Local Existence and Blow Up of Solutions for a Coupled Viscoelastic Kirchhoff-Type Equations with Degenerate Damping

Abstract

In this paper, we consider the initial boundary value problem of a coupled viscoelastic Kirchhofftype equations with degenerate damping. Firstly, we prove a local existence theorem by using the Faedo-Galerkin approximations. Then, we study blow up of solutions when initial energy is possitive.

Keywords

• Local existence
• Blow up
• Degenerate damping

References

1. 1 E. Pişkin, F. Ekinci, General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms, Math. Method Appl. Sci. 42(16) (2019), 1-21.
2. 2 S.T. Wu, General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms, J. Math. Anal. Appl. 406 (2013), 34-48.
3. 3 E. Pişkin, F. Ekinci, K. Zennir, Local existence and blow-up of solutions for coupled viscoelastic wave equations with degenerate damping terms, Theoret. Appl. Mech. 47(1) (2020), 123-154.
4. 4 B. Feng, Y. Qin, M. Zhang, General decay for a system of nonlinear viscoelastic wave equations with weak damping, Bound. Value Probl. 146 (2012), 1-11.
5. 5 X.S. Han, M.X. Wang, Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal. TMA 71 (2009) 5427-5450.
6. 6 B. Said-Houari, S.A. Messaoudi, A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wave equations, NoDEA 18 (2011), 659-684.
7. 7 S.A. Messaoudi, B. S. 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.
8. 8 E. Pişkin, Global nonexistence of solutions for a system of viscoelastic wave equations with weak damping terms, Malaya J. Mat. 3(2) (2015), 168-174 .

9. 9 E. Pişkin, A lower bound for the blow up time of a system of viscoelastic wave equations with nonlinear damping and source terms, J. Nonlinear Funct. Anal. 2017 (2017), 1-9.
10. 10 A. Benaissa, D. Ouchenane, K. Zennir, Blow up of positive initial energy solutions to system of nonlinear wave equations with degenerate damping and source terms, Nonliner Studies 19(4) (2012), 523-535.
11. 11 K. Zennir, Growth of solutions to system of nonlinear wave equations with degenerate damping and strong sources, Nonlinear Anal. Appl. (2013), 1-11.
12. 12 M.A. Rammaha, S. Sakuntasathien, Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal. TMA 72 (2010), 2658-2683.
13. 13 E. Vitillaro Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (1999), 155-182.