Well-Posedness and Stability Result of the Nonlinear Thermodiffusion Full von Kármán Beam with Thermal Effect and Time-Varying Delay

Abdelbaki Choucha, Djamel Ouchenane, Salah Mahmoud Boulaaras , Bahri Belkacem Cherif , and Mohamed Abdalla 6,7 Laboratory of Operator Theory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, Algeria Laboratory of Pure and Applied Mathematics, Amar Telidji Laghouat University, Algeria Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Algeria Preparatory Institute for Engineering Studies in Sfax, Tunisia Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt


Introduction and Preliminaries
In this paper, we are concerned with the following problem: Here, τðtÞ > 0 represents the time-varying delay, and d 1 , d 2 , δ 1 , δ 2 , c, d, r, and μ 1 are positive constants; μ 2 is a real number, and β 1 and β 2 are the relaxation functions, with the initial data w x, 0 ð Þ= w 0 x ð Þ, where and Neumann-Dirichlet boundary conditions w x, t ð Þ= u x, t ð Þ= P x, t ð Þ= 0, x = 0, L,∀t ≥ 0, 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 d 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 τðtÞ > 0 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 β 2 w t ðx, t − τðtÞÞ; thermal and mass diffusion effects make the problem different from those considered in the literature (see ). 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 θ and P defined by (see [31][32][33][34][35][36]) Differentiating (14) 1 and (14) 2 , we get with the boundary and initial conditions We set Concerning the memory kernels β 1 and β 2 , we set Assuming β 1 ð∞Þ = β 2 ð∞Þ = 0, then from (14), we infer Journal of Function Spaces and therefore, Consequently, the problem is equivalent to where with the initial and boundary conditions where the function τðtÞ 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 Λ is positive definite, where That is, Condition (28) is needed to stabilize the system when diffusion effects are added to thermal effects (see, e.g., [31][32][33][34][35][36][37][38] for more information on this). By virtue of cr > d 2 , we deduce that d/c < r/d. Let, then, ζ be a number chosen in such a way that Thus, Young's inequality leads to (H3). We assume the following set of hypotheses on μ and λ: Let f be a memory kernel satisfying the assumptions (31) and (32). Now, we consider the weighted Hilbert spaces equipped with the inner product 3 Journal of Function Spaces and the norm We also introduce the linear operator T on M f defined by with where Φ σ is the distributional derivative of Φ with respect to the internal variable σ, and then, the operator T is the infinitesimal generator of a C 0 -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) for all η, ν ∈ DðTÞ. Finally, we define the operator L f : with the domain

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 η = η t ðσÞ and ν = ν t ðσÞ and the new dependent variables φ = ω t and ψ = u t ; then, (21)-(23) can be written as with the linear problem where the time-varying operator A is defined by Journal of Function Spaces The energy space H is defined as and the domain of A is We equip H with the inner product with the existence and the uniqueness in the following result.
(3) For all t ∈ ½0, T, AðtÞ generates a strongly continuous semigroup on H and the family A = fAðtÞ: t ∈ ½0, Tg is stable with stability constants C and m independent of t; i.e., the semigroup ðS t ðsÞÞ s≥0 generated by AðtÞ satisfies for any initial datum in DðAð0ÞÞ.
Proof. To prove Theorem 1, we use the method in [1] with the necessary modification.
For u ∈ DðAðtÞÞ, we get from (55) that and by the density of DðAðtÞÞ in H 1 0 ð0, LÞ, we obtain f 3 = 0. For ψ ∈ DðAðtÞÞ, we get from (55) that and by the density of DðAðtÞÞ in H 1 ð0, LÞ, we obtain f 4 = 0.
(2) With our choice, DðAðtÞÞ is independent of t; consequently, (3) Now, we show that the operator AðtÞ generates a C 0 -semigroup in H for a fixed t. We define the time-dependent inner product on H : where ξ satisfies thanks to hypothesis (26). Let us set In this step, we prove the dissipativity of the operator AðtÞ = AðtÞ − τðtÞI.
For a fixed t and U = ðw, φ, u, ψ, θ, η, P, ν, zÞ T ∈ DðAðtÞÞ, we have Observe that Journal of Function Spaces By using Young's inequality and (7), we get under condition (66) which allows to write Consequently, the operator AðtÞ = AðtÞ − κðtÞI is dissipative. Now, we prove the subjectivity of the operator I − AðtÞ for fixed t > 0.
To complete the proof of (3), it suffices to show that where U = ðw, φ, u, ψ, θ, η, P, ν, zÞ T and k:k t is the norm associated with the inner product (56).
(4) It is clear that Then, by (11) and (25), (4) holds exactly as in [1]. Consequently, from the above analysis, we deduce that the problem Now, let with ϑðtÞ = Ð t 0 κðsÞds; then, by using (98), we have Consequently, UðtÞ is the unique solution of (46). It remains to prove that the operator F defined in (48) is locally Lipschitz in H . Let where Adding and subtracting the term ðu 1x + ð1/2Þw 2 1x Þw 2x inside the norm |R | , we find Using the embedding of H 1 ð0, LÞ into L ∞ ð0, LÞ, from (104), one has Using once again the embedding of H 1 ð0, LÞ into L ∞ ð0, LÞ, one also sees that Combining (102), (105), and (106), consequently, FðUÞ is locally Lipschitz continuous in H . This ends the proof of Theorem 1.

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 E by where The following lemma shows that the energy is decreasing. (7), (11), and (25) are satisfied. Then, for ∀C ≥ 0,
In the following, we state and prove our stability result; we introduce and prove several lemmas.

Lemma 4. The functional
satisfies, for any ε 1 > 0, Proof. By differentiating F 1 , 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.
Proof. We take the derivative of F 3 = G 1 + G 2 , 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 C β 0 > 0. Similarly, we obtain where C ′ β 0 > 0. Using (29), we get Then, we obtain Then, ζ satisfies (29).
Now, let us introduce the following functional.

Lemma 7. The functional
satisfies where η 1 is a positive constant.
We are now ready to prove the following result.
Theorem 8. Assume (26)-(31) hold; there exist positive constants C 1 and C 2 such that the energy functional given by (107) satisfies Proof. We define a Lyapunov functional where N and N i , i = 1, 2, 3, are positive constants to be selected later. By differentiating (142) and using (109), (115), (118), (123), and (137), including the relation 13 Journal of Function Spaces we get First, we choose ε 1 small enough such that By setting we obtain Next, we carefully choose our constants so that the terms inside the brackets are positive.

Data Availability
No data were used to support the study.

Conflicts of Interest
This work does not have any conflicts of interest.