一类具有局部记忆阻尼的弱耦合系统的能量衰减估计

数学物理学报 ›› 2015, Vol. 35 ›› Issue (1): 194-209.

一类具有局部记忆阻尼的弱耦合系统的能量衰减估计

章春国, 张海燕, 谷尚武

杭州电子科技大学数学系杭州 310018

收稿日期:2013-11-07 修回日期:2014-10-04 出版日期:2015-02-25 发布日期:2015-02-25
基金资助:
国家自然科学基金(61374096, 11271104)

Energy Decay Estimates for the Weakly Coupled Systems with Local Memory Damping

ZHANG Chun-Guo, ZHANG Hai-Yan, GU Shang-Wu

Department of Mathematics, Hangzhou Dianzi University, Hangzhou Zhejiang 310018

Received:2013-11-07 Revised:2014-10-04 Online:2015-02-25 Published:2015-02-25

摘要/Abstract

摘要：

研究具有局部记忆阻尼弱耦合梁-弦系统.首先在合适的假设条件下, 应用线性算子半群理论证明了系统的适定性;
进而运用线性算子半群的频域定理证明了具有局部记忆阻尼弱耦合梁-弦系统的能量是一致指数衰减的.

关键词: 耦合梁-弦系统 , 线性算子半群 , 局部记忆阻尼 , 一致指数衰减

Abstract:

This paper studies the weakly coupled beam-string systems with local memory damping. First, under the appropriate hypothesis, we proved that the well-posedness of the system by using the theory of linear operator semigroup. And then, we show that the energy of the weakly coupled beam-string system with local memory damping is uniform exponential decay by applying the frequence domain
result on Hilbert space.

Key words: Coupled beam-string system , Linear operator semigroup , Local memory damping , Uniform exponential decay

中图分类号:

35B37

章春国, 张海燕, 谷尚武. 一类具有局部记忆阻尼的弱耦合系统的能量衰减估计[J]. 数学物理学报, 2015, 35(1): 194-209.

ZHANG Chun-Guo, ZHANG Hai-Yan, GU Shang-Wu. Energy Decay Estimates for the Weakly Coupled Systems with Local Memory Damping[J]. Acta mathematica scientia,Series A, 2015, 35(1): 194-209.

参考文献

[1] Aissa Guesmia. Energy decay for a damped nonlinear coupled system. Journal of Mathematical Analysis and Applications,
1999, 239: 38--48
[2] Han X S, Wang M X. Energy decay rate for a coupled hyperbolic system with nonlinear damping.  Nonlinear Analysis, 2009, 70: 3264--3272
[3] Aissa Guesmia, Salim A Messaoudi. General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math Meth Appl Sci, 2009, 32: 2102--2122
[4] Han X S, Wang M X. General decay of energy for a viscoelastic equation with nonlinear damping. Math Meth Appl Sci, 2009, 32: 346--358
[5] Liu K S, Liu Z Y. On the type of $C_{0}$-semigroup associated with the abstract linear viscoelastic system. Z Angew Math Phys, 1996, 47(2): 1--15
[6] Liu K S, Liu Z Y. Exponential decay of energy of vibrating strings with local viscoelasticity. Z Angew Math Phys, 2002, 53(2): 265--280
[7] Rivera J E M, Salverra A P. Asymptotic behavior of energy in partially viscoelastic material. Quart Appl Math, 2001, 1(3): 557--578
[8] Mauro de Lima Santos. Decay rates for solutions of a system of wave equations with memory. Electronic Journal of Differential Equations, 2002, 38(1): 1--17
[9] 章春国. 具有局部记忆阻尼的非均质Timoshenko梁的稳定性. 数学物理学报, 2012, 32(1): 186--200
[10] Jong Yeoul Park, Sun Hye Park. General decay for a quasilinear system of   viscoelastic equations with nonlinear damping. Acta Mathematica Scientia, 2012, 32B(4): 1321--1332
[11] 章春国, 谷尚武, 姜敬华. 具有Boltzmann阻尼Petrovsky系统的稳定性.  系统科学与数学, 2013, 33(7): 807--817
[12] Pazy A. Semigroups of Linear Operators and Applications to Partical Differential Euqations. New York: Springer-Verlag, 1983
[13] Huang F L. Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann of Diff Eqs, 1985, 1(1): 43--56
[14] Adams R A. Sobolev Spaces. New York: Acadamic Press, 1975

[15] Chen S P, Liu K S, Liu Z Y. Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping. SIAM J Appl Math, 1998, 2(59): 651--668
[16] Hartman P. Ordinary Differential Equations (2nd ed). Boston, Basel, Stuttgart: Birkh\"{a}user, 1982
[17] Hille E, Hillips R S. Functional Analysis and Semi-Groups. American Mathematical Society Colloquium Publications. 1957, 31: 141--150