Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access March 10, 2022

The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

  • Penghui Lv EMAIL logo , Guoguang Lin and Yuting Sun
From the journal Open Mathematics

Abstract

In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher-order Kirchhoff equation with variable coefficients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of D k -absorbing sets of the stochastic dynamical system Φ k is obtained, and the asymptotic compactness of Φ k is proved by the decomposition method. Finally, the D k -stochastic attractor family of the stochastic dynamical system Φ k in V m + k ( Ω ) × V k ( Ω ) is obtained.

MSC 2010: 37B55; 35B41; 35G31; 60H15

1 Introduction

Let Ω R N be a bounded domain with smooth boundary (i.e., the derivative of the function at the boundary exists and is continuous). In this paper, we study the asymptotic behavior of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients on Ω :

(1.1) u t t + a ( x ) ( Δ ) m u t + b ( x ) M ( m u ) ( Δ ) m u + g ( x , u ) = f ( x , t ) + h ( x ) w t , u = 0 , i u v i = 0 , i = 1 , 2 , , m 1 , x Γ , t τ , u ( x , τ ) = u τ ( x ) , u t ( x , τ ) = u 1 τ ( x ) , x Ω ,

where Γ is the smooth boundary of Ω , v is the outer normal vector on the boundary Γ , m > 1 , a ( x ) and b ( x ) are variable coefficient functions, f ( x , t ) L loc 2 ( R , V k ( Ω ) ) is a time-dependent external force term, w is a one-dimensional bilateral standard Wiener process, h ( x ) w t describes white noise, and g ( x , u ) is a nonlinear function that satisfies certain growth conditions and dissipation conditions.

The Kirchhoff model was proposed in 1883 to describe the motion of elastic cross-section. Compared with classical wave equations, the Kirchhoff model can describe the motion of elastic rod more accurately. There has been a lot of in-depth research on the Kirchhoff equations. [1,2,3, 4,5,6] studied the long-term dynamics of the autonomous low-order Kirchhoff equation; [7,8,9, 10,11] studied the existence of global solutions and the blow-up of solutions of the higher-order Kirchhoff equations.

Stochastic wave equations are a very important class of stochastic partial differential equations, which are widely used in many fields such as fluid mechanics, physics, electricity, etc. The random attractor is an important tool for studying the long-term asymptotic behavior of stochastic dynamical systems. Using it to characterize the long-term behavior of random dynamical systems has laid a solid foundation for the study of random dynamical systems. After more than 30 years of development, random dynamical systems have also been extensively studied. Many scholars have conducted in-depth studies on the dynamical behavior of random wave equations in unbounded domains [12,13,14, 15,16,17] and bounded domains [18,19,20]. For new trends in functional analysis and random attractors, see also [21,22, 23,24].

Regarding the variable coefficients in the equation, it represents the wave velocity at the space coordinate x , which will appear in the wave phenomena in mathematical physics, marine acoustics, and other fields. It is of great practical significance to study the mathematical and physical equations with variable coefficients. In [25], they studied the global well-posedness and asymptotic behavior of solutions of Kirchhoff-type equations with variable coefficients and weak damping in unbounded domains. More relevant results can also be found in [26,27,28, 29,30,31, 32,33].

In recent years, Lin and Chen [34], Lin and Jin [35] have performed a detailed study on the long-term dynamical behavior of higher-order wave equations and proposed the concept of the family of attractors. Combined with the current research results, there are no relevant research results on the long-time dynamics of the nonautonomous stochastic higher-order Kirchhoff equation, and the asymptotic behavior of the higher-order Kirchhoff equation with variable coefficients has not been studied. By studying the nonautonomous stochastic higher-order Kirchhoff model with variable coefficients, the relevant results of the Kirchhoff model can be generalized, and the theoretical achievements of the Kirchhoff model can be enriched, which lays a theoretical foundation for later application. Therefore, this article will specifically study the family of random attractors of nonautonomous random higher-order Kirchhoff equation with variable coefficients. In the research process, the reasonable assumption and Leibniz formula are used to overcome how to define the L p -weighted space and the difficulty of estimating the absorption sets and asymptotic compactness caused by the variable coefficients.

Section 2 of this article introduces related theories, related definitions, and theories of stochastic dynamical systems; Section 3 presents the family of the continuous cocycle of the problem; In Section 4, the uniform estimation of the solution of problem (1.1) is obtained, and the asymptotic compactness of Φ k is obtained through the decomposition method; in Section 5, we get the family of D k -random attractors of Φ k in X k .

2 Preparatory knowledge

In this section, we mainly give the related theories of nonautonomous stochastic dynamical systems and random attractor (the family of random attractors).

First, the relevant notation needed in this paper is introduced: Define the inner product and norm on H = L 2 ( Ω ) as ( , ) and ( ) , L p = L p ( Ω ) , p = L p , where p 1 . Set variable coefficient a ( x ) , b ( x ) = b 0 a ( x ) , b 0 as a positive constant, satisfying a C 0 ( Ω ) , a ( x ) a 00 > 0 , i a i v Γ = 0 , a 0 = a ( x ) , a ( x ) 1 = μ ( x ) , x Ω , and μ L N 2 ( Ω ) C 0 ( Ω ) .

By D 1 , 2 , we define the closure of the C 0 ( Ω ) functions with respect to the “energy norm” u D 1 , 2 = Ω u 2 d x . It is well known that

D 1 , 2 D 1 , 2 ( Ω ) = { u L 2 N / ( N 2 ) ( Ω ) u ( L 2 ( Ω ) ) N } ,

and for D 1 , 2 L 2 N / ( N 2 ) ( Ω ) , there exists β > 0 such that u 2 N / ( N 2 ) β u D 1 , 2 .

