doi: 10.3934/mcrf.2021058
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

* Corresponding author: Jianghao Hao

Received  June 2021 Revised  September 2021 Early access December 2021

Fund Project: This research was partially supported by Natural Science Foundation of China (grant number 11871315, 61374089), Natural Science Foundation of Shanxi Province of China (grant number 201901D111021)

In this paper we consider a system of viscoelastic wave equations of Kirchhoff type with dynamic boundary conditions. Supposing the relaxation functions $ g_i $ $ (i = 1, 2, \cdots, l) $ satisfy $ g_i(t)\leq-\xi_i(t)G(g_i(t)) $ where $ G $ is an increasing and convex function near the origin and $ \xi_i $ are nonincreasing, we establish some optimal and general decay rates of the energy using the multiplier method and some properties of convex functions. Moreover, we obtain the finite time blow-up result of solution with nonpositive or arbitrary positive initial energy. The results in this paper are obtained without imposing any growth condition on weak damping term at the origin. Our results improve and generalize several earlier related results in the literature.

Citation: Mengxian Lv, Jianghao Hao. General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021058
References:
[1]

A. B. Aissa and M. Ferhat, Stability result for viscoelastic wave equation with dynamic boundary conditions, Z. Angew. Math. Phys., 69 (2018), Paper No. 95, 13 pp. doi: 10.1007/s00033-018-0983-0.

[2]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-2063-1.

[3]

S. Berrimi and S. A. Messaoudi, Exponential Decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differ. Equ., 2004 (2004), No. 88, 10 pp.

[4]

M. M. CavalcantiV. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Method. Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250.

[5]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equation with localized damping, Electron. J. Differ. Equ., 2002 (2002), No. 44, 14 pp.

[6]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[7]

B. Feng, General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history, Mediterr. J. Math., 15 (2018), Paper No. 103, 17 pp. doi: 10.1007/s00009-018-1154-4.

[8]

M. Ferhat and A. Hakem, Global existence and energy decay result for a weak viscoelastic wave equations with a dynamic boundary and nonlinear delay term, Comput. Math. Appl., 71 (2016), 779-804.  doi: 10.1016/j.camwa.2015.12.039.

[9]

S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differ. Equ., 13 (2008), 1051-1074. 

[10]

S. Gerbi and B. Said-Houari, 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.

[11]

G. Kirchhoff, Vorlesungen über Mechanik, Teubner Leipzig, 1883.

[12]

M. O. Korpusov, Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electron. J. Differ. Equ., 2012 (2012), No. 119, 10 pp.

[13]

G. LiL. Hong and W. Liu, Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping, Appl. Anal., 92 (2013), 1046-1062.  doi: 10.1080/00036811.2011.647911.

[14]

G. Li, B. Q. Zhu and D. H. Wang, Existence, general decay and blow-up of solutions for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and dynamic boundary conditions, Mathematics-Basel., 2015.

[15]

W. LiuB. ZhuG. Li and D. Wang, General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term, Evol. Equ. Control The., 6 (2017), 239-260.  doi: 10.3934/eect.2017013.

[16]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.  doi: 10.1016/j.jmaa.2007.11.048.

[17]

S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035.

[18]

C. Mu and J. Ma, 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.

[19]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Method. Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[20]

M. I. Mustafa and M. Kafini, Decay rates for a coupled quasilinear system of nonlinear viscoelastic equations, J. Appl. Anal., 25 (2019), 97-110.  doi: 10.1515/jaa-2019-0011.

[21]

S.-H. ParkM. J. Lee and J.-R. Kang, Blow-up results for viscoelastic wave equations with weak damping, Appl. Math. Lett., 80 (2018), 20-26.  doi: 10.1016/j.aml.2018.01.002.

[22]

E. Pişkin and A. Fidan, Blow up of solutions for viscoelastic wave equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differ. Equ., 2017 (2017), Paper No. 242, 10 pp.

[23]

H. Song and D. Xue, Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., 109 (2014), 245-251.  doi: 10.1016/j.na.2014.06.012.

[24]

H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.  doi: 10.1016/j.nonrwa.2010.02.015.

[25]

S.-T. Wu, On decay and blow-up of solutions for a system of nonlinear wave equations, J. Math. Anal. Appl., 394 (2012), 360-377.  doi: 10.1016/j.jmaa.2012.04.054.

[26]

S.-T. Wu and L.-Y. Tsai, Blow-up of positive-initial-energy solutions for an integro-differential equation with nonlinear damping, Taiwan. J. Math., 14 (2010), 2043-2058.  doi: 10.11650/twjm/1500406031.

