Blow up and Exponential Growth to a Kirchhoff-Type Viscoelastic Equation with Degenerate Damping Term

Yıl 2023, Cilt: 11 Sayı: 3, 153 - 163, 02.09.2023

Fatma Ekinci , Erhan Pişkin

https://doi.org/10.36753/mathenot.1005570

Cited By: 1

Öz

In this paper, we consider a Kirchhoff-type viscoelastic equation with degenerate damping term have initial and Dirichlet boundary conditions. We obtain the blow up and exponential growth of solutions with negative initial energy.

Anahtar Kelimeler

Blow up, Degenerate damping, Exponential growth, Kirchhoff-type equation, Viscoelastic equation

Kaynakça

• [1] Barbu, V., Lasiecka, I., Rammaha, M. A.: Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms. Control Cybernetics. 34(3), 665-687 (2005).
• [2] Nishihara, K., Yamada, Y.: On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms. Funkcialaj Ekvacioj. 33, 151-159 (1990).
• [3] Ikehata, R., Matsuyama, T.: On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms. Journal of Mathematical Analysis and Applications. 204, 729-753 (1996).
• [4] Ono, K.: Global existence, decay, and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings. Journal of Differential Equations. 137, 273-301 (1997).
• [5] Taniguchi, T.: Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms. Journal of Mathematical Analysis and Applications. 361(2), 566-578 (2010).
• [6] Han, X., Wang, M.: Global existence and blow-up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source. Mathematische Nachrichten. 284(5-6), 703-716 (2011).
• [7] Pitts, D. R., Rammaha, M. A.: Global existence and nonexistence theorems for nonlinear wave equations. Indiana University Mathematics Journal. 51(6), 1479-1509 (2002).
• [8] Barbu, V., Lasiecka, I., Rammaha, M. A.: Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms. Indiana University Mathematics Journal. 56(3), 995-1022 (2007).
• [9] Barbu, V., Lasiecka, I., Rammaha, M. A.: On nonlinear wave equations with degenerate damping and source terms. Transactions of the American Mathematical Society. 357(7), 2571-2611 (2005).
• [10] Hu, Q., Zhang, H.: Blow up and asymptotic stability of weak solutions to wave equations with nonlinear degenerate damping and source terms. Electronic Journal of Differential Equations. 2007(76), 1-10 (2007).
• [11] Xiao, S., Shubin,W.: A blow-up result with arbitrary positive initial energy for nonlinear wave equations with degenerate damping terms. Journal of Differential Equations. 32, 181-190 (2019).
• [12] Ekinci, F., Pi¸skin, E.: Nonexistence of global solutions for the Timoshenko equation with degenerate damping. Discovering Mathematics(Menemui Matematik). 43(1), 1-8 (2021).
• [13] Pi¸skin, E.: Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms. Open Mathematics. 13, 408-420 (2005).
• [14] Pi¸skin, E., Irkıl, N.: Blow up positive initial-energy solutions for the extensible beam equation with nonlinear damping and source terms. Facta Universitatis, Series: Mathematics and Informatics. 31(3), 645-654 (2016).
• [15] Pi¸skin, E., Yüksekkaya, H.: Non-existence of solutions for a Timoshenko equations with weak dissipation. Mathematica Moravica. 22(2), 1-9 (2018).
• [16] Pereira, D. C., Nguyen, H., Raposo, C. A., Maranhao, C. H. M.: On the solutions for an extensible beam equation with internal damping and source terms. Differential Equations & Applications. 11(3), 367-377 (2019).
• [17] Pereira, D. C., Raposo, C. A., Maranhao, C. H. M., Cattai, A. P.: Global existence and uniform decay of solutions for a Kirchhoff beam equation with nonlinear damping and source term. Differential Equations and Dynamical Systems.(2021). https://doi.org/10.1007/s12591-021-00563-x
• [18] Pi¸skin, E., Ekinci, F.: General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms. Mathematical Methods in the Applied Sciences. 42(16), 1-21 (2019).
• [19] Pi¸skin, E., Ekinci, F.: Local existence and blow up of solutions for a coupled viscoelastic Kirchhoff-type equations with degenerate damping. Miskolc Mathematical Notes. 22(2), 861-874 (2021).
• [20] Pi¸skin, E., Ekinci, F.: Blow up of solutions for a coupled Kirchhoff-type equations with degenerate damping terms. Applications & Applied Mathematics. 14(2), 942-956 (2019).
• [21] Pi¸skin, E., Ekinci, F.: Global existence of solutions for a coupled viscoelastic plate equation with degenerate damping terms. Tbilisi Mathematical Journal. 14, 195-206 (2021).
• [22] Pi¸skin, E., Ekinci, F., Zhang, H.: Blow up, lower bounds and exponential growth to a coupled quasilinear wave equations with degenerate damping terms. Dynamics of Continuous, Discrete and Impulsive Systems. 29, 321-345 (2022).
• [23] Ekinci, F., Pi¸skin, E., Boulaaras, S. M., Mekawy, I.: Global existence and general decay of solutions for a quasilinear system with degenerate damping terms. Journal of function Spaces. 2021, 4316238 (2021).
• [24] Ekinci, F., Pi¸skin, E.: Blow up and exponential growth to a Petrovsky equation with degenerate damping. Universal Journal of Mathematics and Applications. 4(2), 82-87 (2021).
• [25] Ekinci, F., Pi¸skin, E.: Global existence and growth of solutions to coupled degeneratly damped Klein-Gordon equations. Al-Qadisiyah Journal of Pure Science. 27(1), 29-40 (2022).
• [26] Ekinci, F., Pi¸skin, E.: Growth of solutions for fourth order viscoelastic system. Sigma Journal of Engineering and Natural Sciences. 39(5), 41-47 (2021).
• [27] Ekinci, F., Pi¸skin, E., Zennir, K.: 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. 43(4), 705-733 (2022).
• [28] Pi¸skin, E., Ekinci, F.: Blow up, exponential growth of solution for a reaction-diffusion equation with multiple nonlinearities. Tbilisi Mathematical Journal. 12(4), 61-70 (2019).
• [29] Pi¸skin, E., Ekinci, F., Zennir, K.: Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms. Theoretical and Applied Mechanics. 47(1), 123-154 (2020).
• [30] Cordeiro, S. M. S., Pereira, D. C., Ferreira, J., Raposo, C. A.: Global solutions and exponential decay to a Klein–Gordon equation of Kirchhoff-Carrier type with strong damping and nonlinear logarithmic source term. Partial Differential Equations in Applied Mathematics. 3, 100018 (2021).