Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping

Foued Mtiri; Amina Chaili; Ahmed Bchatnia; Abderrahmane Beniani

doi:10.1515/dema-2024-0091

Article Open Access

Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping

Foued Mtiri , Amina Chaili , Ahmed Bchatnia and Abderrahmane Beniani

Published/Copyright: March 13, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this work, we investigate the stabilization problem for coupled biharmonic Schrödinger equations with fractional internal damping. First, we establish that the system is well-posed using the semigroup theory of linear operators. Second, we demonstrate that the system is strongly stable. Finally, we prove a polynomial decay rate using multiplier techniques combined with the frequency domain method, and we show the lack of exponential decay of the energy in a specific case.

Keywords: biharmonic Schrödinger equation; internal fractional damping; semigroup theory; polynomial stability

MSC 2010: 35B40; 35Q41; 93D20

1 Introduction

In this study, we explore the existence and decay characteristics of solutions to the biharmonic Schrödinger equation. The system under consideration is given by

(1) i u t ( x , t ) + Δ u ( x , t ) − Δ 2 u ( x , t ) + ( − 1 ) j i γ ∂ α , η Δ j u ( x , t ) + i l ( u ( x , t ) − v ( x , t ) ) = 0 , in Ω × R + , i v t ( x , t ) + Δ v ( x , t ) − Δ 2 v ( x , t ) + ( − 1 ) j i γ ∂ α , η Δ j v ( x , t ) + i l ( v ( x , t ) − u ( x , t ) ) = 0 , in Ω × R + ,

where Ω is an open bounded domain in R n , l and γ are positive constants, and j ∈ { 0 , 1 , 2 } . The term ∂ α , η represents the generalized Caputo fractional derivative of order α with respect to the time variable [1], and is defined by

(2) ∂ α , η w ( t ) = 1 Γ ( 1 − α ) ∫ 0 t ( t − s ) − α e − η ( t − s ) w ( s ) d s , 0 < α < 1 , η ≥ 0 ,

where Γ denotes the Gamma function. The system is subject to the boundary conditions

(3) u ( x , t ) = v ( x , t ) = Δ u ( x , t ) = Δ v ( x , t ) = 0 , on ∂ Ω × R + * .

Finally, the initial conditions are given by

(4) u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) ,

where the initial data u 0 and v 0 belong to an appropriate function space. In this work, we delve into the stabilization and energy decay properties of a coupled system involving biharmonic Schrödinger equations with internal fractional damping. Through a mathematical analysis grounded in semigroup theory, we establish the well-posedness of our system, ensuring the existence and uniqueness of solutions under fractional damping conditions. Demonstrating strong stability, we show that solutions asymptotically approach zero, indicating robustness against perturbations for applications in quantum mechanics and wave propagation. Furthermore, employing advanced multiplier techniques and frequency domain methods, we unveil a polynomial decay rate in energy, highlighting the efficacy of fractional damping in swiftly dissipating energy. These findings not only deepen our theoretical understanding of coupled wave systems but also offer practical insights into enhancing stability and control in various physical and engineering contexts.

For the literature, we first quote the result obtained in [2], where Zhu et al. addressed the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L 2 supercritical nonlinearity

i u t − Δ 2 u + ∣ u ∣ p − 1 u = 0 , t ∈ [ 0 , T ) , x ∈ R N , u ( 0 , x ) = u 0 .

They provided a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation. Additionally, they focused on establishing the profile decomposition of bounded sequences in a specific functional space and derived the best constant of a Gagliardo-Nirenberg inequality.

Li and Di [3] investigated the well-posedness and stability of the Cauchy problem for the fourth-order Schrödinger equation with a nonlinear derivative term

i u t + Δ 2 u − u Δ ∣ u ∣ 2 + λ ∣ u ∣ p u = 0 , λ ∈ R , t ∈ R , and x ∈ R n .

The study established local well-posedness for initial data, along with a criterion for finite time blow-up, providing rough estimates for blow-up time and rate. Moreover, for small initial values, they demonstrated global well-posedness using two different methods. They finally examined the stability of the solutions using a priori estimates through short-term and long-term perturbation theories.

Recently, Capistrano-Filho et al. [4] investigated the stabilization of the linear biharmonic Schrödinger equation in an n -dimensional open bounded domain under Dirichlet-Neumann boundary conditions

i ∂ t y ( x , t ) + Δ y ( x , t ) − Δ 2 y ( x , t ) + ( − 1 ) j i ∫ 0 + ∞ f ( s ) Δ j y ( x , t − s ) d s = 0 , ( x , t ) ∈ Ω × R + , y ( x , t ) = ∇ y ( x , t ) = 0 , ( x , t ) ∈ Γ × R + * , y ( x , − t ) = y 0 ( x , t ) , ( x , t ) ∈ Ω × R + .

The study examined the effects of three infinite memory terms used as damping mechanisms. They demonstrated that the solution decays to zero at a polynomial rate, depending on the smoothness of the initial data and the growth properties of the kernel function associated with the infinite memory terms. Specifically, the decay rate follows the pattern t − n , where n is influenced by the assumptions about the kernel function. Previous literature [5–9] present a comprehensive collection of published works that support the mathematical formulation of problems related to fractional differential equations and the decay rate of the associated energy.

This work is organized as follows: In Section 2, we present preliminary results and reformulate the system (1) into an augmented system (6) below. Section 3 demonstrates the well-posedness of the problem using semigroup theory. In Section 4, we analyze the asymptotic stability by proving strong stability through the Arendt-Batty Theorem. Section 5 establishes the polynomial stability. In Section 6, we prove the lack of exponential decay of the energy in a specific case.

2 Preliminary results

In this section, we reformulate (1), (3), and (4) into an augmented system. To achieve this, we require the following auxiliary results.

Theorem 1

[10] Let μ be the function:

μ ( ξ ) = ∣ ξ ∣ ( 2 α − 1 ) ⁄ 2 , ξ ∈ R , 0 < α < 1 .

