Global existence and energy decay for a transmission problem under a boundary fractional derivative type

1.   Introduction and position of problem

A generalization of power series, in which each term has an integer exponent in mathematical analysis, and Laurent series in the theory of functions of a complex variable, was Hadamard series (Hadamard operator) and Frobenius series in mathematical physics with fractional exponents for each term. It is possible to generalize the derivative of an integer order to a derivative of a fractional order (Riemann-Liouville derivatives and Caputo-Gerasimov derivatives). It is natural to write solutions of differential equations of fractional order in terms of Hadamard or Frobenius series. Fractional derivatives appear in new physical, technical and chemical problems arising in research activities. Here, we are interested in taking this phenomenon (fractional derivatives) in the boundary conditions with respect to the time variable, for more detail, please see ^[5,6,7,9,19].

A new class of initial boundary value problems is a transmission problem given by the equations

subject to the initial conditions

transmission conditions

boundary conditions

and compatibility conditions

where $0 < l_0 < L, \varpi _1, \varpi _2$ are positive constants, $\rho_{1}, \rho_{2}, \tau_1, \tau_{2} > 0$ represent the densities and tensions of the strings $u$ and $v$, respectively, $\gamma > 0$, the initial data $\left(u_{0}, u_{1}, v_{0}, v_{1}\right)$ belong to a suitable function space which will be defined later. We will mention some works related to the stabilization of transmission problems with mechanism of damping (see ^[8,20,21]). In ^[18], the authors consider a transmission problem in viscoelasticity. The exponential decay of the solutions is obtained and it is proved that the linear model is well posed. In ^[12], a transmission problem involving two Euler-Bernoulli equations which model the vibrations of a composite beam is considered. By one boundary damping term, the global existence and decay property of the solutions are showed.

Recently in ^[2], Benaissa and Atoui consider the following transmission problem

The lack of exponential decay of the energy is proved and also the polynomial decay rate is showed by using the spectrum method and the Borichev-Tomilov theorem ^[10].

The paper is organized as follows. In section 1, we introduce our model in (1.1) and the actual state-of-the-art is given. In section 2, the well-posedness of strong/weak solutions of the system is given by using the Hille-Yosida theorem. In section 3, we treat the question of stability where we find that the augmented model is strongly stable in the absence of compactness of the resolvent by using a criteria of Arendt-Batty. In section 4, we show the lack of exponential stability by spectral analysis and the polynomial type decay rate is proved which depends on a parameter $\alpha$. We finished our work with section 5 by dealing with the polynomial stability for $\zeta \neq 0$.

2.   Preliminary and well-posedness of solution

The beginning of this section concerns to write the system (1.1) by another way. For this aim, we will use the following result.

Theorem 2.1. ^[14] Let $\varrho$ be a function defined by

Then the relationship between the 'input' $U$ and the 'output' $O$ of the following system

is introduced by

where

Lemma 2.1. ^[1] Define

If $\varpi \in D$, then

and

We need now to reformulate the system (1.1). For this aim, we take $U = \partial_{t} v$ in (2.1) and using (1.4), the system (1.1) becomes for $t\in (0, +\infty)$

where $\varsigma = \left(\pi\right)^{-1}\sin \left(\alpha\pi\right)\gamma$. The energy associated with the solutions $\left(u, v, \Phi\right)$ of (2.2) is defined as follows

Lemma 2.2. Let $\left(u, v, \Phi\right)$ be a regular solution of (2.2). Then, the energy (2.3) satisfies

Proof. By multiplication of (2.2)$_1$ by $\partial_t\overline{u}$ and then integrating by parts over $\left(0, l_{0}\right)$, we obtain

Now, we multiply (2.2)$_2$ by $\partial_t\overline{v}$ and then we integrate by parts over $\left(l_{0}, L\right)$ and we arrive at

Summing, we obtain

From the boundary condition (2.2)$_6$, we get

We multiply (2.2)$_3$ by $\varsigma \Phi$ and then we integrate over $\left(-\infty, +\infty\right)$ and we obtain

Then, using (2.5) and (2.6), we arrive at

Now, we will use a semigroup setting for (2.2). For this aim, we introduce the vector $X = (u, \varphi, v, \psi, \Phi)^{T}$, where $\varphi = \partial_{t} u$ and $\psi = \partial_{t} v$. Then we get that the system (2.2) is equivalent to the following system

