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doi: 10.3934/dcdss.2020160

Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy

1. 

College of Automation, Harbin Engineering University, Heilongjiang, Harbin 150001, China

2. 

College of Mathematical Sciences, Harbin Engineering University, Heilongjiang, Harbin 150001, China

* Corresponding author: Runzhang Xu

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday

Received  August 2018 Revised  December 2018 Published  November 2019

Fund Project: R. Xu is partially supported by the National Natural Science Foundation of China (11871017), the China Postdoctoral Science Foundation(2013M540270) and the Fundamental Research Funds for the Central Universities. M. Zhang is partially supported by the National Natural Science Foundation of China (11871199)

In this paper, we study blow up and blow up time of solutions for initial boundary value problem of Kirchhoff-type wave equations involving the fractional Laplacian
$\left\{ \begin{align} & {{u}_{tt}}+[u]_{s}^{2(\theta -1)}{{(-\Delta )}^{s}}u=f(u),\ \ \ \ \text{in}\ \Omega \times {{\mathbb{R}}^{+}}, \\ & u(x,0)={{u}_{0}},\ \ {{u}_{t}}(x,0)={{u}_{1}},\ \ \ \ \ \ \text{in}\ \Omega , \\ & u=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ ({{\mathbb{R}}^{N}}\backslash \Omega )\times \mathbb{R}_{0}^{+}, \\ \end{align} \right.$
where
$ [u]_s $
is the Gagliardo seminorm of
$ u $
,
$ s\in(0, 1) $
,
$ \theta\in[1, 2_s^*/2) $
with
$ 2_s^* = \frac{2N}{N-2s} $
,
$ (-\Delta)^s $
is the fractional Laplacian operator,
$ f(u) $
is a differential function satisfying certain assumptions,
$ \Omega\subset\mathbb{R}^N $
is a bounded domain with Lipschitz boundary
$ \partial \Omega $
. By introducing a new auxiliary function and an adapted concavity method, we establish some sufficient conditions on initial data such that the solutions blow up in finite time for the arbitrary positive initial energy. Moreover, as
$ f(u) = |u|^{p-1}u $
, we estimate the upper and lower bounds for blow up time with arbitrary positive energy.
Citation: Qiang Lin, Xueteng Tian, Runzhang Xu, Meina Zhang. Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020160
References:
[1]

D. Applebaum, Lévy processes from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[2]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

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G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.  Google Scholar

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G. M. Bisci and V. D. Rǎdulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math., 17 (2015), 1450001, 17 pp. doi: 10.1142/S0219199714500011.  Google Scholar

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G. M. Bisci and V. D. Rǎdulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.  doi: 10.1007/s00526-015-0891-5.  Google Scholar

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G. M. Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088, 23 pp. doi: 10.1142/S0219199715500881.  Google Scholar

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M. BonforteY. Sire and J. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[10]

M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x.  Google Scholar

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P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605.  Google Scholar

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A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[13]

Y. Q. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electron. J. Qual. Theory Differ. Equ., 70 (2016), 17 pp. doi: 10.14232/ejqtde.2016.1.70.  Google Scholar

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F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990.   Google Scholar

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F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

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M. Korpusov, Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electron. J. Differential Equations, 2012 (2012), 1-10.   Google Scholar

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N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[18]

W. LianM. S. Ahmed and R. Z. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239-257.  doi: 10.1016/j.na.2019.02.015.  Google Scholar

[19]

W. Lian, R. Z. Xu, V. D. Rǎdulescu, Y. B. Yang and N. Zhao, Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Adv. Calc. Var., (2019). Google Scholar

[20]

W. Lian and R. Z. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[21]

Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7.  Google Scholar

[22]

Y. C. Liu and R. Z. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.  doi: 10.3934/dcdsb.2007.7.171.  Google Scholar

[23]

T. Matsuyama and R. Ikehata, On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl., 204 (1996), 729-753.  doi: 10.1006/jmaa.1996.0464.  Google Scholar

[24]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[25]

N. PanP. Pucci and B. L. Zhang, Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evol. Equ., 18 (2018), 385-409.  doi: 10.1007/s00028-017-0406-2.  Google Scholar

[26]

L. E. Pany and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[27]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[28]

P. Pucci and V. D. Rǎdulescu, Progress in nonlinear Kirchhoff problems, Nonlinear Anal., 186 (2019), 1-5.  doi: 10.1016/j.na.2019.02.022.  Google Scholar

[29]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879.  Google Scholar

[30]

P. PucciM. Q. Xiang and B. L. Zhang, A diffusion problem of Kirchhoff type involving the nonlocal fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 37 (2017), 4035-4051.  doi: 10.3934/dcds.2017171.  Google Scholar

[31]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[32]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.  Google Scholar

[33]

J. L. Vázquez, Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D, J. Evol. Equ., 16 (2016), 723-758.  doi: 10.1007/s00028-016-0340-8.  Google Scholar

[34]

M. Q. XiangB. L. Zhang and V. D. Rǎdulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 3186-3205.  doi: 10.1088/0951-7715/29/10/3186.  Google Scholar

[35]

R. Z. Xu, Initial boundary value problem of semilinear hyperbolic equations and parabolic equations with critical initial data, Quar. Appl. Math., 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0.  Google Scholar

[36]

R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[37]

R. Z. XuM. Y. ZhangS. H. ChenY. B. Yang and J. H. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[38]

R. Z. Xu, Y. X. Chen, Y. B. Yang, S. H. Chen, J. H. Shen, T. Yu and Z. S. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and schrodinger equations, Electron. J. Differential Equations, 55 (2018), 52 pp. doi: 10.3934/dcds.2017244.  Google Scholar

