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Stability of Solutions for a Krichhoff-Type Plate Equation with Degenerate Damping

Year 2022, Volume: 5 Issue: 3, 131 - 136, 30.09.2022

Fatma Ekinci Erhan Pişkin

https://doi.org/10.33434/cams.1118409

Abstract

We investigate a Kirchhoff type plate equation with degenerate damping term. By potential well theory, we show the asymptotic stability of energy in the presence of a degenerate damping.

Keywords

Stability Degenerate damping. Kirchhoff-type equation

References

  • [1] V. Barbu, I. Lasiecka, M. A. Rammaha, Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms, Control Cybernet., 34(3) (2005), 665-687.
  • [2] S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars, J Appl Mech., 17 (1950), 35-36.
  • [3] J. A. Esquivel-Avila, Dynamic analysis of a nonlinear Timoshenko equation, Abstr Appl Anal., 2010 (2011), 1-36.
  • [4] J. A. Esquivel-Avila, Global attractor for a nonlinear Timoshenko equation with source terms, Math Sci., 7(32) (2013), 1-8.
  • [5] E. Pis¸kin, Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Math., 13 (2005), 408-420.
  • [6] E. Pis¸kin, N. Irkıl, Blow up positive initial-energy solutions for the extensible beam equation with nonlinear damping and source terms, Ser. Math. Inform., 31(3) (2016), 645-654.
  • [7] D. C. Pereira, H. Nguyen, C. A. Raposo, C. H. M. Maranhao, On the solutions for an extensible beam equation with internal damping and source terms, Differential Equations & Applications, 11(3) (2019), 367-377.
  • [8] E. Pis¸kin, H. Y¨uksekkaya, Non-existence of solutions for a Timoshenko equations with weak dissipation, Math Morav., 22(2) (2018), 1-9.
  • [9] H. A. Levine, J. Serrin, Global nonexistence theorems for quasilinear evolution with dissipation, Arch. Rational Mech. Anal, 137 (1997), 341-361.
  • [10] D. R. Pitts, M. A. Rammaha, Global existence and nonexistence theorems for nonlinear wave equations, Indiana Uni. Math. J., 51(6) (2002), 1479-1509.
  • [11] V. Barbu, I. Lasiecka, M. A. Rammaha, Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms, Indiana Uni. Math. J., 56(3) (2007), 995-1022.
  • [12] V. Barbu, I. Lasiecka, M.A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc., 357(7) (2005), 2571-2611.
  • [13] Q. Hu, H. Zhang, Blow up and asymptotic stability of weak solutions to wave equations with nonlinear degenerate damping and source terms, Electron J. Differ. Eq., 2007 (76) (2007), 1-10.
  • [14] S. Xiao, W. Shubin, A blow-up result with arbitrary positive initial energy for nonlinear wave equations with degenerate damping terms, J. Part. Diff. Eq., 32 (2019), 181-190.
  • [15] F. Ekinci, E. Pis¸kin, Nonexistence of global solutions for the Timoshenko equation with degenerate damping, Menemui Mat., 43(1) (2021), 1-8.
  • [16] E. Pis¸kin, F. Ekinci, General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms, Math. Meth. App. Sci., 42(16) (2019), 1-21.
  • [17] E. Pis¸kin, F. Ekinci, Local existence and blow up of solutions for a coupled viscoelastic Kirchhoff-type equations with degenerate damping, Miskolc Math. Notes, 22(2) (2021), 861-874.
  • [18] E. Pis¸kin, F. Ekinci, Blow up of solutions for a coupled Kirchhoff-type equations with degenerate damping terms, Applications & Applied Mathematics, 14(2) (2019), 942-956.
  • [19] E. Pis¸kin, F. Ekinci, K. Zennir, Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms, Theor. Appl. Mech., 47(1) (2020), 123-154.
  • [20] E. Pis¸kin, F. Ekinci, Global existence of solutions for a coupled viscoelastic plate equation with degenerate damping terms, Tbilisi Math. J., 14(2021), 195-206.
  • [21] E. Pis¸kin, F. Ekinci, H. Zhang, Blow up, lower bounds and exponential growth to a coupled quasilinear wave equations with degenerate damping terms, Dynamics of Continuous, Discrete and Impulsive Systems, In press.
  • [22] F. Ekinci, E. Pis¸kin, S. M. Boulaaras, I. Mekawy, Global existence and general decay of solutions for a quasilinear system with degenerate damping terms, J. Funct. Spaces, 2021 (2021), 4316238.
  • [23] F. Ekinci, E. Pis¸kin, Blow up and exponential growth to a Petrovsky equation with degenerate damping, Univers. J. Math. Appl., 4(2) (2021), 82-87.
  • [24] F. Ekinci, E. Pis¸kin, Global existence and growth of solutions to coupled degeneratly damped Klein-Gordon equations, Al-Qadisiyah Journal of Pure Science, 27(1) (2022), 29-40.
  • [25] F. Ekinci, E. Pis¸kin, Growth of solutions for fourth order viscoelastic system, Sigma Journal of Engineering and Natural Sciences, (2021), 1-7.
  • [26] F. Ekinci, E. Pis¸kin, K. Zennir, Existence, blow up and growth of solutions for a coupled quasi-linear viscoelastic Petrovsky equations with degenerate damping terms, Journal of Information and Optimization Sciences, (2021), 1-29.
  • [27] E. Pis¸kin, F. Ekinci, Blow up, exponential growth of solution for a reaction-diffusion equation with multiple nonlinearities, Tbilisi Math. J., 12(4) (2019), 61-70.

Year 2022, Volume: 5 Issue: 3, 131 - 136, 30.09.2022

Fatma Ekinci Erhan Pişkin

https://doi.org/10.33434/cams.1118409

Abstract

References

  • [1] V. Barbu, I. Lasiecka, M. A. Rammaha, Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms, Control Cybernet., 34(3) (2005), 665-687.
  • [2] S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars, J Appl Mech., 17 (1950), 35-36.
  • [3] J. A. Esquivel-Avila, Dynamic analysis of a nonlinear Timoshenko equation, Abstr Appl Anal., 2010 (2011), 1-36.
  • [4] J. A. Esquivel-Avila, Global attractor for a nonlinear Timoshenko equation with source terms, Math Sci., 7(32) (2013), 1-8.
  • [5] E. Pis¸kin, Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Math., 13 (2005), 408-420.
  • [6] E. Pis¸kin, N. Irkıl, Blow up positive initial-energy solutions for the extensible beam equation with nonlinear damping and source terms, Ser. Math. Inform., 31(3) (2016), 645-654.
  • [7] D. C. Pereira, H. Nguyen, C. A. Raposo, C. H. M. Maranhao, On the solutions for an extensible beam equation with internal damping and source terms, Differential Equations & Applications, 11(3) (2019), 367-377.
  • [8] E. Pis¸kin, H. Y¨uksekkaya, Non-existence of solutions for a Timoshenko equations with weak dissipation, Math Morav., 22(2) (2018), 1-9.
  • [9] H. A. Levine, J. Serrin, Global nonexistence theorems for quasilinear evolution with dissipation, Arch. Rational Mech. Anal, 137 (1997), 341-361.
  • [10] D. R. Pitts, M. A. Rammaha, Global existence and nonexistence theorems for nonlinear wave equations, Indiana Uni. Math. J., 51(6) (2002), 1479-1509.
  • [11] V. Barbu, I. Lasiecka, M. A. Rammaha, Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms, Indiana Uni. Math. J., 56(3) (2007), 995-1022.
  • [12] V. Barbu, I. Lasiecka, M.A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc., 357(7) (2005), 2571-2611.
  • [13] Q. Hu, H. Zhang, Blow up and asymptotic stability of weak solutions to wave equations with nonlinear degenerate damping and source terms, Electron J. Differ. Eq., 2007 (76) (2007), 1-10.
  • [14] S. Xiao, W. Shubin, A blow-up result with arbitrary positive initial energy for nonlinear wave equations with degenerate damping terms, J. Part. Diff. Eq., 32 (2019), 181-190.
  • [15] F. Ekinci, E. Pis¸kin, Nonexistence of global solutions for the Timoshenko equation with degenerate damping, Menemui Mat., 43(1) (2021), 1-8.
  • [16] E. Pis¸kin, F. Ekinci, General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms, Math. Meth. App. Sci., 42(16) (2019), 1-21.
  • [17] E. Pis¸kin, F. Ekinci, Local existence and blow up of solutions for a coupled viscoelastic Kirchhoff-type equations with degenerate damping, Miskolc Math. Notes, 22(2) (2021), 861-874.
  • [18] E. Pis¸kin, F. Ekinci, Blow up of solutions for a coupled Kirchhoff-type equations with degenerate damping terms, Applications & Applied Mathematics, 14(2) (2019), 942-956.
  • [19] E. Pis¸kin, F. Ekinci, K. Zennir, Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms, Theor. Appl. Mech., 47(1) (2020), 123-154.
  • [20] E. Pis¸kin, F. Ekinci, Global existence of solutions for a coupled viscoelastic plate equation with degenerate damping terms, Tbilisi Math. J., 14(2021), 195-206.
  • [21] E. Pis¸kin, F. Ekinci, H. Zhang, Blow up, lower bounds and exponential growth to a coupled quasilinear wave equations with degenerate damping terms, Dynamics of Continuous, Discrete and Impulsive Systems, In press.
  • [22] F. Ekinci, E. Pis¸kin, S. M. Boulaaras, I. Mekawy, Global existence and general decay of solutions for a quasilinear system with degenerate damping terms, J. Funct. Spaces, 2021 (2021), 4316238.
  • [23] F. Ekinci, E. Pis¸kin, Blow up and exponential growth to a Petrovsky equation with degenerate damping, Univers. J. Math. Appl., 4(2) (2021), 82-87.
  • [24] F. Ekinci, E. Pis¸kin, Global existence and growth of solutions to coupled degeneratly damped Klein-Gordon equations, Al-Qadisiyah Journal of Pure Science, 27(1) (2022), 29-40.
  • [25] F. Ekinci, E. Pis¸kin, Growth of solutions for fourth order viscoelastic system, Sigma Journal of Engineering and Natural Sciences, (2021), 1-7.
  • [26] F. Ekinci, E. Pis¸kin, K. Zennir, Existence, blow up and growth of solutions for a coupled quasi-linear viscoelastic Petrovsky equations with degenerate damping terms, Journal of Information and Optimization Sciences, (2021), 1-29.
  • [27] E. Pis¸kin, F. Ekinci, Blow up, exponential growth of solution for a reaction-diffusion equation with multiple nonlinearities, Tbilisi Math. J., 12(4) (2019), 61-70.
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Fatma Ekinci

