Optimal Decay Rates for Kirchhoff Plates with Intermediate Damping

In this paper we study the asymptotic behavior of Kirchhoff plates with intermediate damping. The damping considered contemplates the frictional and the Kelvin-Voigt type dampings. We show that the semigroup those equations decays polynomially in time at least with the rate t−1/(2−2θ), where θ is a parameter in the interval [0,1[. Moreover, we prove that this decay rate is optimal.


INTRODUCTION
Consider Ω a bounded open set of R n with smooth boundary. This paper deals with the asymptotic stability of the solutions of the following Kirchhoff plate equation u tt − γ∆u tt + β ∆ 2 u + ε(−∆) θ u t = 0 in Ω×]0, ∞[, (1.1) satisfying the boundary conditions u = 0, ∆u = 0 on ∂ Ω×]0, ∞[, (1.2) and the initial data It is well-known that the semigroup of the system (1.1)-(1.3) is exponentially stable when the damping in equation (1.1) is of Kelvin-Voigt type and that this uniform decay is lost if the damping is of frictional type. We do not find in the literature results about the exact decay rates for the associated semigroups to Kirchhoff plates with intermediate damping between frictional dissipation and the Kelvin-Voigt type. This paper aims to find some answers related to this subject.
There exist many works about the stability of the solutions of plate models with some dissipative mechanisms. A variety of plate models can be found in the books [7,9,10].
Concerning plate models with other dissipative mechanisms, we are going to mention some of them. In [2], Barbosa et al. studied the stability of the thermoelastic plate with boundary conditions u = ∆u = θ = 0. They showed the existence of exponential attractors using semigroup theory.
In [13], the authors considered the viscoelastic plate.
with boundary conditions u = ∆u = 0 and exponentially decreasing kernel g. For regular initial data, it was shown that the associated semigroup is polynomially stable with decay rate t −1/(4−4θ ) when θ < 1.
For plates with boundary damping, we have the work of Rao et al. [14], who studied the stability of the equation Here, Γ 0 ∪ Γ 1 = ∂ Ω and η, ξ are solutions of the equations For any initial data, they showed the strong stability of the system. Moreover, for regular initial data and with boundary Γ 1 satisfying suitable geometric condition, they proved that the energy decays with polynomially rate t −1 .
Other authors studied the stability problem for plates with different dissipative mechanisms. the main results are related to the asymptotic stability of the semigroup with exponential or poly nomial decay. See for instance [1,5,6,8,11,12,15,16].
It is well known that the operator A = −∆ defined in the space D(A) = H 2 (Ω) ∩ H 1 0 (Ω) is a positive self-adjoint operator in the Hilbert space L 2 (Ω). Even more, this operator has compact inverse. Using this notation for the operator −∆ the system (1.1)-(1.3) can be written in the following abstract setting satisfying the initial conditions The main result of this paper deals with the asymptotic behavior of the semigroup for the abstract system (1.4)-(1.5).
To be more precise, we show that the associated semigroup decays polynomially as the time goes to infinity with rate t −1/(2−2θ ) for regular initial data (see Theorem 2). In this same Theorem we also show the optimality of this decay rate.
The remainder of this article is organized as follows: In section 2 we introduce the semigroup of the system (1.4)-(1.5). In section 3 we enunciate and prove the main result of this paper.

EXISTENCE OF SOLUTIONS
We are going to use the semigroup theory to show existence of solutions for the system of equa- , is a positive selfadjoint operator with compact inverse in the Hilbert space H := L 2 (Ω). Therefore, the operator A θ is self-adjoint, positive for θ ∈ R and the embedding · denotes the norm of the Hilbert space H. More details about fractional operators can be found in [4]. Note that, in view of the Riesz representation Theorem we have: where ·, · on the right side of this equation denotes the inner product in the space H.
Now, if we consider the vector U(t) = (u, u t ), then the system (1.4)-(1.5) can be written in an abstract framework as d dt where U 0 = (u 0 , u 1 ) and the operator B is given by To study the abstract system (2.2) through semigroup theory we are going to work in the Hilbert space where the inner product is defined by Here, ·, · on the right side of this equation denotes the inner product in the space H. With theses considerations, the domain of the operator B is defined by This problem can be put in a variational formulation: to find u ∈ D(A) such that Now, applying the Lax-Milgram Theorem we have a unique solution U ∈ H. As this solution satisfies (2.6) we can conclude that U ∈ D(B). Moreover, taking z = u in (2.7) and applying Cauchy-Schwarz and Young inequalities we conclude that This inequality and the first equation of (2.6) imply that U X ≤ C F X , so we have 0 belongs to the resolvent set ρ(B). Consequently, from Theorem 1 we have B is the generator of a contractions semigroup.
Finally, the well-posedness of the system (1.4)-(1.5) is a consequence of the semigroup theory. We state this result in the following theorem

