Asymptotic limits and attractors for a laminated beam model

Abstract

This paper deals with long-time dynamics of nonlinear laminated beams modeled under the assumptions of Timoshenko beam theory. The model considered here is composed of two-layered beams and was proposed by Hansen and Spies. It describes the slip produced by a thin adhesive layer uniting the structure. As the adhesive stiffness $\gamma$ tends to infinity (effectively causing a no-slip condition between the two layers), we get the convergence (in some sense) to the Timoshenko beam system. The existence of smooth finite-dimensional global attractors and exponential attractors is proved using the recent quasi-stability theory. We also established that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals. Finally, we compare the laminated beam model with the Timoshenko model in the sense of the upper semicontinuity of its attractors as $\gamma \rightarrow \infty$.

Appendix A: Long-time behavior of quasi-stable gradient systems

In this appendix, we present some definitions related to global attractors for quasi-stable gradient systems that can be found in recent references such as [11, 16]. A dynamical system is a pair (H, S(t)), where H is a Banach space and S(t) is a continuous semigroup defined on H. We recall that a set $\mathcal {A}\subset H$ is a global attractor for (H, S(t)) if it is compact, invariant, that is, $S(t)\mathcal {A}=\mathcal {A}$ for all $t\geqslant 0$, and uniformly attracting, that is,

$$\begin{aligned} \begin{aligned} \lim _{t\rightarrow \infty } \textrm{dist}_H(S(t)B,\mathcal {A})=0, \end{aligned} \end{aligned}$$

for any bounded set $B\subset H$, where $\textrm{dist}_H$ is the Hausdorff semi-distance in H.

As is well known in the literature, the existence of a global attractor is granted under suitable dissipativeness and compactness conditions. A dynamical system is called dissipative if it admits a bounded absorbing set, that is, a bounded set $\mathcal {B}\subset H$ such that, for any bounded set $B\subset H$, there exists a time $T_B>0$ satisfying

$$\begin{aligned}\begin{aligned} S(t)B\subset \mathcal {B},\quad \forall t\geqslant T_B. \end{aligned}\end{aligned}$$

A dynamical system is called asymptotically smooth if, for any bounded set $B\subset H$ forward invariant $(S(t)B\subset B, t \geqslant 0)$, there exists a compact set $K\subset \overline{B}$ that uniformly attracts B. Then, we have the following classical result (see [11, 16] and references therein).

Theorem A.1

Let (H, S(t)) be a dynamical system dissipative and asymptotically smooth. Then, it possesses a unique compact global attractor.

To obtain the asymptotically smooth property, we shall introduce the concept of quasi-stability [16, Chapter 7, Definition 7.9.2].

Definition A.2

Let X and Y be three reflexive Banach spaces with X compactly embedded in Y. We consider the space $H=X\times Y$ and the dynamical system (H, S(t)) given by

$$\begin{aligned} \begin{aligned} S(t)y=(u(t),u_t(t)),\quad \quad y=(u(0),u_t(0))\in H, \end{aligned} \end{aligned}$$

(A.80)

where the function u has the regularity

$$\begin{aligned} \begin{aligned} u\in C\big ([0,\infty );X\big )\cap C^1\big ([0,\infty );Y\big ). \end{aligned} \end{aligned}$$

(A.81)

The dynamical system (H, S(t)) is called quasi-stable on a set $B\subset H$ if there exists a compact seminorm $n_X$ on the space X and nonnegative scalar functions a and c, locally bounded in $[0,\infty )$, and $b\in L^1(0,\infty )$, with $\displaystyle \lim _{t\rightarrow \infty }b(t)=0$, such that

$$\begin{aligned} \begin{aligned} \Vert S(t)y_1-S(t)y_2\Vert _{H}^2\leqslant a(t)\Vert y_1-y_2\Vert _{H}^2, \end{aligned} \end{aligned}$$

(A.82)

and

$$\begin{aligned} \begin{aligned} \Vert S(t)y_1-S(t)y_2\Vert _{H}^2\leqslant b(t)\Vert y_1-y_2\Vert _{H}^2+c(t)\sup _{0\leqslant s\leqslant t}[n_X(u^1(s)-u^2(s))]^2, \end{aligned} \end{aligned}$$

(A.83)

for any $y_1,y_2\in B$. Here, we denote $S(t)y_i=(u^i(t),u_t^i(t))$, $i=1,2$.

