具有时变时滞和速度相关材料密度的非线性粘弹性方程的整体存在性和一般衰减性

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1684-1704.

具有时变时滞和速度相关材料密度的非线性粘弹性方程的整体存在性和一般衰减性

张再云^1,*(),刘振海²,邓又军³

¹ 湖南理工学院数学学院湖南岳阳 414006
² 广西民族大学理学院南宁 530006
³ 中南大学数学与统计学院长沙 410083

收稿日期:2020-08-06 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 张再云 E-mail:1226@126.com
基金资助:
国家自然科学基金(12071413);国家自然科学基金(11971487);the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie(823731CONMECH);长沙理工大学数学模型与分析重点实验室项目(2018MMAEZD05);湖南省教育厅科学研究项目(18A325);广西自科基金(2018GXNSFDA138002);湖南自科基金(2021JJ30297);湖南自科基金(2020JJ2038);湖南理工学院科研创新团队项目(2019-TD-15);海南省计算与应用重点实验室开放基金(JSKX201905)

Global Existence and General Decay for a Nonlinear Viscoelastic Equation with Time-Varying Delay and Velocity-Dependent Material Density

Zaiyun Zhang^1,*(),Zhenhai Liu²,Youjun Deng³

¹ School of Mathematics, Hunan Institute of Science and Technology, Hunan Yueyang 414006
² College of Science, Guangxi University for Nationalities, Nanning 530006
³ School of Mathematics and Statistics, Central South University, Changsha 410083

Received:2020-08-06 Online:2021-12-26 Published:2021-12-02
Contact: Zaiyun Zhang E-mail:1226@126.com
Supported by:
the NSFC(12071413);the NSFC(11971487);the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie(823731CONMECH);the Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering of Changsha University of Science and Technology Grant(2018MMAEZD05);the Scientific Research Fund of Hunan Provincial Education Department(18A325);the NSF of Guangxi(2018GXNSFDA138002);the NSF of Hunan Province(2021JJ30297);the NSF of Hunan Province(2020JJ2038);the Research and Innovation Team of Hunan Institute of Science and Technology(2019-TD-15);the Open Project of Hainan Key Laboratory of Computing Science and Application(JSKX201905)

摘要/Abstract

摘要：

研究了一类具有时变时滞效应和速度相关材料密度的非线性粘弹性方程.在适当的松弛函数和时变时滞效应假设下，分别用Faedo-Galerkin方法和摄动能量方法证明了弱解的整体存在性和能量的一般衰减性.这一结果改进了早期文献[1, 48-50]中的结果.

关键词: 整体存在性, 非线性粘弹性方程, 一般衰减性, 时变时滞, 速度相关材料密度, 摄动能量法

Abstract:

In this paper, we investigate a nonlinear viscoelastic equation with a time-varying delay effect and velocity-dependent material density. Under suitable assumptions on the relaxation function and time-varying delay effect, we prove the global existence of weak solutions and general decay of the energy by using Faedo-Galerkin method and the perturbed energy method respectively. This result improves earlier ones in the literature, such as Refs.[1, 48-50].

Key words: Global existence, Nonlinear viscoelastic equation, General decay, Time-varying delay, Velocity-dependent material density, Perturbed energy method

中图分类号:

O175.2

张再云,刘振海,邓又军. 具有时变时滞和速度相关材料密度的非线性粘弹性方程的整体存在性和一般衰减性[J]. 数学物理学报, 2021, 41(6): 1684-1704.

Zaiyun Zhang,Zhenhai Liu,Youjun Deng. Global Existence and General Decay for a Nonlinear Viscoelastic Equation with Time-Varying Delay and Velocity-Dependent Material Density[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1684-1704.