Lemma 2.1

[26] Suppose that μ L N 2 ( Ω ) C 0 ( Ω ) , then for all u C 0 ( Ω ) , there exists α > 0 such that

α Ω μ u 2 d x Ω u 2 d x ,

where α = β 2 μ N / 2 1 .

Let μ > 0 be the weight function, and the weighted space L μ p = L μ p ( Ω ) with the following norm:

u L μ p p = Ω μ u p d x = μ 1 p u p p ,

for 1 p < + . Clearly L μ 2 = L μ 2 ( Ω ) is a separable Hilbert space the inner product and norm are respectively:

( u , v ) μ = Ω μ u v d x = μ 1 2 u , μ 1 2 v , u L μ 2 = μ 1 2 u .

For p : 1 p < , the Banach space L μ p is uniformly convex, reflexive space, and ( L μ p ) = L μ p , where p is the conjugate number of p .

Lemma 2.2

[26] Suppose that μ L N 2 ( Ω ) C 0 ( Ω ) , then D 1 , 2 is compactly embedded in L μ 2 . Let

V m = H 0 m ( Ω ) = H m ( Ω ) H 0 1 ( Ω ) , V m + k = H 0 m + k ( Ω ) = H m + k ( Ω ) H 0 1 ( Ω ) , k = 0 , 1 , , m ,

and the corresponding inner product and norm are, respectively,

( u , v ) V m + k = ( m + k u , m + k v ) , u V m + k = m + k u H .

At the same time, a general form of Poincare inequality: λ 1 r u 2 r + 1 u 2 , where λ 1 is the first eigenvalue of Δ . In the text, C i is a positive constant, C ( ) represents a positive constant that depends on the parameters in parentheses, and C m n is the corresponding number of combinations.

Assuming that ( X , X ) is a separable Hilbert space, and B ( X ) is the Borel σ -algebra of X ( Ω 1 , , P ) is the metric probability space.

Definition 2.3

[12] Let θ t : R × Ω 1 Ω 1 be a family of ( B ( X ) × , ) -measurable mappings such that θ 0 ( ) is the identity on Ω 1 t , s R , θ t + s ( ) = θ t ( ) θ s ( ) , P θ t ( ) = P . A mapping Φ : R + × R × Ω 1 × X X is called a continuous cocycle or continuous random dynamical system (RDS) on X over R and ( Ω 1 , , P , ( θ t ) t R ) if for all τ R , w Ω 1 , t , s R + the following conditions are satisfied:

  1. Φ ( , τ , , ) : R + × Ω 1 × X X is a ( B ( R + ) × × B ( X ) , B ( X ) ) -measurable mapping;

  2. Φ ( 0 , τ , w , ) is the identity on X ;

  3. Φ ( t + s , τ , w , ) = Φ ( t , τ + s , θ s w , Φ ( s , τ , w , ) ) ;

  4. Φ ( t , τ , w , ) : X X is continuous.

Let D = { D ( τ , w ) X : τ R , w Ω 1 } be a family of subsets parameterized by ( τ , w ) R × Ω 1 in X .

Definition 2.4

[13] The family D = { D ( τ , w ) X : τ R , w Ω 1 } satisfies:

  1. for all ( τ , w ) R × Ω 1 D ( τ , w ) is a closed nonempty subset of X ;

  2. for every fixed x X and any τ R , the mapping w Ω 1 dist X ( x , B ( τ , w ) ) is ( , B ( R + ) ) measurable, then the family D is measurable with to in Ω 1 .

Definition 2.5

[15] For all σ > 0 , w Ω 1 D = { D ( τ , w ) X : τ R , w Ω 1 } satisfies:

lim t e σ t D ( τ + t , θ t w ) X = 0 ,

then D = { D ( τ , w ) X : τ R , w Ω 1 } is called tempered.

Let D = D ( X ) be the set of all random tempered sets in X .

Definition 2.6

[12] A family K = { K ( τ , w ) X : τ R , w Ω 1 } D of nonempty subsets of X is called a measurable D -pullback attracting(or absorbing) set for { Φ ( t , τ , w ) } t 0 , τ R , w Ω 1 if

  1. K is measurable with respect to the P completion of in Ω 1 ;

  2. for all τ R , w Ω 1 and for every D D , there exists T ( D , τ , w ) > 0 such that

    Φ ( t , τ t , θ t w , D ( τ t , θ t w ) ) K ( τ , w ) , t T ( D , τ , w ) .

Definition 2.7

[15] Φ is said to be asymptotically compact in X if for τ R , w Ω 1 , D = { D ( τ , w ) X : τ R , w Ω 1 } D , x n B ( τ t n , θ t n w ) { Φ ( t n , τ t n , θ t n w , x n ) } n = 1 has a convergent subsequence in X whenever t n .

Definition 2.8

[13] A family A = { A ( τ , w ) X : τ R , w Ω 1 } D is called a D -pullback random attractor for { Φ ( t , τ , w ) } t 0 , τ R , w Ω 1 if

  1. A ( τ , w ) is measurable in Ω 1 with respect to and compact in X for τ R , w Ω 1 ,

  2. A is invariant, i.e., for τ R and w Ω 1 , t 0 ,

    Φ ( t , τ , w , A ( τ , w ) ) = A ( t + τ , θ t w ) ;

  3. A attracts every member of D , i.e., for every D D , τ R and for every w Ω 1 ,

    lim t + dist X ( Φ ( t , τ t , θ t w , B ( τ t , θ t w ) ) , A ( τ , w ) ) = 0 ,

    where dist X ( P , Q ) denotes the Hausdorff semi-distance between two subsets P and Q of X .

