Asymptotic behavior for Timoshenko systems with fractional damping

1. Introduction

Stabilization of the Timoshenko beam with various damping mechanisms has been subject to extensive research over the years. In particular, the systems with frictional damping or Kelvin–Voigt damping were is considered by some authors who have studied these problems and obtained several results on the asymptotic behavior of their solutions (see for example [1,4,14,15,19,21,22]).

In this paper, we are interested in the stabilization of the Timoshenko beams with an intermediate damping among the frictional and the Kelvin–Voigt type. The problem is formulated as it follows: considering a bar of length L, denoting by ϕ ( x , t ) the vertical deflection from equilibrium state and ψ ( x , t ) the angle of rotation of a cross-section, the Timoshenko system to be studied is given by the equations (1) ρ 1 ϕ t t − κ ( ϕ x x + ψ x ) = 0 , x ∈ ] 0 , L [ , t > 0 , (2) ρ 2 ψ t t − b ψ x x + κ ( ϕ x + ψ ) + ω ( − ∂ x x ) θ ψ t = 0 , x ∈ ] 0 , L [ , t > 0 , satisfying the boundary conditions (3) ϕ ( 0 , t ) = 0 , ϕ ( L , t ) = 0 , ψ x ( 0 , t ) = 0 , ψ x ( L , t ) = 0 , t > 0 , and initial data (4) ϕ ( x , 0 ) = ϕ 0 ( x ) , ϕ t ( x , 0 ) = ϕ 1 ( x ) , ψ ( x , 0 ) = ψ 0 ( x ) , ψ t ( x , 0 ) = ψ 1 ( x ) , x ∈ ] 0 , L [ . Here, all the coefficients ρ 1 , ρ 2 , κ , b , ω are positive numbers and θ ∈ [ 0 , 1 ] . The value of the coefficient (5) χ 0 : = κ ρ 1 − b ρ 2 establishes a relation among the wave propagation speeds of equations (1)–(2) which will play an important role in the stabilization of the system.

Note that, if we consider θ = 1 and the constant initial data ϕ 0 ≡ 0 , ϕ 1 ≡ 0 , ψ 0 ≡ a 1 , ψ 1 ≡ a 2 , then the system (1)–(4) has oscillating solutions given by ϕ ≡ 0 , ψ = a 1 cos ( κ / ρ 2 t ) + a 2 ρ 2 / κ sin ( κ / ρ 2 t ) . The constants a 1 , a 2 in these solutions can be rewritten as a 1 = 1 L ∫ 0 L ψ 0 ( x ) d x , a 2 = 1 L ∫ 0 L ψ 1 ( x ) d x . In this sense, in order to avoid solutions that do not decay to zero when t → ∞ , the following condition is additionally imposed (6) ∫ 0 L ψ 0 ( x ) d x = 0 , ∫ 0 L ψ 1 ( x ) d x = 0 .

A reasonable number of publications concerning the stabilization of Timoshenko beams have been found when researching previous literature. Some of them are below mentioned:

Soufyane [21] studied the Timoshenko system with frictional damping in the equation of the rotation angle ρ 1 ϕ t t − κ ( ϕ x x + ψ x ) = 0 , ρ 2 ψ t t − b ψ x x + κ ( ϕ x + ψ ) + ψ t = 0 , with Dirichlet boundary conditions. He showed that this system is exponentially stable if, and only if, the wave propagation speeds of these equations are the same ( χ 0 = 0 ). This same result was obtained by Rivera and Racke [19] for this system, but with boundary conditions (3). Moreover, if the wave propagation speeds are not the same ( χ 0 ≠ 0 ), they proved that the system is polynomially stable. Note that this problem is a particular case of the system (1)–(4) with θ = 0 , and it means the semigroup decays polynomially with the rate t − 1 / 2 .

Malacarne and Rivera [14] considered a Timoshenko system with viscosity in both equations ρ 1 ϕ t t − κ ( ϕ x x + ψ x ) − γ 1 ( ϕ x x t + ψ x t ) = 0 , ρ 2 ψ t t − b ψ x x + κ ( ϕ x + ψ ) − γ 2 ψ x x t = 0 , with boundary conditions (3). They showed that the semigroup is analytic when γ 1 , γ 2 > 0 . On the other hand, if one of the coefficients γ i vanishes, this property is lost and in this case, they showed that the semigroup decays polynomially with optimal rate t − 1 / 2 . Note that, for γ 1 = 0 this system coincides with the system (1)–(4) with θ = 1 .

