Differential and Integral Equations

Global existence and asymptotic stability for viscoelastic problems

M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma, and J. A. Soriano

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Abstract

One considers the damped semilinear viscoelastic wave equation $$u_{tt}-\Delta u+\alpha u+f(u)+\int_0^tg(t-\tau )\Delta u(\tau )\, d\tau +h(u_t)=0\,\,\,\hbox{in}\,\,\,\Omega\times (0,\infty ),$$ where $\Omega$ is any bounded or finite measure domain of ${\bf R}^ n$, $\alpha\geq 0$ and $f,h$ are power like functions. The existence of global regular and weak solutions is proved by means of the Faedo-Galerkin method and uniform decay rates of the energy are obtained following the perturbed energy method by assuming that $g$ decays exponentially.

Article information

Source
Differential Integral Equations, Volume 15, Number 6 (2002), 731-748.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060814

Mathematical Reviews number (MathSciNet)
MR1893844

Zentralblatt MATH identifier
1015.35071

Subjects
Primary: 74H20: Existence of solutions
Secondary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35L70: Nonlinear second-order hyperbolic equations 35Q72 74D10: Nonlinear constitutive equations 74H25: Uniqueness of solutions 74H40: Long-time behavior of solutions

Citation

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Ma, T. F.; Soriano, J. A. Global existence and asymptotic stability for viscoelastic problems. Differential Integral Equations 15 (2002), no. 6, 731--748. https://projecteuclid.org/euclid.die/1356060814


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