Exponential Growth and Lower Bounds for the Blow-up Time of Solutions for a System of Kirchhoff-type Equations

Yıl 2019, Cilt: 8 Sayı: 2, 1 - 8, 31.12.2019

Erhan Pişkin Fatma Ekinci Veysel Butakın

Öz

This paper deals with the system of Kirchhoff-type equations with a bounded domain Ω⊂Rⁿ. We prove exponential growth of solutions with negative initial energy. Later, we give some estimates for lower bounds of the blow up time.

Anahtar Kelimeler

Exponential growth, Lower bounds for the blow up time

Kaynakça

• R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.
• V. Georgiev, D. Todorova, Existence of solutions of the wave equations with nonlinear damping and source terms, J. Differential Equations, 109 (1994) 295-308.
• S.A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math Nachr, 231 (2001) 105-111.
• S.A. Messaoudi, B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J Math. Anal. Appl., 365 (2010) 277-287.
• E. Pişkin, Blow up of positive initial-energy solutions for coupled nonlinear wave equations with degenerate damping and source terms, Boundary Value Problems, 43 (2015) 1-11.
• E. Pişkin, Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms, Open Math., 13 (2015) 408-420.
• T. Taniguchi, 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 (2010) 566-578.
• M. O. Korpusov, Blow up the solution of a nonlinear system of equations with positive energy, Theoretical and Mathematical Physics, 171 (2012) 725-738.
• E. Pişkin, Blow-up of solutions for coupled nonlinear Klein-Gordon equations with weak damping terms, Mathematical Sciences Letters, 3 (2014) 189-191.
• E. Pişkin, Uniform decay and blow-up of solutions for coupled nonlinear Klein-Gordon equations with nonlinear damping terms, Mathematical Methods in the Applied Sciences, 37 (2014) 3036-3047.
• S.T. Wu, Blow-up results for system of nonlinear Klein-Gordon equations with arbitrary positive initial energy, Electronic Journal of Differential Equations, 2012 (2012) 1-13.
• Y. Ye, Global existence and asymptotic stability for coupled nonlinear Klein-Gordon equations with nonlinear damping terms, Dynamical Systems, 28 (2013) 287-298.
• A. Peyravi, Blow up solutions to a system of higher-order Kirchhoff-type equations with positive initial energy, 21(4) (2017) 767-789.
• E. Pişkin, Lower Bounds for Blow-up Time of Coupled Nonlinear Klein-Gordon Equations, Gulf Journal of Mathematics, 5(2) (2017) 56-61.
• E. Pişkin, On decay and blow up of solutions for a system of Kirchhoff-type equations with damping terms, Middle East Journal of Science, 5(1) (2019) 1-12.
• M. M. Miranda, L. A. Medeiros, On the existence of global solutions of a coupled nonlinear Klein-Gordon equations, Funkcialaj Ekvacioj, 30 (1987) 147-161.
• K. Agre, M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential and Integral Equations, 19 (2006) 1235-1270.
• B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differential Integral Equations, 23 (1--2) (2010) 79-92.
• B. Said-Houari, Global existence and decay of solutions of a nonlinear system of wave equations, Applicable Analysis, 91 (2012) 475-489.