Attractors for weakly damped beam equations with $p$-Laplacian

Pages: 525 - 534, Issue special, November 2013

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T. F. Ma - Instituto de Ci^encias Matemáticas e de Computação, Universidade de São Paulo, 13566-590, São Carlos, SP, Brazil (email)
M. L. Pelicer - Departamento de Ciências, Campus Regional de Goioerê, Universidade Estadual de Maringá, 87360-000, Goioerê, PR, Brazil (email)

Abstract: This paper is concerned with a class of weakly damped one-dimensional beam equations with lower order perturbation of $p$-Laplacian type $$ u_{tt} + u_{xxxx} - (\sigma(u_x))_x + ku_t + f(u)= h \quad \hbox{in} \quad (0,L) \times \mathbb{R}^{+} , $$ where $\sigma(z)=|z|^{p-2}z$, $p \ge 2$, $k>0$ and $f(u)$ and $h(x)$ are forcing terms. Well-posedness, exponential stability and existence of a finite-dimensional attractor are proved.

Keywords:  Beam equation, $p$-Laplacian, global attractor, uniqueness of weak solutions.
Mathematics Subject Classification:  Primary: 35L75, 35B40; Secondary: 74J30.

Received: September 2012;      Revised: January 2013;      Published: November 2013.