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Research Article

General decay for a wave equation with Wentzell boundary conditions and nonlinear delay terms

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Pages 2565-2580 | Received 20 Nov 2020, Accepted 18 Mar 2021, Published online: 03 May 2021

References

  • Alabau-Boussouira, F. (2005). Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Applied Mathematics and Optimization, 51(1), 61–105. https://doi.org/10.1007/s00245
  • Arnold, V. I. (1989). Mathematical methods of classical mechanics. Springer-Verlag.
  • Barros, V., Nonato, C., & Rapaso, C. (2020). Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 28(1), 205–220. https://doi.org/10.3934/era.2020014
  • Benaissa, A., & Louhibi, N. (2013). Global existence and energy decay of solutions to a nonlinear wave equation with a delay term. Georgian Mathematical Journal, 20(1), 1–24. https://doi.org/10.1515/gmj-2013-0006
  • Cavalcanti, M. M., & Lasiecka, I. (2007). Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. Journal of Differential Equations, 236(2), 407–459. https://doi.org/10.1016/j.jde.2007.02.004
  • Cavalcanti, M. M., Lasiecka, I., & Toundykov, D. (2012). Geometrically constrained stabilization of wave equations with Wentzell boundary conditions. Applicable Analysis, 91(8), 1427–1452. https://doi.org/10.1080/00036811.2011.647910
  • Cavalcanti, M. M., & Medjden, M. (2007). Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary condition. Journal of Mathematical Analysis and Applications, 328(2), 900–930. https://doi.org/10.1016/j.jmaa.2006.05.070
  • Cavalcanti, M. M., & Oquendo, H. P. (2003). Frictional versus viscoelastic damping in a semilinear wave equation. SIAM Journal on Control and Optimization, 42(4), 1310–1324. https://doi.org/10.1137/S0363012902408010
  • Cavalcanti, M. M., & Toundykov, D. (2012). Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable. Transactions of the American Mathematical Society, 364(11), 5693–5713. https://doi.org/10.1090/tran/2012-364-11
  • Coclite, G. M., Goldstein, G. R., & Goldstein, J. A. (2011). Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions. Advances in Differential Equations, 16(9–10), 895–916.
  • Coclite, G. M., Goldstein, G. R., & Goldstein, J. A. (2013). Stability estimates for nonlinear hyperbolic problems with nonlinear Wentzell boundary conditions. Zeitschrift für Angewandte Mathematik und Physik, 64(3), 733–753. https://doi.org/10.1007/s00033-012-0261-5
  • Daoulatli, M., Lasiecka, I., & Toundykov, D. (2009). Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete and Continuous Dynamical Systems, 2(1), 67–95. https://doi.org/10.3934/dcdss.2009.2.67
  • Eller, M., Lagnese, J. E., & Nicaise, S. (2002a). Decay rates for solutions of a Maxwell system with nonlinear boundary damping. Computational and Applied Mathematics, 21(1), 135–165.
  • Eller, M., Lagnese, J. E., & Nicaise, S. (2002b). Stabilization of heterogeneous Maxwell's equations by linear or nonlinear boundary feedbacks. Electronic Journal of Differential Equations, 21(21), 1–26.
  • Feng, B. (2018). General decay for a viscoelastic wave equation with density and time delay term in Rn. Taiwanese Journal of Mathematics, 22(1), 205–223. https://doi.org/10.11650/tjm/8105
  • Feng, B., & Soufiyane, A. (2020). Optimal decay rates of a nonlinear time-delayed viscoelastic wave equation. Differential Integral Equations, 33(1/2), 43–65.
  • Ferhat, M., & Hakem, A. (2016). Global existence and energy decay result for a weak viscoelastic wave equations with a dynamic boundary and nonlinear delay term. Computers and Mathematics with Applications, 71(3), 779–804. https://doi.org/10.1016/j.camwa.2015.12.039
  • Gerbi, S., & Said-Houari, B. (2008). Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions. Advances in Differential Equations, 13(11–12), 1051–1074.
  • Hadeler, K. P. (1979). Delay equations in biology. In Functional differential equations and approximation of fixed points. Proceedings of the Summer School and Conference, Univeristy of Bonn, Bonn, 1978), volume 730 of Lecture Notes in Mathematics (pp. 136–156). Springer.
  • Heminna, A. (2000). Stabilisation frontière de problèmes de Ventcel. ESAIM: Control, Optimisation and Calculus of Variations, 5, 591–622.
  • Kafini, M., & Messaoudi, S. (2016). A blow-up result in a nonlinear wave equation with delay. Mediterranean Journal of Mathematics, 0, 0–0. https://doi.org/10.1007/s00009-014-0500-4.
  • Kafini, M., & Messaoudi, S. (2019). On the decay and global nonexistence of solutions to a damped wave equation with variable-exponent nonlinearity and delay. Annales Polonici Mathematici, 122(1), 49–70. https://doi.org/10.4064/ap180524-31-10
  • Kafini, M., & Messaoudi, S. (2020). Local existence and blow-up of solutions to a logarithmic nonlinear wave equation with delay. Applicable Analysis, 99(3), 530–547. https://doi.org/10.1080/00036811.2018.1504029.
