Home Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
Article Open Access

Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term

  • Abdelbaki Choucha , Salah Mahmoud Boulaaras EMAIL logo and Rashid Jan
Published/Copyright: December 8, 2023
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

The Lord Shulman swelling porous thermo-elastic soil system with the presence of a distributed delay term is studied in this work. We will establish the well-posedness of the system and the exponential stability of the system is derived.

Keywords: Lord-Shulman; mathematical operators; swelling porous system; partial differential equations; general decay; distributed delay term
MSC 2010: 35B40; 35L70; 74D05; 93D20

1 Introduction and preliminaries

A concept in which a mixture of viscous liquid and particles combined with gas was first introduced by Eringen [1]. We go to the field equations after examining this heat-resistant combination [2]. Expansive (swelling) soils are categorized under the porous media theory, which investigates this kind of issue. That is why there have been several research studies to reduce the damage caused by swelling soil, particularly in architecture and civil engineering; for more information, see [3]–[10]. The linear theory of swelling porous elastic soil fundamental field equations are stated as follows:

(1.1) ℓ y y t t = P 1 x + G 1 + K 1 , ℓ ℏ ℏ t t = P 2 x + G 2 + K 2 ,

where elastic solid material and fluid displacement are denoted by ℏ , y , and ℓ ℏ , ℓ y > 0 are their respective densities. The partial tension, internal body forces, and eternal forces, which are all acting on the displacement are ( P 1 , G 1 , K 1 ), respectively. Acting similarly to ( P 2 , G 2 , K 2 ) but on the elastic solid. The constitutive equations for partial tensions are also provided by

(1.2) P 1 P 2 = a 1 , a 2 a 2 , a 3 ︸ A . y x ℏ x ,

where a 1 , a 3 > 0 , and a 2 ≠ 0 is a real number. A is matrix positive definite with a 1 a 3 > a 2 2 .

Quintanilla [11] studied (1.1) by assuming

G 1 = G 2 = ξ ( y t − ℏ t ) , K 1 = a 3 y x x t , K 2 = 0 .

They discovered that stability occurs exponentially in cases when ξ > 0 . Similar to this, in [12] the writers examined (1.1) under different conditions

G 1 = G 2 = 0 , K 1 = − ℓ y γ ( x ) y t , K 2 = 0 ,

where γ ( x ) is an internal viscous damping function with positive mean. They discovered the exponential stability finding utilizing the spectral approach, for more information, refer [13]–[19].

The majority of natural phenomena and industrial equipment involve time delays, which are significant because time lag causes instability and should be taken into consideration. Additionally, other studies have examined this type of problem, including [20]–[26].

In recent times, there has been a substantial surge of interest among scientists in Lord Shulman’s thermoelasticity, leading to an extensive collection of contributions aimed at elucidating this theory. This theoretical framework encompasses the examination of a system comprising four hyperbolic equations coupled with heat transfer dynamics. Moreover, Lord-Shulman thermoelastic theory was introduced to usher in a more robust heat conduction law as it concerns thermoelastic materials exhibiting elastic vibrations. Notably, the heat equation within this context is itself hyperbolic and parallels the equation initially formulated by Fourier’s law. To delve deeper into the specifics and gain a comprehensive understanding of this theory, it is recommended that the reader consult the following papers: [27,28].

The fundamental equations of evolution for one-dimensional theories of swelling porous thermoelasticity with microtemperature and temperature [29]–[34] are provided by

(1.3) ℓ y y t t = T x , ℓ φ ℏ t t = K x + G , ℓ T 0 η t = q x , ℓ E t = P x * + q − Q ,

where K , T 0 , G , T , q , P * , η , E , Q represent the equilibrated stress, the reference temperature, the equilibrated body force, the stress, the heat flux vector, the first heat flux moment, the entropy, the first moment of energy, and the mean heat flux. To make the calculations easier, let T 0 = 1 .

In this study, we take into account the natural counterpart for the microtemperatures of the Lord-Shulman theory. The constitutive equations must be modified in this case to the following form:

(1.4) T = P 1 + G 1 + K 1 , P * = − k 2 ω x , K = P 2 + P 3 , ℓ η = γ 0 y x + γ 1 ℏ + β 1 ( κ ℜ t + ℜ ) , G = G 2 + K 2 , Q = ( k 1 − k 3 ) ω + ( k − k 1 ) ℜ x , q = k ℜ x + k 1 ω , ℓ E = − β 2 ( κ ω t + ω ) − γ 2 ℏ x ,

where ω is the microtemperature vector, κ > 0 is the relaxation parameter, and ℓ y , ℓ ℏ , a 1 , a 2 , a 3 , β 1 , β 2 > 0 . The coefficients γ 1 , k , γ 0 denote, the coupling between the volume fraction and the temperature, the thermal conductivity, the coupling between temperature and displacement, respectively.

As coupling is considered, a 2 ≠ 0 , the coefficients k 1 , k 2 , k 3 , γ 2 > 0 satisfy the following inequalities:

(1.5) a = a 3 − a 2 2 a 1 > 0 ,

(1.6) k 1 2 < k k 3 .

The focus of this study is on thermal impacts, thus we assume that the heat capacity β 1 > 0 . Additionally, we do not assume the influence of microtemperature, and β 2 = k 1 = k 2 = k 3 = γ 2 = 0 , and by including the distributed delay term, we create a new problem that is distinct from earlier works. We establish the system’s well-posedness under the proper assumptions, and we use the energy approach to demonstrate the outcome of the exponential stability.

In this study, we take into account:

(1.7) G 1 = G 2 = P 3 = 0 , K 1 = − γ 0 ( κ ℜ t + ℜ ) , K 2 = γ 1 ( κ ℜ t + ℜ ) − ϑ 1 ℏ t − ∫ ν 1 ν 2 ϑ 2 ( s ) ℏ t ( x , t − s ) d s .

Now, by substituting (1.4) and (1.7) into (1.3), we arrive at the following problem:

ℓ y y t t − a 1 y x x − a 2 ℏ x x + γ 0 ( κ ℜ t + ℜ ) x = 0 , ℓ ℏ ℏ t t − a 3 ℏ x x − a 2 y x x − γ 1 ( κ ℜ t + ℜ ) + ϑ 1 ℏ t + ∫ ν 1 ν 2 ϑ 2 ( s ) ℏ t ( x , t − s ) d s = 0 , β ( κ ℜ t + ℜ ) t + γ 0 y t x + γ 1 ℏ t − k ℜ x x = 0 ,

where

( x , s , t ) ∈ H = ( 0 , 1 ) × ( ν 1 , ν 2 ) × ( 0 , ∞ ) ,

with initial and boundary conditions

(1.8) y ( x , 0 ) = y 0 ( x ) , y t ( x , 0 ) = y 1 ( x ) , ℜ ( x , 0 ) = ℜ 0 ( x ) , ℏ ( x , 0 ) = ℏ 0 ( x ) , ℏ t ( x , 0 ) = ℏ 1 ( x ) , ℜ t ( x , 0 ) = ℜ 1 ( x ) , x ∈ ( 0 , 1 ) , ℏ t ( x , − t ) = f 0 ( x , t ) , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ν 2 ) , y ( 0 , t ) = y ( 1 , t ) = ℏ ( 0 , t ) = ℏ ( 1 , t ) = ℜ ( 0 , t ) = ℜ ( 1 , t ) = 0 , t ≥ 0 .

