Global existence and asymptotic behavior for a viscoelastic Kirchhoff equation with a logarithmic nonlinearity, distributed delay and Balakrishnan-Taylor damping terms

Abdelbaki Choucha; Salah Boulaaras; Asma Alharbi; Abdelbaki Choucha; Salah Boulaaras; Asma Alharbi

doi:10.3934/math.2022252

AIMS Mathematics

2022, Volume 7, Issue 3: 4517-4539. doi: 10.3934/math.2022252

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Research article

Global existence and asymptotic behavior for a viscoelastic Kirchhoff equation with a logarithmic nonlinearity, distributed delay and Balakrishnan-Taylor damping terms

1.
Laboratory of Operator Theory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, Algeria
2.
Department of Mathematics, Faculty of Sciences, Amar Teledji Laghouat University, Algeria
3.
Department of Mathematics, College of Sciences and Arts, ArRass, Qassim University, Saudi Arabia

Received: 31 August 2021 Revised: 12 December 2021 Accepted: 16 December 2021 Published: 22 December 2021
MSC : 35B40, 35L20, 35L70, 93D20

A nonlinear viscoelastic Kirchhoff-type equation with a logarithmic nonlinearity, Balakrishnan-Taylor damping, dispersion and distributed delay terms is studied. We establish the global existence of the solutions of the problem and by the energy method we prove an explicit and general decay rate result under suitable hypothesis.
- Kirchhoff equation,
- exponential decay,
- distributed delay term,
- viscoelastic term,
- logarithmic nonlinearity
Citation: Abdelbaki Choucha, Salah Boulaaras, Asma Alharbi. Global existence and asymptotic behavior for a viscoelastic Kirchhoff equation with a logarithmic nonlinearity, distributed delay and Balakrishnan-Taylor damping terms[J]. AIMS Mathematics, 2022, 7(3): 4517-4539. doi: 10.3934/math.2022252

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Abstract

A nonlinear viscoelastic Kirchhoff-type equation with a logarithmic nonlinearity, Balakrishnan-Taylor damping, dispersion and distributed delay terms is studied. We establish the global existence of the solutions of the problem and by the energy method we prove an explicit and general decay rate result under suitable hypothesis.

