Skip to main content

Decay rate of the solutions to the Cauchy problem of the Bresse system in thermoelasticity of type III with distributed delay

Abstract

The decay rate of solutions to a Bresse system in thermoelasticity of type III with respect to the distributed delay term is the subject of this study. We demonstrate our major finding utilising the energy approach in the Fourier space.

1 Introduction and preliminaries

Fourier law provides the fundamental principle governing classical heat conduction:

$$\begin{aligned} q(x,t)=-\kappa \nabla \upsilon (x,t), \end{aligned}$$
(1.1)

where t represent the time, x is the Lagrangian coordinates material point, υ is the temperature, measured with respect to a reference temperature, is the gradient operator, q is the heat flux and κ is the thermal conductivity of the material which is a thermodynamic state property. According to equation (1.1), the heat flux is caused by the temperature gradient at the same material point x and at the same time t. Equation (1.1) and the conservation law together (assuming for simplicity that no heat sources are present)

$$\begin{aligned} \jmath \upsilon _{t}+\varrho \operatorname{div} q=0, \end{aligned}$$
(1.2)

produces the classical heat transport equation (of parabolic type)

$$\begin{aligned} \jmath \upsilon _{t}-\varrho \kappa \Delta \upsilon =0, \end{aligned}$$
(1.3)

Green & Naghdi [6, 7] created a thermoelasticity model that incorporates the temperature gradient and thermal displacement gradient among the constitutive variables, and presented a heat conduction law as

$$\begin{aligned} q(x,t)=-\kappa \nabla \upsilon -\kappa ^{*}\nabla r, \end{aligned}$$
(1.4)

where \(r_{t} = \upsilon \) and r is the thermal displacement gradient and the constants κ and \(\kappa ^{*}\) are both positive. The energy balance law (1.2) and equation (1.4) result in the equation

$$\begin{aligned} \jmath \upsilon _{tt}-\kappa \varrho \Delta \upsilon _{t}- \kappa ^{*} \varrho \Delta \upsilon =0, \end{aligned}$$
(1.5)

this allows thermal waves to travel at a finite speed.

Several authors have discussed the interaction between Fourier law of heat conduction and various systems, and there are numerous outcomes. Examples include the Timoshenko system in [9, 13], the Bresse system (Bresse–Fourier) in [5, 10, 1517], the Bresse system combined with the Cattaneo law of heat conduction in [14] and the MGT problem in [1]. We recommend the following papers [24, 8] to the reader for more information.

We would like to demonstrate the general decay result in the Fourier space to the Cauchy issue of the Bresse system in type III thermoelasticity using all of the papers cited above, particularly [15]. This is one of the earliest papers that we are aware of that look at this issue in Fourier space.

Therefore, the primary objective of this paper is to investigate the rate at which the following system’s solutions decay:

$$\begin{aligned} \textstyle\begin{cases} \varsigma _{tt}-(\varsigma _{x}-\hbar -l\Im )_{x}-k_{0}^{2}l(\Im _{x}-l \varsigma ) =0, \\ \hbar _{tt}-a^{2}\hbar _{xx}-(\varsigma _{x}-\hbar -l\Im )+m\upsilon _{x}=0, \\ \Im _{tt}-k_{0}^{2}(\Im _{x}-l\varsigma )_{x}-l(\varsigma _{x}-\hbar -l \Im )+\aleph _{1}\Im _{t} + \int _{\wp _{1}}^{\wp _{2}} \aleph _{2}(s)\Im _{t} ( x, t-s )\,ds=0, \\ \upsilon _{tt}-k_{1}\upsilon _{xx}+\beta \hbar _{ttx}-k_{2}\upsilon _{txx}=0, \end{cases}\displaystyle \end{aligned}$$
(1.6)

where

$$\begin{aligned} (x, s, t)\in \mathbb{R}\times (\wp _{1}, \wp _{2})\times \mathbb{R}_{+}, \end{aligned}$$

with the initial and boundary conditions

$$\begin{aligned} \begin{aligned} &(\varsigma,\varsigma _{t},\hbar,\hbar _{t},\Im,\Im _{t},\upsilon, \upsilon _{t}) (x,0)=( \varsigma _{0},\varsigma _{1},\hbar _{0},\hbar _{1}, \Im _{0},\Im _{1},\upsilon _{0},\upsilon _{1}),\quad x\in \mathbb{R}, \\ &\Im _{t}(x,-t)=f_{0}(x,t),\quad (x,t)\in (0,1)\times (0,\wp _{2}), \end{aligned} \end{aligned}$$
(1.7)

where the functions ς, and ħ denote the vertical displacements of the beam, longitudinal displacements and the rotation angle of the linear filaments material, respectively; \(a, l, m,k_{0}, k_{1}, k_{2},\aleph _{1}\) and β are positive constants and the function υ is the temperature difference; the integral represent the distributed delay terms with \(\wp _{1}, \wp _{2} >0\) being a time delay, \(\aleph _{2}\) is an \(L^{\infty}\) function satisfying:

(H1) \(\aleph _{2}:[\wp _{1}, \wp _{2}]\rightarrow \mathbb{R}\) is a bounded function satisfying

$$\begin{aligned} \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \,ds< \aleph _{1}. \end{aligned}$$
(1.8)

The sections of this paper are as follows: In this section, we apply our assumptions and preliminary findings to the major decay result. We build the Lyapunov functional and determine the estimate for the Fourier image in the following section by employing the energy approach in Fourier space. The conclusion is covered in the final section.

As in [12], we begin by introducing the new variable

$$\begin{aligned} \mathcal{Y}(x, \jmath, s, t)=\Im _{t}(x, t-s\jmath ), \end{aligned}$$

then, we get

$$\begin{aligned} \textstyle\begin{cases} s\mathcal{Y}_{t}(x, \jmath, s, t)+\mathcal{Y}_{\jmath}(x, \jmath, s, t)=0, \\ \mathcal{Y}(x, 0, s, t)=\Im _{t}(x, t), \end{cases}\displaystyle \end{aligned}$$

and utilize the transformation [18]

$$\begin{aligned} \overline{\upsilon}:= \int _{0}^{t}\upsilon (x,s)\,ds+\chi (x), \end{aligned}$$
(1.9)

with a function \(\chi:=\chi (x)\) satisfying

$$\begin{aligned} k_{1}\chi _{xx}=\upsilon _{1}-k_{2} \upsilon _{0xx}+\beta \hbar _{1x}. \end{aligned}$$
(1.10)

We can also write the proposed problem in the form (by writing, υ instead of υ̅)

$$\begin{aligned} \textstyle\begin{cases} \varsigma _{tt}-(\varsigma _{x}-\hbar -l\Im )_{x}-k_{0}^{2}l(\Im _{x}-l \varsigma ) =0, \\ \hbar _{tt}-a^{2}\hbar _{xx}-(\varsigma _{x}-\hbar -l\Im )+m\upsilon _{tx}=0, \\ \Im _{tt}-k_{0}^{2}(\Im _{x}-l\varsigma )_{x}-l(\varsigma _{x}-\hbar -l \Im )+\aleph _{1}\Im _{t} + \int _{\wp _{1}}^{\wp _{2}} \aleph _{2}(s)\mathcal{Y} ( x, 1,s,t )\,ds =0, \\ \upsilon _{tt}-k_{1}\upsilon _{xx}+\beta \hbar _{tx}-k_{2}\upsilon _{txx}=0, \\ s\mathcal{Y}_{t}(x, \jmath, s, t)+\mathcal{Y}_{\jmath}(x, \jmath, s, t)=0, \end{cases}\displaystyle \end{aligned}$$
(1.11)

where

$$\begin{aligned} (x, \jmath, s, t)\in \mathbb{R}\times (0, 1)\times (\wp _{1}, \wp _{2}) \times \mathbb{R}_{+}, \end{aligned}$$

with initial conditions

$$\begin{aligned} \textstyle\begin{cases} (\varsigma,\varsigma _{t},\hbar,\hbar _{t},\Im,\Im _{t},\upsilon, \upsilon _{t})(x,0)=(\varsigma _{0},\varsigma _{1},\hbar _{0},\hbar _{1}, \Im _{0},\Im _{1},\overline{\upsilon}(x,0),\overline{\upsilon}_{t}(x,0)), \\ \mathcal{Y}(x,\jmath,s,0)=f_{0}(x,s\jmath ),\qquad (x,\jmath,s)\in \mathbb{R}\times (0,1)\times (0,\wp _{2}), \end{cases}\displaystyle \end{aligned}$$
(1.12)

In order to get the main result, we require the Hausdorff–Young inequality in the following lemma.

Lemma 1.1

([11])

For any \(k,\alpha \geq 0,c>0\), a constant \(C>0\) exist in such a way that \(\forall t\geq 0\) the following estimate hold:

$$\begin{aligned} \int _{ \vert \imath \vert \leq 1} \vert \imath \vert ^{k}e^{-c \vert \imath \vert ^{\alpha}t} \,d\imath \leq C(1+t)^{-(k+n)/\alpha},\quad \imath \in \mathbb{R}^{n}. \end{aligned}$$
(1.13)

2 Energy method and decay estimates

We will obtain a decay estimate of the Fourier image of the solution for problem (1.11)–(1.12) in this section. This approach enables us to provide the decay rate of the solution in the energy space by utilising Plancherel’s theorem along with some integral estimates, such as Lemma (1.1). Using the energy approach in Fourier space, we create the proper Lyapunov functionals for this problem. Lastly, we prove our major finding.

2.1 The energy method in the Fourier space

.

Now, we introduce the new variables to construct the Lyapunov functional in the Fourier space

$$\begin{aligned} \begin{aligned} &r=(\varsigma _{x}-\hbar -l\Im ),\qquad g=\varsigma _{t},\qquad v=a \hbar _{x},\qquad w=\hbar _{t} \\ &\phi =k_{0}(\Im _{x}-l\varsigma ),\qquad \varpi =\Im _{t}, \qquad\vartheta =\upsilon _{t},\qquad \sigma =\upsilon _{x}. \end{aligned} \end{aligned}$$
(2.1)

Then, the system (1.11) takes the following form

$$\begin{aligned} \textstyle\begin{cases} r_{t}-g_{x}+w+l\varpi =0, \\ g_{t}-r_{x}-k_{0}l\phi =0, \\ v_{t}-ay_{x}=0, \\ w_{t}-az_{x}-r+m\vartheta _{x}=0, \\ \phi _{t}-k_{0}\varpi _{x}+k_{0}lu=0, \\ \varpi _{t}-k_{0}\phi _{x}-lv+\aleph _{1}\varpi + \int _{ \wp _{1}}^{\wp _{2}}\aleph _{2}(s)\mathcal{Y} ( x, 1,s,t )\,ds=0, \\ \vartheta _{t}-k_{1}\sigma _{x}+\beta w_{x}-k_{2}\vartheta _{xx}=0, \\ \sigma _{t}-\vartheta _{x}=0, \\ s\mathcal{Y}_{t}+\mathcal{Y}_{\jmath}=0, \end{cases}\displaystyle \end{aligned}$$
(2.2)

with initial conditions

$$\begin{aligned} (r,g,v,w,\phi,\varpi,\vartheta,\sigma,\mathcal{Y}) (x,0)=(r_{0},g_{0},v_{0},w_{0}, \phi _{0},\varpi _{0},\vartheta _{0},\sigma _{0},f_{0}),\quad x\in \mathbb{R}, \end{aligned}$$
(2.3)

where

$$\begin{aligned} &r_{0}=(\varsigma _{0,x}-\hbar _{0}-l\Im _{0}),\qquad g_{0}=\varsigma _{1}, \qquad v_{0}=a \hbar _{0,x}, \qquad w_{0}=\hbar _{1}, \\ &\phi _{0}=k_{0}(\Im _{0,x}-l\varsigma _{0}), \qquad \varpi _{0}=\Im _{1},\qquad \vartheta _{0}=\upsilon _{1},\qquad \sigma _{0}=\upsilon _{0,x}. \end{aligned}$$

