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, 15–17], the Bresse system combined with the Cattaneo law of heat conduction in [14] and the MGT problem in [1]. We recommend the following papers [2–4, 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, ℑ 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 ℓ 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.