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Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity
Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China |
A transmission problem for Kirchhoff-type wave equations with nonlinear damping and delay term in the internal feedback is considered under a memory condition on one part of the boundary. By virtue of multiplier method, Faedo-Galerkin approximation and energy perturbation technique, we establish the appropriate conditions to guarantee the existence of global solution, and derive a general decay estimate of the energy, which includes exponential, algebraic and logarithmic decay etc.
References:
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Feedback boundary stabilization of wave equations with interior delay, Syst. Control Lett., 59 (2010), 623-628.
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J. J. Bae,
Nonlinear transmission problem for wave equation with boundary condition of memory type, Acta Appl. Math., 110 (2010), 907-919.
doi: 10.1007/s10440-009-9485-6. |
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W. D. Bastos and C. A. Raposo, Transmission problem for waves with frictional damping, Electron. J. Differ. Eq., 2007 (2007), 10 pages. |
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A. Benaissa, A. Benguessoum and S. A. Messaoudi, Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the nonlinear internal feedback, Int. J. Dyn. Syst. Differ. Equ., 5 (2014), 26 pages.
doi: 10.1504/IJDSDE.2014.067080. |
[9] |
A. Benaissa and N. Louhibi, Global existence and energy decay of solutions to a nonlinear wave equation with a delay term, Georgian Math. J., 20(1) (2013), 24 pages.
doi: 10.1515/gmj-2013-0006. |
[10] |
A. Benseghir, Existence and exponential decay of solutions for transmission problems with delay, Electron. J. Differ. Eq., 2014 (2014), 11 pages. |
[11] |
S. Berrimi and S. A. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differ. Eq., 2004 (2004), 10 pages. |
[12] |
M. M. Cavalcanti, E. R. Coelho and V. N. D. Cavalcanti, Exponential stability for a transmission problem of a viscoelastic wave equation, Appl. Math. Optim., (2018), to appear.
doi: 10.1007/s00245-018-9514-9. |
[13] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano,
Existence and uniformdecay of solutions of a degenerate equation with nonlinear boundary damping and memory source term, Nonlinear Anal-Theor., 38 (1999), 281-294.
doi: 10.1016/S0362-546X(98)00195-3. |
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M. M. Cavalcanti, V. N. Domingos Cavalcanti and M. L. Santos,
Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Appl. Math. Comput., 150 (2004), 439-465.
doi: 10.1016/S0096-3003(03)00284-4. |
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M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differ. Eq., 2002 (2002), 14 pages. |
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Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.
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Global existence and energy decay of solutions to a viscoelastic Timoshenko beam system with a nonlinear delay term, Appl. Anal., 95 (2016), 2637-2660.
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S. S. Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs, Victoria University, Australia, 2002. |
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B. Feng, Global well-posedness and stability for a viscoelastic plate equation with a time delay, Math. Probl. Eng., 2015 (2015), 10 pages.
doi: 10.1155/2015/585021. |
[23] |
M. Kirane and B. Said-Houari,
Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.
doi: 10.1007/s00033-011-0145-0. |
[24] |
G. Li, D. Wang and B. Zhu, Well-posedness and decay of solutions for a transmission problem with history and delay, Electron. J. Differ. Eq., 2016 (2016), 21 pages. |
[25] |
J. Li and S. Chai,
Existence and energy decay rates of solutions to the variable-coefficient Euler-Bernoulli plate with a delay in localized nonlinear internal feedback, J. Math. Anal. Appl., 443 (2016), 981-1006.
doi: 10.1016/j.jmaa.2016.05.060. |
[26] |
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 68 pages.
doi: 10.1137/1030001. |
[27] |
G. Liu and L. Diao,
Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta Appl. Math., 155 (2018), 9-19.
doi: 10.1007/s10440-017-0142-1. |
[28] |
W. Liu, Z. Chen and D. Chen, New general decay results for a Moore-Gibson-Thompson equation with memory, Appl. Anal., DOI: 10.1080/00036811.2019.1577390
doi: 10.1080/00036811.2019.1577390. |
[29] |
W. Liu, D. Wang and D. Chen,
General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound, J. Therm. Stresses, 41 (2018), 758-775.
