The existence of time-dependent attractors for abstract evolution equations with fading memory

doi:10.3969/j.issn.1000-5641.2018.01.005

Abstract

Abstract: In this paper, the long-time dynamical behavior of solutions for the abstract evolution equations with fading memory is investigated on time-dependent spaces. By applying the modified pull-back attractors theory, techniques of a priori estimate and operator decomposition, we verify the asymptotic compactness of the process. Furthermore, the existence and regularity of time-dependent global attractors are proved. This paper improves some known results.

Key words: abstract evolution equation, time-dependent attractor, fading memory, existence, regularity

CLC Number:

HU Di-di, WANG Xuan. The existence of time-dependent attractors for abstract evolution equations with fading memory[J]. Journal of East China Normal University(Natural Sc, 2018, 2018(1): 35-49.

References

[1] DI PLINIO F, DUANE G S, TEMAM R. Time-dependent attractor for the oscillation equation[J]. Discrete and Continuous Dynamical Systems, 2010, 29(1):141-167.
[2] CONTI M, PATA V, TEMAM R. Attractors for processes on time-dependent spaces and applications to wave equation[J]. Journal of Differential Equations, 2013, 255(255):1254-1277.
[3] CONTI M, PATA V. Asymptotic structure of the attractor for processes on time-dependent spaces[J]. Nonlinear Analysis:Real World Applications, 2014, 19(1):1-10.
[4] 刘亭亭, 马巧珍. Plate,方程时间依赖全局吸引子的存在性[J].华东师范大学学报,(自然科学版), 2016(2):35-44.
[5] MA Q Z, WANG X P, XU L. Existence and regularity of time-dependent global attractos for the nonclassical reaction-diffusion equations with lower forcing term[J]. Boundary Value Problems, 2016(1):1-11.
[6] COLEMAN B D, NOLL W. Foundations of linear viscoelasticity[J]. Reviews of Modern Physics, 1961, 33(2):239-249.
[7] DAFERMOS C M. Asymptotic stability in viscoelasticity[J]. Archive for Rational Mechanics and Analysis, 1970, 37(4):297-308.
[8] FABRIZIO M, MORRO A. Mathematical Problems in Linear Viscoelasticity[M]. Pennsylvania:Society for Industrial and Applied Mathematics, 1992.
[9] 张玉宝, 汪璇. 无阻尼弱耗散抽象发展方程的强全局吸引子[J]. 华东师范大学学报,(自然科学版), 2017(2):8-19.
[10] WANG X, YANG L, ZHONG C K. Attractors for the nonclassical diffusion equations with fading memory[J]. Journal of Mathematical Analysis and Applications, 2010, 362(2):327-337.
[11] CONTI M, PATA V. Weakly dissipative semilinear equations of viscoelasticity[J]. Communications on Pure and Applied Analysis, 2005, 4(4):705-720.
[12] PATA V, ZUCCHI A. Attractors for a damped hyperbolic equation with linear memory[J]. Advances in Mathematical Sciences and Applications, 2001, 11(2):505-529.
[13] BORINI S, PATA V. Uniform attractors for a strongly damped wave equations with linear memory[J]. Asymptotic Analysis, 1999, 20(3):263-277.
[14] GATTI S, MIRANVILLE A, PATA V, et al. Attractors for semilinear equations of viscoelasticity with very low dissipation[J]. Rocky Mountain Journal of Mathematics, 2008, 38(4):1117-1138.
[15] 汪璇, 段奋霞. 记忆型抽象发展方程全局吸引子的存在性[J]. 兰州大学学报,(自然科学版), 2016, 52(4):523-529.