Taiwanese Journal of Mathematics


Jie Ma and Hongrui Geng

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In this paper, we consider the long time behavior of solutions of the initial value problem for the viscoelastic wave equation under boundary damping \begin{eqnarray*} u_{tt} - \Delta u + \int_0^t g(t-\tau) \text{div}(a(x)\nabla u(\tau)) d\tau + u_t = 0 &\text{in}\, \Omega \times (0,\infty). \end{eqnarray*} For the low initial energy case, which is the non-positive initial energy, based on concavity argument we prove the blow up result. As for the high initial energy case, we give out sufficient conditions of the initial datum such that the solution blows up in finite time.

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Taiwanese J. Math., Volume 16, Number 6 (2012), 2019-2033.

First available in Project Euclid: 18 July 2017

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Primary: 35L15: Initial value problems for second-order hyperbolic equations 35B40: Asymptotic behavior of solutions

viscoelastic equations blow up arbitrary initial energy boundary damping


Ma, Jie; Geng, Hongrui. GLOBAL NONEXISTENCE OF ARBITRARY INITIAL ENERGY SOLUTIONS OF VISCOELASTIC EQUATION WITH NONLOCAL BOUNDARY DAMPING. Taiwanese J. Math. 16 (2012), no. 6, 2019--2033. doi:10.11650/twjm/1500406836. https://projecteuclid.org/euclid.twjm/1500406836

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  • C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var. Partial Differential Equations, 34(3) (2009), 377-411.
  • M. Aassila, M. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38(5) (2000), 1581-1602.
  • M. M. Cavalcanti, V. N. D. Cavalcanti and J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24 (2001), 1043-1053.
  • M. M. Cavalcanti, V. N. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236(2) (2007), 407-459.
  • M. M. Cavalcanti, V. N. D. Cavalcanti and J. A. Soriano, Existence and uniform decay rate for viscoelastic problems with nonlinear boundary damping, Differential Intergral Equation, 14 (2001), 85-116.
  • M. M. Cavalcanti, V. N. D. Cavalcanti and J. A. Soriano, Exponential decay for the solution of the semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations, 44 (2002), 1-14.
  • M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42(4) (2003), 1310-1324.
  • M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, SIAM Stud. Appl. Math, Philadelphia, 1992.
  • K. B. Hannsgen, Indirect abelian theorems and a linear Volterra equation, Trans. Amer. Math. Soc., 142 (1969), 539-555.
  • H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equation of the form $Pu_{tt}=\Delta u+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.
  • H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.
  • F. S. Li and C. L. Zhao, Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping, Nonlinear Anal., 74 (2011), 3465-3477.
  • F. S. Li, Z. Q. Zhao and Y. F. Chen, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear Anal. RWA, 12 (2011), 1759-1773.
  • A. E. H. Love, A treatise on the mathematical theory of elasticity, Dover, New York, 1944.
  • S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66.
  • S. A. Messaoudi, Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902-915.
  • J. Ma, C. L. Mu and R. Zeng, A blow up result for viscoelastic equations with arbitary positive initial energy, Boundary Value Problems, 2011, 2011:6.
  • J. A. Nobel and D. F. Shea, Frequency domain methods for Volterra equations, Adv. Math., 22 (1976), 278-304.
  • M. A. Rammaha, The influence of damping and source terms on solutions of nonlinear wave equations, Bol. Soc. Parana. Math., 25(1-2) (2007), 77-90.
  • M. Renardy, W. J. Hausa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in pure and Applied Mathematics, Vol. 35, John Wiley and Sons, New York, 1987.
  • S. A. Messaoudi and N. -E. Tatar, Global existence and uniform stability of solutions for a quailinear viscoleastic problem, Math. Meth. Appl. Sci., 30 (2007), 665-680.
  • E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186(1) (2002), 259-298.
  • Y. J. Wang and Y. F. Wang, Exponential energy decay of solutions of viscoelastic wave equations, J. Math. Anal. Appl., 347 (2008), 18-25.
  • Y. J. Wang, A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy, Appl. Math. Letters, 22 (2009), 1394-1400.
  • Y. J. Wang, A sufficient condition for finite time for the nonlinear Klein-Gordon equations with arbitarility positive initial energy, Proc. Amer. Math. Soc., 136 (2008), 3477-3482.
  • R. Zeng, C. L. Mu and S. M. Zhou, A blow up result for Kirchhoff type equations with high energy, Math. Meth. Appl. Sci., 34(4) (2011), 479-486.