American Institute of Mathematical Sciences

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March  2020, 28(1): 91-102. doi: 10.3934/era.2020006

Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems

Xiaoqiang Dai 1, , Chao Yang 2,, , Shaobin Huang 2, , Tao Yu 3, and  Yuanran Zhu 3,

1. 

Department of Electronic Information, Jiangsu University of Science and Technology, Zhenjiang, MO 212003, China
2. 

College of Computer Science and Technology, Harbin Engineering University, Harbin, MO 150001, China
3. 

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

* Corresponding author: Chao Yang

Received  November 2019 Published  March 2020

Fund Project: The first author is supported by Natural Science Foundation of Jiangsu Province (BK20160564) and Jiangsu key R & D plan(BE2018007).

We study the initial boundary value problem of linear homogeneous wave equation with dynamic boundary condition. We aim to prove the finite time blow-up of the solution at critical energy level or high energy level with the nonlinear damping term on boundary in control systems.

Keywords: High energy level, critical energy level, dynamical boundary condition, wave equation.
Mathematics Subject Classification: Primary: 35L05.
Citation: 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
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[2]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction, J. Differential Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.  Google Scholar

[3]

C. E. Kenig, The method of energy channels for nonlinear wave equations, Discrete and Continuous Dynamical Systems, 39 (2019), 6979-6993.  doi: 10.3934/dcds.2019240.  Google Scholar

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G. Chen, Energy decay estimates and exact boundary value controllabiity for the wave equation in a bounded domin, J. Math. Pures Appl., 58 (1979), 249-273.   Google Scholar

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G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66-81.  doi: 10.1137/0317007.  Google Scholar

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G. Chen, Control and stabilization for the wave equation in a bounded domain, part Ⅱ, SIAM J. Control Optim., 19 (1981), 114-122.  doi: 10.1137/0319009.  Google Scholar

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G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113.  doi: 10.1137/0319008.  Google Scholar

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F. Gazzola and M. Squassina, Global solutions and finite time blow-up for damped semilinear wave equations, Nonlinear Analysis, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

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S. Gerbi and B. Said-Houari, Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions, Adv. Nonlinear Anal., 2 (2013), 163-193.   Google Scholar

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N. Hoai-Minh, Superlensing using complementary media and reflecting complementary media for electromagnetic waves, Adv. Nonlinear Anal., 7 (2018), 449-467.  doi: 10.1515/anona-2017-0146.  Google Scholar

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E. Iryna, M. Johanna and T. Gerald, Rarefaction waves for the toda equation via nonlinear steepest descent, Discrete and Continuous Dynamical Systems, 38 (2018), 2007-2028.  doi: 10.3934/dcds.2018081.  Google Scholar

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V. Komorkin and E. Zuazua, A direat method for boundary stablization of wave equation, J. Math. Pures Appl., 69 (1990), 33-54.   Google Scholar

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J. Lagnese, Deacy of solutions of wave equations in a bounded region with boundary dissipation, Journal of Differential Equations, 50 (1983), 163-182.  doi: 10.1016/0022-0396(83)90073-6.  Google Scholar

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J. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.  doi: 10.1137/0326068.  Google Scholar

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I. Lasiecka and D. Tataru, Uniform boundary stabilization of wave equation with nonlieary boundary damping, Differential and Integral Equations, 6 (1990), 507-533.   Google Scholar

[16]

M. J. Lee, J. R. Kang and S. H. Park, Blow-up of solution for quasilinear viscoelastic wave equation with boundary nonlinear damping and source terms, Bound. Value Probl., 67 (2019), 11pp. doi: 10.1186/s13661-019-1180-6.  Google Scholar

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M. J. Lee and J. Y. Park, Energy decay of solutions of nonlinear viscoelastic problem with the dynamic and acoustic boundary conditions, Bound. Value Probl., 1 (2018), 26pp. doi: 10.1186/s13661-017-0918-2.  Google Scholar

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H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361.  doi: 10.1007/s002050050032.  Google Scholar

