In this paper, we investigate the stability of the transmission problem for Rayleigh beam model with heat conduction. First, we reformulate our system into an evolution equation and prove our problem’s well-posedness. Next, we demonstrate the resolvent of the operator is compact in the energy space, then by using the general criteria of Arendt–Batty, we prove that the thermal dissipation is enough to stabilize our model. Finally, a polynomial energy decay rate has been obtained which depends on the mass densities and the moments of inertia of the Rayleigh beams.
In this paper, we study the stability of a transmission problem for Rayleigh beam model with heat conduction
with boundary conditions
transmission conditions
and initial data
where, for , is the mass density per unit volume, is the moment of inertia of the cross-sections, is the stiffness constant, while and represent, respectively, the specific heat and the thermal conductivity. Here and γ is a non-zero real number.
The model at hand describes a Rayleigh beam formed of two distinct materials, one of which is sensitive to thermal differences and the other of which is unaffected by temperature changes. In other words, the material has a limited thermoelastic effect [46,47].
The stabilization of the Rayleigh beam equation retains the attention of many authors. In this regard, different types of damping have been introduced to the Rayleigh beam equation and several uniform and polynomial stability results have been obtained. Rao [45] studied the stabilization of Rayleigh beam equation subject to a positive internal viscous damping. Using a constructive approximation, he established the optimal exponential decay rate. There exists many papers concerning the stability with different types of damping [18,32–35,38,51].
In [50], the authors are concerned with the stability of an interconnected system of an Euler–Bernoulli beam and a heat equation with boundary coupling. The boundary temperature of the beam is fed as the boundary moment of the Euler–Bernoulli equation and the boundary angular velocity of the Euler–Bernoulli beam is fed into the boundary heat flux of the heat equation. It is shown that the spectrum of the closed-loop system consists of only two branches: one along the real axis and the other along two parabolas that are symmetric to the real axis and open to the imaginary axis. The asymptotic expressions of both eigenvalues and eigenfunctions are obtained. With a careful estimate of the resolvent operator, the completeness of the root subspaces of the system is verified. The Riesz basis property and exponential stability of the system are then proved. Moreover, it is shown that the semigroup generated by the system operator is of Gevrey class .
In [57], the authors studied the stabilization problem for a coupled PDE system in which the beam (1-dimensional or 2-dimensional) and heat equations are coupled at the boundary conditions. Moreover, a dissipative damping is produced in the heat equation via the boundary connections only. In the first part, the authors considered the asymptotic behavior of the 1-dimensional coupled system mainly by the Riesz basis approach. By using a detailed spectral analysis for the system operator, they obtained asymptotic expressions for the spectrum and the corresponding eigenvectors. The authors further obtained a spectrum-determined growth condition by showing the Riesz basis property of the eigenvectors. Then, based on the spectral distribution, they deduced the Gevrey regularity of the semigroup for the system and the exponential decay rate of the system energy. In the second part, the authors investigated the asymptotic behavior of the 2-dimensional coupled PDE system by using the frequency domain method. By estimating the uniform boundedness of the norm of the resolvent operator along the imaginary axis, they showed that the coupled system is also exponentially stable when an additional dissipation in the boundary of the plate part exists.
We mention some papers studied the stability of different system under heat conduction [1,2,4–6,8–17,20,25–30,39–42,44,48,49,53–55].
Now, we mention some papers concerning a transmission wave-heat system. In [56], the author studied the stability analysis of an interaction system comprised of a wave equation and a heat equation with memory, where the hereditary heat conduction is due to Gurtin–Pipkin law or Coleman–Gurtin law. First, she showed the strong asymptotic stability of solutions to this system. Then, the exponential stability of the interaction system is obtained when the hereditary heat conduction is of Gurtin–Pipkin type. Further, she showed the lack of uniform decay of the interaction system when the heat conduction law is of Coleman–Gurtin type. In [24], the authors extended the result of [56] by proving the optimal polynomial decay rate of type when the heat conduction law is of Coleman–Gurtin type.
