jamp Journal of Applied Mathematics and Physics 2327-4379 2327-4352 Scientific Research Publishing 10.4236/jamp.2026.146103 jamp-151827 Article Physics Mathematics The Stability of a Class of Non-Newtonian Micropolar Fluid Equations with Unbounded Delays Liu Wanjia 1 Yi Luyan 1 School of Mathematical Sciences, Chongqing Normal University, Chongqing, China

The authors declare no conflicts of interest regarding the publication of this paper.

 11 06 2026 06 2026 14 06 2109 2127 05 05 2026 08 06 2026 11 06 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/jamp.2026.146103

We address the stability of stationary solutions to a class of 2D non-newtonian micropolar fluid equations, when the external force contains hereditary characteristics involving unbounded delays. Specifically, when the delay function is continuous with respect to time, the stability of the weak solution with respect to the non-trivial stationary solution is established by the definition of Lyapunov stability, and the asymptotic stability of the weak solution with respect to the trivial stationary solution is established by the method of constructing Lyapunov functional; when the delay function is only continuous with respect to time, the stability of the strong solution with respect to the non-trivial stationary solution is established by using Razumikhin technique; finally, the proportional time delay is introduced to establish the polynomial stability of the weak solution with respect to the trivial stationary solution.

 Non-Newtonian Micropolar Fluid Unbounded Delays Weak (Strong) Solutions Stationary Solution Asymptotic (Polynomial) Stability 1. Introduction

In fluid dynamics, fluids satisfying Newton’s law of viscosity, namely those for which the stress tensor depends linearly on the rate-of-strain tensor, are generally referred to as Newtonian fluids. Typical examples include water, alcohol, and air. In contrast, fluids that do not obey this linear constitutive relation are called non-Newtonian fluids, such as molten plastics, synthetic fibers, paints, and greases. Micropolar fluids form a class of complex fluids that account for microstructural effects and non-symmetric stress tensors; examples include blood, polymer suspensions, and related materials. Non-Newtonian micropolar fluids combine the constitutive characteristics of non-Newtonian fluids with the microrotation effects of micropolar fluids. Therefore, the non-Newtonian micropolar fluid equations can be used to describe the motion of non-Newtonian fluids with microstructures, namely fluids composed of randomly oriented particles suspended in a viscous non-Newtonian medium, without considering the deformation of the particles, such as lubricants and other complex fluids [1].

At present, numerous results have been obtained on various mathematical aspects of the non-Newtonian micropolar fluid equations. For instance, Araújo[2] established the existence and uniqueness of weak solutions in a two-dimensional bounded domain and investigated the existence and upper semicontinuity of pullback attractors. Later, Ai and Tan [3] proved the existence of global and exponential attractors for the equations in a two-dimensional bounded domain and studied the existence of pullback exponential attractors. In another direction, Zhao, Zhang et al. [1] established the existence and degenerate regularity of statistical solutions for the equations in a two-dimensional bounded domain. Subsequently, Chen, Yang et al. [4] proved the existence and degenerate regularity of trajectory statistical solutions for the equations in a three-dimensional bounded domain.

Delay effects reveal the delayed reaction phenomenon of a system and have an important impact on the stability and evolution process of the system. In the real world, delay effects appear naturally, for example, in the motion of fluids in wind tunnel tests [5]. Currently, there have been studies on the long-term dynamical behavior of solutions to a class of non-Newtonian fluid and micropolar fluid equations with delays. For example, on the one hand, for a class of non-Newtonian fluids, when the bounded delay initial value space is C H , Zhao, Park, Lázaro et al. [6]-[9] established the well-posedness of weak solutions, the existence and properties of pullback attractors, Liu et al. [10] established the exponential stability of weak and strong solutions with respect to stationary solutions; when the bounded delay initial value space is C ρ ( H ) ( ρ > 0 ), Liu, Xu et al. [11][12] established the existence and properties of pullback attractors for solutions in two-dimensional unbounded domains; when considering unbounded delays with initial value space C ρ ( H ) ( ρ > 0 ), Zhao et al. [13] established the well-posedness of weak solutions and the existence of pullback attractors in two-dimensional bounded domains; when ρ = 0 , i.e., considering the initial value space C 0 ( H ) , Liu et al. [14] established the stability of weak (strong) solutions with respect to stationary solutions for three special types of unbounded delay functions. On the other hand, for micropolar fluid equations, when the bounded delay initial value space is L 2 ( − h , 0 ; H ^ ) , Sun and Liu [15][16] established the global well-posedness of weak solutions and the existence and properties of pullback attractors in two-dimensional unbounded domains. On this basis, Sun [17] established the existence and uniqueness of stationary solutions and the exponential stability of weak solutions with respect to stationary solutions in two-dimensional unbounded domains. In the case of unbounded delays, with initial value space C ρ ( H ^ ) ( ρ > 0 ), Zhao etal. [13] established the global well-posedness of weak solutions and the existence of pullback attractors in two-dimensional bounded domains. Furthermore, Zhou et al. [18] established the H 2 boundedness of pullback attractors for solutions in two-dimensional bounded domains. In contrast, research results on non-Newtonian micropolar fluid equations with delay effects are still relatively lacking.

This paper mainly studies the following class of non-Newtonian micropolar fluid equations with bounded delay:

∂ u ∂ t − ∇ ⋅ [ μ ( u ) e ( u ) − 2 μ 1 Δ e ( u ) ] + ( u ⋅ ∇ ) u + ∇ p − 2 ν r ∇ × ω = f + g ( t , u t ) , ∇ ⋅ u = 0 , ∂ ω ∂ t − ν 1 Δ ω + ( u ⋅ ∇ ) ω + 4 ν r ω − 2 ν r ∇ × u = f ˜ + g ˜ ( t , ω t ) ,

where x ∈ Ω ⊂ ℝ 2 , t ∈ ℝ + , u = ( u 1 , u 2 ) denotes the velocity field of the fluid, ω denotes the angular velocity of particle rotation, p = p ( x , t ) denotes the pressure of the fluid, f = ( f 1 , f 2 ) and f ˜ denote the external force and torque respectively, the positive constants ν 1 , ν r denote viscosity coefficients, ∇ denotes the gradient operator, Δ denotes the Laplace operator,

∇ × ω = ( ∂ ω ∂ x 2 , − ∂ ω ∂ x 1 ) ,     ∇ × u = ∂ u 2 ∂ x 1 − ∂ u 1 ∂ x 2 ,

μ ( u ) : = 2 μ 0 ( ε + | e ( u ) | 2 ) − α / 2 , denotes the rate-of-strain tensor, where

e i j ( u ) = 1 2 ( ∂ u i ∂ x j + ∂ u j ∂ x i ) ,     i , j = 1 , 2 , ⋯ , n .

∇ ⋅ u = 0 describes the incompressibility of the fluid. g ( t , u t ) = ( g 1 ( t , u t ) , g 2 ( t , u t ) ) , g ˜ ( t , ω t ) denote the external force terms with delays that have some memory or hereditary characteristics. u t , ω t are the delay functions defined as

u t ( θ ) = u ( t + θ ) ,     ω t ( θ ) = ω ( t + θ ) ,     θ ∈ ( − ∞ , 0 ] .

Let ∂ Ω be the boundary of Ω and be sufficiently smooth. The initial-boundary value conditions for Equations (1)-(3) are given as

u ( θ , x ) = ϕ ( θ , x ) ,     ω ( θ , x ) = ϕ 3 ( θ , x ) , u | ( 0 , T ) × ∂ Ω = 0 ,     ω | ( 0 , T ) × ∂ Ω = 0 ,     T i j k u n j n k | ( 0 , T ) × ∂ Ω = 0 ,

where ϕ = ( ϕ 1 , ϕ 2 ) , ϕ 3 are initial velocities defined on the delay interval,

T i j k u : = 2 μ 1 ∂ e i j ( u ) ∂ x k , i , j , k = 1 , 2 , ( n 1 , n 2 ) denotes the unit outward normal vector

on ∂ Ω . In the boundary conditions (5), the first two conditions indicate no-slip phenomenon of the fluid on the boundary, and the third condition indicates that the traction force of the viscous fluid vanishes on the boundary. For the specific physical background, references [19]-[25] can be consulted.

