Stability for Weakly Coupled Wave Equations with a General Internal Control of Diffusive Type

: The present paper deals with well-posedness and asymptotic stability for weakly coupled wave equations with a more general internal control of diffusive type. Owing to the semigroup theory of linear operator, the well-posedness of system is proved. Furthermore, we show a general decay rate result. The method is based on the frequency domain approach combined with multiplier technique


Introduction
When describing the propagation of nonlinear waves with an internal control of diffusive type, the theory of semigroup is often used.It is used in the case, which is quite important for applications, when the internal diffusive mechanism is described by integer derivatives.The large amount of currently available experimental data on the internal structure of nonlinear waves in applications requires the complication and modification of mathematical modeling methods.Here, the main attention is paid to the construction and analysis of stability for nonlinear mathematical models that reflect the influence of internal control of diffusive type.
The authors proved a very general rate depending on the form of the function .Our paper extends all the previous works, and its plan is as follows.In Section 2, we give preliminary results, and we establish the well-posedness of the system (1), owing to the Hille-Yosida Theorem.We show, in Section 3, the lack of exponential stability.In Section 4, an asymptotic stability of our model is studied, where the main results are Theorem 4 and Theorem 7. In Theorem 7, we established a general rate of decay which depends on that of the density function .

Remark 1.
For this topic, we can say that there are many related problems which still are open, such as in the unbounded domain, where one can consider the same model in R n with weighted functions.

Preliminary Results and Well-Posedness
We state hypotheses on the even non-negative measurable function as Now, we recall some definitions which are needed in Section 4 for the application.
Definition 1.Let a ≥ 0, and let M : [a, +∞) → (0, +∞) be a measurable function, then M has a positive increase if there exist α > 0, c ∈ (0, 1] and s 0 ≥ a, such that The next Lemma will be useful (see [1]). and We are now ready to give the existence and uniqueness result for the problem (1) by using semigroup theory.The energy space is defined as equipped with the following inner product where which guarantees the positivity of the norm.
In order to transform the problem (1) to an abstract problem on the Hilbert space H, we introduce the vector function U = (u, v, w, z, φ, ϕ) T , where w = ∂ t u and z = ∂ t v.Then, problem (1) can be rewritten as where and its domain is given by The energy associated to the solution of the problem ( 1) is given by Differentiating E in a formal way, using (1) and integrating by parts, we obtain, after a straightforward computation, the following Lemma.Lemma 2. Let (u, v, w, z, φ, ϕ) be a regular solution of problem (1).Then, the energy functional defined by (7) satisfies We have the following results.
Proposition 1.The operator A is the infinitesimal generator of a contraction semigroup {S(t)} t≥0 .
Proof.First, we prove that the operator A is dissipative.We observe that U ∈ D(A) and by ( 5), ( 8) and the fact that we obtain R AU, In fact, using (3), and integrating by parts, we obtain Hence, taking the real part, then estimate (10) holds.
Consequently, using the Lumer-Philips Theorem [4], we have the following result.
Moreover, if U 0 ∈ D(A), then system (5) has a unique strong solution

Lack of Exponential Stability
Theorem 2 ([5]).Let S(t) = e At be a C 0 -semigroup of contractions on Hilbert space X.Then, S(t) is exponentially stable if, and only if, and lim Our main result in this part is the following Theorem.
Theorem 3. The semigroup generated by the operator A cannot be exponentially stable.
Proof.Let −δ 2 n = (iδ n ) 2 be a sequence of eigenvalues corresponding to the sequence of normalized eigenfunctions u n of the operator ∆ x , such that |δ n | −→ ∞ as n −→ ∞ and Our aim is to prove, under some conditions, that if iδ n satisfies (18), then (19) does not hold.In other words, we want to prove that there exist an infinite number of eigenvalues of A approaching the imaginary axis, which prevents the wave system (1) from being exponentially stable.Indeed, we first compute the characteristic equation that gives the eigenvalues of A. Let κ be an eigenvalue of A with associated eigenvector U = (u, v, w, z, φ, ϕ) T .Then, AU = κU is equivalent to We note that assuming the decomposition given by Φ := u + v, Θ := w + z and Λ := φ + ϕ, we have The problem ( 22) can be rewritten as where V 0 = (Φ 0 , Φ 1 , Λ 0 ) T , and A 1 : Taking Ψ := u − v, Υ := w − z and Ξ := φ − ϕ, we have Moreover, note that and φ := 1 2 (Λ − Ξ).We define the Hilbert space equipped with the following inner product where Now, we need to solve problems ( 22)-(25).From (22) 1 , we have Inserting ( 27) in (22) 2 , we obtain Then, from (27), (22) 3 , and (28), we obtain 2 ( ) it follows that From (20) and (30), we obtain the existence of a sequence of eigenvalues κ n of A corresponding to the sequence δ n , such that then, we obtain reduces to show that, as n −→ ∞, Indeed, using the fact that (see Lemma 4.3 in [6]) and the fact that Φ n is a normalized eigenfunction of the operator ∆ x for each n ∈ N, we obtain the desired limit.Therefore, taking U = (u, v, w, z, φ, ϕ) ∈ D(A), we conclude that This completes the proof.

Strong Stability of the System
Here, we use the general Theorem of Arendt-Batty in [7] to show the strong stability of the C 0 -semigroup e tA associated to the system (1).Our main result is stated in the following.Theorem 4. The C 0 -semigroup e tA is strongly stable in H; i.e, for all U 0 ∈ H, the solution of (5) satisfies lim t→∞ e tA U 0 H = 0.
Hence, u, v are constant in the whole domain Ω, and u = v = 0 on ∂Ω, then we have u = 0, and v = 0 in the whole domain Ω.Therefore, U = (u, v, w, z, φ, ϕ) T = 0. We deduce that, consequently, A has no eigenvalue on the imaginary axis.
Proof of Theorem 4. Following a general Theorem of Arendt-Batty in [7], the C 0 -semigroup of contractions can be taken as strongly stable if A does not have eigenvalues on iR and σ(A) ∩ iR is at most a countable set.Owing to the Lemmas 3 and 4, we find the result.