参考文献 54

1	Kirane M , Said-Houari B . Existence and aymptotic stability of a viscoelastic wave equation with a delay. Z Angew Math Phys, 2011, 62, 1065- 1082 doi: 10.1007/s00033-011-0145-0
2	Cavalcanti M M , Domingos Cavalcanti V , Ferreira J . Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math Methods Appl Sci, 2001, 24, 1043- 1053 doi: 10.1002/mma.250
3	Zhang Z Y , Liu Z H , Gan X Y . Global existence and general decay for a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density. Appl Anal, 2013, 92, 2021- 2048 doi: 10.1080/00036811.2012.716509
4	Zhang Z Y , Miao X J . Global existence and uniform decay for wave equation with dissipative term and boundary damping. Computers & Mathematics with Applications, 2010, 59, 1003- 1018
5	Zhang Z Y , Liu Z H , Miao X J , Chen Y Z . Global existence and uniform stabilization of a generalized dissipative Klein-Gordon equation type with boundary damping. J Math Phys, 2011, 52, 023502 doi: 10.1063/1.3544046
6	Zhang Z Y , Liu Z H , Miao X J . Estimate on the dimension of global attractor for nonlinear dissipative Kirchhoff equation. Acta Appl Math, 2010, 110, 271- 282 doi: 10.1007/s10440-008-9405-1
7	Cavalcanti M M , Domingos Cavalcanti V , Fukuoka R , Soriano J A . Uniform stabilization of the wave equation on compact manifolds and locally distributed damping-a sharp result. J Math Anal Appl, 2009, 351, 661- 674 doi: 10.1016/j.jmaa.2008.11.008
8	Cavalcanti M M , Domingos Cavalcanti V , Fukuoka R , Soriano J A . Asymptotic stability of the wave equation on compact surfaces and locally distributed damping: a sharp result. Arch Ration Mech Anal, 2010, 197, 925- 964 doi: 10.1007/s00205-009-0284-z
9	Cavalcanti M M , Domingos Cavalcanti V , Soriano J A . Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Elec J Diff Equa, 2002, 44, 1- 14
10	Berrimi S , Messaoudi S A . Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Elec J Diff Equa, 2004, 88, 1- 10
11	Cavalcanti M M , Oquendo H P . Frictional verus viscoelastic damping in a semilinear wave equation. SIAM J Control Optim, 2003, 42, 1310- 1324 doi: 10.1137/S0363012902408010
12	Zhang Z Y , Liu Z H , Miao X J , Chen Y Z . A note on decay properties for the solutions of a class of partial differential equation with memory. J Appl Math Comput, 2011, 37, 85- 102 doi: 10.1007/s12190-010-0422-7
13	Liu W J . General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms. J Math Phys, 2009, 50, 113506 doi: 10.1063/1.3254323
14	Liu W J . Uniform decay of solutions for a quasilinear system of viscoelastic equations. Nonlinear Anal TMA, 2009, 71, 2257- 2267 doi: 10.1016/j.na.2009.01.060
15	Liu W J . General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal TMA, 2010, 73, 1890- 1904 doi: 10.1016/j.na.2010.05.023
16	Liu W J . General decay of solutions to a viscoelastic equation with nonlinear localized damping. Ann Acad Sci Fenn Math, 2009, 34, 291- 302
17	Liu W J . General decay of solutions of a nonlinear system of viscoelastic equations. Acta Appl Math, 2010, 110, 53- 165
18	Liu W J . Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping. J Appl Math Comput, 2010, 32, 59- 68 doi: 10.1007/s12190-009-0232-y
19	Han X S , Wang M X . Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Anal TMA, 2009, 70, 3090- 3098 doi: 10.1016/j.na.2008.04.011
20	Han X S , Wang M X . General decay of energy for a viscoelastic equation with nonlinear damping. Journal of the Franklin Institute, 2010, 347, 806- 817 doi: 10.1016/j.jfranklin.2010.02.010
21	Han X S , Wang M X . Global existence and asymptotic behavior for a couple hyperbolic system with localized damping. Nonlinear Anal TMA, 2010, 72, 965- 986 doi: 10.1016/j.na.2009.07.032
22	Han X S , Wang M X . Energy decay rate for a couple hyperbolic system with nonlinear damping. Nonlinear Anal TMA, 2009, 70, 3264- 3272 doi: 10.1016/j.na.2008.04.029
23	Messaoudi S A , Tatar N E . Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math Methods Appl Sci, 2007, 30, 665- 680 doi: 10.1002/mma.804
24	Cavalcanti M M , Domingos Cavalcanti V , Fukuoka R , et al. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differential and Integral Equations, 2001, 14, 85- 116
25	Messaoudi S A , Tatar N E . Exponential or polynomial decay for a quasilinear viscoelastic equation. Nonlinear Anal, 2008, 68, 785- 793 doi: 10.1016/j.na.2006.11.036
26	Tatar N E . Exponential decay for a viscoelastic problem with a singular kenerl. Z Angew Math Phys, 2009, 60, 640- 650 doi: 10.1007/s00033-008-8030-1
