Blow Up and Exponential Growth to a Petrovsky Equation with Degenerate Damping

Fatma Ekinci; Erhan Pişkin

doi:10.32323/ujma.935519

EN

Blow Up and Exponential Growth to a Petrovsky Equation with Degenerate Damping

Abstract

This paper deals with the initial boundary value problem of Petrovsky type equation with degenerate damping. Under some appropriate conditions, we study the finite time blow up and exponential growth of solutions with negative initial energy.

Keywords

Blow up , Exponential growth , Petrovsky equation , Degenerate damping

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 Cybernetics, 34(3) (2005), 665-687.
[2] J. M. Rivera, E. C. Lapa, R. Barreto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44(1) (1996), 61-87.
[3] F. Alabau-Boussouira, P. Cannarsa, D. Sforza, Decay estimates for the second order evolution equation with memory, J. Func. Anal., 245(5) (2008), 1342-1372.
[4] F. Tahamtani, M. Shahrouzi, Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term, Bound. Value Probl., 50(2012) (2012), 1-15.
[5] F. Li, Q. Gao, Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Comput., 274 (2016), 383-392.
[6] L. Liu, F. Sun, Y. Wu, Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level, Bound. Value Probl., 15(2019) (2019), 1-18.
[7] L. Liu, F. Sun, Y. Wu, Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy, SN Partial Differ. Equ. Appl., 1(31) (2020), 1-18.
[8] S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J.Math. Anal. Appl., 265(2) (2002), 296-308.
[9] W. Chen,Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal., 70 (2009), 3203-3208.
[10] E. Pis¸kin, N. Polat, On the Decay of Solutions for a Nonlinear Petrovsky Equation, Math. Sci. Letter, 3(1) (2013), 43-47.

[11] S. Antontsev, J. Ferreira, E. Pis¸kin, Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponent nonlinearities, Electron. J. Diff. Eq., 2021(6) (2021), 1-18.
[12] E. Pis¸kin, T. Uysal, Blow up of the solutions for the Petrovsky equation with fractional damping terms, Malaya J. Math., 6(1) (2018), 85-90.
[13] E. Piskin, Z. C¸ alıs¸ır, Decay and blow up at infinite time of solutions for a logarithmic Petrovsky equation, Tbilisi Math. J., 13(4) (2020), 113-127.
[14] H. Y¨uksekkaya, E. Pis¸kin, Blow up of Solutions for Petrovsky Equation with Delay term, J. Nepal Math. Soc., 4(1) (2021), 76-84.
[15] H. A. Levine, J. Serrin, Global nonexistence theorems for quasilinear evolution with dissipation, Arch. Ration. Mech. Anal., 137 (1997), 341-361.
[16] D. R. Pitts, M. A. Rammaha, Global existence and nonexistence theorems for nonlinear wave equations, Indiana Uni. Math. J., 51(6) (2002), 1479-1509.
[17] V. Barbu, I. Lasiecka, M. A. Rammaha, Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms, IIndiana Uni. Math. J.,, 56(3) (2007), 995-1022.
[18] 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.
[19] Q. Hu, H. Zhang, Blow up and asymptotic stability of weak solutions to wave equations with nonlinear degenerate damping and source terms, Electron. J. Diff. Eq., 2007(76) (2007), 1-10.
[20] S. Xiao, W. Shubin, A blow-up result with arbitrary positive initial energy for nonlinear wave equations with degenerate damping terms, J. Diff. Eq., 32 (2019), 181-190.
[21] F. Ekinci, E. Pis¸kin, Nonexistence of global solutions for the Timoshenko equation with degenerate damping, Discovering Mathematics (Menemui Matematik), 43(1) (2021), 1-8.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Fatma Ekinci
Türkiye

Erhan Pişkin ^*
Türkiye

Publication Date

June 30, 2021

Submission Date

May 10, 2021

Acceptance Date

June 27, 2021

Published in Issue

Year 2021 Volume: 4 Number: 2

DOI

https://doi.org/10.32323/ujma.935519

IZ

https://izlik.org/JA54CT42YL

APA

Ekinci, F., & Pişkin, E. (2021). Blow Up and Exponential Growth to a Petrovsky Equation with Degenerate Damping. Universal Journal of Mathematics and Applications, 4(2), 82-87. https://doi.org/10.32323/ujma.935519

AMA

1.Ekinci F, Pişkin E. Blow Up and Exponential Growth to a Petrovsky Equation with Degenerate Damping. Univ. J. Math. Appl. 2021;4(2):82-87. doi:10.32323/ujma.935519

Chicago

Ekinci, Fatma, and Erhan Pişkin. 2021. “Blow Up and Exponential Growth to a Petrovsky Equation With Degenerate Damping”. Universal Journal of Mathematics and Applications 4 (2): 82-87. https://doi.org/10.32323/ujma.935519.

EndNote

Ekinci F, Pişkin E (June 1, 2021) Blow Up and Exponential Growth to a Petrovsky Equation with Degenerate Damping. Universal Journal of Mathematics and Applications 4 2 82–87.

IEEE

[1]F. Ekinci and E. Pişkin, “Blow Up and Exponential Growth to a Petrovsky Equation with Degenerate Damping”, Univ. J. Math. Appl., vol. 4, no. 2, pp. 82–87, June 2021, doi: 10.32323/ujma.935519.

ISNAD

Ekinci, Fatma - Pişkin, Erhan. “Blow Up and Exponential Growth to a Petrovsky Equation With Degenerate Damping”. Universal Journal of Mathematics and Applications 4/2 (June 1, 2021): 82-87. https://doi.org/10.32323/ujma.935519.

JAMA

1.Ekinci F, Pişkin E. Blow Up and Exponential Growth to a Petrovsky Equation with Degenerate Damping. Univ. J. Math. Appl. 2021;4:82–87.

MLA

Ekinci, Fatma, and Erhan Pişkin. “Blow Up and Exponential Growth to a Petrovsky Equation With Degenerate Damping”. Universal Journal of Mathematics and Applications, vol. 4, no. 2, June 2021, pp. 82-87, doi:10.32323/ujma.935519.

Vancouver

1.Fatma Ekinci, Erhan Pişkin. Blow Up and Exponential Growth to a Petrovsky Equation with Degenerate Damping. Univ. J. Math. Appl. 2021 Jun. 1;4(2):82-7. doi:10.32323/ujma.935519