According to these results we have that, for χ 0 ≠ 0 the semigroup of system (1)–(4) decays with the rate t − 1 / 2 no matter if the damping is of the frictional type ( θ = 0 ) or the Kelvin–Voigt type ( θ = 1 ). This leads one to think that, for intermediate dampings ( θ ∈ ] 0 , 1 [ ), the semigroup should also decay with the rate t − 1 / 2 , but it is not true (see Theorem 8).

Ammar-Khodja et al. [3] studied a linear system of Timoshenko type with memory ρ 1 ϕ t t − κ ( ϕ x x + ψ x ) = 0 , ρ 2 ψ t t − b ψ x x + ∫ 0 t g ( t − s ) ψ x x ( s ) d s + κ ( ϕ x + ψ ) = 0 , satisfying Dirichlet boundary conditions. They considered exponentially and polynomially decreasing kernels g ( t ) . When the kernel is exponentially decreasing and χ 0 = 0 they proved the exponential stability of the system. When the kernel decays to zero polynomially and χ 0 = 0 they proved that the solution also decays polynomially with the same rate.

Danese et al. [10] studied an abstract system that generalizes a thermoelastic Timoshenko beam. The considered system is ρ 1 ϕ t t + κ A 1 / 2 ( A 1 / 2 ϕ + ψ ) = 0 , ρ 2 ψ t t + b A ψ + κ ( A 1 / 2 ϕ + ψ ) − δ A γ θ = 0 , ρ 3 θ t + c A θ + δ A γ ψ t = 0 , where A is a self-adjoint positive operator. They showed that the semigroup is exponentially stable if and only if χ 0 = 0 and γ = 1 / 2 . Moreover, for any value of χ 0 and γ ∈ [ 1 / 2 , 1 ] , they proved the semigroup decays polynomially at least with the rate t − 1 / 10 for regular initial data. Results about optimal rates for polynomial decays were not addressed.

Studies about the stability for Timoshenko systems with other local or global dampings, or damping on the boundary can be found in [2,5–8,11,16,20,23].

As previously mentioned, the asymptotic behavior of Timoshenko system (1)–(4) for the cases θ = 0 and θ = 1 were already studied. Therefore, our interest in this paper is to find exact rates of decay for θ ∈ ] 0 , 1 [ , that is, exact decay rates for intermediate dampings between the frictional and Kelvin–Voigt type.

Our main results are the following: (see Theorem 8 and 9)

The remaining part of this paper is organized as it follows: In Section 2, we study the well-posedness of the system (1)–(4) using a semigroup approach. In Section 3, the asymptotic behavior of the semigroup is studied. This is where our main results are enunciated.

2. Existence of solutions

An approach of semigroup is used in this study to show existence of solutions for the system of equations (1)–(4) that satisfy (6). Before, some notations need to be introduced.

For 1 ⩽ p ⩽ ∞ , L p = L p ( 0 , L ) denotes the usual Lebesgue space with the norm ‖ · ‖ L p . In order to simplify the notation it is used ‖ · ‖ instead of ‖ · ‖ L 2 or ‖ · ‖ H and ⟨ · , · ⟩ instead of ⟨ · , · ⟩ L 2 or ⟨ · , · ⟩ H , where H is a Hilbert space that will be defined later. Let s be a nonnegative number, then H s = H s ( 0 , L ) denotes the usual Sobolev space of L 2 functions, equipped with the norm ‖ · ‖ H s .

The subset of functions with null mean of the Hilbert space L 2 ( 0 , L ) , L ∗ 2 ( 0 , L ) : = { h ∈ L 2 ( 0 , L ) : ∫ 0 L h ( x ) d x = 0 } , is a closed subspace. Consequently, it is a Hilbert space with the L 2 -norm.

It is well-known that the differential operators (7) B = − ∂ x x : D ( B ) ⊂ L 2 ( 0 , L ) → L 2 ( 0 , L ) , B ∗ = − ∂ x x : D ( B ∗ ) ⊂ L ∗ 2 ( 0 , L ) → L ∗ 2 ( 0 , L ) , defined in the subspaces D ( B ) = H 2 ( 0 , L ) ∩ H 0 1 ( 0 , L ) , D ( B ∗ ) = { ψ ∈ H 2 ( 0 , L ) ∩ L ∗ 2 ( 0 , L ) : ψ x ( 0 ) = ψ x ( L ) = 0 } , are positive self-adjoint and have a compact inverse. Therefore, the operators B σ , B ∗ σ are bounded for σ ⩽ 0 , and they are positive self-adjoint for any σ ∈ R . Moreover, the embeddings D ( B σ 1 ) ↪ D ( B σ 2 ) , D ( B ∗ σ 1 ) ↪ D ( B ∗ σ 2 ) are continuous for σ 1 > σ 2 . Here, the norms in D ( B σ ) and D ( B ∗ σ ) , for σ ⩾ 0 , are given by ‖ ϕ ‖ D ( B σ ) : = ‖ B σ ϕ ‖ and ‖ ψ ‖ D ( B ∗ σ ) : = ‖ B ∗ σ ψ ‖ respectively (remember that ‖ · ‖ denotes the L 2 -norm).

With the considerations presented above the spectrum of the operators B and B ∗ is constituted only by positive eigenvalues. The eigenvalues for both operators are given by γ n 2 where γ n = n π L , n ∈ N , and the corresponding unitary eigenfunctions associated to these eigenvalues are (8) e n ( x ) = 2 L sin ( γ n x ) , e n ∗ ( x ) = 2 L cos ( γ n x ) . The sequences { e n } n ∈ N and { e n ∗ } n ∈ N constitute a Hilbert’s base for the spaces L 2 ( 0 , L ) and L ∗ 2 ( 0 , L ) respectively, then for ϕ ∈ L 2 ( 0 , L ) and ψ ∈ L ∗ 2 ( 0 , L ) we have ϕ = ∑ n = 1 ∞ ⟨ ϕ , e n ⟩ e n , ψ = ∑ n = 1 ∞ ⟨ ψ , e n ∗ ⟩ e n ∗ , where ⟨ · , · ⟩ denotes the L 2 -inner product. Note that, for ϕ ∈ D ( B σ + 1 / 2 ) , we have B σ + 1 / 2 ϕ = ∑ n = 1 ∞ γ n 2 σ + 1 ⟨ ϕ , e n ⟩ e n , B ∗ σ ∂ x ϕ = ∑ n = 1 ∞ γ n 2 σ + 1 ⟨ ϕ , e n ⟩ e n ∗ , from where it follows, by Parseval’s identity, that (9) ‖ B σ + 1 / 2 ϕ ‖ = ‖ B ∗ σ ∂ x ϕ ‖ . In particular, for σ = 0 we have ‖ B 1 / 2 ϕ ‖ = ‖ ∂ x ϕ ‖ . In a similar way, for ψ ∈ D ( B ∗ σ + 1 / 2 ) , we get (10) ‖ B ∗ σ + 1 / 2 ψ ‖ = ‖ B σ ∂ x ψ ‖ , and for ϕ ∈ D ( B σ 0 ) , ψ ∈ D ( B ∗ σ 0 ) , with σ 0 = max { σ , 1 / 2 } , we have (11) ⟨ B ∗ σ ψ , ∂ x ϕ ⟩ = − ⟨ ∂ x ψ , B σ ϕ ⟩ .

To write the system (1)–(4) in a semigroup scheme the variables Φ = ϕ t and Ψ = ψ t are introduced. With these notations the system (1)–(4) can be written as (12) d d t U ( t ) = A U ( t ) , U ( 0 ) = U 0 , where U = ( ϕ , ψ , Φ , Ψ ) , U 0 = ( ϕ 0 , ψ 0 , ϕ 1 , ψ 1 ) and the operator A is defined by (13) A U = ( Φ , Ψ , κ ρ 1 ( ϕ x x + ψ x ) , b ρ 2 ψ x x − κ ρ 2 ( ϕ x + ψ ) − ω ρ 2 B ∗ θ Ψ ) . This operator will be studied in the phase space H = H 0 1 ( 0 , L ) × H ∗ 1 ( 0 , L ) × L 2 ( 0 , L ) × L ∗ 2 ( 0 , L ) , where H ∗ 1 ( 0 , L ) : = H 1 ( 0 , L ) ∩ L ∗ 2 ( 0 , L ) . Note that H is a Hilbert space with the inner product ⟨ U 1 , U 2 ⟩ H = ρ 1 ⟨ Φ 1 , Φ 2 ⟩ + ρ 2 ⟨ Ψ 1 , Ψ 2 ⟩ + κ ⟨ ∂ x ϕ 1 + ψ 1 , ∂ x ϕ 2 + ψ 2 ⟩ + b ⟨ ∂ x ψ 1 , ∂ x ψ 2 ⟩ . The natural domain for the operator A is defined by D ( A ) = { U ∈ H : Φ ∈ H 0 1 ( 0 , L ) , Ψ ∈ H ∗ 1 ( 0 , L ) , ϕ ∈ H 2 ( 0 , L ) , b B ∗ ψ + ω B ∗ θ Ψ ∈ L ∗ 2 ( 0 , L ) } .

To show the existence of solutions for the abstract system (12) it suffices to prove that the operator A is the generator of a C 0 -semigroup. For this, we will use a variant of Lumer–Phillips’s Theorem enunciated in Liu and Zheng’s book [13].

We are going to verify that the operator A defined by (13) satisfies the conditions of this theorem. As the set D ( B ) × D ( B ∗ ) × D ( B ) × D ( B ∗ ) is dense in H and this set is contained in D ( A ) , we have that D ( A ) is dense in H . On the other hand, performing a simple computation we obtain (14) Re ⟨ A U , U ⟩ = − ω ‖ B ∗ θ / 2 Ψ ‖ 2 , ∀ U ∈ D ( A ) , that is, the operator A is dissipative. It still remains to show that 0 belongs to the resolvent set ρ ( A ) . For this, it is sufficient to show that for F = ( f 1 , f 2 , f 3 , f 4 ) ∈ H the stationary problem A U = F has a solution U = ( ϕ , ψ , Φ , Ψ ) ∈ D ( A ) and ‖ U ‖ ⩽ C ‖ F ‖ for some positive constant C independent of F. In components, the system A U = F reads (15) Φ = f 1 , κ ( ϕ x x + ψ x ) = ρ 1 f 3 , (16) Ψ = f 2 , b ψ x x − ω B ∗ θ Ψ − κ ( ϕ x + ψ ) = ρ 2 f 4 , that is, ( ϕ , ψ ) must satisfy κ ( ϕ x x + ψ x ) = g 1 , b ψ x x − κ ( ϕ x + ψ ) = g 2 , where g 1 = ρ 1 f 3 , g 2 = ρ 2 f 4 + ω B ∗ θ f 2 . This system can be placed in a variational problem: (17) a ( ϕ , ψ ; ϕ ˜ , ψ ˜ ) = − ⟨ g 1 , ϕ ˜ ⟩ − ⟨ g 2 , ψ ˜ ⟩ where the sesquilinear form in the space H 0 1 ( 0 , L ) × H ∗ 1 ( 0 , L ) is given by a ( ϕ , ψ , ϕ ˜ , ψ ˜ ) = κ ⟨ ϕ x + ψ , ϕ ˜ x + ψ ˜ ⟩ + b ⟨ ψ x , ψ ˜ x ⟩ . As this sesquilinear form is continuous and coercive and ( g 1 , g 2 ) ∈ ( H 0 1 ( 0 , L ) ) ′ × ( H ∗ 1 ( 0 , L ) ) ′ , by Lax–Milgram’s Theorem there exists an unique solution ( ϕ , ψ ) ∈ H 0 1 ( 0 , L ) × H ∗ 1 ( 0 , L ) . Therefore, the equations (15)–(16) are satisfied in a weak sense (variational problem). Thus, from second equations of (15)–(16) we have ϕ x x ∈ L 2 ( 0 , L ) and b B ∗ ψ + ω B ∗ θ Ψ ∈ L ∗ 2 ( 0 , L ) . This implies U = ( ϕ , ψ , Φ , Ψ ) ∈ D ( A ) . On the other hand, taking ( ϕ ˜ , ψ ˜ ) = ( ϕ , ψ ) in (17) we have κ ‖ ϕ x + ψ ‖ 2 + b ‖ ψ x ‖ 2 = − ρ 1 ⟨ f 3 , ϕ ⟩ − ρ 2 ⟨ f 4 , ψ ⟩ − ω ⟨ B ∗ θ − 1 / 2 f 2 , B ∗ 1 / 2 ψ ⟩ from where follows, by application of Young inequality and the continuous embedding D ( B θ 1 ) ↪ D ( B θ 2 ) , θ 1 > θ 2 , that κ ‖ ϕ x + ψ ‖ 2 + b ‖ ψ x ‖ 2 ⩽ ε ( ‖ ϕ x ‖ 2 + ‖ ψ x ‖ 2 ) + K ε ‖ F ‖ 2 . Using the inequality ‖ ϕ x ‖ 2 ⩽ 2 ( ‖ ϕ x + ψ ‖ 2 + ‖ ψ ‖ 2 ) ⩽ C 1 ( ‖ ϕ x + ψ ‖ 2 + ‖ ψ x ‖ 2 ) and fixing ε small we get ‖ ϕ x + ψ ‖ 2 + ‖ ψ x ‖ 2 ⩽ C 2 ‖ F ‖ 2 . This inequality, along with the first equations of (15)–(16), imply that there exist C > 0 such that ‖ U ‖ ⩽ C ‖ F ‖ , that is 0 ∈ ρ ( A ) . Consequently, from Theorem 1 we have that A is the generator of a contractions semigroup.

