Taiwanese Journal of Mathematics

ASYMPTOTIC BEHAVIOR FOR A VISCOELASTIC WAVE EQUATION WITH A DELAY TERM

Shun-Tang Wu

Full-text: Open access

Abstract

The following viscoelastic wave equation with a delay term in internal feedback: \begin{equation*} \left\vert u_{t}\right\vert ^{\rho }u_{tt}-\Delta u-\Delta u_{tt} + \int_{0}^{t} g(t-s) \Delta u(s) ds + \mu _{1} u_{t}(x,t) + \mu _{2} u_{t}(x,t-\tau ) = 0, \end{equation*} is considered in a bounded domain. Under appropriate conditions on $\mu_{1}$, $\mu_{2}$ and on the kernel $g$, we prove the local existence result by Faedo-Galerkin method and establish the decay result by suitable Lyapunov functionals.

Article information

Source
Taiwanese J. Math., Volume 17, Number 3 (2013), 765-784.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499705981

Digital Object Identifier
doi:10.11650/tjm.17.2013.2517

Mathematical Reviews number (MathSciNet)
MR3072260

Zentralblatt MATH identifier
1297.35044

Subjects
Primary: 35L05: Wave equation 35L15: Initial value problems for second-order hyperbolic equations 35L70: Nonlinear second-order hyperbolic equations 93D15: Stabilization of systems by feedback

Keywords
global existence asymptotic behavior general decay delay

Citation

Wu, Shun-Tang. ASYMPTOTIC BEHAVIOR FOR A VISCOELASTIC WAVE EQUATION WITH A DELAY TERM. Taiwanese J. Math. 17 (2013), no. 3, 765--784. doi:10.11650/tjm.17.2013.2517. https://projecteuclid.org/euclid.twjm/1499705981


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