Attractors and their properties for a class of  nonlocal extensible beams

Marcio Antonio Jorge da Silva; Vando Narciso

doi:10.3934/dcds.2015.35.985

Article Contents

2015, Volume 35, Issue 3: 985-1008. Doi: 10.3934/dcds.2015.35.985

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Attractors and their properties for a class of nonlocal extensible beams

Marcio Antonio Jorge da Silva^1, and
Vando Narciso^2,

1.
Department of Mathematics, State University of Londrina, Londrina, 86057-970
2.
Center of exact sciences, State University of Mato Grosso do Sul, Dourados, 79804-970

Received: September 2013

Revised: August 2014

Published: March 2015

Abstract / Introduction Related Papers Cited by

Abstract

This paper is concerned with well-posedness and asymptotic behavior to a class of extensible beams with nonlinear fractional damping and source terms $$ u_{tt} + \Delta^2 u - M\left(\int_{\Omega}|\nabla u|^2 \, dx \right)\Delta u + N\left(\int_{\Omega}|\nabla u|^2 \, dx \right) \left(-\Delta\right)^{\theta} u_t + f(u) = h $$ in $\Omega\times(0,\infty)$, where $\Omega\subset \mathbb{R}^q$ is a bounded domain with smooth boundary $\partial\Omega$, $0\leq \theta \le 1$, $M$ and $N$ are scalar functions, $f(u)$ is a source term and $h$ is a forcing term. When $\theta=1$ we have the classical strong damping $-\Delta u_t$, and if $\theta=0$ then weak damping $u_t$ is considered. Under the mathematical point of view the abstract setting of $\left(-\Delta\right)^{\theta}u_t$ with $0< \theta < 1$ constitutes a way to change the characteristic of the problem in wave and plate vibrations, and maybe the decay rate of solutions mainly when we consider nonconstant coefficient damping $N$. The main purpose in the present paper is to show that the transition from the case $0< \theta \leq 1$ to the case $\theta=0$ does not produce any influence on the asymptotic behavior of solutions, that is, all results are obtained by moving $\theta\in[0,1]$ uniformly. Our main result establishes the existence of finite-dimensional compact global and exponential attractors by using an approach on quasi-stable dynamical systems given by Chueshov and Lasiecka (2010).

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Mathematics Subject Classification: Primary: 35B40, 35B41, 37L30; Secondary: 74H40.

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