Exponential Stability of a Kirchhoff Plate Equation with Structural Damping and Internal Time Delay

Abstract

In this manuscript, we investigate the exponential stability of a Kirchhoff plate equation with free boundary conditions in a bounded domain of ${\mathbb{R}}^2$. First, we consider the case where the model is subjected only to structural damping and, using the energy method, we prove the exponential decay of the energy associated with this model. Second, we consider the case where structural damping with a linear internal time delay term is applied, and we use the same technique to derive the exponential stability of the new model. Finally, we examine the scenario where the model is governed by structural damping with an added fractional time delay term and, by constructing an appropriate Lyapunov function, we show that the energy associated with the system is exponentially stable. This work improves upon prior results concerning the Kirchhoff plate equation with frictional damping, where exponential stability could not be achieved.

1. Introduction

We consider the following Kirchhoff plate equation

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{\displaystyle u_{tt}+{\Delta }^{2}u-\gamma \Delta u_{tt}-{\alpha }_0\Delta u_t+{\alpha }_1{u}_t(x,t-{\tau }_0)+{\alpha }_2{\partial }_t^{\alpha ,\beta }u\left(x,t-{\tau }_0\right)=0}\hfill & \mathrm{in}\hfill & \Omega \times (0,\infty )\hfill \\ u={\partial }_{\nu }u=0\hfill & \mathrm{on}\hfill & {\Gamma }_0\times (0,\infty )\hfill \\ \Delta u+(1-\mu )B_{1}u={\partial }_{\nu }\Delta u+(1-\mu )B_{2}u-\gamma {\partial }_{\nu }{u}_{tt}-{\alpha }_0{\partial }_{\nu }{u}_t=0\hfill & \mathrm{on}\hfill & {\Gamma }_1\times (0,\infty )\hfill \\ u(x,0)=u^0,u_t(x,0)=u^1,\hfill & \mathrm{in}\hfill & \Omega ,\hfill \\ u_t(x,t-{\tau }_0)=f_0(x,t-{\tau }_0),t\in (0,{\tau }_0),\hfill \end{array}\right.\end{array}$$

(1)

where $\Omega$ is an open set of ${\mathbb{R}}^2$ with regular boundary $\partial \Omega ={\Gamma }_0\cup {\Gamma }_1$ (class $C^4$ will be enough) such that ${\overline{\Gamma }}_0\ne \varnothing$ and ${\overline{\Gamma }}_0\cap {\overline{\Gamma }}_1=\varnothing$, $u^0,u^1,f_0$ are the given data, the constant $\gamma >0$ is the rotational inertia of the plate and the constant $0<\mu <{\displaystyle \frac{1}{2}}$ is the Poisson coefficient. The damping coefficients ${\alpha }_0,{\alpha }_1$ and ${\alpha }_2$ will be specified in each section of this paper. The boundary operators $B_1$ and $B_2$ are defined by

$$B_{1}u=2{\nu }_1{\nu }_2{u}_{xy}-{\nu }_1^2{u}_{yy}-{\nu }_2^2{u}_{xx},$$

$$B_{2}u={\partial }_{\tau }\left(({\nu }_1^2-{\nu }_2^2)u_{xy}+{\nu }_1{\nu }_2(u_{yy}-u_{xx})\right),$$

where $\nu =({\nu }_1,{\nu }_2)$ is the unit outer normal vector to $\partial \Omega$ and $\tau =(-{\nu }_2,{\nu }_1)$ is a unit tangent vector.

The constant ${\tau }_0>0$ represents the time delay. The operator ${\partial }_t^{\alpha ,\beta }$ is the exponential fractional derivative of order $\alpha$ and is defined by

$${\partial }_t^{\alpha ,\beta }u\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}\underset{0}{\overset{t}{\int }}{(t-s)}^{-\alpha }{e}^{-\beta (t-s)}{\displaystyle \frac{du}{ds}}\left(s\right)ds,0<\alpha <1,\beta >0,$$

(2)

where $\Gamma$ is the well-known gamma function.

Note that this fractional operator was first introduced by Choi and MacCamy [1].

The exponential fractional integral of order $\alpha$ is defined by the following formula:

$$\left[I^{\alpha ,\beta }u\right]\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}\underset{0}{\overset{t}{\int }}{(t-s)}^{-\alpha }{e}^{-\beta (t-s)}u\left(s\right)ds.$$

(3)

From Formulas (2) and (3), we have

$${\partial }_t^{\alpha ,\beta }u=\left[I^{\alpha ,\beta }{\displaystyle \frac{du}{dt}}\right].$$

We refer the reader to [1,2,3] for more properties of fractional calculus.

Note that (2) is a slightly different version of the Caputo fractional derivative operator of order $\alpha$, which is defined by the following (see [4]):

$$D^{\alpha }u\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}\underset{0}{\overset{t}{\int }}{(t-s)}^{-\alpha }{\displaystyle \frac{du}{ds}}\left(s\right)ds.$$

Model (1) describes a Kirchhoff plate, with rotational forces, that is clamped along ${\Gamma }_0$ and without bending and twisting moments on ${\Gamma }_1$ (please see [5,6,7] for more details about this model). Free boundary conditions and rotational forces make the analysis of stability issues in plate models more difficult. Additionally, in our case, the time delay term (linear or fractional) interacts with the structural dissipation term, making it interesting to explore this interaction.

Let us present some works concerning a Kirchhoff plate in the absence of time delay terms and rotational forces. The first results concerning the Kirchhoff plate equation are due to Lagnese and Lions [8], where the authors provided the model at hand (without delay terms and rotational forces). Afterwards, Magnese [7] established the exponential stability of (1) in a bounded strongly star-shaped domain (with respect to one point $x_0$) of ${\mathbb{R}}^2$ and with linear boundary controls. In [9], Rao proved results on the exponential and polynomial stability of the same model considered in [7], but with nonlinear boundary controls. Later, by assuming a geometric control condition and taking into account rotational forces and dynamic boundary controls, Rao and Wehbe [10] established a polynomial decay for the model. For more results on the Kirchhoff plate equation with different damping terms, we refer the reader to [5,11,12,13] and the references therein.

Delay effects are often encountered in many real-world problems, although they are commonly viewed as a source of instability. By extending the traditional notion of time delay to fractional orders, fractional delay, a concept from fractional calculus, permits more precise and flexible control of dynamic systems. By incorporating fractional delay into damping systems, engineers can achieve enhanced control over the vibration phase and amplitude, leading to more effective stabilization. In fact, fractional calculus in modeling increases the ability to capture the complex dynamics of natural systems and improves control performance beyond what is possible with integer-order controls. This approach is especially pertinent in fields such as engineering, quantum mechanics, nuclear physics, and biological phenomena like fluid flow (see for example [14,15,16]). These derivatives better capture the viscoelastic and nonlinear properties inherent in systems, reflecting the distributed nature of forces and displacements and accounting for memory effects that impact the system’s dynamic response over time.

Let us mention the recent works on the Kirchhoff plate equation in the presence of a time delay term. In [17], Hajjej and Park established an explicit and general decay rate result for a quasilinear viscoelastic Kirchhoff plate equation with a logarithmic source and time delay. Badawi et al. [18] studied a Kirchhoff plate equation that includes delay terms in the boundary control and established its exponential stability through a multiplier geometric control condition. We also cite the recent work [19], where the author proved the polynomial stability of a Kirchhoff plate equation in the presence of a frictional damping, a fractional time delay, and a source term. Finally, we mention the works [20,21], which consider a weakly coupled system (of symmetric type) consisting of a Kirchhoff plate equation with free boundary conditions and a wave equation with Dirichlet boundary conditions in a bounded domain.

We note that in previous works where frictional damping occurs in the Kirchhoff plate equation, the energy of the damped system does not decay exponentially; this is due to the fact that the operator defining the damping is compact in the energy space [22,23].

In the present work, we aim to achieve the following:

• Study the stability of the Kirchhoff plate equation in the presence of a damping term and a time delay term (of linear or fractional type) that have different orders (as carried out in the very recent work [24] for wave equations), which is not the case for all previous works, even for wave equations and other damped systems, where the damping and the time delay terms are of the same orders.
• Prove the exponential decay of the energy of our model, which is not the case in the presence of frictional damping.

All through this paper, we let

$$H_{{\Gamma }_0}^1=\{u\in H^1(\Omega ):u=0\mathrm{on}{\Gamma }_0\},H_{{\Gamma }_0}^2=\{u\in H^2(\Omega ):u={\displaystyle \frac{\partial u}{\partial \nu }}=0\mathrm{on}{\Gamma }_0\}.$$

The symmetric operator $b(·,·)$ is defined as

$$b(\phi ,\varphi )={\int }_{\Omega }\left({\phi }_{x_1{x}_1}{\varphi }_{x_1{x}_1}+{\phi }_{x_2{x}_2}{\varphi }_{x_2{x}_2}+\mu \left({\phi }_{x_1{x}_1}{\varphi }_{x_2{x}_2}+{\phi }_{x_2{x}_2}{\varphi }_{x_1{x}_1}\right)+2(1-\mu ){\phi }_{x_1{x}_2}{\varphi }_{x_1{x}_2}\right)dx.$$

For $(\phi ,\varphi )\in \left(H^4(\Omega )\cap H_{{\Gamma }_0}^2\right)\times H_{{\Gamma }_0}^2$, we know

$$\begin{array}{c}\hfill {\displaystyle {\int }_{\Omega }\left({\Delta }^2\phi \right)\varphi dx=b(\phi ,\varphi )-{\int }_{{\Gamma }_1}{\mathcal{B}}_1\phi \frac{\partial \varphi }{\partial \nu }d{\Gamma }_1+{\int }_{{\Gamma }_1}{\mathcal{B}}_2\phi \varphi d{\Gamma }_1,}\end{array}$$

where

$${\mathcal{B}}_1\phi =\Delta \phi +(1-\mu )B_1\phi ,$$

$${\mathcal{B}}_2\phi ={\partial }_{\nu }\Delta \phi +(1-\mu )B_2\phi .$$

In addition, if $\phi$ and $\varphi \in H_{{\Gamma }_0}^2$ we obtain

$$b(\phi ,\varphi )={\left({\Delta }^2\phi ,\varphi \right)}_{H_{{\Gamma }_0}^{-2}\times H_{{\Gamma }_0}^2},$$

where $H_{{\Gamma }_0}^{-2}$ is the dual space of $H_{{\Gamma }_0}^2$ and ${\left(.,.\right)}_{H_{{\Gamma }_0}^{-2}\times H_{{\Gamma }_0}^2}$ denotes the corresponding duality.

We define the operator $B_{\gamma }$ by

$$B_{\gamma }=I-\gamma \Delta ,$$

with domain

$$D\left(B_{\gamma }\right)=\{u\in H^2(\Omega ):u=0\mathrm{on}{\Gamma }_0\mathrm{and}{\partial }_{\nu }u=0\mathrm{on}{\Gamma }_1\}.$$

$B_{\gamma }$ is a positive definite and self-adjoint operator. We define a space $H_{{\Gamma }_0,\gamma }^1(\Omega )$ equivalent to $H_{{\Gamma }_0}^1(\Omega )$ with its inner product being

$${\displaystyle {⟨\phi ,\varphi ⟩}_{H_{{\Gamma }_0,\gamma }^1(\Omega )}={\int }_{\Omega }\phi \varphi dx+\gamma {\int }_{\Omega }\nabla \phi \nabla \varphi dx={\int }_{\Omega }{B}_{\gamma }^{\frac{1}{2}}\phi B_{\gamma }^{\frac{1}{2}}\varphi dx,\forall \phi ,\varphi \in H_{{\Gamma }_0}^1(\Omega ),}$$

and with its dual (pivotal with respect to $L^2$ inner product) denoted as $H_{{\Gamma }_0,\gamma }^{-1}(\Omega )$. Then, we obtain

$$B_{\gamma }\in \mathcal{L}\left(H_{{\Gamma }_0,\gamma }^1(\Omega ),H_{{\Gamma }_0,\gamma }^{-1}(\Omega )\right).$$

Let $C_p$ and $C_s$ be the embedding constants such that

$$\begin{array}{c}\hfill {\displaystyle {\int }_{\Omega }{|u|}^{2}dx\le C_p{\int }_{\Omega }{|\nabla u|}^{2}dx\mathrm{and}{\int }_{\Omega }{|\nabla u|}^{2}dx\le C_{s}b(u,u),\forall u\in H_{{\Gamma }_0}^2.}\end{array}$$

(4)

The present paper is organized as follows: In the next section, we prove the exponential stability of problem (1) in the case where ${\alpha }_1={\alpha }_2=0$ and ${\alpha }_0>0$. Section 3 is devoted to studying the exponential stability of model (1) in the presence of structural damping and linear time delay, i.e., ${\alpha }_2=0$ and ${\alpha }_0,|{\alpha }_1|>0$. Finally, we prove the exponential stability of system (1) if it is subjected to structural and fractional time delay damping, i.e., ${\alpha }_0,{\alpha }_2>0$ and ${\alpha }_1=0$.

2. Exponential Stability of System (1) in the Case Without Delay

In this section, we will prove the exponential stability of system (1) with only structural damping, i.e., ${\alpha }_0>0$ and ${\alpha }_1={\alpha }_2=0$. Let us define the space energy by

$${\mathcal{H}}_1=H_{{\Gamma }_0}^2(\Omega )\times H_{{\Gamma }_0,\gamma }^1(\Omega ),$$

which is endowed with its inner product

$${\displaystyle {⟨U,\tilde{U}⟩}_{{\mathcal{H}}_1}=b(u,\tilde{u})+{\int }_{\Omega }{B}_{\gamma }^{\frac{1}{2}}v{B}_{\gamma }^{\frac{1}{2}}\tilde{v}dx,\forall U={(u,v)}^T,\tilde{U}={(\tilde{u},\tilde{v})}^T\in {\mathcal{H}}_1.}$$

The following proposition ensures the well posedness of system (1):

Proposition 1.