Hence, the problem (2.2)–(2.3) is written as

$$\begin{aligned} \textstyle\begin{cases} Z_{t}+\mathcal{A}Z_{x}+\mathcal{L}Z=\mathcal{B}Z_{xx}, \\ Z(x,0)=Z_{0}(x), \end{cases}\displaystyle \end{aligned}$$
(2.4)

with \(Z=(r,g,v,w,\phi,\varpi,\vartheta,\sigma,\mathcal{Y})^{T}, Z_{0}=(r_{0},g_{0},v_{0},w_{0},\phi _{0},\varpi _{0},\vartheta _{0}, \sigma _{0},f_{0})\) and

$$\begin{aligned} \begin{aligned} &\mathcal{A}Z= \begin{pmatrix} -g \\ -r \\ -ay \\ -az+m\vartheta \\ -k_{0}\varpi \\ -k_{0}\phi \\ -k_{1}\sigma +\beta w \\ -\vartheta \\ 0 \end{pmatrix},\qquad \mathcal{L}Z= \begin{pmatrix} w+l\varpi \\ -k_{0}l\phi \\ 0 \\ r \\ k_{0}lu \\ -lv+\aleph _{1}\varpi +\int _{\wp _{1}}^{\wp _{2}}\aleph _{2}(s) \mathcal{Y} ( x, 1,s,t )\,ds \\ 0 \\ 0 \\ \frac{1}{s}\mathcal{Y}_{\jmath} \end{pmatrix},\\ &\mathcal{B}Z= \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ k_{2}\vartheta \\ 0 \\ 0 \end{pmatrix}. \end{aligned} \end{aligned}$$
(2.5)

Utilizing the Fourier transform to (2.4), we get

$$\begin{aligned} \textstyle\begin{cases} \widehat{Z}_{t}+i\imath \mathcal{A}\widehat{Z}+\mathcal{L}\widehat{Z}=- \imath ^{2}\mathcal{B}\widehat{Z}, \\ \widehat{Z}(\imath,0)=\widehat{Z}_{0}(\imath ), \end{cases}\displaystyle \end{aligned}$$
(2.6)

where \(\widehat{Z}(\imath,t)=(\widehat{r},\widehat{g},\widehat{v}, \widehat{w},\widehat{\phi},\widehat{\varpi},\widehat{\vartheta}, \widehat{\sigma},\widehat{\mathcal{Y}})^{T}(\imath,t)\). The equation (2.6)1 can be stated as

$$\begin{aligned} \textstyle\begin{cases} \widehat{r}_{t}-i\imath \widehat{g}+\widehat{w}+l\widehat{\varpi}=0, \\ \widehat{g}_{t}-i\imath \widehat{r}-k_{0}l\widehat{\phi}=0, \\ \widehat{v}_{t}-ai\imath \widehat{w}=0, \\ \widehat{w}_{t}-ai\imath \widehat{v}-\widehat{r}+mi\imath \widehat{\vartheta}=0, \\ \widehat{\phi}_{t}-k_{0}i\imath \widehat{\varpi}+k_{0}l\widehat{g}=0, \\ \widehat{\varpi}_{t}-k_{0}i\imath \widehat{\phi}-l\widehat{r}+\aleph _{1} \widehat{\varpi}+ \int _{\wp _{1}}^{\wp _{2}}\aleph _{2}(s) \widehat{\mathcal{Y}} (\imath, 1,s,t )\,ds=0, \\ \widehat{\vartheta}_{t}-k_{1}i\imath \widehat{\sigma}+\beta \widehat{w}+\imath ^{2}k_{2}\widehat{\vartheta}=0, \\ \widehat{\sigma}_{t}-i\imath \widehat{\vartheta}=0, \\ s\widehat{\mathcal{Y}}_{t}+\widehat{\mathcal{Y}}_{\jmath}=0. \end{cases}\displaystyle \end{aligned}$$
(2.7)

Lemma 2.1

Suppose that (1.8) holds. Assume that \(\widehat{Z}(\imath,t)\) is the solution of (2.6), then the energy functional \(\widehat{V}(\imath,t)\) is stated as

$$\begin{aligned} \widehat{V}(\imath,t)={}&\frac{\beta}{2} \biggl\{ \vert \widehat{r} \vert ^{2}+ \vert \widehat{g} \vert ^{2}+ \vert \widehat{v} \vert ^{2}+ \vert \widehat{w} \vert ^{2} + \vert \widehat{\phi} \vert ^{2}+ \vert \widehat{\varpi} \vert ^{2}+\frac{m}{\beta} \vert \widehat{\vartheta} \vert ^{2}+\frac{mk_{1}}{\beta} \vert \widehat{\sigma} \vert ^{2} \biggr\} \\ &{}+\frac{\beta}{2} \int _{0}^{1} \int _{ \wp _{1}}^{\wp _{2}}s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath, \end{aligned}$$
(2.8)

satisfies

$$\begin{aligned} \frac{d\widehat{V}(\imath,t)}{dt}\leq - C_{1} \vert \widehat{\varpi} \vert ^{2}-k_{2}m\imath ^{2} \vert \widehat{\vartheta} \vert ^{2}\leq 0, \end{aligned}$$
(2.9)

where \(C_{1}=\beta (\aleph _{1}-\int _{\wp _{1}}^{\wp _{2}}\vert \aleph _{2}(s)\vert \,ds )>0\).

Proof

First of all, multiplying (2.7)1,2,3,4,5,6 by \(\beta \overline{\widehat{r}},\beta \overline{\widehat{g}},\beta \overline{\widehat{v}},\beta \overline{\widehat{w}},\beta \overline{\widehat{\phi}}\), and \(\beta \overline{\widehat{\varpi}}\), respectively. Further, multiplying (2.7)7,8 by \(m\overline{\widehat{\vartheta}}\) and \(k_{1}m\overline{\widehat{\sigma}}\). Then by adding these equalities and taking the real part, we obtain

$$\begin{aligned} &\frac{\beta}{2}\frac{d}{dt} \biggl[ \vert \widehat{r} \vert ^{2}+ \vert \widehat{g} \vert ^{2}+ \vert \widehat{v} \vert ^{2}+ \vert \widehat{w} \vert ^{2} + \vert \widehat{\phi} \vert ^{2}+ \vert \widehat{\varpi} \vert ^{2}+\frac{m}{\beta} \vert \widehat{\vartheta} \vert ^{2}+\frac{mk_{1}}{\beta} \vert \widehat{\sigma} \vert ^{2} \biggr] \,dx \\ &\quad{}+k_{2}m\imath ^{2} \vert \widehat{\vartheta} \vert ^{2}+\beta \aleph _{1} \vert \widehat{\varpi} \vert ^{2}+\Re e \biggl\{ \beta \int _{\wp _{1}}^{ \wp _{2}} \aleph _{2}(s) \overline{ \widehat{\varpi}} \widehat{\mathcal{Y}} ( \imath, 1, s, t )\,ds \biggr\} =0. \end{aligned}$$
(2.10)

In second step, by multiplying (2.7)9 by \(\overline{\widehat{\mathcal{Y}}}\vert \aleph _{2}(s)\vert \) and integrating the result over \((0, 1)\times (\wp _{1}, \wp _{2})\)

$$\begin{aligned} &\frac{d}{dt }\frac{\beta}{2} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}}s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath, \jmath, s, t) \bigr\vert ^{2}\,ds \,d\jmath \\ &\quad=-\frac{\beta }{2 } \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \frac{d}{ \,d\jmath} \bigl\vert \widehat{\mathcal{Y}}(\imath, \jmath, s, t) \bigr\vert ^{2}\,ds \,d\jmath \\ &\quad =\frac{\beta }{2 } \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl( \bigl\vert \widehat{\mathcal{Y}}(\imath, 0, s, t) \bigr\vert ^{2} - \bigl\vert \widehat{\mathcal{Y}}(\imath, 1, s, t) \bigr\vert ^{2} \bigr)\,ds \\ &\quad=\frac{\beta}{2 } \biggl( \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \,ds \biggr) \vert \widehat{\varpi} \vert ^{2}- \frac{\beta}{2} \int _{ \wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath,1, s, t) \bigr\vert ^{2} \,ds, \end{aligned}$$
(2.11)

utilizing Young’s inequality, we get

$$\begin{aligned} &\Re e \biggl\{ \beta \int _{\wp _{1}}^{\wp _{2}} \aleph _{2}(s) \overline{ \widehat{\varpi}}\widehat{\mathcal{Y}} ( \imath, 1, s, t )\,ds \biggr\} \\ &\quad\leq \frac{\beta}{2 } \biggl( \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \,ds \biggr) \vert \widehat{\varpi} \vert ^{2}+ \frac{\beta}{2} \int _{ \wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath,1, s, t) \bigr\vert ^{2} \,ds, \end{aligned}$$
(2.12)

by substituting (2.11) and (2.12) into (2.10), we find

$$\begin{aligned} \frac{d\widehat{V}(\imath,t)}{dt}\leq - \beta \biggl(\aleph _{1}- \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \,ds \biggr) \vert \widehat{\varpi} \vert ^{2}-k_{2}m \imath ^{2} \vert \widehat{\vartheta} \vert ^{2}, \end{aligned}$$

then, by (1.8), \(\exists C_{1}=\beta (\aleph _{1}-\int _{\wp _{1}}^{\wp _{2}}\vert \aleph _{2}(s)\vert \,ds)>0\) such that

$$\begin{aligned} \frac{d\widehat{V}(\imath,t)}{dt}\leq - C_{1} \vert \widehat{\varpi} \vert ^{2}-k_{2}m\imath ^{2} \vert \widehat{\vartheta} \vert ^{2}\leq 0. \end{aligned}$$
(2.13)

Hence, we get the required result. □

The following Lemma is required in order to get the main result.