|
[30] |
W. Liu and W. Zhao,
Stabilization of a thermoelastic laminated beam with past history, Appl. Math.Optim., 80 (2019), 103-133.
doi: 10.1007/s00245-017-9460-y. |
[31] |
W. J. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, J. Math. Phys., 54 (2013), 9 pages.
doi: 10.1063/1.4799929. |
[32] |
T. F. Ma and J. E. Muñoz Rivera,
Positive solutions for a nonlinear nonlocal elliptic transmission problem, App. Math. Lett., 16 (2003), 243-248.
doi: 10.1016/S0893-9659(03)80038-1. |
[33] |
A. Marzocchi, J. E. Muñoz Rivera and M. G. Naso,
Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity, Math. Method. Appl. Sci., 25 (2002), 955-980.
doi: 10.1002/mma.323. |
[34] |
A. Marzocchi and M. G. Naso,
Transmission problem in thermoelasticity with symmetry, IMA J. Appl. Math., 68 (2003), 23-46.
doi: 10.1093/imamat/68.1.23. |
[35] |
J. E. Muñoz Rivera and H. P. Oquendo, The transmission problem of viscoelastic waves, Acta Appl. Math., 62 (2000), 21 pages.
doi: 10.1023/A:1006449032100. |
[36] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[37] |
S. Nicaise and C. Pignotti,
Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21 (2008), 935-958.
|
[38] |
J. Y. Park, J. J. Bae and I. H. Jung,
Uniform decay of solution for wave equation of Kirchhoff type with nonlinearboundary damping and memory term, Nonlinear Anal-Theor., 50 (2002), 871-884.
doi: 10.1016/S0362-546X(01)00781-7. |
[39] |
S. H. Park,
General decay of a transmission problem for Kirchhoff type wave equations with boundary memory condition, Acta Math. Sci., 34 (2014), 1395-1403.
doi: 10.1016/S0252-9602(14)60091-6. |
[40] |
S. H. Park,
Stability of a transmission problem for Kirchhoff type wave equations with memory on the boundary, Math. Method. Appl. Sci., 40 (2017), 3528-3537.
doi: 10.1002/mma.4242. |
[41] |
M. L. Santos, J. Ferreira, D. C. Pereira and C. A. Raposo,
Global existence and stability for the wave equationof Kirchhoff type with memory condition at the boundary, Nonlinear Anal-Theor., 54 (2003), 959-976.
doi: 10.1016/S0362-546X(03)00121-4. |
[42] |
J. Simon,
Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pura Appl., 146(4) (1987), 65-96.
doi: 10.1007/BF01762360. |
[43] |
D. Wang, G. Li and B. Zhu,
Well-posedness and general decay of solution for a transmission problem with viscoelastic term and delay, J. Nonlinear Sci. Appl., 9 (2016), 1202-1215.
doi: 10.22436/jnsa.009.03.46. |
[44] |
G. Q. Xu, S. P. Yung and L. K. Li,
Stabilization of wave systems with input delay in the boundary control, ESAIM Contr. Optim. Ca., 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[45] |
Y. Yamada,
Some nonlinear degenerate wave equations, Nonlinear Anal-Theor., 11 (1987), 1155-1168.
doi: 10.1016/0362-546X(87)90004-6. |
show all references
References:
[1] |
K. Ammari, S. Nicaise and C. Pignotti,
Feedback boundary stabilization of wave equations with interior delay, Syst. Control Lett., 59 (2010), 623-628.
doi: 10.1016/j.sysconle.2010.07.007. |
[2] |
D. Andrade, L. H. Fatori and J. E. Muñoz Rivera, Nonlinear transmission problem with a dissipative boundary condition of memory type, Electron. J. Differ. Eq., 2006 (2006), 16 pages. |
[3] |
T. A. Apalara, S. A. Messaoudi and M. I. Mustafa, Energy decay in thermoelasticity type Ⅲ with viscoelastic dampingand delay term, Electron. J. Differ. Equations, 2012 (2012), 15 pages. |
[4] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[5] |
J. J. Bae,
On uniform decay of coupled wave equation of Kirchhoff type subject to memory condition onthe boundary, Nonlinear Anal-Theor., 61 (2005), 351-372.
