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

Journal of Inequalities and Applications volume 2007, Article number: 056350 (2007) Cite this article

Abstract

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.

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References

  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)00312-6

    Article  MathSciNet  MATH  Google Scholar 

  3. Miettinen M, Panagiotopoulos PD: On parabolic hemivariational inequalities and applications. Nonlinear Analysis 1999,35(7):885–915. 10.1016/S0362-546X(97)00720-7

    Article  MathSciNet  MATH  Google Scholar 

  4. Panagiotopoulos PD: Inequality Problems in Mechanics and Applications Convex and Nonconvex Energy Functions. Birkhäuser, Boston, Mass, USA; 1985.

    Book  MATH  Google 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)00216-5

    Article  MathSciNet  MATH  Google 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/j.na.2004.02.024

    Article  MathSciNet  MATH  Google 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/S0002-9939-1977-0442453-6

    Article  MathSciNet  MATH  Google 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.2005.89

    Article  MATH  Google 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/03040747

    Article  MathSciNet  MATH  Google 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.

    Article  MathSciNet  MATH  Google 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/j.na.2005.06.009

    Article  MathSciNet  MATH  Google 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.2001.7697

    Article  MathSciNet  MATH  Google 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.1997.5697

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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.1999.6528

    Article  MathSciNet  MATH  Google 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.

    MATH  Google Scholar 

  18. Nakao M: Energy decay for the quasilinear wave equation with viscosity. Mathematische Zeitschrift 1995,219(2):289–299.

    Article  MathSciNet  MATH  Google 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 

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Authors and Affiliations

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

    Jong Yeoul Park & Sun Hye Park

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  1. Jong Yeoul Park
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  2. Sun Hye Park
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Correspondence to Jong Yeoul Park.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Park, J.Y., Park, S.H. Existence and Asymptotic Stability of Solutions for Hyperbolic Differential Inclusions with a Source Term. J Inequal Appl 2007, 056350 (2007). https://doi.org/10.1155/2007/56350

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  • DOI: https://doi.org/10.1155/2007/56350

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