Lemma 2.2

The functional

$$\begin{aligned} \mathcal{D}_{1}(\imath,t):= \Re e \bigl\{ i\imath ( \widehat{\varpi} \overline{\widehat{\phi}} +l\widehat{\phi} \overline{\widehat{w}} ) \bigr\} , \end{aligned}$$
(2.14)

satisfies the following for any \(\varepsilon _{1}>0\)

$$\begin{aligned} \frac{d\mathcal{D}_{1}(\imath,t)}{dt}\leq {}& {-}\frac{k_{0}}{2}\imath ^{2} \vert \widehat{\phi} \vert ^{2} +2\varepsilon _{1} \frac{\imath ^{2}}{1+\imath ^{2}} \vert \widehat{g} \vert ^{2}+c( \varepsilon _{1}) \bigl(1+\imath ^{2}\bigr) \vert \widehat{\varpi} \vert ^{2} \\ &{}+c(\varepsilon _{1}) \bigl(1+\imath ^{2}\bigr) \vert \widehat{w} \vert ^{2} +c \vert \widehat{\vartheta} \vert ^{2} \\ &{}+c \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath,1, s, t) \bigr\vert ^{2} \,ds. \end{aligned}$$
(2.15)

Proof

By differentiating \(\mathcal{D}_{1}\) and using (2.7), we get

$$\begin{aligned} \frac{d\mathcal{D}_{1}(\imath,t)}{dt}={}&\Re e \{i\imath \widehat{\varpi}_{t}\overline{ \widehat{\phi}} -i\imath \widehat{\phi}_{t} \overline{\widehat{\varpi}} +i\imath l\widehat{\phi}_{t} \overline{\widehat{w}} -i\imath l \widehat{w}_{t} \overline{\widehat{\phi}} \} \\ ={}&{-}k_{0}\imath ^{2} \vert \widehat{\phi} \vert ^{2}+k_{0}\imath ^{2} \vert \widehat{\varpi} \vert ^{2}-\Re e \{i\aleph _{1}\imath \widehat{\varpi} \overline{\widehat{\phi}} \}+\Re e \bigl\{ al \imath ^{2}\widehat{v} \overline{\widehat{\phi}} \bigr\} \\ &{}+\Re e \{ik_{0}l\imath \widehat{g}\overline{\widehat{\varpi}} \}- \Re e \bigl\{ k_{0}l\imath ^{2}\widehat{\varpi} \overline{ \widehat{w}} \bigr\} -\Re e \bigl\{ ik_{0}l^{2}\imath \widehat{g}\overline{\widehat{w}} \bigr\} \\ &{}-\Re e \bigl\{ ml\imath ^{2}\widehat{\vartheta} \overline{\widehat{ \phi}} \bigr\} -\Re e \biggl\{ i\imath \int _{\wp _{1}}^{ \wp _{2}} \aleph _{2}(s) \overline{ \widehat{\phi}} \widehat{\mathcal{Y}}(\imath,1, s, t) \,ds \biggr\} . \end{aligned}$$
(2.16)

The terms in the RHS of (2.16) are obtained by utilizing the Young’s inequality. For any \(\varepsilon _{1},\delta _{1},\delta _{2}>0\), we have

$$\begin{aligned} \begin{aligned} &{-}\Re e \{i\aleph _{1}\imath \widehat{\varpi} \overline{\widehat{ \phi}} \}\leq \delta _{1}\imath ^{2} \vert \widehat{ \phi} \vert ^{2} +c(\delta _{1}) \vert \widehat{\varpi} \vert ^{2}, \\ &\Re e \{ik_{0}l\imath \widehat{g}\overline{\widehat{\varpi}} \}\leq \varepsilon _{1}\frac{\imath ^{2}}{1+\imath ^{2}} \vert \widehat{g} \vert ^{2} +c(\varepsilon _{1}) \bigl(1+\imath ^{2} \bigr) \vert \widehat{\varpi} \vert ^{2}, \\ &{-}\Re e \{lk_{0}\imath \widehat{\varpi}\overline{\widehat{w}} \}\leq c\imath ^{2} \vert \widehat{w} \vert ^{2} +c \vert \widehat{\varpi} \vert ^{2}, \\ &{-}\Re e \bigl\{ ik_{0}l^{2}\imath \widehat{g}\overline{ \widehat{w}} \bigr\} \leq \varepsilon _{1}\frac{\imath ^{2}}{1+\imath ^{2}} \vert \widehat{g} \vert ^{2} +c(\varepsilon _{1}) \bigl(1+\imath ^{2}\bigr) \vert \widehat{w} \vert ^{2}, \\ &\Re e \bigl\{ al\imath ^{2}\widehat{v}\overline{\widehat{\phi}} \bigr\} \leq \delta _{1}\imath ^{2} \vert \widehat{\phi} \vert ^{2} +c( \delta _{1})\imath ^{2} \vert \widehat{v} \vert ^{2}, \\ &{-}\Re e \bigl\{ ml\imath ^{2}\widehat{\vartheta} \overline{\widehat{ \phi}} \bigr\} \leq \delta _{1}\imath ^{2} \vert \widehat{\phi} \vert ^{2} +c(\delta _{1})\imath ^{2} \vert \widehat{\vartheta} \vert ^{2}, \\ &{-}\Re e \biggl\{ i\imath \int _{\wp _{1}}^{\wp _{2}} \aleph _{2}(s) \overline{ \widehat{\phi}}\widehat{\mathcal{Y}}(\imath,1, s, t) \,ds \biggr\} \\ &\quad\leq \delta _{2}\aleph _{1}\imath ^{2} \vert \widehat{\phi} \vert ^{2} +c(\delta _{2}) \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath,1, s, t) \bigr\vert ^{2} \,ds. \end{aligned} \end{aligned}$$
(2.17)

Inserting the above estimates (2.17) into (2.16) and by letting \(\delta _{1}=\frac{k_{0}}{12},\delta _{2}=\frac{k_{0}}{4\aleph _{1}}\), we get the required (2.15). □

Lemma 2.3

The functional

$$\begin{aligned} \mathcal{D}_{2}(\imath,t):= \Re e \bigl\{ i\imath (ak_{1} \widehat{\vartheta}\overline{\widehat{\sigma}} +a\beta \widehat{\vartheta} \overline{\widehat{w}} +2k_{1}\widehat{v} \overline{\widehat{\sigma}} ) \bigr\} , \end{aligned}$$
(2.18)

satisfies the following for any \(\varepsilon _{2},\varepsilon _{3}>0\)

$$\begin{aligned} \frac{d\mathcal{D}_{2}(\imath,t)}{dt}\leq {}& {-}\frac{ak_{1}^{2}}{2} \imath ^{2} \vert \widehat{\sigma} \vert ^{2} -\frac{a\beta ^{2}}{2} \imath ^{2} \vert \widehat{w} \vert ^{2}+\varepsilon _{2}\imath ^{2} \vert \widehat{r} \vert ^{2} +\varepsilon _{3}\imath ^{2} \vert \widehat{v} \vert ^{2} \\ &{} +c(\varepsilon _{2},\varepsilon _{3}) \bigl(1+\imath ^{2}+\imath ^{4}\bigr) \vert \widehat{\vartheta} \vert ^{2}. \end{aligned}$$
(2.19)

Proof

By differentiating \(\mathcal{D}_{2}\) and using (2.7), we get

$$\begin{aligned} \frac{\mathcal{D}_{2}(\imath,t)}{dt}={}&\Re e \{i\imath ak_{1} \widehat{ \vartheta}_{t}\overline{\widehat{\sigma}} -i\imath ak_{1} \widehat{\sigma}_{t}\overline{\widehat{\vartheta}} -i\imath \beta a \widehat{\vartheta}_{t}\overline{\widehat{w}} +i\imath \beta a \widehat{w}_{t}\overline{\widehat{\vartheta}} \} \\ &{}+\Re e \{2i\imath \beta k_{1}\widehat{v}_{t} \overline{ \widehat{\sigma}} -2i\imath \beta k_{1}\widehat{\sigma}_{t} \overline{\widehat{v}} \} \\ ={}&{-}ak_{1}\imath ^{2} \vert \widehat{\sigma} \vert ^{2}-a\beta ^{2} \imath ^{2} \vert \widehat{w} \vert ^{2}+a(k_{1}+m\beta )\imath ^{2} \vert \widehat{\vartheta} \vert ^{2} \\ &{}+\Re e \bigl\{ \beta \imath ^{2}\bigl(2k_{1}-a^{2} \bigr)\widehat{v} \overline{\widehat{\vartheta}} \bigr\} +\Re e \{ia\beta \imath \widehat{r}\overline{\widehat{\vartheta}} \} \\ &{}-\Re e \bigl\{ iak_{1}k_{2}\imath ^{3}\widehat{ \vartheta} \overline{\widehat{\sigma}} \bigr\} +\Re e \bigl\{ ia\beta k_{2}\imath ^{3} \widehat{\vartheta}\overline{\widehat{w}} \bigr\} . \end{aligned}$$
(2.20)

The terms in the RHS of (2.20) are obtained by utilizing Young’s inequality. Next, for any \(\varepsilon _{2},\varepsilon _{3},\delta _{3},\delta _{4}>0\), we can find

$$\begin{aligned} \begin{aligned} &\Re e \{ia\beta \imath \widehat{r}\overline{\widehat{\vartheta}} \}\leq \varepsilon _{2}\imath ^{2} \vert \widehat{r} \vert ^{2} +c( \varepsilon _{2}) \vert \widehat{\vartheta} \vert ^{2}, \\ &\Re e \bigl\{ \beta \imath ^{2}\bigl(2k_{1}-a^{2} \bigr)\widehat{v} \overline{\widehat{\vartheta}} \bigr\} \leq \varepsilon _{3}\imath ^{2} \vert \widehat{v} \vert ^{2} +c(\varepsilon _{3}) \vert \widehat{\vartheta} \vert ^{2}, \\ &{-}\Re e \bigl\{ iak_{1}k_{2}\imath ^{3}\widehat{ \vartheta} \overline{\widehat{\sigma}} \bigr\} \leq \delta _{3}\imath ^{2} \vert \widehat{\sigma} \vert ^{2} +c(\delta _{3})\imath ^{4} \vert \widehat{\vartheta} \vert ^{2}, \\ &\Re e \bigl\{ ia\beta k_{2}\imath ^{3}\widehat{\vartheta} \overline{\widehat{w}} \bigr\} \leq \delta _{4}\imath ^{2} \vert \widehat{w} \vert ^{2} +c(\delta _{4})\imath ^{4} \vert \widehat{\vartheta} \vert ^{2}. \end{aligned} \end{aligned}$$
(2.21)

By substituting (2.21) into (2.20) and letting \(\delta _{3}=\frac{ak_{1}^{2}}{2},\delta _{4}=\frac{a\beta ^{2}}{2}\), we get (2.19). □

Lemma 2.4

The functional

$$\begin{aligned} \mathcal{D}_{3}(\imath,t):= \Re e \{\widehat{\phi} \overline{ \widehat{g}} \}, \end{aligned}$$
(2.22)

satisfies the below for any \(\varepsilon _{4}>0\)

$$\begin{aligned} \frac{d\mathcal{D}_{3}(\imath,t)}{dt}\leq -\frac{k_{0}l^{2}}{2} \vert \widehat{g} \vert ^{2} +\varepsilon _{4} \vert \widehat{r} \vert ^{2}+c \imath ^{2} \vert \widehat{\varpi} \vert ^{2}+c(\varepsilon _{4}) \bigl(1+ \imath ^{2} \bigr) \vert \widehat{\phi} \vert ^{2}. \end{aligned}$$
(2.23)

Proof

By differentiating \(\mathcal{D}_{3}\) and using (2.7), we have

$$\begin{aligned} \frac{\mathcal{D}_{3}(\imath,t)}{dt}={}&\Re e \{ \widehat{\phi}_{t} \overline{ \widehat{g}} + \widehat{g}_{t}\overline{\widehat{\phi}} \} \\ ={}&-k_{0}l \vert \widehat{g} \vert ^{2}+k_{0}l \vert \widehat{\phi} \vert ^{2} \\ &{}+\Re e \{ik_{0}\imath \widehat{\varpi}\overline{\widehat{g}} \}+\Re e \{i\imath \widehat{r}\overline{\widehat{\phi}} \}. \end{aligned}$$
(2.24)

The last two terms in the RHS of (2.24) are obtained by Young’s inequality, which we solve for any \(\varepsilon _{4},\delta _{5}>0\)

$$\begin{aligned} \begin{aligned} &\Re e \{ik_{0}\imath \widehat{\varpi}\overline{\widehat{g}} \}\leq \delta _{5} \vert \widehat{g} \vert ^{2} +c(\delta _{5}) \imath ^{2} \vert \widehat{\varpi} \vert ^{2}, \\ &\Re e \{i\imath \widehat{r}\overline{\widehat{\phi}} \}\leq \varepsilon _{4} \vert \widehat{r} \vert ^{2} +c(\varepsilon _{4}) \imath ^{2} \vert \widehat{\phi} \vert ^{2}. \end{aligned} \end{aligned}$$
(2.25)

By substituting (2.25) into (2.24) and letting \(\delta _{5}=\frac{k_{0}l^{2}}{2}\), we obtained (2.23). □

Next, we have the following lemma.