To our best knowledge, the transmission problem for Rayleigh beam with heat conduction is not treated in the literature. The goal of this paper is to fix this gap by considering System (1.1)–(1.12).
The paper is organized as follows: In Section 2, we formulate the System (1.1)–(1.12) into an evolution equation , (see (2.14)). Next, Section 3 is divided into two subsections. In Section 3.1 we study the well-posedness of Problem (1.1)–(1.12). According to Lumer–Phillips theorem (see [37,43]), we prove that the operator is m-dissipative. In Section 3.2, we prove the strong stability of (1.1)–(1.12). Firstly, we prove that the operator has a compact resolvent on the energy space. Next, we prove the strong stability of System (1.1)–(1.12) by using Arendt–Batty Theorem. In Section 4, we prove the polynomial stability of System (1.1)–(1.12). The decay rate of the energy depends on the physical coefficients. We obtain the following result:
A polynomial energy decay rate of type if and .
A polynomial energy decay rate of type if or .
We use Borichev–Tomilov Theorem combining with a specific multiplier technics and a particular attention of the sharpness of the estimates to optimize the results.
Formulation of the problem
We start this section by defining the energy of a solution of System (1.1)–(1.12) by
Multiplying (1.1) and (1.3) by and θ, respectively, integrating by parts over with respect to x and taking the sum of the resulting equations, we get
Next, multiplying (1.2) by , integrating by parts over with respect to x, we obtain
Adding (2.1) and (2.2), then using the boundary condition (1.4)–(1.10), we infer that
Hence, System (1.1)–(1.12) is dissipative in the sense that its energy is non increasing with respect to the time t.
We start our study by formulating problem (1.1)–(1.12) in an appropriate Hilbert space:
∙ We introduce the following spaces:
Set
∙ Let be a regular solution of System (1.1)–(1.12). Let . Multiplying (1.1), (1.2), and (1.3) by , , and , respectively, integrating by parts over , , and , respectively and then taking the sum, we derive
Since , then
Using the above boundary conditions in (2.3), we get
Using the boundary conditions (1.9) and (1.10) in the above equation, we obtain
Equivalently, the variational equation of problem (1.1)–(1.12) is given by
∙ We identify with its dual and with its dual , so that we have the following continuous embeddings:
∙ We introduce the following sesquilinear forms:
Here and below, and denote the usual inner product of and , respectively, and and their correspsesquilinearonding norms. The form (resp. ) is a sesquilinear continuous coercive form on (resp. on ), while is a sesquilinear continuous form on and satisfies
∙ We define the operators , , and by
The operator C (resp. ) is an isomorphism of onto (resp. onto ) and is the canonical isomorphism, so we can introduce (resp. as a scalar product on (resp. on ), i.e.,
∙ The variational equation (2.4) can be written in terms of the above operators as an equation in as follows:
Furthermore, assume that , then we obtain that
Defining and , then (2.10) can be written as
∙ We introduce the following energy space:
For all and , such that , , , and , it is easy to check that the space is a Hilbert space over equipped with the following inner product
Hereafter, we use to denote the corresponding norm.
∙ For all , such that and , we define the unbounded linear operator by
with domain
∙ If is a regular solution of System (1.1)–(1.12), then we rewrite this system as the following evolution equation
where .
Well-posedness and strong stability
Well-posedness of the problem
For the well-posedness of Problem (1.1)–(1.12), according to Lumer–Phillips theorem (see [37,43]), we need to prove that the operator is m-dissipative. Hence, we shall prove the following proposition.
The unbounded linear operatoris m-dissipative in the energy space.
