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Decay rate of the solutions to the Cauchy problem of the Bresse system in thermoelasticity of type III with distributed delay
Boundary Value Problems volume 2023, Article number: 67 (2023)
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:
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)
produces the classical heat transport equation (of parabolic type)
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
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
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:
where
with the initial and boundary conditions
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
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
then, we get
and utilize the transformation [18]
with a function \(\chi:=\chi (x)\) satisfying
We can also write the proposed problem in the form (by writing, Ï… instead of Ï…Ì…)
where
with initial conditions
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:
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
Then, the system (1.11) takes the following form
with initial conditions
where
Hence, the problem (2.2)–(2.3) is written as
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
Utilizing the Fourier transform to (2.4), we get
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
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
satisfies
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
In second step, by multiplying (2.7)9 by \(\overline{\widehat{\mathcal{Y}}}\vert \aleph _{2}(s)\vert \) and integrating the result over \((0, 1)\times (\wp _{1}, \wp _{2})\)
utilizing Young’s inequality, we get
by substituting (2.11) and (2.12) into (2.10), we find
then, by (1.8), \(\exists C_{1}=\beta (\aleph _{1}-\int _{\wp _{1}}^{\wp _{2}}\vert \aleph _{2}(s)\vert \,ds)>0\) such that
Hence, we get the required result. □
The following Lemma is required in order to get the main result.
Lemma 2.2
The functional
satisfies the following for any \(\varepsilon _{1}>0\)
Proof
By differentiating \(\mathcal{D}_{1}\) and using (2.7), we get
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
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
satisfies the following for any \(\varepsilon _{2},\varepsilon _{3}>0\)
Proof
By differentiating \(\mathcal{D}_{2}\) and using (2.7), we get
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
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
satisfies the below for any \(\varepsilon _{4}>0\)
Proof
By differentiating \(\mathcal{D}_{3}\) and using (2.7), we have
The last two terms in the RHS of (2.24) are obtained by Young’s inequality, which we solve for any \(\varepsilon _{4},\delta _{5}>0\)
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
where
satisfies
-
(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)
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
and
Now, differentiating \(\mathcal{D}_{4}\) and by (2.30) and (2.31), we have
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
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
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
satisfies
where \(\zeta _{1}>0\).
Proof
By differentiating \(\mathcal{D}_{5}\) with respect to t and utilizing (2.7)9, we have
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
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
By setting
We obtain the following
Next, we fix \(N_{3},N_{4}\) and choose \(N_{1}\) large enough such that
then, we pick \(N_{2}\) and \(N_{5}\) large enough in such a way that
Hence, we have
Secondly, we have
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
Hence, we get
Now, we choose N large enough in such a way that
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
and
then
Therefore, for some positive constant \(\mu _{3}=\frac{\lambda _{1}}{\mu _{2}}>0\), we get
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
Hence, the estimate (2.19) can also be written as
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
By setting
we obtain the following
Next, we fix \(M_{3},M_{4}\) and choose \(M_{1}\) large enough such that
then, we select \(M_{2},M_{5}\) large enough such that
Hence, we arrive at
On the other hand, we have
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
Hence, we get
Now, we choose M large enough in such a way that
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
and
then
Therefore, for some positive constant \(\mu _{6}=\frac{\lambda _{2}}{\mu _{5}}>0\), we get
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
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
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 â„“ 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) -
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.
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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.
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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
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DOI: https://doi.org/10.1186/s13661-023-01753-y