Then, the relationship between the “input” U and the “output” O of the system

∂ t θ ( x , ξ , t ) + ξ 2 θ ( x , ξ , t ) + η θ ( x , ξ , t ) − U ( t ) μ ( ξ ) = 0 , − ∞ < ξ < + ∞ , η ≥ 0 , t > 0 ,

θ ( x , ξ , 0 ) = 0 ,

O ( t ) = ( π ) − 1 sin ( α π ) ∫ − ∞ + ∞ μ ( ξ ) θ ( x , ξ , t ) d ξ ,

is given by

(5) O ( t ) = I 1 − α , η U ( t ) ,

where

[ I α , η f ] ( t ) = 1 Γ ( α ) ∫ 0 t ( t − τ ) α − 1 e − η ( t − τ ) f ( τ ) d τ .

Taking the input U ( x , t ) = y ( x , t ) and combining (2) with (5), we obtain

O ( t ) = I 1 − α , η y ( x , t ) = 1 Γ ( 1 − α ) ∫ 0 t ( t − s ) − α e − η ( t − s ) y ( x , s ) d s = ∂ α , η y ( x , t ) .

Using the previous theorem, systems (1), (3), and (4) can be written as the augmented model:

(6) u t ( x , t ) − i Δ u ( x , t ) + i Δ 2 u ( x , t ) + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ ( ξ ) Δ j θ ( x , ξ , t ) d ξ + l ( u ( x , t ) − v ( x , t ) ) = 0 , ( x , ξ , t ) ∈ Ω × R × R + , v t ( x , t ) − i Δ v ( x , t ) + i Δ 2 v ( x , t ) + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ ( ξ ) Δ j ϑ ( x , ξ , t ) d ξ + ( v ( x , t ) − u ( x , t ) ) = 0 , ( x , ξ , t ) ∈ Ω × R × R + , ∂ t θ ( x , ξ , t ) + ( ξ 2 + η ) θ ( x , ξ , t ) − μ ( ξ ) u = 0 , ( x , ξ , t ) ∈ Ω × R × R + , ∂ t ϑ ( x , ξ , t ) + ( ξ 2 + η ) ϑ ( x , ξ , t ) − μ ( ξ ) v = 0 , ( x , ξ , t ) ∈ Ω × R × R + ,

with the following initial conditions:

u ( x , t ) = v ( x , t ) = Δ u ( x , t ) = Δ v ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω , θ ( x , ξ , 0 ) = θ 0 ( x , ξ ) , ϑ ( x , ξ , 0 ) = ϑ 0 ( x , ξ ) , ( x , ξ ) ∈ Ω × R ,

where γ ˜ = γ π − 1 sin ( α π ) .

We define the energy associated with the solution of problem (6) by

(7) E ( t ) = 1 2 ‖ u ‖ L 2 ( Ω ) 2 + ‖ v ‖ L 2 ( Ω ) 2 + γ ˜ 2 ‖ Δ j 2 θ ‖ L 2 ( Ω × R ) 2 + ‖ Δ j 2 ϑ ‖ L 2 ( Ω × R ) 2 .

Next we analyze the behavior of the energy functional E ( t ) associated with the solution of the augmented system (6). We establish that the energy E ( t ) is non-increasing over time, as stated in the following lemma.

Lemma 1

Let ( u , v , θ , ϑ ) be a regular solution of problem (6). Then, the functional energy defined in equation (7) satisfies

d d t E ( t ) = − γ ˜ ∫ Ω ∫ − ∞ + ∞ ( ξ 2 + η ) ∣ Δ j 2 θ ( x , ξ , t ) ∣ 2 + ∣ Δ j 2 ϑ ( x , ξ , t ) ∣ 2 d ξ d x − l ‖ u − v ‖ L 2 ( Ω ) 2 ≤ 0 .

Proof

Multiplying equations (6)₁ and (6)₂ by u and v , respectively, integrating over Ω , and multiplying equations (6)₃ and (6)₄ by ( − 1 ) j γ ˜ Δ j θ and ( − 1 ) j γ ˜ Δ j ϑ , respectively, integrating over Ω × ( − ∞ , + ∞ ) , and applying the Green formula, we obtain the desired result.□

The following lemmas will be necessary for the sequel.

Lemma 2

[11] If λ ∈ D η = C \ ] − ∞ , − η ] , then

∫ − ∞ + ∞ μ 2 ( ξ ) λ + η + ξ 2 d ξ = π sin α π ( λ + η ) α − 1 .

Lemma 3

If λ ∈ D η = C \ ] − ∞ , − η ] , then

(8) ∫ − ∞ + ∞ ∣ ξ ∣ μ ( ξ ) ( i λ + ξ 2 + η ) 2 d ξ = 1 − 2 α 4 π sin ( 2 α + 3 ) 4 π ( i λ + η ) ( 2 α − 5 ) 4 ,

∫ − ∞ + ∞ 1 ∣ i λ + ξ 2 + η ∣ 2 d ξ ≤ π 2 ∣ i λ + η ∣ − 3 2 a n d ∫ − ∞ + ∞ ∣ ξ ∣ 2 ∣ i λ + ξ 2 + η ∣ 4 d ξ ≤ π 16 ∣ i λ + η ∣ − 5 2 .

Proof

Using Lemma 2, it is easy to see that for 0 < ς < 1 ,

(9) ∫ − ∞ + ∞ ∣ ξ ∣ 2 ς − 1 ( ξ 2 + ω + λ ) 2 d ξ = ( 1 − ς ) π sin ( ς π ) ( λ + ω ) ς − 2 ,

then (8) is a consequence of (9) with ς = 2 α + 3 4 .

Now, coming back to (8) with ς = 1 2 , we obtain

∫ − ∞ + ∞ 1 ∣ i λ + ξ 2 + η ∣ 2 d ξ = π 2 ∣ i λ + η ∣ − 3 2 ≤ π 2 ∣ ∣ λ ∣ + η ∣ − 3 2 .

