Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain

Fang-hong Zhang

doi:10.1515/math-2025-0152

Article Open Access

Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain

Fang-hong Zhang

Published/Copyright: June 2, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this article, we investigate the long-time behavior for the ill-posed problems

∂ 2 u ∂ t 2 + ∂ u ∂ t + λ u − Δ u − Δ ∂ u ∂ t − Δ ∂ 2 u ∂ t 2 = f ( t , u ( x , t − ρ ( t ) ) ) + g ( t , x ) , in ( τ , + ∞ ) × R N ,

with some hereditary characteristics. First, we establish the existence of solutions for the second-order non-autonomous evolution equation by the standard Faedo-Galerkin methods, but without any Lipschitz conditions on the nonlinear term f ( ⋅ ) . Then, by proving the D -pullback asymptotically upper-semicompact property for the multivalued process { U ( t , τ ) } , we establish the existence of pullback attractors A C H 1 ( R N ) , H 1 ( R N ) in the Banach spaces C H 1 ( R N ) , H 1 ( R N ) for the multi-valued process generated by a class of second-order non-autonomous evolution equations with delays and ill-posedness.

Keywords: non-autonomous semi-linear second-order evolution; pullback attractors; variable delays; multi-valued process

MSC 2010: 35B41; 35Q35; 35B40; 37L05

1 Introduction

In this article, we consider the following non-autonomous semi-linear second-order evolution equation with delays:

(1.1) ∂ 2 u ∂ t 2 + ∂ u ∂ t + λ u − Δ u − Δ ∂ u ∂ t − ν Δ ∂ 2 u ∂ t 2 = f ( t , u ( x , t − ρ ( t ) ) ) + g ( t , x ) , in ( τ , + ∞ ) × R N , u ( t , x ) = ϕ ( t − τ , x ) , t ∈ [ τ − h , τ ] , x ∈ R N , ∂ u ( t , x ) ∂ t = ∂ ϕ ( t − τ , x ) ∂ t , t ∈ [ τ − h , τ ] , x ∈ R N ,

where λ > 0 , h > 0 , τ is the initial time, and ϕ is the initial data on the interval [ τ − h , τ ] .

The nonlinearity f ( ⋅ ) and the external force g ( t , x ) satisfy the following conditions, respectively.

( A 1 ) There exist a function α 1 ( ⋅ ) ∈ L 2 ( R N ) , and a positive constant α 2 such that the functions f ∈ C ( R × R N ; R ) , ρ ∈ C 1 ( R ; [ 0 , h ] ) satisfy

(1.2) ∣ f ( t , v ) ∣ 2 ≤ ∣ α 1 ( x ) ∣ 2 + α 2 2 ∣ v ∣ 2 , ∀ t ∈ R , v ∈ R N ;

(1.3) ∣ ρ ′ ( t ) ∣ ≤ ρ * < 1 , ∀ t ∈ R ;

( A 2 ) the external force g ( t , x ) belongs to the space L loc 2 ( R , L 2 ( R N ) ) such that

(1.4) ∫ − ∞ τ ∫ R N e δ r ∣ g ( r , ⋅ ) ∣ 2 d r d x < ∞ , ∀ τ ∈ R , δ > 0 ,

which implies that

(1.5) lim K → ∞ ∫ − ∞ τ ∫ ∣ x ∣ ≥ K e δ r ∣ g ( r , ⋅ ) ∣ 2 d r d x = 0 , ∀ τ ∈ R , δ > 0 .

When ν = 0 and without variable delays, equation (1.1) becomes the usual strongly damped wave equation

(1.6) ∂ 2 u ∂ t 2 − Δ u − Δ ∂ u ∂ t = f ( u ) + g ( t , x ) .

Its asymptotic behavior has been studied extensively in terms of attractors (see [1–9] and references therein). The longtime behavior for the strongly damped wave equation with delays has been investigated in [10,11].

For each fixed ν > 0 and without variable delays, equation (1.1) becomes

(1.7) ∂ 2 u ∂ t 2 − Δ u − Δ ∂ u ∂ t − ν Δ ∂ 2 u ∂ t 2 = f ( u ) + g ( t , x ) .

The nonlinear wave equation (1.7) is a mathematical model that describes the spread of longitudinal strain waves in nonlinear elastic rods and weakly nonlinear ion-acoustic waves, and it is a special form of the so-called improved Boussinesq equation (see, e.g., [12–15] and the references therein). The terms − Δ ∂ 2 u ∂ t 2 and − Δ ∂ u ∂ t are called the dispersive and the viscosity dissipative terms, respectively. The damped term − Δ ∂ u ∂ t was used to describe ion-sound waves in plasma by Makhankov [13,16] and is also known to represent other sorts of “propagation problems” of, for example, lengthway waves in nonlinear elastic rods and ion-sonic waves of space transformations by a weak nonlinear effect, and the term − Δ ∂ 2 u ∂ t 2 introduces a dispersion effect to describe the rotational inertia or nonloc interactions [12–16].

In bounded domains, the longtime behavior, especially the global attractor and exponential attractors, has been extensively studied by many researchers (see, e.g., [17–21] and references therein). For instance, when the nonlinearity f satisfies the subcritical growth, Carvalho and Cholewa [17] presented systematic results including the existence-uniqueness and long-time behavior by using the semigroup approach. For the critical exponential growth nonlinearity, Xie and Zhong [18,19] investigated the existence of global attractors using the new method named “Condition C”; Sun et al. [20] studied the asymptotic regularity of the solutions and obtained the existence of exponential attractors by applying the decomposition technique. For the (non-autonomous) semi-linear second-order evolution equation (1.7) with memory terms, Zhang [21] constructed the existence of a robust family of exponential attractors when the nonlinearity is critical.

On unbounded domain, up to now, there are few results. Jones and Wang [22] applied the cut-off method and a decomposition trick to obtain the existence of a random attractor for the stochastic second-order evolution equations

(1.8) u t t + α u t + λ u − Δ u − Δ u t − β Δ u t t + f ( x , u ) = g ( x ) + h ( x ) d w d t ,

with subcritical nonlinearity. In our previous work [23], the existence of ( H l u 1 ( R N ) × H l u 1 ( R N ) , H ρ 1 ( R N ) × H ρ 1 ( R N ) ) -global attractor is established for the following second-order evolution equations with dispersive and dissipative terms:

(1.9) u t t − Δ u − Δ u t − β Δ u t t + α u t + λ u + f ( u ) = g ( x ) , in R N × R + ,

in locally uniform spaces with critical exponent.

Indeed, for all the aforementioned results, we required the solution operator

S ( t ) : u 0 ↦ u ( t )

to be well-defined and continuous in a proper phase space (we can define U ( t , τ ) = S ( t − τ ) for the non-autonomous case). However, for many interesting problems, the well-posedness of the solution operator S ( t ) is not known or does not hold (e.g., see [11,24–27] and references therein).

The evolution equations with a time delay arise from evolution phenomena in physics, biology, and engineering when after-effect, time lag, or pre-history influence is taken into account in the model of the systems: in this case, the response of a system depends not only on the current state of the system, but also on the past history of the system (we refer the readers to [1,10,11,14,24–26,28–31] for more comments and citations).

As far as the author is concerned, the long-time dynamics for equation (1.1) with hereditary characteristics on the unbounded domain R N remains open. There are some barriers encountered.

Equation (1.1) contains the dispersive term − Δ ∂ 2 u ∂ t 2 and dissipative term − Δ ∂ u ∂ t , which essentially distinguishes it from the damped (or strong damped) wave equation in [1,3–8,10,11]. For example, the wave equation has some smoothing effect, e.g., although the initial data only belong to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. However, for equation (1.1), if the initial data ϕ ∈ C H 1 ( R N ) , H 1 ( R N ) , then the solution u t ( ⋅ ) is always in C H 1 ( R N ) , H 1 ( R N ) and has no higher regularity, it will cause some difficulties.
Since we do not assume any Lipschitz conditions on the nonlinear term f ( ⋅ ) , the uniqueness of the weak solutions for equation (1.1) is lost, i.e., we need to overcome some difficulties brought by ill-posedness.
The compact Sobolev embeddings on the unbounded domain R N are not available.
The delay term f ( t , u ( x , t − ρ ( t ) ) ) also causes some difficulties to obtain the pullback attractors.

We overcome the aforementioned difficulties (less regularity, lack of compactness, the equation itself, delay term) and establish the existence of weak solutions in C H 1 ( R N ) , H 1 ( R N ) (Theorem 3.1). Additionally, we prove the existence of D C H 1 ( R N ) , H 1 ( R N ) -pullback attractor for the multi-valued process { U ( t , τ ) } corresponding to the second-order evolution equations with dispersive and dissipative terms equation (1.1) in C H 1 ( R N ) , H 1 ( R N ) (Theorem 4.1).

This article is organized as follows: in Section 2, we introduce some notations and function spaces, and we recall some useful results on non-autonomous multi-valued dynamical systems and pullback attractors. In Section 3, we prove the existence of solutions for equation (1.1) in C H 1 ( R N ) , H 1 ( R N ) . The existence of the pullback attractor for the multi-valued process { U ( t , τ ) } corresponding to equation (1.1) is proved in Section 4.

2 Preliminaries

To make the article more self-contained, we recall some definitions and abstract results concerning the multi-valued dynamical systems and pullback attractor, which are necessary to obtain our main results. For more details, see [10,11,24,28–31].

Let X be a complete metric space with metric d X ( ⋅ , ⋅ ) , and let P ( X ) be the class of nonempty subsets of X . Denote by H X semi ( ⋅ , ⋅ ) the Hausdorff semidistance between two nonempty subsets of X , which is defined by

H X semi ( A , B ) = sup a ∈ A inf b ∈ B d X ( a , b ) .