If we change D = D ( X ) to D k = D k ( X k ) , where k = 0 , 1 , , m , then A in Definition 2.8 can be a family of random attractors { A k } .

Lemma 2.9

[12] Let D be a neighborhood-closed collection of ( τ , w ) -parametrized families of nonempty subsets of X and Φ be a continuous cocycle on X over R and ( Ω 1 , , P , { θ t } t R ) , then Φ has a pullback D -attract A if and only if Φ is pullback D asymptotically compact in X and Φ has a closed, -measurable pullback D -absorbing set K in D and the unique pullback D -attractor A = { A ( τ , w ) } is given by

A ( τ , w ) = τ 0 t τ Φ ( t , τ t , θ t w , K ( τ t , θ t w ) ) ¯ , τ R , w Ω 1 .

Similarly, Lemma 2.9 can be extended to Lemma 2.10 of the family of pullback attractors.

Lemma 2.10

Let D k be neighborhood-closed collections of ( τ , w ) -parametrized families of nonempty subsets of X k , k = 1 , 2 , , m , and Φ k be the family of continuous cocycles on X k , k = 1 , 2 , , m over R and ( Ω 1 , , P , { θ t } t R ) , then Φ k has the family of pullback D k -attracts { A k } if and only if Φ k is pullback D k -asymptotically compact in X k and D k has closed, -measurable pullback D k -absorbing sets K k in D k and the unique pullback D k -attractor A k = { A k ( τ , w ) } is given by

A k ( τ , w ) = τ 0 t τ Φ k ( t , τ t , θ t w , K k ( τ t , θ t w ) ) ¯ , τ R , w Ω 1 .

3 The family of cocycles of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

Let ( Ω 1 , , P ) be a probability space, where

Ω 1 = { w C ( R , R ) , w ( 0 ) = 0 } .

w is a two-sided real-valued Winner processes on the probability space ( Ω 1 , , P ) . Define θ t w ( ) = w ( + t ) w ( t ) , w Ω 1 , t R , thus, ( Ω 1 , , P , ( θ t ) t R ) is an ergodic metric dynamical system.

For a small positive number ε , let z be a new variable given by z = u t + ε u and then, system (1.1) becomes

(3.1) u t + ε u = z ; z t = ε z a ( x ) ( Δ ) m z + ε a ( x ) ( Δ ) m u ε 2 u b ( x ) M ( m u 2 ) ( Δ ) m u g ( x , u ) + f ( x , t ) + h ( x ) d w d t ; u = 0 , i u v i = 0 , i = 1 , 2 , , m 1 , x Γ , t τ ; u ( x , τ ) = u τ ( x ) , z ( x , τ ) = z τ ( x ) = u 1 τ ( x ) + u τ ( x ) , x Ω ,

where ( M ) M C 1 ( R + ) , M 0 , and M ( s ) M 0 ( 1 + s q ) , 0 < q < 1 / 2 , M 0 = M ( 0 ) is a positive constant s R + , h ( x ) V m + k ( Ω ) , x Ω , t τ , τ R , k = 0 , 1 , , m , f ( x , t ) L loc 2 ( R , V k ( Ω ) ) . In order to get the conclusion of this article, suppose that the nonlinear term g ( x , u ) satisfies the following conditions: for u R , x Ω , there are positive constants c 1 , c 2 , c 3 , c 4 , c 5 > 0 , satisfying

(3.2) g ( x , u ) c 1 p p + ϕ 1 ( x ) , ϕ 1 L μ 2 ( Ω ) ,

(3.3) u g ( x , u ) c 2 G ( x , u ) ϕ 2 ( x ) , ϕ 2 L μ 1 ( Ω ) ,

(3.4) G ( x , u ) c 3 u p + 1 ϕ 3 ( x ) , ϕ 3 L μ 1 ( Ω ) ,

(3.5) g u ( x , u ) c 4 u p 1 + ϕ 4 ( x ) , ϕ 4 V m ( Ω ) ,

(3.6) x k g ( x , u ) c 5 u p + ϕ 5 ( x ) , ϕ 5 V k ( Ω ) ,

where 1 p < + , for N = 1 , 2 ; 1 p < N N 2 N = 3 , 4 ; and G ( x , u ) = 0 u g ( x , s ) d s . From equations (3.2) and (3.3), we can get

(3.7) G ( x , u ) c 6 ( u 2 + u p + 1 + ϕ 1 2 + ϕ 2 ) .

To show that problem (3.1) generates a random dynamical system, we let v ( t , τ , w ) = z ( t , τ , w ) h w ( t ) , and then, (3.1) can be rewritten as the equivalent system with random coefficients but without white noise:

(3.8) u t v + ε u = h w ( t ) ; v t = ε v a ( x ) ( Δ ) m v + ε a ( x ) ( Δ ) m u ε 2 u b ( x ) M ( m u 2 ) ( Δ ) m u g ( x , u ) + f ( x , t ) + ε h ( x ) w ( t ) a ( x ) ( Δ ) m h ( x ) w ( t ) ; u = 0 , i u v i = 0 , i = 1 , 2 , , m 1 , x Γ , t τ ; u ( x , τ ) = u τ ( x ) , v ( x , τ ) = v τ ( x ) = z τ ( x ) h w ( τ ) , x Ω .