doi: 10.1016/j.na.2004.11.014. |
[6] |
J. J. Bae,
Nonlinear transmission problem for wave equation with boundary condition of memory type, Acta Appl. Math., 110 (2010), 907-919.
doi: 10.1007/s10440-009-9485-6. |
[7] |
W. D. Bastos and C. A. Raposo, Transmission problem for waves with frictional damping, Electron. J. Differ. Eq., 2007 (2007), 10 pages. |
[8] |
A. Benaissa, A. Benguessoum and S. A. Messaoudi, Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the nonlinear internal feedback, Int. J. Dyn. Syst. Differ. Equ., 5 (2014), 26 pages.
doi: 10.1504/IJDSDE.2014.067080. |
[9] |
A. Benaissa and N. Louhibi, Global existence and energy decay of solutions to a nonlinear wave equation with a delay term, Georgian Math. J., 20(1) (2013), 24 pages.
doi: 10.1515/gmj-2013-0006. |
[10] |
A. Benseghir, Existence and exponential decay of solutions for transmission problems with delay, Electron. J. Differ. Eq., 2014 (2014), 11 pages. |
[11] |
S. Berrimi and S. A. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differ. Eq., 2004 (2004), 10 pages. |
[12] |
M. M. Cavalcanti, E. R. Coelho and V. N. D. Cavalcanti, Exponential stability for a transmission problem of a viscoelastic wave equation, Appl. Math. Optim., (2018), to appear.
doi: 10.1007/s00245-018-9514-9. |
[13] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho and J. A. Soriano,
Existence and uniformdecay of solutions of a degenerate equation with nonlinear boundary damping and memory source term, Nonlinear Anal-Theor., 38 (1999), 281-294.
doi: 10.1016/S0362-546X(98)00195-3. |
[14] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and M. L. Santos,
Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary, Appl. Math. Comput., 150 (2004), 439-465.
doi: 10.1016/S0096-3003(03)00284-4. |
[15] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differ. Eq., 2002 (2002), 14 pages. |
[16] |
Z. Chen, W. Liu and D. Chen, General decay rates for a laminated beam with memory, Taiwan. J. Math., to appear.
doi: 10.11650/tjm/181109. |
[17] |
R. Datko,
Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.
doi: 10.1137/0326040. |
[18] |
R. Datko, J. Lagnese and M. P. Polis,
An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.
doi: 10.1137/0324007. |
[19] |
R. Dautray and J. L. Lions, Analyse mathematique et calcul numerique pour les sciences et les techniques, Masson, Paris, 1984. |
[20] |
L. Djilali and A. Benaissa,
Global existence and energy decay of solutions to a viscoelastic Timoshenko beam system with a nonlinear delay term, Appl. Anal., 95 (2016), 2637-2660.
doi: 10.1080/00036811.2015.1105961. |
[21] |
S. S. Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs, Victoria University, Australia, 2002. |
[22] |
B. Feng, Global well-posedness and stability for a viscoelastic plate equation with a time delay, Math. Probl. Eng., 2015 (2015), 10 pages.
doi: 10.1155/2015/585021. |
[23] |
M. Kirane and B. Said-Houari,
Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.
doi: 10.1007/s00033-011-0145-0. |
[24] |
G. Li, D. Wang and B. Zhu, Well-posedness and decay of solutions for a transmission problem with history and delay, Electron. J. Differ. Eq., 2016 (2016), 21 pages. |
[25] |
J. Li and S. Chai,
Existence and energy decay rates of solutions to the variable-coefficient Euler-Bernoulli plate with a delay in localized nonlinear internal feedback, J. Math. Anal. Appl., 443 (2016), 981-1006.