For any initial datum $U_0={(u^0,u^1)}^T\in {\mathcal{H}}_1$, there exists a unique solution $U={(u,u_t)}^T\in C([0,\infty );{\mathcal{H}}_1)$ of problem (1).

Proof. 

System (1) can be recast as a first-order equation. In fact, let $v=u_t$ and set $U={(u,v)}^T$. Then, (1) is equivalent to

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& U_t\left(t\right)={\mathcal{A}}_{1}U\left(t\right),\hfill \\ & U(x,0)=U_0={(u^0,u^1)}^T,\hfill \end{array}\right.\end{array}$$

(5)

where

$$\begin{array}{c}\hfill {\mathcal{A}}_{1}U:=\left(\begin{array}{c}v\\ -B_{\gamma }^{-1}{\Delta }^{2}u+{\alpha }_0{B}_{\gamma }^{-1}\Delta v\end{array}\right)\end{array}$$

with domain

$$\begin{array}{ccc}\hfill D\left({\mathcal{A}}_1\right)& =& \left\{{(u,v)}^T\in {\mathcal{H}}_1:u\in H^3(\Omega ),v\in H_{{\Gamma }_0,\gamma }^2(\Omega ),\Delta u+(1-\mu )B_{1}u=0\mathrm{on}{\Gamma }_1\right\}.\hfill \end{array}$$

Let $U={(u,v)}^T\in D\left({\mathcal{A}}_1\right)$. Then, we have

$$\begin{array}{ccc}\hfill {⟨{\mathcal{A}}_{1}U,U⟩}_{{\mathcal{H}}_1}& =& {\displaystyle b(v,u)+{\int }_{\Omega }{B}_{\gamma }^{\frac{1}{2}}\left(-B_{\gamma }^{-1}{\Delta }^{2}u+{\alpha }_0{B}_{\gamma }^{-1}\Delta v\right)B_{\gamma }^{\frac{1}{2}}vdx}\hfill \\ & =& {\displaystyle b(v,u)-{\int }_{\Omega }{B}_{\gamma }^{\frac{1}{2}}{B}_{\gamma }^{-1}{\Delta }^{2}u{B}_{\gamma }^{\frac{1}{2}}vdx+{\alpha }_0{\int }_{\Omega }{B}_{\gamma }^{\frac{1}{2}}{B}_{\gamma }^{-1}\Delta v{B}_{\gamma }^{\frac{1}{2}}vdx}\hfill \\ & =& {\displaystyle b(v,u)-{\left({\Delta }^{2}u,v\right)}_{H_{{\Gamma }_0}^{-2}\times H_{{\Gamma }_0}^2}+{\alpha }_0{\int }_{\Omega }v\Delta vdx}\hfill \\ & =& {\displaystyle -{\alpha }_0{\int }_{\Omega }{|\nabla v|}^{2}dx\le 0,}\hfill \end{array}$$

which means that the operator ${\mathcal{A}}_1$ is dissipative.

Next, we show that $I-{\mathcal{A}}_1$ is onto. Given a vector ${(f,g)}^T\in {\mathcal{H}}_1$, we look for ${(u,v)}^T\in D\left({\mathcal{A}}_1\right)$ that satisfies

$$\left(I-{\mathcal{A}}_1\right){(u,v)}^T={(f,g)}^T.$$

(6)

Referring to the definition of ${\mathcal{A}}_1$, we obtain

$$u-v=f,$$

$$v+B_{\gamma }^{-1}{\Delta }^{2}u-{\alpha }_0{B}_{\gamma }^{-1}\Delta v=g,$$

which can be rewritten as

$$v=u-f,$$

$$B_{\gamma }u+{\Delta }^{2}u-{\alpha }_0\Delta u=B_{\gamma }g+B_{\gamma }f-{\alpha }_0\Delta f.$$

Multiplying the second equation by $\phi \in H_{{\Gamma }_0}^2(\Omega )$ and integrating by parts in $\Omega$, we obtain

$$F(u,\phi )=G\left(\phi \right),\forall \phi \in H_{{\Gamma }_0}^2(\Omega ),$$

(7)

where

$$F(u,\phi )=b(u,\phi )+{\int }_{\Omega }\left\{u\phi +(\gamma +{\alpha }_0)\nabla u\nabla \phi \right\}dx,$$

and

$$G\left(\phi \right)={\int }_{\Omega }\left\{\left(f+g\right)\phi +((\gamma +{\alpha }_0)\nabla f+\gamma \nabla g)\nabla \phi \right\}dx.$$

It can be shown that F is a continuous bilinear coercive form on $H_{{\Gamma }_0}^2(\Omega )\times H_{{\Gamma }_0}^2(\Omega )$ and G is a continuous linear form on $H_{{\Gamma }_0}^2(\Omega )$. Therefore, by applying the Lax–Milgram lemma, problem (7) admits a unique solution $u\in H_{{\Gamma }_0}^2(\Omega )$.

By performing integration by parts, it is straightforward to verify that u satisfies

$$u-(\gamma +{\alpha }_0)\Delta u+{\Delta }^{2}u=B_{\gamma }(g+f)-{\alpha }_0\Delta f\in H_{{\Gamma }_0,\gamma }^{-1}(\Omega ),$$

as well as the boundary conditions in $D\left({\mathcal{A}}_1\right)$. Applying classical elliptic regularity, we obtain that $u\in H^3(\Omega )$. This implies that (6) is satisfied, and therefore $I-{\mathcal{A}}_1$ is onto. Using the Lumer–Phillips theorem, we conclude that the operator ${\mathcal{A}}_1$ generates a $C_0$ semigroup of contractions, and therefore, if $U_0={(u^0,u^1)}^T\in {\mathcal{H}}_1$, system (16) possesses a unique weak solution $U\in C([0,\infty );{\mathcal{H}}_1)$. □

The energy of solutions, in this case, is defined by

$${\displaystyle E_1\left(t\right)=\frac{1}{2}b(u,u)+\frac{1}{2}{\int }_{\Omega }\left(|u_t{|}^2+{|\nabla u_t|}^2\right)dx,}$$

and satisfies

$${\displaystyle E_1^{\prime }\left(t\right)=-{\alpha }_0{\int }_{\Omega }{|\nabla u_t|}^{2}dx.}$$

Now, we define the functional

$${\displaystyle {\phi }_1\left(t\right)={\int }_{\Omega }\left(u{u}_t+\gamma \nabla u\nabla u_t\right)dx.}$$

We can easily check that $|{\phi }_1\left(t\right)|\le {\delta }_1{E}_1\left(t\right)$, for some positive constant ${\delta }_1$. Moreover, by using (1), we find

$$\begin{array}{ccc}\hfill {\phi }_1^{\prime }\left(t\right)& =& {\displaystyle {\int }_{\Omega }\left(|u_t{|}^2+u{u}_{tt}+\gamma {|\nabla u_t|}^2+\gamma \nabla u_t\nabla u_{tt}\right)dx}\hfill \\ & =& {\displaystyle -b(u,u)+{\int }_{\Omega }\left(|u_t{|}^2-{\alpha }_0\nabla u_t\nabla u+\gamma {|\nabla u_t|}^2\right)dx.}\hfill \end{array}$$

(8)

Theorem 1.

There exist positive constants $a_1$ and $b_1$ such that any solution of (1) verifies

$$E_1\left(t\right)\le a_1{E}_1\left(0\right)e^{-b_{1}t},t\ge 0.$$

(9)

Proof. 

Consider the functional

$$F_1\left(t\right)=E_1\left(t\right)+N_1{\phi }_1\left(t\right),$$

where $N_1$ is a positive constant that satisfies for the moment $N_1<{\displaystyle \frac{1}{{\delta }_1}}$. We have

$$\begin{array}{ccc}\hfill F_1^{\prime }\left(t\right)& =& E_1^{\prime }\left(t\right)+N_1{\phi }_1^{\prime }\left(t\right)\hfill \\ & =& {\displaystyle -{\alpha }_0{\int }_{\Omega }{|\nabla u_t|}^{2}dx-N_{1}b(u,u)+N_1{\int }_{\Omega }\left(|u_t{|}^2-{\alpha }_0\nabla u_t\nabla u+\gamma {|\nabla u_t|}^2\right)dx}\hfill \end{array}$$

Using Young’s inequality, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle -{\alpha }_0{\int }_{\Omega }\nabla u_t\nabla udx}& \le & {\displaystyle \frac{{\alpha }_0^2{C}_s}{2}{\int }_{\Omega }|\nabla u_t{|}^{2}dx+\frac{1}{2C_s}{\int }_{\Omega }{|\nabla u|}^{2}dx}\hfill \\ & \le & {\displaystyle \frac{{\alpha }_0^2{C}_s}{2}{\int }_{\Omega }{|\nabla u_t|}^{2}dx+\frac{1}{2}b(u,u).}\hfill \end{array}$$

(10)

Combining (10) with the fact that ${\displaystyle {\int }_{\Omega }|u_t{|}^{2}dx\le C_p{\int }_{\Omega }{|\nabla u_t|}^{2}dx}$ yields

$$\begin{array}{ccc}\hfill F_1^{\prime }\left(t\right)& \le & {\displaystyle -\frac{N_1}{2}b(u,u)-N_1{\int }_{\Omega }|u_t{|}^{2}dx+2N_1{\int }_{\Omega }|u_t{|}^{2}dx-\left({\alpha }_0-N_1\frac{{\alpha }_0^2{C}_s}{2}-N_1\gamma \right){\int }_{\Omega }{|\nabla u_t|}^{2}dx}\hfill \\ & \le & {\displaystyle -\frac{N_1}{2}b(u,u)-N_1{\int }_{\Omega }|u_t{|}^{2}dx-\left({\alpha }_0-2N_1{C}_p-N_1(\gamma +\frac{{\alpha }_0^2{C}_s}{2})\right){\int }_{\Omega }{|\nabla u_t|}^{2}dx.}\hfill \end{array}$$

If we choose $N_1$ small enough so that

$${\alpha }_0-N_1\left(2C_p+\gamma +{\displaystyle \frac{{\alpha }_0^2{C}_s}{2}}\right)>0,$$

we obtain that

$$F_1^{\prime }\left(t\right)\le -C_1{E}_1\left(t\right),$$

(11)

for some positive constant $C_1$.

Using the fact that

$$\left(1-{\delta }_1{N}_1\right)E_1\left(t\right)\le F_1\left(t\right)\le \left(1+{\delta }_1{N}_1\right)E_1\left(t\right),$$

and (11), we infer that

$$F_1^{\prime }\left(t\right)\le -{\displaystyle \frac{C_1}{1+{\delta }_1{N}_1}}{F}_1\left(t\right).$$

This later leads to

$$F_1\left(t\right)\le F_1\left(0\right)e^{-{\displaystyle \frac{C_1}{1+{\delta }_1{N}_1}}t}.$$

Consequently, we arrive at

$$E_1\left(t\right)\le {\displaystyle \frac{1}{1-{\delta }_1{N}_1}}{F}_1\left(t\right)\le {\displaystyle \frac{1}{1-{\delta }_1{N}_1}}{F}_1\left(0\right)e^{-{\displaystyle \frac{C_1}{1+{\delta }_1{N}_1}}t}\le {\displaystyle \frac{1+{\delta }_1{N}_1}{1-{\delta }_1{N}_1}}{E}_1\left(0\right)e^{-{\displaystyle \frac{C_1}{1+{\delta }_1{N}_1}}t},$$

This ends the proof of Theorem 1. □

3. Exponential Stability of System (1) in the Case with Linear Delay

In this section, we take ${\alpha }_2=0$ and ${\alpha }_0,|{\alpha }_1|>0$. In addition, we assume that

$$0<|{\alpha }_1|<{\displaystyle \frac{{\alpha }_0}{C_p}}.$$

(12)

As in [25], we define

$$z(x,\theta ,t)=u_t(x,t-\theta {\tau }_0),(x,\theta ,t)\mathrm{in}\Omega \times (0,1)\times (0,\infty ).$$

(13)

It is easy to see that

$${\tau }_0{z}_t(x,\theta ,t)+z_{\theta }(x,\theta ,t)=0,x\in \Omega ,\theta \in (0,1),t\ge 0.$$

(14)

System (1) becomes

$$\left\{\begin{array}{ccc}{u}_{tt}+{\Delta }^{2}u-\gamma \Delta u_{tt}-{\alpha }_0\Delta u_t+{\alpha }_{1}z(x,1,t)=0\hfill & \mathrm{in}\hfill & \Omega \times (0,\infty )\hfill \\ {\tau }_0{z}_t(x,\theta ,t)+z_{\theta }(x,\theta ,t)=0\mathrm{for}(x,\theta ,t)\in \Omega \times (0,1)\times (0,\infty )\hfill \\ u={\partial }_{\nu }u=0\hfill & \mathrm{on}\hfill & {\Gamma }_0\times (0,\infty )\hfill \\ \Delta u+(1-\mu )B_{1}u={\partial }_{\nu }\Delta u+(1-\mu )B_{2}u-\gamma {\partial }_{\nu }{u}_{tt}-{\alpha }_0{\partial }_{\nu }{u}_t=0\hfill & \mathrm{on}\hfill & {\Gamma }_1\times (0,\infty )\hfill \\ u(x,0)=u^0,u_t(x,0)=u^1\hfill & \mathrm{in}\hfill & \Omega ,\hfill \\ u_t(x,t-{\tau }_0)=f_0(x,t-{\tau }_0)\hfill & \mathrm{in}\hfill & \Omega \times (0,{\tau }_0).\hfill \end{array}\right.$$

(15)

Define

$${\mathcal{H}}_2=H_{{\Gamma }_0}^2(\Omega )\times H_{{\Gamma }_0,\gamma }^1(\Omega )\times L^2(\Omega \times (0,1)).$$

Let $v=u_t$ and $U={(u,v,z)}^T$. Then, system (15) can be rewritten as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& U_t\left(t\right)={\mathcal{A}}_{2}U\left(t\right),\hfill \\ & U(x,0)=U_0={(u^0,u^1,f_0)}^T,\hfill \end{array}\right.\end{array}$$

(16)

where

$$\begin{array}{c}\hfill {\mathcal{A}}_{2}U:=\left(\begin{array}{c}v\\ -B_{\gamma }^{-1}{\Delta }^{2}u+{\alpha }_0{B}_{\gamma }^{-1}\Delta v-{\alpha }_1{B}_{\gamma }^{-1}z(x,1)\\ -{\displaystyle \frac{1}{{\tau }_0}}{z}_{\theta }\end{array}\right)\end{array}$$

with domain

$$\begin{array}{ccc}\hfill D\left({\mathcal{A}}_2\right)& =& \{{(u,v,z)}^T\in {\mathcal{H}}_2:u\in H^3(\Omega ),v\in H_{{\Gamma }_0,\gamma }^2(\Omega ),z_{\theta }\in L^2(\Omega \times (0,1)),v=z(.,0),\hfill \\ & & \Delta u+(1-\mu )B_{1}u=0\mathrm{on}{\Gamma }_1\}.\hfill \end{array}$$

As in Proposition 1, we can show that ${\mathcal{A}}_2$ is maximal dissipative in ${\mathcal{H}}_2$. The well posedness of system (15) is guaranteed by the following proposition:

Proposition 2.

For any initial datum $U_0=(u^0,u^1,f_0)\in {\mathcal{H}}_2$, there exists a unique solution $U=(u,u_t,z)\in C([0,\infty );{\mathcal{H}}_2)$ of problem (15).

We define the energy associated with the solution of problem (15) by

$${\displaystyle E_2\left(t\right)=E_1\left(t\right)+\frac{\xi {\tau }_0}{2}{\int }_{\Omega }{\int }_0^1{|z(x,\theta ,t)|}^{2}d\theta dx,}$$

where $\xi$ is a positive number satisfying

$$|{\alpha }_1|<\xi <{\displaystyle \frac{2{\alpha }_0}{C_p}}-|{\alpha }_1|.$$

(17)

The energy $E_2\left(t\right)$ satisfies the following dissipation law:

Proposition 3.

$${\displaystyle E_2^{\prime }\left(t\right)\le -C_2{\int }_{\Omega }\left(|\nabla u_t{|}^2+{|z(x,1,t)|}^2\right)dx,}$$

(18)

where $C_2=min\left\{{\alpha }_0-{\displaystyle \frac{|{\alpha }_1|C_p}{2}}-{\displaystyle \frac{\xi C_p}{2}},{\displaystyle \frac{\xi }{2}}-{\displaystyle \frac{|{\alpha }_1|}{2}}\right\}.$

Proof. 

Multiplying ${\left(15\right)}_1$ and ${\left(15\right)}_2$ by $u_t$ and $\xi z$, and integrating over $\Omega$ and $\Omega \times (0,1)$, respectively, one has

$${\displaystyle E_2^{\prime }\left(t\right)=-{\alpha }_0{\int }_{\Omega }{|\nabla u_t|}^{2}dx-{\alpha }_1{\int }_{\Omega }z(x,1,t)u_{t}dx-\frac{\xi }{2}{\int }_{\Omega }\left({|z(x,1,t)|}^2-{|z(x,0,t)|}^2\right)dx.}$$

(19)

Applying Young’s inequality, we obtain

$${\displaystyle -{\alpha }_1{\int }_{\Omega }z(x,1,t)u_{t}dx\le \frac{|{\alpha }_1|}{2}{\int }_{\Omega }{|z(x,1,t)|}^{2}dx+\frac{|{\alpha }_1|}{2}{\int }_{\Omega }{|u_t|}^{2}dx.}$$

(20)

Inserting (20) in (19) and using the fact that ${\displaystyle {\int }_{\Omega }|u_t{|}^{2}dx\le C_p{\int }_{\Omega }{|\nabla u_t|}^{2}dx}$, (18) holds true. □

Let $N_2$ be a positive constant that will be specified later, and define the functional

$$F_2\left(t\right)=E_2\left(t\right)+N_2{\phi }_2\left(t\right),$$

where

$${\phi }_2\left(t\right)={\phi }_1\left(t\right)+{\tau }_0{\int }_{\Omega }{\int }_0^1{e}^{-\theta {\tau }_0}{|z(x,\theta ,t)|}^{2}d\theta dx.$$

The main result in this section is as follows.

Theorem 2.

There exist positive constants $a_2$ and $b_2$ such that any solution of (15) verifies

$$E_2\left(t\right)\le a_2{E}_2\left(0\right)e^{-b_{2}t},t\ge 0.$$

(21)

Proof. 

It is easy to see that $F_2\left(t\right)\sim E_2\left(t\right)$, i.e.,

$$C_3{F}_2\left(t\right)\le E_2\left(t\right)\le C_4{F}_2\left(t\right),$$

(22)

for some positive constants $C_3$ and $C_4$. In addition, in this case the term ${\displaystyle -{\alpha }_0{\int }_{\Omega }\nabla u_t\nabla udx}$ will be estimated as

$$\begin{array}{c}\hfill {\displaystyle -{\alpha }_0{\int }_{\Omega }\nabla u_t\nabla udx\le {\alpha }_0^2{C}_s{\int }_{\Omega }{|\nabla u_t|}^{2}dx+\frac{1}{4}b(u,u).}\end{array}$$

In addition, it is easy to see that (see [17])

$${\displaystyle \frac{d}{dt}\left({\tau }_0{\int }_{\Omega }{\int }_0^1{e}^{-\theta {\tau }_0}{|z(x,\theta ,t)|}^{2}d\theta dx\right)\le {\int }_{\Omega }|u_t{|}^{2}dx-{\tau }_0{e}^{-{\tau }_0}{\int }_{\Omega }{\int }_0^1{|z(x,\theta ,t)|}^{2}d\theta dx,}$$

(23)

and by Young’s inequality

$${\displaystyle -{\alpha }_1{\int }_{\Omega }uz(x,1,t)dx\le \frac{1}{4}b(u,u)+{\alpha }_1^2{C}_s{C}_p{\int }_{\Omega }{|z(x,1,t)|}^{2}dx}$$

Proceeding as in the proof of Theorem 1 and using the last inequalities, we find that

$$\begin{array}{ccc}\hfill F_2^{\prime }\left(t\right)& =& E_2^{\prime }\left(t\right)+N_2{\phi }_2^{\prime }\left(t\right)\hfill \\ & \le & {\displaystyle -\frac{N_2}{2}b(u,u)-N_2{\int }_{\Omega }|u_t{|}^{2}dx-\left(C_2-3N_2{C}_p-N_2(\gamma +{\alpha }_0^2{C}_s)\right){\int }_{\Omega }{|\nabla u_t|}^{2}dx}\hfill \\ & & {\displaystyle -\left(C_2-N_2{\alpha }_1^2{C}_s{C}_p\right){\int }_{\Omega }{|z(x,1,t)|}^{2}dx-N_2{\tau }_0{e}^{-{\tau }_0}{\int }_{\Omega }{\int }_0^1{|z(x,\theta ,t)|}^{2}d\theta dx.}\hfill \end{array}$$

If we select $N_2$ to be sufficiently small such that

$$C_2-N_2\left(3C_p+\gamma +{\displaystyle \frac{{\alpha }_0^2{C}_s}{2}}\right)>0\mathrm{and}{C}_2-N_2{\alpha }_1^2{C}_s{C}_p>0,$$

we obtain the existence of a positive constant $C_5$ verifying

$$F_2^{\prime }\left(t\right)\le -C_5{E}_2\left(t\right),$$

(24)

Using Gronwall’s inequality and (22), we deduce that

$$E_2\left(t\right)\le {\displaystyle \frac{C_4}{C_3}}{E}_2\left(0\right)e^{-C_3{C}_{5}t},t\ge 0.$$

This ends the proof of Theorem 2. □

4. Exponential Stability of System (1) in the Case with Fractional Delay

In this section, we take ${\alpha }_1=0$ and ${\alpha }_0,{\alpha }_2>0$ and we assume that the damping coefficients ${\alpha }_0$ and ${\alpha }_2$ satisfy

$${\alpha }_2<min\{{\displaystyle \frac{{\alpha }_0{\beta }^{1-\alpha }}{2C_p}},{\displaystyle \frac{e^{-{\tau }_0}{\beta }^{1-\alpha }}{{\tau }_0}}\}.$$

(25)

Lemma 1.

(see [2]) Let μ be the function:

$$\mu \left(\zeta \right)={\left|\zeta \right|}^{{\displaystyle \frac{2\alpha -1}{2}}},\zeta \in \mathbb{R},0<\alpha <1.$$

Then, the relationship between the ‘input’ U and the ‘output’ O of the system

$$\left\{\begin{array}{c}{\eta }_t(x,\zeta ,t)+({\zeta }^2+\beta )\eta (x,\zeta ,t)-U(x,t)\mu \left(\zeta \right)=0,\zeta \in \mathbb{R},t>0,\beta >0,\hfill \\ \eta (x,\zeta ,0)=0,\hfill \\ {\displaystyle O\left(t\right)={\pi }^{-1}sin\left(\alpha \pi \right){\int }_{-\infty }^{+\infty }\eta (x,\zeta ,t)\mu \left(\zeta \right)d\zeta }\hfill \end{array}\right.$$

(26)

is given by

$$O=I^{1-\alpha ,\beta }U,$$

where

$${\displaystyle I^{\alpha ,\beta }z\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}{e}^{-\beta (t-s)}z\left(s\right)ds.}$$

We also need the next lemma.

Lemma 2.

(see [26]) If $\lambda \in D_{\beta }=\mathbb{C}\setminus \left]-\infty ,-\beta \right[$, then

$$\underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \frac{{\mu }^2\left(\zeta \right)}{\lambda +\beta +{\zeta }^2}}d\zeta ={\displaystyle \frac{\pi }{sin\left(\alpha \pi \right)}}{(\lambda +\beta )}^{\alpha -1}.$$

Consequently, Lemma 1, (13) and (14) give us the following equivalent (to (1)) system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle u_{tt}+{\Delta }^{2}u-\gamma \Delta u_{tt}-{\alpha }_0\Delta u_t+\kappa {\int }_{-\infty }^{+\infty }\eta (x,\zeta ,t)\mu \left(\zeta \right)d\zeta =0}\hfill & x\in \Omega ,t>0\\ \\ {\eta }_t(x,\zeta ,t)+({\zeta }^2+\beta )\eta (x,\zeta ,t)-z(x,1,t)\mu \left(\zeta \right)=0,\hfill & x\in \Omega ,\zeta \in \mathbb{R},t>0\\ \\ {\tau }_0{z}_t(x,\theta ,t)+z_{\theta }(x,\theta ,t)=0,\hfill & x\in \Omega ,\theta \in (0,1),t>0,\\ \\ u={\partial }_{\nu }u=0\hfill & \mathrm{on}{\Gamma }_0\times (0,\infty )\\ \\ \Delta u+(1-\mu )B_{1}u={\partial }_{\nu }\Delta u+(1-\mu )B_{2}u-\gamma {\partial }_{\nu }{u}_{tt}-{\alpha }_0{\partial }_{\nu }{u}_t=0\hfill & \mathrm{on}{\Gamma }_1\times (0,\infty )\\ \\ u(x,0)=u^0\left(x\right),u_t(x,0)=u^1\left(x\right),\hfill & x\in \Omega ,\\ \\ \eta (x,\zeta ,0)=0,\hfill & x\in \Omega ,\zeta \in \mathbb{R},\\ \\ z(x,\theta ,0)=f_0(x,-\theta {\tau }_0),\hfill & x\in \Omega ,\theta \in (0,1)\end{array}\right.\end{array}$$

(27)

where $\kappa ={\displaystyle \frac{{\alpha }_{2}sin\left(\alpha \pi \right)}{\pi }}.$

Denote by $\chi$ the quantity

$${\displaystyle \chi =\underset{-\infty }{\overset{+\infty }{\int }}\frac{{\mu }^2\left(\zeta \right)}{{\zeta }^2+\beta }d\zeta ,}$$

and let $\sigma$ be a positive constant satisfying

$$\kappa \chi <\sigma <{\displaystyle \frac{{\alpha }_0}{C_p}}-\kappa \chi .$$

(28)

Remark 1.

From (25) and Lemma 2, we have that $\kappa \chi ={\alpha }_2{\beta }^{\alpha -1}<{\displaystyle \frac{{\alpha }_0}{2C_p}}$. Therefore, the inequality (28) makes sense, i.e., ${\displaystyle \frac{{\alpha }_0}{C_p}}-\kappa \chi >\kappa \chi >0$.

We define the energy space ${\mathcal{H}}_3$ by

$${\mathcal{H}}_3=H_{{\Gamma }_0}^2(\Omega )\times H_{{\Gamma }_0,\gamma }^1(\Omega )\times L^2(\Omega \times \mathbb{R})\times L^2(\Omega \times (0,1)).$$

Let $v=u_t$ and set $U={(u,v,\eta ,z)}^T$. Then, (27) is equivalent to

$$\left\{\begin{array}{c}{U}_t\left(t\right)={\mathcal{A}}_{3}U\left(t\right),\hfill \\ U(x,0)=U_0=(u^0,u^1,0,f_0(-\theta {\tau }_0)),\hfill \end{array}\right.$$

(29)

where

$$\begin{array}{c}\hfill {\mathcal{A}}_{3}U:=\left(\begin{array}{c}v\\ {\displaystyle -B_{\gamma }^{-1}{\Delta }^{2}u+{\alpha }_0{B}_{\gamma }^{-1}\Delta v-\kappa B_{\gamma }^{-1}{\int }_{-\infty }^{+\infty }\eta (x,\zeta )\mu \left(\zeta \right)d\zeta }\\ -({\zeta }^2+\beta )\eta (x,\zeta )+z(x,1)\mu \left(\zeta \right)\\ -{\displaystyle \frac{1}{{\tau }_0}}{z}_{\theta }\end{array}\right)\end{array}$$

with domain

$$\begin{array}{ccc}\hfill D\left({\mathcal{A}}_3\right)& =& \{(u,v,\eta ,z)\in {\mathcal{H}}_3:u\in H^3(\Omega ),v\in H_{{\Gamma }_0}^2(\Omega ),\zeta \eta \in L^2(\Omega \times \mathbb{R}),z_{\theta }\in L^2(\Omega \times (0,1))\hfill \\ & & -({\zeta }^2+\beta )\eta (x,\zeta )+z(x,1)\mu \left(\zeta \right)\in L^2(\Omega \times \mathbb{R}),v=z(.,0)\hfill \\ & & \Delta z+(1-\mu )B_{1}z=0\mathrm{on}{\Gamma }_1\}.\hfill \end{array}$$

By following the same steps as in the proof of Proposition 1, we find that ${\mathcal{A}}_3$ is maximal dissipative in ${\mathcal{H}}_3$. Consequently, we obtain the following proposition.

Proposition 4.

Suppose that (28) holds true. Hence, if $U_0=(u^0,u^1,0,f_0(-\theta {\tau }_0))\in {\mathcal{H}}_3$, then the system (27) has a unique mild solution $U\in C([0,\infty );{\mathcal{H}}_3)$.

Define the energy of system (27) by

$$\begin{array}{ccc}\hfill E_3\left(t\right)& =& {\displaystyle E_1\left(t\right)+\frac{\kappa }{2}{\int }_{\Omega }{\int }_{-\infty }^{+\infty }{\eta }^2(x,\zeta ,t)d\zeta dx+\sigma {\tau }_0{\int }_{\Omega }{\int }_0^1{z}^2(x,\theta ,t)d\theta dx}\hfill \end{array}$$

We can easily verify that

$$\begin{array}{ccc}\hfill E_3^{\prime }\left(t\right)& \le & {\displaystyle -C_6({\int }_{\Omega }|\nabla u_t{|}^{2}dx+{\int }_{\Omega }{|z(x,1,t)|}^{2}dx}\hfill \\ & & {\displaystyle +{\int }_{\Omega }{\int }_{-\infty }^{+\infty }({\zeta }^2+\beta ){|\eta (x,\zeta ,t)|}^{2}d\zeta dx),}\hfill \end{array}$$

(30)

where $C_6=min\left\{{\alpha }_0-(\kappa \chi +\sigma )C_p,\sigma -\kappa \chi ,{\displaystyle \frac{\kappa }{2}}\right\}$.

We need the following lemma.

Lemma 3.

Let $U={(u,u_t,\eta ,z)}^T$ be a solution of system (27). The functional

$${\displaystyle {\phi }_3\left(t\right)={\phi }_1\left(t\right)+\frac{\kappa }{2}{\int }_{\Omega }{\int }_{-\infty }^{+\infty }({\zeta }^2+\beta ){|M(x,\zeta ,t)|}^{2}d\zeta dx}$$

satisfies

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\phi }_3^{\prime }\left(t\right)& \le & {\displaystyle -\frac{1}{2}b(u,u)+{\int }_{\Omega }|u_t{|}^{2}dx+\left(\frac{{\alpha }_0^2{C}_s}{2}+\gamma \right){\int }_{\Omega }|\nabla u_t{|}^{2}dx+\sigma {\tau }_0^2{\int }_{\Omega }{\int }_0^1{|z(x,\theta ,t)|}^{2}d\theta dx}\hfill \\ & +& {\displaystyle \frac{\kappa }{4}{\int }_{\Omega }{\int }_{-\infty }^{+\infty }({\zeta }^2+\beta )|\eta (x,\zeta ,t){|}^{2}d\zeta dx-\kappa {\int }_{\Omega }{\int }_{-\infty }^{+\infty }{|\eta (x,\zeta ,t)|}^{2}d\zeta dx,}\hfill \end{array}\end{array}$$

(31)

where ${\displaystyle M(x,\zeta ,t)={\int }_0^t\eta (x,\zeta ,s)ds-\frac{{\tau }_0\mu \left(\zeta \right)}{{\zeta }^2+\beta }{\int }_0^1{f}_0(x,-\theta {\tau }_0)d\theta +\frac{u^0\left(x\right)\mu \left(\zeta \right)}{{\zeta }^2+\beta }.}$

Proof. 

By differentiating ${\phi }_3\left(t\right)$ and using ${\left(27\right)}_1$, we obtain

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\phi }_3^{\prime }\left(t\right)& =& {\displaystyle {\int }_{\Omega }|u_t{|}^{2}dx-b(u,u)-{\alpha }_0{\int }_{\Omega }\nabla u_t\nabla udx+\gamma {\int }_{\Omega }{|\nabla u_t|}^{2}dx}\hfill \\ & +& {\displaystyle \kappa {\int }_{\Omega }{\int }_{-\infty }^{+\infty }({\zeta }^2+\beta )\eta (x,\zeta ,t)M(x,\zeta ,t)d\zeta dx.}\hfill \end{array}\end{array}$$

(32)

According to Lemma 6 in [27], we have

$$\begin{array}{ccc}\hfill & & {\displaystyle {\int }_{\Omega }{\int }_{-\infty }^{+\infty }\left({\zeta }^2+\beta \right)\eta (x,\zeta ,t)M(x,\zeta ,t)d\zeta dx}\hfill \\ & =& {\displaystyle {\int }_{\Omega }u(x,t){\int }_{-\infty }^{+\infty }\eta (x,\zeta ,t)\mu \left(\zeta \right)d\zeta dx-{\tau }_0{\int }_{\Omega }{\int }_0^{1}z(x,\theta ,t){\int }_{-\infty }^{+\infty }\eta (x,\zeta ,t)\mu \left(\zeta \right)d\zeta d\theta dx}\hfill \\ & -& {\displaystyle {\int }_{\Omega }{\int }_{-\infty }^{+\infty }{|\eta (x,\zeta ,t)|}^{2}d\zeta dx.}\hfill \end{array}$$

