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 & Control Theory, 2017, 6 (2) : 239-260. doi: 10.3934/eect.2017013
References:
[1]

K. T. AndrewsK. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. M. CavalcantiV. 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.  Google Scholar

[9]

M. M. CavalcantiV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

V. GeorgievB. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. -L. Lions, Quelques Méthodes de Résolution des Problémes Aux Limites non Linéaires Dunod, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

S. Pignotti and C. Nicaise, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar

[31]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations 2011(2011), 20 pp.  Google Scholar

[32]

S. NicaiseC. 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.  Google Scholar

[33]

S. NicaiseJ. 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.  Google Scholar

[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.  Google Scholar

[35]

N. Zaraï and A. Tatar, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping, Demonstratio Math., 44 (2011), 67-90.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[40]

A. ZaraïN. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

K. T. AndrewsK. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. M. CavalcantiV. 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.  Google Scholar

[9]

M. M. CavalcantiV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

V. GeorgievB. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. -L. Lions, Quelques Méthodes de Résolution des Problémes Aux Limites non Linéaires Dunod, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

S. Pignotti and C. Nicaise, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar

[31]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations 2011(2011), 20 pp.  Google Scholar

[32]

S. NicaiseC. 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.  Google Scholar

[33]

S. NicaiseJ. 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.  Google Scholar

[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.  Google Scholar

[35]

N. Zaraï and A. Tatar, Exponential stability and blow up for a problem with Balakrishnan-Taylor damping, Demonstratio Math., 44 (2011), 67-90.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[40]

A. ZaraïN. 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.  Google Scholar

[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.  Google Scholar

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