Definition 2.1

A family of mappings U ( t , τ ) : X → P ( X ) , t ≥ τ , τ ∈ R , is called to be a multi-valued process if

U ( τ , τ ) x = { x } , ∀ τ ∈ R , x ∈ X ;

U ( t , r ) U ( r , τ ) x = U ( t , τ ) x , for all τ ≤ r ≤ t , x ∈ X .

Let D be a nonempty class of parameterized sets D = { D ( t ) } t ∈ R ⊂ P ( X ) .

Definition 2.2

A collection D of some families of nonempty closed subsets of X is said to be inclusion-closed if for each D = { D ( t ) } t ∈ R ∈ D ,

{ D ˜ ( t ) : D ˜ ( t ) is a nonempty subset of D ( t ) , ∀ t ∈ R }

also belongs to D .

Definition 2.3

Let { U ( t , τ ) } be a multi-valued process on X .

Q = { Q ( t ) } t ∈ R ∈ D is called a D -pullback absorbing set for { U ( t , τ ) } if for any ℬ = { B ( t ) } t ∈ R ∈ D and each t ∈ R , there exists a t 0 = t 0 ( ℬ , t ) ∈ R + such that
U ( t , t − s ) B ( t − s ) ⊂ Q ( t ) , ∀ s ≥ t 0 ;
{ U ( t , τ ) } is said to be D -pullback asymptotically upper-semicompact in X with respect to ℬ if for each fixed t ∈ R , any sequence y n ∈ U ( t , t − s n ) x n has a convergent subsequence in X whenever s n → + ∞ ( n → ∞ ) , x n ∈ B ( t − s n ) with ℬ = { B ( t ) } t ∈ R ∈ D .

Theorem 2.1

A family of nonempty compact subsets A = { A ( t ) } t ∈ R ∈ D of X is called to be a D -pullback attractor for the multi-valued process { U ( t , τ ) } if

A = { A ( t ) } t ∈ R is invariant, i.e.,
U ( t , τ ) A ( τ ) = A ( t ) , ∀ t ≥ τ , τ ∈ R ;
A attracts every member of D , i.e., for every ℬ = { B ( t ) } t ∈ R ∈ D and any fixed t ∈ R ,
lim s → + ∞ H X semi ( U ( t , t − s ) B , A ( t ) ) = 0 .

We need the following result to check that the multi-valued process { U ( t , τ ) } on X is D -pullback asymptotically upper-semicompact [11].

Theorem 2.2

Let { U ( t , τ ) } be a multi-valued process on Banach space X, and let Q = { Q ( t ) } t ∈ R be a D -pullback absorbing set for { U ( t , τ ) } in D . Suppose that U can be written as

U ( t , τ ) = U 1 ( t , τ ) + U 2 ( t , τ ) , ∀ t ≥ τ ,

and for any fixed t ∈ R ,

lim s → ∞ ‖ U 1 ( t , t − s ) Q ( t − s ) ‖ X = 0 ;
for any fixed s > 0 , every sequence y n ∈ U 2 ( t , t − s ) Q ( t − s ) is a Cauchy sequence in X.

Then, { U ( t , τ ) } is D -pullback asymptotically upper-semicompact in X.

The following result for the existence of D -pullback attractor can be found in the studies of Caraballo et al. [10,28–30] and Wang et al. [11].

Theorem 2.3

Let D be an inclusion-closed collection of some families of nonempty closed subsets of X and { U ( t , τ ) } be a multi-valued process on X. Also, U has a closed value, and let U ( t , τ ) x be upper semi-continuous in x for fixed t ≥ τ , τ ∈ R . Suppose that { U ( t , τ ) } is D -pullback asymptotically upper-semicompact in X, { U ( t , τ ) } has a D -pullback absorbing set Q = { Q ( t ) } t ∈ R in D , and Q ( t ) is closed for every t ∈ R . Then, the D -pullback attractor A = { A ( t ) } t ∈ R is unique and is given by, for each t ∈ R ,

A ( t ) = ⋂ T ∈ R + ⋃ s ≥ T U ( t , t − s ) Q ( t − s ) ¯ .

Let H = L 2 ( R N ) with norm ∣ ⋅ ∣ and inner product ( ⋅ , ⋅ ) , and let V = H 1 ( R N ) with norm ‖ ⋅ ‖ and associated scalar product ( ( ⋅ , ⋅ ) ) . Let X be a Banach space with norm ‖ ⋅ ‖ X , and let h > 0 be a given positive number, which will denote the delay time, and let C X denote the Banach space C 0 ( [ − h , 0 ] ; X ) with the sup-norm

‖ ψ ‖ C X ≔ sup s ∈ [ − h , 0 ] ‖ ψ ( s ) ‖ X , for ψ ∈ C X .

We denote by C X , X the Banach spaces C X ∩ C 1 ( [ − h , 0 ] ; X ) with the norm ‖ ⋅ ‖ C X , X defined by

‖ ψ ‖ C X , X 2 ≔ ‖ ψ ‖ C X 2 + ‖ ψ ′ ‖ C X 2 , for ψ ∈ C X , X .

Given τ ∈ R , T > τ and u : [ τ − h , T ) → X , for each t ∈ [ τ , T ) , u t : [ − h , 0 ] → X denotes the function defined as

u t ( s ) = u ( t + s ) , s ∈ [ − h , 0 ] .

Without loss of generality, we assume that ν = 1 in the following discussion.

3 Multi-valued processes

In this section, we want to construct the multi-valued evolution processes corresponding to equation (1.1). The existence of solutions can be obtained by the standard Faedo-Galerkin methods (see [1,32–37], and references therein). Here, we only give the sketch of proof, and the details are similar to those in the proof of Theorem 3.1 in [32], Sec. XV.3, and the arguments in [37] Section IV. 4.4.

Theorem 3.1

Suppose that ( A 1 ) – ( A 2 ) hold true, g ∈ L loc 2 ( R ; H ) and ϕ ∈ C V , V . Then, there is a solution u ( t ) of equation (1.1) such that

u ∈ C ( [ τ − h , T ] ; V ) , ∂ u ∂ t ∈ C ( [ τ − h ] ; V ) , ∀ T > τ .

Proof

(Sketch) We divide the proof into three steps:

Step 1. By the a priori estimate given in Lemma 4.1, and integrating (4.6) from τ to t , we have

(3.1) ∣ v ( t ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( t ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( t ) ∣ 2 + ∣ ∇ v ( t ) ∣ 2 + ( 2 − 2 δ − ε 1 − ε 2 ) ∫ τ t ∣ v ( s ) ∣ 2 d s + 2 δ ( λ − δ + δ 2 ) ∫ τ t ∣ u ( s ) ∣ 2 d s + 2 δ ( 1 − δ + δ 2 ) ∫ τ t ∣ ∇ u ( s ) ∣ 2 d s + 2 ( 1 − δ ) ∫ τ t ∣ ∇ v ( s ) ∣ 2 d s ≤ ∣ v ( τ ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( τ ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( τ ) ∣ 2 + ∣ ∇ v ( τ ) ∣ 2 + α 2 2 ε 1 ∫ τ t ∣ u ( s − ρ ( s ) ) ∣ 2 d s + ∣ α 1 ∣ 2 ε 1 ( t − τ ) + 1 ε 2 ∫ τ t ∣ g ( s ) ∣ 2 d s .

Note that ρ ( s ) ∈ [ 0 , h ] and the fact

1 1 − ρ ′ ( s ) ≤ 1 1 − ρ * ,

for all s ∈ R .

Setting r = s − ρ ( s ) , we arrive at

(3.2) α 2 2 ε 1 ∫ τ t ∣ u ( s − ρ ( s ) ) ∣ 2 d s ≤ α 2 2 ε 1 ( 1 − ρ * ) ∫ τ − h τ ∣ u ( r ) ∣ 2 d r + ∫ τ t ∣ u ( r ) ∣ 2 d r .

Hence,

(3.3) ∣ v ( t ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( t ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( t ) ∣ 2 + ∣ ∇ v ( t ) ∣ 2 + ( 2 − 2 δ − ε 1 − ε 2 ) ∫ τ t ∣ v ( s ) ∣ 2 d s + 2 δ ( λ − δ + δ 2 ) ∫ τ t ∣ u ( s ) ∣ 2 d s + 2 δ ( 1 − δ + δ 2 ) ∫ τ t ∣ ∇ u ( s ) ∣ 2 d s + 2 ( 1 − δ ) ∫ τ t ∣ ∇ v ( s ) ∣ 2 d s ≤ ∣ v ( τ ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( τ ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( τ ) ∣ 2 + ∣ ∇ v ( τ ) ∣ 2 + α 2 2 ε 1 ( 1 − ρ * ) ∫ τ − h τ ∣ u ( r ) ∣ 2 d r + ∫ τ t ∣ u ( r ) ∣ 2 d r + ∣ α 1 ∣ 2 ε 1 ( t − τ ) + 1 ε 2 ∫ τ t ∣ g ( s ) ∣ 2 d s .

Step 2. We consider the Dirichlet problem in a bounded domain Ω 2 K

(3.4) ∂ 2 u ∂ t 2 + ∂ u ∂ t + λ u − Δ u − Δ ∂ u ∂ t − Δ ∂ 2 u ∂ t 2 = f ( t , u ( x , t − ρ ( t ) ) ) + g ( t , x ) , i n ( τ , t ) × Ω 2 K , u ∣ ∂ Ω 2 K = 0 , t > τ , u ( t , x ) = ϕ ( t − τ , x ) , ∂ u ( t , x ) ∂ t = ∂ ϕ 2 K ( t − τ , x ) ∂ t , t ∈ [ τ − h , τ ] , x ∈ Ω 2 K ,

where K is a positive integer and Ω 2 K = { x ∈ R N : ∣ x ∣ ≤ 2 K } ,

(3.5) ϕ 2 K ( t , x ) = ϕ ( t , x ) 1 − ξ 2 ∣ x ∣ 2 K 2 , ∀ t ∈ [ − h , 0 ] ,

where ξ ( ⋅ ) is the cutoff function defined in the proof Lemma 4.3.

Let H 2 K = L 2 ( Ω 2 K ) and V 2 K = H 1 ( Ω 2 K ) . Denote A u = − Δ u for any u ∈ D ( A ) , where D ( A ) = { u ∈ V 2 K : A u ∈ H 2 K } . Since A is self-adjoint, positive operator and has a compact inverse, there exists a complete set of eigenvectors { ω i } i = 1 ∞ in H 2 K , and the corresponding eigenvalues { λ i } i = 1 ∞ satisfy

A ω i = λ i ω i , 0 < λ 1 ≤ λ 2 ≤ … ≤ λ i ≤ … → + ∞ , i → + ∞ .

Setting V 2 K m = span { ω 1 , ω 2 , … , ω m } , and P m is the orthogonal projection,

P m u = ∑ i = 1 m ( u , ω i ) ω i , u ∈ H 2 K .

We consider the approximate solutions of equation (3.4) of the form

u m ( t ) = ∑ j = 1 m α j m ( t ) w j ,

then, u m ( t ) satisfies

(3.6) ∂ 2 u m ∂ t 2 + ∂ u m ∂ t + λ u m − Δ u m − Δ ∂ u m ∂ t − Δ ∂ 2 u m ∂ t 2 = f ( t , u m ( x , t − ρ ( t ) ) ) + P m g ( t , x ) , u m ( t , x ) = P m ϕ ( t − τ , x ) , t ∈ [ τ − h , τ ] , ∂ u m ( t , x ) ∂ t = ∂ P m ϕ ( t − τ , x ) ∂ t , t ∈ [ τ − h , τ ] .

By the integral form of the Gronwall lemma, if follows from (3.3) that

(3.7) { u m , u m ′ } is a bounded set of L ∞ ( τ − h , T ; V 2 K × V 2 K ) as m → ∞ .

Thus, we can extract a subsequence, still denoted m , such that

(3.8) u m → u , in L ∞ ( τ − h , T ; V 2 K ) weak-star , as m → ∞ ,

and

(3.9) u m ′ → u ′ , in L ∞ ( τ − h , T ; V 2 K ) weak-star, as m → ∞ .

Furthermore,

(3.10) u m → u , in L 2 ( Ω 2 K × [ τ − h , T ] ) strongly ,

and

(3.11) u m → u , for almost every ( t , x ) ∈ [ τ − h , T ] × Ω 2 K .

Note that f ∈ C ( R × R N ; R ) , then

(3.12) f ( u m ) → f ( u ) , in L 2 ( τ − h , T ; V 2 K ) weakly .

We then pass the limit in equation (3.6), and we can find that u ( t ) is a solution of equation (3.4) such that

(3.13) u ∈ L ∞ ( τ − h , T ; V 2 K ) and u ′ ∈ L ∞ ( τ − h , T ; V 2 K ) .

The continuity properties

u ∈ C ( [ τ − h , T ] ; V 2 K ) , ∂ u ∂ t ∈ C ( [ τ − h , T ] ; V 2 K ) , ∀ T > τ ,

can be established with the methods indicated in Sections II.3 and II.4 in Temam [37] (e.g., Theorems 3.1 and 3.2).

Step 3. Since Ω 2 K represents a sequence of bounded subdomains of R N and Ω 2 K → R N as K → ∞ , similar to the approximation argument of Theorem 5 in [38], we infer the existence of weak solutions associated with equation (1.1) as

u ∈ C ( [ τ − h , T ] ; V ) , ∂ u ∂ t ∈ C ( [ τ − h , T ] ; V ) , ∀ T > τ .

This completes the proof.□

Remark 3.2

According to Theorem 3.1, we can define a family of multi-valued mappings { U ( t , τ ) } on C V , V

U ( t , τ ) : C V , V → C V , V ,

corresponding to equation (1.1) by

U ( t , τ ) ϕ = { u t ( ⋅ ; τ , ϕ ) ∣ u ( ⋅ ) is a solution of equation (1.1) with ϕ ∈ C V , V } .

Then, { U ( t , τ ) } is a multi-valued process on C V , V .

4 Pullback attractors in C V , V

4.1 Existence of D -pullback absorbing set

We denote by R the set of all functions r : R → ( 0 , + ∞ ) such that

(4.1) lim t → − ∞ e δ ′ t r 2 ( t ) = 0 ,

where δ ′ > 0 will be defined in Lemma 4.1, and denote by D C V , V the class of all families D = { D ( t ) } t ∈ R ⊂ P ( C V , V ) such that D ( t ) ⊂ N ¯ ( 0 , r D ( t ) ) , for some r D ∈ R , where P ( C V , V ) denotes the family of all nonempty subsets of C V , V and N ¯ ( 0 , r D ( t ) ) denotes the closed ball in C V , V centered at zero with radius r D ( t ) .

Lemma 4.1

(Existence of D -pullback absorbing set) Suppose that ( A 1 ) – ( A 2 ) hold true, g ∈ L loc 2 ( R ; H ) , ϕ ∈ C V , V . Then, the multi-valued process { U ( t , τ ) } possesses a D C V , V -pullback absorbing set Q C V , V in D C V , V .

Proof

Let v = u ′ + δ u ( 0 < δ < 1 3 ) , where u ′ = d d t u , and we rewrite equation (1.1) as

(4.2) d d t v + ( 1 − δ ) v + ( λ − δ + δ 2 ) u − ( 1 − δ + δ 2 ) Δ u − ( 1 − δ ) Δ v − d d t Δ v = f ( t , u ( t − ρ ( t ) ) ) + g ( t , x ) .

Multiplying equation (4.2) by v in L 2 ( R N ) , we infer

(4.3) 1 2 d d t ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) + ( 1 − δ ) ∣ v ∣ 2 + δ ( λ − δ + δ 2 ) ∣ u ∣ 2 + δ ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ( 1 − δ ) ∣ ∇ v ∣ 2 = ( f ( t , u ( t − ρ ( t ) ) ) , v ) + ( g ( t , x ) , v ) .

Noting that (1.2), using Young’s inequality, for ε 1 , ε 2 > 0 , we have

(4.4) ∣ ( f ( t , u ( t − ρ ( t ) ) ) , v ) ∣ ≤ ε 1 2 ∣ v ∣ 2 + α 2 2 2 ε 1 ∣ u ( t − ρ ( t ) ) ∣ 2 + ∣ α 1 ∣ 2 2 ε 1

and

(4.5) ∣ ( g ( t , x ) , v ) ∣ ≤ ε 2 2 ∣ v ∣ 2 + 1 2 ε 2 ∣ g ( t , x ) ∣ 2 .

Then, it follows that

(4.6) d d t ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) + ( 2 − 2 δ − ε 1 − ε 2 ) ∣ v ∣ 2 + 2 δ ( λ − δ + δ 2 ) ∣ u ∣ 2 + 2 δ ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + 2 ( 1 − δ ) ∣ ∇ v ∣ 2 ≤ α 2 2 ε 1 ∣ u ( t − ρ ( t ) ) ∣ 2 + ∣ α 1 ∣ 2 ε 1 + 1 ε 2 ∣ g ( t , x ) ∣ 2 .

Let δ ′ > 0 be determined later on, and we infer

(4.7) d d t ( e δ ′ t ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) ) = δ ′ e δ ′ t ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) + e δ ′ t d d t ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) ≤ e δ ′ t ( ( δ ′ + 2 δ + ε 1 + ε 2 − 2 ) ∣ v ∣ 2 + ( δ ′ − 2 δ ) ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( δ ′ − 2 δ ) ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ( δ ′ + 2 δ − 2 ) ∣ ∇ v ∣ 2 ) + α 2 2 ε 1 e δ ′ t ∣ u ( t − ρ ( t ) ) ∣ 2 + ∣ α 1 ∣ 2 ε 1 e δ ′ t + 1 ε 2 e δ ′ t ∣ g ( t , x ) ∣ 2 .

Now integrating (4.7) from τ to t , thus

(4.8) e δ ′ t ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) ≤ e δ ′ τ ( ∣ v ( τ ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( τ ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( τ ) ∣ 2 + ∣ ∇ v ( τ ) ∣ 2 ) + ( δ ′ + 2 δ + ε 1 + ε 2 − 2 ) ∫ τ t e δ ′ s ∣ v ( s ) ∣ 2 d s + ( δ ′ + 2 δ − 2 ) ∫ τ t e δ ′ s ∣ ∇ v ( s ) ∣ 2 d s + ( δ ′ − 2 δ ) ( λ − δ + δ 2 ) ∫ τ t e δ ′ s ∣ u ( s ) ∣ 2 d s + ( δ ′ − 2 δ ) ( 1 − δ + δ 2 ) ∫ τ t e δ ′ s ∣ ∇ u ( s ) ∣ 2 d s + α 2 2 ε 1 ∫ τ t e δ ′ s ∣ u ( s − ρ ( s ) ) ∣ 2 d s + ∣ α 1 ∣ 2 ε 1 ∫ τ t e δ ′ s d s + 1 ε 2 ∫ τ t e δ ′ s ∣ g ( s , x ) ∣ 2 d s .

Note that ρ ( s ) ∈ [ 0 , h ] and the fact

1 1 − ρ ′ ( s ) ≤ 1 1 − ρ * ,

for all s ∈ R .

