# American Institute of Mathematical Sciences

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

 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.

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 [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

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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