ON DECAY AND BLOW UP OF SOLUTIONS FOR A SYSTEM OF KIRCHHOFF TYPE EQUATIONS WITH DAMPING TERMS

Abstract

 In this paper, we investigate system of Kirchhoff type equations with bounded domain. We obtain decay of solutions by using multiplier method. Later, we will proved blow up results for negative inital energy.

Keywords

• Decay,Blow up

References

1. 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.
2. Kirchhoff, G. Mechanik, Teubner, (1883).
3. K. Ono, Global existence, decay, and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137 (1997) 273-301.
4. S.T. Wu, L.Y. Tsai, Blow-up solutions for some nonlinear wave equations of Kirchhoff type with some dissipation, Nonlinear Anal., 65 (2006) 243-264.
5. A. Benaissa, S. A. Messaoudi, Blow-up of solutions for the Kirchhoff equation of q-Laplacian type with nonlinear dissipation, Colloquium Mathematicum, 94 (2002) 103-109.
6. T. Matsuyama, R. Ikehata, On global solutions and energy decay for the wave equations of the Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl., 204 (1996) 729-753.
7. T. Taniguchi, Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms, J. Math. Anal. Appl., 361 (2010) 566-578.
8. V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term, J. Differ. Equations, 109 (1994) 295-308.

9. K. Agre, M.A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Diff. Integral Eqns., 19 (11) (2006) 1235-1270.
10. B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Diff. Integral Eqns., 23 (2010) 79-92.
11. B.Said-Houari, Global existence and decay of solutions of a nonlinear system of wave equations, Appl. Anal., 91(3) (2012) 475-489.
12. R. A. Adams, J. J. F. Fournier, Sobolev Spaces, Academic Press, (2003).
13. S.A. Messaoudi, On the decay of solutions for a class of quasilinear hyperbolic equations with non-linear damping and source terms, Mat. Meth. Appl. Sci., 28 (2005) 1819-1828.
14. M.O. Korpusov, Blow up the solution of a nonlinear system of equations with positive energy, Theoretical and Mathematical Physics. 171(3) (2012) 725-738.
15. S.A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachr. 231 (2001) 105-111.
16. M.M. Miranda and L.A. Medeiros, On the existence of global solutions of a coupled nonlinear Klein-Gordon equations, Funkcial Ekvac. 30 (1987) 147-161.
17. E. Pişkin, Uniform decay and blow-up of solutions for coupled nonlinear Klein-Gordon equations with nonlinear damping terms, Math. Meth. Appl. Sci. 37(18) (2014) 3036-3047.
18. E. Pişkin, Blow-up of solutions for coupled nonlinear Klein-Gordon equations with weak damping terms, Math. Sci. Letters, 3(3) (2014) 189-191.
19. 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.
20. Y. Ye, Global existence and asymptotic stability for coupled nonlinear Klein-Gordon equations with nonlinear damping terms. Dynamical Syst. 28(2) (2013) 287-298.
21. S.T. Wu, Blow-up results for system of nonlinear Klein-Gordon equations with arbitrary positive initial energ, Electron J Diff Equations. 2012 (2012) 1-13.