具有时间依赖系数的波方程拉回吸引子的存在性

doi:10.6040/j.issn.1671-9352.0.2024.255

摘要/Abstract

摘要： 研究具有时间依赖系数的非线性波方程。首先,根据一致扇形算子理论证明解的适定性。其次,通过构造恰当的泛函获得过程的有界耗散性。最后,利用压缩函数的方法证明过程的渐近紧性,从而得到其拉回吸引子的存在性。在问题的研究中,由于将时间依赖系数分解为正部与负部,因此在证明过程的渐近紧性时,无法应用算子分解技巧,从而选择了压缩函数的方法。

关键词: 波方程, 适定性, 拉回吸引子, 时间依赖系数, 耗散性, 紧性

Abstract: This paper is concerned with the nonlinear wave equations with time-dependent coefficients. Firstly, the well-posedness of the solution is proved based on the theory of the uniform sector operator. Secondly, the bounded dissipation of the process is obtained by constructing appropriate functionals. Finally, the asymptotic compactness of the process is proved by using the method of the compression function. In the research of the problem, since the time-dependent coefficients α₁ are decomposed into positive and negative parts, the operator decomposition technique cannot be applied when proving the asymptotic compactness of the process. Therefore, the method of compressing the function is chosen.

Key words: contractive function, well-posedness, pullback attractors, time-dependent coefficients, dissipation, compactness

中图分类号:

O175

翟昭,张平,马巧珍. 具有时间依赖系数的波方程拉回吸引子的存在性[J]. 《山东大学学报(理学版)》, 2026, 61(2): 75-87.

ZHAI Zhao, ZHANG Ping, MA Qiaozhen. Existence of pullback attactors for wave equations with time dependent coefficients[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2026, 61(2): 75-87.

参考文献

[1] BABIN A V, VISHIK M I. Attractors of evolution equations[M] //Springer Monographs in Mathematics. New York: Springer, 1992:389-401.
[2] ARRIETA J, CARVALHO A N, HALE J K. A damped hyerbolic equation with critical exponent[J]. Communications in Partial Differential Equations, 1992, 17(5/6):841-866.
[3] BALL J. Global attractors for damped semilinear wave equations[J]. Discrete and Continuous Dynamical Systems, 2004, 10(1/2):31-52.
[4] TEMAM R. Infinite-dimensional dynamical systems in mechanics and physics[M]. New York: Springer, 1997:68
[5] CARABALLO T, CARVALHO A N, LANGA J A, et al. A non-autonomous strongly damped wave equation: existence and continuity of the pullback attractor[J]. Nonlinear Analysis: Theory, Methods & Applications, 2011, 74(6):2272-2283.
[6] 吴晓霞,马巧珍. 带有线性记忆的波方程在Rⁿ上的时间依赖吸引子[J]. 应用数学,2021,34(1):73-85. WU Xiaoxia, MA Qiaozhen. Time-dependent attractors of wave equations with linear memory on Rⁿ[J]. Mathematica Applicata, 2021, 34(1):73-85.
[7] CARVALHO A N, LANGA J A, ROBINSON J C. Attractors for infinite-dimensional non-autonomous dynamical systems[M]. New York: Springer, 2013.
[8] PATA V, ZELIK S. A remark on the damped wave equation[J]. Communications on Pure & Applied Analysis, 2006, 5(3):611-616.
[9] LI D D, SUN C Y, CHANG Q Q. Global attractor for degenerate damped hyperbolic equations[J]. Journal of Mathematical Analysis and Applications, 2017, 453(1):1-19.
[10] LI D D, CHANG Q Q, SUN C Y. Pullback attractors for a critical degenerate wave equation with time-dependent damping[J]. Nonlinear Analysis: Real World Applications, 2022, 63:103421.
[11] ARAGÃO G S, BEZERRA F D M, FIGUEROA-LÓPEZ R N, et al. Continuity of pullback attractors for evolution processes associated with semilinear damped wave equations with time-dependent coefficients[J]. Journal of Differential Equations, 2021, 298:30-67.
[12] UESAKA H. A pointwise oscillation property of semilinear wave equations with time-dependent coefficients II[J]. Nonlinear Analysis: Theory, Methods & Applications, 2001, 47(4):2563-2571.
[13] ISHIDA H, YUZAWA Y. Oscillatory properties for semilinear degenerate hyperbolic equations of second order[J]. Journal of Mathematical Analysis and Applications, 2009, 356(2):624-632.
[14] GRASSELLI M, PATA V. On the damped semilinear wave equation with critical exponent[J]. Conference Publications, 2003:351-358.
[15] EBERT M R, GIRARDI G, REISSIG M. Critical regularity of nonlinearities in semilinear classical damped wave equations[J]. Mathematische Annalen, 2020, 378(3):1311-1326.
[16] CARABALLO T, LANGA J A, RIVERO F, et al. A gradient-like nonautonomous evolution process[J]. International Journal of Bifurcation and Chaos, 2010, 20(9):2751-2760.
[17] MA T F, MARÍN-RUBIO P, SURCO CHUÑO C M. Dynamics of wave equations with moving boundary[J]. Journal of Differential Equations, 2017, 262(5):3317-3342.
[18] HAN J B, WANG K Y, XU R Z, et al. Global quantitative stability of wave equations with strong and weak dampings[J]. Journal of Differential Equations, 2024, 390:228-344.
[19] YAN S L, ZHU X M, ZHONG C K, et al. Long-time dynamics of the wave equation with nonlocal weak damping and super-cubic nonlinearity in 3D domains, part II: nonautonomous case[J]. Applied Mathematics & Optimization, 2023, 88(3):69.
[20] 吕鹏辉,余莎莎,林国广. 具变系数和弱阻尼的非局部高阶波方程的长时间动力学行为[J]. 集美大学学报(自然科学版),2023,28(5):407-414. LÜ Penghui, YU Shasha, LIN Guoguang. Long-time dynamic behavior of nonlocal higher-order wave equations with variable coefficients and weak damping[J]. Journal of Jimei University(Natural Science), 2023, 28(5):407-414.
[21] ZHOU F, ZHU K X, XIE Y Q. Dynamics of the non-autonomous wave equations with nonlocal weak damping and critical nonlinearity[J]. Discrete and Continuous Dynamical Systems-B, 2025, 30(1):1-25.
[22] BEZERRA F D M, LIU L F, NARCISO V. Attractors for a class of wave equations with nonlocal structural energy damping[J]. Nonlinear Differential Equations and Applications NoDEA, 2024, 31(6):114.
[23] NOLASCO DE CARVALHO A, NASCIMENTO M J D. Singularly non-autonomous semilinear parabolic problems withcritical exponents[J]. Discrete & Continuous Dynamical Systems-S, 2009, 2(3):449-471.
[24] CHUESHOV I. Dynamics of quasi-stable dissipative systems[M]. New York: Springer, 2015.
[25] SUN C Y, CAO D M, DUAN J Q. Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity[J]. Nonlinearity, 2006, 19(11):2645-2665.