Throughout this paper, we investigate three special classes of unbounded delay functions and studies the stability of weak and strong solutions with respect to stationary solutions by combining different analytical techniques with stability theory. Specifically, when the delay function is continuous in time, the Lyapunov stability of weak solutions with respect to nontrivial stationary solutions is established via the definition of Lyapunov stability, while the asymptotic stability of weak solutions with respect to the trivial stationary solution is obtained by constructing a suitable Lyapunov functional. When the delay function is only piecewise continuous in time, the stability of strong solutions with respect to nontrivial stationary solutions is proved by means of the Razumikhin technique. Compared with the case of bounded delay [10][17], the unbounded delay considered in this paper does not yield exponential stability of weak or strong solutions with respect to the steady-state solution. Therefore, we introduce a proportional delay and establish the polynomial stability of weak solutions with respect to the trivial steady-state solution. The innovations of this paper are threefold. Firstly, while existing studies focus only on non-Newtonian micropolar fluid equations without delay, this paper considers the delayed case, which is theoretically significant because delay effects strongly influence well-posedness and solution dynamics. Secondly, new phase spaces are introduced for bounded and unbounded delays, respectively, overcoming the analytical difficulties caused by delay terms. Besides, the stability of weak and strong solutions with respect to steady-state solutions is established, enriching the study of the system’s long-time dynamical behavior. The paper is organized as follows. Section 1 presents the preliminaries. By introducing suitable abstract operators, the initial-boundary value problem (1)-(5) is reformulated as an initial value problem for a functional differential equation. The definitions of weak and strong solutions are also given in this section. Section 2 proves the well-posedness of weak solutions and establishes the existence, uniqueness, and regularity of stationary solutions to problem (1)-(5). Section 3 investigates the exponential stability of weak and strong solutions with respect to stationary solutions for problem (1)-(5).

 2. Preliminaries

In this paper, ℝ denotes the set of real numbers, ℤ denotes the set of integers. c , c i , C denote constants, which may take different values in different places. → denotes strong convergence or convergence in norm, ⇀ denotes weak convergence, ⇀ * denotes weak-star convergence. ↪↪ denotes a compact embedding between two spaces. Let L p ( Ω ) and W m , p ( Ω ) denote the classical Lebesgue space and Sobolev space, respectively, with norms

‖ u ‖ p : = ( ∫ Ω | u | p d x ) 1 / p ,       ‖ u ‖ m , p : = ( ∑ | β | ≤ m ∫ Ω | D β u | p d x ) 1 / p .

In particular, denote ‖   ⋅   ‖ : = ‖   ⋅   ‖ 2 , H m ( Ω ) : = W m , 2 ( Ω ) , the dual space of H m ( Ω ) is H − m ( Ω ) , H 0 1 ( Ω ) denotes the closure of C 0 ∞ ( Ω ) in H 1 ( Ω ) . On this basis, introduce vector function spaces:

V : = { u ∈ C 0 ∞ ( Ω ) × C 0 ∞ ( Ω ) | u = ( u 1 , u 2 ) , ∇ ⋅ u = 0 } , H : = closure   of   V   in   ( L 2 ( Ω ) ) 2 ,     norm   ‖   ⋅   ‖ H   and   dual   space   H ′ = H , V : = closure   of   V   in   ( H 2 ( Ω ) ) 2 ,     norm   ‖   ⋅   ‖ V   and   dual   space   V ′ , H ^ : = H × L 2 ( Ω ) ,     norm   ‖   ⋅   ‖ H ^ ,     dual   space   H ^ ′ , V ^ : = V × H 0 1 ( Ω ) ,     norm     ‖   ⋅   ‖ V ^ ,     dual   space   V ^ ′ .

Denote ( ⋅ , ⋅ ) as the inner product in L 2 ( Ω ) , H or H ^ , and 〈 ⋅ , ⋅ 〉 as the duality product between V and V ′ or V ^ and V ^ ′ . When no confusion arises, the norms ‖   ⋅   ‖ H and ‖   ⋅   ‖ H ^ may be abbreviated as ‖   ⋅   ‖ . Furthermore, let L p ( I ; X ) be the set of all p -integrable functions defined on interval I with values in Banach space X , with norm

‖ u ‖ L p ( I , X ) : = ( ∫ I ‖ u ‖ X p d x ) 1 p ,     1 ≤ p < ∞ ;

C ( I ; X ) denotes the set of continuous functions defined on interval I with values in Banach space X , with norm ‖   ⋅   ‖ C ( I ; X ) .

To obtain the weak form of the non-Newtonian micropolar fluid equations, we introduce four abstract operators.

(i) Define operator A : for any v = ( u , ω ) , Ψ = ( ψ , ψ 3 ) ∈ V ^ , where u = ( u 1 , u 2 ) , ψ = ( ψ 1 , ψ 2 ) ∈ V ,

( A v , Ψ ) = 2 μ 1 〈 A 1 u , ψ 〉 + ν 1 〈 A 2 ω , ψ 3 〉 = 2 μ 1 ∑ i , j , k = 1 2 ∫ Ω ∂ e i j ( u ) ∂ x k ∂ e i j ( ψ ) ∂ x k d x + ν 1 ∑ k = 1 2 ∫ Ω ∂ ω ∂ x k ∂ ψ 3 ∂ x k d x .

Remark 1. A 1 = P Δ 2 , where P is the Leray projection operator from ( L 2 ( Ω ) ) 2 to H , and D ( A 1 ) = { u ∈ V : A 1 u ∈ H } = ( H 4 ( Ω ) ) 2 ∩ V is a Hilbert space compactly embedded into V . Moreover, the operator A 1 satisfies ([13]): there exists a positive constant c 1 such that for any u ∈ V ,

c 1 ‖ u ‖ V 2 ≤ 〈 A 1 u , u 〉 ≤ ‖ u ‖ V 2 .

Remark 2. According to the definition of operators A 1 , A 2 and the spectral theory of elliptic operators [25][26], there exists a sequence { λ n , n = 1 , 2 , ⋯ } consisting of eigenvalues of A 1 and a sequence { v n } n = 1 ∞ ⊂ D ( A 1 ) which is an orthonormal basis of H , such that span { v 1 , v 2 , ⋯ , v n , ⋯ } is dense in V . There exists a sequence { λ n * , n = 1 , 2 , ⋯ } consisting of eigenvalues of A 2 and a sequence { v n * } n = 1 ∞ ⊂ D ( A 2 ) which is an orthonormal basis of H , such that span { v 1 * , v 2 * , ⋯ , v n * , ⋯ } is dense in H 0 1 ( Ω ) . Moreover, for any n ∈ ℕ ,

A 1 υ n = λ n υ n ,     0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ ⋯ ≤ λ n ≤ ⋯ ,     n → ∞ ,     λ n → ∞ , A 2 υ n * = λ n * υ n * ,     0 < λ 1 * ≤ λ 2 * ≤ λ 3 * ≤ ⋯ ≤ λ n * ≤ ⋯ ,     n → ∞ ,     λ n * → ∞ .

Thus the following Poincaré inequalities hold:

λ 1 ‖ u ‖ 2 ≤ ‖ u ‖ V 2 ,     λ 1 * ‖ ω ‖ 2 ≤ ‖ ∇ ω ‖ 2 ,     ∀ u ∈ V ,   ω ∈ H 0 1 ( Ω ) .

From (6), for any v = ( u , ω ) ∈ V ^ ,

‖ v ‖ 2 = ‖ u ‖ 2 + ‖ ω ‖ 2 ≤ λ 1 − 1 ‖ u ‖ V 2 + λ 1 * − 1 ‖ ∇ ω ‖ 2 ≤ γ − 1 ‖ v ‖ V ^ 2 , γ = min { λ 1 , λ 1 * }

(ii) Define operator B : for any u = ( u 1 , u 2 ) ∈ V , Ψ = ( ψ 1 , ψ 2 , ψ 3 ) , Φ = ( ϕ 1 , ϕ 2 , ϕ 3 ) ∈ V ^ ,

〈 B ( u , Ψ ) , Φ 〉 = b ( u , Ψ , Φ ) = ∑ j = 1 3 ∑ i = 1 2 ∫ Ω u i ∂ ψ j ∂ x i ϕ j d x .

(iii) Define operator N :

N ( v ) = ( N 1 ( u ) , 0 ) ,     ∀ v = ( u , ω ) .

Here N 1 is defined as: for any u = ( u 1 , u 2 ) , ψ = ( ψ 1 , ψ 2 ) ∈ V ,

〈 N 1 ( u ) , ψ 〉 = n 1 ( u , u , ψ ) = ∑ i , j = 1 2 ∫ Ω 2 μ 0 ( ε + | e ( u ) | 2 ) − α / 2 e i j ( u ) e i j ( ψ ) d x .

If we define for any Ψ = ( ψ , ψ 3 ) ∈ V ^ , n ( v , v , Ψ ) = 〈 N ( v ) , Ψ 〉 , then n ( v , v , Ψ ) = n 1 ( u , u , ψ ) .

(iv) Define operator J : for any v = ( u , ω ) ∈ V ^ ,

J ( v ) = ( − 2 ν r ∇ × ω , − 2 ν r ∇ × u + 4 ν r ω ) .

Lemma 1.The following are some properties of operators A , B , N , J ([4][13]):

(i) A : V ^ → V ^ ′ and D ( A ) → H ^ is a continuous linear operator; N ( ⋅ ) : V ^ → V ^ ′ is a continuous nonlinear operator; J ( ⋅ ) : V ^ → H ^ is a continuous linear operator.