Finally, for ς = 3 2 in (8), we obtain

∫ − ∞ + ∞ ∣ ξ ∣ 2 ∣ i λ + ξ 2 + η ∣ 2 d ξ = π 2 ∣ i λ + η ∣ − 1 2 .

We derive twice with respect to η , we obtain

□ ∫ − ∞ + ∞ ∣ ξ ∣ 2 ∣ i λ + ξ 2 + η ∣ 4 d ξ = π 16 ∣ i λ + η ∣ − 5 2 ≤ π 16 ∣ ∣ λ ∣ + η ∣ − 5 2 .

3 Well-posedness of the problem

In this section, our goal is to demonstrate that system (6) is well-posed in the context of semigroups. Introducing the vector function U = ( u , v , θ , ϑ ) T , then system (6) is equivalent to

(10) ∂ t U = A U , U ( 0 ) = U 0 = ( u 0 , v 0 , θ 0 , ϑ 0 ) T ,

where the operator A is defined by

A u v θ ϑ = i Δ u − i Δ 2 u − ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ ( ξ ) Δ j θ ( x , ξ ) d ξ − l ( u ( x , t ) − v ( x , t ) ) i Δ v − i Δ 2 v − ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ ( ξ ) Δ j ϑ ( x , ξ ) d ξ − l ( v ( x , t ) − u ( x , t ) ) − ( ξ 2 + η ) θ ( x , ξ ) + μ ( ξ ) u − ( ξ 2 + η ) ϑ ( x , ξ ) + μ ( ξ ) v

with domain

D ( A ) = ( u , v , θ , ϑ ) ∈ ℋ : u , v ∈ H 4 ( Ω ) ∩ H 0 j ( Ω ) , j = 0 , 2 ¯ , ∣ ξ ∣ θ , ∣ ξ ∣ ϑ ∈ L 2 ( Ω × R ) , i Δ u − i Δ 2 u − ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ ( ξ ) Δ j θ ( x , ξ ) d ξ − l ( u ( x , t ) − v ( x , t ) ) ∈ L 2 ( Ω ) i Δ v − i Δ 2 v − ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ ( ξ ) Δ j ϑ ( x , ξ ) d ξ − l ( v ( x , t ) − u ( x , t ) ) ∈ L 2 ( Ω ) − ( ξ 2 + η ) θ + u ( x ) μ ( ξ ) ∈ L 2 ( Ω × R ) , − ( ξ 2 + η ) ϑ + v ( x ) μ ( ξ ) ∈ L 2 ( Ω × R ) ,

where

ℋ = ( L 2 ( Ω ) ) 2 × ( H 0 j ( Ω × ( − ∞ , + ∞ ) ) ) 2 , j = 0 , 2 ¯ ,

equipped with the inner product

⟨ U , U 1 ⟩ ℋ = ∫ Ω ( u u 1 ¯ + v v 1 ¯ ) d x + γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j 2 θ Δ j 2 θ 1 ¯ + Δ j 2 ϑ Δ j 2 ϑ 1 ¯ d ξ d x ,

where U = ( u , v , θ , ϑ ) , U 1 = ( u 1 , v 1 , θ 1 , ϑ 1 ) ∈ ℋ .

We establish the following result regarding existence and uniqueness.

Theorem 2

If U 0 ∈ D ( A ) , then system (10) has a unique strong solution
U ∈ C 0 ( R + , D ( A ) ) ∩ C 1 ( R + , ℋ ) .
If U 0 ∈ ℋ , then system (10), has a unique weak solution
U ∈ C 0 ( R + , ℋ ) .

Proof

First, verifying that for every U ∈ D ( A ) , we have

(11) ℜ ⟨ A U , U ⟩ ℋ = − γ ˜ ∫ Ω ∫ − ∞ + ∞ ( ξ 2 + η ) ∣ Δ j 2 θ ( x , ξ ) ∣ 2 + ∣ Δ j 2 ϑ ( x , ξ , t ) ∣ 2 d ξ d x − l ‖ u − v ‖ L 2 ( Ω ) 2 ≤ 0 , j = 0 , 2 ¯ .

Therefore, A is dissipative.

Next we show that the operator I − A is surjective. Given F = ( f 1 , f 2 , f 3 , f 4 ) T ∈ ℋ , we prove that there exists U ∈ D ( A ) satisfying

(12) U − A U = F .

Equation (12) is equivalent to

(13) u − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ ( ξ ) Δ j θ ( x , ξ ) d ξ + l ( u ( x , t ) − v ( x , t ) ) = f 1 , v − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ ( ξ ) Δ j ϑ ( x , ξ ) d ξ + l ( v ( x , t ) − u ( x , t ) ) = f 2 , θ + ( ξ 2 + η ) θ ( x , ξ ) − μ ( ξ ) u = f 3 , ϑ + ( ξ 2 + η ) ϑ ( x , ξ ) − μ ( ξ ) u = f 4 .

Using (13)₃ and (13)₄, we obtain for j ∈ { 0 , 1 , 2 } :

(14) Δ j θ ( x , ξ ) = Δ j f 3 ( x , ξ ) + μ ( ξ ) Δ j u ξ 2 + η + 1 , Δ j ϑ ( x , ξ ) = Δ j f 4 ( x , ξ ) + μ ( ξ ) Δ j v ξ 2 + η + 1 .

By substituting (14)₁ into (13)₁ and (14)₂ into (13)₂, we obtain

(15) u − i Δ u + i Δ 2 u + a 0 Δ j u + l ( u − v ) = f 1 − ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η + 1 μ ( ξ ) d ξ , v − i Δ v + i Δ 2 v + a 0 Δ j v + l ( v − u ) = f 2 − ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η + 1 μ ( ξ ) d ξ .

with

a 0 = ( − 1 ) j γ ˜ π sin ( α π ) ( 1 + η ) α − 1 .

Solving system (15) is equivalent to finding u , v ∈ H 0 j ( Ω ) such that

∫ Ω ( u − i Δ u + i Δ 2 u + a 0 Δ j u + l ( u − v ) ) χ ¯ d x = ∫ Ω f 1 χ ¯ d x − ( − 1 ) j γ ˜ ∫ Ω χ ¯ ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η + 1 μ ( ξ ) d ξ , ∫ Ω ( v − i Δ v + i Δ 2 v + a 0 Δ j v + l ( v − u ) ) ζ ¯ d x = ∫ Ω f 2 ζ ¯ d x − ( − 1 ) j γ ˜ ∫ Ω ζ ¯ ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η + 1 μ ( ξ ) d ξ ,

for all χ , ζ ∈ H 0 j ( Ω ) . Thus, we have

(16) ∫ Ω ( u − i Δ u + i Δ 2 u + a 0 Δ j u ) χ ¯ d x + ∫ Ω ( v − i Δ v + i Δ 2 v + a 0 Δ j v ) ζ ¯ d x + l ∫ Ω ( u − v ) ( χ ¯ − ζ ¯ ) d x = ∫ Ω ( f 1 χ ¯ + f 2 ζ ¯ ) d x − ( − 1 ) j γ ˜ ∫ Ω χ ¯ ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η + 1 μ ( ξ ) d ξ − ( − 1 ) j γ ˜ ∫ Ω ζ ¯ ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η + 1 μ ( ξ ) d ξ .

Consequently, problem (16) is equivalent to the problem

(17) a ( ( u , v ) , ( χ , ζ ) ) = ℒ ( χ , ζ ) ,

where the sesquilinear form a : ( H 0 j ( Ω ) × H 0 j ( Ω ) ) 2 ⟶ C and the antilinear form ℒ : H 0 j ( Ω ) × H 0 j ( Ω ) ⟶ C are defined by

a ( ( u , v ) , ( χ , ζ ) ) = ∫ Ω ( u − i Δ u + i Δ 2 u + a 0 Δ j u ) χ ¯ d x + ∫ Ω ( v − i Δ v + i Δ 2 v + a 0 Δ j v ) ζ ¯ d x + l ∫ Ω ( u − v ) ( χ ¯ − ζ ¯ ) d x

and

ℒ ( χ , ζ ) = ∫ Ω ( f 1 χ ¯ + f 2 ζ ¯ ) d x − ( − 1 ) j γ ˜ ∫ Ω χ ¯ ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η + 1 μ ( ξ ) d ξ − ( − 1 ) j γ ˜ ∫ Ω ζ ¯ ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η + 1 μ ( ξ ) d ξ .

We can easily verify that a is both continuous and coercive, while ℒ is continuous. As a result, according to the Lax-Milgram Lemma, system (17) has a unique solution u , v ∈ H 0 j ( Ω ) . Furthermore, by applying the regularity theory for linear elliptic equations, we deduce that u , v ∈ H 4 ( Ω ) . Consequently, the operator I − A is surjective.□

4 Strong stability

In this section, we will examine the strong stability of the solution associated with problem (10). To this end, we apply a version of the Arendt-Batty and Lyubich-Vu theorems for Hilbert spaces [12,13].

Theorem 3

[12,13] Let A be the generator of a uniformly bounded C 0 -semigroup { S ( t ) } t ≥ 0 on a Hilbert space ℋ . Assume that

A does not have eigenvalues on i R ;
the intersection of the spectrum σ ( A ) with i R is at most a countable set.

Then, the semigroup { S ( t ) } t ≥ 0 is asymptotically stable, i.e., ‖ S ( t ) z ‖ ℋ → 0 as t → ∞ for any z ∈ ℋ .

Our main result is the following theorem.

Theorem 4

The C 0 -semigroup { S ( t ) } t ≥ 0 is strongly stable in ℋ , i.e., for all U 0 ∈ ℋ , the solution of (10) satisfies

lim t → ∞ ‖ S ( t ) U 0 ‖ ℋ = 0 .

To prove Theorem 4, we need the following two lemmas.

Lemma 4

For all λ ∈ R , the operator i λ I − A is injective

ker ( i λ I − A ) = { 0 } .

Proof

Given λ ∈ R where i λ is an eigenvalue of the operator A , let U = ( u , v , θ , ϑ ) ∈ D ( A ) be a corresponding eigenvector such that

(18) A U = i λ U .

Equivalently,

(19) i λ u − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j θ ( x , ξ ) μ ( ξ ) d ξ + l ( u − v ) = 0 , i λ v − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j ϑ ( x , ξ ) μ ( ξ ) d ξ + l ( v − u ) = 0 , i λ θ + ( ξ 2 + η ) θ ( x , ξ ) − μ ( ξ ) u = 0 , i λ ϑ + ( ξ 2 + η ) ϑ ( x , ξ ) − μ ( ξ ) v = 0 .

From (11) and (18), we obtain

0 = ℜ ⟨ A U , U ⟩ ℋ = − γ ˜ ∫ Ω ∫ − ∞ + ∞ ( ξ 2 + η ) ( ∣ Δ j ⁄ 2 θ ( x , ξ , t ) ∣ 2 + ∣ Δ j ⁄ 2 ϑ ( x , ξ , t ) ∣ 2 ) d ξ d x − l ‖ u − v ‖ L 2 ( Ω ) 2 .

It is obvious that

(20) θ ( x , ξ ) = ϑ ( x , ξ ) = 0 in Ω × ( − ∞ , + ∞ ) .

Substituting (20) into (19)₃ and (19)₄, we obtain

u = 0 , v = 0 .

Consequently, U = 0 .□

Lemma 5

If η > 0 and λ ∈ R , or η = 0 and λ ∈ R * , then i λ I − A is surjective.

Proof

We distinguish the following cases:

Case 1: λ ≠ 0 . Let F = ( f 1 , f 2 , f 3 , f 4 ) T ∈ ℋ be given. We seek for U = ( u , v , θ , ϑ ) T ∈ D ( A ) such that

( i λ I − A ) U = F .

Equivalently, we have

(21) i λ u − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j θ ( x , ξ ) μ ( ξ ) d ξ + l ( u − v ) = f 1 ( x ) , i λ v − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j ϑ ( x , ξ ) μ ( ξ ) d ξ + l ( v − u ) = f 2 ( x ) , i λ θ + ( ξ 2 + η ) θ ( x , ξ ) − μ ( ξ ) u = f 3 ( x , ξ ) , i λ ϑ + ( ξ 2 + η ) ϑ ( x , ξ ) − μ ( ξ ) v = f 4 ( x , ξ ) .

Using (21)₃ and (21)₄, we can find Δ j θ and Δ j ϑ as

(22) Δ j θ ( x , ξ ) = Δ j f 3 ( x , ξ ) + μ ( ξ ) Δ j u ξ 2 + η + i λ , Δ j ϑ ( x , ξ ) = Δ j f 4 ( x , ξ ) + μ ( ξ ) Δ j v ξ 2 + η + i λ .

By substituting (22)₁ in (21)₁ and (22)₂ in (21)₂, we obtain

(23) i λ u − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) + μ ( ξ ) Δ j u ξ 2 + η + i λ μ ( ξ ) d ξ + l ( u − v ) = f 1 ( x ) i λ v − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) + μ ( ξ ) Δ j v ξ 2 + η + i λ μ ( ξ ) d ξ + l ( v − u ) = f 2 ( x ) .

Solving system (23) is equivalent to finding u , v ∈ H 4 ( Ω ) ∩ H 0 j ( Ω ) such that

∫ Ω i λ u − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j u ξ 2 + η + i λ d ξ + l ( u − v ) χ d x = ∫ Ω f 1 χ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η + i λ μ ( ξ ) χ d ξ d x , ∫ Ω i λ v − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j v ξ 2 + η + i λ d ξ + l ( v − u ) ζ d x = ∫ Ω f 2 ζ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η + i λ μ ( ξ ) ζ d ξ d x ,

for all χ , ζ ∈ H 0 1 ( Ω ) . Then,

(24) ∫ Ω i λ u − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j u ξ 2 + η + i λ d ξ + l ( u − v ) χ d x + ∫ Ω i λ v − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j v ξ 2 + η + i λ d ξ + l ( v − u ) ζ d x = ∫ Ω f 1 χ + f 2 ζ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η + i λ μ ( ξ ) χ d ξ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η + i λ μ ( ξ ) ζ d ξ d x .

Thus, (24) is equivalent to the problem

(25) a ˜ ( ( u , v ) , ( χ , ζ ) ) = ℒ ˜ ( χ , ζ ) ,

where the sesquilinear form a ˜ : ( H 0 j ( Ω ) × H 0 j ( Ω ) ) 2 → C and the antilinear form ℒ ˜ : H 0 j × : H 0 j ( Ω ) → C are defined by

a ˜ ( ( u , v ) , ( χ , ζ ) ) = ∫ Ω i λ u − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j u ξ 2 + η + i λ d ξ + l ( u − v ) χ d x + ∫ Ω i λ v − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j v ξ 2 + η + i λ d ξ + l ( v − u ) ζ d x

and

ℒ ˜ ( χ , ζ ) = ∫ Ω f 1 χ + f 2 ζ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η + i λ μ ( ξ ) χ d ξ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η + i λ μ ( ξ ) ζ d ξ d x .

It is easy to verify that a ˜ is both continuous and coercive, and ℒ ˜ is continuous. By the Lax-Milgram theorem, we can conclude that for any ( χ , ζ ) ∈ H 0 j ( Ω ) × H 0 j ( Ω ) , problem (25) has a unique solution ( u , v ) ∈ H 0 j ( Ω ) × H 0 j ( Ω ) . Through the application of classical elliptic regularity, it can be deduced that ( u , v ) ∈ H 4 ( Ω ) × H 4 ( Ω ) .

Hence, the operator i λ I − A is surjective.

Case 2: λ = 0 and η ≠ 0 . In this case, system (21) simplifies to the following:

(26) − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j θ ( x , ξ ) μ ( ξ ) d ξ + l ( u − v ) = f 1 ( x ) , − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j ϑ ( x , ξ ) μ ( ξ ) d ξ + l ( v − u ) = f 2 ( x ) , ( ξ 2 + η ) θ ( x , ξ ) − μ ( ξ ) u = f 3 ( x , ξ ) , ( ξ 2 + η ) ϑ ( x , ξ ) − μ ( ξ ) v = f 4 ( x , ξ ) .

From (26)₃ and (26)₄, we have

(27) Δ j θ ( x , ξ ) = Δ j f 3 ( x , ξ ) + μ ( ξ ) Δ j u ξ 2 + η Δ j ϑ ( x , ξ ) = Δ j f 4 ( x , ξ ) + μ ( ξ ) Δ j v ξ 2 + η .

Using (27), we obtain

(28) − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) + μ ( ξ ) Δ j u ξ 2 + η μ ( ξ ) d ξ + l ( u − v ) = f 1 ( x ) − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) + μ ( ξ ) Δ j v ξ 2 + η μ ( ξ ) d ξ + l ( v − u ) = f 2 ( x ) .

Solving system (28) is equivalent to finding u , v ∈ H 4 ( Ω ) ∩ H 0 j ( Ω ) such that

∫ Ω − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j u ξ 2 + η d ξ + l ( u − v ) χ d x = ∫ Ω f 1 χ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η μ ( ξ ) χ d ξ d x , ∫ Ω − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j v ξ 2 + η d ξ + l ( v − u ) ζ d x = ∫ Ω f 2 ζ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η μ ( ξ ) ζ d ξ d x ,

for all χ , ζ ∈ H 0 j ( Ω ) . Then,

(29) ∫ Ω − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j u ξ 2 + η d ξ + l ( u − v ) χ d x + ∫ Ω − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j v ξ 2 + η d ξ + l ( v − u ) ζ d x = ∫ Ω f 1 χ + f 2 ζ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η μ ( ξ ) χ d ξ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η μ ( ξ ) ζ d ξ d x .

Therefore, (29) is equivalent to the problem

(30) a ˜ ( ( u , v ) , ( χ , ζ ) ) = ℒ ˜ ( w ) ,

where the sesquilinear form a ˜ : ( H 0 j ( Ω ) × H 0 j ( Ω ) ) 2 → C and the antilinear form ℒ ˜ : H 0 j ( Ω ) × H 0 j ( Ω ) → C are defined by

a ˜ ( ( u , v ) , ( χ , ζ ) ) = ∫ Ω − i Δ u + i Δ 2 u + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j u ξ 2 + η d ξ + l ( u − v ) χ d x + ∫ Ω − i Δ v + i Δ 2 v + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) Δ j v ξ 2 + η d ξ + l ( v − u ) ζ d x

and

ℒ ˜ ( χ , ζ ) = ∫ Ω f 1 χ + f 2 ζ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 3 ( x , ξ ) ξ 2 + η μ ( ξ ) χ d ξ d x − ( − 1 ) j γ ˜ ∫ Ω ∫ − ∞ + ∞ Δ j f 4 ( x , ξ ) ξ 2 + η μ ( ξ ) ζ d ξ d x .

It is simple to verify that a ˜ is both continuous and coercive, and ℒ ˜ is continuous. Using the the Lax-Milgram theorem, we conclude that for all ( χ , ζ ) ∈ H 0 j ( Ω ) × H 0 j ( Ω ) , problem (30) has a unique solution ( u , v ) ∈ H 0 j ( Ω ) × H 0 j ( Ω ) . Consequently, by applying the classical elliptic regularity, ( u , v ) ∈ H 4 ( Ω ) × H 4 ( Ω ) . Hence, the operator − A is surjective.□

Lemma 6

Assume that η = 0 . Then, the operator − A is not invertible and consequently 0 ∈ σ ( A ) .

Proof

Let U = ( u , v , θ , ϑ ) ∈ D ( A ) such that u ≠ 0 and v ≠ 0 , and we define the vector F = ( f 1 , f 2 , f 3 , f 4 ) ∈ ℋ such that

A U = F .

We seek U = ( u , v , θ , ϑ ) ∈ D ( A ) such that

− A U = G ,

where G = ( − f 1 , − f 2 , 0 , 0 ) . Then,

(31) ∣ ξ ∣ 2 θ + μ ( ξ ) u = 0 on Ω

and

(32) ∣ ξ ∣ 2 ϑ + μ ( ξ ) v = 0 on Ω .

From (31) and (32), we conclude that θ ( x , ξ ) = − ∣ ξ ∣ 2 α − 5 2 u ∉ L 2 ( Ω × ( − ∞ , + ∞ ) ) and ϑ ( x , ξ ) = − ∣ ξ ∣ 2 α − 5 2 v ∉ L 2 ( Ω × ( − ∞ , + ∞ ) ) . Therefore, given that the assumption of the existence of U is incorrect, it follows that the operator − A is not invertible.□

Proof of Theorem 4

Applying a general criterion of Arendt-Batty [12], the C 0 − semigroup of contractions { S ( t ) } t ≥ 0 is strongly stable if σ ( A ) ∩ i R is countable and if no eigenvalue of A lies on the imaginary axis. Initially, by utilizing Lemma 4, we infer that A possesses non-purely imaginary eigenvalues. Subsequently, employing Lemmas 5 and 6, and the Banach closed graph theorem, which implies that σ ( A ) ∩ i R = ∅ if η > 0 and σ ( A ) ∩ i R = { 0 } if η = 0 . The proof is complete.□

5 Polynomial stability

In this section, we will establish a polynomial decay rate. First, we recall the following lemma due to [14].

Lemma 7

[14] Assume that A is the generator of a strongly continuous semigroup of contractions { S ( t ) } t ≥ 0 on a Hilbert space ℋ . If

(33) i R ⊂ ρ ( A ) ,

then for a fixed δ > 0 , the following conditions are equivalent:

(34) lim s ∈ R sup ∣ s ∣ → ∞ 1 ∣ s ∣ δ ‖ ( i s I − A ) − 1 ‖ ℒ ( ℋ ) < ∞ ,

‖ S ( t ) U 0 ‖ ℋ 2 ≤ c t 2 δ ‖ U 0 ‖ D ( A ) 2 , U 0 ∈ D ( A ) , for s o m e c > 0 .

Our main result in the section is the following.

Theorem 5

The semigroup { S ( t ) } t ≥ 0 is polynomially stable and

E ( t ) = ‖ S ( t ) U 0 ‖ ℋ 2 ≤ 1 t 2 1 − α ‖ U 0 ‖ D ( A ) 2 .

Furthermore, the energy decay rate of t 2 ⁄ 1 − α is optimal for general initial data in D ( A ) .

Proof

Based on Lemma 7, the proof of Theorem 5 requires verifying the validity of (33) and (34), where δ = 1 − α . Given that (33) has already been established in Theorem 4, our focus shifts solely to proving (34). Here we employ a contradiction argument. Suppose that (34) is invalid; consequently, there exists a sequence λ n ∈ R , n ∈ N such that λ n → + ∞ as n → + ∞ , and a sequence U n = ( u n , v n , θ n , ϑ n ) ∈ D ( A ) such that

(35) ‖ U n ‖ = 1

and

(36) lim n → ∞ 1 λ n δ ‖ ( i λ n I − A ) − 1 ‖ ℒ ( H ) = 0 .

We have

F n = λ n δ ( i λ n I − A ) U n = ( f 1 n , f 2 n , f 3 n , f 4 n ) .

From (36), we obtain

(37) i λ n u n − i Δ u n + i Δ 2 u n + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j θ n ( x , ξ , t ) μ ( ξ ) d ξ − l ( u n − v n ) = f 1 n λ n δ → 0 , i λ n v n − i Δ v n + i Δ 2 v n + ( − 1 ) j γ ˜ ∫ − ∞ + ∞ Δ j ϑ n ( x , ξ , t ) μ ( ξ ) d ξ − l ( v n − u n ) = f 2 n λ n δ → 0 , i λ n θ n + ( ξ 2 + η ) θ n − μ ( ξ ) u n = f 3 n λ n δ → 0 , i λ n ϑ n + ( ξ 2 + η ) ϑ n − μ ( ξ ) v n = f 4 n λ n δ → 0 .

To deduce from (36) that ‖ U n ‖ ℋ = o ( 1 ) , we require the following results.

Lemma 8

Under (37), we have

(38) ∫ Ω ∫ − ∞ + ∞ ( ξ 2 + η ) ∣ Δ j 2 θ n ( x , ξ ) ∣ 2 + ∣ Δ j 2 ϑ n ( x , ξ ) ∣ 2 d ξ d x + l ‖ u n − v n ‖ L 2 ( Ω ) 2 = o ( 1 ) λ n δ

and

(39) ∫ Ω ∫ − ∞ + ∞ ∣ Δ j 2 θ n ( x , ξ ) ∣ 2 + ∣ Δ j 2 ϑ n ( x , ξ ) ∣ 2 d ξ d x = o ( 1 ) λ n δ .

Proof

From (11) and (36), we have

ℜ ⟨ i λ n U n − A U n , U n ⟩ ℋ = γ ˜ ∫ Ω ∫ − ∞ + ∞ ( ξ 2 + η ) ∣ Δ j 2 θ n ( x , ξ ) ∣ 2 + ∣ Δ j 2 ϑ n ( x , ξ ) ∣ 2 d ξ d x + l ‖ u n − v n ‖ L 2 ( Ω ) 2 = o ( 1 ) λ n δ ,

which implies (38).

Estimation (39) is a consequence of

□ ∫ Ω ∫ − ∞ + ∞ ∣ Δ j 2 θ n ( x , ξ ) ∣ 2 + ∣ Δ j 2 ϑ n ( x , ξ ) ∣ 2 d ξ d x ≤ 1 η ∫ Ω ∫ − ∞ + ∞ ( ξ 2 + η ) ∣ Δ j 2 θ n ( x , ξ ) ∣ 2 + ∣ Δ j 2 ϑ n ( x , ξ ) ∣ 2 d ξ d x .

Lemma 9

We have ∫ Ω ∣ u n ( x ) ∣ 2 d x = o ( 1 ) λ n δ + α − 1 and ∫ Ω ∣ v n ( x ) ∣ 2 d x = o ( 1 ) λ n δ + α − 1 .

Proof

From (37) 3 , we have

( i λ n + ξ 2 + η ) θ n − f 3 n λ n δ = u n ( x ) μ ( ξ ) , on Ω .

By multiplying it by ( i λ n + ξ 2 + η ) − 2 ∣ ξ ∣ , we obtain

(40) ( i λ n + ξ 2 + η ) − 2 ∣ ξ ∣ u n ( x ) μ ( ξ ) = ( i λ n + ξ 2 + η ) − 1 ∣ ξ ∣ θ n − ( i λ n + ξ 2 + η ) − 2 ∣ ξ ∣ f 3 n λ n δ , ∀ x ∈ Ω .

Taking the absolute values of both left- and right sides of (40), integrating over ( − ∞ , + ∞ ) with respect to the variable ξ , and applying Cauchy-Schwarz’s inequality, we obtain

(41) P ∣ u n ( x ) ∣ ≤ ℳ ∫ − ∞ + ∞ ( ξ 2 + η ) ∣ θ n ( x , ξ ) ∣ 2 d ξ 1 2 + N ∫ − ∞ + ∞ f 3 n λ n δ 2 d ξ 1 2 ,

where P , ℳ , and N are defined as

P ≔ ∫ − ∞ + ∞ ( i λ n + ξ 2 + η ) − 2 ξ ∣ μ ( ξ ) d ξ ∣ , ℳ ≔ ∫ − ∞ + ∞ ∣ i λ n + ξ 2 + η ∣ − 2 d ξ 1 2 , N ≔ ∫ − ∞ + ∞ ( ∣ i λ n + ξ 2 + η ∣ ) − 4 ∣ ξ ∣ 2 d ξ 1 2 .

Using Lemma 3, we obtain

P = ∣ 1 − 2 α ∣ 4 π ∣ sin ( 2 α + 3 ) 4 π ∣ ∣ i λ n + η ∣ ( 2 α − 5 ) 4 , ℳ ≤ π ∣ ∣ λ n ∣ + η ∣ − 3 4 ,

and

N ≤ 2 π 16 ∣ ∣ λ n ∣ + η ∣ − 5 2 1 ⁄ 2 .

By applying Young’s inequality and integrating (41) over ( Ω ) , we obtain

∫ Ω ∣ u n ( x ) ∣ 2 d x ≤ 2 ℳ 2 P 2 ∫ Ω ∫ − ∞ + ∞ ( ξ 2 + η ) ∣ θ n ( x , ξ ) ∣ 2 d ξ d x + 2 N 2 P 2 ∫ Ω ∫ − ∞ + ∞ f 3 n λ n δ 2 d ξ d x .

It is simple to check that

P 2 = O ∣ λ n ∣ 2 α − 5 2 , ℳ 2 = O ∣ λ n ∣ − 3 2 , and N 2 = O ∣ λ n ∣ − 5 2 .

Using (37) and (38), we obtain

∫ Ω ∣ u n ( x ) ∣ 2 d x = o ( 1 ) λ n α − 1 + δ + o ( 1 ) λ n α + 2 δ = o ( 1 ) λ n α − 1 + δ .

Using the same argument, we can prove

□ ∫ Ω ∣ v n ( x ) ∣ 2 d x = o ( 1 ) λ n α − 1 + δ .

Returning to the proof of Theorem 5, taking into account Lemmas 8 and 9, we conclude

‖ U n ‖ 2 ∫ Ω ∣ u n ( x ) ∣ 2 d x + ∫ Ω ∣ v n ( x ) ∣ 2 d x + γ ˜ ∫ Ω ∫ − ∞ + ∞ ∣ Δ j 2 θ n ( x , ξ ) ∣ 2 + ∣ Δ j 2 ϑ n ( x , ξ ) ∣ 2 d ξ d x = 2 o ( 1 ) λ n α − 1 + δ + o ( 1 ) λ n δ .

By considering δ = 1 − α , we establish that ‖ U n ‖ = o ( 1 ) , which contradicts (35). Furthermore, we establish the optimality of the decay rate, which corresponds closely to the asymptotic expansion of the eigenvalues. Specifically, it shows a behavior in the real part resembling k ( 1 − α ) . This completes the proof.□

6 Lack of exponential stability

This section is dedicated to demonstrate the lack of exponential stability for j = 0 . We will employ the following theorem.

Theorem 6

[15] Assume that A is the generator of a strongly continuous semigroup of contractions { S ( t ) } t ≥ 0 on a Hilbert space X. Then, S ( t ) is exponentially stable if and only if

(42) ρ ( A ) ⊇ { i β : β ∈ R } ≡ i R

and

(43) lim ¯ ∣ β ∣ → ∞ ∥ ( i β I − A ) − 1 ∥ L ( X ) < ∞ .

Our main result is stated in the following theorem.

Theorem 7

The semigroup generated by the operator A is not exponentially stable for j = 0 .

Proof

Let − β n 2 = ( i β n ) 2 be a sequence of eigenvalues corresponding to the sequence of normalized eigenfunctions u n of the operator Δ , such that

∣ β n ∣ ⟶ ∞ as n ⟶ ∞

and

(44) Δ u n = − β n 2 u n , in Ω , u n = 0 , on ∂ Ω .

Our objective is to demonstrate that, under certain conditions, if i β n satisfies (42), then (43) is not satisfied. In essence, we aim to prove that an infinite number of eigenvalues of A converge to the imaginary axis, which prevents the Schördinger system (6) from achieving exponential stability. First, we calculate the characteristic equation that gives the eigenvalues of A . Let λ be an eigenvalue of A with associated eigenvector U = ( u , v , θ , ϑ ) T . Then, A U = λ U is equivalent to

(45) λ u − i Δ u + i Δ 2 u + γ ˜ ∫ − ∞ + ∞ μ ( ξ ) θ ( x , ξ ) d ξ + l ( u − v ) = 0 , λ v − i Δ v + i Δ 2 v + γ ˜ ∫ − ∞ + ∞ μ ( ξ ) ϑ ( x , ξ ) d ξ + l ( v − u ) = 0 , λ θ ( x , ξ ) + ( ξ 2 + η ) θ ( x , ξ ) − μ ( ξ ) u = 0 , λ ϑ ( x , ξ ) + ( ξ 2 + η ) ϑ ( x , ξ ) − μ ( ξ ) v = 0 .

Using (45)₃ and (45)₄, we can find θ and ϑ as

(46) θ ( x , ξ ) = μ ( ξ ) u ξ 2 + η + λ , ϑ ( x , ξ ) = μ ( ξ ) v ξ 2 + η + λ .

By substituting (46)₁ in (45)₁ and (46)₂ in (45)₂, we obtain

λ u − i Δ u + i Δ 2 u + γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) u ξ 2 + η + λ d ξ + l ( u − v ) = 0 , λ v − i Δ v + i Δ 2 v + γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) v ξ 2 + η + λ d ξ + l ( v − u ) = 0 .

Then,

λ ( u + v ) − i Δ ( u + v ) + i Δ 2 ( u + v ) + γ ˜ ∫ − ∞ + ∞ μ 2 ( ξ ) ξ 2 + η + λ ( u + v ) d ξ = 0 .

Let us use the decomposition of u and v in the Hilbert’s basis { u n , n ∈ N } , (44), and Lemma 2 we obtain for large eigenvalues λ n in the strip − α 0 ≤ ℜ ( λ n ) ≤ 0 , for some α 0 > 0 large enough

λ n + i β n 2 + i β n 4 + γ ( λ n ) 1 − α + o 1 λ n 1 − α = 0 ,

using De Moivre’s formula and taking the real part, we obtain

ℜ ( λ n ) + γ cos ( 1 − α ) θ r 2 ( 1 − α ) + o 1 r ( 1 − α ) = 0 ,

then

ℜ ( λ n ) = − γ cos ( 1 − α ) θ r 2 ( 1 − α ) + o 1 r ( 1 − α ) .

We conclude that

r 2 ( 1 − α ) ℜ λ n ∼ ϱ

with

ϱ = − γ cos ( 1 − α ) θ .

Then, for 0 ≤ θ ≤ π 2 ( 1 − α ) , the operator A has a branch of eigenvalues that do not decay exponentially. The proof is complete.□

7 Conclusion

In this study, we have investigated the stabilization and energy decay properties of a coupled system of biharmonic Schrödinger equations with fractional internal damping. Our analysis began with establishing the well-posedness of the system using semigroup theory for linear operators, ensuring the existence and uniqueness of solutions. Building upon this foundation, we then demonstrated the strong stability of the system, showing that solutions asymptotically approach zero as time tends to infinity. Moreover, leveraging multiplier techniques in conjunction with frequency domain methods, we established a polynomial decay rate for the energy of solutions. Specifically, we found that the L 2 norms of the solution components decay as 1 t 2 ⁄ 1 − α , highlighting the effective damping effect introduced by the fractional internal damping. These results not only confirm the long-term stability of the coupled system but also quantify the rate at which energy dissipates over time. Our findings contribute to the understanding of how fractional damping mechanisms influence the dynamics of biharmonic Schrödinger equations, paving the way for further exploration in related fields of mathematical physics and applied analysis.

Acknowledgements

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khaled University for funding this work through Large Research Project under Grant number RGP2/368/45.

Funding information: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors declare no competing interests.
Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.
Data availability statement: Data on the results of the study may be obtained from the corresponding author upon reasonable request.

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Received: 2024-07-13

Accepted: 2024-10-30

Published Online: 2025-03-13

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https://doi.org/10.1515/dema-2024-0091

Keywords for this article

biharmonic Schrödinger equation; internal fractional damping; semigroup theory; polynomial stability

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