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

Year 2019, Volume: 8 Issue: 2, 1 - 8, 31.12.2019

Erhan Pişkin , Fatma Ekinci , Veysel Butakın

https://izlik.org/JA46SU82GL

Abstract

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.

Keywords

Exponential growth , Lower bounds for the blow up time

References

• 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.