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. CavalcantiV. 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. IrynaM. 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. VuralN. 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. XuM. 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. CavalcantiV. 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. IrynaM. 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. VuralN. 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. XuM. 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

[1]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[2]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[3]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[4]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[5]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[6]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[7]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[8]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[9]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[10]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

[11]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[12]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[13]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[14]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[15]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[16]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[17]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[18]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[19]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[20]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

 Impact Factor: 0.263

Metrics

  • PDF downloads (209)
  • HTML views (454)
  • Cited by (1)

[Back to Top]