Here $X_{0}: = \left(u_{0}, u_{1}, v_{0}, v_{1}, \Phi_{0}\right)^T$. The operator $\mathcal{A}$ given by

is a linear operator. We introduce the following Hilbert space (the energy space)

For $X = (u, \varphi, v, \psi, \Phi)^{T}$ and $\overline{X} = (\overline{u}, \overline{\varphi}, \overline{v}, \overline{\psi}, \overline{\Phi})^{T}$, the inner product in $\mathcal{H}$ is defined as follows

where the domain of $\mathcal{A}$ is defined by

We state now a result for existence and uniqueness.

Theorem 2.2. (1) If $X_{0}\in D\left(\mathcal{A}\right)$, then the system (2.2) has a unique strong solution

(2) If $X_{0}\in\mathcal{H}$, then the system (2.2) has a unique weak solution

Proof. Firstly, we will show the dissipativity of the operator $\mathcal{A}$. In fact, we have $\forall X \in D(\mathcal{A})$

Then, by (2.5) and (2.6), we get

Hence, $\mathcal{A}$ is dissipative.

Now, we will prove the surjectivity of the operator $\varpi I -\mathcal{A}$ for $0 < \varpi$. Let $F = (f_1, f_2, f_3, f_{4}, f_{5})\in \mathcal{H}$. We will prove that there exists $X \in D(\mathcal{A})$ such that

Here the Eq (2.11) is equivalent to the equation

Suppose that $u$ and $v$ are found with the appropriate regularity. Then, from $(2.12)_{1}$ and $(2.12)_{3}$, we find that

It is not hard to see that $\varphi\in H^{1}_{*}\left(0, l_{0}\right)$ and $\psi\in H^{1}\left(l_{0}, L\right)$. Furthermore, by (2.12)$_5$, we can find $\Phi$ as follows

By (2.12) and (2.13), we have that $u$ and $v$ satisfy

The solving of the system (2.15) is equivalent to find $u\in H^{2}\cap H^{1}_{*}(0, l_0)$ and $v\in H^{2}\left(l_{0}, L\right)$ so that

for all $w\in H^{1}_{*}\left(0, l_{0}\right)$ and $\chi\in H^{1}\left(l_{0}, L\right)$. By (2.14) and (2.16), we get that $u$ and $v$ satisfying

Adding the Eqs $(2.2)_{6}$ and $(2.17)_{1, 2}$, we obtain

where $\overline{\varsigma} = \varsigma \left(L\right)\int^{+\infty}_{-\infty}\frac{\varrho^{2}\left(\xi\right)}{\xi^{2}+\zeta+\varpi}\, d\xi$. Then, (2.18) is equivalent to

Here the linear form

and the bilinear form

are given as follows

and

respectively. It is not hard to check that $a$ is coercive and continuous and $L$ is continuous. By using the Lax-Milgram theorem, we find that $\forall\left(w, \chi\right)\in H^{1}_{*}\left(0, l_{0}\right)\times H^{1}\left(l_{0}, L\right)$. Then the problem (2.19) has a unique solution

Using the classical elliptic regularity and (2.20), we find that

Then, $\varpi I-\mathcal{A}$ is surjective $\forall 0 < \varpi$. Owing to the Hille-Yosida theorem, the result in Theorem (2.2) yields.

3.   On the strong stability of solution

In this part, as in ^[4,11], we use the Arendt-Batty theorem and we see that a $C_{0}$-semigroup of contractions $e^{\mathcal{A} t}$ in a Banach space is strongly stable whenever $\sigma\left(\mathcal{A}\right)\cap i\mathbb{R}$ contains only a countable number of elements and $\mathcal{A}$ has no pure imaginary eigenvalues. The following theorem is our next main result.

Theorem 3.1. ^[3] The $C_{0}$-semigroup $e^{\mathcal{A} t}$ is strongly stable in $\mathcal{H}$, i.e, $\forall X_{0}\in \mathcal{H}$, and the solution of (2.7) satisfies

To prove this result, we will have a need of the following lemma.

Lemma 3.1. $\mathcal{A}$ has no eigenvalues on $i\mathbb{R}$.