The existence of solutions of the system (1)–(4) that satisfy the condition (6) is consequence of the semigroup theory. We enunciate this result in the following theorem:

Theorem 2.

Let U 0 = ( ϕ 0 , ψ 0 , ϕ 1 , ψ 1 ) ∈ H , then there is a unique mild solution U = ( ϕ , ψ , ϕ t , ψ t ) of the system ( 1 )–( 4 ) which belongs to C ( [ 0 , ∞ [ ; H ) . Furthermore, if U 0 ∈ D ( A ) , then the solution is more regular, i.e. U ∈ C ( [ 0 , ∞ [ ; D ( A ) ) ∩ C 1 ( [ 0 , ∞ [ ; H ) .

3. Stability results

In this section we are going to study the asymptotic behavior of the solutions of Timoshenko system (1)–(4) satisfying (6). Our strategy is going to be based on obtaining appropriate estimates in order to apply some results about characterizations of stability from spectral properties. These results are enunciated below.

In view of the above theorems we will find some estimates for the solutions of the following stationary system (18) ( i λ I − A ) U = F , where λ ∈ R , i λ ⊂ ρ ( A ) and F = ( f 0 , g 0 , f ˙ 0 , g ˙ 0 ) ∈ H . If the coordinates of the solution are given by U = ( ϕ , ψ , Φ , Ψ ) ∈ D ( A ) they must satisfy the equations (19) i λ ϕ − Φ = f 0 , ρ 1 i λ Φ − κ ( ϕ x x + ψ x ) = ρ 1 f ˙ 0 , (20) i λ ψ − Ψ = g 0 , ( ρ 2 i λ + ω B ∗ θ ) Ψ − b ψ x x + κ ( ϕ x + ψ ) = ρ 2 g ˙ 0 . Substituting Φ and Ψ from the first equations to the seconds ones, this system becomes (21) ρ 1 λ 2 ϕ + κ ( ϕ x x + ψ x ) = f , (22) ( ρ 2 λ 2 − ω i λ B ∗ θ ) ψ + b ψ x x − κ ( ϕ x + ψ ) = g , where the functions f and g are given by (23) f = − ρ 1 ( f ˙ 0 + i λ f 0 ) , g = − ρ 2 ( g ˙ 0 + i λ g 0 ) − ω B ∗ θ g 0 .

Now, we are going to find some estimates for the solutions of system (19)–(20). In what follows, let’s recall that the parameter θ is in the interval [ 0 , 1 ] and we are going to use the notation f ≲ g to mean 0 ⩽ f ⩽ C g for some constant C > 0 .

Let λ ∈ R , F = ( f 0 , g 0 , f ˙ 0 , g ˙ 0 ) ∈ H and U ∈ D ( A ) the solution of the system ( i λ − A ) U = F , then from equality (14) it follows directly that (24) ω ‖ B ∗ θ / 2 Ψ ‖ 2 = Re ⟨ ( i λ − A ) U , U ⟩ = Re ⟨ F , U ⟩ ⩽ ‖ F ‖ ‖ U ‖ . Using equation of (20)₁ and taking into account the estimates (24) and (10) we have λ 2 ‖ B ∗ θ / 2 ψ ‖ 2 ≲ ‖ B ∗ θ / 2 Ψ ‖ 2 + ‖ B ∗ θ / 2 g 0 ‖ 2 ≲ ‖ B ∗ θ / 2 Ψ ‖ 2 + ‖ B ∗ 1 / 2 g 0 ‖ 2 (25) ≲ ‖ F ‖ ‖ U ‖ + ‖ F ‖ 2 .

Now, we are going to prove three technical lemmas which will be used in the proof of our main results.