Lemma 2.5

The functional

$$\begin{aligned} \mathcal{D}_{4}(\imath,t):=al\mathcal{F}_{1}(\imath,t)- \imath ^{2} \mathcal{F}_{2}(\imath,t), \end{aligned}$$
(2.26)

where

$$\begin{aligned} \mathcal{F}_{1}(\imath,t):= \Re e \bigl\{ i\imath (l\widehat{w} \overline{\widehat{v}} +\widehat{v}\overline{\widehat{\varpi}} ) \bigr\} \quad \textit{and}\quad \mathcal{F}_{2}(\imath,t):= \Re e \bigl\{ (\widehat{w} \overline{\widehat{r}} +a\widehat{g}\overline{\widehat{v}} ) \bigr\} , \end{aligned}$$
(2.27)

satisfies

  1. (1)

    For \(a=1\). Then,

    $$\begin{aligned} \frac{d\mathcal{D}_{4}(\imath,t)}{dt}\leq {}& {-}\frac{a^{2}l^{2}}{2} \imath ^{2} \vert \widehat{v} \vert ^{2} -\frac{1}{2}\imath ^{2} \vert \widehat{r} \vert ^{2}+c \vert \widehat{\varpi} \vert ^{2} +\bigl(1+a^{2}l^{2}\bigr) \imath ^{2} \vert \widehat{w} \vert ^{2} \\ &{} +c\bigl(\imath ^{2}+\imath ^{4}\bigr) \vert \widehat{ \vartheta} \vert ^{2}+c \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath,1, s, t) \bigr\vert ^{2} \,ds. \end{aligned}$$
    (2.28)
  2. (2)

    For \(a\neq 1\). Then, for any \(\varepsilon _{5}>0\)

    $$\begin{aligned} \frac{d\mathcal{D}_{4}(\imath,t)}{dt}\leq {}& {-}\frac{a^{2}l^{2}}{2} \imath ^{2} \vert \widehat{v} \vert ^{2} -\frac{1}{2}\imath ^{2} \vert \widehat{r} \vert ^{2}+\varepsilon _{5} \frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \vert \widehat{g} \vert ^{2} +c( \varepsilon _{5})\imath ^{2}\bigl(1+\imath ^{2} \bigr)^{2} \vert \widehat{w} \vert ^{2} \\ &{}+c\bigl(1+\imath ^{2}\bigr) \vert \widehat{\varpi} \vert ^{2} +c\bigl(\imath ^{2}+ \imath ^{4}\bigr) \vert \widehat{\vartheta} \vert ^{2} \\ &{}+c \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath,1, s, t) \bigr\vert ^{2} \,ds. \end{aligned}$$
    (2.29)

Proof

Firstly, by differentiating \(\mathcal{F}_{1}, \mathcal{F}_{2}\) and using (2.7), we get

$$\begin{aligned} \frac{d\mathcal{F}_{1}(\imath,t)}{dt}={}&\Re e \{i\imath l \widehat{w}_{t}\overline{ \widehat{v}} -i\imath l\widehat{v}_{t} \overline{\widehat{w}} +i\imath \widehat{v}_{t} \overline{\widehat{\varpi}} -i\imath \widehat{ \varpi}_{t} \overline{\widehat{v}} \} \\ ={}&{-}al\imath ^{2} \vert \widehat{v} \vert ^{2}+al\imath ^{2} \vert \widehat{w} \vert ^{2}+\Re e \{i\aleph _{1}\imath \widehat{\varpi} \overline{\widehat{v}} \}-\Re e \bigl\{ a \imath ^{2}\widehat{w} \overline{\widehat{\varpi}} \bigr\} \\ &{}+\Re e \bigl\{ k_{0}\imath ^{2}\widehat{\phi}\overline{ \widehat{v}} \bigr\} +\Re e \bigl\{ ml\imath ^{2}\widehat{\vartheta} \overline{\widehat{v}} \bigr\} \\ &{}+\Re e \biggl\{ i\imath \int _{\wp _{1}}^{\wp _{2}} \aleph _{2}(s) \overline{ \widehat{v}}\widehat{\mathcal{Y}}(\imath,1, s, t) \,ds \biggr\} , \end{aligned}$$
(2.30)

and

$$\begin{aligned} \frac{d\mathcal{F}_{2}(\imath,t)}{dt}={}&\Re e \{\widehat{w}_{t} \overline{\widehat{r}} + \widehat{r}_{t}\overline{\widehat{w}} +a \widehat{v}_{t} \overline{\widehat{g}} +a\widehat{g}_{t} \overline{\widehat{v}} \} \\ ={}&{-} \vert \widehat{w} \vert ^{2}+ \vert \widehat{r} \vert ^{2}+\Re e \bigl\{ i\bigl(a^{2}-1\bigr)\imath \widehat{w} \overline{\widehat{g}} \bigr\} -\Re e \{im\imath \widehat{\vartheta}\overline{ \widehat{r}} \} \\ &{}-\Re e \{l\widehat{\varpi}\overline{\widehat{w}} \}+\Re e \{alk_{0} \widehat{\phi}\overline{\widehat{v}} \}. \end{aligned}$$
(2.31)

Now, differentiating \(\mathcal{D}_{4}\) and by (2.30) and (2.31), we have

$$\begin{aligned} \frac{d\mathcal{D}_{4}(\imath,t)}{dt}={}&{-}a^{2}l^{2}\imath ^{2} \vert \widehat{v} \vert ^{2}-\imath ^{2} \vert \widehat{r} \vert ^{2}+\bigl(1+a^{2}l^{2}\bigr) \imath ^{2} \vert \widehat{w} \vert ^{2} +\Re e \{ial \aleph _{1} \imath \widehat{\varpi}\overline{\widehat{v}} \} \\ &{}+\Re e \bigl\{ i\bigl(1-a^{2}\bigr)\imath ^{3}\widehat{w} \overline{\widehat{g}} \bigr\} +\Re e \bigl\{ im\imath ^{3}\widehat{ \vartheta} \overline{\widehat{r}} \bigr\} +\Re e \bigl\{ l\bigl(1-a^{2} \bigr)\imath ^{2} \widehat{\varpi}\overline{\widehat{w}} \bigr\} \\ &{}+\Re e \bigl\{ aml^{2}\imath ^{2}\widehat{\vartheta} \overline{\widehat{v}} \bigr\} +\Re e \biggl\{ ial\imath \int _{\wp _{1}}^{ \wp _{2}} \aleph _{2}(s) \overline{ \widehat{v}}\widehat{\mathcal{Y}}( \imath,1, s, t) \,ds \biggr\} . \end{aligned}$$
(2.32)

At this point, we discuss two cases:

Case 1. \((a=1)\).

In this case, by applying the Young’s inequality to the terms on the RHS of (2.32). Then, for any \(\delta _{6},\delta _{7},\delta _{8}>0\), we get

$$\begin{aligned} \begin{aligned} &\Re e \{ial\aleph _{1}\imath \widehat{\varpi} \overline{\widehat{v}} \}\leq \delta _{6}\imath ^{2} \vert \widehat{v} \vert ^{2} +c(\delta _{6}) \vert \widehat{\varpi} \vert ^{2}, \\ &\Re e \bigl\{ im\imath ^{3}\widehat{\vartheta}\overline{\widehat{r}} \bigr\} \leq \delta _{7}\imath ^{2} \vert \widehat{r} \vert ^{2} +c( \delta _{7})\imath ^{4} \vert \widehat{\vartheta} \vert ^{2}, \\ &\Re e \bigl\{ aml^{2}\imath ^{2}\widehat{\vartheta} \overline{\widehat{v}} \bigr\} \leq \delta _{6}\imath ^{2} \vert \widehat{v} \vert ^{2} +c(\delta _{6})\imath ^{2} \vert \widehat{\vartheta} \vert ^{2}, \\ &\Re e \biggl\{ i\imath \int _{\wp _{1}}^{\wp _{2}} \aleph _{2}(s) \overline{ \widehat{v}}\widehat{\mathcal{Y}}(\imath,1, s, t) \,ds \biggr\} \\ &\quad\leq \delta _{8}\aleph _{1}\imath ^{2} \vert \widehat{v} \vert ^{2} +c(\delta _{8}) \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath,1, s, t) \bigr\vert ^{2} \,ds. \end{aligned} \end{aligned}$$
(2.33)

Inserting the above estimates of (2.33) into (2.32).

Finally, by letting \(\delta _{6}=\frac{a^{2}l^{2}}{8},\delta _{7}=\frac{1}{2},\delta _{8}= \frac{a^{2}l^{2}}{4\aleph _{1}}\), we obtained (2.28).

Case 2. \((a\neq 1)\).

In this case, using the Young’s inequality to the terms on the RHS of (2.32) for any \(\varepsilon _{5},\delta _{9},\delta _{10},\delta _{11}>0 \) gives

$$\begin{aligned} \begin{aligned} &\Re e \{ial\aleph _{1}\imath \widehat{\varpi} \overline{\widehat{v}} \}\leq \delta _{9}\imath ^{2} \vert \widehat{v} \vert ^{2} +c(\delta _{9}) \vert \widehat{\varpi} \vert ^{2}, \\ &\Re e \bigl\{ i\bigl(1-a^{2}\bigr)\imath ^{3}\widehat{w} \overline{\widehat{g}} \bigr\} \leq \varepsilon _{5} \frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \vert \widehat{g} \vert ^{2} +c( \varepsilon _{5})\imath ^{2}\bigl(1+\imath ^{2} \bigr)^{2} \vert \widehat{w} \vert ^{2}, \\ &\Re e \bigl\{ im\imath ^{3}\widehat{\vartheta}\overline{\widehat{r}} \bigr\} \leq \delta _{10}\imath ^{2} \vert \widehat{r} \vert ^{2} +c( \delta _{10})\imath ^{4} \vert \widehat{\vartheta} \vert ^{2}, \\ &\Re e \bigl\{ l\bigl(1-a^{2}\bigr)\imath ^{2}\widehat{\varpi} \overline{\widehat{w}} \bigr\} \leq c\imath ^{2} \vert \widehat{ \varpi} \vert ^{2} +c\imath ^{2} \vert \widehat{w} \vert ^{2}, \\ &\Re e \bigl\{ aml^{2}\imath ^{2}\widehat{\vartheta} \overline{\widehat{v}} \bigr\} \leq \delta _{9}\imath ^{2} \vert \widehat{v} \vert ^{2} +c(\delta _{9})\imath ^{2} \vert \widehat{\vartheta} \vert ^{2}, \\ &\Re e \biggl\{ i\imath \int _{\wp _{1}}^{\wp _{2}} \aleph _{2}(s) \overline{ \widehat{v}}\widehat{\mathcal{Y}}(\imath,1, s, t) \,ds \biggr\} \\ &\quad\leq \delta _{11}\aleph _{1}\imath ^{2} \vert \widehat{v} \vert ^{2} +c(\delta _{11}) \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}}(\imath,1, s, t) \bigr\vert ^{2} \,ds. \end{aligned} \end{aligned}$$
(2.34)

Inserting (2.34) into (2.32), and letting \(\delta _{9}=\frac{a^{2}l^{2}}{8},\delta _{10}=\frac{1}{2},\delta _{11}= \frac{a^{2}l^{2}}{4\aleph _{1}}\), we get (2.29. The proof of Lemma 2.5 is completed. □

Now, introducing the following functional.