Proof. The proof has two stages. The first one is $i\varpi = 0$ and the second one is $i\varpi \neq 0$.

Step 1. It is easy to see, using the boundary conditions in domain (2.9), that the equation $\mathcal{A}X = 0$ leads to $X = 0$. Then, $i\varpi = 0$ can not be an eigenvalue of $\mathcal{A}$.

Step 2. We will use a contradiction argument. Suppose that there exists $\varpi \in \mathbb{R}, \varpi\neq 0$, and $X \neq 0$, such that $\mathcal{A}X = i\varpi X$. Then, we have

Using (2.10), we find

Using $(3.1)_{5}$, we get

Hence, applying $(3.1)_{3}$ and $(2.9)_{4}$, we obtain

Inserting $(3.1)_{3}$ into $(3.1)_{4}$, we arrive at

The solution of (3.3) is given by

By (3.2), we get

Due to the transmission and boundary conditions, we obtain

Similarly, we deduce that

By the Picard Theorem, we get $X = 0$. Then, $\mathcal{A}$ has no purely imaginary eigenvalues.

Lemma 3.2. For $\varpi\neq 0$, we have the operator $i\varpi I-\mathcal{A}$ is surjective. If $\varpi = 0$ and $\zeta\neq 0$, then the operator $i\varpi I-\mathcal{A}$ is surjective.

Proof. Case 1. Suppose that $\varpi\neq 0$.

Let $F = (f1, f2, f3, f_{4}, f_{5})\in \mathcal{H}$. We seek $X = (u, \varphi, v, \psi, \Phi)\in D\left(\mathcal{A}\right)$ as a solution of

which is equivalent to

The proof is divided into several steps.

Inserting (3.5)$_{1}$ into (3.5)$_{2}$ and inserting (3.5)$_{3}$ into (3.5)$_{4}$, we get

The solving of the system (3.6) is equivalent to find $(u, v)\in H^{2}\cap H^{1}_{*}\left(0, l_{0}\right)\times H^{2}\left(l_{0}, L\right)$ such that

$\forall \left(w, \chi\right)\in H^{1}_{*}\left(0, l_{0}\right)\times H^{1}\left(l_{0}, L\right)$. By using $(3.1)_{5}$, $(2.2)_{3}$ and $(3.5)_{5}$, the functions $u$ and $v$ satisfy the following equation

where $\overline{\varsigma} = \varsigma \int^{+\infty}_{-\infty}\frac{\varrho^{2}\left(\xi\right)}{i\varpi +\xi^{2}+\zeta}\, d\xi$. We can rewrite (3.7) as follows

where

with

and

Using the principle of compactness embedding from $\left(L^{2}\left(0, l_{0}\right)\times L^{2}\left(l_{0}, L\right)\right)$ into $\left(H^{1}_{R}\left(0, L\right)\right)^{'}$ and from $H^{1}_{R}\left(0, L\right)$ into $L^{2}\left(0, l_{0}\right)\times L^{2}\left(l_{0}, L\right)$, we find that $L_{\varpi}$ is compact from $L^{2}\left(0, l_{0}\right)\times L^{2}\left(l_{0}, L\right)$ into $L^{2}\left(0, l_{0}\right)\times L^{2}\left(l_{0}, L\right)$. Consequently, by using the Fredholm alternative, to prove that $X$ is a solution of (3.8) we will prove that $1$ can not be an eigenvalue of $L_{\varpi}$. Thus, if $1$ is an eigenvalue, then $\exists X\neq 0$ and

In particular, if $Y = X$, then we have

From the definition of a null complex number, we find

By (3.9), we have

and

where $n_{1} = \sqrt{\rho^{2}_{1}\varpi^{2}+\varpi_{1}}, n_{2} = \sqrt{\rho^{2}_{2}\varpi^{2}+\varpi_{2}}, s_{1} = \tau_{1}\rho_{1}, s_{2} = \tau_{2}\rho_{2}$. We deduce now that the general solutions of (3.11) are given in the form

With the boundary conditions $u(0) = 0$ and $v(L) = v_{x}\left(L\right) = 0$, we have

Under the transmission conditions, $u\left(l_{0}\right) = v\left(l_{0}\right)$ and $s_{1}u_{x}\left(l_{0}\right) = s_{2}v_{x}\left(l_{0}\right)$, we have

Then $c_{2} = 0$. So, $X = 0$.

