doi: 10.3934/mcrf.2021058
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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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