American Institute of Mathematical Sciences

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 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. Cavalcanti, V. 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. Li, L. 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. Liu, B. Zhu, G. 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. Park, M. 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.

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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. Cavalcanti, V. 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. Li, L. 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. Liu, B. Zhu, G. 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. Park, M. 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.
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