We first prove that is monotone. For this aim, let , such that and , using the definitions (2.9), (2.11), (2.13), and (2.12), we have
Since ; i.e., and , then using (2.5), the last equation of (2.8), and second-third equations of (2.8) in the above equation, we obtain
Finally, taking the real parts of the above equation, then using (2.7), we get
We next prove the maximality. For , such that and , we show the existence of , such that and , unique solution of the equation
that is
Since the operator C is an isomorphism of onto , then the above system is equivalent to
Inserting the first equation of (3.3) in the second equation, we obtain that
Since , , and the operator C (resp. ) is an isomorphism of onto (resp. onto ), then using (2.5) and the definition of (see last equation of (2.8)), we get
Using the above equation in (3.4), we get
We define the operator by
From (2.7) and (2.9), we have
So, by using Lax–Milgram lemma, for all , we get that has a unique solution . Consequently, since , we get that (3.5) has a unique solution . Next, we define . Since , we get . Consequently, is the unique solution of (3.3). In addition, since , , and the operator C is an isomorphism of onto , we get
Thus, (3.2) has a unique solution , completing the proof of the proposition. □
Thanks to Lumer–Phillips theorem (see [37,43]), we deduce that generates a -semigroup of contractions in and therefore Problem (2.14) is well-posed. Then, we have the following result.
For any, the Problem (
2.14
) admits a unique weak solution. Moreover, if, then.
Strong stability of the system
Our main result in this part is the following theorem.
The semigroup of contractionsis strongly stable onin the sense that
For the proof of Theorem 3.3: First we will prove that the operator has a compact resolvent on the energy space . Then, we will establish that has no eigenvalues on the imaginary axis. The proof for Theorem 3.3 relies on the subsequent lemmas.
Let, such that, and. Then, we haveIn particular, the resolventofis compact on the energy space.
The proof is divided into 3 steps.
∙ Step 1. In this step, we write the variational problem and we prove (3.6). For this aim, let and , such that
where , , , and . Equation (3.10) is equivalent to
From the first equation of the above system, we obtain (3.6). For all , using the above equation, (2.5), and (2.8), one gets
Using (2.6) in the above equation, we obtain that for all :
∙ Step 2. In this step, we prove (3.7) and
For this aim, setting , , and in (3.11), we obtain
The left hand side of (3.13) is a sesquilinear continuous coercive form on , while the right hand side is a linear continuous form on . Then, using Lax–Milgram lemma, we deduce that there exists unique solution of the variational problem (3.13). Applying classical regularity arguments, we infer that , hence we get (3.7). Consequently, setting in (3.13), then using integration by parts, we obtain
Thus, by applying Corollary 4.24 in [22], we get
Solving the above differential equation (taking into consideration that ; i.e., ), we obtain (3.12).
∙ Step 3. In this step, we prove (3.8) and (3.9). For this aim, setting , and in (3.11), then using (3.12), we obtain
The left hand side of (3.14) is a sesquilinear continuous coercive form on , while the right hand side is a linear continuous form on . Then, using Lax–Milgram lemma, we deduce that there exists unique solution of the variational problem (3.14). Now, fix such that
For any function , we define
Indeed, the function and
Thus , and consequently, by substituting (3.15) in (3.14), we derive
We have
In the above equation, for the first term, using integration by parts, we get
By the same way, using integration by parts, we get
Next, we have
and
Replacing (3.17)–(3.21) in (3.16), we obtain
where
and
Taking in (3.22), we get that
thus, by using the definition of (see page 202 in [22]), we get , and consequently (3.8) holds true. Back to (3.23), using integration by parts in the left hand side, one derives
consequently, by applying Corollary 4.24 in [22], we obtain
Finally, using integration by parts in the left hand side of (3.22), then using (3.24) and taking , it holds that
thus, we obtain (3.9).
∙ Step 4. In this step, we prove that the resolvent of is compact on the energy space . For this aim, let and , such that . Since is monotone, it follows that
The result follows from the above inequality and (3.6)–(3.8). This completes the proof of the lemma. □
For all, we havewheredenotes the Kernel of.