First, as in [35], we introduce the new variable

Y ( x , ℓ , s , t ) = ℏ t ( x , t − s ℓ ) ,

then we obtain

s Y t ( x , ℓ , s , t ) + Y ℓ ( x , ℓ , s , t ) = 0 , Y ( x , 0 , s , t ) = ℏ t ( x , t ) .

We can also write our problem in the following form:

(1.9) ℓ y y t t − a 1 y x x − a 2 ℏ x x + γ 0 ( κ ℜ t + ℜ ) x = 0 , ℓ ℏ ℏ t t − a 3 ℏ x x − a 2 y x x − γ 1 ( κ ℜ t + ℜ ) + ϑ 1 ℏ t + ∫ ν 1 ν 2 ϑ 2 ( s ) Y ( x , 1 , s , t ) d s = 0 , β ( κ ℜ t + ℜ ) t + γ 0 y t x + γ 1 ℏ t − k ℜ x x = 0 , s Y t ( x , ℓ , s , t ) + Y ℓ ( x , ℓ , s , t ) = 0 ,

where

( x , ℓ , s , t ) ∈ ( 0 , 1 ) × H ,

initial conditions are

(1.10) y ( x , 0 ) = y 0 ( x ) , y t ( x , 0 ) = y 1 ( x ) , ℜ ( x , 0 ) = ℜ 0 ( x ) , ℏ ( x , 0 ) = ℏ 0 ( x ) , ℏ t ( x , 0 ) = ℏ 1 ( x ) , ℜ t ( x , 0 ) = ℜ 1 ( x ) , x ∈ ( 0 , 1 ) , Y ( x , ℓ , s , 0 ) = f 0 ( x , s ℓ ) , ( x , ℓ , s ) ∈ ( 0 , 1 ) × ( 0 , 1 ) × ( 0 , ν 2 ) ,

and boundary conditions are

(1.11) y ( 0 , t ) = y ( 1 , t ) = ℏ ( 0 , t ) = ℏ ( 1 , t ) = ℜ ( 0 , t ) = ℜ ( 1 , t ) = 0 , t ≥ 0 .

The integral denote the distributed delay terms with ν 1 , ν 2 > 0 are a time delay, ϑ 1 > 0 , and ϑ 2 is an L ∞ function satisfying:

(H1) ϑ 2 : [ ν 1 , ν 2 ] → R is a bounded function satisfying

(1.12) ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ d s < ϑ 1 .

Let the solution of system (1.9)–(1.11) be ( y , ℏ , ℜ , Y ) and the organization of this work is as follows: The well-posedness is demonstrated in Section 2, and the exponential stability is demonstrated in Section 3. We declare that c > 0 in each of the statements that follow.

2 Well-posedness

We demonstrated the well-posedness of the system (1.9)–(1.11) in this section.

Introducing the vector function first

U = ( y , y t , ℏ , ℏ t , ℜ , ℜ t , Y ) T ,

and variables v = y t , φ = ℏ t , χ = ℜ t , system (1.9) can also be written as:

(2.1) U t = J U U ( 0 ) = U 0 = ( y 0 , y 1 , ℏ 0 , ℏ 1 , ℜ 0 , ℜ 1 , f 0 ) T ,

where J : D ( J ) ⊂ Z : → Z is the linear operator stated as

(2.2) J U = v 1 ℓ y [ a 1 y x x + a 2 ℏ x x − γ 0 ( κ χ + ℜ ) x ] φ 1 ℓ ℏ a 3 ℏ x x + a 2 y x x + γ 1 ( κ χ + ℜ ) − ϑ 1 φ − ∫ ν 1 ν 2 ϑ 2 ( s ) Y ( x , 1 , s , t ) d s χ − 1 κ β [ γ 0 v x + γ 1 φ − k ℜ x x + β χ ] − 1 s Y ℓ ,

and energy space Z is stated as follows:

Z = K 0 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) × K 0 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) × K 0 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) × L 2 ( ( 0 , 1 ) × ( 0 , 1 ) × ( ν 1 , ν 2 ) ) .

For any

U = ( y , v , ℏ , φ , ℜ , χ , Y ) T ∈ Z , U ^ = ( y ^ , v ^ , ℏ ^ , φ ^ , ℜ ^ , χ ^ , Y ^ ) T ∈ Z ,

we equip Z with the inner product stated as follows:

(2.3) ⟨ U , U ^ ⟩ Z = ℓ y ∫ 0 1 v v ^ d x + a 1 ∫ 0 1 y x y ^ x d x + ℓ ℏ ∫ 0 1 φ φ ^ d x + a 3 ∫ 0 1 ℏ x ℏ ^ x d x + a 2 ∫ 0 1 ( y x ℏ ^ x + y ^ x ℏ x ) d x + β ∫ 0 1 ( κ χ + ℜ ) ( κ χ ^ + ℜ ^ ) d x + k κ ∫ 0 1 ℜ x ℜ ^ x d x + ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y Y ^ d s d ℓ d x .

The domain of J is given by

D ( J ) = U ∈ Z ∕ y , ℏ , ℜ ∈ K 2 ( 0 , 1 ) ∩ K 0 1 ( 0 , 1 ) , v , φ , χ ∈ K 0 1 ( 0 , 1 ) , Y , Y ℓ ∈ L 2 ( ( 0 , 1 ) × ( 0 , 1 ) × ( ν 1 , ν 2 ) ) , Y ( x , 0 , s , t ) = φ .

Therefore, D ( J ) is dense in Z .

Theorem 2.1

Assume that (1.5) and (1.12) hold. Let U 0 ∈ Z , then there exists a unique solution U ∈ C ( R + , Z ) of problem (2.1).

Furthermore, if U 0 ∈ D ( J ) , then U ∈ C ( R + , D ( J ) ) ∩ C 1 ( R + , Z ) .

Proof

We will show that the operator J is dissipative. Let for any U 0 ∈ D ( J ) and by utilizing (2.3), we have

(2.4) ⟨ J U , U ⟩ Z = − ϑ 1 ∫ 0 1 φ 2 d x − ∫ 0 1 ∫ ν 1 ν 2 ϑ 2 ( s ) φ Y ( x , 1 , s , t ) d s d x − ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y ℓ Y d s d ℓ d x − k ∫ 0 1 ℜ x 2 d x .