Setting r = s − ρ ( s ) , we arrive at

(4.9) α 2 2 ε 1 ∫ τ t e δ ′ s ∣ u ( s − ρ ( s ) ) ∣ 2 d s ≤ α 2 2 e δ ′ h ε 1 ( 1 − ρ * ) ∫ τ − h τ e δ ′ r ∣ u ( r ) ∣ 2 d r + ∫ τ t e δ ′ r ∣ u ( r ) ∣ 2 d r ≤ α 2 2 e δ ′ ( h + τ ) h ‖ ϕ ‖ C V 2 ε 1 ( 1 − ρ * ) + α 2 2 e δ ′ h ε 1 ( 1 − ρ * ) ∫ τ t e δ ′ r ∣ u ( r ) ∣ 2 d r .

Thus, we obtain that

(4.10) e δ ′ t ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) ≤ e δ ′ τ ( ∣ v ( τ ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( τ ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( τ ) ∣ 2 + ∣ ∇ v ( τ ) ∣ 2 ) + ( δ ′ + 2 δ + ε 1 + ε 2 − 2 ) ∫ τ t e δ ′ s ∣ v ( s ) ∣ 2 d s + ( δ ′ + 2 δ − 2 ) ∫ τ t e δ ′ s ∣ ∇ v ( s ) ∣ 2 d s + ( δ ′ − 2 δ ) ( λ − δ + δ 2 ) − α 2 2 e δ ′ h ε 1 ( 1 − ρ * ) ∫ τ t e δ ′ s ∣ u ( s ) ∣ 2 d s + ( δ ′ − 2 δ ) ( 1 − δ + δ 2 ) × ∫ τ t e δ ′ s ∣ ∇ u ( s ) ∣ 2 d s + 1 ε 1 δ ′ ∣ α 1 ∣ 2 e δ ′ t + 1 ε 2 ∫ τ t e δ ′ s ∣ g ( s , x ) ∣ 2 d s + α 2 2 e δ ′ ( h + τ ) h ‖ ϕ ‖ C V 2 ε 1 ( 1 − ρ * ) .

Choosing ε 1 = ε 2 = δ and δ ′ = 2 δ , and noting that 0 < δ < 1 3 , then

δ ′ + 2 δ + ε 1 + ε 2 − 2 < 0 , δ ′ + 2 δ − 2 < 0 , ( δ ′ − 2 δ ) ( λ − δ + δ 2 ) − α 2 2 e δ ′ h ε 1 ( 1 − ρ * ) < 0 , ( δ ′ − 2 δ ) ( 1 − δ + δ 2 ) = 0 .

Denote μ = min { 1 , λ − δ + δ 2 , 1 − δ + δ 2 } and ν = max { 1 , λ − δ + δ 2 , 1 − δ + δ 2 } . This yields

(4.11) e 2 δ t ( ∣ v ∣ 2 + ∣ u ∣ 2 + ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) ≤ ν μ e 2 δ τ ( ∣ v ( τ ) ∣ 2 + ∣ u ( τ ) ∣ 2 + ∣ ∇ u ( τ ) ∣ 2 + ∣ ∇ v ( τ ) ∣ 2 ) + 1 2 μ δ 2 ∣ α 1 ∣ 2 e 2 δ t + 1 μ δ ∫ τ t e 2 δ s ∣ g ( s , x ) ∣ 2 d s + α 2 2 e 2 δ ( h + τ ) h ‖ ϕ ‖ C V 2 μ δ ( 1 − ρ * ) .

Setting now t + θ instead of t , where θ ∈ [ − h , 0 ] , multiplying by e − 2 δ ( t + θ ) , we obtain that

(4.12) ∣ v ( t + θ ) ∣ 2 + ∣ u ( t + θ ) ∣ 2 + ∣ ∇ u ( t + θ ) ∣ 2 + ∣ ∇ v ( t + θ ) ∣ 2 ≤ ν μ e − 2 δ ( t + θ ) e 2 δ τ ( ∣ v ( τ ) ∣ 2 + ∣ u ( τ ) ∣ 2 + ∣ ∇ u ( τ ) ∣ 2 + ∣ ∇ v ( τ ) ∣ 2 ) + 1 2 μ δ 2 ∣ α 1 ∣ 2 + 1 μ δ e − 2 δ ( t + θ ) ∫ τ t e 2 δ s ∣ g ( s , x ) ∣ 2 d s + α 2 2 e 2 δ ( h + τ ) h ‖ ϕ ‖ C V 2 μ δ ( 1 − ρ * ) e − 2 δ ( t + θ ) .

Noting that v = u ′ + δ u , by (4.12), we infer that

(4.13) ‖ u t ‖ C V , V 2 = ‖ u t ‖ C V 2 + ‖ u t ′ ‖ C V 2 ≤ sup θ ∈ [ − h , 0 ] ‖ u ( t + θ ) ‖ 2 + sup θ ∈ [ − h , 0 ] ‖ ∇ u ( t + θ ) ‖ 2 + sup θ ∈ [ − h , 0 ] ‖ u ′ ( t + θ ) ‖ 2 + sup θ ∈ [ − h , 0 ] ‖ ∇ u ′ ( t + θ ) ‖ 2 ≤ ( 1 + 2 δ 2 ) sup θ ∈ [ − h , 0 ] ‖ ∇ u ( t + θ ) ‖ 2 + ( 1 + 2 δ 2 ) sup θ ∈ [ − h , 0 ] ‖ u ( t + θ ) ‖ 2 + 2 sup θ ∈ [ − h , 0 ] ‖ ∇ v ( t + θ ) ‖ 2 + 2 sup θ ∈ [ − h , 0 ] ‖ v ( t + θ ) ‖ 2 ≤ 2 ν μ e − 2 δ t e 2 δ τ + 2 δ h ( ∣ v ( τ ) ∣ 2 + ∣ u ( τ ) ∣ 2 + ∣ ∇ u ( τ ) ∣ 2 + ∣ ∇ v ( τ ) ∣ 2 ) + 1 μ δ 2 ∣ α 1 ∣ 2 + 2 e 2 δ h μ δ e − 2 δ t ∫ τ t e 2 δ s ∣ g ( s , x ) ∣ 2 d s + 2 α 2 2 e 2 δ ( 2 h + τ ) h ‖ ϕ ‖ C V 2 μ δ ( 1 − ρ * ) e − 2 δ t ≤ C 1 e − 2 δ t + C 2 + C 3 e − 2 δ t ∫ − ∞ t e 2 δ s ∣ g ( s , x ) ∣ 2 d s ,

where

C 1 = 2 ν μ e 2 δ τ + 2 δ h ( ∣ v ( τ ) ∣ 2 + ∣ u ( τ ) ∣ 2 + ∣ ∇ u ( τ ) ∣ 2 + ∣ ∇ v ( τ ) ∣ 2 ) + 2 α 2 2 e 2 δ ( 2 h + τ ) h ‖ ϕ ‖ C V 2 μ δ ( 1 − ρ * ) ,

C 2 = 1 μ δ 2 ∣ α 1 ∣ 2 , and C 3 = 2 e 2 δ h μ δ .

Now, we denote by R ( t ) the nonnegative number given for each t ∈ R by

R 2 ( t ) = C + C e − 2 δ t ∫ − ∞ t e 2 δ s ∣ g ( x , s ) ∣ 2 d s ,

and consider the family of closed bounded balls Q C V , V = { Q ( t ) } t ∈ R in C V , V defined by

Q ( t ) = { φ ∈ C V , V : ‖ φ ‖ C V , V ≤ R ( t ) } .

Clearly, Q C V , V ∈ D C V , V , and moreover, according to (4.1) and (4.13), the family of Q C V , V is D C V , V -pullback absorbing for the multi-valued process { U ( t , τ ) } on C V , V .

This completes the proof.□

4.2 A priori estimates

Lemma 4.2

Suppose that ( A 1 ) – ( A 2 ) hold true, g ∈ L loc 2 ( R ; H ) and ϕ ∈ C V , V . Then,

(4.14) ∣ v ′ ( t ) ∣ 2 + ∣ ∇ v ′ ( t ) ∣ 2 ≤ C ,

where C is dependent on ∣ α 1 ∣ , α 2 , ‖ ϕ ‖ C V , V , and ∣ g ( x , t ) ∣ .

Proof

Multiplying equation (4.2) by v ′ = d d t v in L 2 ( R N ) , we infer

(4.15) ∣ v ′ ∣ 2 + ∣ ∇ v ′ ∣ 2 = − ( 1 − δ ) ⟨ v , v ′ ⟩ − ( λ − δ + δ 2 ) ⟨ u , v ′ ⟩ − ( 1 − δ + δ 2 ) ⟨ ∇ u , ∇ v ′ ⟩ − ( 1 − δ ) ⟨ ∇ v , ∇ v ′ ⟩ + ⟨ f ( t , u ( t − ρ ( t ) ) ) , v ′ ⟩ + ⟨ g ( t , x ) , v ′ ⟩ .

By Young’s inequality, we have

(4.16) ∣ − ( 1 − δ ) ⟨ v , v ′ ⟩ ∣ ≤ ε ∣ v ′ ∣ 2 + C δ , ε ∣ v ∣ 2 ,

(4.17) ∣ − ( λ − δ + δ 2 ) ⟨ u , v ′ ⟩ ∣ ≤ ε ∣ v ′ ∣ 2 + C λ , δ , ε ∣ u ∣ 2 ,

(4.18) ∣ − ( 1 − δ + δ 2 ) ⟨ ∇ u , ∇ v ′ ⟩ ∣ ≤ ε ∣ ∇ v ′ ∣ 2 + C δ , ε ∣ ∇ u ∣ 2 .

