Structural Stability on the Boundary Coefﬁcient of the Thermoelastic Equations of Type III

: This paper investigates the spatial behavior of the solutions of thermoelastic equations of type III in a semi-inﬁnite cylinder by using the partial differential inequalities. By setting an arbitrary positive constant in the energy expression, the fast decay rate of the solutions is obtained. Based on the results of decay, the continuous dependence and the convergence results on the boundary coefﬁcient are established by using the differential inequality technique and the energy analysis method. The main work of this paper is to extend the study of continuous dependence to a semi-inﬁnite cylinder, which can be used as a reference for the study of other types of partial differential equations.


Introduction
Since Hirsch and Smale [1] put forward the concept of structural stability in 1974, this type of structural stability research has attracted a lot of attention. Liu and Zheng [2] considered the exponential stability of the thermoelastic plate model where Ω is a bounded region in R n with smooth boundary Γ and Γ = Γ 0 Γ 1 , Γ 0 Γ 1 = ∅. h, α, σ, µ, µ 1 are positive constants, and u 0 (x, t), u 1 (x, t), θ 0 (x, t) are given functions. u and θ are unknown functions that represent the vertical deflection and the temperature of the plate, respectively. In another paper [3], Avalos and Lesiecka proved that the solution of the Equations (1)-(5) decayed exponentially as t → ∞ under the boundary conditions ∂θ ∂n + λθ = 0, x ∈ Γ, t > 0, (7) k(u + (1 − µ)B 1 u + αθ) = 0, x ∈ Γ, t > 0. (8) Here, k is either 0 or 1. Meyvacı [4] obtained the continuous dependence on coefficients h and β, in the case of σ = 0 in Equations (1) and (2) under the boundary conditions u = ∂u ∂n = θ = 0, x ∈ Γ, t > 0, θ tt − κ∆θ − δ∆θ t + αdivu tt = 0, x ∈ R, t > 0, (10) u, ∇u, θ, ∇θ → 0, as x 3 → ∞, with the initial-boundary conditions u(x, 0) = u t (x, 0) = 0, x ∈ R, t > 0, (12) θ(x, 0) = θ t (x, 0) = 0, x ∈ R, t > 0, (13) u(x 1 , x 2 , 0, t) = g(x 1 x 2 , t), θ(x 1 , x 2 , 0, t) = h(x 1 x 2 , t), (x 1 , x 2 ) ∈ D, t > 0, (14) u = 0, ∂θ ∂n + βθ = F(x, t), x ∈ ∂D × {x 3 ≥ 0}, t > 0, (15) where µ, λ, δ, κ, α, and β are positive constants. By the end of the 20th century, Green and Naghdi [25][26][27] introduced three types of thermoelastic theories. They were respectively called thermoelasticity type I, type II, and type III based on different constitutive assumptions. Since then, thermoelastic equations of type III have attracted a lot of attention. Quintanilla [28] obtained the existence in thermoelasticity without energy dissipation. Ding and Zhou [29] proved the global existence and finite time blow-up for the solutions of the thermoelastic system with p-Laplacian. Zhang and Zuazua [30] have studied the long-time behavior of the solutions of the system. Quintanilla [31] proved that solutions of thermoelasticity of type III converged to solutions of the classical thermoelasticity and to the solution of thermoelasticity without energy dissipation. Quintanilla [32] obtained the structural stability results on the coupling coefficients and the external data in thermoelasticity of type III in a bounded domain. Yan et al. [33] further extended the convergence result of thermoelasticity of type III to a semi-infinite pipe, but in the lateral of the pipe, they assumed that the solutions satisfied This paper studies the structural stability on the coefficient β of system (9)- (15). Different from the continuous dependence on the initial data, the so-called structural stability studies the continuous dependence and convergence of the solutions of the equations on the coefficients in the equations, the parameters in the boundary conditions, and the equations themselves. In the process of model building, simplification, and numerical cal-culation, some errors will inevitably appear. Different from mistakes, the errors will not be completely avoided with the progress of measurement methods. Therefore, it is important for us to study the influence of these errors on the solution of the equations (see [34]). In this paper, we first derive the spatial decay bounds of the solutions by using the partial differential inequalities, and then, we study the effect of the coefficient β by using the total energy bounds obtained in the derivation of spatial decay. By setting an arbitrary positive constant, we also obtain the fast decay rate of the solutions. Obviously, our research is a generalization of [32,33]. Our innovation is to extend the study of continuous dependence to a semi-infinite cylinder. This type of study can be used as a reference for the study of other types of partial differential equations and has not received sufficient attention. Therefore, the research of this paper is very meaningful.

