Asymptotic stability of a quasi-linear viscoelastic Kirchhoff plate equation with logarithmic source and time delay

Zayd Hajjej; Sun-Hye Park; Zayd Hajjej; Sun-Hye Park

doi:10.3934/math.20231228

AIMS Mathematics

2023, Volume 8, Issue 10: 24087-24115. doi: 10.3934/math.20231228

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Research article

Asymptotic stability of a quasi-linear viscoelastic Kirchhoff plate equation with logarithmic source and time delay

Zayd Hajjej ¹,
Sun-Hye Park ^{2
,
,}

1.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Office for Education Accreditation, Pusan National University, Busan 46241, Republic of Korea

Received: 07 June 2023 Revised: 18 July 2023 Accepted: 28 July 2023 Published: 09 August 2023
MSC : 35B40, 93D15, 93D20

In this paper, a quasi-linear viscoelastic Kirchhoff plate equation with logarithmic source and time delay involving free boundary conditions in a bounded domain is considered. The local existence and global existence are proved, respectively. Under the assumptions on a more general type of relaxation functions and suitable conditions on the coefficients between damping term and delay term, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions. As the considered assumption here on the kernel is more general than earlier papers, our result improves and generalizes earlier result in the literature.
- Kirchhoff plate equation,
- logarithmic source,
- time delay, general decay,
- relaxation function
Citation: Zayd Hajjej, Sun-Hye Park. Asymptotic stability of a quasi-linear viscoelastic Kirchhoff plate equation with logarithmic source and time delay[J]. AIMS Mathematics, 2023, 8(10): 24087-24115. doi: 10.3934/math.20231228

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Abstract

In this paper, a quasi-linear viscoelastic Kirchhoff plate equation with logarithmic source and time delay involving free boundary conditions in a bounded domain is considered. The local existence and global existence are proved, respectively. Under the assumptions on a more general type of relaxation functions and suitable conditions on the coefficients between damping term and delay term, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions. As the considered assumption here on the kernel is more general than earlier papers, our result improves and generalizes earlier result in the literature.

