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