具记忆和变时滞项的二阶发展方程能量衰减率的凸性估计

摘要/Abstract

摘要： 研究了如下一类具有记忆和变时滞项的抽象发展方程#br#u_tt（t）+Au（t）-∫₀^t g（t-s）Au（s）ds+μ₁h₁（u_t（t））+μ₂h₂（u_t（x，t-τ（t）））=▽F（u（t））.#br#通过构造合适的能量泛函和Lyapunov泛函，利用凸函数的一些性质，得到了依赖于h₁及记忆核g的能量衰减估计.此衰减估计可以应用于一些具体的模型.

关键词: 二阶发展方程, 能量估计, 时滞, 记忆核

Abstract: In this paper, we consider the following abstract evolution equation with memory and time-varying delay of the form#br#u_tt(t) + Au(t)-∫₀^t g(t-s)Au(s)ds + μ₁h₁(u_t(t)) + μ₂h₂(u_t(x, t-τ(t)))=▽F(u(t)).#br#By introducing suitable energy and Lyapunov functionals, and making use of some properties of the convex functions, we establish decay estimate for the energy, which depends on the behavior of h₁ and the relaxation g. The decay estimate can be applied to various concrete models. We shall also give some applications to illustrate our result.

Key words: Second order evolution, Energy decay, Delay, Memory

中图分类号:

O175.2

刘功伟, 刁林. 具记忆和变时滞项的二阶发展方程能量衰减率的凸性估计[J]. 数学物理学报, 2018, 38(3): 527-542.

Liu Gongwei, Diao Lin. On Convexity for Energy Decay Rates of the Second-Order Evolution Equation with Memory and Time-Varying Delay[J]. Acta mathematica scientia,Series A, 2018, 38(3): 527-542.

参考文献

[1] Alabau-Boussouira F, Cannarsa P. Sforza D. Decay estimates for second order evolution equations with memory. J Funct Anal, 2008, 254(5):1342-1372
[2] Ammari K, Nicaise S, Pignotti C. Stability of an abstract-wave equation with delay and a Kelvin-Voigt damping. Asymptot Anal, 2015, 95:21-38
[3] Arnold V L. Mathematical Methods of Classical Mechanics. New York:Springer, 1989
[4] Berrimi S, Messaoudi S A. Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal, 2006, 64(10):2314-2331
[5] Benaissa A, Benaissa A, Messaoudi S A. Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks. J Math Phys, 2012, 53(12):123514
[6] Cavalcanti M M, Domingos Cavalcanti V N, Soriano J A. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron J Differ Eq, 2002, 2002(44):1-14
[7] Cavalcanti M M, Domingos Cavalcanti V N, Martinez P. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal TMA, 2008, 68(1):177-193
[8] Chen H, Liu G W. Well-posedness for a class of Kirchhoff equations with damping and memory terms. IMA J Appl Math, 2015, 80(6):1808-1836
[9] Chen M M, Liu W J, Zhou W C. Existence and general stabilization of the Timoshenko system of thermoviscoelasticity of type Ⅲ with frictional damping and delay terms. Advances in Nonlinear Analysis, 2016, DOI:10.1515/anona-2016-0085
[10] Dafermos C M. Asymptotic stability in viscoelasticity. Arch Ration Mech Anal, 1970, 37(4):297-308
[11] 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(5):885-903
[12] Datko R, Lagnese J, Polis M P. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J Control Optim, 1986, 24(1):152-156
[13] 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(3):697-713
[14] Guesmia A. Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay. IMA J Math Control Inform, 2013, 30(4):507-526
[15] Kirane M, Said-Houari B. Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z Angew Math Phys, 2011, 62(6):1065-1082
[16] Liu G W, Zhang H W. Blow up at infinity of solutions for integro-differential equation. Appl Math Comput, 2014, 230(2):303-314
[17] Liu G W, Zhang H W. Well-posedness for a class of wave equation with past history and a delay. Z Angew Math Phys, 2016, 67(1):1-14
[18] 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(4):043504, 9pp
[19] Liu W J. General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term. Taiwanese J Math, 2013, 17(6):2101-2115
[20] Messaoudi S A, Mustafa M I. On the control of solutions of viscoelastic equations with boundary feedback. Nonlinear Anal RWA, 2009, 10(5):3132-3140
[21] Messaoudi S A, Mustafa M I. On convexity for energy decay rates of a viscoelastic equation with boundary feedback. Nonlinear Anal TMA, 2010, 72(9/10):3602-3611
[22] Messaoudi S A, Fareh A, Doudi N. Well posedness and exponential satbility in a wave equation with a strong damping and a strong delay. J Math Phys, 2016, 57(11):111501
[23] Mustafa M I. Asymptotic behavior of second sound thermoelasticity with internal time-varying dealy. Z Angew Math Phys, 201364(4):1353-1362
[24] 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(5):1561-1585
[25] Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Differential Integral Equations, 2008, 21(9/10):935-958
[26] Nicaise S, Valein J, Fridman E. Stabilization of the heat and the wave equations with boundary timevarying delays. Discrete Contin Dyn Syst Ser S, 2009, 2(3):559-581
[27] Nicaise S, Pignotti C. Interior feedback stabilization of wave equations with time dependent delay. Electron J Differ Eq, 2011, 2011(41):1-20
[28] Nicaise S, Pignotti C. Stability for second-order evolution equations with switching time-delay. J Dyn Diff Eq, 2014, 26(3):781-803
[29] Nicaise S, Pignotti C. Exponential stability of abstract evolution equations with time delay. J Evol Eq, 2015, 15(1):107-129
[30] Nicaise S, Pignotti C. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete Contin Dyn Syst Ser S, 2016, 9(3):791-813
[31] Pignotti C. Stability for second-order evolution equations with memory and switching time-delay. J Dyn Diff Eq, 2016, DOI:10.1007/s10884-016-9545-3
[32] Park S H. Energy decay for a von Karman equation with time-varying delay. Applied Mathematics Letters, 2016, 55(5):10-17
[33] Peralta G R. Stabilization of viscoelastic wave equations with distributed or boundary delay. Z Anal Anwend, 2016, 35(3):359-381
[34] Wu S T. Energy decay rates via convexity for some second-order evolution equation with memory and nonlinear time-dependent dissipation. Nonlinear Anal TMA, 2011, 74(2):532-543
[35] Xu G Q, Yung S P, Li L K. Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim Calc Var, 2006, 12(4):770-785
[36] Yang Z F. Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay. Z Angew Math Phys, 2015, 66(3):727-745