For the third term of the right-hand side (RHS) of (2.4), we have

(2.5) − ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y ℓ Y d s d ℓ d x = − 1 2 ∫ 0 1 ∫ ν 1 ν 2 ∫ 0 1 ∣ ϑ 2 ( s ) ∣ d d ℓ Y 2 d ℓ d s d x = − 1 2 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x + 1 2 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 0 , s , t ) d s d x .

By utilizing Young’s inequality, we obtain

(2.6) − ∫ 0 1 ∫ ν 1 ν 2 ϑ 2 ( s ) φ Y ( x , 1 , s , t ) d s d x ≤ 1 2 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ d s ∫ 0 1 φ 2 d x + 1 2 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x .

Substituting (2.5) and (2.6) into (2.4), utilizing Y ( x , 0 , s , t ) = φ ( x , t ) and (1.12), we find

(2.7) ⟨ J U , U ⟩ Z ≤ − η 0 ∫ 0 1 φ 2 d x − k ∫ 0 1 ℜ x 2 d x ≤ 0 ,

where η 0 = ( ϑ 1 − ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ d s ) > 0 and J is a dissipative operator.

We then establish that J is a maximal operator. It is sufficient to demonstrate that ( λ I − J ) is a surjective operator.

In fact, we demonstrate that for any F = ( f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 ) T ∈ Z , there exists a unique U = ( y , v , ℏ , φ , ℜ , χ , Y ) T ∈ D ( J ) such that

(2.8) ( λ I − J ) U = F .

That is

(2.9) λ y − v = f 1 ∈ K 0 1 ( 0 , 1 ) ℓ y λ v − a 1 y x x − a 2 ℏ x x + γ 0 ( κ χ + ℜ ) x = ℓ y f 2 ∈ L 2 ( 0 , 1 ) λ ℏ − φ = f 3 ∈ K 0 1 ( 0 , 1 ) ℓ ℏ λ φ − a 3 ℏ x x − a 2 y x x − γ 1 ( κ χ + ℜ ) + ϑ 1 φ + ∫ ν 1 ν 2 ϑ 2 ( s ) Y ( x , 1 , s , t ) d s = ℓ ℏ f 4 ∈ L 2 λ ℜ − χ = f 5 ∈ K 0 1 ( 0 , 1 ) β κ λ χ + γ 0 v x + γ 1 φ − k ℜ x x + β χ = β κ f 6 ∈ L 2 ( 0 , 1 ) s λ Y t ( x , ℓ , s , t ) + Y ℓ ( x , ℓ , s , t ) = s f 7 ∈ L 2 ( ( 0 , 1 ) × ( 0 , 1 ) × ( ν 1 , ν 2 ) ) .

We observe that equation (2.9)7 with Y ( x , 0 , s , t ) = φ ( x , t ) has a unique solution defined by

(2.10) Y ( x , ℓ , s , t ) = e − λ ℓ s φ + s e s ℓ λ ∫ 0 ℓ e λ s ϱ f 7 ( x , ϱ , s , t ) d ϱ ,

then

(2.11) Y ( x , 1 , s , t ) = e − λ s φ + s e λ s ∫ 0 1 e λ s ϱ f 7 ( x , ϱ , s , t ) d ϱ ,

and we have

(2.12) v = λ y − f 1 , φ = λ ℏ − f 3 , χ = λ ℜ − f 5 .

Inserting (2.11) and (2.12) in (2.9)2, (2.9)4, and (2.9)6, we obtain

(2.13) ℓ y λ 2 y − a 1 y x x − a 2 ℏ x x + γ 0 ( κ λ + 1 ) ℜ x = h 1 ϑ 3 ℏ − a 3 ℏ x x − a 2 y x x − γ 1 ( κ λ + 1 ) ℜ = h 2 ϑ 4 ℜ + γ 0 λ y x + γ 1 λ ℏ − k ℜ x x = h 3 ,

where

(2.14) h 1 = ℓ y ( λ f 1 + f 2 ) + γ 0 κ f 5 x , h 2 = ℓ ℏ f 4 + ( ℓ ℏ λ + ϑ 1 + ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ e − s λ d s ) f 3 − γ 1 κ f 5 , − ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ e s λ ∫ 0 1 e λ s ϱ f 7 ( x , ϱ , s , t ) d ϱ d s , h 3 = κ β f 6 + ( γ 0 f 1 x + γ 1 f 3 + β ( 1 + κ λ ) f 5 ) , ϑ 3 = ℓ ℏ λ 2 + ϑ 1 λ + λ ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ e − λ s d s , ϑ 4 = β λ ( κ λ + 1 ) .

We multiply (2.13) by y ^ , ℏ ^ , ℜ ^ , respectively, and integrating their sum over ( 0 , 1 ) , we obtain that

(2.15) B ( ( y , ℏ , ℜ ) , ( y ^ , ℏ ^ , ℜ ^ ) ) = Γ ( y ^ , ℏ ^ , ℜ ^ ) ,

where

B : ( K 0 1 ( 0 , 1 ) × K 0 1 ( 0 , 1 ) × K 0 1 ( 0 , 1 ) ) 2 → R

is the bilinear form given by

(2.16) B ( ( y , ℏ , ℜ ) , ( y ^ , ℏ ^ , ℜ ^ ) ) = ℓ y λ 2 ∫ 0 1 y y ^ d x + a 1 ∫ 0 1 y x y ^ x d x + a 2 ∫ 0 1 ℏ x y ^ x d x + γ 0 ( κ λ + 1 ) ∫ 0 1 ℜ x y ^ d x + ϑ 3 ∫ 0 1 ℏ ℏ ^ d x + a 3 ∫ 0 1 ℏ x ℏ ^ x d x + a 2 ∫ 0 1 y x ℏ ^ x d x − γ 1 ( κ λ + 1 ) ∫ 0 1 ℜ ℏ ^ d x + ϑ 4 ( κ λ + 1 ) λ ∫ 0 1 ℜ ℜ ^ d x + γ 0 ( κ λ + 1 ) ∫ 0 1 y x ℜ ^ d x + γ 1 ( κ λ + 1 ) ∫ 0 1 ℏ ℜ ^ d x + k ( κ λ + 1 ) λ ∫ 0 1 ℜ x ℜ ^ x d x

and

Γ : ( K 0 1 ( 0 , 1 ) × K 0 1 ( 0 , 1 ) × K 0 1 ( 0 , 1 ) ) → R

is the linear functional defined by

(2.17) Γ ( y ^ , ℏ ^ , ℜ ^ ) = ∫ 0 1 h 1 y ^ d x + ∫ 0 1 h 2 ℏ ^ d x + ( κ λ + 1 ) λ ∫ 0 1 h 3 ℜ ^ d x .