Lemma 2.6

The functional

$$\begin{aligned} \mathcal{D}_{5} (\imath, t ):= \int _{0}^{1} \int _{\wp _{1}}^{ \wp _{2}} s e^{-s\jmath } \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath, \end{aligned}$$

satisfies

$$\begin{aligned} \frac{d\mathcal{D}_{5} (\imath, t )}{dt} \leq {}&{ -}\zeta _{1} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath +\aleph _{1} \vert \widehat{\varpi} \vert ^{2} \\ &{}-\zeta _{1} \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, 1, s, t ) \bigr\vert ^{2} \,ds , \end{aligned}$$
(2.35)

where \(\zeta _{1}>0\).

Proof

By differentiating \(\mathcal{D}_{5}\) with respect to t and utilizing (2.7)9, we have

$$\begin{aligned} \frac{d\mathcal{D}_{5} (\imath, t )}{dt} ={}&{-} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s e^{-s\jmath} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath \\ &{}- \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl[e^{-s} \bigl\vert \widehat{\mathcal{Y}} ( \imath, 1, s,t ) \bigr\vert ^{2}- \bigl\vert \widehat{\mathcal{Y}} ( \imath, 0, s, t ) \bigr\vert ^{2}\bigr] \,ds. \end{aligned}$$

Using \(\mathcal{Y}(\imath, 0, s, t)=\Im _{t}(\imath, t)=\varpi \), & \(e^{-s}\leq e^{-s\jmath}\leq 1\), \(0<\jmath <1\), we have

$$\begin{aligned} \frac{d\mathcal{D}_{5} (\imath, t )}{dt} \leq {}&{-} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} se^{-s} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath \\ &{}- \int _{\wp _{1}}^{\wp _{2}} e^{-s} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, 1, s, t ) \bigr\vert ^{2} \,ds +\biggl( \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \,ds\biggr) \vert \widehat{\varpi} \vert ^{2}. \end{aligned}$$

Next, we have \(-e^{-s}\leq -e^{-\wp _{2}}\), for all \(s\in [\wp _{1}, \wp _{2}]\), since \(-e^{-s}\) is an increasing function. Assuming that \(\zeta _{1}=e^{-\wp _{2}}\) and remembering (1.8), we obtain (2.35). □

We define the Lyapunov functionals at this point

  • For \(a=1\):

    $$\begin{aligned} \mathcal{K}_{1}(\imath,t):={}&N\widehat{V}(\imath,t)+N_{1} \frac{\imath ^{4}}{(1+\imath ^{2})^{3}}\mathcal{D}_{1}(\imath,t)+N_{2} \frac{\imath ^{2}}{(1+\imath ^{2})^{2}}\mathcal{D}_{2}(\imath,t) \\ &{} +N_{3}\frac{\imath ^{6}}{(1+\imath ^{2})^{4}}\mathcal{D}_{3}( \imath,t)+N_{4} \frac{\imath ^{2}}{(1+\imath ^{2})^{2}}\mathcal{D}_{4}( \imath,t)+N_{5} \mathcal{D}_{5}(\imath,t). \end{aligned}$$
    (2.36)
  • For \(a\neq 1\):

    $$\begin{aligned} \mathcal{K}_{2}(\imath,t):={}&M\widehat{V}(\imath,t)+M_{1} \frac{\imath ^{4}}{(1+\imath ^{2})^{6}}\mathcal{D}_{1}(\imath,t)+M_{2} \frac{\imath ^{2}}{(1+\imath ^{2})^{3}}\mathcal{D}_{2}(\imath,t) \\ &{} +M_{3}\frac{\imath ^{6}}{(1+\imath ^{2})^{7}}\mathcal{D}_{3}( \imath,t)+M_{4} \frac{\imath ^{2}}{(1+\imath ^{2})^{5}}\mathcal{D}_{4}( \imath,t)+M_{5} \mathcal{D}_{5}(\imath,t), \end{aligned}$$
    (2.37)

where \(N,M,N_{i},M_{i}, i=1,\ldots,5\) are positive constants and will be selected later.

Lemma 2.7

There exist \(\mu _{i}>0,i=1,\ldots,6\) such that the functionals \(\mathcal{K}_{1}(\imath,t)\) and \(\mathcal{K}_{2}(\imath,t)\) given by (2.36) and (2.37) satisfies

  • For \(a=1\):

    $$\begin{aligned} \textstyle\begin{cases} \mu _{1}\widehat{V}(\imath,t)\leq \mathcal{K}_{1}(\imath,t)\leq \mu _{2}\widehat{V}(\imath,t), \\ \mathcal{K}_{1}'(\imath,t)\leq -\mu _{3}\jmath _{1}(\imath ) \mathcal{K}_{1}(\imath,t),\quad \forall t>0. \end{cases}\displaystyle \end{aligned}$$
    (2.38)
  • For \(a\neq 1\):

    $$\begin{aligned} \textstyle\begin{cases} \mu _{4}\widehat{V}(\imath,t)\leq \mathcal{K}_{2}(\imath,t)\leq \mu _{5}\widehat{V}(\imath,t), \\ \mathcal{K}_{2}'(\imath,t)\leq -\mu _{6}\jmath _{2}(\imath ) \mathcal{K}_{2}(\imath,t),\quad \forall t>0, \end{cases}\displaystyle \end{aligned}$$
    (2.39)

    where

    $$\begin{aligned} \jmath _{1}(\imath )=\frac{\imath ^{6}}{(1+\imath ^{2})^{4}} \quad\textit{and}\quad \jmath _{2}(\imath )=\frac{\imath ^{6}}{(1+\imath ^{2})^{7}}. \end{aligned}$$
    (2.40)

Proof

First, by differentiating (2.36) and using (2.9), (2.15), (2.19), (2.23), (2.28) and (2.35) with the fact that \(\frac{\imath ^{2}}{1+\imath ^{2}}\leq \min \{1,\imath ^{2}\}\) and \(\frac{1}{1+\imath ^{2}}\leq 1\), we have

$$\begin{aligned} \mathcal{K}_{1}'(\imath,t)\leq {}&{- }\frac{\imath ^{6}}{(1+\imath ^{2})^{4}} \biggl[\frac{k_{0}l^{2}}{2}N_{3}-2 \varepsilon _{1}N_{1} \biggr] \vert \widehat{g} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \biggl[\frac{1}{2}N_{4}- \varepsilon _{2}N_{2}-\varepsilon _{4}N_{3} \biggr] \vert \widehat{r} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \biggl[\frac{a\beta ^{2}}{2}N_{2}-c( \varepsilon _{1})N_{1}-cN_{4} \biggr] \vert \widehat{w} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \biggl[\frac{a^{2}l^{2}}{2}N_{4}- \varepsilon _{3}N_{2} \biggr] \vert \widehat{v} \vert ^{2} \\ &{}-\frac{\imath ^{6}}{(1+\imath ^{2})^{3}} \biggl[\frac{k_{0}}{2}N_{1}-c( \varepsilon _{4})N_{3} \biggr] \vert \widehat{\phi} \vert ^{2}- \frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \biggl[\frac{ak_{1}^{2}}{2}N_{2} \biggr] \vert \widehat{\sigma} \vert ^{2} \\ &{}-\imath ^{2} \bigl[mk_{2}N-cN_{1}-c(\varepsilon _{2},\varepsilon _{3})N_{2}-cN_{4} \bigr] \vert \widehat{\vartheta} \vert ^{2} \\ &{}- \bigl[C_{1}N-c(\varepsilon _{1}) N_{1}-cN_{3}-cN_{4}- \aleph _{1}N_{5} \bigr] \vert \widehat{\varpi} \vert ^{2} \\ &{}-\zeta _{1}N_{5} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath \\ &{}- [\zeta _{1}N_{5}-cN_{1}-cN_{4} ] \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, 1, s, t ) \bigr\vert ^{2} \,ds. \end{aligned}$$
(2.41)

By setting

$$\begin{aligned} \varepsilon _{1}=\frac{k_{0}l^{2} N_{3}}{8 N_{1}},\qquad \varepsilon _{2}= \frac{N_{4}}{8N_{2}},\qquad \varepsilon _{3}=\frac{a^{2}l^{2}N_{4}}{4N_{2}}, \qquad\varepsilon _{4}=\frac{N_{4}}{8N_{3}}. \end{aligned}$$

We obtain the following

$$\begin{aligned} \mathcal{K}_{1}'(\imath,t)\leq {}&{- }\frac{\imath ^{6}}{(1+\imath ^{2})^{4}} \biggl[\frac{k_{0}l^{2}}{4}N_{3} \biggr] \vert \widehat{g} \vert ^{2}- \frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \biggl[\frac{1}{4}N_{4} \biggr] \vert \widehat{r} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \biggl[\frac{a\beta ^{2}}{2}N_{2}-c(N_{1},N_{3})N_{1}-cN_{4} \biggr] \vert \widehat{w} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \biggl[\frac{a^{2}l^{2}}{4}N_{4} \biggr] \vert \widehat{v} \vert ^{2}- \frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \biggl[ \frac{ak_{1}^{2}}{2}N_{2} \biggr] \vert \widehat{\sigma} \vert ^{2} \\ &{}-\frac{\imath ^{6}}{(1+\imath ^{2})^{3}} \biggl[\frac{k_{0}}{2}N_{1}-c(N_{3},N_{4})N_{3} \biggr] \vert \widehat{\phi} \vert ^{2} \\ &{}-\imath ^{2} \bigl[mk_{2}N-cN_{1}-c(N_{2},N_{4})N_{2}-cN_{4} \bigr] \vert \widehat{\vartheta} \vert ^{2} \\ &{}- \bigl[C_{1}N-c(N_{1},N_{3})N_{1}-cN_{3}-cN_{4}- \aleph _{1}N_{5} \bigr] \vert \widehat{\varpi} \vert ^{2} \\ &{}-\zeta _{1}N_{5} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath \\ &{}- [\zeta _{1}N_{5}-cN_{1}-cN_{4} ] \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, 1, s, t ) \bigr\vert ^{2} \,ds . \end{aligned}$$
(2.42)

Next, we fix \(N_{3},N_{4}\) and choose \(N_{1}\) large enough such that

$$\begin{aligned} \frac{k_{0}}{2}N_{1}-c(N_{3},N_{4})N_{3}>0, \end{aligned}$$

then, we pick \(N_{2}\) and \(N_{5}\) large enough in such a way that

$$\begin{aligned} &\frac{a\beta ^{2}}{2}N_{2}-c(N_{1},N_{3})N_{1}-cN_{4}>0, \\ &\zeta _{1}N_{5}-cN_{1}-cN_{4}>0. \end{aligned}$$

Hence, we have

$$\begin{aligned} \mathcal{K}_{1}'(\imath,t)\leq{}& -\alpha _{0} \frac{\imath ^{6}}{(1+\imath ^{2})^{4}} \vert \widehat{g} \vert ^{2}- \alpha _{5}\frac{\imath ^{6}}{(1+\imath ^{2})^{3}} \vert \widehat{\phi} \vert ^{2}-\imath ^{2} [mk_{2}N-c ] \vert \widehat{ \vartheta} \vert ^{2} \\ &{} -\frac{\imath ^{4}}{(1+\imath ^{2})^{2}} \bigl(\alpha _{1} \vert \widehat{r} \vert ^{2}+\alpha _{2} \vert \widehat{w} \vert ^{2} + \alpha _{3} \vert \widehat{v} \vert ^{2}+\alpha _{4} \vert \widehat{\sigma} \vert ^{2} \bigr)- [C_{1}N-c ] \vert \widehat{\varpi} \vert ^{2} \\ &{} -\alpha _{6} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath. \end{aligned}$$
(2.43)