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H. A. Levine and L. E. Payn, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations, 16 (1974), 319-334.  doi: 10.1016/0022-0396(74)90018-7.  Google Scholar

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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

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G. Olivier and M. Imen, Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach, Adv. Nonlinear Anal., 8 (2019), 253-266.  doi: 10.1515/anona-2016-0274.  Google Scholar

[23]

C. Shane and S. Anton, Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations, Adv. Nonlinear Anal., 9 (2020), 745-787.  doi: 10.1515/anona-2020-0024.  Google Scholar

[24]

E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, Rend. Istit. Mat. Univ. Trieste, 31 (2000), 245-275.   Google Scholar

[25]

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source term, J. Diffrential Equations, 186 (2002), 259-298.  doi: 10.1016/S0022-0396(02)00023-2.  Google Scholar

[26]

B. Vural, N. Emil and O. Ibrahim, Local-in-space blow-up crireria for two-component nonlinear dispersive wave sysytem, Discrete and Continuous Dynamical Systems, 39 (2019), 6023-6037.  doi: 10.3934/dcds.2019263.  Google Scholar

[27]

R. Z. Xu, M. Y. Zhang and S. H. Chen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete and Continuous Dynamical Systems, 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[28]

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

[29]

H. W. Zhang and Q. Y. Hu, Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition, Commun. Pure Appl. Anal., 4 (2005), 861-869.  doi: 10.3934/cpaa.2005.4.861.  Google Scholar

[30]

H. W. Zhang, C. S. Hou and Q. Y. Ho, Energy decay and blow-up of solution for a Kirchhoff equation with dynamic boundary condition, Bound. Value Probl., 2013 (2013), 12pp. doi: 10.1186/1687-2770-2013-166.  Google Scholar

[31]

X. Zhao and W. P. Yan, Existence of standing waves for quasi-linear Schrödinger equations on $ {\rm T^n} $, Adv. Nonlinear Anal., 9 (2020), 978-933.  doi: 10.1515/anona-2020-0038.  Google Scholar

[32]

W. P. Ziemer, Weakly Differently Functions, Graduate Text in Mathematicas, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

[33]

E. Zuazua, Uniform stabilization of the wave equations by nonlinear boundary feedback, SIAM. J. Control Optim., 28 (1990), 466-477.  doi: 10.1137/0328025.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[2]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction, J. Differential Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.  Google Scholar

[3]

C. E. Kenig, The method of energy channels for nonlinear wave equations, Discrete and Continuous Dynamical Systems, 39 (2019), 6979-6993.  doi: 10.3934/dcds.2019240.  Google Scholar

[4]

G. Chen, Energy decay estimates and exact boundary value controllabiity for the wave equation in a bounded domin, J. Math. Pures Appl., 58 (1979), 249-273.   Google Scholar

[5]

G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66-81.  doi: 10.1137/0317007.  Google Scholar

[6]

G. Chen, Control and stabilization for the wave equation in a bounded domain, part Ⅱ, SIAM J. Control Optim., 19 (1981), 114-122.  doi: 10.1137/0319009.  Google Scholar

[7]

G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113.  doi: 10.1137/0319008.  Google Scholar

[8]

F. Gazzola and M. Squassina, Global solutions and finite time blow-up for damped semilinear wave equations, Nonlinear Analysis, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

[9]

S. Gerbi and B. Said-Houari, Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions, Adv. Nonlinear Anal., 2 (2013), 163-193.   Google Scholar

[10]

N. Hoai-Minh, Superlensing using complementary media and reflecting complementary media for electromagnetic waves, Adv. Nonlinear Anal., 7 (2018), 449-467.  doi: 10.1515/anona-2017-0146.  Google Scholar

[11]

E. Iryna, M. Johanna and T. Gerald, Rarefaction waves for the toda equation via nonlinear steepest descent, Discrete and Continuous Dynamical Systems, 38 (2018), 2007-2028.  doi: 10.3934/dcds.2018081.  Google Scholar

