ABSTRACT
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.
Keywords:
plate equation; polynomial decay; optimal decay; frictional damping; Kelvin-Voigt type damping
RESUMO
Neste trabalho estudamos o comportamento assintótico das placas de Kirchhoff com amortecimento intermediário. O amortecimento considerado contempla as dissipações de tipo friccional e de Kelvin-Voigt. Mostramos que o semigrupo dessas equações decai polinomialmente no tempo com a taxa t −1 /(2−2θ ), onde θ é um parâmetro no intervalo [0, 1[. Além disso, provamos que a taxa de decaimento encontrada é ótima.
Palavras-chave:
equação de placa; decaimento polinomial; decaimento ótimo; amortecimento friccional; dissipação de tipo Kelvin-Voigt
1 INTRODUCTION
Consider Ω a bounded open set of ℝn with smooth boundary. This paper deals with the asymptotic stability of the solutions of the following Kirchhoff plate equation
satisfying the boundary conditions
and the initial data
Here, the rotational inertia coefficient γ, the elasticity coefficient β and the damping coefficient ε, are positive and the exponent θ is considered in the interval [0, 1]. The term (−∆)θ u t in equation (1.1) sets up an intermediate dissipation which includes the frictional damping (θ = 0) and the Kelvin-Voigt damping (θ = 1).
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 77 V. Komornik. “Exact controllability and stabilization. The multiplier method. RAM: Research in Applied Mathematics”. Masson, Paris; John Wiley & Sons, 1994.), (99 J. Lagnese. Boundary stabilization of thin plates. SIAM Studies in Applied Mathematics 10.Philadelphia, PA, 1989.), (1010 Z. Liu & S. Zheng. “Semigroups associated with dissipative systems”. Chapman & Hall, 1999..
Concerning plate models with other dissipative mechanisms, we are going to mention some of them. In 22 A.R.A. Barbosa & T.F. Ma. Long-time dynamics of an extensible plate equation with thermal memory. J. Math. Anal. Appl. 416 (2014), 143-165., Barbosa et al. studied the stability of the thermoelastic plate
with boundary conditions . They showed the existence of exponential attractors using semigroup theory.
In 1313 H.P. Oquendo & M. Astudillo. Optimal decay for plates with rotational inertia and memory. Preprint., the authors considered the viscoelastic plate.
with boundary conditions 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. 1414 B. Rao & A. Wehbe. Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent. J. Evol. Equ. 5 (2005), 137-152., who studied the stability of the equation
satisfying the boundary conditions
Here, 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 Γ1 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 11 K. Ammari, M. Dimassi & M. Zerzeri. Rate of decay of some abstract Petrowsky-like dissipative semi-groups. Semigroup Forum 93 (2016), 1-16.), (55 F. Franchi, B. Lazzari & R. Nibbi. On the asymptotic stability for Kirchhoff plates with viscoelastic dissipation. Meccanica 53 (2018), 295-304.), (66 M. Grasselli & M. Squassina. Exponential stability and singular limit for a linear thermoelastic plate with memory effects. Adv. Math. Sci. Appl. 16 (2006), 15-31.), (88 I. Lasiecka & R. Triggiani. Exact controllability and uniform stabilization of Kirchhoff plates with boundary control only on ∆w|Σ and homogeneous boundary displacement. J. Differential Equations 93 (1991), 62-101.), (1111 M.I. Mustafa & M. Kafini. Decay rates for memory-type plate system with delay and source term. Math. Methods Appl. Sci. 40 (2017), 883-895.), (1212 H.P. Oquendo. Uniform decay for a plate equation partially dissipative. Appl. Math. Comput. 146 (2003), 121-134.), (1515 J.E.M. Rivera, H.P. Oquendo & M.L. Santos. Asymptotic behavior to a von Kármán plate with boundary memory conditions. Nonlinear Analysis 6 (2005) 1183-1205.), (1616 M.A.J. Silva, J.E.M. Rivera & R. Racke. On a class of nonlinear viscoelastic Kirchhoff plates: wellposedness and general decay rates. Appl. Math. Optim. 73 (2016), 165-194..
It is well known that the operator defined in the space 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.
2 EXISTENCE OF SOLUTIONS
We are going to use the semigroup theory to show existence of solutions for the system of equations (1.4)-(1.5). It is important recalling that , is a positive selfadjoint operator with compact inverse in the Hilbert space . Therefore, the operator A θ is self-adjoint, positive for and the embedding
is compact for . Here, the norm in the space D(A θ ) is given by , where ║·║ denotes the norm of the Hilbert space ℍ. More details about fractional operators can be found in 44 K.J. Engel & R. Nagel. “One-parameter semigroups for linear evolution equations”. Springer 2000.. Note that, in view of the Riesz representation Theorem we have: for ,
where 〈·,·〉 on the right side of this equation denotes the inner product in the space ℍ.
Now, if we consider the vector , then the system (1.4)-(1.5) can be written in an abstract framework as
where and the operator 𝔹 is given by
for . 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
for . 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
To show the existence of solutions of the system (2.2) we invoke a result from Zheng-Liu’ book 1010 Z. Liu & S. Zheng. “Semigroups associated with dissipative systems”. Chapman & Hall, 1999.:
Theorem 1 (see Theorem 1.2.4 in1010 Z. Liu & S. Zheng. “Semigroups associated with dissipative systems”. Chapman & Hall, 1999.).Let 𝔹 a linear operator with dense domain D(𝔹) in a Hilbert space 𝕏. If 𝔹 is dissipative and, the resolvent set of 𝔹, then 𝔹 is the generator of a C 0 -semigroup of contractions on 𝕏.
We are going to verify that the operator 𝔹 defined by (2.3) satisfies the conditions of this theorem. From definition (2.4), we easily see that D(𝔹) is dense in 𝕏. On the other hand, performing a simple computation we obtain
that is, the operator 𝔹 is dissipative. To complete the conditions of the above theorem, it remains to show that . Let , let us see that the stationary problem has a solution . From definition of the operator 𝔹 in (2.3), this system can be written by
This problem can be put in a variational formulation: to find such that
Now, applying the Lax-Milgram Theorem we have a unique solution . As this solution satisfies (2.6) we can conclude that . 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 , 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
Theorem 2. For there exist a unique solution of the system (1.4)-(1.5) in the space
3 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. Then we have
if and only if,
Let and . In what follows, the stationary problem will be considered several times in the course of this article. Note that 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.
Theorem 2. Consider the semigroup of the system (1.4)-(1.5). For we have the following polynomial decay
Moreover, this decay rate is optimal in the following sense: it does not decay with the rate t−κ, for.
Proof. We are going to use Theorem 1 to show this theorem. Since there exists such that , so to show that ρ(𝔹) contains the imaginary axis is sufficed to show that for real λ such that . Simultaneously we are going to prove the estimative for the solution U of the stationary equation .
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
Applying the Cauchy-Schwarz inequality, Young inequality and taking into account that we get that there exists a positive constant C such that
On the other hand, from the first equation of (3.1), we have
Now we are going to use an interpolation inequality. Since
using inequalities (3.3) and (3.4) we get that
Therefore, in view of inequality (3.2), we conclude that
for . 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
Finally, from this inequality and (3.5) we can conclude that
As we have from where follows the desired inequality . 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 . 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 , with . Let us denote by (en ) the corresponding eigenvectors, that is
We consider where , then the solution of the system 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 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
If we introduce the notation meaning is a positive real number, then from (3.6) and (3.7) we can assert that and . Therefore, if is the solution of the system , we obtain
for some and n large enough. From this estimate, we conclude that
Finally, let us suppose the semigroup decays with the rate t −κ for some . From Theorem 1 we have is bounded. On the other hand, the above inequality implies that
which is absurd. Therefore the decay rate t −1/(2−2θ) is optimal. □
REFERENCES
-
1K. Ammari, M. Dimassi & M. Zerzeri. Rate of decay of some abstract Petrowsky-like dissipative semi-groups. Semigroup Forum 93 (2016), 1-16.
-
2A.R.A. Barbosa & T.F. Ma. Long-time dynamics of an extensible plate equation with thermal memory. J. Math. Anal. Appl. 416 (2014), 143-165.
-
3A. Borichev & Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Math.Ann. 347 (2010), 455-478.
-
4K.J. Engel & R. Nagel. “One-parameter semigroups for linear evolution equations”. Springer 2000.
-
5F. Franchi, B. Lazzari & R. Nibbi. On the asymptotic stability for Kirchhoff plates with viscoelastic dissipation. Meccanica 53 (2018), 295-304.
-
6M. Grasselli & M. Squassina. Exponential stability and singular limit for a linear thermoelastic plate with memory effects. Adv. Math. Sci. Appl. 16 (2006), 15-31.
-
7V. Komornik. “Exact controllability and stabilization. The multiplier method. RAM: Research in Applied Mathematics”. Masson, Paris; John Wiley & Sons, 1994.
-
8I. Lasiecka & R. Triggiani. Exact controllability and uniform stabilization of Kirchhoff plates with boundary control only on ∆w|Σ and homogeneous boundary displacement. J. Differential Equations 93 (1991), 62-101.
-
9J. Lagnese. Boundary stabilization of thin plates. SIAM Studies in Applied Mathematics 10.Philadelphia, PA, 1989.
-
10Z. Liu & S. Zheng. “Semigroups associated with dissipative systems”. Chapman & Hall, 1999.
-
11M.I. Mustafa & M. Kafini. Decay rates for memory-type plate system with delay and source term. Math. Methods Appl. Sci. 40 (2017), 883-895.
-
12H.P. Oquendo. Uniform decay for a plate equation partially dissipative. Appl. Math. Comput. 146 (2003), 121-134.
-
13H.P. Oquendo & M. Astudillo. Optimal decay for plates with rotational inertia and memory. Preprint.
-
14B. Rao & A. Wehbe. Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent. J. Evol. Equ. 5 (2005), 137-152.
-
15J.E.M. Rivera, H.P. Oquendo & M.L. Santos. Asymptotic behavior to a von Kármán plate with boundary memory conditions. Nonlinear Analysis 6 (2005) 1183-1205.
-
16M.A.J. Silva, J.E.M. Rivera & R. Racke. On a class of nonlinear viscoelastic Kirchhoff plates: wellposedness and general decay rates. Appl. Math. Optim. 73 (2016), 165-194.
Publication Dates
-
Publication in this collection
03 Aug 2020 -
Date of issue
May-Aug 2020
History
-
Received
06 May 2019 -
Accepted
13 Feb 2020