By (1.2), we infer

(4.19) ∣ ( f ( t , u ( t − ρ ( t ) ) ) , v ′ ) ∣ ≤ ε ∣ v ′ ∣ 2 + C ε α 2 2 ‖ u ‖ C V , V 2 + C ε ∣ α 1 ∣ 2

and

(4.20) ∣ ( g ( t , x ) , v ′ ) ∣ ≤ ε ∣ v ′ ∣ 2 + C ε ∣ g ( t , x ) ∣ 2 .

It follows from (4.15)–(4.20) and from combining (4.11) with (4.13) that

(4.21) ∣ v ′ ∣ 2 + ∣ ∇ v ′ ∣ 2 ≤ C ∣ α 1 ∣ , α 2 , ‖ ϕ ‖ C V , V + C ∣ g ( t , x ) ∣ 2 + C ∫ − ∞ t e 2 δ s ∣ g ( s , x ) ∣ 2 d s .

This completes the proof.□

We now establish the following skillful estimates, which are crucial for proving that the multi-valued process is D C V , V -pullback asymptotically upper-semicompact.

Lemma 4.3

Under the assumptions of Lemma 4.1, for any t ∈ R , any ε > 0 , and every ℬ = { B ( t ) } t ∈ R ∈ D C V , V , there exist T = T ( ε , t , ℬ ) > 0 and K = K ( ε , t , ℬ ) > 0 such that for any solution u t ∈ U ( t , t − s ) B ( t − s ) satisfies

(4.22) sup θ ∈ [ − h , 0 ] ∫ Ω K C ( ∣ ∇ u ′ ( t + θ ) ∣ 2 + ∣ u ′ ( t + θ ) ∣ 2 + ∣ ∇ u ( t + θ ) ∣ 2 + ∣ u ( t + θ ) ∣ 2 ) d x ≤ ε , ∀ t ≥ T ,

where Ω K C = { x ∈ R N : ∣ x ∣ ≥ K } .

Proof

Choose a smooth function ξ ( s ) (see Remark 4.4) with

(4.23) ξ ( s ) = 0 , 0 ≤ s ≤ 1 , 1 , s ≥ 2 ,

where 0 ≤ ξ ( s ) ≤ 1 , s ∈ R + , and for which there is a constant c such that

(4.24) ∣ ξ ′ ( s ) ∣ + ∣ ξ ″ ( s ) ∣ ≤ c , ∀ s ∈ R + .

Multiplying (4.2) by ξ 2 ∣ x ∣ 2 K 2 v = ξ 2 ∣ x ∣ 2 K 2 ( u ′ + δ u ) ( 0 < δ < 1 3 ) , where u ′ = d d t u , we obtain

(4.25) d d t v , ξ 2 ∣ x ∣ 2 K 2 v + ( 1 − δ ) v , ξ 2 ∣ x ∣ 2 K 2 v + ( λ − δ + δ 2 ) u , ξ 2 ∣ x ∣ 2 K 2 v − ( 1 − δ + δ 2 ) Δ u , ξ 2 ∣ x ∣ 2 K 2 v − ( 1 − δ ) Δ v , ξ 2 ∣ x ∣ 2 K 2 v − d d t Δ v , ξ 2 ∣ x ∣ 2 K 2 v = f ( t , u ( t − ρ ( t ) ) ) , ξ 2 ∣ x ∣ 2 K 2 v + g ( t , x ) , ξ 2 ∣ x ∣ 2 K 2 v .

Next, we deal with each term of (4.25) one by one as follows:

(4.26) d d t v , ξ 2 ∣ x ∣ 2 K 2 v = 1 2 d d t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x ,

(4.27) ( 1 − δ ) v , ξ 2 ∣ x ∣ 2 K 2 v = ( 1 − δ ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x ,

(4.28) ( λ − δ + δ 2 ) u , ξ 2 ∣ x ∣ 2 K 2 v = λ − δ + δ 2 2 d d t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ∣ 2 d x + δ ( λ − δ + δ 2 ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ∣ 2 d x ,

(4.29) − ( 1 − δ + δ 2 ) Δ u , ξ 2 ∣ x ∣ 2 K 2 v = 1 − δ + δ 2 2 d d t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ u ∣ 2 d x + ( 1 − δ + δ 2 ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 u ′ ∇ u d x + δ ( 1 − δ + δ 2 ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ u ∣ 2 d x + δ ( 1 − δ + δ 2 ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 u ∇ u d x ,

(4.30) − ( 1 − δ ) Δ v , ξ 2 ∣ x ∣ 2 K 2 v = ( 1 − δ ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ v ∣ 2 d x + ( 1 − δ ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 v ∇ v d x ,

(4.31) − d d t Δ v , ξ 2 ∣ x ∣ 2 K 2 v = 1 2 d d t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ v ∣ 2 d x + ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 v ∇ v ′ d x .

By (1.2), using Young’s inequality, for ε 1 , ε 2 > 0 , we have

(4.32) f ( t , u ( t − ρ ( t ) ) ) , ξ 2 ∣ x ∣ 2 K 2 v ≤ ε 1 2 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x + α 2 2 2 ε 1 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ( t − ρ ( t ) ) ∣ 2 d x + 1 2 ε 1 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x

and

(4.33) g ( x , t ) , ξ 2 ∣ x ∣ 2 K 2 v ≤ ε 2 2 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x + 1 2 ε 2 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( x , t ) ∣ 2 d x .

It follows from (4.25)–(4.33) that

(4.34) 1 2 d d t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x + 1 − δ − ε 1 2 − ε 2 2 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x + δ ( λ − δ + δ 2 ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ∣ 2 d x + δ ( 1 − δ + δ 2 ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ u ∣ 2 d x + ( 1 − δ ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ v ∣ 2 d x ≤ − ( 1 − δ + δ 2 ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 u ′ ∇ u d x − δ ( 1 − δ + δ 2 ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 u ∇ u d x − ( 1 − δ ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 v ∇ v d x − ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 v ∇ v ′ d x + α 2 2 2 ε 1 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ( t − ρ ( t ) ) ∣ 2 d x + 1 2 ε 1 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x + 1 2 ε 2 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( x , t ) ∣ 2 d x .

Noting that ∣ ξ ′ ( s ) ∣ ≤ c , we have

(4.35) − ( 1 − δ + δ 2 ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 u ′ ∇ u d x ≤ ( 1 − δ + δ 2 ) ∫ K ≤ ∣ x ∣ ≤ 2 K 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 u ′ ∇ u d x ≤ C K ∫ K ≤ ∣ x ∣ ≤ 2 K ξ ∣ x ∣ 2 K 2 ( ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 ) d x ≤ C K ( ∣ u ′ ∣ 2 + ‖ u ‖ 2 ) .

Similar to (4.35), we infer

(4.36) − δ ( 1 − δ + δ 2 ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 u ∇ u d x ≤ C K ( ∣ u ∣ 2 + ‖ u ‖ 2 ) ,

(4.37) − ( 1 − δ ) ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 v ∇ v d x ≤ C K ( ∣ v ∣ 2 + ‖ v ‖ 2 ) .

By Lemmas 4.1 and 4.2, similar to (4.35), we have

(4.38) − ∫ R N 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 v ∇ v ′ d x ≤ C K ( ∣ v ∣ 2 + ‖ v ′ ‖ 2 ) .

Thus, we obtain that

(4.39) d d t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x + ( 2 − 2 δ − ε 1 − ε 2 ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x + 2 δ ( λ − δ + δ 2 ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ∣ 2 d x + 2 δ ( 1 − δ + δ 2 ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ u ∣ 2 d x + 2 ( 1 − δ ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ v ∣ 2 d x ≤ C K ( ∣ u ∣ 2 + ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 + ∣ v ∣ 2 + ∣ ∇ v ∣ 2 + ∣ ∇ v ′ ∣ 2 ) + α 2 2 ε 1 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ( t − ρ ( t ) ) ∣ 2 d x + 1 ε 1 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x + 1 ε 2 ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( t , x ) ∣ 2 d x .

Let δ ′ > 0 be determined later on. We infer

(4.40) d d t e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x = δ ′ e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x + e δ ′ t d d t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x ≤ ( δ ′ + 2 δ + ε 1 + ε 2 − 2 ) e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x + ( δ ′ − 2 δ ) ( λ − δ + δ 2 ) e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ∣ 2 d x + ( δ ′ − 2 δ ) ( 1 − δ + δ 2 ) e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ u ∣ 2 d x + ( δ ′ + 2 δ − 2 ) e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ v ∣ 2 d x + C K e δ ′ t ( ∣ u ∣ 2 + ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 + ∣ v ∣ 2 + ∣ ∇ v ∣ 2 + ∣ ∇ v ′ ∣ 2 ) + α 2 2 ε 1 e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ( t − ρ ( t ) ) ∣ 2 d x + 1 ε 1 e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x + 1 ε 2 e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( t , x ) ∣ 2 d x .

Integrating over [ t − s , t ] , we infer that

(4.41) e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x ≤ e δ ′ ( t − s ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ( t − s ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( t − s ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( t − s ) ∣ 2 + ∣ ∇ v ( t − s ) ∣ 2 ) d x + ( δ ′ + 2 δ + ε 1 + ε 2 − 2 ) ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x d s + ( δ ′ − 2 δ ) ( λ − δ + δ 2 ) ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ∣ 2 d x d s + ( δ ′ − 2 δ ) ( 1 − δ + δ 2 ) ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ u ∣ 2 d x d s + ( δ ′ + 2 δ − 2 ) ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ v ∣ 2 d x d s + C K ∫ t − s t e δ ′ s ( ∣ u ∣ 2 + ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 + ∣ v ∣ 2 + ∣ ∇ v ∣ 2 + ∣ ∇ v ′ ∣ 2 ) d s + α 2 2 ε 1 ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ( s − ρ ( s ) ) ∣ 2 d x d s + 1 ε 1 ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x d s + 1 ε 2 ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( s , x ) ∣ 2 d x d s .

Note that ρ ( s ) ∈ [ 0 , h ] and the fact

1 1 − ρ ′ ( s ) ≤ 1 1 − ρ * ,

for all s ∈ R .

Setting r = s − ρ ( s ) , we arrive at

(4.42) α 2 2 ε 1 ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ( s − ρ ( s ) ) ∣ 2 d x d s ≤ α 2 2 e δ ′ h ε 1 ( 1 − ρ * ) ∫ t − s − h t e δ ′ r ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ( r ) ∣ 2 d r d x ≤ α 2 2 e δ ′ h e δ ′ ( t − s ) h ‖ ϕ ‖ V , V 2 ε 1 ( 1 − ρ * ) + α 2 2 e δ ′ h ε 1 ( 1 − ρ * ) ∫ t − s t e δ ′ r ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ( r ) ∣ 2 d r d x .

Thus, we infer that

(4.43) e δ ′ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x ≤ e δ ′ ( t − s ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ( t − s ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( t − s ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( t − s ) ∣ 2 + ∣ ∇ v ( t − s ) ∣ 2 ) d x + ( δ ′ + 2 δ + ε 1 + ε 2 − 2 ) ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ v ∣ 2 d x d s + ( δ ′ − 2 δ ) ( λ − δ + δ 2 ) − α 2 2 e δ ′ h ε 1 ( 1 − ρ * ) ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ u ∣ 2 d x d s + ( δ ′ − 2 δ ) ( 1 − δ + δ 2 ) ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ u ∣ 2 d x d s + ( δ ′ + 2 δ − 2 ) ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ ∇ v ∣ 2 d x d s + C K ∫ t − s t e δ ′ s ( ∣ u ∣ 2 + ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 + ∣ v ∣ 2 + ∣ ∇ v ∣ 2 + ∣ ∇ v ′ ∣ 2 ) d s + 1 ε 1 ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x d s + 1 ε 2 ∫ t − s t e δ ′ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( s , x ) ∣ 2 d x d s + α 2 2 e δ ′ h e δ ′ ( t − s ) h ‖ ϕ ‖ V , V 2 ε 1 ( 1 − ρ * ) .

Choosing ε 1 = ε 2 = δ and δ ′ = 2 δ , and noting that 0 < δ < 1 3 , then

Thus,

(4.44) e 2 δ t ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ∣ 2 + ( λ − δ + δ 2 ) ∣ u ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ∣ 2 + ∣ ∇ v ∣ 2 ) d x ≤ e 2 δ ( t − s ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ( t − s ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( t − s ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( t − s ) ∣ 2 + ∣ ∇ v ( t − s ) ∣ 2 ) d x + C K ∫ t − s t e 2 δ s ( ∣ u ∣ 2 + ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 + ∣ v ∣ 2 + ∣ ∇ v ∣ 2 + ∣ ∇ v ′ ∣ 2 ) d x d s + 1 δ ∫ t − s t e 2 δ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x d s + 1 δ ∫ t − s t e 2 δ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( s , x ) ∣ 2 d x d s + α 2 2 e 2 δ h e 2 δ ( t − s ) h ‖ ϕ ‖ V , V 2 δ ( 1 − ρ * ) .

Setting now t + θ instead of t (where θ ∈ [ − h , 0 ] ), and multiplying by e − 2 δ ( t + θ ) , we find that

(4.45) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ( t + θ ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( t + θ ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( t + θ ) ∣ 2 + ∣ ∇ v ( t + θ ) ∣ 2 ) d x ≤ e − 2 δ ( t + θ ) e 2 δ ( t − s ) ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ( t − s ) ∣ 2 + ( λ − δ + δ 2 ) ∣ u ( t − s ) ∣ 2 + ( 1 − δ + δ 2 ) ∣ ∇ u ( t − s ) ∣ 2 + ∣ ∇ v ( t − s ) ∣ 2 ) d x + C K e − 2 δ ( t + θ ) ∫ t − s t e 2 δ s ( ∣ u ∣ 2 + ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 + ∣ v ∣ 2 + ∣ ∇ v ∣ 2 + ∣ ∇ v ′ ∣ 2 ) d s + 1 δ e − 2 δ ( t + θ ) ∫ t − s t e 2 δ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x d s + 1 δ e − 2 δ ( t + θ ) ∫ t − s t e 2 δ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( s , x ) ∣ 2 d x d s + α 2 2 e 2 δ h e − 2 δ ( t + θ ) e 2 δ ( t − s ) h ‖ ϕ ‖ V , V 2 δ ( 1 − ρ * ) .

Similar to the arguments in equation (4.13), we have

(4.46) sup s ∈ [ − h , 0 ] ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ( t + θ ) ∣ 2 + ∣ u ( t + θ ) ∣ 2 + ∣ ∇ u ( t + θ ) ∣ 2 + ∣ ∇ v ( t + θ ) ∣ 2 ) d x ≤ C e − 2 δ s ( ∣ v ( t − s ) ∣ 2 + ∣ u ( t − s ) ∣ 2 + ∣ ∇ u ( t − s ) ∣ 2 + ∣ ∇ v ( t − s ) ∣ 2 ) + C K e − 2 δ t ∫ t − s t e 2 δ s ( ∣ u ∣ 2 + ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 + ∣ v ∣ 2 + ∣ ∇ v ∣ 2 + ∣ ∇ v ′ ∣ 2 ) d s + C e − 2 δ t ∫ t − s t e 2 δ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x d s + C e − 2 δ t ∫ t − s t e 2 δ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( s , x ) ∣ 2 d x d s + C e − 2 δ s ‖ ϕ ‖ V , V 2 .

Since ϕ ∈ B ( t − s ) and ℬ = { B ( t ) } t ∈ R ∈ D C V , V , we infer that

(4.47) limsup s → + ∞ C e − 2 δ s ( ∣ v ( t − s ) ∣ 2 + ∣ u ( t − s ) ∣ 2 + ∣ ∇ u ( t − s ) ∣ 2 + ∣ ∇ v ( t − s ) ∣ 2 ) + C e − 2 δ s ‖ ϕ ‖ V , V 2 = limsup s → + ∞ C e − 2 δ s ( ∣ v ( t − s ) ∣ 2 + ∣ u ( t − s ) ∣ 2 + ∣ ∇ u ( t − s ) ∣ 2 + ∣ ∇ v ( t − s ) ∣ 2 + ‖ ϕ ‖ V , V 2 ) ≤ limsup s → + ∞ C e − 2 δ s ‖ B ( t − s ) ‖ C V , V 2 = 0 .

Noting that α 1 ( ⋅ ) ∈ L 2 ( R N ) , by (1.4)–(1.5), we have

(4.48) C e − 2 δ t ∫ t − s t e 2 δ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ α 1 ∣ 2 d x d s < ε

and

(4.49) C e − 2 δ t ∫ t − s t e 2 δ s ∫ R N ξ 2 ∣ x ∣ 2 K 2 ∣ g ( s , x ) ∣ 2 d x d s < ε .

By (4.13), Lemma 4.2, and (4.47), when K and s sufficiently large, we infer that

(4.50) C K e − 2 δ t ∫ t − s t e 2 δ s ( ∣ u ∣ 2 + ∣ u ′ ∣ 2 + ∣ ∇ u ∣ 2 + ∣ v ∣ 2 + ∣ ∇ v ∣ 2 + ∣ ∇ v ′ ∣ 2 ) d s ≤ C K ≤ ε .

It follows from (4.46)–(4.50), when K and s are large enough, that

(4.51) sup θ ∈ [ − h , 0 ] ∫ Ω K C ( ∣ ∇ u ′ ( t + θ ) ∣ 2 + ∣ u ′ ( t + θ ) ∣ 2 + ∣ ∇ u ( t + θ ) ∣ 2 + ∣ u ( t + θ ) ∣ 2 ) d x ≤ sup θ ∈ [ − h , 0 ] ∫ R N ξ 2 ∣ x ∣ 2 K 2 ( ∣ v ( t + θ ) ∣ 2 + ∣ u ( t + θ ) ∣ 2 + ∣ ∇ u ( t + θ ) ∣ 2 + ∣ ∇ v ( t + θ ) ∣ 2 ) d x ≤ ε .

This completes the proof.□

Remark 4.4

Choose a smooth function ψ ( s ) with

ψ ( s ) = 0 , 0 ≤ s ≤ 1 , s − 1 , 1 ≤ s ≤ 2 , 1 , s > 2 ,

and

ψ δ ( s ) = ∫ R ρ δ ( s − r ) ψ ( r ) d r ,

where ρ δ ( s ) is the standard mollifier on R with supp ρ δ ⊂ [ − δ , δ ] .

From [1,34–37], for 0 < δ ≪ 1 , we infer that ψ δ ( ⋅ ) ∈ C ∞ ( R ) , 0 ≤ ψ δ ( ⋅ ) ≤ 1 , and

ξ ( s ) = ψ δ ( s ) = 0 , 0 ≤ s ≤ 1 , 1 , s ≥ 2 .

4.3 Pullback asymptotically upper-semicompact

Lemma 4.5

Under the assumptions of Lemma 4.1, the multi-valued process { U ( t , τ ) } on C V , V is D C V , V -pullback asymptotically upper-semicompact.

Proof

Note that for any T ≥ t − s with s ≥ 0 ,

U ( T , t − s ) ϕ = { u T ( ⋅ ; t − s , ϕ ) ∣ u ( ⋅ ) is a solution of equation (1.1) with ϕ ∈ Q ( t − s ) } ,

where Q C V , V = { Q ( t ) } t ∈ R is D C V , V -pullback absorbing for the multi-valued process { U ( t , τ ) } on C V , V .