Let X k = V m + k × V k , k = 0 , 1 , , m , when k = 0 V 0 = L μ 2 , endowed with the usual norm ( u , v ) X k 2 = u V m + k 2 + v V k 2 . By the standard Galerkin method: If the assumptions ( M ) h ( x ) V m + k ( Ω ) , x Ω , t τ , τ R , f ( x , t ) L loc 2 ( R , V k ( Ω ) ) conditions (3.2)–(3.6) hold the problem (3.8) is well posed in X k = V m + k × V k , i.e., for all τ R and P a . e . w Ω 1 , ( u τ , v τ ) X k , the problem (3.8) has a unique global solution ( u ( t , τ , w , u τ ) , v ( t , τ , w , v τ ) ) C ( [ τ , ) , X k ) and ( u ( τ , τ , w , u τ ) , v ( τ , τ , w , v τ ) ) = ( u τ , v τ ) . Moreover, for t τ , ( u ( t , τ , w , u τ ) , v ( t , τ , w , v τ ) ) is ( , B ( X k ) ) measurable in w and continuous in ( u τ , v τ ) with respect to the X k norm. Thus, the solution mapping can be used to define a family of continuous cocycles for (3.8). Let Φ k : R + × R × Ω 1 × X k X k be mappings given by

(3.9) Φ k ( t , τ , w , ( u τ , v τ ) ) = ( u ( t + τ , τ , θ τ w , u τ ) , v ( t + τ , τ , θ τ w , v τ ) ) ,

where ( t , τ , w , ( u τ , v τ ) ) R + × R × Ω 1 × X k , then Φ k is a family of continuous cocycles over ( R , τ + t ) and ( Ω 1 , , P , { θ t } t R ) on X k . For P a . e . w Ω 1 and t , s 0 , τ R :

(3.10) Φ k ( t + s , τ , w , ( u τ , v τ ) ) = Φ k ( t , s + τ , w , Φ k ( s , τ , w , ( u τ , v τ ) ) ) .

For any bounded nonempty subset B k of X k denote by B k = sup Φ k R Φ X k . Let D k = { D k ( τ , w ) : τ R , w Ω 1 } be a family of bounded nonempty subsets of X k , and for all τ R , w Ω 1 ,

(3.11) lim s e σ s D k ( τ + s , θ s w ) X k 2 = 0 .

Remember that D k is the set of the aforementioned subset family D k , that is, D k = { D k = { D k ( τ , w ) : τ R , w Ω 1 } : D k satisfies (3.11) } .

4 Uniform estimates of solutions

To prove the existence of the family of random attractors, we conduct uniform estimates on the solutions of the problem (3.8) defined on Ω , for the purposes of showing the existence of a family of D k pullback absorbing sets and the pullback D k asymptotic compactness of the random dynamical system. Let ε > 0 be small enough and satisfy α λ 1 m 1 3 ε > 0 , 2 a 00 λ 1 m ( a 00 λ 1 m + 12 ) ε > 0 , M 0 5 2 ε > 0 , b 0 M 0 8 ε > 0 ,

(4.1) σ = 1 2 min α λ 1 m 1 3 ε , ε 2 , ε c 2 2 , σ 1 = 1 2 min { 2 a 00 λ 1 m ( a 00 λ 1 m + 12 ) ε , ε } .

To obtain uniform estimates of the solutions, f ( x , t ) needs to satisfy ( F 1 ) t e σ s f ( , s ) V k 2 d s < .

Lemma 4.1

Suppose M satisfies ( M ) , h ( x ) V m + k ( Ω ) , k = 0 , 1 , , m , (3.2)–(3.6) hold, f ( x , t ) satisfies ( F 1 ) , and B k = { B k ( τ , w ) : τ R , w Ω 1 } D k for P a . e . w Ω 1 , τ R initial value satisfies ( u τ t , v τ t ) B k ( τ t , θ τ w ) , there exists T k = T k ( τ , w , B k ) > 0 such that for all t T k , the solution ( u ( τ , τ , w , u τ t ) , v ( τ , τ , w , v τ t ) ) = ( u τ t , v τ t ) of problem (3.8) satisfies

v ( τ , τ t , θ τ w , v τ t ) V k 2 + u ( τ , τ t , θ τ w , v τ t ) V m + k 2 r 1 k ( τ , w ) ,

where r 1 k ( τ , w ) will be given in detail later.

Proof

Taking the inner product of (3.8) with v in L μ 2 ( Ω ) , we find that

(4.2) 1 2 d d t v L μ 2 2 = ε v L μ 2 2 m v 2 + ε ( ( Δ ) m u , v ) ε 2 ( u , v ) L μ 2 b 0 ( M ( m u 2 ) ( Δ ) m u , v ) ( g ( x , u ) , v ) L μ 2 + ( f ( x , t ) , v ) L μ 2 + ε w ( t ) ( h , v ) L μ 2 w ( t ) ( ( Δ ) m h , v ) ,

for each term on the right-hand side of (4.2):

(4.3) ε ( ( Δ ) m u , v ) = ε ( ( Δ ) m u , u t + ε u + h w ( t ) ) = ε 2 d d t m u 2 + ε 2 m u 2 ε w ( t ) ( ( Δ ) m u , h ) ,

(4.4) ε 2 ( u , v ) L μ 2 = ε 2 ( u , u t + ε u h w ( t ) ) L μ 2 = ε 2 2 d d t v L μ 2 2 + ε 3 v L μ 2 2 ε 2 w ( t ) ( u , h ) L μ 2 ,

(4.5) b 0 ( M ( m u 2 ) ( Δ ) m u , v ) = b 0 ( M ( m u 2 ) ( Δ ) m u , u t + ε u h w ( t ) ) = b 0 2 d d t 0 m u 2 M ( s ) d s + ε b 0 M ( m u 2 ) m u 2 b 0 M ( m u 2 ) w ( t ) ( ( Δ ) m u , h ) ,

(4.6) ( g ( x , u ) , v ) L μ 2 = ( g ( x , u ) , u t + ε u h w ( t ) ) L μ 2 = d d t Ω μ G ( x , u ) d x + ε ( g ( x , u ) , u ) L μ 2 w ( t ) ( g ( x , u ) , h ) L μ 2 .