27	Berrimi S , Messaoudi S A . Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal, 2006, 64, 2314- 2331 doi: 10.1016/j.na.2005.08.015
28	Messaoudi S A . General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal, 2008, 69, 2589- 2598 doi: 10.1016/j.na.2007.08.035
29	Han X S , Wang M X . General decay of energy for a viscoelastic equation with nonlinear damping. Math Methods Appl Sci, 2009, 32, 346- 358 doi: 10.1002/mma.1041
30	Liu W J . Asmptotic behavior of solutions of the time-delayed Burger's equation. Discrete Contin Dyn Syst Ser B, 2002, 2, 47- 56
31	Shang Y F , Xu G Q , Chen Y L . Stability analysis of Euler-Bernoulli beam with input delay in the boundary control. Asian J Control, 2012, 14, 186- 196 doi: 10.1002/asjc.279
32	Datko R . Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J Control Optim, 1988, 26, 697- 713 doi: 10.1137/0326040
33	Datko R , Lagnese J , Polis M P . An example on the effect of time delays in boundary feedback stabilization of wave equation. SIAM J Control Optim, 1986, 24, 152- 156 doi: 10.1137/0324007
34	Nicaise S , Pignotti C . Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J Control Optim, 2006, 45, 1561- 1585 doi: 10.1137/060648891
35	Nicaise S , Pignotti C . Stabilization of the wave equation with boundary or internal distributed delay. Differential and Integral Equations, 2008, 21, 935- 958
36	Xu G Q , Yung S P , Li L K . Stabilization of the wave system with input delay in the boundary control. ESIAM: Control Optim Calc Var, 2006, 12, 770- 785 doi: 10.1051/cocv:2006021
37	Ait Benhassi E M , Ammari K , Boulite S , Maniar L . Feedback stabilization of a class of evolution equation with delay. J Evol Equa, 2009, 9, 103- 121 doi: 10.1007/s00028-009-0004-z
38	Nicaise S , Pignotti C , Valein J . Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin Dyn Syst Ser S, 2011, 4, 693- 722
39	Caraballo T , Real J , Shaiklet L . Method of Lyapunov functionals construction in stability of delay evolution equations. J Math Anal Appl, 2007, 334, 1130- 1145 doi: 10.1016/j.jmaa.2007.01.038
40	Zhang Z Y , Liu Z H , Miao X J , Chen Y Z . Stability analysis of heat flow with boundary time-varying delay effect. Nonlinear Anal, 2010, 73, 1878- 1889 doi: 10.1016/j.na.2010.05.022
41	Nicaise S , Pignotti C . Interior feedback stabilization of wave equations with time dependent delay. Elec J Diff Equa, 2011, 41, 1- 20
42	Nicaise S , Pignotti C , Fridman E . Stability of the heat and the wave equations with boundary time-varying delays. Disc Conti Dyna Syst, 2009, 2, 559- 581
43	Nicaise S , Valein J . Stabilization of the wave equation on 1-d networks with a delay term in thenodal feedbacks. Networks & Heterogeneous Media, 2007, 2, 425- 479
44	Fridman E , Nicaise S , Valein J . Stabilization of seconder order evolution equations with inbounded feedback with time-dependent delay. SIAM J Control Optim, 2010, 48, 5028- 5052 doi: 10.1137/090762105
45	Nicaise S , Valein J . Stabilization of seconder order evolution equations with inbounded feedback with delay. ESIAM: Control Optim Calc Var, 2010, 16, 420- 456 doi: 10.1051/cocv/2009007
46	Lions J L . Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires. Paris: Dunod, 1969
47	Zheng S M . Nonlinear Evolution Equations. Boca Raton: CRC Press, 2004
48	Liu W J . General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback. J Math Phys, 2013, 54, 043504 doi: 10.1063/1.4799929
49	Dai Q Y , Yang Z F . Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay. Z Angew Math Phys, 2014, 65, 885- 903 doi: 10.1007/s00033-013-0365-6
50	Zhang Z Y, Huang J H, Liu Z H, Sun M B. Boundary stabilization of a nonlinear viscoelastic equation with interior time-varying delay and nonlinear dissipative boundary feedback. Abstract and Applied Analysis, 2014, Article ID: 102594
51	Cavalcanti M M , Domingos Cavalcanti V , Lasiecka I , Webler C M . Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density. Adv Nonlinear Anal, 2017, 6, 121- 145 doi: 10.1515/anona-2016-0027
52	Messaoudi S A , Said-Houari B . Energy decay in a transmission problem in thermoelasticity of type Ⅲ. IMA J Appl Math, 2009, 74, 344- 360 doi: 10.1093/imamat/hxp020
53	Messaoudi S A , Bonfoh A , Mukiawa S E , Enyi C D . The global attractor for a suspension bridge with memory and partially hinged boundary conditions. Z Angew Math Mech, 2017, 97, 159- 172 doi: 10.1002/zamm.201600034
54	Yang Z F . Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay. Z Angew Math Phys, 2015, 66, 727- 745 doi: 10.1007/s00033-014-0429-2

