Abstract
In this work, we consider a new full von Kármán beam model with thermal and mass diffusion effects according to the Gurtin-Pinkin model combined with time-varying delay. Heat and mass exchange with the environment during thermodiffusion in the von Kármán beam. We establish the well-posedness and the exponential stability of the system by the energy method under suitable conditions.
1. Introduction and Preliminaries
In this paper, we are concerned with the following problem: where
Here, represents the time-varying delay, and , , , , , , , and are positive constants; is a real number, and and are the relaxation functions, with the initial data where and Neumann-Dirichlet boundary conditions
The case of time-varying delay in the wave equation has been studied recently by Nicaise et al. [1]; they proved the exponential stability under the condition where is a constant that satisfies
For the wave equation with a time-varying delay, in [1], the authors consider the system where the time-varying delay satisfies
They proved the exponential stability under suitable conditions.
The purpose of this work is to study problem (1)–(5), with a delay term appearing in the control term at the first equation, introducing the time-varying delay term ; thermal and mass diffusion effects make the problem different from those considered in the literature (see [2–30]).
This paper is organized as follows: in the rest of this section, we put the preliminaries necessary for problem (1); in Section 2, we establish the well-posedness. As for Section 3, we prove the exponential stability result by the energy method and Lyapunov function.
In order to prove the existence of a unique solution of problem (1)–(5), we introduce the new variable
Then, we obtain
And it is more convenient to work in the history space setting by introducing the so-called summed past history of and defined by (see [31–36])
Differentiating (14)1 and (14)2, we get with the boundary and initial conditions
We set
Concerning the memory kernels and , we set
Assuming , then from (14), we infer and therefore,
Consequently, the problem is equivalent to where with the initial and boundary conditions where the function satisfies (7), (11), and the condition
In this paper, we establish the well-posedness and prove the exponential stability by using the variable of Kato under some restrictions and assumptions:
(H1).
(H2). The symmetric matrix is positive definite, where
That is, implies that
Condition (28) is needed to stabilize the system when diffusion effects are added to thermal effects (see, e.g., [31–38] for more information on this). By virtue of , we deduce that . Let, then, be a number chosen in such a way that
Thus, Young’s inequality leads to
(H3). We assume the following set of hypotheses on and :
Let be a memory kernel satisfying the assumptions (31) and (32).
Now, we consider the weighted Hilbert spaces equipped with the inner product and the norm
We also introduce the linear operator on defined by with where is the distributional derivative of with respect to the internal variable , and then, the operator is the infinitesimal generator of a -semigroup of contractions. Following Ref. [39], there holds
Integration by parts yields
Hence, from (31), we obtain
As a direct consequence, we deduce from (32) and (40) that for all . Finally, we define the operator by with the domain
2. Well-Posedness
In this section, we give sufficient conditions that guarantee the well-posedness of this problem. Let
For the sake of simplicity, we write and and the new dependent variables and ; then, (21)–(23) can be written as with the linear problem where the time-varying operator is defined by
The energy space is defined as and the domain of is
We equip with the inner product with the existence and the uniqueness in the following result.
Theorem 1. Let (7), (11), and (25) be satisfied and assume that (26)–(31) hold. Then, for all , there exists a unique solution of problem (21)–(23) satisfying
In order to prove Theorem 1, we will use the variable norm technique developed by Kato in [40]. The following theorem is proved in [40].
Theorem 2. Assume that (1) is a dense subset of (2)(3)For all , generates a strongly continuous semigroup on and the family is stable with stability constants and independent of ; i.e., the semigroup generated by satisfies (4), where is the space of equivalent classes of essentially bounded, strongly measurable functions from into the set of bounded operators from into Then, problem (46) has a unique solution for any initial datum in .
Proof. To prove Theorem 1, we use the method in [1] with the necessary modification.
(1)First, we show that is dense in Let be orthogonal to all elements of with respect to the inner product :
For all , our goal is to prove that . Let us first take and , so the vector , and therefore, from (55), we deduce that
Since is dense in , it follows then that .
Similarly, let ; then, , which implies from (55) that
So, as above, .
And let ; then, we obtain from (55) that
It is obvious that only if is dense in , with respect to the inner product
We get . By the same ideas as above, we can also show that .
For , we get from (55) that
and by the density of in , we obtain .
For , we get from (55) that
and by the density of in , we obtain .
Next, let ; then, we obtain from (55) that
It is obvious that only if is dense in ; we get ; for , we get from (55) that
which gives . Similarly, for and . This completes the proof of (1).
(2)With our choice, is independent of ; consequently,
(3)Now, we show that the operator generates a -semigroup in for a fixed . We define the time-dependent inner product on :where satisfies
thanks to hypothesis (26).
Let us set
In this step, we prove the dissipativity of the operator .
For a fixed and , we have
Observe that
whereupon
By using Young’s inequality and (7), we get
under condition (66) which allows to write
Consequently, the operator is dissipative.
