Acta mathematica scientia, Series B >
TRAJECTORY ATTRACTORS FOR NONCLASSICAL DIFFUSION EQUATIONS WITH FADING MEMORY
Received date: 2011-12-15
Revised date: 2012-08-02
Online published: 2013-05-20
Supported by
The first author is supported by NSFC Grant (11031003) and the Fundamental Research Funds for the Central Universities; the second author is support by Fund of excellent young teachers in Shanghai (shgcjs008) and Initial Fund of SUES (A–0501–11–016).
In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7, 8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.
Key words： Trajectory attractor; global attractor; memory kernel
WANG Yong-Hai , WANG Ling-Zhi . TRAJECTORY ATTRACTORS FOR NONCLASSICAL DIFFUSION EQUATIONS WITH FADING MEMORY[J]. Acta mathematica scientia, Series B, 2013 , 33(3) : 721 -737 . DOI: 10.1016/S0252-9602(13)60033-8
[1] Aifantis E C. On the problem of diffusion in solids. Acta Mech, 1980, 37: 265–296
[2] Borini S, Pata V. Uniform attractors for a strongly damped wave equation with linear memory. Asymptotic Anal, 1999, 20: 263–277
[3] Caraballo T, Chueshov I D, Real J. Pullback attractors for stochastic heat equations in materials with memory. Discrete Cont Dyn Syst Series B, 2008, 9: 525–539
[4] Caraballo T, Garrido–Atienza M J, Schmalfuß B, Valero J. Global attractor for a non–autonomous integro–differential equation in materials with memory. Nonlinear Anal, TMA, 2010, 73: 183–201
[5] Chepyzhov V V, Gatti S, Grasselli M, et al. Trajectory and global attractors for evolution equations with memory. Appl Math Lett, 2006, 19: 87–96
[6] Chepyzhov V V, Mainini E, Pata V. Stability of abstract linear semigroups arising from heat conduction with memory. Asymptotic Analysis, 2006, 50: 269–291
[7] Chepyzhov V V, Miranville A. Trajectory and global attractors of dissipative hyperbolic equations with memory. Commun Pure Appl Anal, 2005, 4: 115–142
[8] Chepyzhov V V, Miranville A. On trajectory and global attractors for semilinear heat equations with fading memory. Indiana University Math J, 2006, 55: 119–167
[9] Chepyzhov V V, Vishik M I. Attractors for equations of mathematical physics. Amer Math Soc Colloq Publ 49. Providence, RI: Amer Math Soc, 2002
[10] Conti M, Pata V, Squassina M. Singular limit of differential systems with memory. Indiana University Math J, 2006, 55: 169–215
[11] Dafermos C M. Asymptotic stability in viscoelasticity. Arch Rational Mech Anal, 1970, 37: 297–308
[12] Giorgi C, Grasselli M, Pata V. Uniform attractors for a phase–field model with memory and quadratic nonlinearity. Indiana Univ Math J, 1999, 48: 1395–1445
[13] Giorgi C, Grasselli M, Pata V.Well–posedness and longtime behavior of the phase–field model with memory
in a history space setting. Quart Appl Math, 2001, 59: 701–736
[14] Giorgi C, Pata V, Marzocchi A. Asymptotic behavior of a semilinear problem in heat conduction with memory. Nonlinear Diff Eq Appl, 1998, 5: 333–354
[15] Grasselli M, Pata V. Upper semicontinuous attractor for a hyperbolic phase–field model with memory. Indiana Univ Math J, 2001, 50: 1281–1308
[16] Kloeden P E, Real J, Sun C Y. Robust exponential attractors for non–autonomous equations with memory. Commun Pure Appl Anal, 2011, 10: 885–915
[17] Lions J L, Magenes E. Non–Homogeneous Boundary Value Problems and Applications. Springer–Verlag, 1972
[18] Pata V. Attractors for a damped wave equation on R3 with linear memory. Math Meth Appl Sci Appl, 2000, 23: 633–653
[19] Pata V, Zucchi A. Attractors for a damped hyperbolic equation with linear memory. Adv Math Sci Appl, 2001, 11: 505–529
[20] Pata V. Hyperbolic limit of parabolic semilinear heat equations with fading memory. Z Anal Anwendungen, 2001, 20: 359–377
[21] Sun C Y, Cao D M, Duan J Q. Non–autonomous wave dynamics with memory–Asymptotic regularity and uniform attractor. Discrete Cont Dyn Syst, 2008, 9B: 743–761
[22] Sun C Y, Wang S Y, Zhong C K. Global attractors for a nonclassical diffusion equation. Acta Mathematica Sinica, English Series, 2007, 23: 1271–1280
[23] Sun C Y, Yang M H. Dynamics of the nonclassical diffusion equations. Asymptotic Anal, 2008, 59: 51–81
[24] Temam R. Infinite–dimensional dynamical systems in mechanics and physics. Springer-Verlag, 1997
[25] Wang X, Zhong C K. Attractors for the non–autonomous nonclassical diffusion equations with fading memory. Nonlinear Anal, 2009, 71: 5733–5746
[26] Wang Y H, Qin Y M. Upper semicontinuity of pullback attractors for nonclassical diffusion equations. J Math Phys, 2010, 51: 022701
[27] Wang S Y, Li D S, Zhong C K. On the dynamics of a class of nonclassical parabolic equations. J Math Anal Appl, 2006, 317: 565–582
[28] Xiao Y L. Attractors for a nonclassical diffusion equation. Acta Mathematicae Applicatae Sinica, English Series, 2002, 18: 273–276
[29] Zhao C D, Jia X L, Yang X B. Uniform attractor for nonautonomous in compressible non-Newtonian fluid with a new class of external force. Acta Mathematica Scientia, 2011, 31B(5): 1803–1812
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