## Differential and Integral Equations

- Differential Integral Equations
- Volume 17, Number 5-6 (2004), 495-510.

### Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains

M. M. Cavalcanti, V. N. Domingos Cavalcanti, and T. F. Ma

#### Abstract

The viscoelastic Euler-Bernoulli equation with nonlinear and nonlocal damping $$u_{tt}+\Delta^2u-\int_0^tg(t-\tau )\Delta^2u(\tau )\,d\tau +a(t )u_t=0\,\,\,\hbox{in}\,\,\,\Omega\times {\bf R}^{+},$$ where $a(t)=M\left(\int_{\Omega}\left|\nabla u(x,t)\right|^2dx\right )$, is considered in bounded or unbounded domains $\Omega$ of ${\bf R}^ n$. The existence of global solutions and decay rates of the energy are proved.

#### Article information

**Source**

Differential Integral Equations, Volume 17, Number 5-6 (2004), 495-510.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060344

**Mathematical Reviews number (MathSciNet)**

MR2054931

**Zentralblatt MATH identifier**

1174.74320

**Subjects**

Primary: 74D05: Linear constitutive equations

Secondary: 35B40: Asymptotic behavior of solutions 35R10: Partial functional-differential equations 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 74H40: Long-time behavior of solutions 74K10: Rods (beams, columns, shafts, arches, rings, etc.)

#### Citation

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Ma, T. F. Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains. Differential Integral Equations 17 (2004), no. 5-6, 495--510. https://projecteuclid.org/euclid.die/1356060344