Optimal polynomial decay for a coupled system of wave with past history

Abstract: This work deals with a coupled system of wave with past history effective just in one of the equations. We show that the dissipation given by the memory effect is not strong enough to produce exponential decay. On the other hand, we show that the solution of this system decays polynomially with rate t− 1 2 . Moreover by recent result due to A. Borichev and Y. Tomilov, we show that the rate is optimal. To the best of our knowledge, there is no result for optimal rate of polynomial decay for coupled wave systems with memory in the previous literature.


Introduction
I n this paper we consider a coupled system of wave with past history given by (u(x, 0), v(x, 0)) = (u 0 (x), v 0 (x)), in Ω, (4) (u t (x, 0), v t (x, 0)) = (u 1 (x), v 1 (x)), in Ω, where Ω is an open bounded set of R n with smooth boundary Γ. The above model can be used to describe the evolution of a system consisting of two elastic membranes subject to an elastic force that attracts one membrane to the other with coefficient α > 0. Note that the term 50 in [8] for wave equations coupled in parallel with coupling distributed springs and viscous dampers due to different boundary conditions and wave propagation speeds.
For weak damping acting only one equation, the optimal polynomial decay to coupled wave equations was studied in [9]. In [10], it was proved that the energy of associated coupled system weakly dissipative decays polynomially with explicit polynomial decay rates for sufficiently smooth solutions. In [11], under new compatibility assumptions, the authors proved polynomial decay for the energy of solutions and optimized previous results by interpolation techniques introduced in [10].
On the asymptotic behavior of the coupled system (1)-(5) we refer the work [12] where the authors proved by method introduced in [11] that the solution has a polynomial rate of decay. The central question of this work is to analyze what is the best decay rate of the system (1)- (5). In this direction, we prove that the associated semigroup decays with rate t − 1 2 . Moreover we show that the rate is optimal. For what we know in the literature the optimal rate of polynomial decay for coupled wave systems with memory was not previously considered.
The mathematical structure of the paper is organized as follows: In Section 2 we discuss the existence, regularity and uniqueness of strong solutions of the system (1)-(5) by semigroup technique, see [13]. In Section 3 we study the lack of exponential decay using Prüss's results [14]. Finally in section 4 we show that the system is polynomially stable giving an optimal decay rate. That is, this rate cannot be improved. For this we use the recent result due to Borichev and Tomilov [15].

Semigroup Setup
Following the approach of Dafermos [16] and Fabrizio and Morro [17], we consider η = η t (s), the relative history of u, defined as Hence, putting the system (1)-(5) turns into the system where the third equation is obtained differentiating (6) with respect to s and the condition (13) means that the history is considered as an initial value. We study the existence and uniqueness of solutions for the system (7)-(13) using the semigroup techniques. As in [18], we use the following hypotheses on g In view of (14), let L 2 g (R + ; H 1 0 (Ω)) be the Hilbert space of H 1 0 (Ω)-value functions on R + , endowed with the inner product To give an accurate formulation of the evolution problem we introduce the product Hilbert spaces endowed with the following inner product where where η s is the distributional derivative of η with respect to the internal variable s. Therefore, the system (7)-(13) is equivalent to With the above notations, we have the following result.
Theorem 1. The operator A generate a C 0 -semigroup S(t) of contraction on H. Thus, for any initial data U 0 ∈ H, the problem (7)-(13) has a unique weak solution U(t) and using the inner product (15), we get Integrating by parts and using (14), we have Therefore, A is a dissipative operator. Next, we show that (I − A) is maximal. For this, let us consider the equation Then, in terms of its components, we can write Integrating (22), we have Substituting ϕ and η from (18) and (23) into (19), we get where Note that β g is a positive constant in virtue of (14). Moreover, it can be shown that the right-had side of (24) is in H −1 (Ω).
Again from (23), we have Thus, I − A is maximal. Then, thanks to the Lumer-Phillips theorem (see [13], Theorem 4.3), the operator A generates a C 0 -semigroup of contractions e tA on H. The proof is now complete.

Lack of exponential decay
Our starting point is to show that the semigroup associated to the system (7)-(13) is not exponential stable. To show this, we assume that g(t) = e −µt , with t ∈ R + and µ > 1. We will use the Prüss's theorem [14] to prove the lack of exponential stability. To do this, let us consider the spectral problem: where lim m→∞ λ m = +∞.
The following theorem describes the main results of this section.

Theorem 3. Let S(t) be C 0 -semigroup of contractions generated by A. Then S(t) is not exponentially stable.
Proof. Here we will use the Theorem 2. That is, we will show that there exists a sequence of values λ m such that It is equivalent to prove that there exist a sequence of data F m ∈ H and a sequence of complex numbers λ m ∈ iR, with ||F m || H ≤ 1 such that where with U m not bounded.
To simplify the notation we will omit the subindex m. Then, the Equation (31) becomes (32) Let us consider f 1 = f 3 = f 5 = 0 and f 2 = f 4 = w m to obtain ϕ = iλu e ψ = iλv. Then, the system (32) becomes We look for solutions of the form with a, b, c, d ∈ C and γ(s) depend on λ and will be determined explicitly in the sequel. From (33), we get a and b satisfy Recalling that Using theorem 3 follows that S(t) is not exponentially stable. The proof is now complete.

Polynomial decay and optimally result
In this section we study the polynomial decay associated to the system (7)-(13) and subsequently we find the optimal rate of decay. Then, let us consider the resolvent equation In the next step we shall show three lemmas important to proof the main result. Proof. Multiplying the equality (39) by ϕ and integrating by parts on Ω, we get Substituting ϕ given in (38) into I 1 and I 2 , we have and Now, substituting ϕ given in (42) into I 3 and integrating by parts, we obtain Taking the real part on the left side of the above equality and using the hypotheses (14) on g, our conclusion follows.
Using the Lemma 1, follows the first inequality.
To show the second inequality, we substitute the equation (38) into (42). This gives, Then taking AU 0 = F, we get Therefore the solution decays polynomially.
To prove that the rate of decay is optimal, we will argue by contradiction. Suppose that the rate t − 1 2 can be improved; for example that the rate is t − 1 2− for some 0 < < 2. From Theorem 5.3 in [20], the operator |λ| −2 + 2 ||(λI − A) −1 || should be limited, but this does not happen. For this, let us suppose that there exist a sequence (λ µ ) ⊂ R with lim µ→∞ |λ µ | = ∞ and (U µ ) ⊂ D(A) for (F µ ) ⊂ H such that So, following the same steps as in the proof of Theorem 3 we can conclude that Therefore the rate cannot be improved. The proof is now complete.