Existence of Exponential Attractors for Suspension Bridge Equations with State Delay

Ruonan Wang^*, Qiaozhen Ma^#

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China.

DOI: 10.4236/jamp.2025.1311208                

Abstract

This paper studies the nonlocal suspension bridge equation with state-dependent delay in 2D space and the existence of attractors. First, by using Banach fixed point theorem and operator semigroup theory, the existence and uniqueness of mild solutions and continuous dependence on initial values of suspension bridge equations with state delay are proved. Then the bounded dissipativity of the related semi-group and the quasi-stability of the system are verified to obtain a global attractor with finite fractal dimension the existence of the attractor in the generalized index.

Keywords

Contractive Function, Well-Posedness, Time-Dependent Global Attractors

1. Introduction

In recent years, some physical problems in the model of suspension bridges have been studied by many people, see [1]-[3]. Moreover, the suspension bridge equation with time delay has become a research hotspot, as the existence of time delay can affect the existence and stability of attractors in the system. [4]-[8] have investigated related issues. [5] introduced the existence of strong solutions and strong global attractors for the coupled suspension bridge equations. [6] studied the case where the damping coefficient satisfies ${\gamma }_1>\frac{3}{2}\left|{\gamma }_2\right|$ , when the nonlinear and external force terms satisfy specific conditions, the equation possesses a unique global solution. Furthermore, the existence of a uniform attractor has been demonstrated. It is worth noting that, in order to describe the system process more naturally, state-dependent models have been proposed and studied (see References [9]-[11]). In Reference [9], to handle the delay term in the energy functional, a compensation term for the delay term is introduced to obtain the uniformly bounded estimate of the solution. When addressing the uniqueness of the solution, the solution space is restricted to $Y\equiv C\left(\left[-h,0\right];V_2\right)\cap C^1\left(\left[-h,0\right];H\right)$ , and the treatment is carried out by combining the Lipschitz continuity of the delay term. [11] investigates the dynamic behavior of equations governing suspension bridges. By employing the principle of contraction mapping, it establishes the well-posedness of these equations. Furthermore, utilizing quasi-stability methods, it demonstrates the existence of global attractors and exponential attractors. Compared with Reference [11], the boundary conditions in this paper have been modified to make the system more consistent with engineering practice. However, this modification may undermine the original dissipativity and stability of the system, necessitating a reanalysis of the existence and structure of the attractor. This also implies that the definitions of the inner product and norm in the solution space are more complex compared to those under hinged or fixed boundary conditions, which renders the verification of compactness more challenging. Therefore, it is both meaningful and intriguing to continue exploring such problems.

For the reasons mentioned above, this paper considers the following suspension bridge equation with state time delay

$\{\begin{array}{ll}{u}_{tt}+{\Delta }^{2}u-\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u+g\left(u\right)+{\delta }_1{u}_t-{\delta }_2\Delta u_t\hfill & \hfill \\ +u\left(x,y,t-\pi \left[u^t\right]\right)=f\left(x,y\right),\hfill & \left(x,y\right)\in \Omega ,t\in \left[0,+\infty \right),\hfill \\ u\left(x,y,t\right)=\phi \left(x,y,t\right),\hfill & \left(x,y\right)\in \Omega ,t\in \left[-h,0\right],\hfill \\ u_t\left(x,y,t\right)={\phi }_t\left(x,y,t\right),\hfill & \left(x,y\right)\in \Omega ,t\in \left[-h,0\right],\hfill \end{array}$ (1.1)

with the following boundary conditions imposed

$\{\begin{array}{ll}u\left(0,y,t\right)={\partial }_{xx}u\left(0,y,t\right)=0,\hfill & \left(y,t\right)\in \left[-l,l\right]\times \left[0,+\infty \right),\hfill \\ u\left(\pi ,y,t\right)={\partial }_{xx}u\left(\pi ,y,t\right)=0,\hfill & \left(y,t\right)\in \left[-l,l\right]\times \left[0,+\infty \right),\hfill \\ {\partial }_{yy}u\left(x,\pm \ell ,t\right)+\sigma {\partial }_{xx}u\left(x,\pm \ell ,t\right)=0,\hfill & \left(x,t\right)\in \left[0,\pi \right]\times \left[0,+\infty \right),\hfill \\ {\partial }_{yyy}u\left(x,\pm \ell ,t\right)+\left(2-\sigma \right){\partial }_{xxy}u\left(x,\pm \ell ,t\right)=0,\hfill & \left(x,t\right)\in \left[0,\pi \right]\times \left[0,+\infty \right),\hfill \end{array}$ (1.2)

where $\Omega =\left[0,\pi \right]\times \left[-l,l\right]$ . $u=u\left(x,y,t\right)$ describes the deformation of the bridge in the vertical plane; ${\delta }_1,{\delta }_2>0$ ; $h>0$ represents the maximum delay time, $u\left(t-\pi \left[u^t\right]\right)$ is the state delay term, $\phi$ is the initial value on the interval $\left[-h,0\right]$ , $\pi$ is a mapping with values in the interval $\left[0,h\right]$ . Among them, $g\left(u\right)$ denotes the nonlinear term inside the bridge deck, while $\varphi \left({\Vert \nabla u\Vert }^2\right)\text{Δ}u$ , first proposed by S. Woinwsky-Krieger [12] is used to describe the transverse deflection of a stretchable beam.

Without loss of generality, let $A={\text{Δ}}^2,A^{\frac{1}{2}}=-\text{Δ}$ , whose domain is

$D\left(A\right)=\left\{u\in H^4\left(\Omega \right):u\left(0,y,t\right)=u\left(\pi ,y,t\right)=u_{xx}\left(0,y,t\right)=u_{xx}\left(\pi ,y,t\right)=0\right\},$

In particular, let $H=V_0=L^2\left(\Omega \right)$ , where its inner product and norm are defined respectively as

${\left(u,v\right)}_H=\left(u,v\right),{\Vert u\Vert }_H^2={\Vert u\Vert }^2.$

Analogously to Reference [13], we introduce the following phase space:

$H_{*}^2\left(\Omega \right)=\left\{w\in H^2\left(\Omega \right):w\left(0,y\right)=w\left(\pi ,y\right)=0,\forall y\in \left(-l,l\right)\right\},$

For the convenience of calculation, we set $V_2=H_{*}^2\left(\Omega \right)$ , and their corresponding inner products and norms are respectively defined as

${\left(u,v\right)}_{V_2}={\displaystyle {\int }_{\Omega }\left[\text{Δ}u\text{Δ}v+\left(1-\sigma \right)\left(2u_{xy}{v}_{xy}-u_{xx}{v}_{yy}-u_{yy}{v}_{xx}\right)\right]\text{d}x\text{d}y},$

${\Vert u\Vert }_2^2={\Vert u\Vert }_{V_2}^2={\left[{\displaystyle {\int }_{\Omega }\left[{\left(\Delta u\right)}^2+2\left(1-\sigma \right)\left(u_{xy}^2-u_{xx}{v}_{yy}\right)\right]\text{d}x\text{d}y}\right]}^{\frac{1}{2}}.$

According to Lemma 4.1 in Reference [13], it follows that the norm ${\Vert \cdot \Vert }_{H_{\ast }^2}$ and ${\Vert \cdot \Vert }_{H^2}$ are equivalent. Furthermore, by Sobolev the compact embedding theorem, $V_2$ ↪$V_0$ . According to Poincaré the inequality, we have

${\Vert u\Vert }_2^2\ge {\lambda }_1{\Vert u\Vert }_0^2,\forall u\in V_2{\Vert u\Vert }_{D\left(A\right)}^2\ge {\lambda }_1^2{\Vert u\Vert }_2^2,u\in D\left(A\right)$

Here ${\lambda }_1$ is $A$ the first eigenvalue.

Define the phase space

$Y\equiv C\left(\left[-h,0\right];V_2\right)\cap C^1\left(\left[-h,0\right];H\right),$

whose norm is given by

${\Vert \phi \Vert }_Y={\Vert \phi \Vert }_{C_{V_2}}+{\Vert {\phi }_t\Vert }_{C_H},{\Vert v\Vert }_{C_X}=\underset{\theta \in \left[-h,0\right]}{\text{sup}}{\Vert v\left(\theta \right)\Vert }_X,\forall v\in C_X.$

Assume that the nonlinear term $g\in C^2\left(ℝ, ℝ\right)$ satisfies the conditions:

$\begin{array}{l}\underset{\left|s\right|\to \infty }{\lim \inf }\frac{G\left(s\right)}{s^2}\ge 0,G\left(s\right)={\displaystyle {\int }_0^{s}g\left(\tau \right)\text{d}\tau },\forall s\in ℝ,\hfill \end{array}$ (1.3)

$\begin{array}{l}\underset{\left|s\right|\to \infty }{\lim \inf }\frac{sg\left(s\right)-C_{0}G\left(s\right)}{s^2}\ge 0,C_0>0,\forall s\in ℝ,\hfill \end{array}$ (1.4)

$\begin{array}{l}\underset{\left|s\right|\to \infty }{\lim \sup }\frac{g^{\prime }\left(s\right)}{|s{|}^p}=0,\forall p\ge 0,\forall s\in ℝ.\hfill \end{array}$ (1.5)

