Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights

  • Received: 01 January 2020 Revised: 01 February 2020
  • Primary: 35D05, 35E15; Secondary: 35Q35

  • We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by

    $ \begin{eqnarray*} \label{NLS} u_{tt}(x,t) - u_{xx}(x,t)+\mu_1(t)u_t(x,t) +\mu_2(t)u_t(x,t-\tau(t)) = 0 \end{eqnarray*} $

    in a bounded domain. Under proper conditions on nonlinear weights $ \mu_1(t), \mu_2(t) $ and non-constant delay $ \tau(t) $, we prove global existence and estimative the decay rate for the energy.

    Citation: Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights[J]. Electronic Research Archive, 2020, 28(1): 205-220. doi: 10.3934/era.2020014

    Related Papers:

  • We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by

    $ \begin{eqnarray*} \label{NLS} u_{tt}(x,t) - u_{xx}(x,t)+\mu_1(t)u_t(x,t) +\mu_2(t)u_t(x,t-\tau(t)) = 0 \end{eqnarray*} $

    in a bounded domain. Under proper conditions on nonlinear weights $ \mu_1(t), \mu_2(t) $ and non-constant delay $ \tau(t) $, we prove global existence and estimative the decay rate for the energy.



    加载中


    [1] F. A. Mehmeti, Nonlinear Waves in Networks, Vol. 80, Mathematical Research, Akademie-Verlag, Berlin, 1994.
    [2] A. Benaissa, A. Benguessoum and S. A. Messaoudi, Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback, Electron. J. Qual. Theory Differ. Equ., 11 (2014), 13 pp. doi: 10.14232/ejqtde.2014.1.11
    [3] Global existence and asymptotic stability for a coupled viscoelastic wave equation with a time-varying delay term. Electron. J. Math. Anal. Appl. (2018) 6: 119-156.
    [4] H. Brézis, Opérators Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York.
    [5] Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim. (1979) 17: 66-81.
    [6] Control and stabilization for the wave equation in a bounded domain. Ⅱ. SIAM J. Control Optim. (1981) 19: 114-122.
    [7] An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. (1986) 24: 152-156.
    [8] 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.
    [9] Long-time dynamics for a nonlinear Timoshenko system with delay. Appl. Anal. (2017) 96: 606-625.
    [10] Well-posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks. Math. Methods Appl. Sci. (2018) 41: 1162-1174.
    [11] Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete Contin. Dyn. Syst. Ser. B (2017) 22: 491-506.
    [12] Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay. IMA J. Math. Control Inform. (2013) 30: 507-526.
    [13] Two remarks on hyperbolic dissipative problems. Res. Notes in Math. (1985) 122: 161-179.
    [14] Nonlinear semigroups and evolution equations. J. Math. Soc. Japan (1967) 19: 508-520.
    [15] T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, Lezioni Fermiane, [Fermi Lectures], Scuola Normale Superiore, Pisa; Accademia NAzionale dei Lincei, Rome, 1985.
    [16] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.
    [17] Uniform exponential energy decay of wave equations in a bounded region with $L_2(0, \infty; L_2(\Gamma))$-feedback control in the Dirichlet boundary conditions. J. Differential Equations (1987) 66: 340-390.
    [18] G. Liu, Well-posedness and exponential decay of solutions for a transmission problem with distributed delay, Electron. J. Differential Equations, 174 (2017), 13 pp.
    [19] General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term. Taiwanese J. Math. (2013) 17: 2101-2115.
    [20] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC, Boca Raton, FL, 1999.
    [21] Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl. (1977) 60: 542-549.
    [22] 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.
    [23] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 41 (2011), 20 pp.
    [24] Stabilization of the wave equation with boundary or internal distributed delay. Differential Integral Equations (2008) 21: 935-958.
    [25] Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin. Dyn. Syst. Ser. S (2011) 4: 693-722.
    [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44, Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1
    [27] C. A. Raposo, H. Nguyen, J. O. Ribeiro and V. Barros, Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition, Electron. J. Differential Equations, 279 (2017), 11 pp.
    [28] Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term. Facta Univ. Ser. Math. Inform (2017) 32: 485-502.
    [29] Asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic wave equation with boundary dissipation. Taiwanese J. Math. (2013) 17: 1921-1943.
    [30] Stabilization of the cascaded ODE-Schrödinger equations subject to observation with time delay. IEEE/CAA J. Autom. Sin. (2019) 6: 1027-1035.
    [31] Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. (2006) 12: 770-785.
    [32] K.-Y. Yang and J.-M. Wang, Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay, ESAIM Control Optim. Calc. Var., 25 (2019), 23 pp. doi: 10.1051/cocv/2017080
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1877) PDF downloads(273) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog