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.