# American Institute of Mathematical Sciences

June  2017, 6(2): 239-260. doi: 10.3934/eect.2017013

## General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term

 College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author:Wenjun Liu

Received  February 17, 2016 Revised  January 29, 2017 Published  April 2017

In this paper, we consider a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term acting on the boundary. By using the Faedo-Galerkin approximation method, we first prove the well-posedness of the solutions. By introducing suitable energy and perturbed Lyapunov functionals, we then prove the general decay results, from which the usual exponential and polynomial decay rates are only special cases. To achieve these results, we consider the following two cases according to the coefficient α of the strong damping term: for the presence of the strong damping term (α>0), we use the strong damping term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the strong damping term; for the absence of the strong damping term (α=0), we use the viscoelasticity term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the kernel function.

Citation: Wenjun Liu, Biqing Zhu, Gang Li, Danhua Wang. General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term. Evolution Equations and Control Theory, 2017, 6 (2) : 239-260. doi: 10.3934/eect.2017013
##### References:
 [1] K. T. Andrews, K. L. Shillor and M. Kuttler, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795.  doi: 10.1006/jmaa.1996.0053. [2] A. V. Balakrishnan and L. W. Taylor, Distributed parameter nonlinear damping models for flight structures, in: Proceedings "Daming 8" Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. [3] R. W. Bass and D. Zes, Spillover nonlinearity, and flexible structures in: The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems Taylor, L. W. (ed. ), NASA Conference Publication, 10065, (1991), 1-14. doi: 10.1109/CDC.1991.261683. [4] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071. [5] A. Bahlil and M. Benaissa, Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term, Taiwanese J. Math., 18 (2014), 1411-1437. [6] A. Benaissa and S. A. Messaoudi, Global existence and energy decay of solutions for a nondissipative wave equation with a time-varying delay term, in Progress in partial differential equations, Springer Proc. Math. Stat., Springer, Cham., 44 (2013), 1-26. doi: 10.1007/978-3-319-00125-8_1. [7] B. M. Budak, A. A. Samarskii and A. N. Tikhonov, A Collection of Problems on Mathematical Physics Translated by A. R. M. Robson; translation edited by D. M. Brink. A Pergamon Press Book, Macmillan, New York, 1964. [8] M. M. Cavalcanti, V. N. Ferreira and J. Domingos Cavalcanti, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250. [9] M. M. Cavalcanti, V. N. Martinez and P. Cavalcanti, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193.  doi: 10.1016/j.na.2006.10.040. [10] Q. Yang and Z. Dai, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6. [11] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040. [12] M. Hakem and A. Ferhat, On convexity for energy decay rates of a viscoelastic wave equation with a dynamic boundary and nonlinear delay term, Facta Univ. Ser. Math. Inform., 30 (2015), 67-87. [13] V. Lucente and S. Georgiev, Decay for nonlinear Klein-Gordon equations, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 529-555.  doi: 10.1007/s00030-004-2027-z. [14] V. Georgiev, B. Sampalmieri and R. Rubino, Global existence for elastic waves with memory, Arch. Ration. Mech. Anal., 176 (2005), 303-330.  doi: 10.1007/s00205-004-0345-2. [15] S. Said-Houari and B. Gerbi, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations, 13 (2008), 1051-1074. [16] S. Said-Houari and B. Gerbi, Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions, Nonlinear Anal., 74 (2011), 7137-7150.  