Secondly, we have

$$\begin{aligned} \bigl\vert \mathcal{K}_{1}(\imath,t)-N\widehat{V}(\imath,t) \bigr\vert ={}&N_{1}\frac{\imath ^{4}}{(1+\imath ^{2})^{3}} \bigl\vert \mathcal{D}_{1}( \imath,t) \bigr\vert +N_{2} \frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \bigl\vert \mathcal{D}_{2}( \imath,t) \bigr\vert \\ &{} +N_{3}\frac{\imath ^{6}}{(1+\imath ^{2})^{4}} \bigl\vert \mathcal{D}_{3}( \imath,t) \bigr\vert +N_{4}\frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \bigl\vert \mathcal{D}_{4}(\imath,t) \bigr\vert +N_{5} \bigl\vert \mathcal{D}_{5}(\imath,t) \bigr\vert \\ \leq {}&aN_{1}\frac{\imath ^{4}}{(1+\imath ^{2})^{3}} \bigl\vert \Re e \bigl\{ i\imath ( \widehat{\varpi}\overline{\widehat{\phi}} +l \widehat{\phi}\overline{\widehat{w}} ) \bigr\} \bigr\vert \\ &{}+N_{2}\frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \bigl\vert \Re e \bigl\{ i\imath (ak_{1}\widehat{\vartheta} \overline{\widehat{\sigma}} +a\beta \widehat{\vartheta} \overline{\widehat{w}} +2k_{1}\widehat{v}\overline{ \widehat{\sigma}} ) \bigr\} \bigr\vert \\ &{}+N_{3}\frac{\imath ^{6}}{(1+\imath ^{2})^{4}} \bigl\vert \Re e \{\widehat{\phi} \overline{\widehat{g}} \} \bigr\vert \\ &{}+N_{4}\frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \bigl\vert \Re e \bigl\{ i\imath (l \widehat{w}\overline{\widehat{v}} +\widehat{v} \overline{\widehat{\varpi}} ) \bigr\} \bigr\vert \\ &{}+N_{4}\frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \bigl\vert \Re e \bigl\{ (\widehat{w} \overline{\widehat{r}} +a\widehat{g} \overline{\widehat{v}} ) \bigr\} \bigr\vert \\ &{}+N_{5} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s e^{-s\jmath } \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath. \end{aligned}$$

By utilizing Young’s inequality, the fact that \(\frac{\imath ^{2}}{1+\imath ^{2}}\leq \min \{1,\imath ^{2}\}\) and \(\frac{1}{1+\imath ^{2}}\leq 1\), we find

$$\begin{aligned} \bigl\vert \mathcal{K}_{1}(\imath,t)-N\widehat{V}(\imath,t) \bigr\vert \leq c\widehat{V}(\imath,t). \end{aligned}$$

Hence, we get

$$\begin{aligned} (N-c)\widehat{V}(\imath,t)\leq \mathcal{K}_{1}(\imath,t)\leq (N+c) \widehat{V}(\imath,t). \end{aligned}$$
(2.44)

Now, we choose N large enough in such a way that

$$\begin{aligned} N-c>0,\qquad C_{1}N-c>0,\qquad mk_{2}N-c>0, \end{aligned}$$

and utilizing (2.8), estimates (2.43) and (2.44), respectively.

One can find a positive constant \(\alpha >0\), then \(t>0\) & \(\imath \in \mathbb{R}\), we obtain

$$\begin{aligned} \mu _{1}\widehat{V}(\imath,t)\leq \mathcal{K}_{1}( \imath,t)\leq \mu _{2}\widehat{V}(\imath,t). \end{aligned}$$
(2.45)

and

$$\begin{aligned} \mathcal{K}_{1}'(\imath,t)\leq {}&-\alpha \frac{\imath ^{6}}{(1+\imath ^{2})^{4}} \biggl( \vert \widehat{g} \vert ^{2}+ \vert \widehat{\phi} \vert ^{2}+ \vert \widehat{\vartheta} \vert ^{2} + \vert \widehat{r} \vert ^{2}+ \vert \widehat{w} \vert ^{2} + \vert \widehat{v} \vert ^{2}+ \vert \widehat{\sigma} \vert ^{2}+ \vert \widehat{\varpi} \vert ^{2} \\ &{}+ \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath \biggr), \end{aligned}$$
(2.46)

then

$$\begin{aligned} \mathcal{K}_{1}'(\imath,t)\leq -\lambda _{1} \jmath _{1}(\imath ) \widehat{V}(\imath,t), \quad\forall t\geq 0. \end{aligned}$$
(2.47)

Therefore, for some positive constant \(\mu _{3}=\frac{\lambda _{1}}{\mu _{2}}>0\), we get

$$\begin{aligned} \mathcal{K}_{1}'(\imath,t)\leq -\mu _{3} \jmath _{1}(\imath ) \mathcal{K}_{1}(\imath,t), \quad\forall t \geq 0, \end{aligned}$$
(2.48)

where \(\jmath _{1}(\imath )=\frac{\imath ^{6}}{(1+\imath ^{2})^{4}}\), for some \(\lambda _{1},\mu _{i}>0, i=1,2,3\). The proof of the first result (2.38) is finished.

Before the proof of the second result (2.39). In the estimates (2.21), we used the inequalities

$$\begin{aligned} &\Re e \{ia\beta \imath \widehat{r}\overline{\widehat{\vartheta}} \}\leq \varepsilon _{2} \frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \vert \widehat{r} \vert ^{2} +c( \varepsilon _{2}) \bigl(1+\imath ^{2} \bigr)^{2} \vert \widehat{\vartheta} \vert ^{2}, \\ &\Re e \bigl\{ \beta \imath ^{2}\bigl(2k_{1}-a^{2} \bigr)\widehat{v} \overline{\widehat{\vartheta}} \bigr\} \leq \varepsilon _{3} \frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \vert \widehat{v} \vert ^{2} +c( \varepsilon _{3}) \bigl(1+\imath ^{2}\bigr)^{2} \vert \widehat{\vartheta} \vert ^{2}. \end{aligned}$$
(2.49)

Hence, the estimate (2.19) can also be written as

$$\begin{aligned} \frac{d\mathcal{D}_{2}(\imath,t)}{dt}\leq {}& -\frac{ak_{1}^{2}}{2} \imath ^{2} \vert \widehat{\sigma} \vert ^{2} -\frac{a\beta ^{2}}{2} \imath ^{2} \vert \widehat{w} \vert ^{2}+\varepsilon _{2} \frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \vert \widehat{r} \vert ^{2} \\ &{}+\varepsilon _{3}\frac{\imath ^{2}}{(1+\imath ^{2})^{2}} \vert \widehat{v} \vert ^{2} +c(\varepsilon _{2},\varepsilon _{3}) \bigl(1+\imath ^{2}\bigr)^{2} \vert \widehat{\vartheta} \vert ^{2}. \end{aligned}$$
(2.50)

Similarly, we can prove the second result.

So, we derive (2.37) and by using (2.9), (2.15), (2.50), (2.23), (2.29) and (2.35) with the fact that \(\frac{\imath ^{2}}{1+\imath ^{2}}\leq \min \{1,\imath ^{2}\}\) and \(\frac{1}{1+\imath ^{2}}\leq 1\), we get

$$\begin{aligned} \mathcal{K}_{2}'(\imath,t)\leq {}&- \frac{\imath ^{6}}{(1+\imath ^{2})^{7}} \biggl[\frac{k_{0}l^{2}}{2}M_{3}-2 \varepsilon _{1}M_{1}- \varepsilon _{5}M_{4} \biggr] \vert \widehat{g} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{5}} \biggl[\frac{1}{2}M_{4}- \varepsilon _{2}M_{2}-\varepsilon _{4}M_{3} \biggr] \vert \widehat{r} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{3}} \biggl[\frac{a\beta ^{2}}{2}M_{2}-c( \varepsilon _{1})M_{1}-cM_{4} \biggr] \vert \widehat{w} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{5}} \biggl[\frac{a^{2}l^{2}}{2}M_{4}- \varepsilon _{3}M_{2} \biggr] \vert \widehat{v} \vert ^{2} \\ &{}-\frac{\imath ^{6}}{(1+\imath ^{2})^{6}} \biggl[\frac{k_{0}}{2}M_{1}-c( \varepsilon _{4})M_{3} \biggr] \vert \widehat{\phi} \vert ^{2}- \frac{\imath ^{4}}{(1+\imath ^{2})^{3}} \biggl[\frac{ak_{1}^{2}}{2}M_{2} \biggr] \vert \widehat{\sigma} \vert ^{2} \\ &{}-\imath ^{2} \bigl[mk_{2}M-cM_{1}-c(\varepsilon _{2},\varepsilon _{3})M_{2}-cM_{4} \bigr] \vert \widehat{\vartheta} \vert ^{2} \\ &{}- \bigl[C_{1}M-c(\varepsilon _{1}) M_{1}-cM_{3}-cM_{4}- \aleph _{1}M_{5} \bigr] \vert \widehat{\varpi} \vert ^{2} \\ &{}-\zeta _{1}M_{5} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath \\ &{}- [\zeta _{1}M_{5}-cM_{1}-cM_{4} ] \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, 1, s, t ) \bigr\vert ^{2} \,ds. \end{aligned}$$
(2.51)

By setting

$$\begin{aligned} \varepsilon _{1}=\frac{k_{0}l^{2} M_{3}}{16 M_{1}},\qquad \varepsilon _{2}= \frac{M_{4}}{8M_{2}},\qquad \varepsilon _{3}=\frac{a^{2}l^{2}M_{4}}{4M_{2}},\qquad \varepsilon _{4}=\frac{M_{4}}{8M_{3}}, \qquad\varepsilon _{5}= \frac{k_{0}l^{2} M_{3}}{8 M_{4}}, \end{aligned}$$

we obtain the following

$$\begin{aligned} \mathcal{K}_{2}'(\imath,t)\leq{} &{-} \frac{\imath ^{6}}{(1+\imath ^{2})^{7}} \biggl[\frac{k_{0}l^{2}}{4}M_{3} \biggr] \vert \widehat{g} \vert ^{2}- \frac{\imath ^{4}}{(1+\imath ^{2})^{5}} \biggl[\frac{1}{4}M_{4} \biggr] \vert \widehat{r} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{3}} \biggl[\frac{a\beta ^{2}}{2}M_{2}-c(M_{1},M_{3})M_{1}-cM_{4} \biggr] \vert \widehat{w} \vert ^{2} \\ &{}-\frac{\imath ^{4}}{(1+\imath ^{2})^{5}} \biggl[\frac{a^{2}l^{2}}{4}M_{4} \biggr] \vert \widehat{v} \vert ^{2}- \frac{\imath ^{4}}{(1+\imath ^{2})^{3}} \biggl[ \frac{ak_{1}^{2}}{2}M_{2} \biggr] \vert \widehat{\sigma} \vert ^{2} \\ &{}-\frac{\imath ^{6}}{(1+\imath ^{2})^{6}} \biggl[\frac{k_{0}}{2}M_{1}-c(M_{3},M_{4})M_{3} \biggr] \vert \widehat{\phi} \vert ^{2} \\ &{}-\imath ^{2} \bigl[mk_{2}M-cM_{1}-c(M_{2},M_{4})M_{2}-cM_{4} \bigr] \vert \widehat{\vartheta} \vert ^{2} \\ &{}- \bigl[C_{1}M-c(M_{1},M_{3}) M_{1}-cM_{3}-cM_{4}-\aleph _{1}M_{5} \bigr] \vert \widehat{\varpi} \vert ^{2} \\ &{}-\zeta _{1}M_{5} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath \\ &{}- [\zeta _{1}M_{5}-cM_{1}-cM_{4} ] \int _{\wp _{1}}^{\wp _{2}} \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, 1, s, t ) \bigr\vert ^{2} \,ds. \end{aligned}$$
(2.52)

