Taiwanese Journal of Mathematics

GLOBAL NONEXISTENCE OF ARBITRARY INITIAL ENERGY SOLUTIONS OF VISCOELASTIC EQUATION WITH NONLOCAL BOUNDARY DAMPING

Jie Ma and Hongrui Geng

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406836

Digital Object Identifier
doi:10.11650/twjm/1500406836

Mathematical Reviews number (MathSciNet)
MR3001832

Zentralblatt MATH identifier
1282.35094

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

Keywords
viscoelastic equations blow up arbitrary initial energy boundary damping

Citation

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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