doi: 10.1016/j.jmaa.2016.05.060. |
[26] |
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 68 pages.
doi: 10.1137/1030001. |
[27] |
G. Liu and L. Diao,
Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta Appl. Math., 155 (2018), 9-19.
doi: 10.1007/s10440-017-0142-1. |
[28] |
W. Liu, Z. Chen and D. Chen, New general decay results for a Moore-Gibson-Thompson equation with memory, Appl. Anal., DOI: 10.1080/00036811.2019.1577390
doi: 10.1080/00036811.2019.1577390. |
[29] |
W. Liu, D. Wang and D. Chen,
General decay of solution for a transmission problem in infinite memory-type thermoelasticity with second sound, J. Therm. Stresses, 41 (2018), 758-775.
|
[30] |
W. Liu and W. Zhao,
Stabilization of a thermoelastic laminated beam with past history, Appl. Math.Optim., 80 (2019), 103-133.
doi: 10.1007/s00245-017-9460-y. |
[31] |
W. J. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, J. Math. Phys., 54 (2013), 9 pages.
doi: 10.1063/1.4799929. |
[32] |
T. F. Ma and J. E. Muñoz Rivera,
Positive solutions for a nonlinear nonlocal elliptic transmission problem, App. Math. Lett., 16 (2003), 243-248.
doi: 10.1016/S0893-9659(03)80038-1. |
[33] |
A. Marzocchi, J. E. Muñoz Rivera and M. G. Naso,
Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity, Math. Method. Appl. Sci., 25 (2002), 955-980.
doi: 10.1002/mma.323. |
[34] |
A. Marzocchi and M. G. Naso,
Transmission problem in thermoelasticity with symmetry, IMA J. Appl. Math., 68 (2003), 23-46.
doi: 10.1093/imamat/68.1.23. |
[35] |
J. E. Muñoz Rivera and H. P. Oquendo, The transmission problem of viscoelastic waves, Acta Appl. Math., 62 (2000), 21 pages.
doi: 10.1023/A:1006449032100. |
[36] |
S. Nicaise and C. Pignotti,
Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[37] |
S. Nicaise and C. Pignotti,
Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21 (2008), 935-958.
|
[38] |
J. Y. Park, J. J. Bae and I. H. Jung,
Uniform decay of solution for wave equation of Kirchhoff type with nonlinearboundary damping and memory term, Nonlinear Anal-Theor., 50 (2002), 871-884.
doi: 10.1016/S0362-546X(01)00781-7. |
[39] |
S. H. Park,
General decay of a transmission problem for Kirchhoff type wave equations with boundary memory condition, Acta Math. Sci., 34 (2014), 1395-1403.
doi: 10.1016/S0252-9602(14)60091-6. |
[40] |
S. H. Park,
Stability of a transmission problem for Kirchhoff type wave equations with memory on the boundary, Math. Method. Appl. Sci., 40 (2017), 3528-3537.
doi: 10.1002/mma.4242. |
[41] |
M. L. Santos, J. Ferreira, D. C. Pereira and C. A. Raposo,
Global existence and stability for the wave equationof Kirchhoff type with memory condition at the boundary, Nonlinear Anal-Theor., 54 (2003), 959-976.
doi: 10.1016/S0362-546X(03)00121-4. |
[42] |
J. Simon,
Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pura Appl., 146(4) (1987), 65-96.
doi: 10.1007/BF01762360. |
[43] |
D. Wang, G. Li and B. Zhu,
Well-posedness and general decay of solution for a transmission problem with viscoelastic term and delay, J. Nonlinear Sci. Appl., 9 (2016), 1202-1215.
doi: 10.22436/jnsa.009.03.46. |
[44] |
G. Q. Xu, S. P. Yung and L. K. Li,
Stabilization of wave systems with input delay in the boundary control, ESAIM Contr. Optim. Ca., 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[45] |
Y. Yamada,
Some nonlinear degenerate wave equations, Nonlinear Anal-Theor., 11 (1987), 1155-1168.
doi: 10.1016/0362-546X(87)90004-6. |

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