Let , such that is an eigenvalue of the operator and a corresponding eigenvector, where and . Therefore, we have
Similar to (3.1), we get
Consequently, we deduce that
Since (i.e., ), we get
Next, writing (3.25) in a detailed form gives
Since the operator C is an isomorphism of onto , then the above system is equivalent to
Inserting the first equation of (3.27) in the second one, then using (3.26), we obtain that
For all , using the above equation, (2.5), and (2.8), we get
Using (2.6) in the above equation, we obtain
Now, setting , , and in (3.28), we get
In the above equation, using integration by parts, we see that
Therefore, by applying Corollary 4.24 in [22], we have
Since and , one derives
Since , then from (3.29) and by the help of Lemma 3.4, we obtain
Next, setting and in (3.28), then using (3.29), one has
Using integration by parts, after that using (3.30) and the fact that , we get
Taking in (3.22), we find that
thus, by using the definition of (see page 202 in [22]), we get . Again, using integration by parts in the right hand side of (3.32), we infer
Consequently, by applying Corollary 4.24 in [22], then using (3.30), we obtain
Using integration by parts in the left hand side of (3.31), after that using (3.33) and the fact that , then taking , one has
Therefore, from (3.30), (3.33) and (3.34), we get
Multiplying (3.35) by , integrating by parts over , then taking the real parts, we find
Using integration by parts and the boundary conditions of (3.36), we arrive at
Consequently, from the above equation and the boundary conditions of (3.36), we obtain
Finally, from the above equation, first equation of (3.27), (3.26), and (3.29), we get . The proof is thus complete. □
From Lemma 3.4, we have that the operator has a compact resolvent. In addition, from Lemma (3.5), we get that the operator has no pure imaginary eigenvalues. Thus, we get the conclusion by applying Arendt and Batty theorem (see Theorem A.2 and Corollary A.3). □
Polynomial stability
In this section, we will prove the polynomial stability of System (1.1)–(1.12). Our main results in this part are the following theorems.
Ifthen for all initial data, there exists a constantindependent ofsuch that the energy of System (
1.1
)–(
1.12
) satisfies the following estimation
Ifthen for all initial data, there exists a constantindependent ofsuch that the energy of System (
1.1
)–(
1.12
) satisfies the following estimation
From Lemma 3.4 and Lemma 3.5, we have seen that , then for the proof of Theorems 4.1 and 4.2, according to Borichev and Tomilov [21] (see Theorem A.4), we need to prove that
where (resp. ) if condition (4.1) (resp. condition (4.2)) holds. We will argue by contradiction. We suppose that there exists
such that
and there exists a sequence , such that
where , , , and . We will check condition (4.3) by finding a contradiction with such as . By detailing Equation (4.5), we get the following system
The proof of Theorems 4.1 and 4.2 is divided into several lemmas.
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimationsIn addition, we haveand
For all , using the second equation of (4.6) and equations (2.5) and (2.9), we get
Since ; i.e., and , then using (2.5), the last equation of (2.8), and second-third equations of (2.8) in the above equation, we obtain
Consequently, from the above equation and (2.6), we find
On the other hand, from first equation of (4.6), we have
Inserting the last equations in (4.12), we infer that
In the above equation, using integration by parts and equation (3.8), we obtain
Since and , then from (2.13) and (3.9), we have the following boundary conditions
Substituting the above boundary conditions in (4.14), we derive that
∙ Taking in equation (4.15), one finds
and consequently, we get
Hence, (4.9) holds true.
∙ Taking in equation (4.15), we get
Since , we get
Thus, we get the first estimation of (4.10). Consequently, integrating by parts (4.16), we obtain
Thus, one has
Hence, we derive (4.7).
∙ By the same way, taking in equation (4.15), one gets
and
Thus, we find (4.8) and the second estimation of (4.10).