Let the mapping $\pi :Y\to \left[0,h\right]$ be locally Lipschitz, that is, for any $N>0$ , there exists $L_N>0$ , For any ${\beta }_1,{\beta }_2\in Y$ , ${\Vert {\delta }_i\Vert }_Y\le N$ , $i=1,2$ , there holds

$\left|\pi \left({\beta }_1\right)-\pi \left({\beta }_2\right)\right|\le L_N{\Vert {\beta }_1-{\beta }_2\Vert }_Y.$ (1.6)

To obtain the compactness of the semigroup, we further assume that there exists $ϵ>0$ such that the delay term satisfies the subcritical local Lipschitz condition, that is, for any $\rho >0$ , there exists $L_{\rho }>0$ , such that for any ${\beta }_i,i=1,2$ , with ${\Vert {\beta }_i\Vert }_Y\le \rho$ , there holds

$\begin{array}{l}\left|\pi \left({\beta }_1\right)-\pi \left({\beta }_2\right)\right|\le L_{\rho }\underset{\theta \in \left[-h,0\right]}{\max }\Vert A^{\frac{1}{2}-ϵ}\left({\beta }_1-{\beta }_2\right)\Vert \hfill \end{array}.$ (1.7)

Finally, according to conditions (1.3)-(1.4) and Poincaré inequality, there exist constants $K_1,K_2>0$ , such that

$\begin{array}{l}{\displaystyle {\int }_{\Omega }G\left(u\right)\text{d}x}+\frac{1}{8}{\Vert \text{Δ}u\Vert }^2\ge -K_1,\forall u\in V,\hfill \end{array}$ (1.8)

$\begin{array}{l}\left(g\left(u\right),u\right)-C_0{\displaystyle {\int }_{\Omega }G\left(u\right)\text{d}x}+\frac{1}{8}{\Vert \text{Δ}u\Vert }^2\ge -K_2,\forall u\in V.\hfill \end{array}$ (1.9)

Assume that the nonlinear function $\varphi (\cdot )\in C^1\left(ℝ\right)$ satisfies

$\begin{array}{l}\varphi \left(s\right)\ge 0,\varphi \left(s\right)s\ge \frac{1}{2}{\displaystyle {\int }_0^s\varphi \left(\tau \right)\text{d}\tau }+\beta s^2,\hfill \end{array}$ (1.10)

where $0<\beta <\frac{1}{2}$ , $\varphi \left(0\right)=0$ .

2. Well-Posedness

2.1. Priori Estimate

Theorem 2.1 Assume that conditions (1.3) and (1.4) hold, $f\in L^2\left(\Omega \right)$ . Then the solution to Equation (1.1) satisfies the following estimate

${\Vert z\Vert }^2+{\Vert \Delta u\Vert }^2+{\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau {\Vert \nabla u\Vert }^2\le 4{\text{e}}^{-\alpha t}\left(E\left(0\right)+\mu h{\Vert \psi \Vert }_Y\right)+\frac{4C}{\alpha },$ (2.1)

where $C=\frac{4}{{\delta }_1}{\Vert g\Vert }^2+\gamma K_1+2\epsilon K_2$ , $E\left(t\right)={\Vert z\Vert }^2+{\Vert \text{Δ}u\Vert }^2+2{\displaystyle {\int }_{\Omega }G\left(u\right)\text{d}x}+\left({\delta }_2\epsilon +{\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau \right){\Vert \nabla u\Vert }^2+2K_1$ .

Proof Taking the inner product of $z={\partial }_{t}u+\epsilon u\left(\epsilon >0\right)$ with (1.1) in $H$ , we obtain

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({\Vert z\left(t\right)\Vert }^2+{\Vert \Delta u\Vert }^2\right)+\epsilon {\Vert \Delta u\Vert }^2+\left({\delta }_1-\epsilon \right){\Vert z\left(t\right)\Vert }^2-\epsilon \left({\delta }_1-\epsilon \right)\left(u,z\right)\\ =\left(f,z\right)-\left(g\left(u\right),z\right)+\left({\delta }_2\Delta u_t,z\right)-\left(u\left(t-\pi \left[u^t\right]\right),z\right)+\left(\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u,z\right).\end{array}$ (2.2)

By applying Young inequality, Hölder and Poincaré inequality, and choosing a sufficiently small $\epsilon >0$ , we have

$\left({\delta }_2\Delta u_t,z\right)=-{\delta }_2{\Vert \nabla u_t\Vert }^2-{\delta }_2\epsilon \frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla u\Vert }^2.$ (2.3)

$\epsilon {\Vert \Delta u\Vert }^2+\left({\delta }_1-\epsilon \right){\Vert z\left(t\right)\Vert }^2-\epsilon \left({\delta }_1-\epsilon \right)\left(u,z\right)\ge \frac{3{\delta }_1}{4}{\Vert z\left(t\right)\Vert }^2+\frac{\epsilon }{2}{\Vert \Delta u\Vert }^2.$ (2.4)

$\left(f,z\right)\le \frac{2}{{\delta }_1}{\Vert f\Vert }^2+\frac{{\delta }_1}{8}{\Vert z\Vert }^2.$ (2.5)

$\begin{array}{c}-\left(u\left(t-\pi \left[u^t\right]\right),z\right)\le \Vert u\left(t-\pi \left[u^t\right]\right)\Vert \Vert z\Vert \\ \le \Vert u\left(t\right)-{\displaystyle {\int }_{t-\pi \left[u^t\right]}^t{\partial }_{t}u\left(s\right)\text{d}s}\Vert \Vert z\Vert \\ \le \left[\Vert u\left(t\right)\Vert +2\Vert {\displaystyle {\int }_0^h{\partial }_{t}u\left(t-s\right)\text{d}s}\Vert \right]\Vert z\Vert \\ \le 2\Vert u\left(t\right)\Vert \Vert z\Vert +2\Vert {\displaystyle {\int }_0^h{\partial }_{t}u\left(t-s\right)\text{d}s}\Vert \Vert z\Vert \\ \le \frac{2}{\sqrt{{\lambda }_1}}\Vert \Delta u\left(t\right)\Vert \Vert z\Vert +2\Vert {\displaystyle {\int }_0^h{\partial }_{t}u\left(t-s\right)\text{d}s}\Vert \Vert z\Vert \\ \le \frac{\epsilon }{8}{\Vert \Delta u\left(t\right)\Vert }^2+\frac{4}{\epsilon {\lambda }_1}{\Vert z\Vert }^2+\frac{8}{{\delta }_1}{\displaystyle {\int }_0^h{\Vert {\partial }_{t}u\left(t-s\right)\Vert }^2\text{d}s}+\frac{{\delta }_1}{8}{\Vert z\Vert }^2.\end{array}$ (2.6)

From (1.8) and (1.9)

$\begin{array}{l}-\left(g\left(u\right),z\right)\le -\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{\Omega }G\left(u\right)\text{d}x}-\epsilon C_0{\displaystyle {\int }_{\Omega }G\left(u\right)\text{d}x}+\frac{\epsilon }{8}{\Vert \Delta u\Vert }^2+\epsilon K_2,\hfill \end{array}$ (2.7)

$\begin{array}{l}\left(\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u,z\right)=\left[-\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert \nabla u\Vert }^2-\epsilon {\Vert \nabla u\Vert }^2\right]\varphi \left({\Vert \nabla u\Vert }^2\right).\hfill \end{array}$ (2.8)

Substituting (2.3)-(2.8) into (2.2), we obtain

$\begin{array}{c}\frac{\text{d}}{\text{d}t}E\left(t\right)\le \frac{16}{{\delta }_1}{\displaystyle {\int }_0^h{\Vert u_t\left(t-s\right)\Vert }^2\text{d}s}+\frac{4}{{\delta }_1}{\Vert f\Vert }^2+2\epsilon K_2\\ -\frac{\epsilon }{2}{\Vert \Delta u\Vert }^2-\left({\delta }_1-\frac{4}{\epsilon {\lambda }_1}\right){\Vert z\Vert }^2-2\epsilon C_0{\displaystyle {\int }_{\Omega }G\left(u\right)\text{d}x}\\ -2{\delta }_2{\Vert \nabla u_t\Vert }^2-2\varphi \left({\Vert \nabla u\Vert }^2\right)\epsilon {\Vert \nabla u\Vert }^2.\end{array}$ (2.9)

Take

$\begin{array}{l}E\left(t\right)={\Vert z\Vert }^2+{\Vert \Delta u\Vert }^2+2{\displaystyle {\int }_{\Omega }G\left(u\right)\text{d}x}+{\delta }_2\epsilon {\Vert \nabla u\Vert }^2+{\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau {\Vert \nabla u\Vert }^2+2K_1,\hfill \end{array}$

define

$V\left(t\right)=E\left(t\right)+\frac{\mu }{h}{\displaystyle {\int }_0^h{\displaystyle {\int }_{t-s}^t{\Vert u_t\left(\xi \right)\Vert }^2\text{d}\xi \text{d}s}}.$

Obviously

$E\left(t\right)\le V\left(t\right)\le E\left(t\right)+\mu {\displaystyle {\int }_0^h{\Vert u_t\left(t-\xi \right)\Vert }^2\text{d}\xi },$ (2.10)

where $0<\mu <\frac{{\delta }_1}{4}$ , $\frac{\mu }{h}{\displaystyle {\int }_0^h{\displaystyle {\int }_{t-s}^t{\Vert {\partial }_{t}u\left(\xi \right)\Vert }^2\text{d}\xi \text{d}s}}$ serves as the compensation term for the delay term in the equation. Choosing a sufficiently large. Choosing a sufficiently large ${\delta }_1$ , ensures that ${\delta }_1-\frac{4}{\epsilon {\lambda }_1}>\frac{{\delta }_1}{2}$ , That is, when ${\delta }_1>\frac{32}{\epsilon {\lambda }_1}$ he system reaches equilibrium. This is because a longer delay time makes the system more unstable, while increasing the damping coefficient can achieve balance. Taking the derivative of $V\left(t\right)$ , we obtain

$\frac{\text{d}}{\text{d}t}V\left(t\right)=\frac{\text{d}}{\text{d}t}E\left(t\right)+\mu {\Vert u_t\left(t\right)\Vert }^2-\frac{\mu }{h}{\displaystyle {\int }_0^h{\Vert u_t\left(t-s\right)\Vert }^2\text{d}s}.$ (2.11)

Since

${\Vert u_t\Vert }^2={\Vert u_t+\epsilon u-\epsilon u\Vert }^2\le 2{\Vert u_t+\epsilon u\Vert }^2+2{\epsilon }^2{\Vert u\Vert }^2\le 2{\Vert z\Vert }^2+\frac{2{\epsilon }^2}{{\lambda }_1}{\Vert \Delta u\Vert }^2,$

Substituting (2.9) into (2.12) and then using (1.10), we can obtain

$\begin{array}{l}\frac{\text{d}}{\text{d}t}V\left(t\right)+\left(\frac{\epsilon }{2}-\frac{2\mu {\epsilon }^2}{{\lambda }_1}\right){\Vert \Delta u\Vert }^2+\left({\delta }_1-2\mu -\frac{4}{\epsilon {\lambda }_1}\right){\Vert z\Vert }^2\\ +2\epsilon C_0{\displaystyle {\int }_{\Omega }G\left(u\right)\text{d}x}+\epsilon {\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau {\Vert \nabla u\Vert }^2+2\epsilon \beta {\Vert \nabla u\Vert }^2\\ \le \left(\frac{16}{{\delta }_1}-\frac{\mu }{h}\right){\displaystyle {\int }_0^h{\Vert u_t\left(t-s\right)\Vert }^2\text{d}s}+\frac{4}{{\delta }_1}{\Vert f\Vert }^2+2\epsilon K_2.\end{array}$

Furthermore, by applying (2.10), where $\mu <\frac{{\delta }_1}{4}$ , we choose a sufficiently small $\epsilon$ , such that $\frac{\epsilon }{2}-\frac{2\mu {\epsilon }^2}{{\lambda }_1}>0$ . Take $\gamma =\text{min}\left\{{\delta }_1-\frac{4}{\epsilon {\lambda }_1}-2\mu ,\frac{\epsilon }{2}-\frac{2\mu {\epsilon }^2}{{\lambda }_1},\epsilon C_0,\epsilon ,2\beta \right\}$ , It follows that

$\frac{\text{d}}{\text{d}t}V\left(t\right)+\gamma V\left(t\right)\le C.$

Applying Gronwall lemma to the above equation, we have

$V\left(t\right)\le V\left(0\right){\text{e}}^{-\gamma t}+\frac{C}{\gamma }\left(1-{\text{e}}^{-\gamma t}\right).$ (2.12)

According to (1.8), (2.10), Combining with (2.12), we have

${\Vert z\Vert }^2+{\Vert \Delta u\Vert }^2+{\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau {\Vert \nabla u\Vert }^2\le E\left(t\right)\le 4{\text{e}}^{-\alpha t}\left(E\left(0\right)+\mu h{\Vert \psi \Vert }_Y^2\right)+\frac{4C}{\alpha },$

where $C=\frac{4}{{\delta }_1}{\Vert f\Vert }^2+\gamma K_1+2\epsilon K_2$ .

2.2. Existence and Uniqueness

Let $U={\left(u,v\right)}^{\text{T}}$ , then Equation (1.1) can be written in the following abstract form in the space $\Re =V_2\times H$ :

$\{\begin{array}{ll}\frac{\text{d}}{\text{d}t}U\left(t\right)=LU\left(t\right)+\aleph \left(U\left(t\right)\right),\hfill & \left(x,y,t\right)\in \Omega \times \left[0,+\infty \right),\hfill \\ U\left(x,y,t\right)=\Psi \left(x,y,t\right),\hfill & \left(x,y,t\right)\in \Omega \times \left[-h,0\right],\hfill \end{array}$

where $\Psi =\left(\psi ,{\psi }^{\prime }\right),\psi \in Y$ , the operator $L$ is defined as

$L\left(\begin{array}{c}u\\ v\end{array}\right)=\left(\begin{array}{c}v\\ -{\text{Δ}}^{2}u-{\delta }_{1}v+{\delta }_2\text{Δ}v\end{array}\right)$

whose domain is

$D\left(L\right)=\left\{\left(u,v\right)\in \Re |u\in H_{*}^2\left(\Omega \right),v\in H\right\},$

$\aleph \left(U\right)={\left(0;-g\left(u\right)-u\left(t-\pi \left[u^t\right]\right)+\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u+f\right)}^{\text{T}}.$

Definition 2.2 [11] A mild solution of Equation (1.1) refers to a function $u\in C\left(\left[-h,T\right];D\left(A^{\frac{1}{2}}\right)\right)\cap C^1\left(\left[-h,T\right];H\right)$ defined on the interval $\left[0,T\right]$ , such that $u\left(\theta \right)=\phi \left(\theta \right)$ , $\theta \in \left[-h,0\right]$ and $U\left(t\right)=\left(u\left(t\right);{\partial }_{t}u\left(t\right)\right)$

satisfies

$U\left(t\right)={\text{e}}^{-tL}U\left(0\right)+{\displaystyle {\int }_0^t{\text{e}}^{-\left(t-s\right)L}\aleph \left(U\left(s\right)\right)\text{d}s},t\in \left[0,T\right].$

Lemma 2.3 $L$ is the infinitesimal generator of the $C_0$ -semigroup ${\text{e}}^{-tL}$ in $\Re$ .

Proof Since for $U\in D\left(A\right)$

${\langle LU,U\rangle }_{\Re }=-{\delta }_1{\Vert v\Vert }^2-{\delta }_2{\Vert \nabla v\Vert }^2<0,$

Thus, the operator $L$ is dissipative. Note that

$\overline{D\left(A\right)}=H_{\text{*}}^2\times H.$

Since

$\left(H_0^1\cap H^2\right)\times \left(H_0^1\cap H^2\right)\subseteq D\left(L\right)\subseteq H_{\text{*}}^2\times H$

Thus, $\left(H_0^1\cap H^2\right)\times \left(H_0^1\cap H^2\right)$ is dense in $H_{\text{*}}^2\times H$ .

Next, $L$ is maximal, that is, for any fixed $\alpha >0$ , it is necessary to prove that $\alpha I-L$ is surjective. To this end, given $\left(k,l\right)\in \Re$ , we seek the following system of equations

$\left(\alpha I-L\right)\left(\begin{array}{c}u\\ v\end{array}\right)=\left(\begin{array}{c}k\\ l\end{array}\right),$

that is, we verify the following system of equations

$\{\begin{array}{l}\alpha u-v=k,\\ \alpha v+{\Delta }^{2}u+{\delta }_{1}v-{\delta }_2\Delta u=l.\end{array}$ (2.13)

admits a unique solution $U={\left(u,v\right)}^{\text{T}}\in D\left(L\right)$ . From the first equation, we obtain $v=\alpha u-k$ , substituting it into the second equation, we have

${\alpha }^{2}u+{\Delta }^{2}u+{\delta }_1\alpha u-{\delta }_2\alpha \Delta u=\alpha k+l+{\delta }_{1}k.$ (2.14)

Then Problem (2.14) can be rephrased as

${\displaystyle {\int }_{\Omega }\left({\alpha }^{2}u+{\Delta }^{2}u+{\delta }_1\alpha u-{\delta }_2\alpha \Delta u\right)\omega \text{d}x}={\displaystyle {\int }_{\Omega }\left(\alpha k+l+{\delta }_{1}k\right)\omega \text{d}x},\forall \omega \in \text{Ł}\left(\Omega \right).$ (2.15)

Define the following bilinear operator and linear operator

$B\left(u,\omega \right)={\displaystyle {\int }_{\Omega }\left({\alpha }^{2}u+{\Delta }^{2}u+{\delta }_1\alpha u-{\delta }_2\alpha \Delta u\right)\omega \text{d}x},F\left(\omega \right)={\displaystyle {\int }_{\Omega }\left(\alpha k+l+{\delta }_{1}k\right)\omega \text{d}x}.$

From the proof results of Theorem 2.1, it is evident that $B$ is coercive and bounded, and $F$ is bounded. Thus, by the Lax-Milgram lemma, it can be guaranteed that there exists a unique solution $U={\left(u,v\right)}^{\text{T}}\in D\left(L\right)$ to (2.15). According to the Lumer-Phillips theorem in Reference [14], it follows that $L$ is an infinitesimal generator in $\Re$

Theorem 2.4 Suppose that Conditions (1.3)-(1.7) and (1.10) hold. Then, for

any initial values ${\phi }_i\in Y,i=1,2$ , there exists $0<T_{\phi }<\infty$ , such that Problem (1.1) has a unique mild solution $U\left(t\right)\equiv \left(u\left(t\right);{\partial }_{t}u\left(t\right)\right)$ on the interval $\left[-h,T_{\phi }\right]$ .

Proof Fix a constant $\sigma >0$ , and define the ball $B_{\sigma }=\left\{U\in C\left(\left[0,T\right];\Re \right):{\Vert U-\overline{V}\Vert }_{C\left(\left[0,T\right];\Re \right)}\le \sigma \right\}$ , where $\overline{V}={\text{e}}^{-tL}\phi \left(0\right)$. Define the mapping $M :C\left(\left[0,T\right];\Re \right)\to C\left(\left[0,T\right];\Re \right)$ :

$\left[MU\right]\left(t\right)=\overline{V}\left(T\right)+{\displaystyle {\int }_0^t{\text{e}}^{-\left(t-s\right)L}\aleph \left(U\left(s\right)\right)\text{d}s},t\in \left[0,T\right].$

If $U$ is a fixed point of the mapping $M$ , then $U$ is a mild solution to equation (3.0.1) on $\left[0,T\right]$ . Next, we prove that $M$ is a contraction mapping.

1) For any $\forall t\in \left[0,T\right]$ , $U_1,U_2\in B_{\sigma }$ , we have

$\begin{array}{l}{\Vert \left[M{U}_1\right]\left(t\right)-\left[M{U}_2\right]\left(t\right)\Vert }_{C\left(\left[0,T\right];\Re \right)}\\ \le {\displaystyle {\int }_0^t{\Vert {\text{e}}^{-\left(t-s\right)L}\left(g\left(u_2\left(s\right)\right)-g\left(u_1\left(s\right)\right)\right)\Vert }_{C\left(\left[0,T\right];H\right)}\text{d}s}\\ +{\displaystyle {\int }_0^t{\Vert {\text{e}}^{-\left(t-s\right)L}\left(u\left(s-\pi \left[u_2^s\right]\right)-u\left(s-\pi \left[u_1^s\right]\right)\right)\Vert }_{C\left(\left[0,T\right];H\right)}\text{d}s}\\ +{\displaystyle {\int }_0^t{\Vert {\text{e}}^{-\left(t-s\right)L}\left(\varphi \left({\Vert \nabla u_1\Vert }^2\right)\Delta u_1-\varphi \left({\Vert \nabla u_2\Vert }^2\right)\Delta u_2\right)\Vert }_{C\left(\left[0,T\right];H\right)}\text{d}s}\\ \le {\displaystyle {\int }_0^t{\Vert g\left(u_2\left(s\right)\right)-g\left(u_1\left(s\right)\right)\Vert }_{C\left(\left[0,T\right];H\right)}\text{d}s}\\ +{\displaystyle {\int }_0^t{\Vert u\left(s-\pi \left[u_2^s\right]\right)-u\left(s-\pi \left[u_1^s\right]\right)\Vert }_{C\left(\left[0,T\right];H\right)}\text{d}s}\\ +{\displaystyle {\int }_0^t{\Vert \varphi \left({\Vert \nabla u_1\Vert }^2\right)\Delta u_1-\varphi \left({\Vert \nabla u_2\Vert }^2\right)\Delta u_2\Vert }_{C\left(\left[0,T\right];H\right)}\text{d}s},\end{array}$ (2.16)

According to Condition (1.4), combined with (2.1) and the Sobolev embedding theorem, there exists a constant $K_3>0$ , such that

${\left|g\left(u\right)\right|}_{L^{\infty }}\le K_3,{\left|g^{\prime }\left(u\right)\right|}_{L^{\infty }}\le K_3.$ (2.17)

The following estimate applies the Mean Value Theorem for Differentiation, first,

$\Vert g\left(u_2\left(s\right)\right)-g\left(u_1\left(s\right)\right)\Vert \le K_3\Vert u_2-u_1\Vert \le \frac{K_3}{{\lambda }_1}\Vert A^{\frac{1}{2}}\left(u_2-u_1\right)\Vert ,$

$\left|\varphi \left({\Vert \nabla u_n\Vert }^2\right)-\varphi \left({\Vert \nabla u\Vert }^2\right)\right|\le C{\Vert \nabla u_n-\nabla u\Vert }^2.$

From the previous estimates, it follows that ${\Vert \nabla u_n-\nabla u\Vert }^2$ is bounded and $\varphi \left({\Vert \nabla u\Vert }^2\right)$ is bounded, combining this with the above equation, we obtain that ${\varphi }^{\prime }\left({\Vert \nabla u\Vert }^2\right)$ is bounded. Next, there exist constants $K_4,K_5$ and $C_3$ such that

$\begin{array}{l}\Vert \varphi \left({\Vert \nabla u_1\Vert }^2\right)\text{Δ}{u}_1-\varphi \left({\Vert \nabla u_2\Vert }^2\right)\text{Δ}{u}_2\Vert \\ \le \Vert \varphi \left({\Vert \nabla u_1\Vert }^2\right)\left(\text{Δ}{u}_1-\text{Δ}{u}_2\right)\Vert +\Vert \varphi \left({\Vert \nabla u_1\Vert }^2\right)-\varphi \left({\Vert \nabla u_2\Vert }^2\right)\text{Δ}{u}_2\Vert \\ \le K_4\Vert \text{Δ}{u}_1-\text{Δ}{u}_2\Vert +\Vert K_5\left({\Vert \nabla u_1\Vert }^2-{\Vert \nabla u_2\Vert }^2\right)\text{Δ}{u}_2\Vert \\ \le K_4\Vert \text{Δ}{u}_1-\text{Δ}{u}_2\Vert +K_{5}C\left(R\right)\left(\Vert \nabla u_1-\nabla u_2\Vert \right)\\ \le C_3\Vert A^{\frac{1}{2}}\left(u_2-u_1\right)\Vert .\end{array}$

Given that $U_i\in B_{\sigma }$ and ${\Vert U_i-\overline{V}\Vert }_{C\left(\left[0,T\right];\Re \right)}\le \sigma$ , we have

$\begin{array}{c}{\Vert U_i\Vert }_{C\left(\left[0,T\right];\Re \right)}=\underset{t\in \left[0,T\right]}{\max }\left(\Vert A^{\frac{1}{2}}{u}_i\left(t\right)\Vert +\Vert {\partial }_t{u}_i\left(t\right)\Vert \right)\\ \le \sigma +{\Vert \overline{V}\Vert }_{C\left(\left[0,T\right];\Re \right)}\\ \le \sigma +\underset{t\in \left[0,T\right]}{\max }\left(\Vert A^{\frac{1}{2}}{\text{e}}^{-tA}\beta \left(0\right)\Vert +\Vert {\text{e}}^{-tA}{\beta }_t\phi \left(0\right)\Vert \right)\\ \le \sigma +\left(\Vert A^{\frac{1}{2}}\beta \left(0\right)\Vert +\Vert {\partial }_t\beta \left(0\right)\Vert \right)\triangleq \tilde{N},\end{array}$ (2.18)

so $\Vert A^{\frac{1}{2}}{u}_i\left(t\right)\Vert \le \tilde{N},t\in \left[0,T\right],i=1,2$ . Thus

$\begin{array}{l}{\Vert g\left(u_2\left(s\right)\right)-g\left(u_1\left(s\right)\right)\Vert }_{C\left(\left[0,T\right];H\right)}\\ \le \frac{K_3}{{\lambda }_1}\underset{t\in \left[\tau ,T\right]}{\max }\Vert A^{\frac{\alpha }{2}}\left(u_2-u_1\right)\Vert \le Z_R{\Vert U_2-U_1\Vert }_{C\left(\left[\tau ,T\right];\Re \right)}.\end{array}$ (2.19)

and

$\begin{array}{l}{\Vert \varphi \left({\Vert \nabla u_1\Vert }^2\right)\text{Δ}{u}_1-\varphi \left({\Vert \nabla u_2\Vert }^2\right)\text{Δ}{u}_2\Vert }_{C\left(\left[0,T\right];H\right)}\\ \le C_3\underset{t\in \left[\tau ,T\right]}{\text{max}}\Vert A^{\frac{1}{2}}\left(u_2-u_1\right)\Vert \le Z_{\tilde{N}}{\Vert U_2-U_1\Vert }_{C\left(\left[\tau ,T\right];ℝ\right)}\end{array}$ (2.20)

From (2.16), for any $\forall 0\le s\le T$ ,

$\begin{array}{c}{\Vert u_i^s\Vert }_Y=\underset{\kappa \in \left[-h,0\right]}{\max }\Vert A^{\frac{1}{2}}{u}_i^s\left(\kappa \right)\Vert +\underset{\kappa \in \left[-h,0\right]}{\max }\Vert {\partial }_t{u}_i^s\left(\kappa \right)\Vert \\ =\underset{a\in \left[s-h,s\right]}{\max }\Vert A^{\frac{1}{2}}{u}_i\left(a\right)\Vert +\underset{a\in \left[s-h,s\right]}{\max }\Vert {\partial }_t{u}_i\left(a\right)\Vert \\ \le \underset{a\in \left[-h,T\right]}{\max }\Vert A^{\frac{1}{2}}{u}_i\left(a\right)\Vert +\underset{a\in \left[-h,T\right]}{\max }\Vert {\partial }_t{u}_i\left(a\right)\Vert \\ =\underset{a\in \left[-h,0\right]}{\max }\Vert A^{\frac{1}{2}}{u}_i\left(a\right)\Vert +\underset{a\in \left[0,T\right]}{\max }\Vert A^{\frac{1}{2}}{u}_i\left(a\right)\Vert \\ +\underset{a\in \left[-h,0\right]}{\max }\Vert {\partial }_t{u}_i\left(a\right)\Vert +\underset{a\in \left[0,T\right]}{\max }\Vert {\partial }_t{u}_i\left(a\right)\Vert \\ \le 2{\Vert \phi \Vert }_Y+2\underset{a\in \left[0,T\right]}{\max }\left(\Vert A^{\frac{1}{2}}{u}_i\left(a\right)\Vert +\Vert {\partial }_t{u}_i\left(a\right)\Vert \right)\\ \le 2{\Vert \phi \Vert }_Y+2{\Vert U_i\Vert }_{C\left(\left[\tau ,T\right];\Re \right)}\\ \le 2{\Vert \phi \Vert }_Y+2\tilde{N}\triangleq \widehat{N}.\end{array}$ (2.21)

From $u\left(t-\pi \left[u^t\right]\right)=u\left(t\right)-{\displaystyle {\int }_{t-\pi \left[u^t\right]}^t{\partial }_{t}u\left(s\right)\text{d}s}$ , combined with Condition (1.6) and Reference [9], we obtain

$\Vert u_2\left(s-\pi \left[u_2^s\right]\right)-u_1\left(s-\pi \left[u_1^s\right]\right)\Vert \le \left(\widehat{N}\cdot L_N+1\right){\Vert u_2^s-u_1^s\Vert }_Y,$

$\begin{array}{c}{\Vert u_2^s-u_1^s\Vert }_Y\le \underset{a\in \left[s-h,0\right]}{\max }\left(\Vert A^{\frac{1}{2}}\left(u_2\left(a\right)-u_1\left(a\right)\right)\Vert +\Vert {\partial }_t{u}_2\left(a\right)-{\partial }_t{u}_1\left(a\right)\Vert \right)\\ +\underset{a\in \left[0,s\right]}{\max }\left(\Vert A^{\frac{1}{2}}\left(u_2\left(a\right)-u_1\left(a\right)\right)\Vert +\Vert {\partial }_t{u}_2\left(a\right)-{\partial }_t{u}_1\left(a\right)\Vert \right)\\ \le 2{\Vert U_2-U_1\Vert }_{C\left(\left[\tau ,T\right];\Re \right)}.\end{array}$

Through the above inequality, it holds that

$\begin{array}{l}{\Vert u_2\left(s-\pi \left[u_2^s\right]\right)-u_1\left(s-\pi \left[u_1^s\right]\right)\Vert }_{C\left(\left[\tau ,T\right];\Re \right)}\le 2\left(\widehat{N}\cdot L_N+1\right){\Vert U_2-U_1\Vert }_{C\left(\left[\tau ,T\right];\Re \right)}.\hfill \end{array}$ (2.22)

Substituting (2.19), (2.20) and (2.22) into (2.16), we obtain

$\begin{array}{c}{\Vert \left[M{U}_1\right]\left(t\right)-\left[M{U}_2\right]\left(t\right)\Vert }_{C\left(\left[0,T\right];\Re \right)}\le {\displaystyle {\int }_0^t\left(2Z_{\tilde{N}}+2\left(\widehat{N}\cdot L_N+1\right)\right){\Vert U_2-U_1\Vert }_{C\left(\left[\tau ,T\right];\Re \right)}}\\ \le T\cdot \left(2Z_{\tilde{N}}+2\left(\widehat{N}\cdot L_N+1\right)\right){\Vert U_2-U_1\Vert }_{C\left(\left[\tau ,T\right];\Re \right)},\end{array}$

Choose $T$ ufficiently small, such that $T\cdot \left(2Z_{\tilde{N}}+\left(\widehat{N}\cdot L_R+1\right)\right)<1$ .

(2) For any $\forall t\in \left[0,T\right]$ and $z\in B_{\sigma }$ , combined with (2.17)-(2.21) we have

$\begin{array}{l}{\Vert \left[MU\right]\left(t\right)-\overline{V}\left(t\right)\Vert }_{C\left(\left[0,T\right];\Re \right)}\\ \le {\Vert {\displaystyle {\int }_0^t{\text{e}}^{-\left(t-s\right)L}\left(-g\left(u\left(s\right)\right)-u\left(s-\pi \left[u^s\right]\right)\right)}\Vert }_{C\left(\left[0,T\right];H\right)}\text{d}s+\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u+f\\ \le {\displaystyle {\int }_0^t\left({\Vert g\left(u\left(s\right)\right)\Vert }_{C\left(\left[0,T\right];H\right)}+{\Vert u\left(s-\pi \left[u^s\right]+\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u\right)\Vert }_{C\left(\left[0,T\right];H\right)}+{\Vert f\Vert }_{C(\left[0,T\right];H}\right)\text{d}s}\\ \le {\displaystyle {\int }_0^t\left(2Z_{\tilde{N}}+2\left(\widehat{N}\cdot L_N+1\right)\right){\Vert U\Vert }_{C\left(\left[0,T\right];\Re \right)}\text{d}s}+{\Vert f\Vert }_{C\left(\left[0,T\right];\Re \right)}\\ \le T\cdot \left(2Z_{\tilde{N}}+2\left(\widehat{N}\cdot L_N+1\right)\right)\cdot \tilde{N}+{\Vert f\left(0\right)\Vert }_{C\left(\left[0,T\right];\Re \right)},\end{array}$

Choose an appropriate $T$ , such that $T\cdot \left(2Z_{\tilde{N}}+2\left(\widehat{N}\cdot L_R+1\right)\right)\tilde{N}\le \sigma$ . From (1) and (2), it follows that $M:B_{\sigma }\to B_{\sigma }$ contraction mapping. According to the Banach Contraction Fixed Point Theorem, there exists a unique fixed point $U\in C\left(\left[0,T\right];\Re \right)$ . Let

$\overline{u}=\{\begin{array}{ll}u\left(t\right),\hfill & t\in \left[0,T\right],\hfill \\ \phi \left(t\right),\hfill & t\in \left[-h,0\right],\hfill \end{array}$

and $\overline{u}\in C\left(\left[-h,T\right];D\left(A^{\frac{1}{2}}\right)\right)\cap C\left(\left[-h,T\right];H\right)$ , therefore, $\overline{u}$ is a mild solution to Equation (3.0.1) on the interval $\left[-h,T\right]$ .

Theorem 2.5 Suppose Conditions (1.3)-(1.7) and (1.10) hold. Then, for any initial values ${\phi }_i\in Y$ , ${\Vert \phi \Vert }_Y\le \varpi$ , $i=1,2$ , there exists $0<T_{\phi }<\infty$ , such that Problem (1.1) has a unique global mild solution $U\left(t\right)\equiv \left(u\left(t\right);{\partial }_{t}u\left(t\right)\right)$ . on the interval $\left[0,+\infty \right)$ . Furthermore, for any $\varpi >0$ , $T>0$ , there exists a positive constant $C$ , such that

$\begin{array}{l}\underset{a\in \left[0,t\right]}{\text{max}}\left({\Vert A^{\frac{1}{2}}u\left(a\right)\Vert }^2+{\Vert {\partial }_{t}u\left(a\right)\Vert }^2\right)\le C\left(1+E_1\left(0\right)+{\Vert \phi \Vert }_Y^2+{\Vert f\Vert }^2\right){\text{e}}^{{\gamma }_{1}t}.\hfill \end{array}$

Proof Taking the inner product of $u_t$ with Equation (1.1) in $H$ , we have

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({\Vert u_t\Vert }^2+{\Vert \Delta u\Vert }^2+{\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau \cdot {\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }G}\left(u\right)\text{d}x\right)\\ +{\delta }_1{\Vert u_t\Vert }^2+{\delta }_2{\Vert \nabla u_t\Vert }^2=\left(f,u_t\right)-\left(u\left(t-\pi \left[u^t\right]\right),u_t\right).\end{array}$ (2.23)

Let

$E_1\left(t\right)={\Vert u_t\Vert }^2+{\Vert \Delta u\Vert }^2+{\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau \cdot {\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }G}\left(u\right)\text{d}x\ge 0.$

Integrating Equation (2.23) over the interval $\left[0,t\right]$ , we obtain

$\begin{array}{l}{E}_1\left(t\right)+{\delta }_1{\displaystyle {\int }_0^t{\Vert u_t\left(s\right)\Vert }^2\text{d}s}+{\delta }_2{\displaystyle {\int }_0^t{\Vert \nabla u_t\left(s\right)\Vert }^2\text{d}s}\\ \le E_1\left(0\right)+\frac{2t}{{\delta }_1}{\Vert f\Vert }^2+\frac{2}{{\delta }_1}{\displaystyle {\int }_0^t{\Vert u^s\Vert }_Y^2\text{d}s}.\end{array}$ (2.24)