STABILITY RESULTS
In this section we study the asymptotic behavior of the semigroup e tB associated with the system (1.4)-(1.5). To do this, we are going to use the following spectral characterization for the polynomial stability of semigroups due to Borichev and Tomilov: Theorem 1 (see [3]). Let B be the generator of a C 0 -semigroup of bounded operators on a Hilbert space such that iR ⊂ ρ(B). Then we have if and only if, Let λ ∈ R and F = ( f , g) ∈ X. In what follows, the stationary problem (iλ − B)U = F will be considered several times in the course of this article. Note that U = (u, v) is a solution of this problem if the following equations are satisfied: In what follows, C denotes a positive constant that assume different values in different places.
The main result of this paper is given by the following theorem. Moreover, this decay rate is optimal in the following sense: it does not decay with the rate t −κ , for κ > 1/(2 − 2θ ).
Proof. We are going to use Theorem 1 to show this theorem. Since 0 ∈ ρ(B) there exists δ > 0 such that ] − iδ , iδ [⊂ ρ(B), so to show that ρ(B) contains the imaginary axis is sufficed to show that iλ ∈ ρ(B) for real λ such that |λ | ≥ δ . Simultaneously we are going to prove the estimative U X ≤ C|λ | 2−2θ F X for the solution U of the stationary equation (iλ − B)U = F.
Note that, from identity (2.5) we easily obtain Applying the duality product to the second equation of (3.1) with A θ u and taking into account that the fractional powers of the operator A are self-adjoint, we obtain Therefore, in view of inequality (3.2), we conclude that for |λ | ≥ δ . On the other hand, applying the duality product to the second equation of (3.1) with u and using the first equation we have Applying the Cauchy-Schwarz and Young inequalities, and using the estimates (3.5), we obtain BRAVO, OQUENDO and RIVERA 267 Finally, from this inequality and (3.5) we can conclude that As |λ | ≥ δ we have U 2 X ≤ Cλ 2−2θ F X U X + F 2 X from where follows the desired inequality U X ≤ Cλ 2−2θ F X . Note that, if F = 0 then U = 0, consequently B does not have eigenvalues in the imaginary axis. As the operator B −1 is compact its spectrum is constituted only by eigenvalues, so iR ⊂ ρ(B). This completes the proof of the polynomial decay of the semigroup with rate t −1/(2−2θ ) .
Optimality of the decay rate: Since A is a positive self-adjoint operator with compact resolvent, its spectrum is constituted by positive eigenvalues α n , n ∈ N, with α n → ∞. Let us denote by (e n ) the corresponding eigenvectors, that is Ae n = α n e n , n ∈ N.
We consider F n = (0, −ẽ n ) ∈ X whereẽ n = e n / A 1/2 e n , then the solution U = (u, v) of the system (iλ I − B)U = F n satisfies the following conditions: By substituting the first identity in the second equation we obtain Now, we are going to look for by solutions of the form u = ηẽ n for some complex number η. Therefore, the coefficient η must satisfy the equation Solving this equation we have In this point, taking the above formula becomes η = η n = i 1 + γα n ελ n α θ n . (3.7) If we introduce the notation x n ≈ y n meaning lim n→∞ |x n | |y n | is a positive real number, then from (3.6) and (3.7) we can assert that λ n ≈ α 1/2 n and η n ≈ λ 1−2θ n . Therefore, if U n = (u n , v n ) is the solution of the system (iλ n − A)U = F n , we obtain A 1/2 v n = λ n A 1/2 u n = λ n |η n | A 1/2ẽ n ≥ δ λ 2−2θ n , for some δ > 0 and n large enough. From this estimate, we conclude that U n X ≥ γ A 1/2 v n ≥ γδ λ 2−2θ n .
Finally, let us suppose the semigroup decays with the rate t −κ for some κ > 1/(2 − 2θ ). From Theorem 1 we have λ −1/κ n U n X is bounded. On the other hand, the above inequality implies that λ −1/κ n U n X ≥ γδ λ which is absurd. Therefore the decay rate t −1/(2−2θ ) is optimal.