For V = K 0 1 ( 0 , 1 ) × K 0 1 ( 0 , 1 ) × K 0 1 ( 0 , 1 ) , with the norm

‖ ( y , ℏ , ℜ ) ‖ V 2 = ‖ y ‖ 2 2 + ‖ y x ‖ 2 2 + ‖ ℏ ‖ 2 2 + ‖ ℏ x ‖ 2 2 + ‖ ℜ ‖ 2 2 + ‖ ℜ x ‖ 2 2 ,

then, we have

(2.18) B ( ( y , ℏ , ℜ ) , ( y , ℏ , ℜ ) ) = ℓ y λ 2 ∫ 0 1 y 2 d x + a 1 ∫ 0 1 y x 2 d x + ϑ 3 ∫ 0 1 ℏ 2 d x + a 3 ∫ 0 1 ℏ x 2 d x + ϑ 4 ( κ λ + 1 ) λ ∫ 0 1 ℜ 2 d x + 2 a 2 ∫ 0 1 y x ℏ x d x + k ( κ λ + 1 ) λ ∫ 0 1 ℜ x 2 d x .

On the other hand, we can write

a 1 y x 2 + 2 a 2 y x ℏ x + a 3 ℏ x 2 = 1 2 a 1 y x + a 2 a 3 ℏ x 2 + a 3 ℏ x + a 2 a 1 y x 2 + y x 2 a 1 − a 2 2 a 3 + ℏ x 2 a 3 − a 2 2 a 1 .

Since (1.5), we deduce

(2.19) a 1 y x 2 + 2 a 2 y x ℏ x + a 3 ℏ x 2 > 1 2 y x 2 a 1 − a 2 2 a 3 + ℏ x 2 a 3 − a 2 2 a 1 ,

then, for some M 0 > 0

(2.20) B ( ( y , ℏ , ℜ ) , ( y , ℏ , ℜ ) ) ≥ M 0 ‖ ( y , ℏ , ℜ ) ‖ V 2 .

B is therefore coercive. Therefore, we deduce from the Lax-Milgram theorem that (2.15) has a unique solution.

(2.21) y , ℏ , ℜ ∈ K 0 1 ( 0 , 1 ) .

Substituting y , ℏ , and ℜ into (2.9)1,3,5, we obtain

v , φ , χ ∈ K 0 1 ( 0 , 1 ) .

Also, the compensation of φ in (2.10) with (2.9)7, yields

Y , Y ℓ ∈ L 2 ( ( 0 , 1 ) × ( 0 , 1 ) × ( ν 1 , ν 2 ) ) .

In addition, if we assume y ^ = ℜ ^ = 0 in (2.16), we obtain

a 3 ∫ 0 1 ℏ x ℏ ^ x d x + ϑ 3 ∫ 0 1 ℏ ℏ ^ d x + a 2 ∫ 0 1 y x ℏ ^ x d x − γ 1 ( κ λ + 1 ) ∫ 0 1 ℜ ℏ ^ d x = ∫ 0 1 h 2 ℏ ^ d x , ∀ ℏ ^ ∈ K 0 1 ( 0 , 1 ) ,

which implies

a 3 ∫ 0 1 ℏ x ℏ ^ x d x = ∫ 0 1 ( h 2 − ϑ 3 ℏ + a 2 y x x + γ 1 ( κ λ + 1 ) ℜ ) ℏ ^ d x , ∀ ℏ ^ ∈ K 0 1 ( 0 , 1 ) ,

that is

(2.22) a 3 ℏ x x + a 2 y x x = ϑ 3 ℏ − γ 1 ( κ λ + 1 ) ℜ − h 2 ∈ L 2 ( 0 , 1 ) .

Similarly, if we take ℏ ^ = ℜ ^ = 0 in (2.16), we obtain

(2.23) a 2 ℏ x x + a 1 y x x = ℓ y λ 2 y + γ 0 ( κ λ + 1 ) ℜ x − h 1 ∈ L 2 ( 0 , 1 ) .

Combining (2.22) and (2.23) and by using (1.5) yields that

(2.24) y , ℏ ∈ K 2 ( 0 , 1 ) ∩ K 0 1 ( 0 , 1 ) .

In the same way, if we let y ^ = ℏ ^ = 0 in (2.16), we obtain

ϑ 4 ( κ λ + 1 ) λ ∫ 0 1 ℜ ℜ ^ d x + k ( κ λ + 1 ) λ ∫ 0 1 ℜ x ℜ ^ x d x + γ 0 ( κ λ + 1 ) ∫ 0 1 y x ℜ ^ d x + γ 1 ( κ λ + 1 ) ∫ 0 1 ℏ ℜ ^ d x = ( κ λ + 1 ) λ ∫ 0 1 h 3 ℜ ^ d x , ∀ ℜ ^ ∈ K 0 1 ( 0 , 1 ) ,

which implies

k ℜ x x = ϑ 4 ℜ + γ 0 λ y x + γ 1 λ ℏ − h 3 ∈ L 2 ( 0 , 1 ) .

Consequently,

ℜ ∈ K 2 ( 0 , 1 ) ∩ K 0 1 ( 0 , 1 ) .

Finally, the existence of a unique U ∈ D ( J ) such that (2.8) holds is guaranteed by the use of regularity theory for the linear elliptic equations.

As a result, we draw the conclusion that J is a maximal dissipative operator. As a result, we have the well-posedness result according to the Lumer-Philips theorem (see [36]).□

3 Exponential decay

We will demonstrate the system (1.9)–(1.11) stability in this section. For the required result we will discuss the following lemmas.

Lemma 3.1

The energy functional E, stated by

(3.1) E ( t ) = 1 2 ∫ 0 1 [ ℓ y y t 2 + a 1 y x 2 + ℓ ℏ ℏ t 2 + a 3 ℏ x 2 + 2 a 2 y x ℏ x + β ( κ ℜ t + ℜ ) 2 + k κ ℜ x 2 ] d x + 1 2 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x

satisfies

(3.2) E ′ ( t ) ≤ − k ∫ 0 1 ℜ x 2 d x − η 0 ∫ 0 1 ℏ t 2 d x ≤ 0 ,

where η 0 = ϑ 1 − ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ d s > 0 .

Proof

Multiplying the equations (1.9)1,2,3 by y t , ℏ t , and ( κ ℜ t + ℜ ) , integrating by parts over ( 0 , 1 ) , and utilizing (1.11), we obtain

(3.3) 1 2 d d t ∫ 0 1 [ ℓ y y t 2 + a 1 y x 2 + ℓ ℏ ℏ t 2 + a 3 ℏ x 2 + 2 a 2 y x ℏ x + β ( κ ℜ t + ℜ ) 2 + k κ ℜ x 2 ] d x + ϑ 1 ∫ 0 1 ℏ t 2 d x + ∫ 0 1 ℏ t ∫ ν 1 ν 2 ϑ 2 ( s ) Y ( x , 1 , s , t ) d s d x + k ∫ 0 1 ℜ x 2 d x = 0 .

