Open Access

Global solutions for a nonlinear hyperbolic equation with boundary memory source term

Journal of Inequalities and Applications20062006:60734

https://doi.org/10.1155/JIA/2006/60734

Received: 21 January 2005

Accepted: 17 August 2005

Published: 20 April 2006

Abstract

We study a nonlinear hyperbolic equation with boundary memory source term. By the use of Galerkin procedure, we prove the global existence and the decay property of solution.

[12345678910111213141516]

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Tianjin University of Technology and Education
(2)
Department of Mathematics, Southeast University

References

  1. Aassila M, Cavalcanti MM, Domingos Cavalcanti VN: Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term. Calculus of Variations and Partial Differential Equations 2002,15(2):155–180. 10.1007/s005260100096MATHMathSciNetView ArticleGoogle Scholar
  2. Adams RA: Sobolev Spaces. Academic Press, New York; 1975:xviii+268.MATHGoogle Scholar
  3. Bae JJ: Uniform decay for the unilateral problem associated to the Kirchhoff type wave equations with nonlinear boundary damping. Applied Mathematics and Computation 2004,156(1):41–57. 10.1016/j.amc.2003.07.003MATHMathSciNetView ArticleGoogle Scholar
  4. Cavalcanti MM: Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete and Continuous Dynamical Systems 2002,8(3):675–695.MATHMathSciNetView ArticleGoogle Scholar
  5. Cavalcanti MM, Domingos Cavalcanti VN, Martinez P: Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term. Journal of Differential Equations 2004,203(1):119–158. 10.1016/j.jde.2004.04.011MATHMathSciNetView ArticleGoogle Scholar
  6. Cavalcanti MM, Domingos Cavalcanti VN, Soriano JA: On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions. Journal of Mathematical Analysis and Applications 2003,281(1):108–124.MATHMathSciNetView ArticleGoogle Scholar
  7. Giorgi C, Pata V: Asymptotic behavior of a nonlinear hyperbolic heat equation with memory. Nonlinear Differential Equations and Applications 2001,8(2):157–171. 10.1007/PL00001443MATHMathSciNetView ArticleGoogle Scholar
  8. Graffi D: Qualche problema di elettromagnetismo. In Trends in Applications of Pure Mathematics to Mechanics (Conf., Univ. Lecce, Lecce, 1975), Monographs and Studies in Math.. Volume 2. Edited by: Fichera G. Pitman, London; 1976:129–144.Google Scholar
  9. Gurtin ME, Pipkin AC: A general theory of heat conduction with finite wave speeds. Archive for Rational Mechanics and Analysis 1968, 31: 113–126. 10.1007/BF00281373MATHMathSciNetView ArticleGoogle Scholar
  10. Lions J-L: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris; 1969:xx+554.MATHGoogle Scholar
  11. Park JY, Bae JJ: On coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term. Applied Mathematics and Computation 2002,129(1):87–105. 10.1016/S0096-3003(01)00031-5MATHMathSciNetView ArticleGoogle Scholar
  12. Park JY, Park SH: On solutions for a hyperbolic system with differential inclusion and memory source term on the boundary. Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2004,57(3):459–472.MATHView ArticleMathSciNetGoogle Scholar
  13. Pata V: Attractors for a damped wave equation onwith linear memory. Mathematical Methods in the Applied Sciences 2000,23(7):633–653. 10.1002/(SICI)1099-1476(20000510)23:7<633::AID-MMA135>3.0.CO;2-CMATHMathSciNetView ArticleGoogle Scholar
  14. Vitillaro E: A potential well theory for the wave equation with nonlinear source and boundary damping terms. Glasgow Mathematical Journal 2002,44(3):375–395. 10.1017/S0017089502030045MATHMathSciNetView ArticleGoogle Scholar
  15. Vitillaro E: Global existence for the wave equation with nonlinear boundary damping and source terms. Journal of Differential Equations 2002,186(1):259–298. 10.1016/S0022-0396(02)00023-2MathSciNetView ArticleMATHGoogle Scholar
  16. Yang Z, Chen G: Global existence of solutions for quasi-linear wave equations with viscous damping. Journal of Mathematical Analysis and Applications 2003,285(2):604–618. 10.1016/S0022-247X(03)00448-7MATHMathSciNetView ArticleGoogle Scholar

Copyright

© F. Sun and M.Wang 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.