相关文章 15

[1]	桑彦彬. 一类带扰动的分数阶临界Choquard方程的正规解[J]. 《山东大学学报(理学版)》, 2024, 59(8): 48-55,66.
[2]	任倩,杨和. 一类Riemann-Liouville分数阶发展包含mild解的存在性[J]. 《山东大学学报(理学版)》, 2022, 57(4): 76-84.
[3]	孟旭东. 集合优化问题解集的稳定性和扩展适定性[J]. 《山东大学学报(理学版)》, 2022, 57(2): 98-110.
[4]	吴晓霞,马巧珍. 带有强阻尼的波方程在Rⁿ上的时间依赖吸引子[J]. 《山东大学学报(理学版)》, 2021, 56(6): 22-29.
[5]	杜亚利,汪璇. 带非线性阻尼的非自治Berger方程的时间依赖拉回吸引子[J]. 《山东大学学报(理学版)》, 2021, 56(6): 30-41.
[6]	戚斌,程浩. Neumann边界条件分数阶扩散波方程的源项辨识[J]. 《山东大学学报(理学版)》, 2021, 56(6): 64-73.
[7]	方刚,栾锡武,方建会. 弹性介质Hamilton正则方程与声波方程辛几何算法[J]. 山东大学学报（理学版）, 2016, 51(11): 99-106.
[8]	蒋晓宇. Bloch-type空间到Zygmund-type空间的微分复合算子[J]. J4, 2013, 48(4): 51-56.
[9]	陈翠. 加权型空间上的n阶微分复合算子[J]. J4, 2013, 48(4): 57-59.
[10]	常胜伟,刘瑞. 双连续n次积分C-半群与抽象Cauchy问题的C-适定性[J]. J4, 2012, 47(6): 107-110.
[11]	张国强. 加权Bergman空间里加权复合算子的紧差分[J]. J4, 2012, 47(12): 78-81.
[12]	张晓燕. Banach空间中非线性Volterra型积分方程整体解的存在性定理[J]. J4, 2010, 45(8): 57-61.
[13]	吕海玲,刘希强,牛磊. 一类推广的[G′/G]展开方法及其在非线性数学物理方程中的应用[J]. J4, 2010, 45(4): 100-105.
[14]	李耀红1,2,张海燕1. Banach空间中二阶非线性混合型积分-微分方程初值问题的解[J]. J4, 2010, 45(12): 93-97.
[15]	娄国久张兴秋王建国. Banach空间一阶非线性混合型积微分方程正解的存在性[J]. J4, 2009, 44(8): 74-79.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed