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Stability result of a suspension bridge Problem with time-varying delay and time-varying weight

Abstract

In this paper, we study a plate equation as a model for a suspension bridge with time-varying delay and time-varying weights. Under some conditions on the delay and weight functions, we establish a stability result for the associated energy functional. The present work extends and generalizes some similar results in the case of wave or plate equations.

Introduction

In this paper, we consider \(\Omega =(0,\pi )\times (-\ell ,\ell )\subset {\mathbb {R}}^2\) to be a thin rectangular plate with suspension hangers and suppose the plate is partially hinged on the vertical edges and free on the horizontal edges, see [7,8,9] for details on suspension bridge models through partially hinged plates. We are interested in studying the vertical displacement of the thin plate \(u=u(x,y,t)\) in the presence of time-varying delay and time-varying weights, namely

$$\begin{aligned} u_{tt}(x,y,t) + \Delta ^{2} u(x,y,t) +h(u(x,y,t))+\delta _1(t)u_t(x,y,t)+\delta _2(t)u_t(x,y,t-\tau (t))=0, \end{aligned}$$
(1.1)

where \((x,t)\in \Omega \times (0,T)\) and subjected to the partially hinged boundary conditions

$$\begin{aligned} \left\{ \begin{array}{ll} u(0,y,t) = u_{xx}(0, y, t) =0, &{} \ \mathrm{for} \,\,(y,t)\in (-\ell ,\ell )\times (0,T), \\ u(\pi , y, t) = u_{xx}(\pi ,y,t)=0, &{} \ \mathrm{for} \,\,(y,t)\in (-\ell ,\ell )\times (0,T), \\ u_{yy}(x,\pm \ell , t)+\sigma u_{xx}(x,\pm \ell ,t)= 0, &{} \ \mathrm{for} \,\,(x,t)\in \, (0,\pi )\times (0,T), \\ u_{yyy}(x,\pm \ell ,t) +(2-\sigma )u_{xxy}(x,\pm \ell ,t)= 0, &{} \ \mathrm{for} \,\,(x,t)\in \, (0,\pi )\times (0,T),\\ \end{array}\right. \end{aligned}$$
(1.2)

and initial data

$$\begin{aligned} \left\{ \begin{array}{ll} u(x,y, 0)= u_{0}(x, y),\ \ u_{t}(x, y, 0 )= u_{1}(x, y), &{} \quad \mathrm{in}\ \ \ \Omega ,\\ u_t(x,y,t-\tau (0))=g_0(x,y,t-\tau (0)), &{} \quad \mathrm{in}\ \ \ \Omega \times (0,\tau (0)). \end{array}\right. \end{aligned}$$
(1.3)

Here, \( 0<\sigma < \frac{1}{2}\) is the Poisson ratio, h is a non-linear function which represents the hangers restoring force, \(\tau (t)>0\) is the time-varying delay, \(\delta _1(t)\) and \(\delta _2(t) \) are nonlinear damping weights functions. Damping and internal feedback are important properties of great interest when modelling systems of natural phenomena in areas such as, physics, thermal science, biosciences and structural engineering. To have a better understanding of these phenomena, the present state and past history of occurrences are of great importance. A substantial number of articles have investigated wave or plate equations in the presence of constant time delay \((\tau (t)\equiv \tau )\) and constant weights \((\delta _1(t)\equiv \delta _1,\ \delta _2(t)\equiv \delta _2)\) and proved exponential or polynomial decay results for the associated energy functional, see [1, 13, 14, 22] and the references cited in them. When \(\delta _2\equiv 0\) and \(\delta _1(t)\equiv \delta >0\) in (1.1), Messaoudi and Mukiawa [17] showed that the associated energy functional decays exponentially. See similar results for wave equations in [4, 10, 11] and the references therein. If \(\delta _2(t)=\delta >0\) and \(\delta _1(t)\equiv 0\) in (1.1), then the system becomes unstable, see for instance the results in [5, 18, 19] and the references cited.In the case of constant weights (\(\delta _1\) and \(\delta _2\) constants) and time-varying delay equations, not so much has been done. We mention among others, the works of Nicaise and Pignotti [20, 21], where they considered a wave equation with boundary or internal time-varying delay and established an exponential stability result. Recently, Enyi and Mukiawa [6] considered a viscoelastic plate equation when \(\delta _1\) and \(\delta _2\) are constants and time-varying delay and proved a general decay result provided \(|\delta _2|<|\delta _1|\sqrt{1-d}\). For existence and stability results for the wave equation with time-varying weights and constant time feedback, see [2, 3, 12] and references therein.

Motivated by the result in [3], where an energy decay estimate is established for a wave equation with constant feedback (\(\tau \equiv \) constant) and \(h\equiv 0,\) we establish a stability result for Problem (1.1)–(1.3). The present paper improved and extended the results in [6, 17] for plate equations and the results in [3, 18] for wave equations. This paper is organized as follows: in Sect. 2, we recall some fundamental materials for the clarity of the reader. In Sect. 3, we state and prove our main stability result for Problem (1.1)–(1.3).

Functional setting and hypotheses

Here, we assume the hangers restoring force h satisfies the following, as given in [17]:

(A1):

\( h:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a locally Lipschitz non-decreasing function such that \( h(0)=0\) and the positive function H defined by \( H(s) =\displaystyle \int _0^{s} h(\tau )d \tau \) satisfying

$$\begin{aligned} sh(s)- H(s)\ge 0, \quad \forall s\in {\mathbb {R}}. \end{aligned}$$

Using the ideas of [3], we assume the time varying weights \(\delta _1(t)\) and \(\delta _2(t)\) satisfy

(A2):

\(\delta _1:[0,+\infty )\longrightarrow (0,+\infty )\) is a non-increasing \(C^1-\) function and there exists a positive constant \(B_1\) such that

$$\begin{aligned} \left| \dfrac{\delta _1'(t)}{\delta _1(t)}\right| \le B_1, \ \ \forall t\ge 0. \end{aligned}$$
(2.1)

\(\delta _2:[0,+\infty )\longrightarrow {\mathbb {R}}\) is a \(C^1-\) function (which is not necessarily monotone or positive) and there exist some positive constants \(0<\alpha <\sqrt{1-d}, B_2\) such that

$$\begin{aligned} |\delta _2(t)|\le \alpha \delta _1(t), \end{aligned}$$
(2.2)

and

$$\begin{aligned} |\delta _2'(t)|\le B_2 \delta _1(t). \end{aligned}$$
(2.3)

As in [19], we assume the time varying internal feedback \(\tau (t)\) satisfies

(A3):

There exist \(\tau _0,\tau _1>0\) such that

$$\begin{aligned}&0<\tau _0\le \tau (t)\le \tau _1, \ \forall \ t>0, \end{aligned}$$
(2.4)
$$\begin{aligned}&\tau \in W^{2,\infty }(0,T), \ \forall T>0, \end{aligned}$$
(2.5)
$$\begin{aligned}&\tau '(t)\le d<1, \ \forall \ t>0. \end{aligned}$$
(2.6)

As in [7], we introduce the Hilbert space

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

endowed with the inner product

$$\begin{aligned} (v,w)_{H_{*}^2}(\Omega )=\int _{\Omega }[(\Delta v\Delta w +(1-\sigma )(2v_{xy}w_{xy}-v_{xx}w_{yy}-v_{yy}w_{xx})]\mathrm{d}x\mathrm{d}y \end{aligned}$$

and denote by \({\mathcal {H}}(\Omega )\) the dual of \(H_{*}^2(\Omega ).\) We now state some needed Lemmas:

Lemma 2.1

[7] Suppose \(1< \mathrm{p} < +\infty \). Then, for any \( w \in H_{*}^2(\Omega )\), there exists an embedding constant \( C_e =C_e(\Omega ,p)> 0\) such that

$$\begin{aligned} \Vert w\Vert _{L^p(\Omega )} \le C_e\Vert w\Vert _{H_{*}^2(\Omega )}. \end{aligned}$$

Lemma 2.2

[16] Let \(u\in H^4(\Omega )\cap H_{*}^2(\Omega )\) and \(v\in H_{*}^2(\Omega ).\) Then,

$$\begin{aligned} \left( \Delta ^2u,v\right) _{L^2(\Omega )} =\left( u,v \right) _{H_{*}^2(\Omega )}. \end{aligned}$$
(2.7)

Lemma 2.3

[15] Let \(E:[0,+\infty )\longrightarrow [0,+\infty )\) be a non-increasing function and \(\psi :[0,+\infty )\longrightarrow [0,+\infty )\) an increasing \(C^1-\) function satisfying

$$\begin{aligned} \psi (0)=0\ \ \ \mathrm{and} \ \ \ \lim _{t\longrightarrow +\infty } \psi (t)=+\infty . \end{aligned}$$
(2.8)

Suppose there exist \(\beta >-1\) and \(\gamma >0\) so that

$$\begin{aligned} \int _s^{+\infty }E^{1+\beta }(t)\psi '(t)dt \le \dfrac{1}{\gamma }E^{\beta }(s) , \ \ 0\le s<+\infty . \end{aligned}$$
(2.9)

Then

$$\begin{aligned} {\left\{ \begin{array}{ll} \ E(t)=0, \ \ \forall \ t\ge \dfrac{E^{\beta }(0)}{\gamma |\beta |}, \ \mathrm{if}\ -1<\beta <0\\ \\ \ E(t)\le E(0)\left( \dfrac{1+\beta }{1+\gamma \beta \psi (t)}\right) ^{\frac{1}{\beta }}, \ \forall \ t\ge 0, \ \ \mathrm{if}\ \beta >0\\ \\ \ E(t)\le E(0)e^{1-\gamma \psi (t)}, \ \forall \ t\ge 0, \ \ \mathrm{if}\ \beta =0. \end{array}\right. } \end{aligned}$$
(2.10)

Similarly, as in Nicaise and Pignotti [18], we introduce the following change of variable:

$$\begin{aligned} z(x,y,\rho , t)=u_t(x,y,t-\tau (t)\rho ), \ \ \mathrm{for} \ (x,y,\rho ,t)\in \Omega \times (0,1)\times (0,T). \end{aligned}$$
(2.11)

Then simple calculations gives

$$\begin{aligned} \tau (t)z_t(x,y,\rho ,t)+(1-\tau '(t)\rho )z_{\rho }(x,y,\rho ,t)=0, \ \ \mathrm{for}\ (x,y,\rho ,t)\in \Omega \times (0,1)\times (0,T). \end{aligned}$$
(2.12)

Therefore, Problem (1.1)–(1.3) becomes

$$\begin{aligned} \left\{ \begin{array}{ll} u_{tt} + \Delta ^{2} u +h(u)+\delta _1(t)u_t +\delta _2(t)z(x,y,1,t)=0, &{} \ \mathrm{in}\ \ \Omega \times (0,T).\\ \tau (t)z_t(x,y,\rho ,t)+(1-\tau '(t)\rho )z_{\rho }(x,y,\sigma ,t)=0, &{} \ \mathrm{in}\ \ \Omega \times (0,1)\times (0,T)\\ \end{array}\right. \end{aligned}$$
(2.13)

with boundary conditions

$$\begin{aligned} \left\{ \begin{array}{ll} u(0,y,t) = u_{xx}(0, y, t) =0, &{} \ \mathrm{for} \,\,(y,t)\in (-\ell ,\ell )\times (0,T), \\ u(\pi , y, t) = u_{xx}(\pi ,y,t)=0,&{} \ \mathrm{for} \,\,(y,t)\in (-\ell ,\ell )\times (0,T), \\ u_{yy}(x,\pm \ell , t)+\sigma u_{xx}(x,\pm \ell ,t)= 0, &{} \ \mathrm{for} \,\,(x,t)\in \, (0,\pi )\times (0,T), \\ u_{yyy}(x,\pm \ell ,t) +(2-\sigma )u_{xxy}(x,\pm \ell ,t)= 0,&{} \ \mathrm{for} \,\,(x,t)\in \, (0,\pi )\times (0,T),\\ z(x,y,0, t)=u_t(x,y,t),&{} \ \mathrm{for}\ (x,y,t)\in \Omega \times (0,T) \end{array}\right. \end{aligned}$$
(2.14)

and initial data

$$\begin{aligned} \left\{ \begin{array}{ll} u(x,y, 0)= u_{0}(x, y),\,\,\,\,\,\, u_{t}(x, y, 0 )= u_{1}(x, y),&{} \ \mathrm{in}\,\,\,\,\,\Omega ,\\ z(x,y,\rho ,0)= u_t(x,y,-\tau (0)\rho )=g_0(x,y,-\tau (0)\rho ), &{} \ \mathrm{in}\ \ \Omega \times (0,1). \end{array}\right. \end{aligned}$$
(2.15)

Let \(\bar{\xi }\) be a positive constant satisfying

$$\begin{aligned} \dfrac{\alpha }{\sqrt{1-d}}<\bar{\xi }<2-\dfrac{\alpha }{\sqrt{1-d}}. \end{aligned}$$
(2.16)

The energy function of Problem (2.13)–(2.15) is defined by

$$\begin{aligned} E(t)=\frac{1}{2}\Vert u_t\Vert ^2_{L^2(\Omega )} +\frac{1}{2}\Vert u\Vert ^2_{H^2_{*}(\Omega )} +\int _{\Omega }H(u)dxdy+\frac{\xi (t)\tau (t)}{2} \Vert z\Vert ^2_{L^2(\Omega \times (0,1))}, \end{aligned}$$
(2.17)

where \(\xi (t)=\bar{\xi }\delta _1(t)\) is a non-increasing \(C^1-\) function. The main existence and uniqueness result of (2.13)–(2.15) is the following.

Theorem 2.4

Assume (A1)(A3) hold and let

$$\begin{aligned} (u_0,u_1,g_0)\in H^4(\Omega )\cap H^2_{*}(\Omega ) \times H^2_{*}(\Omega )\times H^1(\Omega ;H^1_0(0,1)) \end{aligned}$$

be given and satisfies the compatibility condition \(g_0(.,0)=u_1\). Then Problem (1.1)–(1.3) has a global unique solution u satisfying

$$\begin{aligned}&u\in L^{\infty }_\mathrm{loc}\left( (-\tau (0),T);H^4(\Omega ) \cap H^2_{*}(\Omega )\right) , \ u_t \in L^{\infty }_\mathrm{loc} \left( (-\tau (0),T); H^2_{*}(\Omega )\right) \nonumber \\&u_{tt}\in L^{\infty }_\mathrm{loc}\left( (-\tau (0),T);L^2(\Omega )\right) . \end{aligned}$$
(2.18)

Proof

The result can be obtain by using the Galerkin approximation method and combining the ideas in ([17], Theorem 3.1) and ([3], Theorem 2.2). \(\square \)

Lemma 2.5

Assume (A1)(A3) hold. Then the energy functional defined by (2.17) satisfies along the solution of Problem (2.13)

$$\begin{aligned} \dfrac{\mathrm{d}E}{\mathrm{d}t}&\le -\delta _1(t)\left( 1-\dfrac{\bar{\xi }}{2} -\dfrac{\alpha }{2\sqrt{1-d}}\right) \Vert u_t\Vert ^2_{L^2(\Omega )} \nonumber \\&\quad -\delta _1(t)\left( \dfrac{\bar{\xi }}{2}-\frac{\bar{\xi } \tau '(t)}{2} -\dfrac{\alpha \sqrt{1-d}}{2}\right) \Vert z(.,1) \Vert ^2_{L^2(\Omega )} \nonumber \\&\le 0. \end{aligned}$$
(2.19)

Proof

First, we multiple (2.13)\(_1\) by \(u_t\), integrating over \(\Omega \) and using condition (A1) and Lemma 2.2, we arrive at

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{1}{2}\Vert u_t\Vert ^2_{L^2(\Omega )} +\frac{1}{2}\Vert u\Vert ^2_{H^2_{*}(\Omega )} +\int _{\Omega }H(u)\mathrm{d}x\mathrm{d}y \right) \nonumber \\&\quad =-\delta _1(t)\Vert u_t\Vert ^2_{L^2(\Omega )} -\delta _2(t)\int _{\Omega }u_tz(x,y,1,t)\mathrm{d}x\mathrm{d}y. \end{aligned}$$
(2.20)

Next, we multiple (2.13)\(_2\) by \(\xi (t)z\), integrate over \(\Omega \times (0,1)\), we get

$$\begin{aligned}&\xi (t)\tau (t) \int _{\Omega }\int _0^1 z(x,y,\rho ,t) z_t(x,y,\rho ,t)\mathrm{d} \rho \mathrm{d}x\mathrm{d}y\nonumber \\&\quad =-\xi (t)\int _{\Omega }\int _0^1(1-\tau '(t)\rho ) z(x,y,\rho ,t)z_{\rho }(x,y,\rho ,t)\mathrm{d}\rho \mathrm{d}x\mathrm{d}y, \end{aligned}$$
(2.21)

from which we obtain

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\xi (t)\tau (t)}{2} \Vert z\Vert ^2_{L^2(\Omega \times (0,1))}\right) \nonumber \\&\quad =-\frac{\xi (t)}{2}\int _{\Omega }\int _0^1 \frac{\partial }{\partial \rho }\left[ (1-\tau '(t)\rho ) z^2(x,y,\rho ,t)\right] \mathrm{d}\rho \mathrm{d}x\mathrm{d}y\nonumber \\&\qquad + \frac{\xi '(t)\tau (t)}{2}\int _{\Omega } \int _0^1 z^2(x,y,\rho ,t)d\rho \mathrm{d}x\mathrm{d}y\nonumber \\&\quad =-\frac{\xi (t)(1-\tau '(t))}{2}\Vert z(.,1) \Vert ^2_{L^2(\Omega )}+\frac{\xi (t)}{2}\Vert u_t\Vert ^2_{L^2(\Omega )} \nonumber \\&\qquad + \frac{\xi '(t)\tau (t)}{2}\Vert z\Vert ^2_{L^2(\Omega \times (0,1))}. \end{aligned}$$
(2.22)

Adding (2.20) and (2.22) leads to

$$\begin{aligned} \dfrac{\mathrm{d}E(t)}{\mathrm{d}t}&=-\left( \delta _1(t)-\frac{\xi (t)}{2}\right) \Vert u_t\Vert ^2_{L^2(\Omega )}-\delta _2(t)\int _{\Omega }u_tz(x,y,1,t)\mathrm{d}x\mathrm{d}y\\&\quad -\frac{\xi (t)(1-\tau '(t))}{2}\Vert z(.,1)\Vert ^2_{L^2(\Omega )} +\frac{\xi '(t)\tau (t)}{2}\Vert z\Vert ^2_{L^2(\Omega \times (0,1))}. \end{aligned}$$

Using Young’s inequality, the fact that \(\xi (t)\) is non-increasing and recalling (2.16), it follows that

$$\begin{aligned} \dfrac{\mathrm{d}E(t)}{\mathrm{d}t}&\le -\left( \delta _1(t)-\frac{\xi (t)}{2} -\frac{|\delta _2(t)|}{2\sqrt{1-d}}\right) \Vert u_t\Vert ^2_{L^2(\Omega )}\\&\quad -\left( \frac{\xi (t)(1-\tau '(t))}{2}-\frac{|\delta _2(t)| \sqrt{1-d}}{2}\right) \Vert z(.,1)\Vert ^2_{L^2(\Omega )}\\&\le -\delta _1(t)\left( 1-\frac{\bar{\xi }}{2} -\frac{\alpha }{2\sqrt{1-d}}\right) \Vert u_t\Vert ^2_{L^2(\Omega )}\\&\quad -\delta _1(t)\left( \frac{\bar{\xi }(1-\tau '(t))}{2} -\frac{\alpha \sqrt{1-d}}{2} \right) \Vert z(.,1)\Vert ^2_{L^2(\Omega )} \\&\le 0. \end{aligned}$$

\(\square \)

Stability result

The main stability result is the following.

Theorem 3.1

Under the assumptions of Theorem 2.4, there exist positive constants \(m,\lambda \) such that the energy functional (2.17) satisfies

$$\begin{aligned} E(t)\le mE(0)\exp \left( -\lambda \int _0^t\delta _1(s)\mathrm{d}s\right) , \ \forall \ t\ge 0. \end{aligned}$$
(3.1)

Proof

We start by defining the function \(\psi \) as follows:

$$\begin{aligned} \psi (t)=\int _0^t\delta _1(s)ds. \end{aligned}$$
(3.2)

It follows from (A2) that \(\psi \) is non-decreasing and bounded \(C^1\)-function such that

$$\begin{aligned} \psi (t)\longrightarrow +\infty \quad \mathrm{as}\ t\longrightarrow +\infty . \end{aligned}$$

Hence, \(\psi \) satisfies the conditions of Lemma 2.3. Now, multiplying (2.13)\(_1\) by \(E^q(t)\psi '(t) u\) and integrating over \(\Omega \times (s,T), \ T>s\), we obtain

$$\begin{aligned} \int _s^T \int _{\Omega } E^q(t)\psi '(t)u\left( u_{tt} +\Delta ^2 u + h(u)+\delta _1(t)u_t+\delta _2(t)z(x,y,1,t)\right) \mathrm{d}x\mathrm{d}y\mathrm{d}t=0. \end{aligned}$$

Using Lemma 2.2 and some calculations, we obtain

$$\begin{aligned}&\left[ E^q(t)\psi '(t)\int _{\Omega }uu_t \mathrm{d}x\mathrm{d}y \right] _s^T -\int _s^T\left[ qE'(t)E^{q-1}(t)\psi '(t)+E^q(t)\psi ''(t) \right] \int _{\Omega }uu_t\mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad + \int _s^T E^q(t)\psi '(t)\left( \frac{1}{2} \Vert u_t\Vert ^2_{L^2(\Omega )}+\frac{1}{2}\Vert u\Vert ^2_{H^2_{*}(\Omega )} +\int _{\Omega }H(u)dxdy\right) dt\nonumber \\&\quad -\frac{3}{2}\int _s^TE^q(t)\psi '(t) \Vert u_t\Vert ^2_{L^2(\Omega )}\mathrm{d}t +\frac{1}{2}\int _s^TE^q(t)\psi '(t)\Vert u\Vert ^2_{H^2_{*}(\Omega )}\mathrm{d}t\nonumber \\&\quad + \int _s^TE^q(t)\psi '(t)\int _{\Omega }\left( uh(u)-H(u) \right) \mathrm{d}x\mathrm{d}y\mathrm{d}t + \int _s^TE^q(t)\psi '(t)\delta _1(t)\int _{\Omega }uu_t \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad + \int _s^TE^q(t)\psi '(t)\delta _2(t) \int _{\Omega }uz(x,y,1,t)\mathrm{d}x\mathrm{d}y\mathrm{d}t=0. \end{aligned}$$
(3.3)

Next, we multiple (2.13)\(_2\) by \(E^q(t)\psi '(t)\xi (t)e^{-2\tau (t)\rho }z\) and integrate over \(\Omega \times (0,1)\times (s,T)\) to get

$$\begin{aligned} \int _s^T\int _{\Omega }\int _0^1E^q(t)\psi '(t)\xi (t) e^{-2\tau (t)\rho } z\left[ \tau (t)z_t +(1-\tau '(t)\rho )z_{\rho } \right] d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t=0. \end{aligned}$$

This gives us

$$\begin{aligned} 0&=\left[ \frac{1}{2} E^q(t)\psi '(t)\xi (t)\tau (t) \int _{\Omega }\int _0^1e^{-2\tau (t)\rho } z^2d\rho \mathrm{d}x\mathrm{d}y\right] _s^T\\&\quad -\frac{1}{2} \int _s^T\frac{\mathrm{d}}{\mathrm{d}t} \left[ E^q(t)\psi '(t)\xi (t)\tau (t)\right] \int _{\Omega }\int _0^1 e^{-2\tau (t)\rho } z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\\&\quad +\int _s^TE^q(t)\psi '(t)\xi (t)\tau (t)\tau '(t) \int _{\Omega }\int _0^1\rho e^{-2\tau (t)\rho } z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\\&\quad + \int _s^TE^q(t)\psi '(t)\xi (t)\int _{\Omega } \int _0^1\left( \frac{1}{2}\frac{\partial }{\partial \rho } e^{-2\tau (t)\rho }(1-\tau (t)\rho ) z^2 \right) d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\\&\quad + \int _s^TE^q(t)\psi '(t)\xi (t)\tau (t)\int _{\Omega } \int _0^1e^{-2\tau (t)\rho }(1-\tau (t)\rho ) z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\\&\quad +\int _s^TE^q(t)\psi '(t)\xi (t)\tau '(t)\int _{\Omega } \int _0^1e^{-2\tau (t)\rho } z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} 0&=\left[ \frac{1}{2} E^q(t)\psi '(t)\xi (t)\tau (t) \int _{\Omega }\int _0^1e^{-2\tau (t)\rho } z^2d\rho \mathrm{d}x\mathrm{d}y\right] _s^T\nonumber \\&\quad -\frac{1}{2} \int _s^T\frac{\mathrm{d}}{\mathrm{d}t} \left[ E^q(t)\psi '(t)\xi (t)\tau (t)\right] \int _{\Omega }\int _0^1 e^{-2\tau (t)\rho } z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad +\int _s^TE^q(t)\psi '(t)\xi (t)\tau (t)\tau '(t) \int _{\Omega }\int _0^1\rho e^{-2\tau (t)\rho } z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad + \frac{1}{2}\int _s^TE^q(t)\psi '(t)\xi (t) e^{-2\tau (t)}(1-\tau (t))\int _{\Omega } z^2(x,y,1,t) \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad - \frac{1}{2}\int _s^TE^q(t)\psi '(t)\xi (t) \Vert u_t\Vert ^2_{L^2(\Omega )}dt\nonumber \\&\quad + \int _s^TE^q(t)\psi '(t)\xi (t)\tau (t) \int _{\Omega }\int _0^1e^{-2\tau (t)\rho }(1-\tau (t)\rho ) z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad +\int _s^TE^q(t)\psi '(t)\xi (t)\tau '(t) \int _{\Omega }\int _0^1e^{-2\tau (t)\rho } z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t. \end{aligned}$$
(3.4)

Recalling conditions (A1) and setting \(k=\min \left\{ 1,e^{-2\tau _0},\frac{d}{\tau _0} \right\} \), the summation of (3.3) and (3.4) leads to

$$\begin{aligned} k\int _s^T E^{q+1}(t)\psi '(t)\mathrm{d}t&\le \int _s^T \left[ qE'(t)E^{q-1}(t)\psi '(t)+E^q(t)\psi ''(t)\right] \int _{\Omega }uu_t\mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad -\left[ E^q(t)\psi '(t)\int _{\Omega }uu_t \mathrm{d}x\mathrm{d}y \right] _s^T +\frac{3}{2}\int _s^TE^q(t)\psi '(t)\Vert u_t\Vert ^2_{L^2(\Omega )}\mathrm{d}t\nonumber \\&\quad -\int _s^TE^q(t)\psi '(t)\delta _1(t)\int _{\Omega }uu_t \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad - \int _s^TE^q(t)\psi '(t)\delta _2(t) \int _{\Omega }uz(x,y,1,t) dxdydt\nonumber \\&\quad -\left[ \frac{1}{2} E^q(t)\psi '(t)\xi (t)\tau (t) \int _{\Omega }\int _0^1e^{-2\tau (t)\rho } z^2d\rho \mathrm{d}x\mathrm{d}y\right] _s^T\nonumber \\&\quad +\frac{1}{2} \int _s^T\frac{d}{dt} \left[ E^q(t)\psi '(t)\xi (t)\tau (t)\right] \int _{\Omega }\int _0^1 e^{-2\tau (t)\rho } z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad - \frac{1}{2}\int _s^TE^q(t)\psi '(t)\xi (t) e^{-2\tau (t)}(1-\tau (t))\int _{\Omega } z^2(x,y,1,t) \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad + \frac{1}{2}\int _s^TE^q(t)\psi '(t)\xi (t) \Vert u_t\Vert ^2_{L^2(\Omega )}\mathrm{d}t. \end{aligned}$$
(3.5)

Now, we estimate the terms on the right-hand side of (3.5). Using (A2)(A3), Cauchy–Schwarz inequality, the embedding Lemma 2.1, (2.19) and assuming that \(\psi '\) is a non-negative and bounded function on \([0,+\infty )\), we obtain

$$\begin{aligned}&\left| \int _s^T q E'(t)E^{q-1}(t) \psi '(t) \int _{\Omega }uu_t \mathrm{d}x\mathrm{d}y\mathrm{d}t \right| \le -C\int _s^T E^{q}(t)E'(t)\mathrm{d}t\le CE^{q+1}(s),\nonumber \\&\left| \int _s^T E^q(t)\psi ''(t) \int _{\Omega }uu_t\mathrm{d}x\mathrm{d}y\mathrm{d}t\right| \le -C\int _s^TE^{q+1}(t)\psi ''(t)dt\le CE^{q+1}(s),\nonumber \\&\left| E^q(t)\psi '(t)\int _{\Omega }uu_t \mathrm{d}x\mathrm{d}y \right| \le CE^{q+1}(t),\nonumber \\&\left| E^q(t)\psi '(t)\xi (t)\tau (t)\int _{\Omega } \int _0^1e^{-2\tau (t)\rho } z^2d\rho \mathrm{d}x\mathrm{d}y \right| \le CE^{q+1}(t) \end{aligned}$$
(3.6)

and

$$\begin{aligned} \frac{3}{2}\int _s^TE^q(t)\psi '(t)\Vert u_t\Vert ^2_{L^2(\Omega )}\mathrm{d}t&\le C\int _s^TE^q(t)\frac{\psi '(t)}{\delta _1(t)}\delta _1(t) \Vert u_t\Vert ^2_{L^2(\Omega )}\mathrm{d}t\nonumber \\&\le -C\int _s^TE^q(t)\frac{\psi '(t)}{\delta _1(t)}E'(t)\mathrm{d}t. \end{aligned}$$
(3.7)

Thus, using (3.7) and the definition of \(\psi \) in (3.2), we get

$$\begin{aligned} \frac{3}{2}\int _s^TE^q(t)\psi '(t)\Vert u_t\Vert ^2_{L^2(\Omega )}\mathrm{d}t \le -C\int _s^TE^q(t)E'(t)\mathrm{d}t\le CE^{q+1}(s). \end{aligned}$$
(3.8)

Using (A2), Young’s inequality, Lemma 2.1 and (2.20), we have for any \(\epsilon >0\)

$$\begin{aligned}&\left| -\int _s^TE^q(t)\psi '(t)\delta _1(t) \int _{\Omega }uu_t \mathrm{d}x\mathrm{d}y\mathrm{d}t \right| \nonumber \\&\quad \le \ \epsilon \int _s^TE^q(t)\psi '(t)\Vert u\Vert ^2_{L^2(\Omega )}\mathrm{d}t +C_{\epsilon }\int _s^TE^q(t)\psi '(t)\Vert u_t\Vert ^2_{L^2(\Omega )}dt\nonumber \\&\quad \le \epsilon C_e\int _s^TE^q(t)\psi '(t) \Vert u\Vert ^2_{H_{*}^2(\Omega )}dt+ C_{\epsilon } \int _s^TE^q(t)\psi '(t)\Vert u_t\Vert ^2_{L^2(\Omega )}\mathrm{d}t\nonumber \\&\quad \le \epsilon C_e\int _s^TE^{q+1}(t)\psi '(t)\mathrm{d}t +C_{\epsilon }E^{q+1}(s) \end{aligned}$$
(3.9)

and

$$\begin{aligned}&\left| - \int _s^TE^q(t)\psi '(t)\delta _2(t) \int _{\Omega }uz(x,y,1,t) \mathrm{d}x\mathrm{d}y\mathrm{d}t\right| \nonumber \\&\quad \le \epsilon \int _s^TE^q(t)\psi '(t) \Vert u\Vert ^2_{L^2(\Omega )}\mathrm{d}t+ C_{\epsilon }\int _s^TE^q(t) \psi '(t)\Vert z(.,1)\Vert ^2_{L^2(\Omega )}\mathrm{d}t\nonumber \\&\quad \le \epsilon C_e\int _s^TE^q(t)\psi '(t) \Vert u\Vert ^2_{H_{*}^2(\Omega )}\mathrm{d}t+C_{\epsilon } \int _s^TE^q(t)\psi '(t)\Vert z(.,1)\Vert ^2_{L^2(\Omega )}\mathrm{d}t\nonumber \\&\quad \le \epsilon C_e\int _s^TE^{q+1}(t)\psi '(t)\mathrm{d}t +C_{\epsilon }E^{q+1}(s), \end{aligned}$$
(3.10)

