Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term

Zhiqing Liu; Zhong Bo Fang

doi:10.3934/cpaa.2020043

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February 2020, 19(2): 941-966. doi: 10.3934/cpaa.2020043

Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term

Zhiqing Liu and Zhong Bo Fang ^,

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

^* Corresponding author

Received February 2019 Revised May 2019 Published October 2019

Fund Project: This work is supported by the National Natural Science Foundation of China (No.11671188) and the Fundamental Research Funds for the Central Universities (No.201861002, 201964008).

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.

Keywords: Transmission problem, Kirchhoff-type wave equations, nonlinear delay, global solvability, general decay.

Mathematics Subject Classification: Primary: 35B40, 35L53; Secondary: 93D15, 93D20.

Citation: Zhiqing Liu, Zhong Bo Fang. Global solvability and general decay of a transmission problem for kirchhoff-type wave equations with nonlinear damping and delay term. Communications on Pure and Applied Analysis, 2020, 19 (2) : 941-966. doi: 10.3934/cpaa.2020043

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.

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.

Figure 1. An example of $ \Omega $

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