(ii) B ( ⋅ , ⋅ ) : V × V ^ → V ^ ′ is a continuous nonlinear operator, and

〈 B ( u , Ψ ) , Ψ 〉 = 0 ,     〈 B ( u , Ψ ) , Φ 〉 = − 〈 B ( u , Φ ) , Ψ 〉 .

(iii) For any v = ( u , ω ) , Ψ = ( ψ , ψ 3 ) ∈ V ^ , u = ( u 1 , u 2 ) , ψ = ( ψ 1 , ψ 2 ) ∈ V , there exists a positive constant c 2 such that

| 〈 A v , Ψ 〉 | ≤ c 2 ‖ v ‖ V ^ ‖ Ψ ‖ V ^ ,

and

〈 A Ψ , Ψ 〉 + 〈 J ( Ψ ) , Ψ 〉 ≥ δ 1 ‖ Ψ ‖ V ^ 2 , δ 1 = min { 2 c 1 μ 1 − ν r , ν 1 }

(iv) There exist positive constants c i , i = 3 , 4 , 5 , such that for any v = ( u , ω ) , Φ = ( ψ , ϕ 3 ) , Ψ = ( ψ , ψ 3 ) ∈ V ^ with u = ( u 1 , u 2 ) , ψ = ( ψ 1 , ψ 2 ) , ϕ = ( ϕ 1 , ϕ 2 ) ∈ V ,

| 〈 B ( u , Φ ) , Ψ 〉 | ≤ c 3 ‖ u ‖ 1 2 ‖ u ‖ V 1 2 ‖ Φ ‖ V ^ ‖ Ψ ‖ 1 2 ‖ Ψ ‖ V ^ 1 2 . | 〈 J ( v ) , Ψ 〉 | ≤ c ‖ v ‖ V ^ ‖ Ψ ‖ ≤ c 4 ‖ v ‖ V ^ ‖ Ψ ‖ V ^ . | 〈 N ( v ) , Ψ 〉 | ≤ c ‖ u ‖ V ‖ Ψ ‖ V ≤ c 5 ‖ v ‖ V ^ ‖ Ψ ‖ V ^ .

Based on the above operators, in the divergence-free field, the weak form of the initial-boundary value problem (1)-(5) in the sense of D ′ ( 0 , T ) can be written as:

d v d t + A v + B ( u , v ) + N ( v ) + J ( v ) = F ( t ) + G ( t , v t ) , v ( θ ) = Φ ( θ ) ,     θ ∈ ( − ∞ , 0 ] .

where v = ( u , ω ) , Φ = ( ϕ , ϕ 3 ) , ϕ = ( ϕ 1 , ϕ 2 ) , F = ( f , f ˜ ) , G ( t , v t ) = ( g ( t , u t ) , g ˜ ( t , ω t ) ) . The variables in this equation have the same physical meanings as those in Equations (1)-(5). To deal with unbounded delays, we consider the initial value space C 0 ( H ^ )

C 0 ( H ^ ) = { Φ ∈ C ( ( − ∞ , 0 ] ; H ^ ) : ∃ lim s → − ∞ Φ ( s ) ∈ H ^ } ,

which is a Banach space with norm

‖ Φ ‖ C 0 ( H ^ ) = sup s ∈ ( − ∞ , 0 ] ‖ Φ ( s ) ‖ .

Now we give the definitions of weak and strong solutions.

Definition 1. Given T > 0 and v 0 = Φ ∈ C 0 ( H ^ ) , Ψ ∈ V ^ . A function

v ∈ C ( ( − ∞ , T ] ; H ^ ) ∩ L 2 ( 0 , T ; V ^ )

is called a weak solution of the initial value problem (14)-(15) if it satisfies in the sense of D ′ ( 0 , T ) :

d d t ( v ( t ) , Ψ ) + 〈 A v , Ψ 〉 + 〈 B ( u ( t ) , v ( t ) ) , Ψ 〉 + 〈 N ( v ( t ) ) , Ψ 〉 + 〈 J ( v ( t ) ) , Ψ 〉 = 〈 F ( t ) , Ψ 〉 + ( G ( t , v t ) , Ψ ) .

If a weak solution satisfies v ∈ C ( ( − ∞ , 0 ] ; V ^ ) ∩ L 2 ( 0 , T ; D ( A ) ) , then it is called a strong solution.

3. Existence and Uniqueness of Weak Solutions and Stationary Solutions

In this section, we prove the global well-posedness of weak solutions to the initial value problem (14)-(15). For that purpose, the following assumption on the delay external force term G ( t , v t ) is needed.

Assumption 1. Let G : [ 0 , T ] × C 0 ( H ^ ) → H ^ satisfy:

(i) For any ξ ∈ C 0 ( H ^ ) , the map [ 0 , T ] ∋ t ↦ G ( t , ξ ) ∈ H ^ is measurable.

(ii) G ( ⋅ , 0 ) = 0 .

(iii) There exists a constant L G > 0 such that for any t ∈ [ 0 , T ] , ξ , η ∈ C 0 ( H ^ ) ,

‖ G ( t , ξ ) − G ( t , η ) ‖ ≤ L G ‖ ξ − η ‖ C 0 ( H ^ ) .

Remark 3. From (ii) and (iii),

‖ G ( t , ξ ) ‖ ≤ L G ‖ ξ ‖ C 0 ( H ^ ) ,     ∀ ξ ∈ C 0 ( H ^ ) .

Theorem 1. For any T > 0 , F ( t , x ) ∈ L 2 ( 0 , T ; V ^ ) , Φ ∈ C 0 ( H ^ ) , and G ( t , v t ) satisfying Assumption 1, the initial value problem (14)-(15) has a unique weak solution v = ( u , ω ) on the interval ( − ∞ , T ] , and this weak solution depends continuously on the initial value. If F ( t , x ) ∈ L 2 ( 0 , T ; H ^ ) , Φ ∈ C 0 ( H ^ ) and Φ ( 0 ) ∈ V ^ , then the weak solution v = ( u , ω ) is actually a strong solution of (14)-(15).

Proof. The proof uses the classical Faedo-Galerkin approximation method and the energy method, it consists of three steps: constructing Galerkin approximate solutions, deriving a priori estimates, and passing to the limit, which is similar to [7][13], only minor adjustments are neededthe detailed proof is omitted here. □

The stationary equation corresponding to the initial value problem (14)-(15) is

A v * + B ( u * , v * ) + N ( v * ) + J ( v * ) = F + G ( v * ) .

Definition 2. A function v * = ( u * , ω * ) ∈ V ^ is called a stationary solution of the initial value problem (14)-(15) if it satisfies

〈 A v * , Ψ 〉 + 〈 B ( u * , v * ) , Ψ 〉 + 〈 N ( v * ) , Ψ 〉 + 〈 J ( v * ) , Ψ 〉 = 〈 F , Ψ 〉 + ( G ( v * ) , Ψ ) ,     ∀ Ψ ∈ V ^ .

Theorem 2. Assume that G satisfies Assumption 1 and δ 1 > γ − 1 L G . Then,

(a) For all F ∈ V ^ ′ , Equation (17) has at least one solution.

(b) If ( δ 1 − γ − 1 L G ) 2 > c 3 γ − 1 2 ‖ F ‖ V ^ ′ , then Equation (17) has a unique solution.

(c) If F ∈ H ^ , then the solution v * ∈ D ( A ) .

Proof. The proof mainly relies on the Lax–Milgram theorem and the Schauder fixed-point theorem, which is similar to references [17][27], the detailed steps are omitted here. □

 4. Stability of Weak and Strong Solutions with Respect to Stationary Solutions

This section considers several special forms of unbounded delay functions and further establishes the stability of stationary solutions to the initial value problems (14)-(15). First, when the unbounded delay function is continuously differentiable with respect to time, the stability of weak solutions with respect to nontrivial stationary solutions and the asymptotic stability of weak solutions with respect to trivial stationary solutions are established by applying the definition of Lyapunov stability and by constructing a Lyapunov functional, respectively. Next, when the unbounded delay function is only continuous with respect to time, the stability of strong solutions with respect to nontrivial stationary solutions is proved by means of the Razumikhin technique. Finally, when the unbounded delay function is a proportional delay, the polynomial stability of weak solutions with respect to trivial stationary solutions is obtained. The definitions of stability of weak and strong solutions with respect to stationary solutions are given below.

Definition 3.([28]) Let v ( t ) and v * be a weak (strong) solution and a stationary solution of the initial value problem (14)-(15), respectively.

(i) If for any ε > 0 there exists δ > 0 such that whenever the initial value Φ ∈ C 0 ( H ^ ) satisfies ‖ Φ − v * ‖ C 0 ( H ^ ) < δ , we have

‖ v ( t ) − v * ‖ < ε ,     ∀ t > 0 ,

then v ( t ) is said to be stable with respect to the stationary solution v * .

(ii) If there exists δ ˜ > 0 such that whenever Φ ∈ C 0 ( H ^ ) satisfies ‖ Φ − v * ‖ C 0 ( H ^ ) < δ ˜ , we have

lim t → ∞ ‖ v ( t ) − v * ‖ = 0 ,

then v ( t ) is said to be attractive with respect to v * .

(iii) If v ( t ) is both stable and attractive with respect to v * , then v ( t ) is said to be asymptotically stable with respect to v * .

4.1. Stability and Asymptotic Stability: The Unbounded Variable Delay Driven by a Continuously Differential Function

In this subsection, we consider delay functions satisfying the following condition.

Condition 1. Let

G ( t , ξ ) = G ^ ( ξ ( − ρ ( t ) ) ) ,

where ( t , ξ ) ∈ ℝ + × C 0 ( H ^ ) , G ^ : H ^ → H ^ is a Lipschitz map with Lipschitz constant L G ^ , G ^ ( 0 ) = 0 , and ρ ∈ C 1 ( ℝ + ; ℝ + ) , ρ * = sup t ≥ 0 ρ ′ ( t ) < 1 .

Remark 4. From Remark 3, G ( t , ξ ) satisfying Condition 1 also satisfies Assumption 1.

Theorem 3. Let F ∈ H ^ , Condition 1 hold, δ 1 > γ − 1 L G ^ , and

δ 1 > c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ + ( 2 − ρ * ) γ − 1 L G ^ 2 ( 1 − ρ * ) ,

then the initial value problem (14)-(15) has a stationary solution v * ∈ D ( A ) . When F ( t ) ≡ F , Φ ∈ C 0 ( H ^ ) , there exists a unique weak solution v ( t ) of (14)-(15) satisfying

‖ v ( t ) − v * ‖ 2 ≤ C ( ‖ Φ ( 0 ) − v * ‖ 2 + ‖ Φ − v * ‖ L 2 ( ( − ρ ( 0 ) , 0 ) ; V ^ ) 2 ) ,     ∀ t ≥ 0.

Proof. By Condition 1, F ∈ H ^ , δ 1 > γ − 1 L G ^ and Theorem 1, when F ( t ) ≡ F , Φ ∈ C 0 ( H ^ ) , there exists a unique weak solution v = ( u , ω ) of (14)-(15). From Theorem 2, there exists a unique stationary solution v * ∈ D ( A ) . Let v ˜ ( t ) = v ( t ) − v * . Then

d d t v ˜ ( t ) + A v ˜ ( t ) + B ( u ( t ) , v ( t ) ) − B ( u * , v * ) + N ( v ( t ) ) − N ( v * ) + J ( v ˜ ( t ) ) = G ( v ( t − ρ ( t ) ) ) − G ( v * ) .

Taking the duality product of (19) with v ˜ ( t ) yields

d d t ‖ v ˜ ( t ) ‖ 2 = − 2 〈 A v ˜ ( t ) , v ˜ ( t ) 〉 − 2 〈 B ( u ( t ) , v ( t ) ) − B ( u * , v * ) , v ˜ ( t ) 〉       − 2 〈 N ( v ( t ) ) − N ( v * ) , v ˜ ( t ) 〉 − 2 〈 J ( v ˜ ( t ) ) , v ˜ ( t ) 〉       + 2 ( G ( v ( t − ρ ( t ) ) ) − G ( v * ) , v ˜ ( t ) ) .

Considering Condition 1, we obtain

d d t ‖ v ˜ ( t ) ‖ 2 ≤ − 2 δ 1 ‖ v ˜ ( t ) ‖ V ^ 2 + 2 c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ ‖ v ˜ ( t ) ‖ V ^ 2 + γ − 1 L G ^ ‖ v ˜ ( t ) ‖ V ^ 2       + L G ^ ‖ v ˜ ( t − ρ ( t ) ) ‖ 2 = ( 2 c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ + γ − 1 L G ^ − 2 δ 1 ) ‖ v ˜ ( t ) ‖ V ^ 2 + L G ^ ‖ v ˜ ( t − ρ ( t ) ) ‖ 2 ,

Integrating (21) over [ 0 , t ] gives

‖ v ˜ ( t ) ‖ 2 ≤ ‖ v ˜ ( 0 ) ‖ 2 + ( 2 c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ + γ − 1 L G ^ − 2 δ 1 ) ∫ 0 t ‖ v ˜ ( s ) ‖ V ^ 2 d s       + L G ^ ∫ 0 t ‖ v ˜ ( s − ρ ( s ) ) ‖ 2 d s .

For the delay integral term, set η = s − ρ ( s ) = τ ( s ) , then

L G ^ ∫ 0 t ‖ v ˜ ( s − ρ ( s ) ) ‖ 2 d s ≤ L G ^ 1 − ρ * ∫ − ρ ( 0 ) t ‖ v ˜ ( η ) ‖ 2 d η ,

From (22)-(23) and (7), we obtain

‖ v ˜ ( t ) ‖ 2 ≤ ‖ v ˜ ( 0 ) ‖ 2 + ( 2 c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ + γ − 1 L G ^ − 2 δ 1 + γ − 1 L G ^ 1 − ρ * ) ∫ 0 t ‖ v ˜ ( s ) ‖ V ^ 2 d s       + γ − 1 L G ^ 1 − ρ * ∫ − ρ ( 0 ) 0 ‖ v ˜ ( s ) ‖ V ^ 2 d s = ‖ v ˜ ( 0 ) ‖ 2 + ( 2 c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ − 2 δ 1 + ( 2 − ρ * ) γ − 1 L G ^ 1 − ρ * ) ∫ 0 t ‖ v ˜ ( s ) ‖ V ^ 2 d s       + γ − 1 L G ^ 1 − ρ * ∫ − ρ ( 0 ) 0 ‖ v ˜ ( s ) ‖ V ^ 2 d s .

By (18), we have

2 c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ − 2 δ 1 + ( 2 − ρ * ) γ − 1 L G ^ 1 − ρ * < 0 ,

thus

‖ v ˜ ( t ) ‖ 2 ≤ ‖ v ˜ ( 0 ) ‖ 2 + γ − 1 L G ^ 1 − ρ * ∫ − ρ ( 0 ) 0 ‖ v ˜ ( s ) ‖ V ^ 2 d s ≤ C ( ‖ Φ ( 0 ) − v * ‖ 2 + ‖ Φ − v * ‖ L 2 ( ( − ρ ( 0 ) , 0 ) ; V ^ ) 2 ) ,

where C = max { 1 , γ − 1 L G ^ 1 − ρ * } . This completes the proof of Theorem 3. □

Next, we establish the asymptotic stability of weak solutions with respect to the trivial stationary solution for the initial value problem (14)-(15) by constructing a Lyapunov functional.

Theorem 4. Let F ≡ 0 , Condition 1 hold, Φ ∈ C 0 ( H ^ ) , δ 1 > γ − 1 L G ^ . Then the initial value problem (14)-(15) has a unique trivial stationary solution v * ≡ 0 and a unique weak solution v ( t ) . Moreover, if

δ 1 > γ − 1 L G ^ 1 − ρ * ,

then v ( t ) is asymptotically stable with respect to v * ≡ 0 .

Proof. By the above conditions and Theorem 2(a)-(b), the initial value problem (14)-(15) has a unique trivial stationary solution v * ≡ 0 and a unique weak solution v ( t ) . We prove the asymptotic stability in three steps.

Step 1 (Construction of a Lyapunov functional). Define W ( t , v ( t ) ) : ℝ + × C 0 ( H ^ ) → ℝ + by

W ( t , v ( t ) ) = ‖ v ( t ) ‖ 2 + C ′ 1 − ρ * ∫ t − ρ ( t ) t ‖ v ( s ) ‖ 2 d s ,

where C ′ > 0 will be determined later. From (26) and the nonnegativity of the integral term,

W ( t , v ( t ) ) ≤ ‖ v ( t ) ‖ 2 .

On the other hand, from (26),

W ( 0 , v ( 0 ) ) = ‖ v ( 0 ) ‖ 2 + C ′ 1 − ρ * ∫ − ρ ( 0 ) 0 ‖ v ( s ) ‖ 2 d s ≤ ‖ Φ ‖ C 0 ( H ^ ) 2 + C ′ 1 − ρ * ∫ − ρ ( 0 ) 0 ‖ Φ ( s ) ‖ 2 d s ≤ ‖ Φ ‖ C 0 ( H ^ ) 2 + C ′ ρ ( 0 ) 1 − ρ * ‖ Φ ‖ C 0 ( H ^ ) 2 .

Thus there exists β = 1 + C ′ ρ ( 0 ) 1 − ρ * > 0 such that

W ( 0 , v ( 0 ) ) ≤ β ‖ Φ ‖ C 0 ( H ^ ) 2 .

Step 2 (Proof of stability). From (26), and Condition 1,

d W ( t , v ( t ) ) d t = d d t ‖ v ( t ) ‖ 2 + C ′ 1 − ρ * ‖ v ( t ) ‖ 2 − C ′ ( 1 − ρ ′ ( t ) ) 1 − ρ * ‖ v ( t − ρ ( t ) ) ‖ 2 ≤ d d t ‖ v ( t ) ‖ 2 + C ′ 1 − ρ * ‖ v ( t ) ‖ 2 − C ′ ‖ v ( t − ρ ( t ) ) ‖ 2 .

Since F ≡ 0 and Condition 1 holds, Equation (14), becomes

d v d t = − A v − B ( u , v ) − N ( v ) − J ( v ) + G ^ ( v ( t − ρ ( t ) ) ) ,     ∀ t ≥ 0 ,

Taking the duality product with v ( t ) yields

d d t ‖ v ( t ) ‖ 2 = − 2 〈 − A v − B ( u , v ) − N ( v ) − J ( v ) , v ( t ) 〉 + 2 ( G ^ ( v ( t − ρ ( t ) ) ) , v ( t ) ) .

Using (8), (10), Condition 1, the nonnegativity of N , and Hölder’s inequality, we obtain

d d t ‖ v ( t ) ‖ 2 ≤ − 2 δ 1 ‖ v ( t ) ‖ V ^ 2 + 2 L G ^ ‖ v ( t − ρ ( t ) ) ‖ ‖ v ( t ) ‖ ≤ − 2 δ 1 ‖ v ( t ) ‖ V ^ 2 + L G ^ 2 C ′ ‖ v ( t ) ‖ 2 + C ′ ‖ v ( t − ρ ( t ) ) ‖ 2 .

Substituting (30) into (29), and using (7),

d W ( t , v ( t ) ) d t ≤ − 2 δ 1 ‖ v ( t ) ‖ V ^ 2 + L G ^ 2 C ′ ‖ v ( t ) ‖ 2 + C ′ 1 − ρ * ‖ v ( t ) ‖ 2 ≤ ( − 2 δ 1 + γ − 1 ( L G ^ 2 C ′ + C ′ 1 − ρ * ) ) ‖ v ( t ) ‖ V ^ 2 ,

Choosing C ′ = L G ^ 1 − ρ * , by (25), and (31), we have

d W ( t , v ( t ) ) d t ≤ − 2 ( δ 1 − γ − 1 L G ^ 1 − ρ * ) ‖ v ( t ) ‖ V ^ 2 < 0 ,

Therefore, W ( t , v ( t ) ) is strictly decreasing. From (27)-(28),

‖ v ( t ) ‖ 2 ≤ W ( t , v ( t ) ) < W ( 0 , v ( 0 ) ) ≤ β ‖ Φ ‖ C 0 ( H ^ ) 2 ,

so v * ≡ 0 is stable.

Step 3 (Proof of attractivity). Integrate (32) over [ 0 , t ] :

W ( t , v ( t ) ) − W ( 0 , v ( 0 ) ) ≤ − 2 ( δ 1 − γ − 1 L G ^ 1 − ρ * ) ∫ 0 t ‖ v ( s ) ‖ V ^ 2 d s ,

By (33),

∫ 0 t ‖ v ( s ) ‖ V ^ 2 d s ≤ 1 2 ( δ 1 − γ − 1 L G ^ 1 − ρ * ) ( W ( t , v ( 0 ) ) − W ( v ( t ) ) ) ≤ 1 2 ( δ 1 − γ − 1 L G ^ 1 − ρ * ) ( β ‖ Φ ‖ C 0 ( H ^ ) 2 − ‖ v ( t ) ‖ 2 ) ≤ β 2 ( δ 1 − γ − 1 L G ^ 1 − ρ * ) ‖ Φ ‖ C 0 ( H ^ ) 2 ,

which implies lim t → ∞ ‖ v ( t ) ‖ V = 0 , i.e., v * ≡ 0 is attractive. Thus v ( t ) is asymptotically stable with respect to v * ≡ 0 . □

 4.2. Stability: The Unbounded Variable Delay Driven by a Continuous Function

In this subsection, we consider delay functions satisfying the following condition.

Condition 2. Let

G ( t , ξ ) = G ^ ( ξ ( − ρ ( t ) ) ) ,     ∀ ( t , ξ ) ∈ ℝ + × C 0 ( H ) ,

where G ^ : H ^ → H ^ is a Lipschitz map with Lipschitz constant L G ^ , G ^ ( 0 ) = 0 , ρ ∈ C ( ℝ + ; ℝ + ) .

To establish the stability of strong solutions with respect to non-trivial stationary solutions, we need to modify Assumption 1 (i) as follows:

(i”) For any ξ ∈ C 0 ( H ^ ) , the map [ 0 , T ] ∋ t ↦ G ( t , ξ ) ∈ H ^ is continuous.

Remark 5. From Remark 3, the delay term satisfying Condition 2 satisfies Assumption 1 and (I”).

Then, we can establish an important lemma.

Lemma 2. Let G satisfies Assumption 1 and (i”), F ∈ H ^ , v * ∈ D ( A ) be a stationary solution of (14)-(15). For any Φ = ( ϕ , ϕ 3 ) ∈ C 0 ( H ^ ) with Φ ( 0 ) = ( ϕ ( 0 ) , ϕ 3 ( 0 ) ) ∈ V ^ , where ϕ = ( ϕ 1 , ϕ 2 ) ∈ C 0 ( H ) , ϕ ( 0 ) = ( ϕ 1 ( 0 ) , ϕ 2 ( 0 ) ) ∈ V , and Φ ≠ v * satisfying

‖ Φ − v * ‖ C 0 ( H ^ ) = ‖ Φ ( 0 ) − v * ‖

such that

− ( A ( Φ ( 0 ) − v * ) , Φ ( 0 ) − v * ) − ( B ( ϕ ( 0 ) , Φ ( 0 ) ) − B ( u * , v * ) , Φ ( 0 ) − v * ) − ( N ( Φ ( 0 ) ) − N ( v * ) , Φ ( 0 ) − v * ) − ( J ( Φ ( 0 ) − v * ) , Φ ( 0 ) − v * ) + ( G ( t , Φ ) − G ( t , v * ) , Φ ( 0 ) − v * ) < 0.

then the strong solution v ( t ) of (14)-(15) satisfies

‖ v ( t ) − v * ‖ ≤ ‖ Φ − v * ‖ C 0 ( H ^ ) ,     ∀ t ≥ 0.

Proof. When Φ = v * , the conclusion is trivial. When Φ ≠ v * , we prove by contradiction. Suppose there exists an initial value Φ satisfying Φ ∈ C 0 ( H ^ ) , Φ ( 0 ) ∈ V ^ , Φ ≠ v * such that (36) fails. Then there exists t > 0 with

‖ v ( t ) − v * ‖ > ‖ Φ − v * ‖ C 0 ( H ^ ) .

Define

σ = inf { t > 0 : ‖ v ( t ) − v * ‖ > ‖ Φ − v * ‖ C 0 ( H ^ ) } ,

Then for any 0 ≤ s ≤ σ ,

‖ v ( s ) − v * ‖ ≤ ‖ v ( σ ) − v * ‖ = ‖ Φ − v * ‖ C 0 ( H ^ ) ,

and there exists a subsequence { t k } k ≥ 1 ⊂ ( σ , ∞ ) , t k ↓ σ as k → ∞ such that

‖ v ( t k ) − v * ‖ > ‖ v ( σ ) − v * ‖ .

From (37),

‖ v σ − v * ‖ C 0 ( H ^ ) = sup θ ≤ 0 ‖ v ( σ + θ ) − v * ‖ = ‖ v ( σ ) − v * ‖ .

By (34)-(35) and (39), taking Φ = v σ we have

− ( A ( v ( σ ) − v * ) , v ( σ ) − v * ) − ( B ( u ( σ ) , v ( σ ) ) − B ( u * , v * ) , v ( σ ) − v * ) − ( N ( v ( σ ) ) − N ( v * ) , v ( σ ) − v * ) − ( J ( v ( σ ) − v * ) , v ( σ ) − v * ) + ( G ( σ , v σ ( ⋅ ) ) − G ( t , v * ) , v ( σ ) − v * ) < 0.

By the continuity of operators A , B , N , G , there exists ϵ > 0 such that for any t ∈ [ σ , σ + ϵ ] ,

− ( A ( v ( t ) − v * ) , v ( t ) − v * ) − ( B ( u ( t ) , v ( t ) ) − B ( u * , v * ) , v ( t ) − v * ) − ( N ( v ( t ) ) − N ( v * ) , v ( t ) − v * ) − ( J ( v ( t ) − v * ) , v ( t ) − v * ) + ( G ( t , v t ) − G ( t , v * ) , v ( t ) − v * ) < 0.

Let v ˜ ( t ) = v ( t ) − v * . From (20), for any t ∈ [ σ , σ + ϵ ] ,

1 2 d d t ‖ v ˜ ( t ) ‖ 2 = − ( A v ˜ ( t ) , v ˜ ( t ) ) − ( B ( u ( t ) , v ( t ) ) − B ( u * , v * ) , v ˜ ( t ) )       − ( N ( v ( t ) ) − N ( v * ) , v ˜ ( t ) ) − ( J ( v ˜ ( t ) ) , v ˜ ( t ) )       + ( G ( t , v t ) − G ( t , v * ) , v ˜ ( t ) ) < 0 ,

Take t k ( ϵ ) ∈ ( σ , σ + ϵ ] and integrate (40) over [ σ , t k ( ϵ ) ] to obtain

‖ v ˜ ( t k ( ϵ ) ) ‖ 2 − ‖ v ˜ ( σ ) ‖ 2 ≤ 2 ∫ σ t k ( ϵ ) − ( A v ˜ ( t ) , v ˜ ( t ) ) − ( B ( u ( t ) , v ( t ) ) − B ( u * , v * ) , v ˜ ( t ) ) d t       + 2 ∫ σ t k ( ϵ ) − ( N ( v ( t ) ) − N ( v * ) , v ˜ ( t ) ) − ( J ( v ˜ ( t ) ) , v ˜ ( t ) ) d t       + 2 ∫ σ t k ( ϵ ) ( G ( t , v t ) − G ( t , v * ) , v ˜ ( t ) ) d t < 0 ,

i.e., ‖ v ˜ ( t k ( ϵ ) ) ‖ < ‖ v ˜ ( σ ) ‖ , contradicting (38). Hence (36) holds. □

Theorem 5. Suppose Condition 2 holds, F ∈ H ^ , δ 1 > γ − 1 L G ^ , and

δ 1 > γ − 1 L G ^ + c 3 ⋅ γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ ,

then for any Φ ∈ C 0 ( H ^ ) with Φ ( 0 ) ∈ V ^ , the initial value problem (14)-(15) has a unique stationary solution v * ∈ D ( A ) and a unique strong solution v ( t ) satisfying

‖ v ( t ) − v * ‖ ≤ ‖ Φ − v * ‖ C 0 ( H ^ ) ,     ∀ t ≥ 0.

Proof. By Condition 2, F ∈ H ^ , δ 1 > γ − 1 L G ^ , Theorem 1 and Theorem 2 imply the existence of a unique stationary solution v * ∈ D ( A ) and a unique strong solution v ( t ) of (14)-(15). We now prove (42). Let Φ = ( ϕ , ϕ 3 ) ∈ C 0 ( H ^ ) with Φ ( 0 ) = ( ϕ ( 0 ) , ϕ 3 ( 0 ) ) ∈ V ^ , where ϕ = ( ϕ 1 , ϕ 2 ) ∈ C 0 ( H ) , ϕ ( 0 ) = ( ϕ 1 ( 0 ) , ϕ 2 ( 0 ) ) ∈ V , and Φ ≠ v * satisfies

‖ Φ − v * ‖ C 0 ( H ^ ) = ‖ Φ ( 0 ) − v * ‖ .

Using (7), (10), (16), (43), we obtain

− ( A ( Φ ( 0 ) − v * ) , Φ ( 0 ) − v * ) − ( B ( ϕ ( 0 ) , Φ ( 0 ) ) − B ( u * , v * ) , Φ ( 0 ) − v * ) − ( N ( Φ ( 0 ) ) − N ( v * ) , Φ ( 0 ) − v * ) − ( J ( Φ ( 0 ) − v * ) , Φ ( 0 ) − v * ) + ( G ( t , Φ ) − G ( t , v * ) , Φ ( 0 ) − v * ) ≤ − δ 1 ‖ Φ ( 0 ) − v * ‖ V ^ 2 + | ( B ( Φ ( 0 ) − u * , v * ) , Φ ( 0 ) − v * ) |         + L G ^ ‖ Φ − v * ‖ C 0 ( H ^ ) ‖ Φ ( 0 ) − v * ‖ ≤ − δ 1 ‖ Φ ( 0 ) − v * ‖ V ^ 2 + c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ ‖ Φ ( 0 ) − v * ‖ V ^ 2 + γ − 1 L G ^ | ‖ Φ ( 0 ) − v * ‖ | V ^ 2 .

By (41) and (42),

− ( A ( Φ ( 0 ) − v * ) , Φ ( 0 ) − v * ) − ( B ( ϕ ( 0 ) , Φ ( 0 ) ) − B ( u * , v * ) , Φ ( 0 ) − v * ) − ( N ( Φ ( 0 ) ) − N ( v * ) , Φ ( 0 ) − v * ) − ( J ( Φ ( 0 ) − v * ) , Φ ( 0 ) − v * ) + ( G ( t , Φ ) − G ( t , v * ) , Φ ( 0 ) − v * ) ≤ ( − δ 1 + c 3 γ − 1 ‖ F ‖ δ 1 − γ − 1 L G ^ + γ − 1 L G ^ ) ‖ Φ ( 0 ) − v * ‖ V ^ 2 < 0.

Thus (35) holds. By Lemma 2, Theorem 5 follows. □

 4.3. Polynomial Stability: Proportional Delay

In this subsection, we consider delay functions satisfying the following condition.

Condition 3. Let

G ( t , ξ ) = G ^ ( ξ ( − ( 1 − γ 0 ) t ) ) ,     ( t , ξ ) ∈ ℝ + × C 0 ( H ^ ) ,

where γ 0 ∈ ( 0 , 1 ) , G ^ : H ^ → H ^ is a Lipschitz map with Lipschitz constant L G ^ and G ^ ( 0 ) = 0 .

Remark 6. From Remark 3, the delay term satisfying Condition 3 satisfies Assumption 1.

We first recall the following important result.

Lemma 3. ([10][29]) Consider the initial value problem for the Pantograph equation

{ x ′ ( t ) = a x ( t ) + b x ( γ 0 t ) ,     ∀ t ≥ 0 ,     γ 0 ∈ ( 0 , 1 ) , x ( 0 ) = x 0 .

Let a < 0 , b ∈ ℝ , and x be the solution of (45). Then there exists C = C ( a , b , γ 0 ) > 0 such that

| x ( t ) | ≤ C | x ( 0 ) | ( 1 + t ) μ ′ ,     t ≥ 0 ,

where μ ′ ∈ ℝ satisfies 0 = a + | b | γ 0 μ ′ .

Theorem 6. Suppose Condition 3 holds, F ≡ 0 , δ 1 > γ − 1 L G ^ . Then for any Φ ∈ C 0 ( H ^ ) , the initial value problem (14)-(15) has a unique trivial stationary solution v * = 0 and a unique weak solution v ( t ) , and there exist C = C ( δ 1 , L G ^ , γ 0 ) > 0 , μ ′ < 0 such that

‖ v ( t ) ‖ 2 < C ‖ Φ ( 0 ) ‖ 2 ( 1 + t ) μ ′ ,     ∀ t ≥ 0.

Proof. By F ≡ 0 , δ 1 > γ − 1 L G ^ , Condition 3 and Theorem 2(a)-(b), the initial value problem (14)-(15) has a unique trivial stationary solution and a unique weak solution v ( t ) . Taking the duality product of (14) with v ( t ) yields

1 2 d d t ‖ v ( t ) ‖ 2 + 〈 A v ( t ) , v ( t ) 〉 + 〈 B ( u ( t ) , v ( t ) ) , v ( t ) 〉 + 〈 N ( v ( t ) ) , v ( t ) 〉 + 〈 J ( v ( t ) ) , v ( t ) 〉 = ( G ^ ( v ( γ 0 t ) ) , v ( t ) ) ,

Using (8), (10), the nonnegativity of N , and Condition 3, together with the Cauchy inequality, we obtain

d d t ‖ v ( t ) ‖ 2 + 2 δ 1 ‖ v ( t ) ‖ V ^ 2 ≤ 2 L G ^ ‖ v ( γ 0 t ) ‖ ‖ v ( t ) ‖ ≤ L G ^ ‖ v ( γ 0 t ) ‖ 2 + L G ^ ‖ v ( t ) ‖ 2 ,

By (7),

d d t ‖ v ( t ) ‖ 2 + 2 γ δ 1 ‖ v ( t ) ‖ 2 ≤ L G ^ ‖ v ( γ 0 t ) ‖ 2 + L G ^ ‖ v ( t ) ‖ 2 ,

which simplifies to

d d t ‖ v ( t ) ‖ 2 ≤ ( − 2 γ δ 1 + L G ^ ) ‖ v ( t ) ‖ 2 + L G ^ ‖ v ( γ 0 t ) ‖ 2 .

Since δ 1 > γ − 1 L G ^ , we have − 2 γ δ 1 + L G ^ < 0 . By Lemma 2 and (47), there exist C = C ( δ 1 , L G ^ , γ 0 ) > 0 and μ ′ ∈ ℝ such that

‖ v ( t ) ‖ 2 ≤ C ‖ Φ ( 0 ) ‖ 2 ( 1 + t ) μ ′ ,

where μ ′ satisfies

− 2 δ 1 γ + L G ^ + L G ^ γ 0 μ ′ = 0 ,

Solving,

μ ′ = log γ 0 ( 2 δ 1 γ L G ^ − 1 ) .

Since δ 1 γ > L G ^ , we have 2 δ 1 γ L G ^ − 1 > 1 , and with γ 0 ∈ ( 0 , 1 ) , it follows that μ ′ < 0 . This completes the proof of Theorem 6. □

 5. Conclusion and Outlook

At present, the proposed method is limited to two-dimensional bounded domains, and the admissible unbounded delay functions are required to possess certain specific structures. Future work will focus on extending the analysis to three-dimensional problems, considering more general boundary conditions (e.g., free boundaries), investigating stochastic perturbations, and establishing stability results for broader classes of delay functions.

 References Zhao, C., Zhang, Y., Caraballo, T. and Łukaszewicz, G. (2023) Statistical Solutions and Degenerate Regularity for the Micropolar Fluid with Generalized Newton Constitutive Law. MathematicalMethodsintheAppliedSciences, 46, 10311-10331. https://doi.org/10.1002/mma.9123 10.1002/mma.9123https://doi.org/10.1002/mma.9123 Zhao, C. Zhang, Y. Caraballo, T. 2023 Statistical Solutions and Degenerate Regularity for the Micropolar Fluid with Generalized Newton Constitutive Law Mathematical Methods in the Applied Sciences 46 10.1002/mma.9123 de Araújo, G.M., de Araújo, M.A.F. and Lucena, E.F.L. (2015) On a System of Equations of a Non-Newtonian Micropolar Fluid. JournalofAppliedMathematics, 2015, 1-11. https://doi.org/10.1155/2015/481754 10.1155/2015/481754https://doi.org/10.1155/2015/481754 Lucena, E.F.L. 2015 On a System of Equations of a Non-Newtonian Micropolar Fluid Journal of Applied Mathematics 2015 10.1155/2015/481754 Ai, C. and Tan, Z. (2021) Pullback Exponential Attractors for a Class of Non-Newtonian Micropolar Fluids. JournalofMathematicalAnalysisandApplications, 503, Article 125320. https://doi.org/10.1016/j.jmaa.2021.125320 10.1016/j.jmaa.2021.125320https://doi.org/10.1016/j.jmaa.2021.125320 Ai, C. Tan, Z. 2021 Pullback Exponential Attractors for a Class of Non-Newtonian Micropolar Fluids Journal of Mathematical Analysis and Applications 503 125320 10.1016/j.jmaa.2021.125320 Chen, X., Yang, H., Han, X. and Zhao, C. (2024) Existence and Degenerate Regularity of Trajectory Statistical Solution for the 3D Incompressible Non-Newtonian Micropolar Fluids. DiscreteandContinuousDynamicalSystems— S, 17, 1101-1118. https://doi.org/10.3934/dcdss.2023122 10.3934/dcdss.2023122https://doi.org/10.3934/dcdss.2023122 Chen, X. Yang, H. Han, X. Zhao, C. 2024 Existence and Degenerate Regularity of Trajectory Statistical Solution for the 3D Incompressible Non-Newtonian Micropolar Fluids Discrete and Continuous Dynamical Systems—S 17 10.3934/dcdss.2023122 Manitius, A. (1984) Feedback Controllers for a Wind Tunnel Model Involving a Delay: Analytical Design and Numerical Simulation. IEEETransactionsonAutomaticControl, 29, 1058-1068. https://doi.org/10.1109/tac.1984.1103436 10.1109/tac.1984.1103436https://doi.org/10.1109/tac.1984.1103436 Manitius, A. 1984 Feedback Controllers for a Wind Tunnel Model Involving a Delay: Analytical Design and Numerical Simulation IEEE Transactions on Automatic Control 29 10.1109/tac.1984.1103436 Zhao, C., Zhou, S. and Li, Y. (2009) Existence and Regularity of Pullback Attractors for an Incompressible Non-Newtonian Fluid with Delays. Quarterly of AppliedMathematics, 67, 503-540. https://doi.org/10.1090/s0033-569x-09-01146-2 10.1090/s0033-569x-09-01146-2https://doi.org/10.1090/s0033-569x-09-01146-2 Zhao, C. Zhou, S. Li, Y. 2009 Existence and Regularity of Pullback Attractors for an Incompressible Non-Newtonian Fluid with Delays Quarterly of Applied Mathematics 67 10.1090/s0033-569x-09-01146-2 Park, J.Y. and Jeong, J.U. (2016) Pullback Attractors for a 2D-Non-Autonomous Incompressible Non-Newtonian Fluid with Variable Delays. DiscreteandContinuousDynamicalSystems— B, 21, 2687-2702. https://doi.org/10.3934/dcdsb.2016068 10.3934/dcdsb.2016068https://doi.org/10.3934/dcdsb.2016068 Park, J.Y. Jeong, J.U. 2016 Pullback Attractors for a 2D-Non-Autonomous Incompressible Non-Newtonian Fluid with Variable Delays Discrete and Continuous Dynamical Systems—B 21 10.3934/dcdsb.2016068 Liu, L., Caraballo, T. and Fu, X. (2017) Dynamics of a Non-Autonomous Incompressible Non-Newtonian Fluid with Delay. DynamicsofPartialDifferentialEquations, 14, 375-402. https://doi.org/10.4310/dpde.2017.v14.n4.a4 10.4310/dpde.2017.v14.n4.a4https://doi.org/10.4310/dpde.2017.v14.n4.a4 Liu, L. Caraballo, T. Fu, X. 2017 Dynamics of a Non-Autonomous Incompressible Non-Newtonian Fluid with Delay Dynamics of Partial Differential Equations 14 10.4310/dpde.2017.v14.n4.a4 López-Lázaro, H.L., Marín-Rubio, P. and Planas, G. (2024) Non-Newtonian Incompressible Fluids with Nonlinear Shear Tensor and Hereditary Conditions. CommunicationsinNonlinearScienceandNumericalSimulation, 138, Article 108204. https://doi.org/10.1016/j.cnsns.2024.108204 10.1016/j.cnsns.2024.108204https://doi.org/10.1016/j.cnsns.2024.108204 Rubio, P. Planas, G. 2024 Non-Newtonian Incompressible Fluids with Nonlinear Shear Tensor and Hereditary Conditions Communications in Nonlinear Science and Numerical Simulation 138 108204 10.1016/j.cnsns.2024.108204 Liu, L., Caraballo, T. and Fu, X. (2018) Exponential Stability of an Incompressible Non-Newtonian Fluid with Delay. Discrete&ContinuousDynamicalSystems— B, 23, 4285-4303. https://doi.org/10.3934/dcdsb.2018138 10.3934/dcdsb.2018138https://doi.org/10.3934/dcdsb.2018138 Liu, L. Caraballo, T. Fu, X. 2018 Exponential Stability of an Incompressible Non-Newtonian Fluid with Delay Discrete & Continuous Dynamical Systems—B 23 10.3934/dcdsb.2018138 Liu, G., Xu, H. and Zhao, C. (2023) Upper Semi-Continuity of Pullback Attractors for Bipolar Fluids with Delay. ElectronicResearchArchive, 31, 5996-6011. https://doi.org/10.3934/era.2023305 10.3934/era.2023305https://doi.org/10.3934/era.2023305 Liu, G. Xu, H. Zhao, C. 2023 Upper Semi-Continuity of Pullback Attractors for Bipolar Fluids with Delay Electronic Research Archive 31 10.3934/era.2023305 Liu, G., Xu, H. and Zhao, C. (2024) Existence and Regularity of Pullback Attractors for a Non-Newtonian Fluid with Delays in 2D Unbounded Domains. EvolutionEquationsandControlTheory, 13, 382-399. https://doi.org/10.3934/eect.2023050 10.3934/eect.2023050https://doi.org/10.3934/eect.2023050 Liu, G. Xu, H. Zhao, C. 2024 Existence and Regularity of Pullback Attractors for a Non-Newtonian Fluid with Delays in 2D Unbounded Domains Evolution Equations and Control Theory 13 10.3934/eect.2023050 Zhao, C. and Sun, W. (2017) Global Well-Posedness and Pullback Attractors for a Two-Dimensional Non-Autonomous Micropolar Fluid Flows with Infinite Delays. CommunicationsinMathematicalSciences, 15, 97-121. https://doi.org/10.4310/cms.2017.v15.n1.a5 