[39]

R. Z. XuX. C. Wang and Y. B. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[40]

Y. B. Yang and R. Z. Xu, Finite time blowup for nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Appl. Math. Lett., 77 (2018), 21-26.  doi: 10.1016/j.aml.2017.09.014.  Google Scholar

[41]

R. Z. Xu, X. C. Wang, Y. B. Yang and S. H. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[42]

R. Z. Xu, W. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., (2019), https://doi.org/10.1007/s11425-017-9280-x. doi: 10.1007/s11425-017-9280-x.  Google Scholar

[43]

T. Yamazaki, Scattering for a quasilinear hyperbolic equation of Kirchhoff type, J. Differential Equations, 143 (1998), 1-59.  doi: 10.1006/jdeq.1997.3372.  Google Scholar

[44]

B. L. ZhangV. D. Rǎdulescu and L. Wang, Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 1061-1081.  doi: 10.1080/17476933.2015.1005612.  Google Scholar

show all references

References:
[1]

D. Applebaum, Lévy processes from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[2]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

[3]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.  Google Scholar

[4]

G. M. Bisci and V. D. Rǎdulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math., 17 (2015), 1450001, 17 pp. doi: 10.1142/S0219199714500011.  Google Scholar

[5]

G. M. Bisci and V. D. Rǎdulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.  doi: 10.1007/s00526-015-0891-5.  Google Scholar

[6] G. M. BisciV. D. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.  Google Scholar
[7]

G. M. Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088, 23 pp. doi: 10.1142/S0219199715500881.  Google Scholar

[8]

M. BonforteY. Sire and J. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.  Google Scholar

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[10]

M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x.  Google Scholar

[11]

P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605.  Google Scholar

[12]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[13]

Y. Q. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electron. J. Qual. Theory Differ. Equ., 70 (2016), 17 pp. doi: 10.14232/ejqtde.2016.1.70.  Google Scholar

[14]

F. Gazzola and T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990.   Google Scholar

[15]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

[16]

M. Korpusov, Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electron. J. Differential Equations, 2012 (2012), 1-10.   Google Scholar

[17]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[18]

W. LianM. S. Ahmed and R. Z. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239-257.  doi: 10.1016/j.na.2019.02.015.  Google Scholar

[19]

W. Lian, R. Z. Xu, V. D. Rǎdulescu, Y. B. Yang and N. Zhao, Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Adv. Calc. Var., (2019). Google Scholar

[20]

W. Lian and R. Z. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[21]

Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.  doi: 10.1016/S0022-0396(02)00020-7.  Google Scholar

[22]

Y. C. Liu and R. Z. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.  doi: 10.3934/dcdsb.2007.7.171.  Google Scholar

[23]

T. Matsuyama and R. Ikehata, On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl., 204 (1996), 729-753.  doi: 10.1006/jmaa.1996.0464.  Google Scholar

[24]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[25]

N. PanP. Pucci and B. L. Zhang, Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evol. Equ., 18 (2018), 385-409.  doi: 10.1007/s00028-017-0406-2.  Google Scholar

[26]

L. E. Pany and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[27]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[28]

P. Pucci and V. D. Rǎdulescu, Progress in nonlinear Kirchhoff problems, Nonlinear Anal., 186 (2019), 1-5.  doi: 10.1016/j.na.2019.02.022.  Google Scholar

[29]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879.  Google Scholar

[30]

P. PucciM. Q. Xiang and B. L. Zhang, A diffusion problem of Kirchhoff type involving the nonlocal fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 37 (2017), 4035-4051.  doi: 10.3934/dcds.2017171.  Google Scholar

[31]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[32]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.  Google Scholar

[33]

J. L. Vázquez, Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D, J. Evol. Equ., 16 (2016), 723-758.  doi: 10.1007/s00028-016-0340-8.  Google Scholar

[34]

M. Q. XiangB. L. Zhang and V. D. Rǎdulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 3186-3205.  doi: 10.1088/0951-7715/29/10/3186.  Google Scholar

[35]

R. Z. Xu, Initial boundary value problem of semilinear hyperbolic equations and parabolic equations with critical initial data, Quar. Appl. Math., 68 (2010), 459-468.  doi: 10.1090/S0033-569X-2010-01197-0.  Google Scholar

[36]

R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[37]

R. Z. XuM. Y. ZhangS. H. ChenY. B. Yang and J. H. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[38]

R. Z. Xu, Y. X. Chen, Y. B. Yang, S. H. Chen, J. H. Shen, T. Yu and Z. S. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and schrodinger equations, Electron. J. Differential Equations, 55 (2018), 52 pp. doi: 10.3934/dcds.2017244.  Google Scholar

[39]

R. Z. XuX. C. Wang and Y. B. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.  doi: 10.1016/j.aml.2018.03.033.  Google Scholar

[40]

Y. B. Yang and R. Z. Xu, Finite time blowup for nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Appl. Math. Lett., 77 (2018), 21-26.  doi: 10.1016/j.aml.2017.09.014.  Google Scholar

[41]

R. Z. Xu, X. C. Wang, Y. B. Yang and S. H. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[42]

R. Z. Xu, W. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., (2019), https://doi.org/10.1007/s11425-017-9280-x. doi: 10.1007/s11425-017-9280-x.  Google Scholar

[43]

T. Yamazaki, Scattering for a quasilinear hyperbolic equation of Kirchhoff type, J. Differential Equations, 143 (1998), 1-59.  doi: 10.1006/jdeq.1997.3372.  Google Scholar

[44]

B. L. ZhangV. D. Rǎdulescu and L. Wang, Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 1061-1081.  doi: 10.1080/17476933.2015.1005612.  Google Scholar

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