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Existence and asymptotic behavior for systems of nonlinear hyperbolic equations

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Abstract

The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.

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References

  1. K Agre, M A Rammaha. Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 2006, 19: 1235–1270.

    MathSciNet  MATH  Google Scholar 

  2. A B Aliev, A A Kazimov. Existence, non-existence and asymptotic behavior of global solutions to the Cauchy problem for systems of semilinear hyperbolic equations with damping terms, Nonlinear Anal, 2012, 75: 91–102.

    Article  MathSciNet  MATH  Google Scholar 

  3. A B Aliev, B H Lichaei. Existence and non-existence of global solutions of the Cauchy problem for higher order semilinear pseudo-hyperbolic equations, Nonlinear Anal, 2010, 72: 3275–3288.

    Article  MathSciNet  MATH  Google Scholar 

  4. C O Alves, M M Cavalcanti, V N Domingos, M Rammaha, D Toundykov. On existence uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete Contin Dyn Syst Ser S, 2009, 2: 583–608.

    Article  MathSciNet  MATH  Google Scholar 

  5. P Bernner, W V Whal. Global classical solutions of nonlinear wave equations, Math Z, 1981, 176: 87–121.

    Article  MathSciNet  Google Scholar 

  6. G Li, Y N Sun, W J Liu. Global existence, uniform decay and blow-up of solutions for a system of Petrovsky equations, Nonlinear Anal, 2011, 74: 1523–1538.

    MathSciNet  MATH  Google Scholar 

  7. S A Messaoudi. Global existence and nonexistence in a system of Petrovsky, J Math Anal Appl, 2002, 265: 296–308.

    Article  MathSciNet  MATH  Google Scholar 

  8. S A Messaoudi, B Said-Houari. Global nonexistence of positive initial energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J Math Anal Appl, 2010, 365: 277–287.

    Article  MathSciNet  MATH  Google Scholar 

  9. C X Miao. The time space estimates and scattering at low energy for nonlinear higher order wave equations, Acta Math Sinica, Ser A, 1995, 38: 708–717.

    Google Scholar 

  10. M Nakao. Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term, J Math Soc Japan, 1978, 30: 375–394.

    Article  MathSciNet  MATH  Google Scholar 

  11. M Nakao. A difference inequality and its application to nonlinear evolution equations, J Math Soc Japan, 1978, 30: 747–762.

    Article  MathSciNet  MATH  Google Scholar 

  12. M Nakao, H Kuwahara. Decay estimates for some semilinear wave equations with degenerate dissipative terms, Funkcial Ekvac, 1987, 30: 135–145.

    MathSciNet  MATH  Google Scholar 

  13. L E Payne, D H Sattinger. Saddle points and instability of nonlinear hyperbolic equations, Israel J Math, 1975, 22: 273–303.

    Article  MathSciNet  Google Scholar 

  14. H Pecher. Die existenz reguläer Lösungen für Cauchy-und anfangs-randwertproble-me michtlinear wellengleichungen, Math Z, 1974, 140: 263–279.

    Article  MathSciNet  MATH  Google Scholar 

  15. M A Rammaha, S Sakuntasathien. Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal, 2010, 72: 2658–2683.

    Article  MathSciNet  MATH  Google Scholar 

  16. M Reed. Abstract Nonlinear Wave Equations, Springer-Verlag 1976.

    Google Scholar 

  17. B Said-Houari, S A Messaoudi, A Guesmia. General decay of solutions of a nonlinear system of viscoelastic wave equations, Nonlinear Differ Equ Appl, 2011, 18: 659–684.

    Article  MathSciNet  MATH  Google Scholar 

  18. D H Sattinger. On global solutions for nonlinear hyperbolic equations, Arch Ration Mech Anal, 1968, 30: 148–172.

    Article  MathSciNet  MATH  Google Scholar 

  19. W A Strauss. On continuity of functions with values in various Banach spaces, Pacific J Math, 1966, 19: 543–551.

    Article  MathSciNet  MATH  Google Scholar 

  20. H Taskesen, N Polat, A Ertas. Global existence and decay of solutions for the generalized bad Boussinesq equation, Appl Math J Chinese Univ Ser B, 2013, 28: 253–268.

    Article  MathSciNet  MATH  Google Scholar 

  21. B X Wang. Nonlinear scattering theory for a class of wave equations in Hs, J Math Anal Appl, 2004, 296: 74–96.

    Article  MathSciNet  MATH  Google Scholar 

  22. Y J Ye. Existence and asymptotic behavior of global solutions for a class of nonlinear higher-order wave equation, J Inequal Appl, 2010, 2010: 1–14.

    Article  Google Scholar 

  23. Y J Ye. Global existence and blow-up of solutions for higher-order viscoelastic wave equation with a nonlinear source term, Nonlinear Anal, 2015, 112: 129–146.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou, 310023, China

    Yao-jun Ye

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  1. Yao-jun Ye
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Correspondence to Yao-jun Ye.

Additional information

This Research was supported by National Natural Science Foundation of China (61273016), The Natural Science Foundation of Zhejiang Province (Y6100016), The Public Welfare Technology Application Research Project of Zhejiang Province Science and Technology Department (2015C33088).

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Ye, Yj. Existence and asymptotic behavior for systems of nonlinear hyperbolic equations. Appl. Math. J. Chin. Univ. 30, 453–465 (2015). https://doi.org/10.1007/s11766-015-3083-3

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  • DOI: https://doi.org/10.1007/s11766-015-3083-3

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