On the extensible viscoelastic beam

Abstract

This work is focused on the equationL 2 (0,1) ∂ xx u = f describing the motion of an extensible viscoelastic beam. Under suitable boundary conditions, the related dynamical system in the history space framework is shown to possess a global attractor of optimal regularity. The result is obtained by exploiting an appropriate decomposition of the solution semigroup, together with the existence of a Lyapunov functional.

Buckling and nonlinear dynamics of elastically coupled double-beam systems

“…where > a 0 is large enough that the following inequality holds (see [19]) 24 1 2 2 2 2 2 1 2 2 2 2 for all ∈ u H 2 and for some = (ℓ ) ≥ m m a , 1 . The proof of Lemma 5 is carried out in the next two subsections, and the last part of Theorem 5 follows from the compact embedding ⋐ 2 .…”

“…In spite of a relatively wide literature concerning statics and dynamics of a single WK-beam (see e.g. [2,3,5,7,8,14,15,19,22,31,37] and references therein), we are not aware of analytic studies which consider the elastic coupling of two or more nonlinear beams of this type. On the other hand, mathematical models of sandwich beam-type and plate-type structures raised a wide interest in the literature, due to their relevance in many branches of modern civil, mechanical and aerospace engineering.…”

Asymptotic dynamics of nonlinear coupled suspension bridge equations

“…In Section 4 we present the main result, namely the existence of the regular global attractor for a general value of p. This is done by exploiting the existence of a Lyapunov functional and without require any assumption on the strength of the dissipation terms. The asymptotic compactness property is shown by using a particular decomposition of the semigroup, as devised in [15], but a final bootstrap argument is needed here, unlike [15]. The Appendix at the end of the paper is devoted to recall some technical lemmas which are required to obtain main results.…”

“…All these facts allow us to prove the existence of the global attractor by showing a suitable (really, exponential) asymptotic compactness property of the semigroup. Such a property will be obtained exploiting a particular decomposition of S(t) devised in [15] and then applying a general result (see, for instance, [10]), tailored to our particular case. The existence of a bounded absorbing set in H can be obtained as a byproduct.…”

Global Attractors of the Extensible Plate Equations with Nonlinear Damping and Memory

“…Shen and Ma [21] studied the existence of the random attractor for plate equations with memory and additive 2 Journal of Function Spaces white noise. On the other hand, the asymptotic behavior of solutions for the extensible plate equations without memory affection was studied by several authors in [24][25][26][27].…”