Multiplying the equation (1.9)4 by Y ∣ ϑ 2 ( s ) ∣ , and integrating the result over ( 0 , 1 ) × ( 0 , 1 ) × ( ν 1 , ν 2 )

(3.4) d d t 1 2 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x = − ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y Y ℓ ( x , ℓ , s , t ) d s d ℓ d x = − 1 2 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ d d ℓ Y 2 ( x , ℓ , s , t ) d s d ℓ d x = 1 2 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ ( Y 2 ( x , 0 , s , t ) − Y 2 ( x , 1 , s , t ) ) d s d x = 1 2 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ d s ∫ 0 1 ℏ t 2 d x − 1 2 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x .

Now, by putting (3.4) in (3.3), and utilizing Young’s inequality, we obtain

E ′ ( t ) ≤ − k ∫ 0 1 ℜ x 2 d x − ϑ 1 − ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ d s ∫ 0 1 ℏ t 2 d x ,

then, by (1.12), ∃ η 0 > 0 so that

(3.5) E ′ ( t ) ≤ − k ∫ 0 1 ℜ x 2 d x − η 0 ∫ 0 1 ℏ t 2 d x ,

then we obtain (3.2) ( E is a non-increasing function).□

Remark 3.2

Using (1.5) and (2.19), we conclude that E ( t ) satisfies

E ( t ) > 1 2 ∫ 0 1 [ ℓ y y t 2 + a 4 y x 2 + ℓ ℏ ℏ t 2 + a 5 ℏ x 2 + β ( κ ℜ t + ℜ ) 2 + k κ ℜ x 2 ] d x + 1 2 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x ,

where

(3.6) a 4 = 1 2 a 1 − a 2 2 a 3 > 0 , a 5 = 1 2 a 3 − a 2 2 a 1 > 0 .

Then, the function E ( t ) is non-negative.

Lemma 3.3

The functional

(3.7) D 1 ( t ) ≔ ℓ ℏ ∫ 0 1 ℏ t ℏ d x − a 2 a 1 ℓ y ∫ 0 1 ℏ y t d x + ϑ 1 2 ∫ 0 1 ℏ 2 d x

satisfies, for any ε 1 > 0

(3.8) D 1 ′ ( t ) ≤ − a 2 ∫ 0 1 ℏ x 2 d x + ε 1 ∫ 0 1 y t 2 d x + c 1 + 1 ε 1 ∫ 0 1 ℏ t 2 d x + c ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x + c ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x .

Proof

Direct computation utilizing Young’s inequality and integration by parts produce

(3.9) D 1 ′ ( t ) = − a 3 ∫ 0 1 ℏ x 2 d x + ℓ ℏ ∫ 0 1 ℏ t 2 d x + a 2 2 a 1 ∫ 0 1 ℏ x 2 d x − a 2 a 1 ℓ y ∫ 0 1 ℏ t y t d x + γ 1 ∫ 0 1 ( κ ℜ t + ℜ ) ℏ d x + a 2 γ 0 a 1 ∫ 0 1 ( κ ℜ t + ℜ ) x ℏ d x − ∫ 0 1 ℏ ∫ ν 1 ν 2 ϑ 2 ( s ) Y ( x , 1 , s , t ) d s d x = − a 3 − a 2 2 a 1 ∫ 0 1 ℏ x 2 d x + ℓ ℏ ∫ 0 1 ℏ t 2 d x − a 2 a 1 ℓ y ∫ 0 1 ℏ t y t d x + γ 1 ∫ 0 1 ( κ ℜ t + ℜ ) ℏ d x − a 2 γ 0 a 1 ∫ 0 1 ( κ ℜ t + ℜ ) ℏ x d x − ∫ 0 1 ℏ ∫ ν 1 ν 2 ϑ 2 ( s ) Y ( x , 1 , s , t ) d s d x ,

by using Poincare’s and Young’s inequalities, for δ 1 , δ 2 , δ 3 , ε 1 > 0 , we obtain

(3.10) D 1 ′ ( t ) ≤ − a 3 − a 2 2 a 1 − ( ϑ 1 c δ 1 + c δ 2 + δ 3 ) ∫ 0 1 ℏ x 2 d x + ε 1 ∫ 0 1 y t 2 d x + c 1 + 1 ε 1 ∫ 0 1 ℏ t 2 d x + c δ 2 + c δ 3 ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x + c δ 1 ∫ 0 t ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x .

Bearing in mind (1.5), and letting δ 1 = a 6 ϑ 1 c , δ 2 = a 6 c , δ 3 = a 6 , we obtain the estimate (3.8).□

Lemma 3.4

The functional

D 2 ( t ) ≔ a 2 ∫ 0 1 ℏ t y d x − ∫ 0 1 ℏ y t d x

satisfies

(3.11) D 2 ′ ( t ) ≤ − a 2 2 2 ℓ ℏ ∫ 0 1 y x 2 d x + c ∫ 0 1 ℏ x 2 d x + c ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x + c ∫ 0 1 ℏ t 2 d x + c ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x .

Proof

By differentiating D 2 , then utilizing (1.9), integration by parts, and using (1.11) we obtain

(3.12) D 2 ′ ( t ) = − a 2 2 ℓ ℏ ∫ 0 1 y x 2 d x + a 2 2 ℓ y ∫ 0 1 ℏ x 2 d x − a 2 a 3 ℓ ℏ − a 1 a 2 ℓ y ∫ 0 1 ℏ x y x d x − a 2 γ 0 ℓ y ∫ 0 1 ( κ ℜ t + ℜ ) ℏ x d x + a 2 γ 1 ℓ ℏ ∫ 0 1 ( κ ℜ t + ℜ ) y d x − a 2 ϑ 1 ℓ ℏ ∫ 0 1 y ℏ t d x − a 2 ℓ ℏ ∫ 0 1 y ∫ ν 1 ν 2 ϑ 2 ( s ) Y ( x , 1 , s , t ) d s d x .

Now, utilizing Poincare’s and Young’s inequality, we evaluate the final six terms in the RHS of (3.12). For δ 4 , δ 5 > 0 , we have

− a 2 a 3 ℓ ℏ − a 1 a 2 ℓ y ∫ 0 1 ℏ x y x d x ≤ δ 4 ∫ 0 1 y x 2 d x + c δ 4 ∫ 0 1 ℏ 2 d x , − a 2 ϑ 1 ℓ ℏ ∫ 0 1 y ℏ t d x ≤ c δ 5 ∫ 0 1 y x 2 d x + c δ 5 ∫ 0 1 ℏ t 2 d x , a 2 γ 1 ℓ ℏ ∫ 0 1 ( κ ℜ t + ℜ ) y d x ≤ c δ 5 ∫ 0 1 y x 2 d x + c δ 5 ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x

and

− a 2 ℓ ℏ ∫ 0 1 y ∫ ν 1 ν 2 ϑ 2 ( s ) Y ( x , 1 , s , t ) d s d x ≤ c δ 5 ∫ 0 1 y x 2 d x + c δ 5 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x .

By letting δ 4 = a 2 4 ℓ ℏ , δ 5 = a 2 12 c ℓ ℏ , and substituting in (3.12), we obtain (3.11).□

Lemma 3.5

The functional

D 3 ( t ) ≔ − ℓ y ∫ 0 1 y t y d x

satisfies

(3.13) D 3 ′ ( t ) ≤ − ℓ y ∫ 0 1 y t 2 d x + 3 a 1 ∫ 0 1 y x 2 d x + a 3 4 ∫ 0 1 ℏ x 2 d x + γ 0 2 4 a 1 ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x .

Proof

Direct computations give

D 3 ′ ( t ) = − ℓ y ∫ 0 1 y t 2 d x + a 1 ∫ 0 1 y x 2 d x + a 2 ∫ 0 1 y x ℏ x d x − γ 0 ∫ 0 1 y x ( κ ℜ t + ℜ ) d x .

Estimate (3.13) easily follows by utilizing Young’s inequality.□

Lemma 3.6

The functional

D 4 ( t ) ≔ − β κ 2 ∫ 0 1 ℜ t ℜ d x − β κ 2 ∫ 0 1 ℜ 2 d x

satisfies, for any ε 2 > 0

(3.14) D 4 ′ ( t ) ≤ − β 2 ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x + ε 2 ∫ 0 1 y t 2 d x + c ∫ 0 1 ℏ t 2 d x + c 1 + 1 ε 2 ∫ 0 1 ℜ x 2 d x .

Proof

Direct computations give

D 4 ′ ( t ) = − β κ ∫ 0 1 ( κ ℜ t + ℜ ) t ℜ d x − β κ 2 ∫ 0 1 ℜ t 2 d x = γ 0 κ ∫ 0 1 ℜ y t x d x + γ 1 κ ∫ 0 1 ℏ t ℜ d x + k κ ∫ 0 1 ℜ x 2 d x − β κ 2 ∫ 0 1 ℜ t 2 d x .

Estimate (3.14) easily followed by utilizing Young’s inequality for ε 2 > 0 and the fact that

(3.15)□ − ∫ 0 1 ( κ ℜ t ) 2 d x ≤ − 1 2 ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x + ∫ 0 1 ℜ 2 d x .

Now introducing the functional given by

Lemma 3.7

The functional

D 5 ( t ) ≔ ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s e − s ℓ ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x ,

satisfies

(3.16) D 5 ′ ( t ) ≤ − η 1 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x + ϑ 1 ∫ 0 1 ℏ t 2 d x − η 1 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x ,

where η 1 > 0 .

Proof

By differentiating D 5 , with respect to t and utilizing the last equation in (1.9), we obtain

D 5 ′ ( t ) = − 2 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 e − s ℓ ∣ ϑ 2 ( s ) ∣ Y Y ℓ ( x , ℓ , s , t ) d s d ℓ d x = − ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s e − s ℓ ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x − ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ [ e − s Y 2 ( x , 1 , s , t ) − Y 2 ( x , 0 , s , t ) ] d s d x .

Utilizing that Y ( x , 0 , s , t ) = ℏ t ( x , t ) , and e − s ≤ e − s ℓ ≤ 1 , ∀ 0 < ℓ < 1 , we obtain

□ D 5 ′ ( t ) ≤ − ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s e − s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x − ∫ 0 1 ∫ ν 1 ν 2 e − s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x + ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ d s ∫ 0 1 ℏ t 2 d x .

We have − e − s ≤ − e − ν 2 , ∀ s ∈ [ ν 1 , ν 2 ] since − e − s is an increasing function. Setting η 1 = e − ν 2 and remembering (1.12), we find (3.16).

We are now prepared to demonstrate the major finding.

Theorem 3.8

Consider that (1.5) and (1.12) holds. Then, there exist ζ 1 , ζ 2 > 0 such that the energy functional given by (3.1) holds the following:

(3.17) E ( t ) ≤ ζ 1 e − ζ 2 t , ∀ t ≥ 0 .

Proof

We define the functional of Lyapunov

(3.18) ℒ ( t ) ≔ N E ( t ) + N 1 D 1 ( t ) + N 2 D 2 ( t ) + D 3 ( t ) + N 4 D 4 ( t ) + N 5 D 5 ( t ) ,

where N , N 1 , N 2 , N 4 , N 5 > 0 we will assign them later.

By differentiating (3.18) and utilizing (3.2), (3.8), (3.11), (3.13), (3.14), and (3.16), we have

ℒ ′ ( t ) ≤ − a N 1 2 − c N 2 − a 3 4 ∫ 0 1 ℏ x 2 d x − [ ℓ y − ε 1 N 1 − ε 2 N 4 ] ∫ 0 1 y t 2 d x − a 2 2 N 2 2 ℓ ℏ − 3 a 1 ∫ 0 1 y x 2 d x − k N − c 1 + 1 ε 2 N 4 ∫ 0 1 ℜ x 2 d x − η 0 N − c N 1 1 + 1 ε 1 − c N 2 − c N 4 − ϑ 1 N 5 ∫ 0 1 ℏ t 2 d x − β 2 N 4 − c N 1 − c N 2 − γ 0 4 a 1 ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x − [ N 5 η 1 − c N 1 − c N 2 ] ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x − N 5 η 1 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x .

By setting

ε 1 = ℓ y 4 N 1 , ε 2 = ℓ y 4 N 4 ,

we obtain

ℒ ′ ( t ) ≤ − a N 1 2 − c N 2 − a 3 4 ∫ 0 1 ℏ x 2 d x − ℓ y 2 ∫ 0 1 y t 2 d x − a 2 2 N 2 2 ℓ ℏ − 3 a 1 ∫ 0 1 y x 2 d x − [ k N − c N 4 ( 1 + N 4 ) ] ∫ 0 1 ℜ x 2 d x − [ η 0 N − c N 1 ( 1 + N 1 ) − N 2 c − N 4 c − ϑ 1 N 5 ] ∫ 0 1 ℏ t 2 d x − β 2 N 4 − c N 1 − c N 2 − γ 0 2 4 a 1 ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x − [ N 5 η 1 − c N 1 − c N 2 ] ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x − N 5 η 1 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x .

Now, choosing our constants.

We select N 2 large enough in such a way that

α 1 = a 2 2 N 2 2 ℓ ℏ − 3 a 1 > 0 ,

then we pick N 1 large enough in such a way that

α 2 = a N 1 2 − c N 2 − a 3 4 > 0 ,

then we pick N 4 and N 5 large enough in such a way that

α 3 = β 2 N 4 − c N 1 − c N 2 − γ 0 2 4 a 1 > 0 α 4 = N 5 η 1 − c N 1 − c N 2 > 0 .

Thus, we obtain

(3.19) ℒ ′ ( t ) ≤ − α 2 ∫ 0 1 ℏ x 2 d x − ℓ y 2 ∫ 0 1 y t 2 d x − α 1 ∫ 0 1 y x 2 d x − [ η 0 N − c ] ∫ 0 1 ℏ t 2 d x − [ k N − c ] ∫ 0 1 ℜ x 2 d x − α 3 ∫ 0 1 ( κ ℜ t + ℜ ) 2 d x − α 4 ∫ 0 1 ∫ ν 1 ν 2 ∣ ϑ 2 ( s ) ∣ Y 2 ( x , 1 , s , t ) d s d x − α 5 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x ,

where α 5 = η 1 N 5 > 0 .

If we take

L ( t ) = N 1 D 1 ( t ) + N 2 D 2 ( t ) + D 3 ( t ) + N 4 D 4 ( t ) + N 5 D 5 ( t ) ,

then

∣ L ( t ) ∣ ≤ N 1 ℓ ℏ ∫ 0 1 ∣ ℏ ℏ t ∣ d x + N 1 a 2 a 1 ℓ y ∫ 0 1 ∣ ℏ y t ∣ d x + N 1 ϑ 1 2 ∫ 0 1 ℏ 2 d x + N 2 a 2 ∫ 0 1 ∣ ℏ y t − y ℏ t ∣ d x + ℓ y ∫ 0 1 ∣ y t y ∣ d x + N 4 β κ 2 ∫ 0 1 ∣ ℜ t ℜ ∣ d x + N 4 β κ 2 ∫ 0 1 ℜ 2 d x + N 5 ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s e − s ℓ ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ d x .

According to Poincaré, Young’s, and Cauchy-Schwartz inequalities, we find

∣ L ( t ) ∣ ≤ c ∫ 0 1 ( y t 2 + ℏ t 2 + ℏ x 2 + y x 2 + ( κ ℜ t + ℜ ) 2 + ℜ x 2 ) d x + c ∫ 0 1 ∫ 0 1 ∫ ν 1 ν 2 s ∣ ϑ 2 ( s ) ∣ Y 2 ( x , ℓ , s , t ) d s d ℓ .

On the other hand, by (2.19) we have

a 1 y x 2 + 2 a 2 ℏ x y x + a 3 ℏ x 2 > 1 2 a 1 − a 2 2 a 3 y x 2 + a 3 − a 2 2 a 1 ℏ x 2 .

Hence, we obtain

∣ L ( t ) ∣ = ∣ ℒ ( t ) − N E ( t ) ∣ ≤ c E ( t ) ,

that is

(3.20) ( N − c ) E ( t ) ≤ ℒ ( t ) ≤ ( N + c ) E ( t ) .

At this point, we pick N large enough in such a way that

N − c > 0 , N η 0 − c > 0 , N k − c > 0 > 0

and exploiting (3.1), the estimates (3.19) and (3.20), respectively, give

(3.21) c 1 E ( t ) ≤ ℒ ( t ) ≤ c 2 E ( t ) , ∀ t ≥ 0 ,

and

(3.22) ℒ ′ ( t ) ≤ − d 1 E ( t ) , ∀ t ≥ 0 ,

for some d 1 , c 1 , c 2 > 0 .

Consequently, for some ζ 2 > 0 , we find

(3.23) ℒ ′ ( t ) ≤ − ζ 2 ℒ ( t ) , ∀ t ≥ 0 .

Integration of (3.23) over ( 0 , t ) yields

(3.24) ℒ ( t ) ≤ ℒ ( 0 ) e − ζ 2 t , ∀ t ≥ 0 .

Therefore, (3.17) is achieved by virtue of (3.21) and (3.24).□

4 Conclusion

This work studies a swelling-porous elastic system coupled with thermoelasticity of the Lord-Shulman type and distributed delay which is more general than classical thermoelasticity. Furthermore, the problem circumvented the absurd situation of the infinite propagation of the effect of a thermal or mechanical disturbance in the medium. We established the well-posedness of our problem using the semigroup method. Additionally, we used the energy method to prove the stability result for the system. It is intriguing to know that the result was obtained independently of the wave velocities of the system or any form of interactions between coefficients of the system other than hypotheses (1.5) and (1.12), which guarantees the positivity of the energy of the system. The present result contributes significantly to the existing literature on swelling porous elastic problems. In the future studies to so what happens if the system contains some dampings and sources terms.

Acknowledgment

The authors would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.

  1. Funding information: The authors would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.

  2. Author contributions: The authors contributed equally in this work.

  3. Conflict of interest: The authors state no conflicts of interest.

  4. Data availability statement: No data were used to support the study.

References

[1] A. C. Eringen, A continuum theory of swelling porous elastic soils, Internat. J. Engrg. Sci. 32 (1994), no. 8, 1337–1349. 10.1016/0020-7225(94)90042-6Search in Google Scholar

[2] A. Bedford and D. S. Drumheller, Theories of immiscible and structured mixtures, Internat. J. Engrg. Sci. 21 (1983), no. 8, 863–960. 10.1016/0020-7225(83)90071-XSearch in Google Scholar

[3] J. E. Bowels, Foundation Design and Analysis, McGraw Hill Inc, New York, 1988. Search in Google Scholar

[4] R. L. Handy, A stress path model for collapsible Loess, in: E. Derbyshire, T. Dijkstra, I. J. Smalley (Eds.), Genesis and Properties of Collapsible Soils, NATO ASI Series, Vol. 468, Springer, Dordrecht, 1995, pp. 33–47. 10.1007/978-94-011-0097-7_3Search in Google Scholar

[5] V. Q. Hung, Hidden Disaster, University News, University of Saska Techwan, Saskatoon, 2003. Search in Google Scholar

[6] D. Iesan, On the theory of mixtures of thermoelastic solids, J. Therm. Stresses 14 (1991), no. 4, 389–408. 10.1080/01495739108927075Search in Google Scholar

[7] L. D. Jones and I. Jefferson, Expansive Soils, ICE Publishing, London, 2012, pp. 413–441. Search in Google Scholar

[8] T. K. Karalis, On the elastic deformation of non-saturated swelling soils, Acta Mech. 84 (1990), no. 1–4, 19–45. 10.1007/BF01176086Search in Google Scholar

[9] B. Kalantari, Engineering significant of swelling soils, Res. J. Appl. Sci. Eng. Technol. 4 (2012), no. 17, 2874–2878. Search in Google Scholar

[10] R. L. Leonard, Expansive Soils Shallow Foundation, Regent Centre, University of Kansas, Kansas, 1989. Search in Google Scholar

[11] R. Quintanilla, Exponential stability for one-dimensional problem of swelling porous elastic soils with fluid saturation, J. Comput. Appl. Math. 145 (2002), no. 2, 525–533. 10.1016/S0377-0427(02)00442-9Search in Google Scholar

[12] J. M. Wang and B. Z. Guo, On the stability of swelling porous elastic soils with fluid saturation by one internal damping, IMA J. Appl. Math. 71 (2006), no. 4, 565–582. 10.1093/imamat/hxl009Search in Google Scholar

[13] T. A. Apalara, General stability result of swelling porous elastic soils with a viscoelastic damping, Z. Angew. Math. Phys. 71 (2020), 200. 10.1007/s00033-020-01427-0Search in Google Scholar

[14] F. Bofill and R. Quintanilla, Anti-plane shear deformations of swelling porous elastic soils, Int. J. Eng. Sci. 41 (2003), no. 8, 801–816. 10.1016/S0020-7225(02)00281-1Search in Google Scholar

[15] T. A. Apalara, M. O. Yusuf, S. E. Mukiawa, and O. B. Almutairi, Exponential stabilization of swelling porous systems with thermoelastic damping, J. King Saud Univ. Sci. 35 (2023), no. 1, 102460. 10.1016/j.jksus.2022.102460Search in Google Scholar

[16] M. A. Murad and J. H. Cushman, Thermomechanical theories for swelling porous media with microstructure, Internat. J. Engrg. Sci. 38 (2000), no. 5, 517–564. 10.1016/S0020-7225(99)00054-3Search in Google Scholar

[17] R. Quintanilla, Existence and exponential decay in the linear theory of viscoelastic mixtures, Eur. J. Mech. A Solids 24 (2005), no. 2, 311–324. 10.1016/j.euromechsol.2004.11.008Search in Google Scholar

[18] R. Quintanilla, Exponential stability of solutions of swelling porous elastic soils, Meccanica 39 (2004), no. 2, 139–145. 10.1023/B:MECC.0000005105.45175.61Search in Google Scholar

[19] R. Quintanilla, On the linear problem of swelling porous elastic soils with incompressible fluid, Internat. J. Engrg. Sci. 40 (2002), no. 13, 1485–1494. 10.1016/S0020-7225(02)00021-6Search in Google Scholar

[20] A. Arbi, J. Cao, M. Es-saiydy, M. Zarhouni, and M. Zitane, Dynamics of delayed cellular neural networks in the Stepanov pseudo almost automorphic space, Discrete Contin. Dyn. Syst. Ser. S 15 (2022), no. 11, 3097–3109. 10.3934/dcdss.2022136Search in Google Scholar

[21] A. Choucha, D. Ouchenane, and Kh. Zennir, General decay of solutions in one-dimensional porous-elastic with memory and distributed delay term, Tamkang J. Math. 52 (2021), no. 4, 479–495, DOI: https://doi.org/10.5556/j.tkjm.52.2021.3519. 10.5556/j.tkjm.52.2021.3519Search in Google Scholar

[22] A. Choucha, S. M. Boulaaras, D. Ouchenane, B. B. Cherif, and M. Abdalla, Exponential stability of swelling porous elastic with a viscoelastic damping and distributed delay term, J. Funct. Spaces 2021 (2021), 5581634, DOI: https://doi.org/10.1155/2021/5581634. 10.1155/2021/5581634Search in Google Scholar

[23] A. Choucha, D. Ouchenane, Kh. Zennir, and B. Feng, Global well-posedness and exponential stability results of a class of Bresse-Timoshenko-type systems with distributed delay term, Math. Methods Appl. Sci. (2020), DOI: https://doi.org/10.1002/mma.6437. 10.1002/mma.6437Search in Google Scholar

[24] A. Choucha, D. Ouchenane, and S. Boulaaras, Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term, Math. Methods Appl. Sci. 43 (2020), no. 17, 9983–10004, DOI: https://doi.org/10.1002/mma.6673. 10.1002/mma.6673Search in Google Scholar

[25] A. Choucha, D. Ouchenane, S. M. Boulaaras, B. B. Cherif, and M. Abdalla, Well-posedness and stability result of the nonlinear thermodiffusion full von Kármán beam with thermal effect and time-varying delay, J. Funct. Spaces 2021 (2021), 9974034, DOI: https://doi.org/10.1155/2021/9974034. 10.1155/2021/9974034Search in Google Scholar

[26] M. J. Dos Santos, B. Feng, D. S. Almeida Jr., and M. L. Santos, Global and exponential attractors for a nonlinear porous elastic system with delay term, Discrete Contin. Dyn. Syst. Ser. B 26 (2021), no. 5, 2805–2828, DOI: https://doi.org/10.3934/dcdsb.2020206. 10.3934/dcdsb.2020206Search in Google Scholar

[27] N. Bazarra, J. R. Fernández, and R. Quintanilla, Lord-Shulman thermoelasticity with microtemperatures, Appl. Math. Optim. 84 (2021), 1667–1685, DOI: https://doi.org/10.1007/s00245-020-09691-2. 10.1007/s00245-020-09691-2Search in Google Scholar

[28] H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967), 299–309. 10.1016/0022-5096(67)90024-5Search in Google Scholar

[29] A. Choucha, S. M. Boulaaras, D. Ouchenane, B. B. Cherif, M. Hidan, and M. Abdalla, Exponential stabilization of a swelling porous-elastic system with microtemperature effect and distributed delay, J. Funct. Spaces 2021 (2021), 5513981, DOI: https://doi.org/10.1155/2021/5513981. 10.1155/2021/5513981Search in Google Scholar

[30] H. Dridi and A. Djebabla, On the stabilization of linear porous elastic materials by microtemperature effect and porous damping, Ann. dell’Universita di Ferrara 66 (2020), 13–25, DOI: https://doi.org/10.1007/s11565-019-00333-2. 10.1007/s11565-019-00333-2Search in Google Scholar

[31] D. Iesan, Thermoelasticity of bodies with microstructure and microtemperatures, Int. J. Solids Struct. 44 (2007), no. 25–26, 8648–8662. 10.1016/j.ijsolstr.2007.06.027Search in Google Scholar

[32] D. Iesan, On a theory of micromorphic elastic solids with microtemperatures, J. Therm. Stresses 24 (2001), no. 8, 737–752. 10.1080/014957301300324882Search in Google Scholar

[33] D. Iesan and R. Quintanilla, On a theory of thermoelasticity with microtemperature, J. Therm. Stresses 23 (2000), no. 3, 199–215. 10.1080/014957300280407Search in Google Scholar

[34] B. Feng, L. Yan, and D. S. Almeida Jr., Stabilization for an inhomogeneous porous-elastic system with temperature and microtemperature, Z. Angew. Math. Mech. 101 (2021), no. 6, e202000058, DOI: https://doi.org/10.1002/zamm.202000058. 10.1002/zamm.202000058Search in Google Scholar

[35] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations 21 (2008), no. 9–10, 935–958. 10.57262/die/1356038593Search in Google Scholar

[36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. 10.1007/978-1-4612-5561-1Search in Google Scholar
Received: 2023-09-13
Revised: 2023-11-17
Accepted: 2023-11-27
Published Online: 2023-12-08

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2023-0165
Keywords for this article
Lord-Shulman; mathematical operators; swelling porous system; partial differential equations; general decay; distributed delay term
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 12.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2023-0165/html
Scroll to top button