In this case the operator $i\varpi-\mathcal{A}$ is surjective $\forall \varpi \in \mathbb{R}^*$.

Case 2. Assume that $\varpi = 0$ and $\zeta \neq 0$.

Then, the problem (3.5) can be reduced to the problem

which gives the following system

With (3.12)$_2$ and (3.12)$_4$, using that $u\left(x\right) = 0$, we see that

From $(2.2)_{6}$, $(3.12)_{3}$, Lemma 1 and $(3.12)_{5}$, we arrive at

where $\theta = \varsigma \frac{\pi}{\sin \alpha\pi}$.

We have,

We substitute into the equation (3.13) and we find

Using the boundary transmission conditions, we get

Finally, we get that $\mathcal{A}$ is surjective and $\sigma\left(\mathcal{A}\right)\cap i\mathbb{R} = \Phi$. The proof is now completed.

4.   The absence of exponential stability

This section is devoted to the study of the absence of exponential decay of the solutions associated with (2.7). We will need some results and useful lemmas.

Theorem 4.1. ^{[11,12,13,14,15,16,17,18,19]} Let $S\left(t\right) = e^{\mathcal{A}t}$ be a $C_{0}$-semigroup of contractions on a Hilbert space. Then $S(t)$ is exponentially stable if and only if

and

Our main result is given in the following theorem.

Theorem 4.2. The semigroup generated by the operator $\mathcal{A}$ can not be exponentially stable.

Proof. We have the following two cases.

Case 1. Let $\zeta = 0$. We will show that $i\varpi = 0$ can not be in the resolvent set of $\mathcal{A}$. Note that $(\sin x, 0, \sin x, 0, 0)\in \mathcal{H}$, and let $(u, \varphi, v, \psi, \Phi)$ be the image of $(\sin x, 0, \sin x, 0, 0)$ with the operator $\mathcal{A}^{-1}$. We see that $\Phi (\xi) = - \vert \xi\vert^{\frac{2 \alpha -5}{2}}\sin L$. Then $\Phi \notin L^{2}\left(-\infty, +\infty \right)$, since $0 < \alpha < 1$. So $(u, \varphi, v, \psi, \Phi)\notin D\left(\mathcal{A}\right)$.

Case 2. Assume that $\zeta\neq 0$. We aim to show that an infinite number of eigenvalues of $\mathcal{A}$ approach the imaginary axis which prevents the wave system (1.1) from being exponentially stable. Indeed, we first compute the characteristic equation that gives the eigenvalues of $\mathcal{A}$. Let $\varpi$ be an eigenvalue of $\mathcal{A}$ with associated eigenvector $(u, \varphi, v, \psi, \Phi)$. Then $\mathcal{A}X = \varpi X$ is equivalent to

Inserting (4.1)$_{1}$, (4.1)$_{3}$ into (4.1)$_{2}$, (4.1)$_{4}$ and (4.1)$_{5}$, respectively, we get

By $(4.2)_{3}$, Lemma2.1, $(2.2)_{6}$ and the boundary conditions, we have

where $\theta = \varsigma \frac{\pi}{\sin \alpha\pi}$.

By the fact that $u(0) = 0, u(l_{0}) = v(l_{0}), \tau_{1}\rho_{2}u_{x}\left(l_{0}\right) = \tau_{2}\rho_{1}v_{x}\left(l_{0}\right)$ and (4.3), we get

The general solutions of the equations $(4.4)_{1}$ and $(4.4)_{2}$ are given by

where $t_{1} = \sqrt{\frac{\rho_{1}\varpi^{2}+\varpi \varpi_{1}}{\tau_{1}}}, \, t_{2} = -t_{1},$ $t_{3} = \sqrt{\frac{\rho_{2}\varpi^{2}+\varpi \varpi_{1}}{\tau_{2}}},$ and $t_{4} = -t_{3}$.

