EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE Hk

doi:10.1016/S0252-9602(09)60094-1

Abstract

Abstract:

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation u_t=Δ_u+λ_u-u³ possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k(Ω) in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k∈ [0,1] to the case k∈ [0, ∞).

Key words: semigroup of operator, global attractor, evolution equation, regularity of attractor

CLC Number:

35B40

ZHANG Yin-Di, LI Kai-Tai. EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE H^k[J].Acta mathematica scientia,Series B, 2009, 29(5): 1165-1172.

Trendmd

References

[1] Hale J. Asymptotic Behavior of Dissipative Systems. Providence RI: AMS, 1988

[2] Lu S, Wu H, Zhong C K. Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces. Dist Cont Dyna Syst, 2005, 13(3): 701--719

[3] Ma Q F, Wang S H and Zhong C K. Necessary and sufficient conditions for the existence of global attractors for
semigroups and applications. Indiana Univ Math J, 2002, 51(6): 1541--1559

[4] Ma T, Wang S H. Bifurcation Theory and Applications. Nonlinear Science Ser A, Vol 53. Singapore: World Scientific, 2005

[5] Ma T, Wang S H. Stability and Bifurcation of Nonlinear Evolution Equations. Beijing: Science Press, 2006 (in Chinese)

[6] Nicolaenko B, Scheurer B, Temam R. Some global dynamical properties of a class of pattern formation equations. Comm Partial Differ Equ, 1989, 14: 245--297

[7] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl Math Sci, Vol 44. Springer-Verlag, 1983

[8] Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, Vol 68. 2nd ed. New York: Springer-Verlag, 1997

[9] Zhong C K, Yang M, Sun C. The existence of global attractors for the norm-to-weak continuous semigroup and
application to the nonlinear reaction-diffusion equation. J Differ Equa, 2006, 223(2): 367--399

[10] Zhong C K, Sun C, Niu M. On the existence of global attractor for a class of infinite dimensional nonlinear dissipative dynamical systems. Chinese Ann Math, 2005, 26B(3): 1--8

[11] Yang Ling'e, Guo Boling. Global attractor for Camassa-Holm type equations with dissipative term. Acta Math Sci, 2005, 25B(4): 621--628

[12] Shao Zhiqiang, Chen Shuxing. The mixed problem for a class of nonlinear symmetric hyperbolic systems with discontinuous data. Acta Math Sci, 2005, 25B(4): 610--620

[13] Li Kaitai, Xu Zhongfeng, Yang Xiaozhong. A new approximate inertial manifold and associated algorithm.
Acta Math Sci, 2006, 26B(1): 1--16

[14] Guo X L, Li K T. Asymptotic behavior of the drift-diffusion semiconductor equations. Acta Math Sci, 2004, 24B(3): 385--394

[15] Wang H Y, Li K T Numerical approximations of a semi-linear elliptic problem. Acta Math Sci, 2000, 20(2): 175--180

[16] He Yinnian, Li Kaitai. Oseen coupling method for the exterior flow part I: oseen coupling approximation. Acta Math Sin (Chinese Series), 2000, 43(6): 969--974

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE H^k

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Azer KHANMAMEDOV, Sema YAYLA. LONG-TIME DYNAMICS OF THE STRONGLY DAMPED SEMILINEAR PLATE EQUATION IN R^N [J]. Acta mathematica scientia,Series B, 2018, 38(3): 1025-1042.
[2]	Lei ZHANG, Bin LIU. THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM [J]. Acta mathematica scientia,Series B, 2018, 38(2): 651-672.
[3]	Jing CUI, Litan YAN. CONTROLLABILITY OF NEUTRAL STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION [J]. Acta mathematica scientia,Series B, 2017, 37(1): 108-118.
[4]	Abdul-Majeed AL-IZERI, Khalid LATRACH. WELL-POSEDNESS OF A NONLINEAR MODEL OF PROLIFERATING CELL POPULATIONS WITH INHERITED CYCLE LENGTH [J]. Acta mathematica scientia,Series B, 2016, 36(5): 1225-1244.
[5]	Mina JIANG, Jianlin XIANG. ASYMPTOTIC STABILITY OF TRAVELING WAVES FOR A DISSIPATIVE NONLINEAR EVOLUTION SYSTEM [J]. Acta mathematica scientia,Series B, 2015, 35(6): 1325-1338.
[6]	PENG Yan. BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS [J]. Acta mathematica scientia,Series B, 2014, 34(4): 1271-1286.
[7]	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.
[8]	ZHANG Fa-Yong, GUO Bai-Ling. ATTRACTORS FOR FULLY DISCRETE FINITE DIFFERENCE SCHEME OF DISSIPATIVE ZAKHAROV EQUATIONS [J]. Acta mathematica scientia,Series B, 2012, 32(6): 2431-2452.
[9]	Tujin Kim, CHANG Qian-Shun, XU Jing. NOTE ON AN OPEN PROBLEM OF HIGHER ORDER NONLINEAR EVOLUTION EQUATIONS [J]. Acta mathematica scientia,Series B, 2012, 32(6): 2369-2376.
[10]	LUO Hong, PU Zhi-Lin. EXISTENCE AND REGULARITY OF SOLUTIONS TO MODEL FOR LIQUID MIXTURE OF ³HE-⁴HE [J]. Acta mathematica scientia,Series B, 2012, 32(6): 2161-2175.
[11]	CHEN Xue-Yong, LIU Wei-An. GLOBAL ATTRACTOR FOR A DENSITY-DEPENDENT SENSITIVITY CHEMOTAXIS MODEL [J]. Acta mathematica scientia,Series B, 2012, 32(4): 1365-1375.
[12]	YANG Ling-E, GUO Bai-Ling. GLOBAL ATTRACTOR FOR CAMASSA-HOLM TYPE EQUATIONS WITH DISSIPATIVE TERM [J]. Acta mathematica scientia,Series B, 2005, 25(4): 621-628.
[13]	CHEN Zhong-Hua, ZHANG Chu-Jing, CHEN Lan-Sun. ANALYSIS OF AN SI EPIDEMIC MODEL WITH NONLINEAR TRANSMISSION AND STAGE STRUCTURE 1 [J]. Acta mathematica scientia,Series B, 2003, 23(4): 440-.
[14]	HAO Bai-Lin, LIN Guo-An. A REMARK ON THE ATTRACTORS OF NON-NEWTONIAN FLUIDS [J]. Acta mathematica scientia,Series B, 2003, 23(3): 316-.
[15]	LIU Zhen-Hai, Ivan Szanto. MULTIVALUED DIFFERENTIAL EQUATIONS IN BANACH SPACES AND THEIR APPLICATIONS [J]. Acta mathematica scientia,Series B, 2002, 22(2): 213-221.