Erhan Pişkin

Publication Date September 30, 2022
Submission Date May 18, 2022
Acceptance Date August 2, 2022
Published in Issue Year 2022 Volume: 5 Issue: 3

Cite

APA Ekinci, F., & Pişkin, E. (2022). Stability of Solutions for a Krichhoff-Type Plate Equation with Degenerate Damping. Communications in Advanced Mathematical Sciences, 5(3), 131-136. https://doi.org/10.33434/cams.1118409
AMA Ekinci F, Pişkin E. Stability of Solutions for a Krichhoff-Type Plate Equation with Degenerate Damping. Communications in Advanced Mathematical Sciences. September 2022;5(3):131-136. doi:10.33434/cams.1118409
Chicago Ekinci, Fatma, and Erhan Pişkin. “Stability of Solutions for a Krichhoff-Type Plate Equation With Degenerate Damping”. Communications in Advanced Mathematical Sciences 5, no. 3 (September 2022): 131-36. https://doi.org/10.33434/cams.1118409.
EndNote Ekinci F, Pişkin E (September 1, 2022) Stability of Solutions for a Krichhoff-Type Plate Equation with Degenerate Damping. Communications in Advanced Mathematical Sciences 5 3 131–136.
IEEE F. Ekinci and E. Pişkin, “Stability of Solutions for a Krichhoff-Type Plate Equation with Degenerate Damping”, Communications in Advanced Mathematical Sciences, vol. 5, no. 3, pp. 131–136, 2022, doi: 10.33434/cams.1118409.
ISNAD Ekinci, Fatma - Pişkin, Erhan. “Stability of Solutions for a Krichhoff-Type Plate Equation With Degenerate Damping”. Communications in Advanced Mathematical Sciences 5/3 (September 2022), 131-136. https://doi.org/10.33434/cams.1118409.
JAMA Ekinci F, Pişkin E. Stability of Solutions for a Krichhoff-Type Plate Equation with Degenerate Damping. Communications in Advanced Mathematical Sciences. 2022;5:131–136.
MLA Ekinci, Fatma and Erhan Pişkin. “Stability of Solutions for a Krichhoff-Type Plate Equation With Degenerate Damping”. Communications in Advanced Mathematical Sciences, vol. 5, no. 3, 2022, pp. 131-6, doi:10.33434/cams.1118409.
Vancouver Ekinci F, Pişkin E. Stability of Solutions for a Krichhoff-Type Plate Equation with Degenerate Damping. Communications in Advanced Mathematical Sciences. 2022;5(3):131-6.
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