Through Equation (2.21), for any $\forall s\in \left[0,T_{\psi }\right)$ ,

$\begin{array}{l}{\Vert u_i^s\Vert }_Y\le {\Vert \phi \Vert }_Y+2\sqrt{\underset{a\in \left[0,s\right]}{\text{max}}\left({\Vert A^{\frac{1}{2}}u\left(a\right)\Vert }^2+{\Vert {\partial }_{t}u\left(a\right)\Vert }^2\right)}.\hfill \end{array}$ (2.25)

Substituting (2.25) into (2.24), we obtain

$\begin{array}{l}{E}_1\left(t\right)+{\delta }_1{\displaystyle {\int }_0^t{\Vert u_t\left(s\right)\Vert }^2\text{d}s}+{\delta }_2{\displaystyle {\int }_0^t{\Vert \nabla u_t\left(s\right)\Vert }^2\text{d}s}\\ \le E_1\left(0\right)+\frac{2t}{{\delta }_1}{\Vert f\Vert }^2+\frac{2t}{{\delta }_1}{\Vert \phi \Vert }_Y^2+\frac{8}{{\delta }_1}{\displaystyle {\int }_0^t\underset{a\in \left[0,s\right]}{\text{max}}}\left({\Vert A^{\frac{1}{2}}u\left(a\right)\Vert }^2+{\Vert {\partial }_{t}u\left(a\right)\Vert }^2\right)\text{d}s.\end{array}$ (2.26)

From $E_1\left(t\right)={\Vert u_t\Vert }^2+{\Vert \Delta u\Vert }^2+{\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau \cdot {\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }H}\left(u\right)\text{d}x$ , we have

$\begin{array}{l}\underset{a\in \left[0,t\right]}{\text{max}}\left({\Vert A^{\frac{1}{2}}u\left(a\right)\Vert }^2+{\Vert {\partial }_{t}u\left(a\right)\Vert }^2\right)\le E_1\left(t\right).\hfill \end{array}$

Substituting the above equation into (2.26), we obtain

$\begin{array}{l}\underset{a\in \left[0,t\right]}{\text{max}}\left({\Vert A^{\frac{1}{2}}u\left(a\right)\Vert }^2+{\Vert {\partial }_{t}u\left(a\right)\Vert }^2\right)\\ \le C\left(E_1\left(0\right)+t{\Vert f\Vert }^2+t{\Vert \phi \Vert }_Y^2+{\displaystyle {\int }_0^t\underset{a\in \left[0,s\right]}{\text{max}}}\left({\Vert A^{\frac{1}{2}}u\left(a\right)\Vert }^2+{\Vert {\partial }_{t}u\left(a\right)\Vert }^2\right)\text{d}s\right),\end{array}$

where $C>0$ . Applying the integral form of Gronwall’s Lemma to the above equation, for $t<T_{\phi }$ ,

$\begin{array}{l}\underset{a\in \left[0,t\right]}{\text{max}}\left({\Vert A^{\frac{1}{2}}u\left(a\right)\Vert }^2+{\Vert {\partial }_{t}u\left(a\right)\Vert }^2\right)\le C\left(1+E_1\left(0\right)+{\Vert \phi \Vert }_Y^2+{\Vert f\Vert }^2\right){\text{e}}^{{\gamma }_{1}t},\hfill \end{array}$

where ${\gamma }_1>0$ . $\forall T\ge 0$ , the above equation holds identically on $\left[0,T_{\phi }\right)\subset \left[0,T\right)$ , therefore, the solution to Equation (1.1) can be extended to the interval $\left[0,+\infty \right)$ . Thus, the continuous dependence of the solution on the initial value and the uniqueness of the solution are proved.

3. Attractors for Suspension Bridge Equations with State-Dependent Delays

3.1. Existence of Global Attractors

According to Theorem 2.5, define a semigroup $\left\{S_t:Y\to Y\right\}$ , that is, for any $t\ge 0$ , $S_t\phi =u^t$ , where $u\left(t\right)$ the mild solution to Equation (1.1) and satisfies $u^0=\phi$ . Denote $\left\{S_t,Y\right\}$ as the dynamical system generated by the solution semigroup corresponding to Equation (1.1).

Lemma 3.1 (Dissipativity) Suppose Conditions (1.3)-(1.7) and (1.10) hold, and $f\in L^2\left(\Omega \right)$ . Then, for any ${\alpha }_0,$ there exists $h_0=h\left({\alpha }_0\right)$ such that for each $\left(\alpha ,h\right)\in \left[{\alpha }_0,+\infty \right)\times \left(0,h_0\right]$ , the dynamical system $\left\{S_t,Y\right\}$ is dissipative. That is, for any $\rho >0$ , there exists $R>0$ , such that

${\Vert S_t\phi \Vert }_Y\le R,\forall \phi \in Y,{\Vert \phi \Vert }_Y\le \rho ,t\ge t_{\rho },$

and for any ${\mu }_0>0$ , the dissipative radius $R$ is independent of the damping coefficient $\alpha \ge {\alpha }_0$ and the delay time $h\in \left(0,h_0\right]$ .

Proof Similar to the a priori estimates in Section 2.1, we have

${\Vert z\Vert }^2+{\Vert \text{Δ}u\Vert }^2+{\displaystyle {\int }_0^{{\Vert \nabla u\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau {\Vert \nabla u\Vert }^2\le 4{\text{e}}^{-\gamma t}\left(E\left(0\right)+\mu h{\Vert \phi \Vert }_Y\right)+\frac{4C}{\gamma },$ (3.1)

Now, replacing $t$ in the above equation with $t+\theta$ (where $\theta \in \left[-h,0\right]$ ), the following equation holds

$\begin{array}{l}{\Vert z\left(t+\theta \right)\Vert }^2+{\Vert \text{Δ}u\left(t+\theta \right)\Vert }^2+{\displaystyle {\int }_0^{{\Vert \nabla u\left(t+\theta \right)\Vert }^2}\varphi }\left(\tau \right)\text{d}\tau {\Vert \nabla u\left(t+\theta \right)\Vert }^2\\ \le 4{\text{e}}^{-\gamma \left(t-h\right)}\left(E\left(0\right)+\mu h{\Vert \phi \Vert }_Y^2\right)+\frac{4C}{\gamma }.\end{array}$ (3.2)

Therefore, from (3.2), we obtain

$\begin{array}{c}{\Vert u^t\Vert }_Y^2=\underset{\theta \in \left[-h,0\right]}{\text{max}}{\Vert z\left(t+\theta \right)\Vert }^2+\underset{\theta \in \left[-h,0\right]}{\text{max}}{\Vert \text{Δ}u\left(t+\theta \right)\Vert }^2\\ \le 2\underset{\theta \in \left[-h,0\right]}{\text{max}}\left({\Vert z\left(t+\theta \right)\Vert }^2+{\Vert \text{Δ}u\left(t+\theta \right)\Vert }^2\right)\\ \le 8{\text{e}}^{-\delta \left(t-h\right)}\left(E\left(0\right)+\mu h{\Vert \phi \Vert }_Y\right)+\frac{8C}{\alpha }.\end{array}$ (3.3)

Through the above equation, it can be concluded that there exists $t\ge t_{\rho }$ , such that the ball $B_0=B\left(0,R\right)$ is a bounded absorbing set for the dynamical system $\left\{S_t,Y\right\}$ , where $R>\frac{2\sqrt{2C}}{\delta }$ .

Lemma 3.2 (Quasi-stability) Suppose Conditions (1.3)-(1.7) and (1.10) hold, $f\in L^2\left(\Omega \right)$ . Then there exist constants $C_1\left(R\right)>0$ , $C_2\left(R\right)>0$ and $\lambda >0$ , such that the solutions ${\phi }_1,{\phi }_2$ to Problem (1.1) with initial values $u_1,u_2$ , satisfy the following property:

${\Vert {\partial }_t{u}_i\left(t\right)\Vert }^2+{\Vert \text{Δ}{u}_i\left(t\right)\Vert }^2\le R^2,t\ge -h,i=1,2,$ (3.4)

and the quasi-stability estimate

$\begin{array}{l}{\Vert {\partial }_t{u}_1\left(t\right)-{\partial }_t{u}_2\left(t\right)\Vert }^2+{\Vert \Delta u_1\left(t\right)-\Delta u_2\left(t\right)\Vert }^2\\ \le C_1\left(R\right){\text{e}}^{-\lambda t}{\Vert {\phi }_1-{\phi }_2\Vert }_Y^2+C_2\left(R\right)\underset{r\in \left[0,t\right]}{\text{max}}{\Vert A^{\frac{1}{2}-ϵ}\left(u_1\left(r\right)-u_2\left(r\right)\right)\Vert }^2,\end{array}$ (3.5)

where $0<ϵ<\frac{1}{2}$ .

