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

Year 2021, Volume: 4 Issue: 2, 82 - 87, 30.06.2021

Fatma Ekinci Erhan Pişkin

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

Cited By: 2

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.