[27]

Z. Yang and Z. Gong, Blow-up of solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differ. Equ., 2016 (2016), Paper No. 332, 8 pp.

show all references

References:
[1]

A. B. Aissa and M. Ferhat, Stability result for viscoelastic wave equation with dynamic boundary conditions, Z. Angew. Math. Phys., 69 (2018), Paper No. 95, 13 pp. doi: 10.1007/s00033-018-0983-0.

[2]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-2063-1.

[3]

S. Berrimi and S. A. Messaoudi, Exponential Decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differ. Equ., 2004 (2004), No. 88, 10 pp.

[4]

M. M. CavalcantiV. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Method. Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250.

[5]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equation with localized damping, Electron. J. Differ. Equ., 2002 (2002), No. 44, 14 pp.

[6]

M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310-1324.  doi: 10.1137/S0363012902408010.

[7]

B. Feng, General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history, Mediterr. J. Math., 15 (2018), Paper No. 103, 17 pp. doi: 10.1007/s00009-018-1154-4.

[8]

M. Ferhat and A. Hakem, Global existence and energy decay result for a weak viscoelastic wave equations with a dynamic boundary and nonlinear delay term, Comput. Math. Appl., 71 (2016), 779-804.  doi: 10.1016/j.camwa.2015.12.039.

[9]

S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differ. Equ., 13 (2008), 1051-1074. 

[10]

S. Gerbi and B. Said-Houari, 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.

[11]

G. Kirchhoff, Vorlesungen über Mechanik, Teubner Leipzig, 1883.

[12]

M. O. Korpusov, Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electron. J. Differ. Equ., 2012 (2012), No. 119, 10 pp.

[13]

G. LiL. Hong and W. Liu, Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping, Appl. Anal., 92 (2013), 1046-1062.  doi: 10.1080/00036811.2011.647911.

[14]

G. Li, B. Q. Zhu and D. H. Wang, Existence, general decay and blow-up of solutions for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and dynamic boundary conditions, Mathematics-Basel., 2015.

[15]

W. LiuB. ZhuG. Li and D. Wang, General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term, Evol. Equ. Control The., 6 (2017), 239-260.  doi: 10.3934/eect.2017013.

[16]

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl., 341 (2008), 1457-1467.  doi: 10.1016/j.jmaa.2007.11.048.

[17]

S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035.

[18]

C. Mu and J. Ma, 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.

[19]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Method. Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[20]

M. I. Mustafa and M. Kafini, Decay rates for a coupled quasilinear system of nonlinear viscoelastic equations, J. Appl. Anal., 25 (2019), 97-110.  doi: 10.1515/jaa-2019-0011.

[21]

S.-H. ParkM. J. Lee and J.-R. Kang, Blow-up results for viscoelastic wave equations with weak damping, Appl. Math. Lett., 80 (2018), 20-26.  doi: 10.1016/j.aml.2018.01.002.

[22]

E. Pişkin and A. Fidan, Blow up of solutions for viscoelastic wave equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differ. Equ., 2017 (2017), Paper No. 242, 10 pp.

[23]

H. Song and D. Xue, Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., 109 (2014), 245-251.  doi: 10.1016/j.na.2014.06.012.

[24]

H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.  doi: 10.1016/j.nonrwa.2010.02.015.

[25]

S.-T. Wu, On decay and blow-up of solutions for a system of nonlinear wave equations, J. Math. Anal. Appl., 394 (2012), 360-377.  doi: 10.1016/j.jmaa.2012.04.054.

[26]

S.-T. Wu and L.-Y. Tsai, Blow-up of positive-initial-energy solutions for an integro-differential equation with nonlinear damping, Taiwan. J. Math., 14 (2010), 2043-2058.  doi: 10.11650/twjm/1500406031.

[27]

Z. Yang and Z. Gong, Blow-up of solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differ. Equ., 2016 (2016), Paper No. 332, 8 pp.

[1]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[2]

Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106

[3]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671

[4]

Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations and Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669

[5]

Mingqi Xiang, Die Hu. Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4609-4629. doi: 10.3934/dcdss.2021125

[6]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

[7]

Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113

[8]

Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020

[9]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399

[10]

Enzo Vitillaro. Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4575-4608. doi: 10.3934/dcdss.2021130

[11]

Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63

[12]

Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370

[13]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011

[14]

Tayeb Hadj Kaddour, Michael Reissig. Blow-up results for effectively damped wave models with nonlinear memory. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2687-2707. doi: 10.3934/cpaa.2020239

[15]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021

[16]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[17]

Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315

[18]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333

[19]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

[20]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54

2021 Impact Factor: 1.141

Article outline

[Back to Top]