IJMNTA  Vol.2 No.4 , December 2013
Notes on the Global Attractors for Semigroup
Author(s) Lan Xu*, Yuhua Shi*
ABSTRACT

First we introduce two necessary and sufficient conditions which ensure the existence of the global attractors for semigroup. Then we recall the concept of measure of noncompactness of a set and recapitulate its basic properties. Finally, we prove that these two conditions are equivalent directly.


Cite this paper
L. Xu and Y. Shi, "Notes on the Global Attractors for Semigroup," International Journal of Modern Nonlinear Theory and Application, Vol. 2 No. 4, 2013, pp. 219-222. doi: 10.4236/ijmnta.2013.24030.
References
[1]   Q. Ma, S. Wang and C. Zhong, “Necessary and Sufficient Conditions for the Existence of Global Attractors for Semigroups and Applications,” Indiana University Mathematics Journal, Vol. 51, No. 6, 2002, pp. 1542-1558.
http://dx.doi.org/10.1512/iumj.2002.51.2255

[2]   A. V. Babin and M. I. Vishik, “Attractors of Evolution Equations,” North-Hollard, Amsterdam, 1992.

[3]   J. K. Hale, “Asymptotic Behavior of Dissipative Systems,” American Mathematical Society, Providence, 1988.

[4]   J. M. Ball, “Continuity Propertics and Global Attractors of Generalized Semiflows and the Navier-Stokes Equations,” Journal of Nonlinear Science, Vol. 7, No. 5, 1997, pp. 475-502. http://dx.doi.org/10.1007/s003329900037

[5]   R. Sell and Y. You, “Dynamics of Evolutionary Equations,” Springer, New York, 2002.
http://dx.doi.org/10.1007/978-1-4757-5037-9

[6]   R. Temam, “Infinite-Dimensional Dynamical Systems in Mechanics and Physics,” Springer, New York, 1997.

[7]   B. Wang, “Dynamics of Systems on Infinite Lattices,” Journal of Differential Equations, Vol. 221, No. 1, 2006, pp. 224-245. http://dx.doi.org/10.1016/j.jde.2005.01.003

[8]   H. Crauel and F. Flandoli, “Attractors for Random Dynamical Systems,” Probability Theory and Related Fields, Vol. 100, No. 3, 1994, pp. 365-393.
http://dx.doi.org/10.1007/BF01193705

[9]   F. Flandoli and B. Schmalfuss, “Random Attractors for the 3D Stochastic Navier-Stokes Equation with Multiplicative White Noise,” Stochastics and Stochastic Reports, Vol. 59, No. 1-2, 1996, pp. 21-45.

[10]   B. Wang, “Suffcient and Necessary Criteria for Existence of Pullback Attractors for Non-Compact Random Dynamical Systems,” Journal of Differential Equations, Vol. 253, No. 5, 2012, pp. 1544-1583.
http://dx.doi.org/10.1016/j.jde.2012.05.015

[11]   K. Deimling, “Nonlinear Functional Analysis,” SpringerVerlag, Berlin, 1985.
http://dx.doi.org/10.1007/978-3-662-00547-7

 
 
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