Substitute (4.3)–(4.6) into (4.2) to obtain

(4.7) d d t v L μ 2 2 + b 0 0 m u 2 M ( s ) d s ε m u 2 + ε 2 u L μ 2 2 + 2 Ω μ G ( x , u ) d x + 2 m v 2 2 ε v L μ 2 2 + 2 ε b 0 M ( m u 2 ) m u 2 2 ε 2 m u 2 + 2 ε 3 u L μ 2 2 + 2 ε ( g ( x , u ) , u ) L μ 2 = 2 ( f ( x , t ) , v ) L μ 2 + ( 2 b 0 M ( m u 2 ) 2 ε ) w ( t ) ( ( Δ ) m u , h ) 2 ε 2 w ( t ) ( u , h ) L μ 2 + 2 w ( t ) ( g ( x , u ) , h ) L μ 2 + 2 w ( t ) ( h , v ) L μ 2 2 w ( t ) ( ( Δ ) m h , v ) .

Using the Cauchy-Schwarz inequality, Young’s inequality and Holder’s inequality, we have

(4.8) 2 ε 2 w ( t ) ( u , h ) L μ 2 ε 3 u L μ 2 2 + ε w ( t ) 2 h L μ 2 2 ,

(4.9) 2 b 0 M ( m u 2 ) w ( t ) ( ( Δ ) m u , h ) 2 b 0 M 0 ( 1 + m u 2 q ) w ( t ) m u m h 2 b 0 M 0 w ( t ) m u m h + 2 b 0 M 0 w ( t ) m u 2 q + 1 m h ε b 0 M 0 4 m u 2 + 4 ε 1 b 0 M 0 w ( t ) 2 m h 2 + ε b 0 M 0 4 m u 2 + 1 2 q 2 ε 8 q + 4 2 q + 1 2 q 1 b 0 M 0 w ( t ) 2 1 2 q m h 2 1 2 q .

By (3.2) and (3.4), we get

(4.10) 2 w ( t ) ( g ( x , u ) , h ) L μ 2 2 w ( t ) ϕ 1 L μ 2 h L μ 2 + 2 c 1 w ( t ) Ω μ u p + 1 d x p p + 1 h L μ p + 1 2 w ( t ) ϕ 1 L μ 2 h L μ 2 + 2 c 1 w ( t ) Ω ( μ G ( x , u ) + μ ϕ 3 ) d x p p + 1 h L μ p + 1 2 w ( t ) ϕ 1 L μ 2 h L μ 2 + ε c 2 Ω μ G ( x , u ) d x + ε c 2 Ω μ ϕ 3 ( x ) d x + ( 2 c 1 ) p + 1 ( ε c 2 ) p p + 1 p + 1 p p w ( t ) p + 1 h L μ p + 1 p + 1 ,

(4.11) 2 ( f ( x , t ) , v ) L μ 2 + 2 w ( t ) ( h , v ) L μ 2 2 w ( t ) ( ( Δ ) m h , v ) 2 f L μ 2 v L μ 2 + 2 ε w ( t ) h L μ 2 v L μ 2 + 2 w ( t ) m h m v 2 α 1 2 λ 1 m 1 2 f L μ 2 m v + ε v L μ 2 2 + ε w ( t ) 2 h L μ 2 2 + 1 2 m v 2 + 2 w ( t ) 2 m h 2 m v 2 + ε v L μ 2 2 + 2 α 1 λ 1 1 m f L μ 2 2 + 2 w ( t ) 2 m h 2 + ε w ( t ) 2 h L μ 2 2 ,

(4.12) 2 ε w ( t ) ( ( Δ ) m u , h ) 2 ε w ( t ) m h m u ε 2 m u 2 + w ( t ) 2 m h 2 .

Substitute (4.8)–(4.12) into (4.7) to obtain

(4.13) d d t v L μ 2 2 + b 0 0 m u 2 M ( s ) d s ε m u 2 + ε 2 u L μ 2 2 + 2 Ω μ G ( x , u ) d x + m v 2 3 ε v L μ 2 2 + 2 ε M ( m u 2 ) ε M 0 2 b 0 m u 2 3 ε 2 m u 2 + ε 3 u L μ 2 2 + 2 ε ( g ( x , u ) , u ) L μ 2

2 α 1 λ 1 1 m f L μ 2 2 + ( 3 + 4 ε 1 M 0 b 0 ) w ( t ) 2 m h 2 + 2 ε w ( t ) 2 h L μ 2 2 + 1 2 q 2 ε 8 q + 4 2 q + 1 2 q 1 b 0 M 0 w ( t ) 2 1 2 q m h 2 1 2 q + 2 w ( t ) ϕ 1 L μ 2 h L μ 2 + ε c 2 Ω μ G ( x , u ) d x + ε c 2 Ω μ ϕ 3 ( x ) d x + ( 2 c 1 ) p + 1 ( ε c 2 ) p p + 1 p + 1 p p w ( t ) p + 1 h L μ p + 1 p + 1 .

By condition (3.3), we have

(4.14) 2 ε ( g ( x , u ) , u ) L μ 2 2 ε c 2 Ω μ G ( x , u ) d x + Ω μ ϕ 2 ( x ) d x .