where \(C_e\) is the embedding constant in Lemma 2.1. By assumption (A3) and recalling that \(\xi (t)\) is a non-increasing function, we obtain

$$\begin{aligned}&\frac{1}{2} \int _s^T\frac{\mathrm{d}}{\mathrm{d}t}\left[ E^q(t) \psi '(t)\xi (t)\tau (t)\right] \int _{\Omega } \int _0^1 e^{-2\tau (t)\rho } z^2 d\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad \le \frac{1}{2} \int _s^T\frac{\mathrm{d}}{\mathrm{d}t} \left[ E^q(t)\psi '(t)\right] \tau '(t)\int _{\Omega } \int _0^1 e^{-2\tau (t)\rho } z^2 \mathrm{d}\rho \mathrm{d}x\mathrm{d}y\mathrm{d}t\nonumber \\&\quad \le C\int _s^T E^q(t)(-E'(t))\mathrm{d}t\nonumber \\&\quad \le CE^{q+1}(s) \end{aligned}$$
(3.11)

and using Cauchy–Schwarz inequality, we get

$$\begin{aligned}&\left| - \frac{1}{2}\int _s^TE^q(t)\psi '(t)\xi (t)e^{-2\tau (t)} (1-\tau (t))\int _{\Omega } z^2(x,y,1,t) \mathrm{d}x\mathrm{d}y\mathrm{d}t\right| \nonumber \\&\quad \le C\int _s^TE^q(t)\xi (t)\Vert z(.,1)\Vert ^2_{L^2(\Omega )}\mathrm{d}t\nonumber \\&\quad \le CE^{q+1}(s), \end{aligned}$$
(3.12)
$$\begin{aligned}&\frac{1}{2}\int _s^TE^q(t)\psi '(t)\xi (t) \Vert u_t\Vert ^2_{L^2(\Omega )}\mathrm{d}t \le CE^{q+1}(s). \end{aligned}$$
(3.13)

A combination of (3.6)–(3.13) leads to

$$\begin{aligned} (k-\epsilon C_e)\int _s^TE^{q+1}(t)\psi '(t)\mathrm{d}t \le (C_{\epsilon }+C)E^{q+1}(s). \end{aligned}$$
(3.14)

By selecting \(\epsilon \) small enough so that \(k-\epsilon C_e>0\), we get

$$\begin{aligned} \int _s^TE^{q+1}(t)\psi '(t)\mathrm{d}t\le CE^{q+1}(s). \end{aligned}$$
(3.15)

Letting \(T\longrightarrow +\infty \) and applying Lemma 2.3, we obtain the result. \(\square \)

Examples

Let

$$\begin{aligned} \begin{aligned} \delta _1(t)=b(1+t)^{a-1}, \ \ \delta _2(t) =\dfrac{b}{b+1}(1+t)^{a-2}, \ \ 0<a<1, \ b>0, \end{aligned} \end{aligned}$$

where a and b are constants. Then

$$\begin{aligned} \left| \dfrac{\delta '_1(t)}{\delta _1(t)}\right|= & {} \left| \dfrac{b(a-1)(1+t)^{a-2}}{b(1+t)^{a-1}}\right| =\dfrac{(1-a)}{1+t}\le (1-a),\\ |\delta _2(t)|= & {} \left| \dfrac{b}{b+1}(1+t)^{a-1} \dfrac{1}{(1+t)} \right| \le \dfrac{1}{1+b}\delta _1(t) \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} |\delta '_2(t)|=\left| \dfrac{(a-2)b}{1+b}(1+t)^{a-3}\right| \le \left| \dfrac{(a-2)b}{1+b}\right| \delta _1(t). \end{aligned} \end{aligned}$$

Therefore, \(\delta _1\) and \(\delta _2\) satisfy the assumptions in \(\mathbf (A2) \). It follows from (3.1) and (3.2) that

$$\begin{aligned} E(t)\le mE(0)e^{-\lambda _1(1+t)^a}, \ \forall \ t\ge 0. \end{aligned}$$
(3.16)

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The author appreciates the University of Hafr AL Batin for its continuous support.

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Mukiawa, S.E. Stability result of a suspension bridge Problem with time-varying delay and time-varying weight. Arab. J. Math. 10, 659–668 (2021). https://doi.org/10.1007/s40065-021-00345-x

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