[12]

V. Komorkin and E. Zuazua, A direat method for boundary stablization of wave equation, J. Math. Pures Appl., 69 (1990), 33-54.   Google Scholar

[13]

J. Lagnese, Deacy of solutions of wave equations in a bounded region with boundary dissipation, Journal of Differential Equations, 50 (1983), 163-182.  doi: 10.1016/0022-0396(83)90073-6.  Google Scholar

[14]

J. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.  doi: 10.1137/0326068.  Google Scholar

[15]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of wave equation with nonlieary boundary damping, Differential and Integral Equations, 6 (1990), 507-533.   Google Scholar

[16]

M. J. Lee, J. R. Kang and S. H. Park, Blow-up of solution for quasilinear viscoelastic wave equation with boundary nonlinear damping and source terms, Bound. Value Probl., 67 (2019), 11pp. doi: 10.1186/s13661-019-1180-6.  Google Scholar

[17]

M. J. Lee and J. Y. Park, Energy decay of solutions of nonlinear viscoelastic problem with the dynamic and acoustic boundary conditions, Bound. Value Probl., 1 (2018), 26pp. doi: 10.1186/s13661-017-0918-2.  Google Scholar

[18]

H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361.  doi: 10.1007/s002050050032.  Google Scholar

[19]

H. A. Levine and A. Smith, A potential well theory for the wave equation with a nonlinear boundary conditions, J. Reine angew. Math., 374 (1987), 1-23.  doi: 10.1515/crll.1987.374.1.  Google Scholar

[20]

H. A. Levine and L. E. Payn, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time, J. Differential Equations, 16 (1974), 319-334.  doi: 10.1016/0022-0396(74)90018-7.  Google Scholar

[21]

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

[22]

G. Olivier and M. Imen, Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach, Adv. Nonlinear Anal., 8 (2019), 253-266.  doi: 10.1515/anona-2016-0274.  Google Scholar

[23]

C. Shane and S. Anton, Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations, Adv. Nonlinear Anal., 9 (2020), 745-787.  doi: 10.1515/anona-2020-0024.  Google Scholar

[24]

E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, Rend. Istit. Mat. Univ. Trieste, 31 (2000), 245-275.   Google Scholar

[25]

E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source term, J. Diffrential Equations, 186 (2002), 259-298.  doi: 10.1016/S0022-0396(02)00023-2.  Google Scholar

[26]

B. Vural, N. Emil and O. Ibrahim, Local-in-space blow-up crireria for two-component nonlinear dispersive wave sysytem, Discrete and Continuous Dynamical Systems, 39 (2019), 6023-6037.  doi: 10.3934/dcds.2019263.  Google Scholar

[27]

R. Z. Xu, M. Y. Zhang and S. H. Chen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete and Continuous Dynamical Systems, 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[28]

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

[29]

H. W. Zhang and Q. Y. Hu, Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition, Commun. Pure Appl. Anal., 4 (2005), 861-869.  doi: 10.3934/cpaa.2005.4.861.  Google Scholar

[30]

H. W. Zhang, C. S. Hou and Q. Y. Ho, Energy decay and blow-up of solution for a Kirchhoff equation with dynamic boundary condition, Bound. Value Probl., 2013 (2013), 12pp. doi: 10.1186/1687-2770-2013-166.  Google Scholar

[31]

X. Zhao and W. P. Yan, Existence of standing waves for quasi-linear Schrödinger equations on $ {\rm T^n} $, Adv. Nonlinear Anal., 9 (2020), 978-933.  doi: 10.1515/anona-2020-0038.  Google Scholar

[32]

W. P. Ziemer, Weakly Differently Functions, Graduate Text in Mathematicas, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

[33]

E. Zuazua, Uniform stabilization of the wave equations by nonlinear boundary feedback, SIAM. J. Control Optim., 28 (1990), 466-477.  doi: 10.1137/0328025.  Google Scholar

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