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Lithuanian Mathematical Journal

, Volume 57, Issue 1, pp 80–108 | Cite as

A Robin–Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral

  • Nguyen Huu Nhan
  • Le Thi Phuong Ngoc
  • Tran Minh Thuyet
  • Nguyen Thanh Long
Article

Abstract

In this paper, we consider the Robin–Dirichlet problem for a nonlinear wave equation with the source term containing a nonlinear integral. Using the Faedo–Galerkin method and the linearization method for nonlinear terms, we prove the existence and uniqueness of a weak solution. We also discuss an asymptotic expansion of high order in a small parameter of a weak solution.

Keywords

Faedo–Galerkinmethod linear recurrent sequence Robin–Dirichlet conditions asymptotic expansion in many small parameters 

MSC

35L20 35L70 35Q72 

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References

  1. 1.
    S.A. Beilin, On a Mixed nonlocal problem for a wave equation, Electron. J. Differ. Equ., 2006(103):1–10, 2006.MATHGoogle Scholar
  2. 2.
    H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.CrossRefGoogle Scholar
  3. 3.
    T. Caughey and J. Ellison, Existence, uniqueness and stability of solutions of a class of nonlinear differential equations, J. Math. Anal. Appl., 51:1–32, 1975.CrossRefMATHGoogle Scholar
  4. 4.
    C. Corduneanu, Integral Equations and Applications, Cambridge University Press, New York, 1991.CrossRefMATHGoogle Scholar
  5. 5.
    K. Deimling, Nonlinear Functional Analysis, Springer, New York, 1985.CrossRefMATHGoogle Scholar
  6. 6.
    F. Ficken and B. Fleishman, Initial value problems and time periodic solutions for a nonlinear wave equation, Commun. Pure Appl. Math., 10:331–356, 1957.CrossRefMATHGoogle Scholar
  7. 7.
    J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Gauthier-Villars, Paris, 1969.Google Scholar
  8. 8.
    N.T. Long, On the nonlinear wave equation u tt − b(t,u2 ,u x2)u xx = f(x, t, u, u x , u t ,u2 ,u x2) associated with the mixed homogeneous conditions, J. Math. Anal. Appl., 306(1):243–268, 2005.CrossRefMATHGoogle Scholar
  9. 9.
    N.T. Long and T.N. Diem, On the nonlinear wave equation u tt − u xx = f(x, t, u, u x , u t) associated with the mixed homogeneous condition, Nonlinear Anal., Theory Methods Appl., 29(11):1217–1230, 1997.Google Scholar
  10. 10.
    N.T. Long and L.T.P. Ngoc, On a nonlinear wave equation with boundary conditions of two-point type, J. Math. Anal. Appl., 385(2):1070–1093, 2012.CrossRefMATHGoogle Scholar
  11. 11.
    N.T. Long and L.X. Truong, Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary, Electron. J. Differ. Equ., 2007(48):1–19, 2007.Google Scholar
  12. 12.
    L.T.P. Ngoc, L.N.K. Hang, and N.T. Long, On a nonlinear wave equation associated with the boundary conditions involving convolution, Nonlinear Anal., Theory Methods Appl., 70(11):3943–3965, 2009.Google Scholar
  13. 13.
    L.T.P. Ngoc and N.T. Long, Existence and exponential decay for a nonlinear wave equation with a nonlocal boundary condition, Commun. Pure Appl. Anal., 12(5):2001–2029, 2013.CrossRefMATHGoogle Scholar
  14. 14.
    L.T.P. Ngoc, L.K. Luan, T.M. Thuyet, and N.T. Long, On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions, Nonlinear Anal., Theory Methods Appl., 71(11):5799–5819, 2009.Google Scholar
  15. 15.
    L.T.P. Ngoc, N.A. Triet, and N.T. Long, On a nonlinear wave equation involving the term \( -\frac{\partial }{\partial x}\left(\mu \left(x,\ t,\ u,{\left\Vert {u}_x\right\Vert}^2\right){u}_x\right) \): Linear approximation and asymptotic expansion of solution in many small parameters, Nonlinear Anal., Real World Appl., 11(4):2479–2510, 2010.Google Scholar
  16. 16.
    P.H. Rabinowitz, Periodic solutions of nonlinear hyperbolic differential equations, Commun. Pure Appl. Math., 20(1):145–205, 1967.CrossRefMATHGoogle Scholar
  17. 17.
    M.L. Santos, Asymptotic behavior of solutions to wave with a memory condition at the boundary, Electron. J. Differ. Equ., 73:1–11, 2001.Google Scholar
  18. 18.
    R.E. Showater, Hilbert Space Methods for Partial Differential Equations, Electron. J. Differ. Equ., Monogr. 01, Southwest Texas State University, San Marcos, TX, 1994.Google Scholar
  19. 19.
    L.X. Truong, L.T.P. Ngoc, and N.T. Long, High-order iterative schemes for a nonlinear Kirchhoff–Carrier wave equation associated with the mixed homogeneous conditions, Nonlinear Anal., Theory Methods Appl., 71(1–2):467–484, 2009.Google Scholar
  20. 20.
    L.X. Truong, L.T.P. Ngoc, and N.T. Long, The N-order iterative schemes for a nonlinear Kirchhoff–Carrier wave equation associated with the mixed inhomogeneous conditions, Appl. Math. Comput., 215(5):1908–1925, 2009.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Nguyen Huu Nhan
    • 1
    • 4
  • Le Thi Phuong Ngoc
    • 2
  • Tran Minh Thuyet
    • 3
  • Nguyen Thanh Long
    • 4
  1. 1.Dong Nai UniversityBien Hoa CityVietnam
  2. 2.University of Khanh HoaNha Trang CityVietnam
  3. 3.Department of MathematicsUniversity of Economics of Ho Chi Minh CityHo Chi Minh CityVietnam
  4. 4.Department of Mathematics and Computer Science, University of Natural ScienceVietnam National University – Ho Chi Minh CityHo Chi Minh CityVietnam

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