Substitute (4.14) into (4.12) to obtain

(4.15) d d t v L μ 2 2 + b 0 0 m u 2 M ( s ) d s ε m u 2 + ε 2 u L μ 2 2 + 2 Ω μ G ( x , u ) d x + m v 2 3 ε v L μ 2 2 + 2 ε M ( m u 2 ) ε M 0 2 b 0 m u 2 3 ε 2 m u 2 + ε 3 u L μ 2 2 + ε c 2 Ω μ G ( x , u ) d x + 2 ε Ω μ ϕ 2 ( x ) d x 2 α 1 λ 1 1 m f L μ 2 2 + ( 3 + 4 ε 1 M 0 b 0 ) w ( t ) 2 m h 2 + 2 ε w ( t ) 2 h L μ 2 2 + 1 2 q 2 ε 8 q + 4 2 q + 1 2 q 1 b 0 M 0 w ( t ) 2 1 2 q m h 2 1 2 q + 2 w ( t ) ϕ 1 L μ 2 h L μ 2 + ε c 2 Ω μ ϕ 3 ( x ) d x + ( 2 c 1 ) p + 1 ( ε c 2 ) p p + 1 p + 1 p p w ( t ) p + 1 h L μ p + 1 p + 1 .

According to (4.1), we get

(4.16) d d t v L μ 2 2 + b 0 0 m u 2 M ( s ) d s ε m u 2 + ε 2 u L μ 2 2 + 2 Ω μ G ( x , u ) d x + σ v L μ 2 2 + b 0 0 m u 2 M ( s ) d s ε m u 2 + ε 2 u L μ 2 2 + 2 Ω μ G ( x , u ) d x 2 α 1 λ 1 1 m f L μ 2 2 + C 01 ( 1 + w ( t ) 2 + w ( t ) 2 1 2 q + w ( t ) p + 1 ) ,

where

C 01 = max ( 3 + 4 ε 1 M 0 b 0 ) m h 2 + 2 ε h L μ 2 2 + ϕ 1 L μ 2 2 h L μ 2 2 , ε c 2 Ω μ ϕ 3 ( x ) d x , 1 2 q 2 ε 8 q + 4 2 q + 1 2 q 1 b 0 M 0 m h 2 1 2 q , ( 2 c 1 ) p + 1 ( ε c 2 ) p p + 1 p + 1 p p h L μ p + 1 p + 1 .

Using the Gronwall inequality to integrate (4.16) over [ τ t , τ ] with t 0 and replacing w by θ τ w , we obtain

(4.17) e σ τ v L μ 2 2 + b 0 0 m u 2 M ( s ) d s ε m u 2 + ε 2 u L μ 2 2 + 2 Ω μ G ( x , u ) d x e σ ( τ t ) v ( τ t ) L μ 2 2 + b 0 0 m u ( τ t ) 2 M ( s ) d s ε m u ( τ t ) 2 + ε 2 u ( τ t ) L μ 2 2 + 2 Ω μ G ( x , u ( τ t ) ) d x + 2 α 1 λ 1 1 m τ t τ e σ ξ f ( , ξ ) L μ 2 2 d ξ + C 01 τ t τ e σ ξ ( 1 + w ( ξ ) 2 + w ( ξ ) 2 1 2 q + w ( ξ ) p + 1 ) d ξ ,

then

(4.18) v ( τ , τ t , θ τ w , v τ t ) L μ 2 2 + b 0 0 m u ( τ , τ t , θ τ w , v τ t ) 2 M ( s ) d s ε m u ( τ , τ t , θ τ w , v τ t ) 2 + ε 2 u ( τ , τ t , θ τ w , v τ t ) L μ 2 2 + 2 Ω μ G ( x , u ( τ , τ t , θ τ w , v τ t ) ) d x e σ t v ( τ t ) L μ 2 2 + b 0 0 m u ( τ t ) 2 M ( s ) d s ε m u ( τ t ) 2 + ε 2 u ( τ t ) L μ 2 2 + 2 Ω μ G ( x , u ( τ t ) ) d x + 2 α 1 λ 1 1 m e σ τ τ t τ e σ ξ f ( , ξ ) L μ 2 2 d ξ + C 01 e σ τ τ t τ e σ ξ ( 1 + θ τ w ( ξ ) 2 + θ τ w ( ξ ) 2 1 2 q + θ τ w ( ξ ) p + 1 ) d ξ .

By (3.7), we have

(4.19) Ω μ G ( x , u ( τ t ) ) d x c 6 ( ϕ 1 L μ 2 2 + ϕ 2 L μ 1 + u L μ 2 2 + u L μ p + 1 p + 1 ) C 02 ( 1 + u L μ 2 2 + u V m p + 1 ) .

Since ( u ( τ t ) , v ( τ t ) ) B 0 ( τ t , θ τ w ) when t +

(4.20) e σ t v ( τ t ) L μ 2 2 + b 0 0 m u ( τ t ) 2 M ( s ) d s ε m u ( τ t ) 2 + ε 2 u ( τ t ) L μ 2 2 + 2 Ω μ G ( x , u ( τ t ) ) d x C 03 e σ t ( 1 + v ( τ t ) L μ 2 2 + m u ( τ t ) 2 + m u ( τ t ) p + 1 ) 0 ,

and there exists T 0 = T 0 ( τ , w , B 0 ) such that for all t T 0 ,

(4.21) C 03 e σ t ( 1 + v ( τ t ) L μ 2 2 + m u ( τ t ) 2 + m u ( τ t ) p + 1 ) 1 .

By (3.4), it is easy to get to any t 0 ,

(4.22) 2 Ω G ( x , u ) d x 2 c 3 Ω u p + 1 d x + 2 Ω ϕ 3 d x 2 Ω ϕ 3 d x .

When ξ , w ( ξ ) at most polynomial growth,

C 01 e σ τ τ e σ ξ ( 1 + θ τ w ( ξ ) 2 + θ τ w ( ξ ) 2 1 2 q + θ τ w ( ξ ) p + 1 ) d ξ r 00 ( τ , w ) .

We get from (4.18) and (4.21) that

(4.23) v ( τ , τ t , θ τ w , v τ t ) L μ 2 2 + m u ( τ , τ t , θ τ w , v τ t ) 2 C 04 1 + 2 α 1 λ 1 1 m e σ τ τ e σ ξ f ( , ξ ) L μ 2 2 d ξ + r 00 ( τ , w ) r 10 ( τ , w ) ,

and r 10 ( τ , w ) is bounded.□

Taking the inner product of (3.8) with ( Δ ) k v , k = 1 , 2 , , m 1 in L 2 ( Ω ) , we find that

(4.24) 1 2 d d t k v 2 = ε k v 2 ( a ( x ) ( Δ ) m v , ( Δ ) k v ) + ε ( a ( x ) ( Δ ) m u , ( Δ ) k v ) ε 2 ( u , ( Δ ) k v ) ( b ( x ) M ( m u 2 ) ( Δ ) m u , ( Δ ) k v ) ( g ( x , u ) , ( Δ ) k v ) + ( f ( x , t ) , ( Δ ) k v ) + ε w ( t ) ( h , ( Δ ) k v ) w ( t ) ( a ( x ) ( Δ ) m h , ( Δ ) k v ) .

For each term on the right-hand side of (4.24):

(4.25) ( a ( x ) ( Δ ) m v , ( Δ ) k v ) = ( a ( x ) m + k v , m + k v ) + i = 1 m k C m k i m + k i v i a ( x ) , m + k v ,

(4.26) ε ( a ( x ) ( Δ ) m u , ( Δ ) k v ) = ε ( a ( x ) m + k u , m + k v ) + ε i = 1 m k C m k i m + k i v i a ( x ) , m + k u = ε 1 2 d d t ( a ( x ) m + k u , m + k u ) + ε 2 ( a ( x ) m + k u , m + k u ) ε w ( t ) ( a ( x ) m + k u , m + k h ) + ε i = 1 m k C m k i m + k i v i a ( x ) , m + k u ,

(4.27) ε 2 ( u , ( Δ ) k v ) = ε 2 ( u , ( Δ ) k ( u t + ε u h w ( t ) ) ) = ε 2 2 d d t k u 2 + ε 3 k u 2 ε 2 w ( t ) ( k u , k h ) ,

(4.28) ( b ( x ) M ( m u 2 ) ( Δ ) m u , ( Δ ) k v ) = b 0 M ( m u 2 ) ( a ( x ) m + k u , m + k v ) + M ( m u 2 ) i = 1 m k C m k i m + k i v i b ( x ) , m + k u = b 0 2 M ( m u 2 ) d d t ( a ( x ) m + k u , m + k u ) + ε b 0 M ( m u 2 ) ( a ( x ) m + k u , m + k u ) b 0 M ( m u 2 ) w ( t ) ( a ( x ) m + k u , m + k h ) + M ( m u 2 ) i = 1 m k C m k i m + k i v i b ( x ) , m + k u ,

(4.29) ( g ( x , u ) , ( Δ ) k v ) = ( x k g ( x , u ) , k v ) Ω ( c 5 u p + ϕ 5 ( x ) ) k v d x c 5 Ω u p k v d x + Ω ϕ 5 ( x ) k v d x c 5 u L 2 p p k v + ϕ 5 ( x ) k v a 00 8 m + k v 2 + C k 1 ( r 01 p ( τ , w ) + ϕ 5 ( x ) 2 ) ,

(4.30) ( f ( x , t ) , ( Δ ) k v ) = ( k f ( x , t ) , k v ) k f ( x , t ) k v a 00 8 m + k v 2 + 2 λ 1 m a 00 k f ( x , t ) 2 ,

moreover,

(4.31) ( m + k i v i a ( x ) , m + k v ) a i m + k i v m + k v , i = 1 , 2 , , m k , a i = i a ( x ) .

According to the interpolation inequality, we have

m + k i v C i m + k v α i v 1 α i , α i = m + k i m + k ,

then

(4.32) ( C m k i m + k i v i a ( x ) , m + k v ) C m k i C i a i v 1 α i m + k v 1 + α i a 00 8 ( m k ) m + k v 2 + 1 α i 2 a 00 4 ( 1 α i ) ( m k ) 1 + α i 1 α i ( C m k i C i a i ) 2 1 α i v 2 ,

(4.33) ( C m k i m + k i v i a ( x ) , m + k u ) a 00 b 0 M 0 8 ( m k ) m + k u 2 + 2 ( m k ) a 00 b 0 M 0 ( C m k i a i ) 2 m + k i v 2 a 00 b 0 M 0 8 ( m k ) m + k u 2 + a 00 8 ( m k ) m + k v 2 + ( 1 α i ) a 00 8 α i ( m k ) α i α i 1 2 ( m k ) a 00 b 0 M 0 ( C m k i a i C i ) 2 1 1 α i v 2 ,

(4.34) b 0 M ( m u 2 ) w ( t ) ( a ( x ) m + k u , m + k h ) b 0 a 0 M 0 ( 1 + m u 2 q ) w ( t ) m + k u m + k h ε b 0 a 00 M 0 8 m + k u 2 + 2 ε 1 a 00 1 b 0 a 0 2 M 0 ( 1 + m u 2 q ) 2 w ( t ) 2 m + k h 2 ,

(4.35) M ( m u 2 ) i = 1 m k C m k i m + k i v i b ( x ) , m + k u M 0 b 0 ( 1 + m u 2 q ) i = 1 m k C m k i a i m + k i v m + k u ε b 0 a 00 M 0 8 m + k u 2 + ε 1 a 00 1 b 0 M 0 i = 1 m k ( C m k i ( 1 + m u 2 q ) a i ) 2 m + k i v 2 ε b 0 a 00 M 0 8 m + k u 2 + a 00 8 m + k v 2 + ( ε 1 a 00 1 b 0 M 0 ) 1 1 α i i = 1 m k ( 1 α i ) a 00 8 α i ( m k ) α i α i 1 ( C m k i a i C i ( 1 + m u 2 q ) ) 2 1 α i v 2 .

