hilhorst, Finite dimensional exponential attractor for the phase-field model, Appl. Anal. 49 (1993), 197-212. MR 1289743 (95g:35097)
- D. Brochet, D. Hilhorst, Universal attractor and inertial sets for the phase-field model, Appl. Math. Lett. 4 (1991), 59-62. MR 1136614 (92g:35102)
- M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Appl. Math. Sci. 121, Springer, New York, 1996. MR 1411908 (97g:35127)
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- R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal. 53 (2003), 1017-1039. MR 1978032 (2004d:34103)
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- M. Efendiev, A. Miranville, S. Zelik, Infinite dimensional exponential attractors for a non-autonomous reaction-diffusion system, Math. Nachr. 248/249 (2003), 72-96. MR 1950716 (2003i:37082)
- M. Efendiev, S. Zelik, A. Miranville, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems. Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 703-730. MR 2173336 (2007a:37098)
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- M.Fabrizio, Ginzburg-Landau equations and first and second order phase transitions, Internat. J. Engrg. Sci. 44 (2006), 529–539. MR 2234088 (2007f:82118)
- G.J. Fix, Phase field models for free boundary problems, in “Free boundary problems: theory and applications, Vol. II”(A. Fasano and M. Primicerio, eds.), Pitman Res. Notes Math. Ser. 79, 580-589, Longman, London, 1983.
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- S. Gatti, A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in “Differential Equations Inverse and Direct Problems”(A. Favini and A. Lorenzi, eds.), Ser. Lect. Notes Pure Appl. Math. 251, 149-170, Chapman & Hall/CRC, Boca Raton, 2006. MR 2275977 (2007g:35087)
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- M. Grasselli, H. Petzeltová, G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwendungen 25 (2006), 51-72. MR 2216881 (2007b:35159)
- M. Grasselli, H. Petzeltová, G. Schimperna, Convergence to stationary solutions for a parabolic-hyperbolic phase-field system, Commun. Pure Appl. Anal. 5 (2006), 827-838. MR 2246010 (2007g:35088)
- J.K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys Monogr. 25, Amer. Math. Soc., Providence, 1988. MR 941371 (89g:58059)
- A. Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées 17, Masson, Paris, 1991. MR 1084372 (92b:35002)
- A. Haraux, M.A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations 9 (1999), 95-124. MR 1714129 (2000h:35110)
- S.-Z. Huang, P. Takáč, Convergence in gradient–like systems which are asymptotically autonomous and analytic, Nonlinear Anal. 46 (2001), 675-698. MR 1857152 (2002f:35125)
- M.A. Jendoubi, A simple unified approach to some convergence theorem of L. Simon, J. Funct. Anal. 153 (1998), 187-202. MR 1609269 (99c:35101)
- A. Jiménez-Casas, A. Rodríguez-Bernal, Asymptotic behaviour for a phase field model in higher order Sobolev spaces, Rev. Mat. Complut. 15 (2002), 213-248. MR 1915223 (2003f:35148)
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- O.V. Kapustyan, An attractor of a semiflow generated by a system of phase-field equations without uniqueness of the solution (Ukrainian), Ukraïn. Mat. Zh. 51 (1999), 1006-1009 [Translation in Ukrainian Math. J. 51 (1999), no. 7, 1135-1139 (2000)]. MR 1727706 (2001b:37113)
- Ph. Laurençot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 167-185. MR 1378839 (97a:35115)
- S. Ma, C. Zhong, The attractor for weakly damped non-autonomous hyperbolic equations with a new class of external forces, Discrete Contin. Dynam. Syst. 18 (2007), 53-70. MR 2276486
- G.B. McFadden, Phase-field models of solidification, Contemp. Math. 306 (2002), 107-145. MR 1940624 (2003k:80004)
- A. Miranville, Exponential attrators for nonautonomous evolution equations, Appl. Math. Lett. 11 (1998), 19-22. MR 1609661
- N. Sato, T. Aiki, Phase field equations with constraints under nonlinear dynamic boundary conditions, Commun. Appl. Anal. 5 (2001), 215-234. MR 1844192 (2002f:35131)
- G. Schimperna, Abstract approach to evolution equations of phase field type and applications, J. Differential Equations 164 (2000), 395-430. MR 1765571 (2001h:35110)
- H. Wu, M. Grasselli, S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions, Math. Models Methods Appl. Sci. 17 (2007) 125-153. MR 2290411
- Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Commun. Pure Appl. Anal. 4 (2005), 683-693. MR 2167193 (2006i:35142)
- S. Zheng, Nonlinear Evolution Equations, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 133, Chapman & Hall/CRC, Boca Raton, Florida, 2004. MR 2088362 (2006a:35001)
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Additional Information
Maurizio Grasselli
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Milano 20133, Italy
Email:
maurizio.grasselli@polimi.it
Hao Wu
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Email:
haowufd@yahoo.com
Songmu Zheng
Affiliation:
Institute of Mathematics, Fudan University, Shanghai 200433, China
Email:
songmuzheng@yahoo.com
Keywords:
Ginzburg-Landau systems,
phase-field equations,
nonautonomous dynamical systems,
global attractors,
convergence to equilibria,
Łojasiewicz-Simon inequality,
exponential attractors.
Received by editor(s):
July 15, 2007
Published electronically:
October 3, 2008
Additional Notes:
Dr. Songmu Zheng is the corresponding author.
Article copyright:
© Copyright 2008
Brown University
The copyright for this article reverts to public domain 28 years after publication.