Thus,

where $h\left(r\right) = \frac{\tau_{2}}{\rho_{2}}r+\theta\varpi\left(\varpi+\zeta\right)^{\alpha-1}$,

and

Hence, a non-trivial solution $\varphi$ exists if and only if the determinant of $M (\varpi)$ vanishes. Let $f (\varpi) = \det M(\varpi)$. Thus, the characteristic equation is $f (\varpi) = 0$. Our purpose in the sequel is to prove by Rouché's theorem that there is a subsequence of eigenvalues for which their real parts tend to $0$. Since $\mathcal{A}$ is dissipative, we treat the asymptotic behavior of the large eigenvalues $\varpi$ of $\mathcal{A}$ in the strip $-\alpha_{0} \leq \mathcal{R}e(\varpi) \leq 0$, for some $0 < \alpha_{0}$ large enough and for such $\varpi$, we remark that $e^{t_{i}}, \, i = 1, 2$ remains bounded.

The operator $\mathcal{A}$ has no exponential decaying branch of eigenvalues. Thus, the proof is now completed. Case $\frac{\tau_{1}}{\rho_{1}} = \frac{\tau_{2}}{\rho_{2}}:$

Lemma 4.1. ^[2] There exists $N \in \mathbb{N}$ such that

where

Moreover for all $\left|k\right| \geq N$, the eigenvalues $\lambda_{k}$ are simple.

Proof.

We set

where

and

Note that $f_{0}$ and $f_{1}$ remain bounded in the strip $-\alpha_{0}\leq \mathcal{R}e\left(\lambda\right)\leq 0$.

Setep 2. We look at the roots of $f_{0}$. From (4.7), $f_{0}$ has one familie of roots that we denote $\lambda^{k}_{0}$.

Hence

i.e.,

Now with the help of Rouche's Theorem, we will show that the roots of $\widetilde{f}$ are close to those of $f_{0}$. Changing in (4.6) the unknown $\lambda$ by $u = 2\sqrt{\frac{\rho_{1}}{\tau_{1}}}\lambda L$ then (4.6) becomes

The roots of $f_{0}$ are $u_{k} = \frac{i\left(k+\frac{1}{2}\right)}{rL}\pi, k\in \mathbb{Z}.$, and setting $u = u_{k}+re^{it}, t\in \left[0, 2\pi\right]$, we can easily check that there exists a constant $C > 0$ independent of $k$ such that $\left|e^{u}+1\right|\geq Cr$ for $r$ small enough. This allows to apply Rouche's Theorem. Consequently, there exists a subsequence of roots of $\widetilde{f}$ which tends to the roots $u_{k}$ of $f_{0}$. Equivalently, it means that there exists $N \in \mathbb{N}$ and a subsequence $\left\{\lambda_{k}\right\}_{\left|k\right|\geq N}$ of roots of $f\left(\lambda\right)$, such that $\lambda_{k} = \lambda^{0}_{k}+\circ \left(1\right)$ which tends to the roots $\frac{i\left(k+\frac{1}{2}\right)}{rL}\pi$ of $f_{0}$. Finally for $\left|k\right|\geq N, \lambda_{k}$ is simple since $\lambda^{0}_{k}$ is.

Setep3. From Step 2, we can write

Using (4.9), we get

Substituting (4.10) into (4.6), using the fact that $\widetilde{f}\left(\lambda_{k}\right) = 0,$, we get:

and hence

From (4.12) we have in that case $\left|k\right|^{1-\alpha}\mathcal{R}e\lambda_{k}\approx \beta,$ with

Case $\frac{\tau_{1}}{\rho_{1}}\neq \frac{\tau_{2}}{\rho_{2}}:$

Lemma 4.2. ^[2] There exists $N \in \mathbb{N}$ such that

where

Moreover for all $\left|k\right| \geq N$, the eigenvalues $\lambda_{k}$ are simple.

Proof.

We set

where

and

We look at the roots of $f_{0}$. From (4.14), $f_{0}$ has one familie of roots that we denote $\lambda^{k}_{0}$. Indeed, $f_{0}\left(\lambda\right) = 0$ corresponds to the eigenvalues problem to the conservative problem associated with $\left(P^{'}\right)$:

where $\varsigma = \left(\pi\right)^{-1}\sin \left(\alpha\pi\right)\gamma$. For a solution $\left(u, v, \phi\right)$ of (4.15). The abstract formulation of ($P^{'}$) is

The domain of $\mathcal{A}_{0}$

We introduce the following Hilbert space (the energy space):