∙ Taking and in (4.15), then using integration by parts for the first and third terms, we obtain
Substituting (4.7) and (4.8) in the above equation, we obtain (4.11). The proof is thus complete. □
From (4.4) and (4.13), we remark that
From now, we denote by a positive constant number, such that independent of n and , also we denote by a positive constant number independent of n and for , is a positive number depending on n such that .
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimation
Taking the inner product of (4.5) with in , then using Cauchy Schwarz inequality, we get
Now, similar to Equation (3.1), we have
where . Using (4.4) and (4.5), we get . Hence, from (4.19), we obtain the first asymptotic estimate of (4.18). Since , using (4.19) and Poincaré’s inequality, we infer that
where is the Poincaré constant and . Hence, from (4.20), we get the second asymptotic estimate of (4.18). The proof is thus complete. □
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimations
For the proof of Lemma 4.5, we need the following lemmas.
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimation
First, since , then applying (A.1), we obtain
Thus
In the above inequality, using the first estimation of (A.6), we get
Consequently, one derives
Next, from (4.7) and (4.9), we obtain
and
Now, substituting (4.26) in (4.25), we find
Since with , using Poincaré’s inequality, we obtain that there exists independent of n, such that
Inserting the above inequality into (4.27), we derive
where
Since , , and in , then . Substituting (4.28) in (4.24), we get
where
From (4.17), (4.20), the fact that and in , we obtain . Next, taking and in (A.3), then using (4.20), one gets
Inserting the above inequality in (4.29), we have
where . In the above inequality, let
it holds that
Hence, we get (4.23). □
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimationwhere.
First, using the first estimation of (A.6) for (4.23), we get
Consequently, we have
From (4.19), we obtain
Next, taking and in (A.4), then using (4.19), we derive
Substituting (4.32) and (4.33) in (4.31), we get (4.30), where . □
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimationwhere.
First, let’s take
Then, we have
Next, from (4.9), we have
Multiplying the above equation by in , taking the real parts, then using integration by parts and (4.35), we get
Consequently, we obtain
Thus, using (4.35) and the first estimation of (A.6) in the above inequality, one has
Consequently, we get
where
Using (4.19)–(4.20) and the fact that in , we find
Hence, from (4.36), we get (4.48). □
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimation
First, since , then applying (A.2), we obtain
consequently, using the second estimation of (A.6), one has
Substituting (4.23) in (4.38), we derive
Taking and in (A.3), we obtain
Substituting the above equation in (4.39), we see that
Hence (4.37) holds true, with and . □
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimationwhere.
First, from (4.37) and (4.34), we get
Now, taking and in (A.3), we obtain
Next, taking , , and in (A.5), then using (4.19), we find
Also, taking , , and in (A.5), then using (4.19), we have
Substituting (4.42)–(4.44) in (4.41), we get (4.40), where
□
First, from (4.9), we have
Multiplying the above equation by in , then taking the real parts and using integration by parts, we get
Consequently, using Cauchy Schwarz inequality, we obtain
Using the second estimation of (A.6) in (4.45), one finds
Taking and in (A.4), then using (4.20), we get
On the other hand, we have
Substituting (4.47) and (4.48) in (4.46), we infer that
Substituting (4.30) and (4.40) in (4.49), we obtain
where
Using (4.17) and in , we get . Thus, from (4.50), we get
Taking , it holds that
Thus, we obtain (4.21). Finally, substituting (4.21) in (4.37), we obtain (4.22). The proof is thus complete. □
For all, the solutionof System (
4.6
) satisfies the following asymptotic behavior estimationswhere
The proof will be split into several steps:
Step 1. In this step, we prove the following asymptotic behavior estimates:
In fact, since with , then using (4.21), Poincaré’s inequality and trace theorem, we get (4.53).
