Indian Journal of Pure and Applied Mathematics

, Volume 49, Issue 4, pp 729–742 | Cite as

General Decay and Blow-Up Results for Nonlinear Fourth-Order Integro-Differential Equation

  • Mohammad ShahrouziEmail author


This study aims at considering an initial-boundary value problem for nonlinear fourth-order viscoelastic equation in a bounded domain. Under suitable conditions of the initial data and of the relaxation function, it is proved that the solution energy is generally decayed. It is also shown that regarding arbitrary positive initial energy, certain solutions blow-up in a finite time.

Key words

General decay blow up viscoelastic fourth-order 


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  1. 1.
    L. J. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136–155.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D. Andrade, L. H. Fatori, and J. M. Rivera, Nonlinear transmission problem with a dissipative boundary condition of memory type, Electron. J. Diff. Equ., 53 (2006), 1–16.MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma, and J. A. Soriano, Global existence and asymptotic stability for viscoelastic problems, Diff. Integ. Equ., 15(6) (2002), 731–748.MathSciNetzbMATHGoogle Scholar
  4. 4.
    C. S. Chen and L. Ren, Weak solution for a fourth order nonlinear wave equation, J. Southeast Univ. (English Ed.), 21(3) (2005), 369–374.MathSciNetzbMATHGoogle Scholar
  5. 5.
    V. K. Kalantarov and O. A. Ladyzhenskaya, Formation of collapses in quasilinear equations of parabolic and hyperbolic types, zap. Nauchn. Semin. LOMI, 69 (1977), 77–102.MathSciNetzbMATHGoogle Scholar
  6. 6.
    H. A. Levine and S. Ro. Park, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. App., 228(1) (1998), 181–205.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equation with dissipation, Arch. Ration. Mech. Anal., 137(4) (1997), 341–361.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    F. Li and Q. Gao, Blow-up of solution for a nonlinear Petrovsky type equations, Appl. Math. Comput., 274 (2016), 383–392.MathSciNetGoogle Scholar
  9. 9.
    G. Li, Y. Sun and W. Liu, Global existence, uniform decay and blow-up of solutions for a system of Petrovsky equations, Nonlinear Anal., 74(4) (2011), 1523–1538.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320(2) (2006), 902–915.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    S. A. Messaoudi, General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69(8) (2008), 2589–2598.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265(2) (2002), 296–308.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. A. Messaoudi and W. Al-khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Appl. Math. Lett., 66 (2017), 16–22.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. E. Munoz Rivera, E. C. Lapa, and R. Baretto, Decay rates for viscoelastic plates with memory, J. Elasticity, 44(1) (1996), 61–87.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. I. Mustafa, General decay result for nonlinear viscoelastic equations, J. Math. Anal. Apl., 457(1)(2018), 134–152.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. Shahrouzi, Blow-up of solutions for a class of fourth-order equation involving dissipative boundary condition and positive initial energy, J. Part. Diff. Eq., 27(4) (2014), 347–356.MathSciNetzbMATHGoogle Scholar
  17. 17.
    F. Tahamtani and M. Shahrouzi, Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term, Boundary Value Problems, 50 (2012), 1–15.MathSciNetzbMATHGoogle Scholar
  18. 18.
    S. T. Wu and L. Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese. J. Math., 13(2A) (2009), 545–558.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    S. T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source, Electron. J. Differ. Eq., 2006(45) (2006), 1–9.MathSciNetGoogle Scholar
  20. 20.
    S. Q. Yu, On the strongly damped wave equation with nonlinear damping and source terms, Electron J. Qual. Theory Differ. Eqns., 39 (2009), 1–18.Google Scholar
  21. 21.
    W. Zhao and W. Liu, A note on blow-up of solutions for a class of fourth-order wave equation with viscous damping term, Appl. Anal., 97(9) (2018), 1496–1504.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Indian National Science Academy 2018

Authors and Affiliations

  1. 1.Department of MathematicsJahrom UniversityJahromIran

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