doi: 10.1016/j.na.2011.07.026. [17] S. Said-Houari and B. Gerbi, Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term, Appl. Math. Comput., 218 (2012), 11900-11910.  doi: 10.1016/j.amc.2012.05.055. [18] S. Said-Houari and B. Gerbi, Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions, Adv. Nonlinear Anal., 2 (2013), 163-193.  doi: 10.1515/anona-2012-0027. [19] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. [20] P. J. Said-Houari and B. Graber, Existence and asymptotic behavior of the wave equation with dynamic boundary conditions, Appl. Math. Optim., 66 (2012), 81-122.  doi: 10.1007/s00245-012-9165-1. [21] M. Said-Houari and B. Kirane, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0. [22] W. J. Chen and K. Liu, Existence and general decay for nondissipative distributed systems with boundary frictional and memory dampings and acoustic boundary conditions, Z. Angew. Math. Phys., 66 (2015), 1595-1614.  doi: 10.1007/s00033-014-0489-3. [23] W. J. Liu Chen and W. K., Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions, Math. Nachr., 289 (2016), 300-320.  doi: 10.1002/mana.201400343. [24] W. J. Liu, K. W. Chen and J. Yu, Existence and general decay for the full von Karman beam with a thermo-viscoelastic damping, frictional dampings and a delay term IMA J. Math. Control Inform. in press. Advance access. doi: 10.1093/imamci/dnv056. [25] W. J. Sun and Y. Liu, General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions, Z. Angew. Math. Phys., 65 (2014), 125-134.  doi: 10.1007/s00033-013-0328-y. [26] W. J. Liu, Y. Sun and G. Li, On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term Topol. Methods Nonlinear Anal. (2016), in press. doi: 10.12775/TMNA.2016.077. [27] J. -L. Lions, Quelques Méthodes de Résolution des Problémes Aux Limites non Linéaires Dunod, 1969. [28] C. Ma and J. Mu, On a system of nonlinear wave equations with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 65 (2014), 91-113.  doi: 10.1007/s00033-013-0324-2. [29] S. Pignotti and C. Nicaise, 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 (electronic).  doi: 10.1137/060648891. [30] S. Pignotti and C. Nicaise, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958. [31] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations 2011(2011), 20 pp. [32] S. Nicaise, C. Valein and J. Pignotti, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693. [33] S. Nicaise, J. Fridman and E. Valein, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559. [34] F. Peyravi and A. Tahamtani, Asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math., 17 (2013), 1921-1943.  doi: 10.11650/tjm.17.2013.3034. [35] N. Zaraï and A. Tatar, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping, Demonstratio Math., 44 (2011), 67-90. [36] S.-T. Wu, Asymptotic behavior for a viscoelastic wave equation with a delay term, Taiwanese J. Math., 17 (2013), 765-784.  doi: 10.11650/tjm.17.2013.2517. [37] S.-T. Wu, General decay of solutions for a viscoelastic equation with Balakrishnan-Taylor damping, Taiwanese J. Math., 19 (2015), 553-566.  doi: 10.11650/tjm.19.2015.4631. [38] S. Yu, On the strongly damped wave equation with nonlinear damping and source terms, Electron. J. Qual. Theory Differ. Equ. 2009(2009), 18 Pp. [39] A. Tatar and N.-E. Zaraï, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math. (Brno), 46 (2010), 157-176. [40] A. Zaraï, N. Abdelmalek and S. Tatar, Elastic membrane equation with memory term and nonlinear boundary damping: global existence, decay and blowup of the solution, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 84-106.  doi: 10.1016/S0252-9602(12)60196-9. [41] Z. Huang and J. Zhang, On solvability of the dissipative Kirchhoff equation with nonlinear boundary damping, Bull. Korean Math. Soc., 51 (2014), 189-206.  doi: 10.4134/BKMS.2014.51.1.189.

