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Acta Mathematica Scientia

, Volume 39, Issue 2, pp 567–579 | Cite as

Blow-Up Phenomena for a Class of Generalized Double Dispersion Equations

  • Huafei DiEmail author
  • Yadong Shang
Article
  • 3 Downloads

Abstract

In this article, we study the blow-up phenomena of generalized double dispersion equations uttuxxuxxt + uxxxxuxxtt = f(ux)x. Under suitable conditions on the initial data, we first establish a blow-up result for the solutions with arbitrary high initial energy, and give some upper bounds for blow-up time T* depending on sign and size of initial energy E(0). Furthermore, a lower bound for blow-up time T* is determined by means of a differential inequality argument when blow-up occurs.

Key words

double dispersion blow up upper bound lower bound 

2010 MR Subject Classification

35L35 31A35 35B44 

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Notes

Acknowledgements

Dr. Huafei DI also specially appreciates Prof. Yue LIU for his invitation of visit to UTA.

References

  1. [1]
    An L J, Peirce A. A weakly nonlinear analysis of elasto-plastic-microstructure models. SIAM J Appl Math, 1975, 55(1): 136–155CrossRefzbMATHGoogle Scholar
  2. [2]
    Guenther R B, Lee J W. Partial Differential Equations of Mathematical Physics and Integral Equations. Englewood Cliffs, NJ: Prentice Hall, 1988Google Scholar
  3. [3]
    Chen G W, Lu B. The initial-boundary value problems for a class of nonlinear wave equations with damping term. J Math Anal Appl, 2009, 351: 1–15MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Xu R Z, Wang S, Yang Y B, Ding Y H. Initial boundary value problem for a class of fourth-order wave equation with viscous damping term. Appli Anal, 2013, 92: 1403–1416MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Khelghati A, Baghaei K. Blow-up phenomena for a class of fourth-order nonlinear wave equations with a viscous damping term. Math Meth Appl Sci, 2018, 41: 490–494MathSciNetzbMATHGoogle Scholar
  6. [6]
    Yang Z J. Global existence asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term. J Differ Equ, 2003, 187: 520–540CrossRefzbMATHGoogle Scholar
  7. [7]
    Zhu W Q. Nonlinear waves in elastic rods. Acta Mechanica Solida Sinica (Chinese), 1980, 2: 247–253Google Scholar
  8. [8]
    Chen X Y. Existence and nonexistence of global solutions for nonlinear evolution equation of fourth-order. Appl Math J Chinese Univ Ser B, 2001, 16(3): 251–258MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Chen G W, Hou C S. Initial value problem for a class of fourth-order nonlinear wave equations. Appl Math Meth Engl Ed, 2009, 30(3): 391–401MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Chen X Y, Chen G W. Asymptotic behavior and blow-up of solutions to a nonlinear evolution equation of fourth-order. Nonlinear Anal, 2008, 68: 892–904MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Zhuang W, Yang G T. The propagation of solitary waves in a nonlinear elastic rod. Appl Math Mech, 1986, 7(7): 615–626CrossRefzbMATHGoogle Scholar
  12. [12]
    Zhang S Y, Zhuang W. Strain solitary waves in the nonlinear rods. Acta Mechanica Sinia (Chinese), 1998, 20(1): 58–66Google Scholar
  13. [13]
    Xu R Z, Zhang M Y, Chen S H, et al. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete Contin Dyn Syst A, 2017, 37(11): 5631–5649MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Samsonov A M, Sokurinskaya E V. Energy exchange between nonlinear waves in elastic waveguides and external media//Nonlinear Waves in Active Media. Berlin Heidelberg: Springer, 1989: 99–104Google Scholar
  15. [15]
    Samsonov A M. Nonlinear strain waves in elastic waveguide//Nonlinear Waves in Solids. Vienna: Springer, 1994: 349–382Google Scholar
  16. [16]
    Wang S B, Chen G W. Cauchy problem of the generalized double dispersion equation. Nonlinear Anal, 2006, 64: 159–173MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Chen G W, Wang Y P, Wang S B. Initial boundary value problem of the generalized cubic double dispersion equation. J Math Anal Appl, 2004, 299: 563–577MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Polat N, Ertas A. Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation. J Math Anal Appl, 2009, 349: 10–20MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Xu R Z, Liu Y C, Yu T. Global existence of solution for Cauchy problem of multidimentional generelized double dispersion equations. Nonlinear Anal, 2009, 71: 4977–4983MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Wang S B, Da F. On the asymptotic behavior of solution for the generalized double dispersion equation. Appli Anal, 2013, 92(6): 1179–1193MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Di H F, Shang Y D. Global existence and nonexistence of solutions for a fourth-order wave equation with nonlinear damping and source terms. Acta Math Sci, 2015, 35A(3): 618–633MathSciNetzbMATHGoogle Scholar
  22. [22]
    Khanmamedov A, Yayla S. Long-time dynamics of the strongly damped semilinear plate equation in ℝN. Acta Math Sci, 2018, 38B(3): 1025–1042MathSciNetCrossRefGoogle Scholar
  23. [23]
    Shang Y D. The large time behavior of spectral approximation for a class of pseudoparabolic viscous diffusion equation. Acta Math Sci, 2007, 27B(1): 153–168MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Lions J L. Quelques Méthodes de Résolutions des Probléms aux Limites non Linéaires. Paris: Dunod, 1969zbMATHGoogle Scholar
  25. [25]
    Li M R, Tsai L Y. Existence and nonexistence of global solutions of some system of semilinear wave equations. Nonlinear Anal, 2003, 54: 1397–1415MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Liu W J, Yu J. On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Anal, 2011, 74: 2175–2190MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceGuangzhou UniversityGuangzhouChina
  2. 2.Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education InstitutesGuangzhou UniversityGuangzhouChina

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