10.4310/cms.2017.v15.n1.a5https://doi.org/10.4310/cms.2017.v15.n1.a5 Zhao, C. Sun, W. 2017 Global Well-Posedness and Pullback Attractors for a Two-Dimensional Non-Autonomous Micropolar Fluid Flows with Infinite Delays Communications in Mathematical Sciences 15 10.4310/cms.2017.v15.n1.a5 Liu, G., Yi, L. and Zhao, C. (2024) The Stability of a Class of 2D Non-Newtonian Fluid Equations with Unbounded Delays. ApplicableAnalysis, 103, 2224-2240. https://doi.org/10.1080/00036811.2023.2291504 10.1080/00036811.2023.2291504https://doi.org/10.1080/00036811.2023.2291504 Liu, G. Yi, L. Zhao, C. 2024 The Stability of a Class of 2D Non-Newtonian Fluid Equations with Unbounded Delays Applicable Analysis 103 10.1080/00036811.2023.2291504 Sun, W. and Liu, G. (2019) Pullback Attractor for the 2D Micropolar Fluid Flows with Delay on Unbounded Domains. BulletinoftheMalaysianMathematicalSciencesSociety, 42, 2807-2833. https://doi.org/10.1007/s40840-018-0634-9 10.1007/s40840-018-0634-9https://doi.org/10.1007/s40840-018-0634-9 Sun, W. Liu, G. 2019 Pullback Attractor for the 2D Micropolar Fluid Flows with Delay on Unbounded Domains Bulletin of the Malaysian Mathematical Sciences Society 42 10.1007/s40840-018-0634-9 Sun, W. (2020) The Boundedness and Upper Semicontinuity of the Pullback Attractors for a 2D Micropolar Fluid Flows with Delay. ElectronicResearchArchive, 28, 1343-1356. https://doi.org/10.3934/era.2020071 10.3934/era.2020071https://doi.org/10.3934/era.2020071 Sun, W. 2020 The Boundedness and Upper Semicontinuity of the Pullback Attractors for a 2D Micropolar Fluid Flows with Delay Electronic Research Archive 28 10.3934/era.2020071 Sun, W. (2018) Micropolar Fluid Flows with Delay On 2D Unbounded Domains. JournalofAppliedAnalysis&Computation, 8, 356-378. https://doi.org/10.11948/2018.356 10.11948/2018.356https://doi.org/10.11948/2018.356 Sun, W. 2018 Micropolar Fluid Flows with Delay On 2D Unbounded Domains Journal of Applied Analysis & Computation 8 10.11948/2018.356 Zhou, G., Liu, G. and Sun, W. (2017) H2-Boundedness of the Pullback Attractor of the Micropolar Fluid Flows with Infinite Delays. BoundaryValueProblems, 2017, Article No. 133. https://doi.org/10.1186/s13661-017-0866-x 10.1186/s13661-017-0866-xhttps://doi.org/10.1186/s13661-017-0866-x Zhou, G. Liu, G. Sun, W. 2017 H2-Boundedness of the Pullback Attractor of the Micropolar Fluid Flows with Infinite Delays Boundary Value Problems 2017 No 10.1186/s13661-017-0866-x Bellout, H., Bloom, F. and Nečas, J. (1994) Young Measure‐Valued Solutions for Non-Newtonian Incompressible Fluids. Communications in Partial DifferentialEquations, 19, 1763-1803. https://doi.org/10.1080/03605309408821073 10.1080/03605309408821073https://doi.org/10.1080/03605309408821073 Bellout, H. Bloom, F. 1994 Young Measure‐Valued Solutions for Non-Newtonian Incompressible Fluids Communications in Partial Differential Equations 19 10.1080/03605309408821073 Bellout, H., Bloom, F. and Nečas, J. (1995) Existence, Uniqueness, and Stability of Solutions to the Initial-Boundary Value Problem for Bipolar Viscous Fluids. DifferentialandIntegralEquations, 8, 225-464. https://doi.org/10.57262/die/1369083480 10.57262/die/1369083480https://doi.org/10.57262/die/1369083480 Bellout, H. Bloom, F. Existence, U 1995 Existence, Uniqueness, and Stability of Solutions to the Initial-Boundary Value Problem for Bipolar Viscous Fluids Differential and Integral Equations 8 10.57262/die/1369083480 Bloom, F. and Hao, W. (2001) Regularization of a Non-Newtonian System in an Unbounded Channel: Existence and Uniqueness of Solutions. NonlinearAnalysis: Theory, Methods&Applications, 44, 281-309. https://doi.org/10.1016/s0362-546x(99)00264-3 10.1016/s0362-546x(99)00264-3https://doi.org/10.1016/s0362-546x(99)00264-3 Bloom, F. Hao, W. Theory, M 2001 Regularization of a Non-Newtonian System in an Unbounded Channel: Existence and Uniqueness of Solutions Nonlinear Analysis: Theory 44 10.1016/s0362-546x(99)00264-3 Bloom, F. and Hao, W. (2001) Regularization of a Non-Newtonian System in an Unbounded Channel: Existence of a Maximal Compact Attractor. NonlinearAnalysis: Theory, Methods&Applications, 43, 743-766. https://doi.org/10.1016/s0362-546x(99)00232-1 10.1016/s0362-546x(99)00232-1https://doi.org/10.1016/s0362-546x(99)00232-1 Bloom, F. Hao, W. Theory, M 2001 Regularization of a Non-Newtonian System in an Unbounded Channel: Existence of a Maximal Compact Attractor Nonlinear Analysis: Theory 43 10.1016/s0362-546x(99)00232-1 Pokorný, M. (1996) Cauchy Problem for the Non-Newtonian Viscous Incompressible Fluid. ApplicationsofMathematics, 41, 169-201. https://doi.org/10.21136/am.1996.134320 10.21136/am.1996.134320https://doi.org/10.21136/am.1996.134320 1996 Cauchy Problem for the Non-Newtonian Viscous Incompressible Fluid Applications of Mathematics 41 10.21136/am.1996.134320 Bellout, H., Bloom, F. and Nečas, J. (1992) Phenomenological Behavior of Multipolar Viscous Fluids. QuarterlyofAppliedMathematics, 50, 559-583. https://doi.org/10.1090/qam/1178435 10.1090/qam/1178435https://doi.org/10.1090/qam/1178435 Bellout, H. Bloom, F. 1992 Phenomenological Behavior of Multipolar Viscous Fluids Quarterly of Applied Mathematics 50 10.1090/qam/1178435 Málek, J., Nečas, J., Rokyta, M. and Ružička, M. (1996) Weak and Measure-Valued Solutions to Evolutionary PDE. Champman-Hall. Rokyta, M. 1996 Weak and Measure-Valued Solutions to Evolutionary PDE Boyer, F. and Fabrie, P. (2012) Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer. Boyer, F. Fabrie, P. 2012 Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models Yi, L. and Liu, G. (2025) Stability of Stationary Solutions to Micropolar Fluid Equations with Unbounded Delay. Applied Mathematics and Mechanics, 46, 551-562. https://doi.org/10.21656/1000-0887.450300 10.21656/1000-0887.450300https://doi.org/10.21656/1000-0887.450300 Yi, L. Liu, G. 2025 Stability of Stationary Solutions to Micropolar Fluid Equations with Unbounded Delay Applied Mathematics and Mechanics 46 10.21656/1000-0887.450300 Liu, L., Caraballo, T. and Marín-Rubio, P. (2018) Stability Results for 2D Navier-Stokes Equations with Unbounded Delay. Journal of Differential Equations, 265, 5685-5708. https://doi.org/10.1016/j.jde.2018.07.008 10.1016/j.jde.2018.07.008https://doi.org/10.1016/j.jde.2018.07.008 Liu, L. Caraballo, T. Rubio, P. 2018 Stability Results for 2D Navier-Stokes Equations with Unbounded Delay Journal of Differential Equations 265 10.1016/j.jde.2018.07.008 Caraballo, T. and Han, X. (2015) A Survey on Navier-Stokes Models with Delays: Existence, Uniqueness and Asymptotic Behavior of Solutions. DiscreteandContinuousDynamicalSystems— S, 8, 1079-1101. https://doi.org/10.3934/dcdss.2015.8.1079 10.3934/dcdss.2015.8.1079https://doi.org/10.3934/dcdss.2015.8.1079 Caraballo, T. Han, X. Existence, U 2015 A Survey on Navier-Stokes Models with Delays: Existence, Uniqueness and Asymptotic Behavior of Solutions Discrete and Continuous Dynamical Systems—S 8 10.3934/dcdss.2015.8.1079