Next, we fix \(M_{3},M_{4}\) and choose \(M_{1}\) large enough such that

$$\begin{aligned} \frac{k_{0}}{2}M_{1}-c(M_{3},M_{4})M_{3}>0, \end{aligned}$$

then, we select \(M_{2},M_{5}\) large enough such that

$$\begin{aligned} &\frac{a\beta ^{2}}{2}M_{2}-c(M_{1},M_{3})M_{1}-cM_{4}>0, \\ &\zeta _{1}M_{5}-cM_{1}-cM_{4}>0. \end{aligned}$$

Hence, we arrive at

$$\begin{aligned} \mathcal{K}_{2}'(\imath,t)\leq{}&{ -}\kappa _{0} \frac{\imath ^{6}}{(1+\imath ^{2})^{7}} \vert \widehat{g} \vert ^{2}- \kappa _{5}\frac{\imath ^{6}}{(1+\imath ^{2})^{6}} \vert \widehat{\phi} \vert ^{2}- \imath ^{2} [mk_{2}M-c ] \vert \widehat{\vartheta} \vert ^{2} \\ &{} -\frac{\imath ^{4}}{(1+\imath ^{2})^{5}} \bigl(\kappa _{1} \vert \widehat{r} \vert ^{2} +\kappa _{3} \vert \widehat{v} \vert ^{2} \bigr) - \frac{\imath ^{4}}{(1+\imath ^{2})^{3}} \bigl(\kappa _{2} \vert \widehat{w} \vert ^{2} +\kappa _{4} \vert \widehat{\sigma} \vert ^{2} \bigr) \\ &{} - [C_{1}M-c ] \vert \widehat{\varpi} \vert ^{2}- \kappa _{6} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath. \end{aligned}$$
(2.53)

On the other hand, we have

$$\begin{aligned} \bigl\vert \mathcal{K}_{2}(\imath,t)-M\widehat{V}(\imath,t) \bigr\vert ={}&M_{1}\frac{\imath ^{4}}{(1+\imath ^{2})^{6}} \bigl\vert \mathcal{D}_{1}( \imath,t) \bigr\vert +M_{2} \frac{\imath ^{2}}{(1+\imath ^{2})^{3}} \bigl\vert \mathcal{D}_{2}( \imath,t) \bigr\vert \\ &{} +M_{3}\frac{\imath ^{6}}{(1+\imath ^{2})^{7}} \bigl\vert \mathcal{D}_{3}( \imath,t) \bigr\vert +M_{4}\frac{\imath ^{2}}{(1+\imath ^{2})^{5}} \bigl\vert \mathcal{D}_{4}(\imath,t) \bigr\vert +M_{5} \bigl\vert \mathcal{D}_{5}(\imath,t) \bigr\vert \\ \leq {}&aM_{1}\frac{\imath ^{4}}{(1+\imath ^{2})^{6}} \bigl\vert \Re e \bigl\{ i\imath ( \widehat{\varpi}\overline{\widehat{\phi}} +l \widehat{\phi}\overline{\widehat{w}} ) \bigr\} \bigr\vert \\ &{}+M_{2}\frac{\imath ^{2}}{(1+\imath ^{2})^{3}} \bigl\vert \Re e \bigl\{ i\imath (ak_{1}\widehat{\vartheta} \overline{\widehat{\sigma}} +a\beta \widehat{\vartheta} \overline{\widehat{w}} +2k_{1}\widehat{v}\overline{ \widehat{\sigma}} ) \bigr\} \bigr\vert \\ &{}+M_{3}\frac{\imath ^{6}}{(1+\imath ^{2})^{7}} \bigl\vert \Re e \{\widehat{\phi} \overline{\widehat{g}} \} \bigr\vert \\ &{}+M_{4}\frac{\imath ^{2}}{(1+\imath ^{2})^{5}} \bigl\vert \Re e \bigl\{ i\imath (l \widehat{w}\overline{\widehat{v}} +\widehat{v} \overline{\widehat{\varpi}} ) \bigr\} \bigr\vert \\ &{}+M_{4}\frac{\imath ^{2}}{(1+\imath ^{2})^{5}} \bigl\vert \Re e \bigl\{ (\widehat{w} \overline{\widehat{r}} +a\widehat{g} \overline{\widehat{v}} ) \bigr\} \bigr\vert \\ &{}+M_{5} \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s e^{-s\jmath } \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath. \end{aligned}$$

Utilizing Young’s inequality, and the fact that \(\frac{\imath ^{2}}{1+\imath ^{2}}\leq \min \{1,\imath ^{2}\}\) and \(\frac{1}{1+\imath ^{2}}\leq 1\), we find

$$\begin{aligned} \bigl\vert \mathcal{K}_{2}(\imath,t)-M\widehat{V}(\imath,t) \bigr\vert \leq c\widehat{V}(\imath,t). \end{aligned}$$

Hence, we get

$$\begin{aligned} (M-c)\widehat{V}(\imath,t)\leq \mathcal{K}_{2}(\imath,t)\leq (M+c) \widehat{V}(\imath,t). \end{aligned}$$
(2.54)

Now, we choose M large enough in such a way that

$$\begin{aligned} M-c>0,\qquad C_{1}M-c>0,\qquad mk_{2}M-c>0, \end{aligned}$$

using (2.8), we get (2.53) and (2.54), respectively. One can find a positive constant \(\kappa >0\), then \(t>0\) & \(\imath \in \mathbb{R}\), we get

$$\begin{aligned} \mu _{4}\widehat{V}(\imath,t)\leq \mathcal{K}_{2}(\imath,t) \leq \mu _{5}\widehat{V}(\imath,t). \end{aligned}$$
(2.55)

and

$$\begin{aligned} \mathcal{K}_{2}'(\imath,t)\leq {}&{-}\kappa \frac{\imath ^{6}}{(1+\imath ^{2})^{7}} \biggl( \vert \widehat{g} \vert ^{2}+ \vert \widehat{\phi} \vert ^{2}+ \vert \widehat{\vartheta} \vert ^{2} + \vert \widehat{r} \vert ^{2}+ \vert \widehat{w} \vert ^{2} + \vert \widehat{v} \vert ^{2}+ \vert \widehat{\sigma} \vert ^{2}+ \vert \widehat{\varpi} \vert ^{2} \\ &{} + \int _{0}^{1} \int _{\wp _{1}}^{\wp _{2}} s \bigl\vert \aleph _{2}(s) \bigr\vert \bigl\vert \widehat{\mathcal{Y}} ( \imath, \jmath, s, t ) \bigr\vert ^{2} \,ds \,d\jmath \biggr). \end{aligned}$$
(2.56)

then

$$\begin{aligned} \mathcal{K}_{2}'(\imath,t)\leq -\lambda _{2} \jmath _{2}(\imath ) \widehat{V}(\imath,t), \quad\forall t\geq 0. \end{aligned}$$
(2.57)

Therefore, for some positive constant \(\mu _{6}=\frac{\lambda _{2}}{\mu _{5}}>0\), we get

$$\begin{aligned} \mathcal{K}_{2}'(\imath,t)\leq -\mu _{6}\jmath _{2}(\imath ) \mathcal{K}_{2}(\imath,t), \quad\forall t\geq 0, \end{aligned}$$
(2.58)

where \(\jmath _{2}(\imath )=\frac{\imath ^{6}}{(1+\imath ^{2})^{7}}\), for some \(\lambda _{2},\mu _{i}>0, i=4,5,6\). The proof of the second result (2.39) is finished. □

The pointwise estimates of the functional \(\widehat{V}(\imath,t)\) are given in the following result.

Proposition 2.8

Suppose (1.8) holds. Then, for any \(t\geq 0\) and \(\imath \in \mathbb{R}\), there exist a positive constants \(d_{1},d_{2}>0\) such that the energy functional stated by (2.8) holds

$$\begin{aligned} \textstyle\begin{cases} \widehat{V}(\imath,t)\leq d_{1}\widehat{V}(\imath,0)e^{-\mu _{3} \jmath _{1}(\imath )t}& \textit{if } a=1, \\ \widehat{V}(\imath,t)\leq d_{2}\widehat{V}(\imath,0)e^{-\mu _{6} \jmath _{2}(\imath )t}& \textit{if } a\neq 1, \end{cases}\displaystyle \end{aligned}$$
(2.59)

where \(\jmath _{1}(\imath )=\frac {\imath ^{6}}{(1+\imath ^{2})^{4}}, \jmath _{2}(\imath )=\frac {\imath ^{6}}{(1+\imath ^{2})^{7}}\).

Proof

From (2.38)2 and (2.39)2, we have

$$\begin{aligned} &\mathcal{K}_{1}(\imath,t)\leq \mathcal{K}_{1}( \imath,0)e^{-\mu _{3} \jmath _{1}(\imath )t}, \quad\forall t\geq 0, \text{ if } a=1 \end{aligned}$$
(2.60)
$$\begin{aligned} &\mathcal{K}_{2}(\imath,t)\leq \mathcal{K}_{2}( \imath,0)e^{-\mu _{6} \jmath _{2}(\imath )t},\quad \forall t\geq 0, \text{ if } a\neq 1. \end{aligned}$$
(2.61)

Hence, according of (2.38)1, (2.39)1 and (2.60), (2.61), we established (2.59). □

2.2 Decay estimates

Now, we will show the following important result.

Theorem 2.9

Let s be a nonnegative integer, and \(Z_{0}\in H^{s}(\mathbb{R})\cap L^{1}(\mathbb{R})\). Then, the solution Z of problem (2.2)(2.3) holds, \(t\geq 0\) the following decay estimates

  • For \(a=1\)

    $$\begin{aligned} \bigl\Vert \partial _{x}^{k}Z(t) \bigr\Vert _{2}\leq C \Vert Z_{0} \Vert _{1}(1+t)^{- \frac{1}{12}-\frac{k}{6}} +C(1+t)^{-\frac{\ell}{2}} \bigl\Vert \partial _{x}^{k+ \ell}Z_{0} \bigr\Vert _{2} \end{aligned}$$
    (2.62)
  • For \(a\neq 1\)

    $$\begin{aligned} \bigl\Vert \partial _{x}^{k}Z(t) \bigr\Vert _{2}\leq C \Vert Z_{0} \Vert _{1}(1+t)^{- \frac{1}{12}-\frac{k}{6}} +C(1+t)^{-\frac{\ell}{8}} \bigl\Vert \partial _{x}^{k+ \ell}Z_{0} \bigr\Vert _{2}, \end{aligned}$$
    (2.63)

where and k are nonnegative integers \(k+\ell \leq s\) and \(C>0\) is a positive constant.

Proof

From (2.8), we get \(\vert \widehat{Z}(\imath,t)\vert ^{2}\sim \widehat{V}(\imath,t)\).