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##### References:
 [1] K. T. Andrews, K. L. Shillor and M. Kuttler, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795.  doi: 10.1006/jmaa.1996.0053. [2] A. V. Balakrishnan and L. W. Taylor, Distributed parameter nonlinear damping models for flight structures, in: Proceedings "Daming 8" Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. [3] R. W. Bass and D. Zes, Spillover nonlinearity, and flexible structures in: The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems Taylor, L. W. (ed. ), NASA Conference Publication, 10065, (1991), 1-14. doi: 10.1109/CDC.1991.261683. [4] J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071. [5] A. Bahlil and M. Benaissa, Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term, Taiwanese J. Math., 18 (2014), 1411-1437. [6] A. Benaissa and S. A. Messaoudi, Global existence and energy decay of solutions for a nondissipative wave equation with a time-varying delay term, in Progress in partial differential equations, Springer Proc. Math. Stat., Springer, Cham., 44 (2013), 1-26. doi: 10.1007/978-3-319-00125-8_1. [7] B. M. Budak, A. A. Samarskii and A. N. Tikhonov, A Collection of Problems on Mathematical Physics Translated by A. R. M. Robson; translation edited by D. M. Brink. A Pergamon Press Book, Macmillan, New York, 1964. [8] M. M. Cavalcanti, V. N. Ferreira and J. Domingos Cavalcanti, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250. [9] M. M. Cavalcanti, V. N. Martinez and P. Cavalcanti, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal., 68 (2008), 177-193.  doi: 10.1016/j.na.2006.10.040. [10] Q. Yang and Z. Dai, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6. [11] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040. [12] M. Hakem and A. Ferhat, On convexity for energy decay rates of a viscoelastic wave equation with a dynamic boundary and nonlinear delay term, Facta Univ. Ser. Math. Inform., 30 (2015), 67-87. [13] V. Lucente and S. Georgiev, Decay for nonlinear Klein-Gordon equations, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 529-555.  doi: 10.1007/s00030-004-2027-z. [14] V. Georgiev, B. Sampalmieri and R. Rubino, Global existence for elastic waves with memory, Arch. Ration. Mech. Anal., 176 (2005), 303-330.  doi: 10.1007/s00205-004-0345-2. [15] S. Said-Houari and B. Gerbi, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations, 13 (2008), 1051-1074. [16] S. Said-Houari and B. Gerbi, Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions, Nonlinear Anal., 74 (2011), 7137-7150.  doi: 10.1016/j.na.2011.07.026. [17] S. Said-Houari and B. Gerbi, Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term, Appl. Math. Comput., 218 (2012), 11900-11910.  doi: 10.1016/j.amc.2012.05.055. [18] S. Said-Houari and B. Gerbi, Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions, Adv. Nonlinear Anal., 2 (2013), 163-193.  doi: 10.1515/anona-2012-0027. [19] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. [20] P. J. Said-Houari and B. Graber, Existence and asymptotic behavior of the wave equation with dynamic boundary conditions, Appl. Math. Optim., 66 (2012), 81-122.  doi: 10.1007/s00245-012-9165-1. [21] M. Said-Houari and B. Kirane, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0. [22] W. J. Chen and K. Liu, Existence and general decay for nondissipative distributed systems with boundary frictional and memory dampings and acoustic boundary conditions, Z. Angew. Math. Phys., 66 (2015), 1595-1614.  doi: 10.1007/s00033-014-0489-3. [23] W. J. Liu Chen and W. K., Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions, Math. Nachr., 289 (2016), 300-320.  doi: 10.1002/mana.201400343. [24] W. J. Liu, K. W. Chen and J. Yu, Existence and general decay for the full von Karman beam with a thermo-viscoelastic damping, frictional dampings and a delay term IMA J. Math. Control Inform. in press. Advance access. doi: 10.1093/imamci/dnv056. [25] W. J. Sun and Y. Liu, General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions, Z. Angew. Math. Phys., 65 (2014), 125-134.  doi: 10.1007/s00033-013-0328-y. [26] W. J. Liu, Y. Sun and G. Li, On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term Topol. Methods Nonlinear Anal. (2016), in press. doi: 10.12775/TMNA.2016.077. [27] J. -L. Lions, Quelques Méthodes de Résolution des Problémes Aux Limites non Linéaires Dunod, 1969. [28] C. Ma and J. Mu, On a system of nonlinear wave equations with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 65 (2014), 91-113.  doi: 10.1007/s00033-013-0324-2. [29] S. Pignotti and C. Nicaise, 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 (electronic).  doi: 10.1137/060648891. [30] S. Pignotti and C. Nicaise, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958. [31] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations 2011(2011), 20 pp. [32] S. Nicaise, C. Valein and J. Pignotti, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693. [33] S. Nicaise, J. Fridman and E. Valein, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559. [34] F. Peyravi and A. Tahamtani, Asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math., 17 (2013), 1921-1943.  doi: 10.11650/tjm.17.2013.3034. [35] N. Zaraï and A. Tatar, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping, Demonstratio Math., 44 (2011), 67-90. [36] S.-T. Wu, Asymptotic behavior for a viscoelastic wave equation with a delay term, Taiwanese J. Math., 17 (2013), 765-784.  doi: 10.11650/tjm.17.2013.2517. [37] S.-T. Wu, General decay of solutions for a viscoelastic equation with Balakrishnan-Taylor damping, Taiwanese J. Math., 19 (2015), 553-566.  doi: 10.11650/tjm.19.2015.4631. [38] S. Yu, On the strongly damped wave equation with nonlinear damping and source terms, Electron. J. Qual. Theory Differ. Equ. 2009(2009), 18 Pp. [39] A. Tatar and N.-E. Zaraï, Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping, Arch. Math. (Brno), 46 (2010), 157-176. [40] A. Zaraï, N. Abdelmalek and S. Tatar, Elastic membrane equation with memory term and nonlinear boundary damping: global existence, decay and blowup of the solution, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 84-106.  doi: 10.1016/S0252-9602(12)60196-9. [41] Z. Huang and J. Zhang, On solvability of the dissipative Kirchhoff equation with nonlinear boundary damping, Bull. Korean Math. Soc., 51 (2014), 189-206.  doi: 10.4134/BKMS.2014.51.1.189.
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