Proof Let $u_1$ and $u_2$ be two solutions to Problem (1.1). Then $\omega =u_1\left(t\right)-u_2\left(t\right)$ is a solution to the following equation

$\begin{array}{l}{\omega }_{tt}+{\text{Δ}}^2\omega -\varphi \left({\Vert \nabla \omega \Vert }^2\right)\text{Δ}\omega +{\delta }_1{\omega }_t-{\delta }_2\text{Δ}{\omega }_t\\ =-\left(g\left(u_1\right)-g\left(u_2\right)\right)-\left(u_1\left(t-\pi \left[u_1^t\right]\right)-u_2\left(t-\pi \left[u_2^t\right]\right)\right).\end{array}$ (3.6)

According to Lemma 3.2, the dynamical system $\left\{S_t,Y\right\}$ is dissipative, and thus it is obvious that (3.1) holds.

Define the energy functional

$E_{\omega }\left(t\right)=\frac{1}{2}\left({\Vert \Delta \omega \Vert }^2+{\Vert {\omega }_t\Vert }^2+\varphi \left({\Vert \nabla w\Vert }^2\right){\Vert w\Vert }^2\right).$ (3.7)

Taking the inner product of (3.6) with ${\omega }_t\left(t\right)$ in $H$ and integrating over the interval $\left[t,T\right]$ we have

$\begin{array}{l}{E}_{\omega }\left(T\right)-E_{\omega }\left(t\right)+{\delta }_1{\displaystyle {\int }_t^T{\Vert {\omega }_t\left(s\right)\Vert }^2\text{d}s}+{\delta }_2{\displaystyle {\int }_t^T{\Vert \nabla {\omega }_t\left(s\right)\Vert }^2\text{d}s}\\ \le {\displaystyle {\int }_t^T\left(g\left(u_2\left(s\right)\right)-g\left(u_1\left(s\right)\right),{\omega }_t\left(s\right)\right)\text{d}s}\\ +{\displaystyle {\int }_t^T\left(u_2\left(s-\pi \left[u_2^s\right]\right)-u_1\left(s-\pi \left[u_1^s\right]\right),{\omega }_t\left(s\right)\right)\text{d}s},\end{array}$ (3.8)

Using (2.17), we have

$\begin{array}{l}\left|{\displaystyle {\int }_{\Omega }\left(g\left(u_2\left(t\right)\right)-g\left(u_1\left(t\right)\right){\omega }_t\left(t\right)\right)\text{d}x}\right|\\ \le {\displaystyle {\int }_{\Omega }\left|g^{\prime }\left(u_2+\xi \left(u_2-u_1\right)\right)\Vert u_1\left(t\right)-u_2\left(t\right)\Vert {\omega }_t\left(t\right)\right|\text{d}x}\\ \le K_3{\displaystyle {\int }_{\Omega }\left|\Vert u_1\left(t\right)-u_2\left(t\right)\Vert {\omega }_t\left(t\right)\right|\text{d}x}\\ \le \frac{\epsilon }{2}{\Vert \Delta \omega \left(t\right)\Vert }^2+\frac{C_R}{2\epsilon }{\Vert {\omega }_t\Vert }^2,\end{array}$ (3.9)

where $0<\xi <1$ , $\epsilon >0$ . Using (1.7), we have

$\begin{array}{l}{\displaystyle {\int }_{\Omega }\left(u_2\left(t-\pi \left[u_2^t\right]\right)-u_1\left(t-\pi \left[u_1^t\right]\right),{\omega }_t\left(t\right)\right)\text{d}x}\\ \le \Vert u_2\left(t-\pi \left[u_2^t\right]\right)-u_1\left(t-\pi \left[u_1^t\right]\right)\Vert \Vert {\omega }_t\left(t\right)\Vert \\ \le \underset{\theta \in \left[-h,0\right]}{\text{max}}{\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert }^2+C_R{\Vert {\omega }_t\left(t\right)\Vert }^2.\end{array}$ (3.10)

Substituting (3.9) and (3.10) into (3.8), we obtain

$\begin{array}{l}\left|E_{\omega }\left(T\right)-E_{\omega }\left(t\right)+{\delta }_1{\displaystyle {\int }_t^T{\Vert {\omega }_t\left(s\right)\Vert }^2\text{d}s}+{\delta }_2{\displaystyle {\int }_t^T{\Vert \nabla {\omega }_t\left(s\right)\Vert }^2\text{d}s}\right|\\ \le \frac{\epsilon }{2}{\displaystyle {\int }_t^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}+{\displaystyle {\int }_t^T\underset{\theta \in \left[-h,0\right]}{\text{max}}{\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert }^2\text{d}s}\\ +C_R\left(1+\frac{1}{2\epsilon }\right){\displaystyle {\int }_t^T{\Vert {\omega }_t\left(s\right)\Vert }^2\text{d}s},\end{array}$ (3.11)

For any $\epsilon >0,$ choose $\alpha$ sufficiently large such that it satisfies the following relation

$C_R\left(1+\frac{1}{2\epsilon }\right)<\frac{{\delta }_1}{2}.$ (3.12)

Taking the inner product of $\omega \left(t\right)$ with (3.6) in $H$ and integrating over the interval $\left[0,T\right]$ , we obtain

$\begin{array}{l}\left({\partial }_t\omega \left(T\right),\omega \left(T\right)\right)-\left({\partial }_t\omega \left(0\right),\omega \left(0\right)\right)-{\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}\\ +{\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}+{\delta }_1{\displaystyle {\int }_0^T\left({\partial }_t\omega \left(s\right),\omega \left(s\right)\right)\text{d}s}\\ \le \frac{1}{2}{\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}+{\overline{C}}_R{\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}{\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert }^2\text{d}s}+{\overline{C}}_R{\displaystyle {\int }_0^T{\Vert \omega \left(s\right)\Vert }^2\text{d}s},\end{array}$ (3.13)

Furthermore, by the Hölder and Young inequality, we have

${\delta }_1{\displaystyle {\int }_0^T\left({\partial }_t\omega \left(s\right),\omega \left(s\right)\right)\text{d}s}\le \frac{1}{2}{\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}+\frac{{\delta }_1^2}{2}{\displaystyle {\int }_0^T{\Vert \omega \left(s\right)\Vert }^2\text{d}s}.$

According to the definition of the energy functional $E_{\omega }\left(t\right)$ , we have

$\begin{array}{c}\frac{1}{2}{\displaystyle {\int }_0^T{\Vert \text{Δ}\omega \left(s\right)\Vert }^2\text{d}s}\le \frac{3}{2}{\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}+C\left(E_{\omega }\left(0\right)+E_{\omega }\left(T\right)\right)\\ +{\overline{C}}_R\left({\delta }_1\right){\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}{\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert }^2\text{d}s.\end{array}$

Setting $t=0$ in (3.11) and combining it with (3.12), we obtain

$\begin{array}{c}{E}_{\omega }\left(0\right)\le E_{\omega }\left(T\right)+\frac{3{\delta }_1}{2}{\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}+\epsilon {\displaystyle {\int }_0^T{\Vert \text{Δ}\omega \left(s\right)\Vert }^2\text{d}s}\\ +{\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s.\end{array}$ (3.14)

Integrating (3.11) over the interval $\left[0,T\right]$ and combining it with (3.12) we obtain

$T{E}_{\omega }\left(T\right)\le {\displaystyle {\int }_0^T{E}_{\omega }\left(s\right)\text{d}s}+\epsilon T{\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}+T{\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s.$ (3.15)

Setting $t=0$ in (3.11) and combining it with (3.12), we can obtain

$\frac{{\delta }_1}{2}{\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}\le E_{\omega }\left(0\right)+\epsilon T{\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}+{\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s.$ (3.16)

Adding (3.13) and (3.16), and setting ${\delta }_1>8$ ,

$\begin{array}{l}\left(\frac{{\delta }_1}{2}-2\right){\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}+{\displaystyle {\int }_0^T{E}_{\omega }\left(s\right)\text{d}s}\\ \le \epsilon {\displaystyle {\int }_0^T{\Vert \text{Δ}\omega \left(s\right)\Vert }^2\text{d}s}+C\left(E_{\omega }\left(0\right)+E\omega \left(T\right)\right)+{\overline{C}}_R\left(\alpha \right){\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s.\end{array}$(3.17)

Adding $\frac{1}{2}T{E}_{\omega }\left(T\right)$ to both sides of (3.17) and substituting (3.15) into the result, we obtain

$\begin{array}{l}\left(\frac{{\delta }_1}{2}-2\right){\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}+\frac{1}{2}{\displaystyle {\int }_0^T{E}_{\omega }\left(s\right)\text{d}s}+\frac{1}{2}TE\omega \left(T\right)\\ \le \epsilon \left(T+1\right){\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}+C\left(E_{\omega }\left(0\right)+E\omega \left(T\right)\right)\\ +{\overline{C}}_R\left({\delta }_1\right)\left(1+\frac{T}{2}\right){\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s.\end{array}$ (3.18)

Now, we estimate the value of $E_{\omega }\left(0\right)+E_{\omega }\left(T\right)$ . From (3.14), we have

$\begin{array}{c}{E}_{\omega }\left(0\right)+E_{\omega }\left(T\right)\le 2E_{\omega }\left(T\right)+\frac{3{\delta }_1}{2}{\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}\\ +\epsilon {\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}+{\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s.\end{array}$ (3.19)

