JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (6): 22-29.doi: 10.6040/j.issn.1671-9352.0.2020.359

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Time-dependent attractors of the wave equations with strong damping on Rn

WU Xiao-xia1,2, MA Qiao-zhen1*   

  1. 1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China;
    2. College of Information Engineering, Tarim University, Alaer 843300, Xinjiang, China
  • Published:2021-06-03

Abstract: Based on the tail estimation technique and condition (Ct), the asymptotic compactness of the process related to the problem is obtained, and the difficulty of Sobolev embedding noncompactness and Poincaré inequality is not established in the entire space is overcome, thus the existence of time-dependent attractors for the wave equations with strong damping and decay coefficients over unbounded domains is proved.

Key words: wave equations, strong damping, condition (Ct), unbounded domain, time-dependent attractors

CLC Number: 

  • O175.27
[1] 复旦大学数学系.数学物理方程[M].上海:上海科学技术出版社,1961. Department of Mathematics, Fudan University. Mathematical physics equation[M]. Shanghai: Shanghai Science and Technology Press, 1961.
[2] FEIREISL E. Global attractors for damped wave equations with supercritical exponent[J]. Journal of Differential Equations, 1995, 116(5):431-447.
[3] MENG Fengjuan. Strong global attractor for 3D wave equations with weakiy damping[J]. Abstract and Applied Analysis, 2012, 1155(10):1-12.
[4] WANG Yonghai. Pullback attractors for non-autonomous wave equations with critical exponent[J]. Nonlinear Analysis, 2008, 68:365-376.
[5] YANG Meihua, SUN Chunyou. Attractors for strongly damped wave equations[J]. Nonlinear Analysis: Real World Applications, 2009, 10(2):1097-1100.
[6] ZELIK S V. Asymptotic regularity of solutions of a non-autonomous damped wave equations with a critical growth exponent[J]. Communications on Pure and Applied Analysis, 2004, 3(4):921-934.
[7] CONTI M, PATA V, TEMAM R. Attractors for processes on time-dependent spaces: applications to wave equation[J]. Journal of Differential Equations, 2013, 255(6):1254-1277.
[8] PLINIO D F, DUANE G S, TEMAM R. Time-dependent attractor for the oscillon equation[J]. Discrete and Continuous Dynamical Systems, 2011, 29:141-167.
[9] CONTI M, PATA V. Asymptotic structure of the attractors for processes on time-dependent spaces[J]. Nonlinear Analysis: Real World Applications, 2014, 19:1-10.
[10] CONTI M, PATA V. On the time-dependent Cattaneo law in space dimension one[J]. Applied Mathematical Computers, 2015, 259:32-44.
[11] MENG Fengjuan, LIU Cuncai. Necessary and sufficient conditions for the existence of time-dependent global attractor and application[J]. Journal of Mathematical Physics, 2017, 58(3):1-9.
[12] MENG Fengjuan, YANG Meihua, ZHONG Chenkui. Attractors for wave equations with nonlinear damping on time-dependent space[J]. Discrete and Continuous Dynamical Systems, 2016, 25(1):205-225.
[13] BELLERI V, PATA V. Attractors for semilinear strongly damped wave equations on R3[J]. Discrete and Continuous Dynamical Systems, 2001 7(4):719-735.
[14] KARACHALIOS N I, STAVRAKAKIS N M. Existence of a global attractors for semilinear dissipative wave equation on Rn[J]. Journal of Differential Equations, 1999, 157(80):183-205.
[15] WANG Bixiang. Asymptotic behavior of stochastic wave equations critical exponents on Rn[J]. Transactions of American Mathematical Society, 2011, 363(7):3639-3663.
[16] YOU Yuncheng. Global attractors for nonlinear wave equations with critical exponent on unbounded domain[J]. American Mathematical and Nonlinear Science, 2016, 1(2):581-602.
[17] YIN Jinyan, LI Yangrong, GU Anhui. Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain[J]. Computers and Mathematical Applications, 2017, 74(4):744-758.
[18] LIU Tingting, MA Qiaozhen. Time-dependent attractors for plate equations on Rn[J]. Journal of Mathematical Analysis and Applications, 2019, 479:315-332.
[19] DEIMLING K. Nonlinear functional analysis[M]. Berlin: Springer-Verlag, 1985.
[20] PATA V. Attractors for a damped wave equation on Rn with linear memory[J]. Mathematical Methods and Applied Science, 2000, 23(7):633-653.
[21] YANG Zhijian, DING Pengyan. Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on Rn[J]. Journal of Mathematical Analysis and Applications, 2016, 434:1826-1851.
[22] BABIN A V, VISHHIK M I. Attractors of evolution equations[M]. Amsterdam: North-Holland, 1992.
[23] ROBINSON J C. Infinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors[M]. London: Cambridge University Press, 2001.
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