  • Kasri, H., & Heminna, A. (2016). Exponential stability of a couple system with Wentzell conditions. Evolution Equations & Control Theory, 5(2), 235–250. https://doi.org/10.3934/eect.2016003
  • Khemmoudj, A., & M. E. Aries (2019). Stabilisation of a wave equation with localised memory term and boundary frictional damping. International Journal of Control, 92(10), 2383–2395. https://doi.org/10.1080/00207179.2018.1438669
  • Khemmoudj, A., & Medjden, M. (2004). Exponential decay for the semilinear damped Cauchy-Ventcel problem. Boletim da Sociedade Paranaense de Matemática, 22(2), 97–116. https://doi.org/10.5269/bspm.v22i2.7486
  • Khemmoudj, A., & Seghour, L. (2015). Exponential stabilization of a viscoelastic wave equation with dynamic boundary conditions. Nonlinear Differential Equations and Applications NoDEA, 22(5), 1259–1286. https://doi.org/10.1007/s00030-015-0322-5
  • Komornik, V. (1994). Exact controllability and stabilization. The multiplier method. Masson Wiley.
  • Lasiecka, I., & Doundykov, D. (2006). Energy decay rates for the semilinear wave equation with nonlinear localized damping and source term. Nonlinear Analysis: Theory, Methods & Applications, 64(8), 1757–1797. https://doi.org/10.1016/j.na.2005.07.024
  • Lasiecka, I., & Tataru, D. (1993). Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential and Integral Equations, 6, 507–533.
  • Lasiecka, I., Triggiani, R., & Yao, P. F. (1997). Exact controllability for second-order hyperbolic equations with variable coefficients-principal part and first-order term. Nonlinear Analysis Theory: Methods and Application, 30(1), 111–122. Proceedings of 2nd World Congress of nonlinear analysts. https://doi.org/10.1016/S0362-546X(97)00004-7
  • Li, C., & Xiao, T. (2016). Asymptotics for wave equations with Wentzell boundary conditions and boundary damping. Semigroup Forum, 94(3), 520–531. https://doi.org/10.1007/s00233-016-9779-8
  • Lions, J. L. (1969). Quelques methodes de resolution des problemes aux limites non lineaires (in French). Dunod.
  • Liu, W. J., & Zhuang, H. (2021). Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete and Continuous Dynamical Systems, 26(2), 907–942. https://doi.org/10.3934/dcdsb.2020147
  • Liu, W. J., & Zuazua, E. (1999). Decay rates for dissipative wave equations. Ricerche di Matematica, 48, 61–75.
  • Nicaise, S., & Laoubi, K. (2010). Polynomial stabilization of the wave equation with ventcel boundary conditions. Mathematische Nachrichten, 283(10), 1428–1438. https://doi.org/10.1002/mana.200710162
  • Nicaise, S., & Pignotti, C. (2006). Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM Journal on Control and Optimization, 45(5), 1561–1585. https://doi.org/10.1137/060648891
  • Nicaise, S., & Pignotti, C. (2008). Stabilization of the wave equation with boundary or internal distributed delay. Differential and Integral Equations, 21(9–10), 935–958.
  • Nicaise, S., & Valein, J. (2007). Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Networks and Heterogeneous Media, 2(3), 425–479 (electronic). https://doi.org/10.3934/nhm.2007.2.425
  • Park, S. H. (2020). Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source. Advances in Difference Equations.2020, 631. https://doi.org/10.1186/s13662-020-03037-6
  • Remil, M., & Hakem, A. (2017). Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term. Facta Universitatis (NIS), Series Mathematics and Informatics, 32(4), 485–502. https://doi.org/10.22190/FUMI1704485R
  • Shinskey, F. G. (1967). Process control systems. McGraw-Hill.
  • Vazquez, J. L., & Vitillaro, E. (2008). Wave equation with second-order non-standard dynamical boundary conditions. Mathematical Models and Methods in Applied Sciences, 18(12), 2019–2054. https://doi.org/10.1142/S0218202508003285
  • Xiao, T. J., & Liang, J. (2004). Complete second order differential equations in Banach spaces with dynamic boundary conditions. Journal of Differential Equations, 200(1), 105–136. https://doi.org/10.1016/j.jde.2004.01.011
  • Xiao, T. J., & Liang, J. (2008). Second order differential operators with Feller-Wentzell type boundary conditions. Journal of Functional Analysis, 254(6), 1467–1486. https://doi.org/10.1016/j.jfa.2007.12.012
  • Xu, C. Q., Yung, S. P., & Li, L. K. (2006). Stabilization of the wave system with input delay in the boundary control. ESAIM: Control, Optimisation and Calculus of Variations, 12(4), 770–785. https://doi.org/10.1051/cocv:2006021
  • Yao, P. F. (1999). On the observability inequality for exact controllability of wave equations with variable coefficients. SIAM Journal on Control and Optimization, 37(5), 1568–1599. https://doi.org/10.1137/S0363012997331482
  • Zuazua, E. (1990). Exponential decay for the semilinear wave equation with locally distributed damping. Communications in Partial Differential Equations, 15(2), 205–235. https://doi.org/10.1080/03605309908820684

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