Substituting (3.19) into (3.18), one obtains

$\begin{array}{l}\frac{1}{2}{\displaystyle {\int }_0^T{E}_{\omega }\left(s\right)\text{d}s}+\left(\frac{1}{2}T-2C\right)E_{\omega }\left(T\right)\\ \le \left({\delta }_1+2\right){\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}+\epsilon {\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}\\ +{\overline{C^{\prime }}}_R\left({\delta }_1\right)\left(2+\frac{T}{2}\right){\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s.\end{array}$ (3.20)

Assuming $\frac{1}{2}T-2C>1$ , we then have

$\begin{array}{l}{E}_{\omega }\left(T\right)+\frac{1}{2}{\displaystyle {\int }_0^T{E}_{\omega }\left(s\right)\text{d}s}\\ \le {\overline{C^{\prime }}}_R\left({\delta }_1\right)\left(2+\frac{T}{2}\right){\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s\\ +\left({\delta }_1+2\right){\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}+\epsilon \left(T+1\right){\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s},\end{array}$ (3.21)

Similarly, setting $t=0$ in (3.11), we have

$\begin{array}{c}\frac{{\delta }_1}{2}{\displaystyle {\int }_0^T{\Vert {\partial }_t\omega \left(s\right)\Vert }^2\text{d}s}\le E_{\omega }\left(0\right)-E_{\omega }\left(T\right)+\epsilon {\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}\\ +{\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s,\end{array}$ (3.22)

Substituting the above expression into (3.21), we can obtain

$\begin{array}{l}{E}_{\omega }\left(T\right)+\frac{1}{2}{\displaystyle {\int }_0^T{E}_{\omega }\left(s\right)\text{d}s}\\ \le C_{{\delta }_1}\left(E_{\omega }\left(0\right)-E_{\omega }\left(T\right)\right)+2C_{{\delta }_1}\epsilon \left(T+1\right)\epsilon {\displaystyle {\int }_0^T{\Vert \Delta \omega \left(s\right)\Vert }^2\text{d}s}\\ +C_{\mu }{\overline{C^{\prime }}}_R\left(2+\frac{T}{2}\right){\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s,\end{array}$

where $C_{{\delta }_1}>0$ denotes a constant dependent on ${\delta }_1$ . According to the definition of $E_{\omega }\left(t\right)$ , we have ${\Vert \text{Δ}\omega \left(s\right)\Vert }^2\le 2E_{\omega }\left(s\right)$ , Choosing $\epsilon >0$ sufficiently small, such that

$E_{\omega }\left(T\right)\le \frac{C_{{\delta }_1}}{1+C_{{\delta }_1}}{E}_{\omega }\left(0\right)+{\overline{C^{\prime }}}_R\left(T,{\delta }_1\right)\left(2+\frac{T}{2}\right){\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s,$

Obviously, $\omega =\frac{C_{{\delta }_1}}{1+C_{{\delta }_1}}<1$ , thus, there exists a constant $\eta >0$ such that

$E_{\omega }\left(T\right)\le {\text{e}}^{-\eta t}{E}_{\omega }\left(0\right)+{\overline{C^{\prime }}}_R\left(T,{\delta }_1\right)\left(2+\frac{T}{2}\right){\displaystyle {\int }_0^T\underset{\theta \in \left[-h,0\right]}{\text{max}}}\Vert A^{\frac{1}{2}-ϵ}\omega \left(t+\theta \right)\Vert \text{d}s.$ (3.23)

By applying Reference ([14], Remark 3.30) and repeating the steps for the interval $\left(mT\to \left(m+1\right)T\right)$ , we can derive from the relation (3.23) that the conclusion (3.4) holds.

Theorem 3.3 (Global Attractor) Assume that conditions (1.3)-(1.7) and (1.10) hold. Then the dynamical system $\left\{S_t,Y\right\}$ generated by Problem (1.1) possesses a compact global attractor with finite fractal dimension.

Proof It follows from Lemma 3.1 that the dynamical system $\left\{S_t,Y\right\}$ is dissipative, Furthermore, by Lemma 3.2, the dynamical system $\left\{S_t,Y\right\}$ is quasi-stable on any positively invariant bounded set $B$ . According to the proof of Theorem 13 in Reference [14], if the generated system $\left\{S_t,Y\right\}$ satisfies $\phi \in C\left(\left[-h,0\right];V_2\right)\cap C^1\left(\left[-h,0\right];H\right)$ , and $\left\{S_t,Y\right\}$ is quasi-stable on every positively invariant set $B$ in $Y$ , then $\left\{S_t,Y\right\}$ is asymptotically smooth. Thus, we obtain the existence of a compact global attractor.

3.2. Fractal Dimension of Global Attractors

This section considers the fractal dimension of the global attractor, and an auxiliary space is introduced.

$Y\left(-h,T\right)=C\left(\left[-h,T\right];V_2\right)\cap C^1\left(\left[-h,T\right];H\right),T>0,$

A norm is endowed on it

${\Vert \phi \Vert }_{Y\left(-h,T\right)}=\underset{s\in \left[-h,T\right]}{\text{max}}\Vert \text{Δ}\phi \left(s\right)\Vert +\underset{s\in \left[-h,T\right]}{\text{max}}\Vert {\partial }_t\phi \left(s\right)\Vert .$

Moreover, when $T=0$ , it holds that $Y\left(-h,T\right)=Y$ . Therefore, the space $Y\left(-h,T\right)$ is an extension of the space $Y$ .

Let Φ be a set in the phase space $Y.$ Denote by ${\Phi }_T$ the set of functions $u\in Y\left(-h,T\right)$ , where $u$ is a solution to Equation (1.1) corresponding to the initial value ${u^t|}_{t\in \left[-h,0\right]}=\phi \in \Phi$ . Define the translation operator $S_T:{\Phi }_T\mapsto Y\left(-h,T\right)$ , $\left(S_{T}u\right)\left(t\right)=u\left(T+t\right)$ , $t\in \left[-h,T\right]$ .

Lemma 3.4 Let Φ be a forward-invariant set of the dynamical system $\left\{S_t,Y\right\}$ , where for $R>0$ , $\Phi \in \left\{\phi :{\Vert \phi \Vert }_Y\le R\right\}$ . Suppose $T>h$ , then ${\Phi }_T$ is forward-invariant with respect to the translation operator, and for any ${\phi }_1,{\phi }_2\in {\Phi }_T$ , we have $\begin{array}{c}{\Vert S_t{\phi }_1-S_t{\phi }_2\Vert }_{Y\left(-h,T\right)}\le c_1\left(R\right){\text{e}}^{-\varsigma \left(T-h\right)}{\Vert {\phi }_1-{\phi }_2\Vert }_{Y\left(-h,T\right)}\\ +c_2\left(R\right)\left[n\left({\phi }_1-{\phi }_2\right)+n\left({\Re }_T{\phi }_1-{\Re }_T{\phi }_2\right)\right],\end{array}$ (3.24)

where $n\left(\phi \right)={\text{sup}}_{s\in \left[0,T\right]}\Vert A^{\frac{1}{2}-ϵ}\phi \left(s\right)\Vert$ is a compact seminorm on the space $Y\left(-h,T\right)$ . Choose an appropriate $T>h$ , such that ${\zeta }_T=c_1\left(R\right){\text{e}}^{-\varsigma \left(T-h\right)}<1$ , and set $\Phi =\Lambda$ , where $\Lambda$ is the global attractor. It is obvious that the set ${\Lambda }_T$ is strictly positively invariant. Thus, we can obtain that the set ${\Lambda }_T$ has finite dimension in the space $Y\left(-h,T\right)$ . Consider the restriction mapping

$r_h:u\left(t\right),t\in \left[-h,T\right]\mapsto u\left(t\right),t\in \left[-h,0\right],$

and it is obvious that $r_h$ is Lipschitz continuous from $Y\left(-h,T\right)$ to $Y$ . Since $r_h{\Lambda }_T=\Lambda$ and a Lipschitz mapping cannot increase the fractal dimension of a set, the following conclusion holds,

${\dim }_f^Y\Lambda \le {\dim }_f^{Y\left(-h,T\right)}{\Lambda }_T<\infty .$

Thus, the fractal dimension of the global attractor is finite.

4. Conclusion and Suggestion

This paper considers the dynamic behavior of the suspension bridge equation with state delays. Compared with constant delays or time-varying delays, the state delay case exhibits higher complexity, which makes it rather challenging to verify the existence and uniqueness of solutions. To address this issue, by selecting an appropriate phase space, this paper employs the semigroup theory of operators and the Banach fixed point theorem to prove the existence and uniqueness of local solutions. Furthermore, the existence of the global attractor along with its fractal dimension, as well as the existence of the generalized exponential attractor, is established. Beyond the results presented in this work, there remain numerous interesting open problems worthy of further investigation regarding the topic discussed herein: This paper only discusses the existence of the exponential attractor for solutions to the single suspension bridge equation. Future research could further investigate the dynamical behavior of coupled suspension bridge equations. Additionally, it remains to be explored whether the boundary conditions of the suspension bridge equation discussed in this paper can be replaced with other mixed boundary conditions.

Acknowledgements

Sincere thanks to the members of JAMP for their professional performance, and special thanks to managing editor Hellen XU for a rare attitude of high quality.

NOTES

*First author.

^#Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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