Preliminary
In this section, we give some preliminary work, which will be used frequently.
This inequality is usually named as the Hölder inequality.
This inequality is usually named as the Young inequality.

Lemma 3.
Assume that n is a positive integer and x i > 0 (i = 1, 2, . . . , n), then This inequality is usually named as the arithmetic-geometric mean inequality.

Lemma 4.
Assume that u is a vector function in a bounded region Ω, then where ∂Ω is the boundary surface of Ω and n is the outward facing unit normal vector on ∂Ω. This inequality is usually named as the divergence theorem.

The Decay Results
We first give the notations where z is a moving point on the coordinate axis x 3 and 0 ≤ z < ∞.
To get the decay result of the solutions to (9)-(15), using (9), we begin with the following identities where ω is a positive constant. In (16) and in the following, we use commas for derivation, repeated English subscripts for summation from 1 to 3, and repeated Greek subscripts for summation from 1 to 2, e.g., Using the divergence theorem and Equations (9)- (15) in (16) and (17), we have then from (18) and (19), we have From (20), we also have Using the Hölder inequality and the Young inequality, we have where δ 1 , δ 2 are positive constants. Inserting (23)- (29) into (21), choosing δ 1 = δ 2 = 1 and combining (22), we have Integrating (31) from 0 to z, we have Combining (20) and (32), we can obtain the following theorem.

Remark 2.
Obviously, the decay bound in Theorem 1 depends on the total energy E(0, t). To make the decay bound explicit, we have to derive the bound for E(0, t). We write the result in the following theorem. Theorem 2. Let (u, θ) be solutions of Equations (9)-(15) in R, then, for fixed t, where Q 2 (0, t) is a positive function, which will be defined in (53).
Proof. We choose z = 0 in (20) and (21) to have Now, we define two new auxiliary functions where σ 1 , σ 2 are positive constants. Obviously, G i and H have the same boundary conditions with u i and θ, respectively.
Using Equation (9) and the divergence theorem, we have Using the Hölder inequality and the Young inequality in (36), we have where ε i (i = 1, 2, . . . , 5) are positive constants. Using Equation (10) and the divergence theorem, we have Using the Hölder inequality and the Young inequality in (42), we have where ε i (i = 6, 7, . . . , 12) are positive constants. For the last two terms on the right of (34), we can refer to the results that have been derived in (28) and (29), and we have the following inequalities Inserting (37) We have where Based on (33) and (52), we can obtain Theorem 2.
Combining Theorems 1 and 2, we can get the following theorem.

Continuous Dependence on the Boundary Coefficient
We will use the results obtained in Section 2 to investigate the effect of the small change on the coefficient β in Equations (9)- (15). To do this, we let u * and θ * be the solutions of (9)-(15) with the boundary coefficient β replaced by the constant β * and allow u and u * to satisfy same conditions on the entrance D. It is worth noting that if u and u * satisfy different boundary conditions at the entrance, our results are still valid because our problem is linear, and we can decompose and deal with the two effects, respectively. If we let v i and Π be the differences between u i , p and u * i , p * , respectively, i.e., then v and Π satisfy the following equations with the initial boundary conditions v(x 1 , x 2 , 0, t) = 0, θ(x 1 , x 2 , 0, t) = 0, (x 1 , We have the following theorem. where b 1 , b 2 are positive constants. This demonstrates continuous dependence of (u, θ) on the parameter β.

Convergence on the Boundary Coefficient
In this section, we derive the convergence result on the boundary coefficient which is different from the continuous dependence result. We let u * and θ * be the solutions of (9)-(15) with the boundary coefficient β = 0 and v and Π are also defined as (54). So, v and Π also satisfy (55)-(60), but the condition (61) can be replaced by To get our main result, we will use the following lemma.
where b 1 , b 3 are positive constants. This demonstrates convergence of (u, θ) on the parameter β.
Theorem 5 shows that when z → 0, it will not have a significant impact on the solution of the system of equations. It shows the stability of the equations.

Conclusions
In this paper, Equations (9)-(15) are reconsidered in a new semi-infinite cylinder. The structural stability of the solution is obtained by using the differential inequality technique and energy analysis method. In a two-dimensional pipe, Payne and Schaefer [36] obtained Phragmén-Lindelöf alternative results of biharmonic equation. As far as we know, there are a few results in this type of three-dimensional cylinder region. Therefore, it is very interesting to replace the pipe R by where D x 3 can be defined as On the other hand, if the boundary conditions (13)-(15) are replaced by (3), (4)