Step 1. Decomposition of the equations

Let Ω 2 K = { x ∈ R N : ∣ x ∣ ≤ 2 K } . Let K ≥ 1 , and define

(4.52) u ˆ ( T , x ) = 1 − ξ 2 ∣ x ∣ 2 K 2 u ( T , x ) , ∀ T ≥ t − s − h , x ∈ R N ,

where ξ is the cutoff function defined in (4.23)–(4.24).

Multiplying equation (1.1) by 1 − ξ 2 ∣ x ∣ 2 K 2 , then

(4.53) ∂ 2 u ˆ ∂ T 2 + ∂ u ˆ ∂ T + λ u ˆ − Δ u ˆ − Δ ∂ u ˆ ∂ T − Δ ∂ 2 u ˆ ∂ T 2 = 1 − ξ 2 ∣ x ∣ 2 K 2 f ( T , u ( x , T − ρ ( T ) ) ) + 1 − ξ 2 ∣ x ∣ 2 K 2 g ( T , x ) + u Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ξ 2 ∣ x ∣ 2 K 2 ∇ u + ∂ u ∂ T Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ∂ u ∂ T ∇ ξ 2 ∣ x ∣ 2 K 2 + ∂ 2 u ∂ T 2 Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ∂ 2 u ∂ T 2 ∇ ξ 2 ∣ x ∣ 2 K 2 , u ˆ ( T , x ) = 1 − ξ 2 ∣ x ∣ 2 K 2 ϕ ( T − t + s , x ) , t − s − h ≤ T ≤ t − s , x ∈ Ω 2 K , ∂ u ˆ ( T , x ) ∂ T = 1 − ξ 2 ∣ x ∣ 2 K 2 ∂ ϕ ( T − t + s , x ) ∂ T , t − s − h ≤ T ≤ t − s , x ∈ Ω 2 K , u ˆ ( T , x ) = 0 , T ≥ t − s , ∣ x ∣ = 2 K .

Now, we decompose equation (4.53) as follows:

u ˆ ( T , x ) = w ˆ ( T , x ) + y ˆ ( T , x ) ,

where w ˆ ( T , x ) and y ˆ ( T , x ) satisfy the following equations, respectively:

(4.54) ∂ 2 w ˆ ∂ T 2 + ∂ w ˆ ∂ T + λ w ˆ − Δ w ˆ − Δ ∂ w ˆ ∂ T − Δ ∂ 2 w ˆ ∂ T 2 = 0 , w ˆ ( T , x ) = 1 − ξ 2 ∣ x ∣ 2 K 2 ϕ ( T − t + s , x ) , t − s − h ≤ T ≤ t − s , x ∈ Ω 2 K , ∂ w ˆ ( T , x ) ∂ T = 1 − ξ 2 ∣ x ∣ 2 K 2 ∂ ϕ ( T − t + s , x ) ∂ T , t − s − h ≤ T ≤ t − s , x ∈ Ω 2 K , w ˆ ∣ ∂ Ω = 0 , T ≥ t − s , ∣ x ∣ = 2 K .

and

(4.55) ∂ 2 y ˆ ∂ T 2 + ∂ y ˆ ∂ T + λ y ˆ − Δ y ˆ − Δ ∂ y ˆ ∂ T − Δ ∂ 2 y ˆ ∂ T 2 = 1 − ξ 2 ∣ x ∣ 2 K 2 f ( T , u ( x , T − ρ ( T ) ) ) + 1 − ξ 2 ∣ x ∣ 2 K 2 g ( T , x ) + u Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ξ 2 ∣ x ∣ 2 K 2 ∇ u + ∂ u ∂ T Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ∂ u ∂ T ∇ ξ 2 ∣ x ∣ 2 K 2 + ∂ 2 u ∂ T 2 Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ∂ 2 u ∂ T 2 ∇ ξ 2 ∣ x ∣ 2 K 2 , y ˆ ( T , x ) = 0 , t − s − h ≤ T ≤ t − s , x ∈ Ω 2 K , ∂ y ˆ ( t , x ) ∂ T = 0 , t − s − h ≤ T ≤ t − s , x ∈ Ω 2 K , y ˆ ∣ ∂ Ω = 0 , T ≥ t − s , ∣ x ∣ = 2 K .

Step 2. Verifying Theorem 2.2 (1)

For equation (4.54), similar to the proof of Theorem 4.1 (note that f = g = 0 ), we can easily obtain

(4.56) ‖ w ˆ t ‖ C V , V 2 = ‖ w ˆ t ‖ C V 2 + ‖ w ˆ t ′ ‖ C V 2 ≤ C e − 2 δ s ‖ ϕ ‖ C V , V 2 ,

i.e.,

lim s → ∞ ‖ U 1 ( t , t − s ) Q ( t − s ) ‖ C V , V = 0 .

Then, Theorem 2.2 (1) is proved.

Step 3. Verifying Theorem 2.2 (2)

We investigate two solutions of u T 1 and u T 2 for equation (1.1) corresponding to the initial data ϕ 1 and ϕ 2 , respectively. For equation (4.55), let z ( T ) = y 1 ( T ) − y 2 ( T ) , then z ( T ) satisfies

(4.57) ∂ 2 z ∂ T 2 + ∂ z ∂ T + λ z − Δ z − Δ ∂ z ∂ T − Δ ∂ 2 z ∂ T 2 = 1 − ξ 2 ∣ x ∣ 2 K 2 ( f ( T , u 1 ( x , T − ρ ( T ) ) ) − f ( T , u 2 ( x , T − ρ ( T ) ) ) ) + ( u 1 − u 2 ) Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ξ 2 ∣ x ∣ 2 K 2 ( ∇ u 1 − ∇ u 2 ) + ∂ u 1 ∂ T − ∂ u 2 ∂ T Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ∂ u 1 ∂ T − ∇ ∂ u 2 ∂ T ∇ ξ 2 ∣ x ∣ 2 K 2 + ∂ 2 u 1 ∂ T 2 − ∂ 2 u 2 ∂ T 2 Δ ξ 2 ∣ x ∣ 2 K 2 + 2 ∇ ∂ 2 u 1 ∂ T 2 − ∇ ∂ 2 u 2 ∂ T 2 ∇ ξ 2 ∣ x ∣ 2 K 2 , z ( T , x ) = 0 , t − s − h ≤ T ≤ t − s , x ∈ Ω 2 K , ∂ z ( t , x ) ∂ T = 0 , t − s − h ≤ T ≤ t − s , x ∈ Ω 2 K , z ∣ ∂ Ω = 0 , T ≥ t − s , ∣ x ∣ = 2 K .

Multiplying equation (4.57) by z ′ ( T ) , we infer

(4.58) 1 2 d d T ( ∣ ∇ z ′ ( T ) ∣ 2 + ∣ z ′ ( T ) ∣ 2 + ∣ ∇ z ∣ 2 + λ ∣ z ∣ 2 ) + ∣ ∇ z ′ ( T ) ∣ 2 = 1 − ξ 2 ∣ x ∣ 2 K 2 ( f ( T , u 1 ( x , T − ρ ( T ) ) ) − f ( T , u 2 ( x , T − ρ ( T ) ) ) , z ′ ( T ) ) + ( u 1 − u 2 ) Δ ξ 2 ∣ x ∣ 2 K 2 , z ′ ( T ) + 2 ∇ ξ 2 ∣ x ∣ 2 K 2 ( ∇ u 1 − ∇ u 2 ) , z ′ ( T ) + ∂ u 1 ∂ T − ∂ u 2 ∂ T Δ ξ 2 ∣ x ∣ 2 K 2 , z ′ ( T ) + 2 ∇ ∂ u 1 ∂ T − ∇ ∂ u 2 ∂ T ∇ ξ 2 ∣ x ∣ 2 K 2 , z ′ ( T ) + ∂ 2 u 1 ∂ T 2 − ∂ 2 u 2 ∂ T 2 Δ ξ 2 ∣ x ∣ 2 K 2 , z ′ ( T ) + 2 ∇ ∂ 2 u 1 ∂ T 2 − ∇ ∂ 2 u 2 ∂ T 2 ∇ ξ 2 ∣ x ∣ 2 K 2 , z ′ ( T ) .

Note that

(4.59) ∇ ξ 2 ∣ x ∣ 2 K 2 = 4 x K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2

and

(4.60) Δ ξ 2 ∣ x ∣ 2 K 2 = 8 x 2 K 4 ξ ′ ∣ x ∣ 2 K 2 2 + 8 x 2 K 4 ξ ∣ x ∣ 2 K 2 ξ ″ ∣ x ∣ 2 K 2 + 4 K 2 ξ ∣ x ∣ 2 K 2 ξ ′ ∣ x ∣ 2 K 2 .

Integrating from t − s to t + θ (where θ ∈ [ − h , 0 ] ), and by (4.23), (4.24), (4.59), and (4.60), we obtain that

(4.61) ∣ ∇ z ′ ( t + θ ) ∣ 2 + ∣ z ′ ( t + θ ) ∣ 2 + ∣ ∇ z ( t + θ ) ∣ 2 + λ ∣ z ( t + θ ) ∣ 2 ≤ 2 ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) ‖ 1 − ξ 2 ∣ x ∣ 2 K 2 f ( T , u 1 ( x , T − ρ ( T ) ) ) − f ( T , u 2 ( x , T − ρ ( T ) ) ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ‖ u 1 ( T ) − u 2 ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ‖ ∇ u 1 ( T ) − ∇ u 2 ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ∂ u 1 ( T ) ∂ T − ∂ u 2 ( T ) ∂ T L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ∇ ∂ u 1 ( T ) ∂ T − ∇ ∂ u 2 ( T ) ∂ T L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ∂ 2 u 1 ( T ) ∂ T 2 − ∂ 2 u 2 ( T ) ∂ T 2 L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ∇ ∂ 2 u 1 ( T ) ∂ T 2 − ∇ ∂ 2 u 2 ( T ) ∂ T 2 L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) .

