Mathematical behavior of solutions of the Kirchhoff type equation with logarithmic nonlinearity
DOI:
https://doi.org/10.5269/bspm.51971Abstract
We consider the existence and decay estimates of solutions for Kirchhoff type equation with damping logarithmic source term. We proved global existence of solutions under suitable conditons by potential well method and the decay estimates result of the solutions for subcritical energy level.
References
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18. E. Piskin, N. Irkıl, Mathematical behaviour of solutions of the Kirchhoff type equation with logarithmic nonlinearity, AIP Conference Proceedings, 2183 (1) (2019), 090004-090008. https://doi.org/10.1063/1.5136208
19. J. Sun, On the Kirchhoff type equations in RN , arXiv: 1908.01326v1
20. H. Xu, Existence of positive solutions for the nonlinear Kirchhoff type equations in RN , J. Math. Anal. Appl., 482 (2) (2020), 1-15. https://doi.org/10.1016/j.jmaa.2019.123593
21. Y. Yang, J. Li, T. Yu, Qualitative analysis of solutions for a class of Kirchhoff equation with linear strong damping term, nonlinear weak damping term and power-type logarithmic source term, Appl. Numer. Math., 141 (2019), 263-285. https://doi.org/10.1016/j.apnum.2019.01.002
22. Y. Ye, Logarithmic viscoelastic wave equation in three dimensional space, Appl. Anal., (2019) 1-18 (in press).
23. J. Wang, Existence and uniqueness of positive solutions for Kirchhoff type beam equations, arXiv:2003.04746v1. https://doi.org/10.14232/ejqtde.2020.1.61
24. S. Woinowsky- Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. https://doi.org/10.1115/1.4010053
25. H. Zhang , G. Liu, Q. Hu, Initial boundary value problem for class wave equation with logarithmic source term, Acta Math. Scientia, (2019), 1-13 (in press).
2. 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
3. W. F. Ames , Nonlinear Partial Differantial Equations in Engineering, Academic Press, (1972).
4. I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys.,100(1-2) (1976), 62-93. https://doi.org/10.1016/0003-4916(76)90057-9
5. S. Boulaaras, A. Draifia, M. Alnegga, Polynomial decay rate for Kirchhoff type in viscoelasticity with logarithmic nonlinearity and not necassarily decreasing kernel, Symmetry, 11(2) (2019), 1-24. https://doi.org/10.3390/sym11020226
6. T. Cazenave, A. Haraux, Equations d'evolution avec non lin'earit'e logarithmique, Ann. Fac. Sci. Toulouse 2(1) (1980), 21-51. https://doi.org/10.5802/afst.543
7. 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., (2020) 1-39 (in press). https://doi.org/10.1016/j.na.2019.111664
8. P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Pol. B 40(1) (2009), 59-66.
9. S. Goyal, P. K. Mishra, K.Sreenadh, n-Kirchhoff type equations with exponential nonlinearities, RACSAM, 110 (2016), 219-245. https://doi.org/10.1007/s13398-015-0230-x
10. G. Li, L. Hong, W. Liu, Global nonexistence of solutions for viscoelastic wave equations of Kirchoff type with high energy, J. Funct. Anal, Appl. Spaces, no: 530861 (2012),1-15. https://doi.org/10.1155/2012/530861
11. T. Matsuyama, R. Ikehata, On global solutions and energy decay for the wave equations of Kirchoff type with nonlinear damping terms, J. Math. Anal. Appl. 204 (3) (1996) , 729-753. https://doi.org/10.1006/jmaa.1996.0464
12. C. N. Le , X. T. Le, Global solution and blow up for a class of Pseudo p-Laplacian evolution equations with logarithmic nonlinearity, Comput. Math. Appl., 73(9) (2017), 2076-2091. https://doi.org/10.1016/j.camwa.2017.02.030
13. H. Li, Blow-up of Solutions to a p-Kirchhoff-Type Parabolic Equation with General Nonlinearity, J. Dyn. Control Syst., 26 (2020), 383-392. https://doi.org/10.1007/s10883-019-09463-4
14. K. Ono, On global solutions and blow up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 216 (1997), 321-342. https://doi.org/10.1006/jmaa.1997.5697
15. 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
16. E. Piskin, N. Irkıl, Mathematical Behavior of Solutions of Fourth-Order Hyperbolic Equation with Logarithmic Source Term, CPOST, 2(1) (2019), 27-36.
17. E. Piskin, N. Irkıl, Well-posedness results for a sixth-order logarithmic Boussinesq equation, Filomat, 33(13) (2019), 3985-4000. https://doi.org/10.2298/FIL1913985P
18. E. Piskin, N. Irkıl, Mathematical behaviour of solutions of the Kirchhoff type equation with logarithmic nonlinearity, AIP Conference Proceedings, 2183 (1) (2019), 090004-090008. https://doi.org/10.1063/1.5136208
19. J. Sun, On the Kirchhoff type equations in RN , arXiv: 1908.01326v1
20. H. Xu, Existence of positive solutions for the nonlinear Kirchhoff type equations in RN , J. Math. Anal. Appl., 482 (2) (2020), 1-15. https://doi.org/10.1016/j.jmaa.2019.123593
21. Y. Yang, J. Li, T. Yu, Qualitative analysis of solutions for a class of Kirchhoff equation with linear strong damping term, nonlinear weak damping term and power-type logarithmic source term, Appl. Numer. Math., 141 (2019), 263-285. https://doi.org/10.1016/j.apnum.2019.01.002
22. Y. Ye, Logarithmic viscoelastic wave equation in three dimensional space, Appl. Anal., (2019) 1-18 (in press).
23. J. Wang, Existence and uniqueness of positive solutions for Kirchhoff type beam equations, arXiv:2003.04746v1. https://doi.org/10.14232/ejqtde.2020.1.61
24. S. Woinowsky- Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36. https://doi.org/10.1115/1.4010053
25. H. Zhang , G. Liu, Q. Hu, Initial boundary value problem for class wave equation with logarithmic source term, Acta Math. Scientia, (2019), 1-13 (in press).
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2022-02-05
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