Now, we prove the subjectivity of the operator for fixed .
Let ; we seek solution of the following system:
Suppose that we have found and . Then,
Furthermore, by (73), we can find as
Following the same approach as in [1], we obtain, by using the last equation in (73),
where . Whereupon, from (74), we obtain
Integrating (73)6 and (73)8 with , we have
Substituting (73)1,3,6,8,9 into the others, we obtain the following system. Now, we have to find , , , and as solutions of the equations:
Solving (79), we get
where
From (77), we have
where and
It is clear from the above formula that depends only on . Consequently, problem (80) is equivalent to
where the bilinear form and the linear form are defined by
Now, for , equipped with the norm
then, we have
Then, for some ,
Thus, is coercive.
By Cauchy-Schwarz’s and Poincaré’s inequalities, we obtain
Similarly, we get
Consequently, applying the Lax-Milgram theorem, problem (84) admits a unique solution , for all . Applying the classical elliptic regularity, it follows from (80) that .
Therefore, the operator is surjective for any fixed . Since and
we deduce that the operator is also surjective for any .
To complete the proof of (3), it suffices to show that
where and is the norm associated with the inner product (56).
For , we have from (56) that
It is clear that . Now, we will prove that for . To do this, we have
where , which implies
By using (11), we deduce that
which proves (92); therefore, this completes the proof of (3).
(4)It is clear thatThen, by (11) and (25), (4) holds exactly as in [1]. Consequently, from the above analysis, we deduce that the problem
has a solution , and if , then
Now, let
with ; then, by using (98), we have
Consequently, is the unique solution of (46).
It remains to prove that the operator defined in (48) is locally Lipschitz in .
Let and . Then, we have
where
Adding and subtracting the term inside the norm , we find
Using the embedding of into , from (104), one has
Using once again the embedding of into , one also sees that
Combining (102), (105), and (106), consequently, is locally Lipschitz continuous in . This ends the proof of Theorem 1.
3. General Decay
In this section, we state and prove the stability of system (21)–(23) using the multiplier technique under the assumptions (26)–(31).
We define the energy functional by where
The following lemma shows that the energy is decreasing.
Lemma 3. Assume that (26)–(31) hold and the hypotheses (7), (11), and (25) are satisfied. Then, for ,
Proof. Multiplying the equations of (21) by , , , , , , and , respectively, then by integration by parts, we get From (110), we find Using Young’s inequality, we have Inserting (112) into (111), we get Then, by using (7), (28)–(31), and (108), we obtain (109).
In the following, we state and prove our stability result; we introduce and prove several lemmas.
Lemma 4. The functional satisfies, for any ,
Proof. By differentiating , then by integration by parts, we obtain In what follows, using Young’s and Poincaré’s inequalities, we obtain (115).
Then, we have the following lemma.
Lemma 5. The functional where , with , satisfies
Proof. For direct computations, we have Using Young’s inequality and integrating by parts, we obtain From (120) and (121), we obtain (118).
Lemma 6. Assuming that assumptions (31) and (32) hold, the functional satisfies where and satisfies (29).
Proof. We take the derivative of , which gives
The first term on the right-hand side of (125) is
and can be controlled in the following way:
Moreover, by integration by parts, we get
where . Similarly, we obtain
where . Using (29), we get
Then, we obtain
where . Then, using the same arguments, we find
Adding (127) and (133), we obtain (123).
We choose in such a way that
which implies
Then, satisfies (29).
Now, let us introduce the following functional.
Lemma 7. The functional satisfies where is a positive constant.
Proof. By differentiating , with respect to , we have By using the last equation of (21), we have Using (137)–(139), we get Then, by using (7), (25), and the fact that and setting , we obtain (137).
We are now ready to prove the following result.
Theorem 8. Assume (26)–(31) hold; there exist positive constants and such that the energy functional given by (107) satisfies
Proof. We define a Lyapunov functional
where and , , are positive constants to be selected later.
By differentiating (142) and using (109), (115), (118), (123), and (137), including the relation
we get
First, we choose small enough such that
By setting
we obtain
Next, we carefully choose our constants so that the terms inside the brackets are positive.
We choose large enough such that
Then, we choose large enough such that
Thus, we arrive at
where and .
On the other hand, we let
Exploiting Young’s, Cauchy-Schwarz’s, and Poincaré’s inequalities, we get
Then,
Consequently, we obtain
that is,
Now, we choose large enough such that
Exploiting (107), estimates (150) and (155), respectively, give
for some , and
for some ; we have
A combination with (157) and (158) gives
where .
Finally, by simple integration of (159) and (160), we obtain the result (141).
Data Availability
No data were used to support the study.
Conflicts of Interest
This work does not have any conflicts of interest.
Acknowledgments
The fifth author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under Grant No. G.R.P-10/42.