References

[1]	E. H. Brito, Decay estimates for the generalized damped extensible string and beam equations, Nonlinear Anal., 8 (1984), 1489–1496. https://doi.org/10.1016/0362-546X(84)90059-2 doi: 10.1016/0362-546X(84)90059-2
[2]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, Global existence and uniform decay rates to the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), 701–730. https://doi.org/10.57262/ade/1357140586 doi: 10.57262/ade/1357140586
[3]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differ. Int. Equ., 17 (2004), 495–510. https://doi.org/10.57262/die/1356060344 doi: 10.57262/die/1356060344
[4]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Commun. Contemp. Math., 6 (2004), 705–731. https://doi.org/10.1142/S0219199704001483 doi: 10.1142/S0219199704001483
[5]	M. A. Jorge Silva, T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of $p$-Laplacian type, IMA J. Appl. Math., 78 (2013), 1130–1146. https://doi.org/10.1093/imamat/hxs011 doi: 10.1093/imamat/hxs011
[6]	S. H. Park, General decay for a viscoleatic von Karman equation with delay and variable exponent nonlinearites, Bound. Value Probl., 2022 (2022). https://doi.org/10.1186/s13661-022-01602-4 doi: 10.1186/s13661-022-01602-4
[7]	T. F. Ma, Boundary stabilization for a non-linear beam on elastic bearings, Math. Meth. Appl. Sci., 24 (2001), 583–594. https://doi.org/10.1002/mma.230 doi: 10.1002/mma.230
[8]	T. F. Ma, V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal., 73 (2010), 3402–3412. https://doi.org/10.1016/j.na.2010.07.023 doi: 10.1016/j.na.2010.07.023
[9]	J. E. Munõz Rivera, L. H. Fatori, Smoothing effect and propagations of singularities for viscoelastic plates, J. Math. Anal. Appl., 206 (1997), 397–427. https://doi.org/10.1006/jmaa.1997.5223 doi: 10.1006/jmaa.1997.5223
[10]	J. E. Munõz Rivera, E. C. Lapa, R. Barreto, Decay rates for viscoelastic plates with memory, J. Elast. 44 (1996), 61–87. https://doi.org/10.1007/BF00042192 doi: 10.1007/BF00042192
[11]	J. Y. Park, S. H. Park, General decay for a nonlinear beam equation with weak dissipation, J. Math. Phys., 51 (2010), 073508. https://doi.org/10.1063/1.3460321 doi: 10.1063/1.3460321
[12]	S. K. Patcheu, On a global solution and asymptotic behavior for the generalized damped extensible beam equation, J. Differ. Equ., 35 (1997), 299–314. https://doi.org/10.1006/jdeq.1996.3231 doi: 10.1006/jdeq.1996.3231
[13]	D. Andrade, M. A. Jorge Silva, T. F. Ma, Exponential stability for a plate equation with $p$-Laplacian and memory terms, Math. Meth. Appl. Sci., 35 (2012), 417–426. https://doi.org/10.1002/mma.1552 doi: 10.1002/mma.1552
[14]	M. M. Cavalcanti, H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42 (2003), 1310–1324. https://doi.org/10.1137/S0363012902408010 doi: 10.1137/S0363012902408010
[15]	J. E. Munõz Rivera, A. Peres Salvatierra, Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59 (2001), 557–578.
[16]	M. A. Jorge Silva, J. E. Munõz Rivera, R. Racke, On a classes of nonlinear viscoelastic Kirchhoff plates: Well-posedness and generay decay rates, Appl. Math. Optim., 73 (2016), 165–194. https://doi.org/10.1007/s00245-015-9298-0 doi: 10.1007/s00245-015-9298-0
[17]	I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507–533. https://doi.org/10.57262/die/1370378427 doi: 10.57262/die/1370378427
[18]	F. Alabau-Boussouira, P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, Comptes Rendus Math., 347 (2009), 867–872. https://doi.org/10.1016/j.crma.2009.05.011 doi: 10.1016/j.crma.2009.05.011
[19]	F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61–105. https://doi.org/10.1007/s00245 doi: 10.1007/s00245
[20]	F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differ. Equ., 248 (2010), 1473–1517. https://doi.org/10.1016/j.jde.2009.12.005 doi: 10.1016/j.jde.2009.12.005
[21]	F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations, In: Control of Partial Differential Equations, Berlin, Heidelberg: Springer, 2012. https://doi.org/10.1007/978-3-642-27893-8_1
[22]	F. Alabau-Boussouira, K. Ammari, Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system, J. Funct. Anal., 260 (2011), 2424–2450. https://doi.org/10.1016/j.jfa.2011.01.003 doi: 10.1016/j.jfa.2011.01.003
[23]	A. M. Al-Mahdi, General stability result for a viscoelastic plate equation with past history and general kernel, J. Math. Anal. Appl., 490 (2020), 124216. https://doi.org/10.1016/j.jmaa.2020.124216 doi: 10.1016/j.jmaa.2020.124216
[24]	I. Bialynicki-Birula, J. Mycielsk, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys., 23 (1975), 461–466.
