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  • Research Article
  • Open Access

Existence and Asymptotic Stability of Solutions for Hyperbolic Differential Inclusions with a Source Term

Journal of Inequalities and Applications20072007:056350

  • Received: 10 October 2006
  • Accepted: 16 January 2007
  • Published:


We study the existence of global weak solutions for a hyperbolic differential inclusion with a source term, and then investigate the asymptotic stability of the solutions by using Nakao lemma.


  • Weak Solution
  • Source Term
  • Asymptotic Stability
  • Differential Inclusion
  • Global Weak Solution


Authors’ Affiliations

Department of Mathematics, Pusan National University, Pusan, 609-735, South Korea


  1. Carl S, Heikkilä S: Existence results for nonlocal and nonsmooth hemivariational inequalities. Journal of Inequalities and Applications 2006., 2006:Google Scholar
  2. Miettinen M: A parabolic hemivariational inequality. Nonlinear Analysis 1996,26(4):725–734. 10.1016/0362-546X(94)00312-6MathSciNetView ArticleMATHGoogle Scholar
  3. Miettinen M, Panagiotopoulos PD: On parabolic hemivariational inequalities and applications. Nonlinear Analysis 1999,35(7):885–915. 10.1016/S0362-546X(97)00720-7MathSciNetView ArticleMATHGoogle Scholar
  4. Panagiotopoulos PD: Inequality Problems in Mechanics and Applications Convex and Nonconvex Energy Functions. Birkhäuser, Boston, Mass, USA; 1985.View ArticleMATHGoogle Scholar
  5. Park JY, Kim HM, Park SH: On weak solutions for hyperbolic differential inclusion with discontinuous nonlinearities. Nonlinear Analysis 2003,55(1–2):103–113. 10.1016/S0362-546X(03)00216-5MathSciNetView ArticleMATHGoogle Scholar
  6. Park JY, Park SH: On solutions for a hyperbolic system with differential inclusion and memory source term on the boundary. Nonlinear Analysis 2004,57(3):459–472. 10.1016/ ArticleMATHGoogle Scholar
  7. Rauch J: Discontinuous semilinear differential equations and multiple valued maps. Proceedings of the American Mathematical Society 1977,64(2):277–282. 10.1090/S0002-9939-1977-0442453-6MathSciNetView ArticleMATHGoogle Scholar
  8. Varga C: Existence and infinitely many solutions for an abstract class of hemivariational inequalities. Journal of Inequalities and Applications 2005,2005(2):89–105. 10.1155/JIA.2005.89View ArticleMATHGoogle Scholar
  9. Nakao M: A difference inequality and its application to nonlinear evolution equations. Journal of the Mathematical Society of Japan 1978,30(4):747–762. 10.2969/jmsj/03040747MathSciNetView ArticleMATHGoogle Scholar
  10. Panagiotopoulos PD: Modelling of nonconvex nonsmooth energy problems. Dynamic hemivariational inequalities with impact effects. Journal of Computational and Applied Mathematics 1995,63(1–3):123–138.MathSciNetView ArticleMATHGoogle Scholar
  11. Liu L, Wang M: Global existence and blow-up of solutions for some hyperbolic systems with damping and source terms. Nonlinear Analysis 2006,64(1):69–91. 10.1016/ ArticleMATHGoogle Scholar
  12. Messaoudi SA: Global existence and nonexistence in a system of Petrovsky. Journal of Mathematical Analysis and Applications 2002,265(2):296–308. 10.1006/jmaa.2001.7697MathSciNetView ArticleMATHGoogle Scholar
  13. Ono K: On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation. Journal of Mathematical Analysis and Applications 1997,216(1):321–342. 10.1006/jmaa.1997.5697MathSciNetView ArticleMATHGoogle Scholar
  14. Park JY, Bae JJ: On existence of solutions of degenerate wave equations with nonlinear damping terms. Journal of the Korean Mathematical Society 1998,35(2):465–490.MathSciNetMATHGoogle Scholar
  15. Todorova G: Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. Journal of Mathematical Analysis and Applications 1999,239(2):213–226. 10.1006/jmaa.1999.6528MathSciNetView ArticleMATHGoogle Scholar
  16. Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 65. Academic Press, New York, NY, USA; 1975.Google Scholar
  17. Lions JL: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris; 1969.MATHGoogle Scholar
  18. Nakao M: Energy decay for the quasilinear wave equation with viscosity. Mathematische Zeitschrift 1995,219(2):289–299.MathSciNetView ArticleMATHGoogle Scholar
  19. Showalter RE: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs. Volume 49. American Mathematical Society, Providence, RI, USA; 1997.Google Scholar


© J. Y. Park and S. H. Park 2007

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.