Step 2. In this step, we prove the following asymptotic behavior estimate:
For this aim, first, multiplying (4.7) by in , then taking the real parts, using integration by parts and the fact that , we obtain
Next, multiplying (4.8) by in , then taking the real parts, using integration by parts and the fact that , one derives
Adding (4.55) and (4.56), then using (3.9), (4.11), and the fact that , , we find
Using (4.17), the fact that in and in , it holds that
On the other hand, from (4.18) and (4.53), we infer
Substituting (4.58) and (4.59) in (4.57), we obtain (4.54).
Step 3. In this step, we prove the following asymptotic behavior estimate:
For this aim, first, multiplying (4.7) by in , taking the real parts, then using integration by parts and the fact that , we get
Next, multiplying (4.8) by in , taking the real parts, then using integration by parts and the fact that , we obtain
Adding (4.61) and (4.62), then using (3.9), (4.11), and the fact that , , we get
From (4.22), it holds that
Using (4.17), the fact that in and in , we derive
On the other hand, using Cauchy–Schwarz inequality, (4.18), (4.21) and (4.53), we obtain
Substituting (4.64)–(4.66) in (4.63), we get (4.60).
Step 4. In this step, we will prove (4.51)–(4.52). First, adding (4.54) and (4.60), we find that
We distinguish two cases:
Case 1. If and , then substituting (4.21) in (4.67), one derives
Substituting (4.21) and (4.68) in (4.54), we get
Consequently, we obtain
Thus, from (4.13), (4.68), (4.69), and the fact that in , we get (4.51)–(4.52).
Case 2. If or , then substituting (4.21) and (4.53) in (4.67), we infer
Consequently, we obtain
Inserting (4.21), (4.70), and (4.53) in (4.54), we get
Thus from (4.13), (4.53), (4.70), (4.71), and the fact that in , we get (4.51)–(4.52). The proof is thus complete. □
When and , we choose , then from Lemmas 4.4, 4.5, and 4.11, we get which contradicts (4.4). This implies that
The result follows from Theorem A.4. □
When or , we choose , then from Lemmas 4.4, 4.5, and 4.11, we get which contradicts (4.4). Hence
The result follows from Theorem A.4. □
Conclusion and open problems
Conclusion
In this paper, we investigate the stability of a transmission Rayleigh beam with heat conduction. A polynomial energy decay rate has been obtained which depends on the physical constant. We obtain the following result:
A polynomial energy decay rate of type if and .
A polynomial energy decay rate of type if or .
Open problems
In this part, we present some open problems:
The optimality of the polynomial decay rate of the System (1.1)–(1.12). But, we conjecture that the polynomial energy decay rate obtained in Theorem 4.1 and Theorem 4.2 is optimal. The idea of the proof is to find a sequence with , as and a sequence of vectors such that is bounded in and
(see for example Theorem 3.1 in [52], Theorem 5.1 in [3] and [23]). Depending on the boundary conditions and the transmission conditions, this approach and the construction of the vector are not feasible and the question is still an open problem.
What happened if we consider a heat conduction with memory, where the hereditary heat conduction is due to Coleman–Gurtin law or Gurtin–Pipkin law? (See for instance [2,23,24,56]).
Footnotes
Acknowledgement
This work was supported by Researchers Supporting Project number (RSPD2023R736), King Saud University, Riyadh, Saudi Arabia.
Declarations
Notions of stability and theorems used
We introduce here the notions of stability that we encounter in this work.
We now look for necessary conditions to show the strong stability of the -semigroup . We will rely on the following result obtained by Arendt and Batty in [7].
For necessary conditions to show the polynomial stability of the -semigroup , we will rely on the frequency domain approach method that has been obtained by Batty in [19], Borichev and Tomilov in [21], Liu and Rao in [36].
We will recall two forms of Gagliardo–Nirenberg inequality (see [37]) which will be used in this work.
We will recall Young inequality and we will prove some inequalities that will be used in this work.
We will recall the relations between p norms on .
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