Research article Special Issues

Energy decay of solution for nonlinear delayed transmission problem

  • Received: 03 October 2022 Revised: 12 March 2023 Accepted: 20 March 2023 Published: 12 April 2023
  • MSC : 35B37, 35L55, 74D05, 93D15, 93D20

  • In this work, we consider a nonlinear transmission problem in the bounded domain with a delay term in the first equation. Under conditions on the weight of the damping and the weight of the delay, we prove general stability estimates by introducing a suitable Lyapunov functional and using the properties of convex functions.

    Citation: Abdelkader Moumen, Abderrahmane Beniani, Tariq Alraqad, Hicham Saber, Ekram. E. Ali, Keltoum Bouhali, Khaled Zennir. Energy decay of solution for nonlinear delayed transmission problem[J]. AIMS Mathematics, 2023, 8(6): 13815-13829. doi: 10.3934/math.2023707

    Related Papers:

  • In this work, we consider a nonlinear transmission problem in the bounded domain with a delay term in the first equation. Under conditions on the weight of the damping and the weight of the delay, we prove general stability estimates by introducing a suitable Lyapunov functional and using the properties of convex functions.



    加载中


    [1] F. Alabau-Boussouira, Convexity and weighted intgral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61–105. https://doi.org/10.1007/s00245 doi: 10.1007/s00245
    [2] A. Benaissa, N. Louhibi, Global existence and energy decay of solutions to a nonlinear wave equation with a delay term, Georgian Math. J., 20 (2013), 1–24. https://doi.org/10.1515/gmj-2013-0006 doi: 10.1515/gmj-2013-0006
    [3] A. Benaissa, M. Bahlil, Global existence and energy decay of solutions to a nonlinear timoshenko beam system with a delay term, Taiwanese J. Math., 18 (2014), 1411–1437. https://doi.org/10.11650/tjm.18.2014.3586 doi: 10.11650/tjm.18.2014.3586
    [4] A. Benseghir, Existence and exponential decay of solutions for transmission problems with delay, Electron. J. Differ. Equ., 2014 (2014), 212.
    [5] W. D. Bastos, C. A. Raposo, Transmission problem for waves with frictional damping, Electron. J. Differ. Equ., 2007 (2007), 60. http://ejde.math.txstate.edu/
    [6] S. Boulaaras, A. Draifia, K. Zennir, General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and logarithmic nonlinearity, Math. Method. Appl. Sci., 42 (2019), 4795–4814. https://doi.org/10.1002/mma.5693 doi: 10.1002/mma.5693
    [7] M. Daoulatli, I. Lasiecka, D. Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete Cont. Dyn. S, 2 (2009), 67–95. https://doi.org/10.3934/dcdss.2009.2.67 doi: 10.3934/dcdss.2009.2.67
    [8] R. Datko, J. Lagnese, M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152–156. https://doi.org/10.1137/0324007 doi: 10.1137/0324007
    [9] B. Feng, K. zennir, L. K. Laouar, Decay of an extensible viscoelastic plate equation with a nonlinear time delay, Bull. Malays. Math. Sci. Soc., 42 (2019), 2265–2285. https://doi.org/10.1007/s40840-018-0602-4 doi: 10.1007/s40840-018-0602-4
    [10] R. Gabasov, F. M. Kirillova, V. T. T. Ha, Optimal real-time control of multidimensional dynamic plant, Automat. Rem. Contr., 76 (2015), 98–110. https://doi.org/10.1134/S0005117915010099 doi: 10.1134/S0005117915010099
    [11] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary dampin, Differ. Integral Equ., 6 (1993), 507–533. https://doi.org/10.57262/die/1370378427
    [12] I. Lasiecka, Stabilization of wave and plate-like equation with nonlinear dissipation on the boundary, J. Differ. Equ., 79, (1989), 340–381. https://doi.org/10.1016/0022-0396(89)90107-1
    [13] I. Lasiecka, D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal. Theor., 64 (2006), 1757–1797. https://doi.org/10.1016/j.na.2005.07.024 doi: 10.1016/j.na.2005.07.024
    [14] L. K. Laouar, K. zennir, s. Boulaaras, The sharp decay rate of thermoelastic transmission system with infinite memories, Rend. Circ. Mat. Palerm., 69 (2020), 403–423. https://doi.org/10.1007/s12215-019-00408-1 doi: 10.1007/s12215-019-00408-1
    [15] W. J. Liu, E. Zuazua, Decay rates for dissipative wave equations, Ric. Mat., 48 (1999), 61–75.
    [16] A. Marzocchi, J. E. Muñoz Rivera, M. G. Naso, Asymptotic behavior and exponential stability for a transmission problem in thermoelasticity, Math. Method. Appl. Sci., 25 (2002), 955–980. https://doi.org/10.1002/mma.323 doi: 10.1002/mma.323
    [17] A. Marzocchi, J. E. Muñoz Rivera, M. G. Naso, Transmission problem in thermoelasticity with symmetry, IMA J. Appl. Math., 68 (2002), 23–46. https://doi.org/10.1093/imamat/68.1.23 doi: 10.1093/imamat/68.1.23
    [18] S. Nicaise, C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561–1585. https://doi.org/10.1137/060648891 doi: 10.1137/060648891
    [19] Y. Z. Tsypkin, Stability of processes in non-linear sampled-data systems, Dokl. Akad. Nauk SSSR, 152 (1963), 302–304.
    [20] K. Zennir, Stabilization for solutions of plate equation with time-varying delay and weak-viscoelasticity in $\mathbf{R}^n$, Russ. Math., 64 (2020), 21–33. https://doi.org/10.3103/S1066369X20090030 doi: 10.3103/S1066369X20090030
    [21] K. Zennir, B. Feng, One spatial variable thermoelastic transmission problem in viscoelasticity located in the second part, Math. Method. Appl. Sci., 41 (2018), 6895–6906. https://doi.org/10.1002/mma.5201 doi: 10.1002/mma.5201
    [22] K. Zennir, General decay of solutions for damped wave equation of Kirchhoff type with density in $R^{n}$, Ann. Univ. Ferrara, 61 (2015), 381–394. https://doi.org/10.1007/s11565-015-0223-x doi: 10.1007/s11565-015-0223-x
  • Reader Comments
  • © 2023 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(1159) PDF downloads(44) Cited by(3)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog