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Initial boundary value problem for a inhomogeneous pseudo-parabolic equation
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 |
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.
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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[12] |
V. Komorkin and E. Zuazua,
A direat method for boundary stablization of wave equation, J. Math. Pures Appl., 69 (1990), 33-54.
|
[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. |
[14] |
J. Lagnese,
Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.
doi: 10.1137/0326068. |
[15] |
I. Lasiecka and D. Tataru,
Uniform boundary stabilization of wave equation with nonlieary boundary damping, Differential and Integral Equations, 6 (1990), 507-533.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[32] |
W. P. Ziemer, Weakly Differently Functions, Graduate Text in Mathematicas, Springer, New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
[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. |
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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[12] |
V. Komorkin and E. Zuazua,
A direat method for boundary stablization of wave equation, J. Math. Pures Appl., 69 (1990), 33-54.
|
[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. |
[14] |
J. Lagnese,
Note on boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.
doi: 10.1137/0326068. |
[15] |
I. Lasiecka and D. Tataru,
Uniform boundary stabilization of wave equation with nonlieary boundary damping, Differential and Integral Equations, 6 (1990), 507-533.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[32] |
W. P. Ziemer, Weakly Differently Functions, Graduate Text in Mathematicas, Springer, New York, 1989.
doi: 10.1007/978-1-4612-1015-3. |
[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. |
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