带有时滞和强阻尼的梁方程的时间依赖吸引子
The Time-Dependent Attractorsfor Beam Equation withTime Delay and StrongDamping
DOI: 10.12677/PM.2025.1510257, PDF,    国家自然科学基金支持
作者: 汪璇, 赵文佩*：西北师范大学，数学与统计学院，甘肃兰州
关键词: 时滞；强阻尼；时间依赖全局吸引子；梁方程；Time Delay； Strong Damping； Time-Dependent Global Attractors； Beam Equation
摘要: 本文研究了带有时滞和强阻尼的梁方程:∂t2 +△2u -γ△∂tu +f(u)=g(x, uθ)解的渐近性态。当非线性项满足最优增长指数1≤p*= N+2/N-4,N≥5时，应用Faedo-Galerkin逼近方法、能量估计和时间平移方法，得到了解的适定性。进一步应用收缩函数方法，验证了过程的渐近紧性，最后获得了时间依赖全局吸引子在时间依赖空间CH的存在性。
Abstract: In this article, We consider the asymptotic behavior of the solutions to the beam equation with time delay and strong damping: ∂t2 +△2u -γ△∂tu +f(u)=g(x, uθ). First of all, when the growth exponent of nonlinear terms satisfies the optimal growth exponent 1≤p*= N+2/N-4,N≥5, by applying Faedo-Galerkin approximation method, energy estimation and time translation method, we obtain the well-posedness of solutions; Then, using the contraction function method, the asymptotic compactness of the solution process is verified; Finally, the existence of time-dependent global attractor is obtained in the time-dependent space CH.
文章引用：汪璇, 赵文佩. 带有时滞和强阻尼的梁方程的时间依赖吸引子[J]. 理论数学, 2025, 15(10): 137-155. https://doi.org/10.12677/PM.2025.1510257

参考文献

[1] Babin, A.V. and Vishik, M.I. (1992) Attractors of Evolutionary Equations. North-Holland.
[2] Carvalho, A.N., Langa, J.A. and Robinson, J.C. (2013) Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems. Springer.
[3] Chueshov, I. and Lasiecka, I. (2008) Long-Time Behavior of Second Order Evolution Equa- tions with Nonlinear Damping. In: Memoirs of the American Mathematical Society, Vol. 195, American Mathematical Society.[CrossRef] 
[4] Chueshov, I. (2015) Dynamics of Quasi-Stable Dissipative Systems. Springer.
[5] Xu, J., Zhang, Z. and Caraballo, T. (2021) Non-Autonomous Nonlocal Partial Differential Equations with Delay and Memory. Journal of Differential Equations, 270, 505-546.[CrossRef] 
[6] Qin, Y. and Yang, B. (2023) Existence and Regularity of Pullback Attractors for a Non- Autonomous Diffusion Equation with Delay and Nonlocal Diffusion in Time-Dependent Spaces. Applied Mathematics & Optimization, 88, Article No. 10.[CrossRef] 
[7] Yang, Z. (2013) On an Extensible Beam Equation with Nonlinear Damping and Source Terms. Journal of Differential Equations, 254, 3903-3927. [Google Scholar] [CrossRef] 
[8] Zhao, C., Zhao, C. and Zhong, C. (2020) The Global Attractor for a Class of Extensible Beams with Nonlocal Weak Damping. Discrete & Continuous Dynamical Systems|B, 25, 935-955.[CrossRef] 
[9] Antonio Jorge da Silva, M. and Narciso, V. (2015) Attractors and Their Properties for a Class of Nonlocal Extensible Beams. Discrete & Continuous Dynamical Systems|A, 35, 985-1008.[CrossRef] 
[10] Ding, P. and Yang, Z. (2021) Longtime Behavior for an Extensible Beam Equation with Ro- tational Inertia and Structural Nonlinear Damping. Journal of Mathematical Analysis and Applications, 496, Article 124785.[CrossRef] 
[11] Conti, M. and Pata, V. (2014) Asymptotic Structure of the Attractor for Processes on Time- Dependent Spaces. Nonlinear Analysis: Real World Applications, 19, 1-10.[CrossRef] 
[12] Meng, F., Yang, M. and Zhong, C. (2015) Attractors for Wave Equations with Nonlinear Damping on Time-Dependent Space. Discrete and Continuous Dynamical Systems|Series B, 21, 205-225.[CrossRef] 
[13] Li, J. and Huang, J. (2009) Uniform Attractors for Non-Autonomous Parabolic Equations with Delays. Nonlinear Analysis: Theory, Methods & Applications, 71, 2194-2209.[CrossRef] 
[14] Ma, Q., Wang, X. and Xu, L. (2016) Existence and Regularity of Time-Dependent Global At- tractors for the Nonclassical Reaction-Diffusion Equations with Lower Forcing Term. Boundary Value Problems, 2016, Article No. 10.[CrossRef] 
[15] Harraga, H. and Yebdri, M. (2016) Pullback Attractors for a Class of Semilinear Nonclassical Diffusion Equations with Delay. Electronic Journal of Differential Equations, 2016, 1072-6691.
[16] Liu, T. and Ma, Q. (2019) Time-Dependent Attractor for Plate Equations on Rn. Journal of Mathematical Analysis and Applications, 479, 315-332. [Google Scholar] [CrossRef] 
[17] Garcia-Luengo, J. and Marin-Rubio, P. (2014) Reaction-Diffusion Equations with Non- Autonomous Force in H􀀀1 and Delays under Measurability Conditions on the Driving Delay Term. Journal of Mathematical Analysis and Applications, 417, 80-95.[CrossRef] 
[18] Meng, F., Wang, Y. and Zhao, C. (2021) Attractor for a Model of Extensible Beam with Damping on Time-Dependent Space. Topological Methods in Nonlinear Analysis, 57, 365-393.[CrossRef] 
[19] Yang, B., Qin, Y., Miranville, A. and Wang, K. (2025) Pullback Attractors for Nonclassical Diffusion Equations with a Delay Operator. Studies in Applied Mathematics, 154, e70039.[CrossRef] 

为你推荐