By Lemma 4.3 and (4.56), for any ε > 0 and every ℬ = { B ( t ) } t ∈ R ∈ D C V , V , there exist T = T ( ε , t , ℬ ) > 0 and K 0 = K ( ε , t , ℬ ) > 0 such that for any solution u t ∈ U ( t , t − s ) B ( t − s ) satisfies

(4.62) sup θ ∈ [ − h , 0 ] ∫ Ω K C ( ∣ ∇ u ′ ( t + θ ) ∣ 2 + ∣ u ′ ( t + θ ) ∣ 2 + ∣ ∇ u ( t + θ ) ∣ 2 + ∣ u ( t + θ ) ∣ 2 ) d x ≤ ε , ∀ s ≥ T ,

and

(4.63) ∣ w ˆ t ∣ C V , V 2 = ∣ w ˆ t ∣ C V 2 + ∣ w ˆ t ′ ∣ C V 2 ≤ ε , ∀ s ≥ T .

In order to prove the pullback asymptotically upper-semicompactness of { U ( t , τ ) } , let ℬ = { B ( t ) } t ∈ R ∈ D C V , V , and the sequence s n → ∞ ( n → ∞ ) , and u n t ∈ U ( t , t − s n ) B ( t − s n ) be given arbitrarily, we need to show that { u n t } n = 1 ∞ is precompact in C V , V .

Note that there exists N > 0 such that for all n ≥ N , we have

(4.64) U ( t , t − s n ) B ( t − s n ) = U ( t , t − T 0 ) U ( t − T 0 , t − s n ) B ( t − s n ) ⊂ U ( t , t − T 0 ) Q ( t − T 0 ) .

Hence u n t ∈ U ( t , t − T 0 ) Q ( t − T 0 ) for all n ≥ N .

According to Lemmas 4.1 and 4.2, we can find that

(4.65) { u n ( T ) } n = N ∞ is bounded in L ∞ ( t − T 0 − h , t ; H 1 ( R N ) ) ,

(4.66) { u n ′ ( T ) } n = N ∞ is bounded in L ∞ ( t − T 0 − h , t ; H 1 ( R N ) ) ,

(4.67) { ∇ u n ′ ( T ) } n = N ∞ is bounded in L ∞ ( t − T 0 − h , t ; H 1 ( R N ) ) ,

(4.68) { u n ″ ( T ) } n = N ∞ is bounded in L ∞ ( t − T 0 − h , t ; H 1 ( R N ) ) ,

and

(4.69) { ∇ u n ″ ( T ) } n = N ∞ is bounded in L ∞ ( t − T 0 − h , t ; H 1 ( R N ) ) .

Hence, there exist K 1 > K 0 such that for all n , m > N , and we have

(4.70) C K 2 ‖ u 1 ( T ) − u 2 ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ‖ ∇ u 1 ( T ) − ∇ u 2 ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ∂ u 1 ( T ) ∂ T − ∂ u 2 ( T ) ∂ T L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ∇ ∂ u 1 ( T ) ∂ T − ∇ ∂ u 2 ( T ) ∂ T L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ∂ 2 u 1 ( T ) ∂ T 2 − ∂ 2 u 2 ( T ) ∂ T 2 L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) + C K 2 ∇ ∂ 2 u 1 ( T ) ∂ T 2 − ∇ ∂ 2 u 2 ( T ) ∂ T 2 L 2 ( Ω 2 K × [ t − s , t ] ) ‖ z ′ ( T ) ‖ L 2 ( Ω 2 K × [ t − s , t ] ) < ε .

On the other hand, without loss of generality, it follows from (4.65)–(4.69) that

(4.71) u n → u weakly star, in L ∞ ( t − T 0 − h , t ; H 1 ( Ω 2 K 1 ) ) ,

and

(4.72) u n ′ → u ′ weakly star, in L ∞ ( t − T 0 − h , t ; H 1 ( Ω 2 K 1 ) ) .

Then, we infer

(4.73) u n → u , in L 2 ( t − s − h , t ; H 1 ( Ω 2 K 1 ) ) ,

and

(4.74) u n ( T , x ) → u ( T , x ) n → ∞ , for a.e. ( T , x ) ∈ [ t − T 0 − h , t ] × Ω 2 K 1 .

Note that f ∈ C ( R × R N ; R ) , and we have

(4.75) f ( T , u n ( T , x ) ) → f ( T , u ( T , x ) ) n → ∞ , for a.e. ( T , x ) ∈ [ t − T 0 − h , t ] × Ω 2 K 1 .

Applying the Lebesgue-dominated convergence theorem, we infer

(4.76) lim n → ∞ lim m → ∞ ∣ f ( T , u n ( x , T − ρ ( T ) ) ) − f ( T , u m ( x , T − ρ ( T ) ) ) ∣ L 2 ( Ω × [ t − s , t ] ) = 0 .

It follows from (4.61) and (4.70) that

(4.77) ∣ y n t − y m t ∣ C V , V 2 = ∣ y n t − y m t ∣ C V 2 + ∣ y n t ′ − y m t ′ ∣ C V 2 = sup s ∈ [ − h , 0 ] ∣ ∇ y n ( t + θ ) − ∇ y m ( t + θ ) ∣ 2 + sup s ∈ [ − h , 0 ] ∣ ∇ y n ′ ( t + θ ) − ∇ y m ′ ( t + θ ) ∣ 2 ≤ 2 ∣ y n ′ ( T ) − y m ′ ( T ) ∣ L 2 ( Ω × [ t − s , t ] ) × ∣ f ( T , u n ( x , T − ρ ( T ) ) ) − f ( T , u m ( x , T − ρ ( T ) ) ) ∣ L 2 ( Ω × [ t − s , t ] ) .

Combining (4.62) with (4.63), we obtain that

(4.78) lim n → ∞ lim m → ∞ ∣ y n t − y m t ∣ C V , V 2 = 0 .

Thus, for any fixed s > 0 , every sequence y n ∈ U 2 ( t , t − s ) Q ( t − s ) is a Cauchy sequence in C V , V , and Theorem 2.2 (2) is proved.

This completes the proof.□

4.4 Existence of D C V , V -pullback attractors

Theorem 4.1

(Existence of D C V , V -pullback attractors) Under the assumptions of Lemma 4.1, the multi-valued process { U ( t , τ ) } on C V , V possesses an unique pullback attractor { A C V , V ( t ) } t ∈ R in D C V , V , where

A C V , V ( t ) = ω t ( Q C V , V ) = ⋂ T ∈ R + ⋃ s ≥ T U ( t , t − s ) Q ( t − s ) ¯ , ∀ t ∈ R .

Proof

By Lemma 4.5, we know that the multi-valued process { U ( t , τ ) } corresponding to equation (1.1) is D C V , V -pullback asymptotically upper-semicompact on C V , V . Furthermore, according to Lemma 4.1, the multi-valued process { U ( t , τ ) } possesses a D C V , V -pullback absorbing set Q C V , V in D C V , V . In order to obtain the existence of D C V , V -pullback attractors, we only need to show the negative invariance of { A C V , V ( t ) } t ∈ R , where

A C V , V ( t ) = ω t ( Q C V , V ) = ⋂ T ∈ R + ⋃ s ≥ T U ( t , t − s ) Q ( t − s ) ¯ , ∀ t ∈ R ,

and Q C V , V = { Q ( t ) } t ∈ R is the D C V , V -pullback absorbing set of { U ( t , τ ) } in C V , V .

Let y ∈ A C V , V ( t ) . Then, there exist sequences s n ∈ R + , s n → + ∞ ( n → ∞ ) , x n ∈ Q ( t − s n ) , and y n ∈ U ( t , t − s n ) x n such that

(4.79) y n → y ∈ C V , V , as n → ∞ .

On the other hand, for n large enough,

(4.80) y n ∈ U ( t , t − s n ) x n = U ( t , τ ) U ( τ , t − s n ) x n .

Lemma 4.4 implies that the multi-valued process { U ( t , τ ) } corresponding to equation (1.1) is D C V , V -pullback asymptotically upper-semicompact on C V , V , then there is a subsequence of x ˜ n ∈ U ( τ , t − s n ) x n = U ( t , τ − ( τ + s n − t ) ) x n , which we still relabel as x ˜ n such that y n ∈ U ( t , τ ) x ˜ n and

(4.81) x ˜ n → x ∈ C V , V , as n → ∞ .

Clearly, x ∈ A C V , V ( τ ) .

By slightly modifying the proof of the existence of solutions of Theorem 3.1, we can see that

y n ( ⋅ ) ⇀ u ( ⋅ + t , τ , x ) , in L 2 ( − h , 0 ; V ) ∩ H 1 ( − h , 0 ; V ) ,

where u ( ⋅ ) is a solution in Theorem 3.1. Together, with (4.80)–(4.81), we can deduce that

y ∈ U ( t , τ ) x ⊂ U ( t , τ ) A C V , V ( τ ) .

This completes the proof.□

Acknowledgment

The authors are grateful to the anonymous referees for their very helpful comments and suggestions that greatly improve the quality and presentation of the original manuscript.

Funding information: This work was supported by Innovation Foundation for University Teachers of Gansu Province [Grant No. 2025B-283].
Author contributions: All authors contributed equally to each part of this work. All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors state no conflicts of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-06-15

Revised: 2025-04-12

Accepted: 2025-04-15

Published Online: 2025-06-02

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

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https://doi.org/10.1515/math-2025-0152

Keywords for this article

non-autonomous semi-linear second-order evolution; pullback attractors; variable delays; multi-valued process

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