$\mathcal{A}_{0}$ is clearly a skew adjoint operator with a compact resolvent, then there is an orthonormal system of eigenvectors of $\mathcal{A}_{0}$ which is complete in $\mathcal{H}_{0}$. All eigenvalues of $\mathcal{A}_{0}$ are of the form. Now $i\mu_{k}, \mu_{k}\in \mathbb{R}$

By representation of graph of the functions tan and cot, we easily have $\mu_{k}\approx ck$ for large $k$ and a constant $c$ depending on parameters $\rho_{1}, \rho_{1}, \tau_{1}, \tau_{2}$, $l_{0}$ and $L$. Moreover, the algebraic multiplicity of $\mu_{k}$ is one. Then, we follow exactly as the case $\frac{\tau_{1}}{\rho_{1}} = \frac{\tau_{1}}{\rho_{1}}$. The operator $\mathcal{A}$ has a non exponential decaying branche of eigenvalues. Thus the proof is completed.

5.   Polynomial stability for $\zeta \neq 0$

In this part, we prove that (2.2) is polynomially stable when $\zeta > 0$.

Theorem 5.1. The semigroup $S_{\mathcal{A}}\left(t\right)_{t\geq 0}$ is polynomially stable and

Proof. We have a need to study the resolvent equation $\left(i\varpi I-\mathcal{A}\right)X = F$, for $\varpi\in \mathbb{R}$, namely,

where $F = \left(f_{1}, f_{2}, f_{3}, f_{4}, f_{5}\right)^{T}.$

The proof is divided into several steps

Step 1. Inserting $(5.1)_{1}$ into $(5.1)_{2}$ and $(5.1)_{3}$ into $(5.1)_{4}$, we get

where $m_{j} = \frac{\varpi^{2}\varpi_j}{\tau_{j}}-i\frac{\varpi\varpi_{j}}{\tau_{j}}$ with $j\in\left\{1, 2\right\}.$ We have

and hence,

Step2. With the fifth equation of (5.1), we get

Inserting (5.4) into the boundary condition $(2.2)_{6}$ and using Lemma 2.1, we deduce that

Using the Eq (5.2) and the Eq (5.3), we arrive at

where $\beta = \gamma\varpi\left(i\varpi+\zeta\right)^{\alpha-1}$. By $u\left(l_{0}\right) = v\left(l_{0}\right)$ and $r_{1}u_{x}\left(l_{0}\right) = r_{2}v_{x}\left(l_{0}\right)$, we get

By (5.6), we note that we can rewrite (5.5) as an equation in the unknown $C$

Step 3. We set

and we have

As $\left(f_{1}, f_{2}\right)\in \left(H^{1}_{*}\right)^{2}$ and $\left(f_{3}, f_{4}\right)\in \left(H^{1}\left(l_{0}, L\right)\right)^{2}$, we have

We can easily prove that

Then, we deduce that

Moreover, the transmission conditions are as follows

We obtain

Hence,

From (5.1)$_1$, (5.1)$_3$ and (5.2), we get

From (5.4), we have

for $\varpi\neq 0$. If $\vert \varpi \vert > 1$, we get

Thus, we conclude that

6.   Conclusions

In this work, the existence and uniqueness result for the transmission problem is proved in a functional framework by means of the semigroup theory, after a reformulation of the system above into an augmented system according to the transformation introduced in reference ^[14]. Besides this, in a series of results concerning the asymptotic behavior the following are proved: (ⅰ) the strong stability of the semigroup, by using a criteria of Arendt-Batty ^[4], (ⅱ) the impossibility of exponential decay, and (ⅲ) a polynomial decay by means of the Borichev-Tomilov theorem ^[10].

Some of previous recent works prove exponential decay but without a fractional derivative in the boundary condition. The recently published article ^[2] has a very strong relationship with our paper. Indeed, the results here are the same than those proved in ^[2] with $\varpi_j = 0, j = 1, 2$; that is without inner damping. Moreover, the same techniques are employed.

The authors in ^[2] proved that the fractional derivative in time can not ensure the exponential stability of the total system, however they shown polynomial stability. In this paper even with the inclusion of linear damping terms in the equations for $u, v$, the exponential stability of the total system is not achieved under an appropriate conditions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by Researchers Supporting Project number (RSPD2023R736), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors agree with the contents of the manuscript, and there is no conflict of interest among the authors.