References

[1]	A. M. Alghamdi, S. Gala, C. Qian, M. A. Ragusa, The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations, Electron. Res. Arch., 28 (2020), 183–193. https://doi.org/10.3934/era.2020012 doi: 10.3934/era.2020012
[2]	R. Adams, J. Fourier, Sobolev space, New York: Academic Press, 2003.
[3]	A. V. Balakrishnan, L. W. Taylor, Distributed parameter nonlinear damping models for flight structures, In: Proceedings: Damping, Washington: Flight Dynamics Lab and Air Force Wright Aeronautical Labs, 1989.
[4]	J. D. Barrow, P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), 5576–5587. https://doi.org/10.1103/PhysRevD.52.5576 doi: 10.1103/PhysRevD.52.5576
[5]	K. Bartkowski, P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A: Math. Theor., 41 (2008), 355201.
[6]	R. W. Bass D. Zes, Spillover nonlinearity, and flexible structures, In: NASA. Langley Research Center, Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, Part 1. Washington: NASA Conference Publication, 1991.
[7]	I. Bialynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Pol. Sci., Ser. Sci., Math., Astron. Phys., 23 (1975), 461–466.
[8]	D. R. Bland, The theory of linear viscoelasticity, Mineola: Courier Dover Publications, 2016.
[9]	S. Boulaaras, A. Choucha, D. Ouchenane, B. Cherif, Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms, Adv. Differ. Equ., 2020 (2020), 310. https://doi.org/10.1186/s13662-020-02772-0 doi: 10.1186/s13662-020-02772-0
[10]	S. Boulaaras, A. Draifia, K. Zennir, General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and logarithmic nonlinearity, Math. Method. Appl. Sci., 42 (2019), 4795–4814. https://doi.org/10.1002/mma.5693 doi: 10.1002/mma.5693
[11]	H. Chen, P. Luo, G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84–98. https://doi.org/10.1016/j.jmaa.2014.08.030 doi: 10.1016/j.jmaa.2014.08.030
[12]	L. Shen, Sign-changing solutions to a N-Kirchhoff equation with critical exponential growth in $R^{N}$, Bull. Malays. Math. Sci. Soc., 44 (2021), 3553–3570. https://doi.org/10.1007/s40840-021-01127-6 doi: 10.1007/s40840-021-01127-6
[13]	S. Boulaaras, A well-posedness and exponential decay of solutions for a coupled Lamé system with viscoelastic term and logarithmic source terms, Appl. Anal., 100 (2021), 1514–1532. https://doi.org/10.1080/00036811.2019.1648793 doi: 10.1080/00036811.2019.1648793
[14]	A. Choucha, D. Ouchenane, S. Boulaaras, Well posedness and stability result for a thermoelastic laminated timoshenko beam with distributed delay term, Math. Method. Appl. Sci., 43 (2020), 9983–10004. https://doi.org/10.1002/mma.6673 doi: 10.1002/mma.6673
[15]	A. Choucha, D. Ouchenane, S. Boulaaras, Blow-up of a nonlinear viscoelastic wave equation with distributed delay combined with strong damping and source terms, J. Nonlinear Funct. Anal., 2020 (2020), 31. https://doi.org/10.23952/jnfa.2020.31 doi: 10.23952/jnfa.2020.31
[16]	A. Choucha, S. Boulaaras, D. Ouchenane, S. Beloul, General decay of nonlinear viscoelastic Kirchhoff equation wit Balakrishnan-Taylor damping, logarithmic nonlinearity and distributed delay terms, Math. Method. Appl. Sci., 44 (2021), 5436–5457. https://doi.org/10.1002/mma.7121 doi: 10.1002/mma.7121
[17]	A. Choucha, S. M. Boulaaras, D. Ouchenane, B. B. Cherif, M. Abdalla, Exponential stability of swelling porous elastic with a viscoelastic damping and distributed delay term, J. Funct. Spaces, 2021 (2021), 5581634. https://doi.org/10.1155/2021/5581634 doi: 10.1155/2021/5581634
[18]	B. D. Coleman, W. Noll, Foundations of linear viscoelasticity, Rev. Mod. Phys., 33 (1961), 239. https://doi.org/10.1103/RevModPhys.33.239 doi: 10.1103/RevModPhys.33.239
[19]	K. Enqvist, J. McDonald, Q-balls and baryogenesis in the MSSM, Phys. Lett. B, 425 (1998), 309–321. https://doi.org/10.1016/S0370-2693(98)00271-8 doi: 10.1016/S0370-2693(98)00271-8
[20]	B. W. Feng, A. Soufyane, Existence and decay rates for a coupled Balakrishnan-Taylor viscoelastic system with dynamic boundary conditions, Math. Method. Sci., 43 (2020), 3375–3391. https://doi.org/10.1002/mma.6127 doi: 10.1002/mma.6127
[21]	B. Gheraibia, N. Boumaza, General decay result of solution for viscoelastic wave equation with Balakrishnan-Taylor damping and a delay term, Z. Angew. Math. Phys., 71 (2020), 198. https://doi.org/10.1007/s00033-020-01426-1 doi: 10.1007/s00033-020-01426-1
[22]	P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Pol. B., 40 (2009), 59–66.
[23]	L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061–1083. https://doi.org/10.2307/2373688 doi: 10.2307/2373688
[24]	M. Khodabakhshi, S. M. Vaezpour, M. R. H. Tavani, Existence results for a Kirchhoff-type problem with singularity, Miskolc Math. Notes, 22 (2021), 351–362. https://doi.org/10.18514/MMN.2021.3429 doi: 10.18514/MMN.2021.3429
[25]	G. Kirchhoff, Vorlesungen uber Mechanik, Leipzig: Tauber, 1883.
[26]	W. J. Liu, B. Q. Zhu, G. Li, D. H. Wang, General decay for a viscoelastic Kirchhoof equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term, Evol. Equ. Control The. 6 (2017), 239–260. https://doi.org/10.3934/eect.2017013
[27]	F. Mesloub, S. Boulaaras, General decay for a viscoelastic problem with not necessarily decreasing kernel, J. Appl. Math Comput., 58 (2018), 647–665. https://doi.org/10.1007/s12190-017-1161-9 doi: 10.1007/s12190-017-1161-9
[28]	N. Doudi, S. Boulaaras, Global existence combined with general decay of solutions for coupled Kirchhoff system with a distributed delay term, RACSAM, 114 (2020), 204. https://doi.org/10.1007/s13398-020-00938-9 doi: 10.1007/s13398-020-00938-9
[29]	N. Mezouar, S. Boulaaras, Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation, Bull. Malays. Math. Sci. Soc., 43 (2020), 725–755. https://doi.org/10.1007/s40840-018-00708-2 doi: 10.1007/s40840-018-00708-2
[30]	N. Mezouar, S. Boulaaras, Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term, Bound. Value. Probl., 2020 (2020), 90. https://doi.org/10.1186/s13661-020-01390-9 doi: 10.1186/s13661-020-01390-9
[31]	C. L. Mu, J. Ma, On a system of nonlinear wave equations with Balakrishnan-Taylor damping, Z. Angew. Math. Phys., 65 (2014), 91–113. https://doi.org/10.1007/s00033-013-0324-2 doi: 10.1007/s00033-013-0324-2
[32]	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
[33]	S. Nicaise, C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21 (2008), 935–958.
[34]	D. Ouchenane, S. Boulaaras, F. Mesloub, General decay for a viscoelastic problem with not necessarily decreasing kernel, Appl. Anal., 98 (2019), 1677–1693. https://doi.org/10.1080/00036811.2018.1437421 doi: 10.1080/00036811.2018.1437421

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