[1]	贾哲,杨作东. 带非线性扩散项和信号产生项的趋化-趋触模型解的整体有界性[J]. 数学物理学报, 2021, 41(5): 1382-1395.
[2]	王娟,原子霞. 具有对数敏感度和混合边界的一维趋化模型解的整体存在性和收敛性[J]. 数学物理学报, 2020, 40(6): 1646-1669.
[3]	邱华, 张颖姝, 房少梅. 不可压粘弹性流体的Leray-α-Oldroyd模型整体解的存在性[J]. 数学物理学报, 2017, 37(1): 102-112.
[4]	张宏伟, 刘功伟, 呼青英. 一类具对数源项波动方程的初边值问题[J]. 数学物理学报, 2016, 36(6): 1124-1136.
[5]	刘洋. 位势井及其对具临界初始能量的非牛顿渗流方程的应用[J]. 数学物理学报, 2016, 36(6): 1211-1220.
[6]	黄祖达. 一类具反馈控制的时滞飞蝇方程的伪概周期解[J]. 数学物理学报, 2016, 36(3): 558-568.
[7]	狄华斐, 尚亚东. 一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性[J]. 数学物理学报, 2015, 35(3): 618-633.
[8]	沈继红, 张明有, 杨延冰, 刘博为, 徐润章. 具阻尼的高维广义Boussinesq 方程的Cauchy 问题的整体适定性[J]. 数学物理学报, 2014, 34(5): 1173-1187.
[9]	米永生, 穆春来, 吴云龙. 具有多重非线性的非牛顿多方渗流方程的整体存在性与非存在性[J]. 数学物理学报, 2014, 34(3): 626-637.
[10]	凌征球, 周泽文. 非局部边界条件的退化抛物方程组解的爆破和整体存在性[J]. 数学物理学报, 2014, 34(2): 338-346.
[11]	曲风龙, 王玉泉. 一类带有趋化性扩散的细菌模型一致有界解的整体存在性[J]. 数学物理学报, 2014, 34(2): 409-418.
[12]	毛剑峰, 黎野平. 三维双极Euler-Poisson方程初值问题光滑解的整体存在性和渐近性[J]. 数学物理学报, 2013, 33(3): 510-522.
[13]	王术, 冯跃红, 李新. 双极可压缩等熵Euler-Maxwell系统光滑周期解的整体存在性[J]. 数学物理学报, 2012, 32(6): 1041-1049.
[14]	凌征球, 卢伟明, 周泽文. 带非局部边界条件的半线性抛物系统的一致爆破模式[J]. 数学物理学报, 2012, 32(2): 433-440.
[15]	李俊民. 非线性参数化时变时滞系统自适应迭代学习控制[J]. 数学物理学报, 2011, 31(3): 682-690.