Well-Posedness and Stability for a System of Klein-Gordon Equations

Main Article Content

Naaima Latioui
Amar Guesmia
Amar Ouaoua

Abstract

In this paper, we study the weak existence of solution for a non-linear hyperbolic coupled system of Klein-Gordon equations with memory and source terms using the Faedo-Galerkin method techniques and compactness results, we have demonstrated the uniqueness of the solution by using the classical technique. In addition, we show that the solution remains stable over time. The reaction of the proper Lyapunov function is the primary tool of the proof.

Article Details

References

  1. R.A. Adams, J.J.F. Fournier, Sobolev spaces, Academic Press, New York (2003).
  2. D. Andrade, A Mognon, Global solutions for a system of Klein-Gordon equations with memory, Bol. Soc. Paran. Mat. Ser. 3, 21 (2003), 127–136.
  3. S. Brrimi, S.A. Messaoudi, Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, Electron. J. Differ. Equ. 2004 (2004), No. 88, pp. 1–10.
  4. Y. Boukhatem and B. Benabdrrahmane, A. Rahmoune, Mèthode de faedo-Galerkin pour un problème aux limites non linéaire. Anale Universităţii Oradea, Fasc. Math. Tom XVI (2009), 167-181.
  5. C.L. Frota, A. Vicente, A hyperbolic system of Klein-Gordon type with acoustic boundary conditions, Int. J. Pure Appl. Math. 47 (2008), 185-198.
  6. V. Komornik, E. A Zuazua, A direct method for boundary stabilization of the wave equation, J. Math. Pure Appl. 69 (1990), 33-54.
  7. A.T. Louredo, M.M. Miranda, Nonlinear boundary dissipation for a coupled system of Klein-Gordon equations, Electron. J. Differ. Equ. 2010 (2010), No. 120, pp. 1-19.
  8. M.M. Miranda, L.A. Medeiros, On existence of global solutions of a coupled nonlinear Klein-Gordon equation, Funkcialaj Ekvacioj 30 (1987), 147-161.
  9. A. Ouaoua, M. Maouni, A. Khaldi, Exponential decay of solutions with L p norm for class to semilinear wave equation with damping and source terms, Open J. Math Anal. 4 (2020), 123-131. https://doi.org/10.30538/psrp-oma2020.0071.
  10. K. Zennir, A. Guesmia, Existence of solutions to nonlinear κth-order coupled Klein-Gordon equations with nonlinear sources and memory terms, Appl. Math. E-Notes, 15 (2015), 121-136.
  11. S. Zheng, Nonlinear evolution equations, Chapman & Hall/ CRC Monographs and Surveys in Pure and Applied Mathematics 133, Chapman & Hall/CRC, Boca Raton, (2004).
  12. Y. Zhijian, Initial boundary value problem for a class of non-linear strongly damped wave equations, Math. Meth. Appl. Sci. 26 (2003), 1047–1066. https://doi.org/10.1002/mma.412.