Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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GLOBAL EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR COUPLED NONLINEAR WAVE EQUATIONS WITH DAMPING AND SOURCE TERMS

  • Ye, Yaojun (Department of Mathematics and Information Science Zhejiang University of Science and Technology)
  • Received : 2013.10.11
  • Published : 2014.11.30
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Abstract

The initial-boundary value problem for a class of nonlinear higher-order wave equations system with a damping and source terms in bounded domain is studied. We prove the existence of global solutions. Meanwhile, under the condition of the positive initial energy, it is showed that the solutions blow up in the finite time and the lifespan estimate of solutions is also given.

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References

  1. A. B. Aliev and 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 Analysis TMA 75 (2012), no. 1, 91-102. https://doi.org/10.1016/j.na.2011.08.002
  2. A. B. Aliev and B. H. Lichaei, Existence and non-existence of global solutions of the Cauchy problem for higher order semilinear pseudo-hyperbolic equations, Nonlinear Anal. 72 (2010), no. 7-8, 3275-3288. https://doi.org/10.1016/j.na.2009.12.006
  3. K. Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006), no. 11, 1235-1270.
  4. P. Brenner and W. Von Whal, Global classical solutions of nonlinear wave equations, Math. Z. 176 (1981), no. 1, 87-121. https://doi.org/10.1007/BF01258907
  5. H. J. Gao and T. F. Ma, Global solutions for a nonlinear wave equation with the PLaplacian operator, Electron. J. Qual. Theorey Differ. Equ. 1999 (1999), no. 11, 1-13.
  6. S. A. Messaoudi and 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. 365 (2010), no. 1, 277-287. https://doi.org/10.1016/j.jmaa.2009.10.050
  7. C. X. Miao, Time-space estimates and scattering at low energy for higher-order wave equations, Acta Math. Sinica 38 (1995), no. 5 708-717.
  8. L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273-303. https://doi.org/10.1007/BF02761595
  9. H. Pecher, Die existenz regulaer Losungen f¨ur Cauchy-und anfangs-randwertproble-me michtlinear wellengleichungen, Math. Z. 140 (1974), 263-279. https://doi.org/10.1007/BF01214167
  10. B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differential Integral Equations 23 (2010), no. 1-2, 79-92.
  11. D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968), 148-172.
  12. E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (1999), no. 2, 155-182. https://doi.org/10.1007/s002050050171
  13. B. X. Wang, Nonlinear scattering theory for a class of wave equations in $H^s$, J. Math. Anal. Appl. 296 (2004), no. 1, 74-96. https://doi.org/10.1016/j.jmaa.2004.03.050
  14. Z. J. Yang and G. W. Chen, Global existence of solutions for quasi-linear wave equations with viscous damping, J. Math. Anal. Appl. 285 (2003), no. 2, 604-618. https://doi.org/10.1016/S0022-247X(03)00448-7
  15. 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.