[25]	I. Bialynicki-Birula, J. Mycielsk, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62–93. https://doi.org/10.1016/0003-4916(76)90057-9 doi: 10.1016/0003-4916(76)90057-9
[26]	T. Cazenave, A. Haraux, Equations d'evolution avec non-linearite logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21–51.
[27]	P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B., 40 (2009), 59–66.
[28]	K. Bartkowski, P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A Math. Theor., 41 (2008), 355201. https://doi.org/10.1088/1751-8113/41/35/355201 doi: 10.1088/1751-8113/41/35/355201
[29]	T. Hiramatsu, M. Kawasaki, F. Takahashi, Numerical study of Q-ball formation in gravity mediation, Prog. Theor. Exp. Phys., 190 (2011), 229–238. https://doi.org/10.1143/PTPS.190.229 doi: 10.1143/PTPS.190.229
[30]	X. Han, Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50 (2013), 275–283. https://doi.org/10.4134/BKMS.2013.50.1.275 doi: 10.4134/BKMS.2013.50.1.275
[31]	Q. Hu, H. Zhang, G. Liu, Asymptotic behavior for a class of logarithmic wave equations with linear damping, Appl. Math. Optim., 79 (2019), 131–144. https://doi.org/10.1007/s00245-017-9423-3 doi: 10.1007/s00245-017-9423-3
[32]	A. Peyravi, General stability and exponential growth for a class of semi-linear wave equations with logarithmic source and memory terms, Appl. Math. Optim., 81 (2020), 545–561. https://doi.org/10.1007/s00245-018-9508-7 doi: 10.1007/s00245-018-9508-7
[33]	M. M. Al-Gharabli, S. A. Messaoudi, The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term, J. Math. Anal. Appl., 454 (2017), 1114–1128. https://doi.org/10.1016/j.jmaa.2017.05.030 doi: 10.1016/j.jmaa.2017.05.030
[34]	M. M. Al-Gharabli, S. A. Messaoudi, Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term, J. Evol. Equ., 18 (2018), 105–125. https://doi.org/10.1007/s00028-017-0392-4 doi: 10.1007/s00028-017-0392-4
[35]	Y. Chen, R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 111664. https://doi.org/10.1016/j.na.2019.111664 doi: 10.1016/j.na.2019.111664
[36]	T. G. Ha, S. H. Park, Existence and general decay for a viscoelastic wave equation with logarithmic nonlinearity, J. Korean Math. Soc., 58 (2021), 1433–1448. https://doi.org/10.4134/JKMS.j210084 doi: 10.4134/JKMS.j210084
[37]	S. H. Park, A general decay result for a von Karman equation with memory and acoustic boundary conditions, J. Appl. Anal. Comput., 12 (2022), 17–30. https://doi.org/10.11948/20200460 doi: 10.11948/20200460
[38]	M. M. Al-Gharabli, M. Balegh, B. Feng, Z. Hajjej, S. A. Messaoudi, Existence and general decay of Balakrhshnan-Talyor viscoelastic equation with nonlinear frictional damping and logarithmic source term, Evol. Equ. Control Theory, 11 (2022), 1149–1173. https://doi.org/10.3934/eect.2021038 doi: 10.3934/eect.2021038
[39]	M. M. Al-Gharabli, A. Guesmia, S. A. Messaoudi, Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity, Appl. Anal., 99 (2018), 50–74. https://doi.org/10.1080/00036811.2018.1484910 doi: 10.1080/00036811.2018.1484910
[40]	S. Nicaise, 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. https://doi.org/10.1137/060648891 doi: 10.1137/060648891
[41]	M. Kirane, B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065–1082. https://doi.org/10.1007/s00033-011-0145-0 doi: 10.1007/s00033-011-0145-0
[42]	Q. Dai, Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885–903. https://doi.org/10.1007/s00033-013-0365-6 doi: 10.1007/s00033-013-0365-6
[43]	S. H. Park, Decay rate estimates for a weak viscoelastic beam equation with time-varying delay, Appl. Math. Lett., 31 (2014), 46–51. https://doi.org/10.1016/j.aml.2014.02.005 doi: 10.1016/j.aml.2014.02.005
[44]	Z. Yang, Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay, Z. Angew. Math. Phys., 66 (2015), 727–745. https://doi.org/10.1007/s00033-014-0429-2 doi: 10.1007/s00033-014-0429-2
[45]	S. H. Park, Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source, Adv. Differ. Equ., 2020 (2020). https://doi.org/10.1186/s13662-020-03037-6 doi: 10.1186/s13662-020-03037-6
[46]	M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Appl., 457 (2018), 134–152. https://doi.org/10.1016/j.jmaa.2017.08.019 doi: 10.1016/j.jmaa.2017.08.019
[47]	L. Gross, Logarithmic Sobolev inequalities, Am. J. Math., 97 (1975), 1061–1083. https://doi.org/10.2307/2373688 doi: 10.2307/2373688
[48]	E. Lieb, M. Loss, Analysis, 2 Eds., American Mathematical Society, 2001.
[49]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Paris: Dunod, 1969.

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