  • If \(a=1\), then by using the Plancherel theorem and (2.59)1, we have

    $$\begin{aligned} \bigl\Vert \partial _{x}^{k}Z(t) \bigr\Vert _{2}^{2}={}& \int _{\mathbb{R}} \vert \imath \vert ^{2k} \bigl\vert \widehat{Z}(\imath,t) \bigr\vert ^{2}\,d\imath \\ \leq {}&c \int _{\mathbb{R}} \vert \imath \vert ^{2k}e^{-\mu _{3}\jmath _{1}( \imath )t} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d\imath \\ \leq {}& \underbrace{c \int _{ \vert \imath \vert \leq 1} \vert \imath \vert ^{2k}e^{-\mu _{3}\jmath _{1}(\imath )t} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d \imath}_{R_{1}} \\ &{}+ \underbrace{c \int _{ \vert \imath \vert \geq 1} \vert \imath \vert ^{2k}e^{-\mu _{3}\jmath _{1}(\imath )t} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d \imath}_{R_{2}}. \end{aligned}$$
    (2.64)

    Now, we estimate \(R_{1},R_{2}\), the low-frequency part \(\vert \imath \vert \leq 1\) and the high-frequency part \(\vert \imath \vert \geq 1\), respectively. First, we have \(\jmath _{1}(\imath )\geq \frac{1}{16}\imath ^{6}\), for \(\vert \imath \vert \leq 1\). Then

    $$\begin{aligned} R_{1}\leq {}&c \int _{ \vert \imath \vert \leq 1} \vert \imath \vert ^{2k}e^{- \frac{\mu _{3}}{16} \vert \imath \vert ^{6}t} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d\imath \\ \leq {}&c\sup_{ \vert \imath \vert \leq 1}\bigl\{ \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\bigr\} \int _{ \vert \imath \vert \leq 1} \vert \imath \vert ^{2k}e^{- \frac{\mu _{3}}{16} \vert \imath \vert ^{6}t} \,d\imath, \end{aligned}$$
    (2.65)

    by utilizing Lemma 1.1, we get

    $$\begin{aligned} R_{1}&\leq c\sup_{ \vert \imath \vert \leq 1}\bigl\{ \bigl\vert \widehat{Z}( \imath,0) \bigr\vert ^{2}\bigr\} (1+t)^{-\frac{k}{3}-\frac{1}{6}} \\ &\leq c \Vert Z_{0} \Vert ^{2}_{1}(1+t)^{-\frac{k}{3}-\frac{1}{6}}. \end{aligned}$$
    (2.66)

    Secondly, we have \(\jmath _{1}(\imath )\geq \frac{1}{16}\imath ^{-2}\), for \(\vert \imath \vert \geq 1\). Then

    $$\begin{aligned} R_{2}\leq c \int _{ \vert \imath \vert \geq 1} \vert \imath \vert ^{2k}e^{- \frac{\mu _{3}}{16} \vert \imath \vert ^{-2}t} \bigl\vert \widehat{Z}( \imath,0) \bigr\vert ^{2}\,d\imath, \quad\forall t\geq 0. \end{aligned}$$
    (2.67)

    Then, through the inequality

    $$\begin{aligned} \sup_{ \vert \imath \vert \geq 1} \bigl\{ \vert \imath \vert ^{-2\ell}e^{-c \frac{1}{16} \vert \imath \vert ^{-2}t} \bigr\} \leq C(1+t)^{-\ell}, \end{aligned}$$
    (2.68)

    we get that

    $$\begin{aligned} R_{2}&\leq c\sup_{ \vert \imath \vert \geq 1} \bigl\{ \vert \imath \vert ^{-2\ell}e^{-\frac{\mu _{3}}{16} \vert \imath \vert ^{-2}t} \bigr\} \int _{ \vert \imath \vert \geq 1} \vert \imath \vert ^{2(k+ \ell )} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d\imath \\ &\leq c(1+t)^{-\ell} \bigl\Vert \partial ^{k+\ell}_{x}Z(x,0) \bigr\Vert _{2}^{2},\quad \forall t\geq 0. \end{aligned}$$
    (2.69)

    Substituting (2.66) and (2.69) into (2.64), we find (2.62).

  • If \(a\neq 1\), similar to the first estimate, we apply the Plancherel theorem and using (2.59)2, we get

    $$\begin{aligned} \bigl\Vert \partial _{x}^{k}Z(t) \bigr\Vert _{2}^{2}={}& \int _{\mathbb{R}} \vert \imath \vert ^{2k} \bigl\vert \widehat{Z}(\imath,t) \bigr\vert ^{2}\,d\imath \\ \leq {}&c \int _{\mathbb{R}} \vert \imath \vert ^{2k}e^{-\mu _{6}\jmath _{2}( \imath )t} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d\imath \\ \leq {}& \underbrace{c \int _{ \vert \imath \vert \leq 1} \vert \imath \vert ^{2k}e^{-\mu _{6}\jmath _{2}(\imath )t} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d \imath}_{R_{3}} \\ &{}+ \underbrace{c \int _{ \vert \imath \vert \geq 1} \vert \imath \vert ^{2k}e^{-\mu _{6}\jmath _{2}(\imath )t} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d \imath}_{R_{4}}. \end{aligned}$$
    (2.70)

    Now, we estimate \(R_{3},R_{4}\), the low-frequency part \(\vert \imath \vert \leq 1\) and the high-frequency part \(\vert \imath \vert \geq 1\), respectively. First, we have \(\jmath _{2}(\imath )\geq \frac{1}{64}\imath ^{6}\), for \(\vert \imath \vert \leq 1\). Then

    $$\begin{aligned} R_{3}&\leq c \int _{ \vert \imath \vert \leq 1} \vert \imath \vert ^{2k}e^{- \frac{\mu _{6}}{64} \vert \imath \vert ^{6}t} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d\imath \\ &\leq c\sup_{ \vert \imath \vert \leq 1}\bigl\{ \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\bigr\} \int _{ \vert \imath \vert \leq 1} \vert \imath \vert ^{2k}e^{- \frac{\mu _{6}}{64} \vert \imath \vert ^{6}t} \,d\imath, \end{aligned}$$
    (2.71)

    by utilizing Lemma 1.1, we get

    $$\begin{aligned} R_{3}&\leq c\sup_{ \vert \imath \vert \leq 1}\bigl\{ \bigl\vert \widehat{Z}( \imath,0) \bigr\vert ^{2}\bigr\} (1+t)^{-\frac{k}{3}-\frac{1}{6}} \\ &\leq c \Vert Z_{0} \Vert ^{2}_{1}(1+t)^{-\frac{k}{3}-\frac{1}{6}}. \end{aligned}$$
    (2.72)

    Secondly, we have \(\jmath _{2}(\imath )\geq \frac{1}{64}\imath ^{-8}\), for \(\vert \imath \vert \geq 1\). Then

    $$\begin{aligned} R_{4}\leq c \int _{ \vert \imath \vert \geq 1} \vert \imath \vert ^{2k}e^{- \frac{\mu _{6}}{64} \vert \imath \vert ^{-8}t} \bigl\vert \widehat{Z}( \imath,0) \bigr\vert ^{2}\,d\imath,\quad \forall t\geq 0. \end{aligned}$$
    (2.73)

    By (2.68), we find

    $$\begin{aligned} R_{4}&\leq c\sup_{ \vert \imath \vert \geq 1} \bigl\{ \vert \imath \vert ^{-2\ell}e^{-\frac{\mu _{6}}{64} \vert \imath \vert ^{-8}t} \bigr\} \int _{ \vert \imath \vert \geq 1} \vert \imath \vert ^{2(k+ \ell )} \bigl\vert \widehat{Z}(\imath,0) \bigr\vert ^{2}\,d\imath \\ &\leq c(1+t)^{-\frac{\ell}{4}} \bigl\Vert \partial ^{k+\ell}_{x}Z(x,0) \bigr\Vert _{2}^{2},\quad \forall t\geq 0. \end{aligned}$$
    (2.74)

    Substituting (2.72) and (2.74) into (2.70), we obtain (2.63).

 □

3 Conclusion

The investigation of the general decay estimate of Bresse–Fourier system solutions with respect to the distributed delay term is the goal of this work, which employs the energy technique in Fourier space.

The different process that results from the distributed delay, which determines the formation of this term in the system in Fourier space, is what concerns us in the current work.

In the upcoming works, we will try the same approach in the same systems, but with various memory types; we anticipate getting results that are comparable.

Availability of data and materials

No new data and materials are related to this research work.

References

  1. Boulaaras, S., Choucha, A., Scapillato, A.: General decay of the Moore–Gibson–Thompson equation with viscoelastic memory of type II. J. Funct. Spaces 4, 1–12 (2022). https://doi.org/10.1155/2022/9015775

    Article  MathSciNet  MATH  Google Scholar 

  2. Bounadja, H., Said-Houari, B.: Decay rates for the Moore–Gibson–Thompson equation with memory. Evol. Equ. Control Theory 10(3), 431–460 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouzettouta, L., Zennir, K., Zitouni, S.: Uniform decay for a viscoelastic wave equation with density and time-varying delay in \(R^{n}\). Filomat 33, 961–970 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouzettouta, L., Zitouni, S., Zennir, K., Sissaoui, H.: Well-posedness and decay of solutions to Bresse system with internal distributed delay. Int. J. Appl. Math. Stat. 56, 153–168 (2017)

    MathSciNet  Google Scholar 

  5. Bresse, J.A.C.: Cours de Mécanique Appliquée, Professé a L’école des Ponts et Chaussées, Par, M. Bresse, pp. 1865–1868. M. Bresse, Gauthier Villars, Paris (2013)

    Google Scholar 

  6. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A 432, 171–194 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253–264 (1992)

    Article  MathSciNet  Google Scholar 

  8. Gurtin, M.E., Pipkin, A.S.: A general decay of a heat condition with finite wave speeds. Arch. Ration. Mech. Anal. 31(2), 113–126 (1968)

    Article  MATH  Google Scholar 

  9. Khader, M., Said-Houari, B.: The decay rate of solution for the Cauchy problem in Timoshenko system with past history. Appl. Math. Optim. (2016). https://doi.org/10.1007/s00245-016-9336-6

    Article  MATH  Google Scholar 

  10. Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Systems & Control: Foundations & Applications. Birkhäuser, Boston (1994)

    MATH  Google Scholar 

  11. Mori, N., Kawashima, S.: Decay property for the Timoshenko system with Fourier’s type heat conduction. J. Hyperbolic Differ. Equ. 11, 135–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nicaise, A.S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9–10), 935–958 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Said-Houari, B., Rahali, R.: Asymptotic behavior of the Cauchy problem of the Timoshenko system in thermoelsaticity of type III. Evol. Equ. Control Theory 2(2), 423–440 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Said-Houari, B., Hamadouche, T.: The asymptotic behavior of the Bresse–Cattanao system. Commun. Contemp. Math. 18, Article ID 1550045 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Said-Houari, B., Hamadouche, T.: The Cauchy problem of the Bresse system in thermoelasticity of type III. Appl. Anal. 95(11), 2323–2338 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Said-Houari, B., Soufyane, A.: The Bresse system in thermoelasticity. Math. Methods Appl. Sci. 38(17), 3642–3652 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Soufyane, A., Said-Houari, B.: The effect of frictional damping terms on the decay rate of the Bresse system. Evol. Equ. Control Theory 3(4), 713–738 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, X., Zuazua, E.: Decay of solutions of the system of thermoelasticity of type III. Commun. Contemp. Math. 5, 25–83 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

No funding is associated with the current research work.

Author information

Authors and Affiliations

Authors

Contributions

A. Choucha conceptualized, investigated, analyzed and validated the research while Salah Boulaaras, Rashid Jan and Rafik Guefaifia formulated, investigated, numerically examined, reviewed and supervised this research work.

Corresponding author

Correspondence to Rafik Guefaifia.

Ethics declarations

Ethics approval and consent to participate

There are no ethical issues in this work. All authors actively participated in this research and approved it for publication.

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Choucha, A., Boulaaras, S., Jan, R. et al. Decay rate of the solutions to the Cauchy problem of the Bresse system in thermoelasticity of type III with distributed delay. Bound Value Probl 2023, 67 (2023). https://doi.org/10.1186/s13661-023-01753-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-023-01753-y

MSC

Keywords