The following result, which can be found in [16, Proposition 7.9.4], shows that the quasi-stability implies the asymptotic smoothness of the dynamical system.

Theorem A.3

Let (H, S(t)) be a dynamical system given by (A.80) and satisfying (A.81). Then, (H, S(t)) is asymptotically smooth if it is quasi-stable on every bounded positively invariant set of H.

The quasi-stability also implies the smoothness and finite dimensionality of the attractor (see [16, Theorems 7.9.6 and 7.9.8]).

Theorem A.4

Let (H, S(t)) be a dynamical system given by (A.80) and satisfying (A.81). If (H, S(t)) possesses a compact global attractor $\mathcal {A}$ and is quasi-stable on $\mathcal {A}$, then $\mathcal {A}$ has finite fractal dimension, i.e., $\dim _f^H\mathcal {A}<+\infty$. Moreover, any full trajectory $\{(u(t),u_t(t)): t\in \mathbb {R}\}\subset \mathcal {A}$ has the following regularity property (in time)

$$\begin{aligned} \begin{aligned} u_t\in L^\infty (\mathbb {R}; X)\cap C(\mathbb {R};Y),\quad u_{tt}\in L^\infty (\mathbb {R};Y), \end{aligned} \end{aligned}$$

with bound

$$\begin{aligned} \begin{aligned} \Vert u_t(t)\Vert _X^2+\Vert u_{tt}(t)\Vert _Y^2\leqslant R^2,\quad \forall t\in \mathbb {R}, \end{aligned} \end{aligned}$$

where the constant R depends on $\displaystyle \sup _{t\geqslant 0}c(t)$.

Definition A.5

A compact set $\mathcal {A}^{\exp }\subset H$ is called an fractal exponential attractor for (H, S(t)) if

• $\mathcal {A}^{\exp }$ is a positively invariant set, that is, $S(t)\mathcal {A}^{\exp }\subset \mathcal {A}^{\exp }$ for all $t\geqslant 0;$

• $\mathcal {A}^{\exp }$ has finite fractal dimension in H; 

• $\mathcal {A}^{\exp }$ attracts bounded sets of H at an exponential rate; that is, for any bounded set $D\subset H$ there exist $t_{D}, C_{D}, \gamma _{D}>0$ such that

  $$\begin{aligned} \textrm{dist}_H(S(t)D,\mathcal {A}^{\exp })\leqslant C_{D} e^{-\gamma _{D}(t-t_{D})},\quad \forall t\geqslant t_{D}. \end{aligned}$$

If there exists an exponential attractor only having finite dimension in some extended space $\widetilde{H}\supseteq H$, i.e., $\dim _f^{\widetilde{H}}\mathcal {A}^{\exp }<+\infty$, then this exponentially attracting set is called generalized fractal exponential attractor.

Remark A.6

The word “generalized” is included to indicate that the finite dimensionality requirement is allowed in a topology weaker than H.

The following theorem concerns fractal exponential attractors to quasi-stable systems.

Theorem A.7

[16, Theorem 7.9.9] Let (H, S(t)) be a dynamical system given by (A.80) and satisfying (A.81). Assume that (H, S(t)) is dissipative and quasi-stable on some bounded absorbing set B. Assume also that there exists an extended space $\widetilde{H}\supseteq H$ such that $t\mapsto S(t)$ is Hölder continuous in $\widetilde{H}$ for every $y\in B$; that is, there exist $0< \sigma \leqslant 1$ and $C_{B,T}>0$ such that

$$\begin{aligned}\begin{aligned}\Vert S(t_1)y-S(t_2)y\Vert _{{\widetilde{H}}}\leqslant C_{B,T}|t_1-t_2|^\sigma ,\quad t_1,t_2\in [0,T], \ y\in B. \end{aligned}\end{aligned}$$

Then, the dynamical system possesses a generalized fractal exponential attractor $\mathcal {A}^{\exp }$ whose dimension is finite in the space $\widetilde{H}$.