(33)

Inserting (33) in (32), one finds that

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\phi }_3^{\prime }\left(t\right)& =& {\displaystyle {\int }_{\Omega }|u_t{|}^{2}dx-b(u,u)-{\alpha }_0{\int }_{\Omega }\nabla u_t\nabla udx+\gamma {\int }_{\Omega }|\nabla u_t{|}^{2}dx-\kappa {\int }_{\Omega }{\int }_{-\infty }^{+\infty }{|\eta (x,\zeta ,t)|}^{2}d\zeta dx}\hfill \\ & -& {\displaystyle \kappa {\tau }_0{\int }_{\Omega }{\int }_0^{1}z(x,\theta ,t){\int }_{-\infty }^{+\infty }\eta (x,\zeta ,t)\mu \left(\zeta \right)d\zeta d\theta dx.}\hfill \end{array}\end{array}$$

(34)

Using Young’s inequality and Hölder’s inequality, it holds that

$$\begin{array}{c}\hfill \begin{array}{ccc}& & {\displaystyle {\tau }_0{\int }_{\Omega }{\int }_0^{1}z(x,\theta ,t)\underset{-\infty }{\overset{+\infty }{\int }}\eta (x,\zeta ,t)\mu \left(\zeta \right)d\zeta dx}\hfill \\ & & \le \chi {\tau }_0^2{\displaystyle {\int }_{\Omega }{\int }_0^1{|z(x,\theta ,t)|}^{2}d\theta dx}+{\displaystyle \frac{1}{4}}{\int }_{\Omega }\underset{-\infty }{\overset{+\infty }{\int }}({\zeta }^2+\beta ){|\eta (x,\zeta ,t)|}^{2}d\zeta dx.\hfill \end{array}\end{array}$$

(35)

Inserting (10) and (35) in (34) and using (28), we obtain the desired inequality (31). □

Theorem 3.

The system (27) is exponentially stable.

Proof. 

Define the functional

$$F_3\left(t\right)=E_3\left(t\right)+N_3\left({\phi }_3+{\tau }_0{\int }_{\Omega }{\int }_0^1{e}^{-\theta {\tau }_0}{|z(x,\theta ,t)|}^{2}d\theta dx\right),$$

which satisfies $F_3\left(t\right)\sim E_3\left(t\right).$

From (23), (30) and (31), we deduce that

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill F_3^{\prime }\left(t\right)& \le & {\displaystyle -{\displaystyle \frac{N_3}{2}}b(u,u)-N_3{\int }_{\Omega }|u_t{|}^{2}dx-\left(C_6-3N_3{C}_p-N_3(\gamma +\frac{{\alpha }_0^2{C}_s}{2})\right){\int }_{\Omega }{|\nabla u_t|}^{2}dx}\hfill \\ & & {\displaystyle -C_6{\int }_{\Omega }{|z(x,1,t)|}^{2}dx-{\tau }_0{N}_3\left(e^{-{\tau }_0}-\sigma {\tau }_0\right){\int }_{\Omega }{\int }_0^1{|z(x,\theta ,t)|}^{2}d\theta dx}\hfill \\ & & {\displaystyle -\left(C_6-{\displaystyle \frac{N_3\kappa }{4}}\right){\int }_{\Omega }{\int }_{-\infty }^{+\infty }({\zeta }^2+\beta ){|\eta (x,\zeta ,t)|}^{2}d\zeta dx}\hfill \\ & & {\displaystyle -N_3\kappa {\int }_{\Omega }{\int }_{-\infty }^{+\infty }{|\eta (x,\zeta ,t)|}^{2}d\zeta dx.}\hfill \end{array}\end{array}$$

(36)

At this point, we choose $N_3$ small enough such that

$$C_6-3N_3{C}_p-N_3(\gamma +{\displaystyle \frac{{\alpha }_0^2{C}_s}{2}})>0\mathrm{and}{C}_6-{\displaystyle \frac{N_3\kappa }{4}}>0,$$

Then, in addition to (28), we pick $\sigma$ such that

$$\sigma <{\displaystyle \frac{e^{-{\tau }_0}}{{\tau }_0}}.$$

Consequently, by these choices and using the fact that $F_3\left(t\right)\sim E_3\left(t\right)$, we arrive at

$$F_3^{\prime }\left(t\right)\le -C_7{E}_3\left(t\right)\le -C_8{F}_3\left(t\right),\forall t\ge 0,$$

where $C_7$ and $C_8$ are positive constants. A simple integration over $(0,t)$ yields

$$F_3\left(t\right)\le F_3\left(0\right)e^{-C_{8}t},\forall t\ge 0.$$

Since $F_3\left(t\right)$ and $E_3\left(t\right)$ are equivalent, we conclude the existence of a positive constant $C_9$ satisfying

$$E_3\left(t\right)\le C_9{e}^{-C_{8}t},\forall t\ge 0.$$

This ends the proof of Theorem 3. □

5. Conclusions

In this manuscript, we have conducted a detailed analysis of the exponential stability of a Kirchhoff plate equation with free boundary conditions in a bounded domain of ${\mathbb{R}}^2$. Our work addresses several important cases, extending previous results on the stability of such systems and providing new insights into the behavior of models governed by different damping and delay mechanisms.

Initially, we considered the Kirchhoff plate equation subjected solely to structural damping. Through the energy method, we demonstrated that the energy of the system decays exponentially over time. This result was achieved by deriving appropriate energy estimates, which reveal that the structural damping is sufficient to ensure the exponential stability of the plate equation. This first case sets a baseline for understanding how energy dissipation behaves in the absence of time delay terms, and it lays the groundwork for investigating more complex scenarios.

We then extended our analysis to the Kirchhoff plate equation with the addition of a linear internal time delay term, while still incorporating structural damping. The inclusion of the time delay adds complexity to the stability analysis because it introduces memory effects that could potentially destabilize the system. Despite these challenges, we used the same energy-based technique to establish that exponential stability still holds in this case. The analysis required a careful handling of the delay term to ensure that the system’s energy continues to decay at an exponential rate.

Lastly, we investigated the most challenging case where the model is governed by structural damping combined with a fractional time delay term. Fractional time delays present additional difficulties due to their nonlocal nature, which affects the system’s response over a continuum of past states. To address this, we constructed a suitable Lyapunov functional that accounts for the fractional nature of the delay term and we proved that the system is exponentially stable.

Moreover, this work improves upon previous results related to the Kirchhoff plate equation with frictional damping, where only polynomial decay rates could be established [19,20].

One of the limitations of the approaches developed here is that the damping coefficients must satisfy certain inequalities, particularly in the second and third cases. Without these conditions, it is not possible to establish the stability of our systems.

Future research could further explore the impact of nonlinear damping or more general forms of delay terms on the stability of Kirchhoff-type models.