By (4.25)–(4.30) and (4.32)–(4.35), we get

(4.36) d d t [ k v 2 ε ( a ( x ) m + k u , m + k u ) + ε 2 k u 2 ] + b 0 M ( m u 2 ) d d t ( a ( x ) m + k u , m + k u ) + 2 ( a ( x ) m + k v , m + k v ) 4 + ε 4 a 00 m + k v 2 2 ε k v 2 + 2 ε b 0 M ( m u 2 ) ( a ( x ) m + k u , m + k u ) 3 ε a 00 b 0 M 0 4 m + k u 2 2 ε 2 ( a ( x ) m + k u , m + k u ) + 2 ε 3 k u 2 i = 1 m k ( 1 α i ) a 00 4 ( 1 α i ) ( m k ) 1 + α i 1 α i ( C m k i a i C i ) 2 1 α i v 2 + 2 ε i = 1 m k ( 1 α i ) a 00 8 α i ( m k ) α i α i 1 2 ( m k ) a 00 b 0 M 0 ( C m k i a i C i ) 2 1 1 α i v 2

+ 2 i = 1 m k ( ε 1 a 00 1 b 0 M 0 ) 1 1 α i ( 1 α i ) a 00 8 α i ( m k ) α i α i 1 ( C m k i a i C i ( 1 + m u 2 q ) ) 2 1 α i v 2 + 4 ε 1 a 00 1 b 0 a 0 2 M 0 ( 1 + m u 2 q ) 2 w ( t ) 2 m + k h 2 + 2 λ 1 m a 00 k f ( x , t ) 2 + 2 C k 1 ( r 01 p ( τ , w ) + ϕ 5 ( x ) 2 ) 2 ε w ( t ) ( a ( x ) m + k u , m + k h ) + 2 ε 2 w ( t ) ( k u , k h ) + 2 ε w ( t ) ( ( Δ ) k u , h ) 2 w ( t ) ( a ( x ) ( Δ ) m h , ( Δ ) k v ) .

Using the Cauchy-Schwarz inequality,Young’s inequality and Holder’s inequality, etc. we have

(4.37) 2 ε w ( t ) ( a ( x ) m + k u , m + k h ) ε 2 a 00 m + k u 2 + a 00 1 a 0 2 w ( t ) 2 m + k h 2 ,

(4.38) 2 ε 2 w ( t ) ( k u , k h ) ε 3 k u 2 + ε w ( t ) 2 k h 2 ,

(4.39) 2 ε w ( t ) ( ( Δ ) k u , h ) ε k v 2 + ε w ( t ) 2 k h 2 ,

(4.40) 2 w ( t ) ( a ( x ) ( Δ ) m h , ( Δ ) k v ) = 2 w ( t ) ( a ( x ) m + k v , m + k h ) + 2 w ( t ) i = 1 m k C m k i m + k i i a ( x ) , m + k h a 00 4 m + k v 2 + 4 a 00 1 a 0 2 w ( t ) 2 m + k 2 + a 00 4 m + k v 2 + 4 a 00 1 w ( t ) 2 i = 1 m k ( C m k i a i ) 2 λ 1 i m + k h 2 a 00 2 m + k v 2 + 4 a 00 1 a 0 2 w ( t ) 2 m + k 2 + 4 a 00 1 w ( t ) 2 i = 1 m k ( C m k i a i ) 2 λ 1 i m + k h 2 .

Substitute (4.37)–(4.40) into (4.36) to obtain

(4.41) d d t [ k v 2 ε ( a ( x ) m + k u , m + k u ) + ε 2 k u 2 ] + b 0 M ( m u 2 ) d d t ( a ( x ) m + k u , m + k u ) + 2 ( a ( x ) m + k v , m + k v ) 6 + ε 4 a 00 m + k v 2 3 ε k v 2 + 2 ε b 0 M ( m u 2 ) ( a ( x ) m + k u , m + k u ) 3 ε a 00 b 0 M 0 4 m + k u 2 2 ε 2 ( a ( x ) m + k u , m + k u ) ε 2 a 00 m + k u 2 + ε 3 k u 2 i = 1 m k ( 1 α i ) a 00 4 ( 1 α i ) ( m k ) 1 + α i 1 α i ( C m k i a i C i ) 2 1 α i v 2 + 2 ε i = 1 m k ( 1 α i ) a 00 8 α i ( m k ) α i α i 1 2 ( m k ) a 00 b 0 M 0 ( C m k i a i C i ) 2 1 1 α i v 2 + 2 i = 1 m k ( 1 α i ) ( ε 1 a 00 1 b 0 M 0 ) 1 1 α i a 00 8 α i ( m k ) α i α i 1 ( C m k i a i C i ( 1 + m u 2 q ) ) 2 1 α i v 2 + 4 ε 1 a 00 1 b 0 a 0 2 M 0 ( 1 + m u 2 q ) 2 w ( t ) 2 m + k h 2 + 2 λ 1 m a 00 k f ( x , t ) 2 + 2 C k 1 ( r 01 p ( τ , w ) + ϕ 5 ( x ) 2 ) + 5 a 00 1 a 0 2 w ( t ) 2 m + k 2 + 2 ε w ( t ) 2 k h 2 + 4 a 00 1 w ( t ) 2 i = 1 m k ( C m k i a i ) 2 λ 1 i m + k h 2 .

When d d t ( a ( x ) m + k u , m + k u ) + 2 ε ( a ( x ) m + k u , m + k u ) 0 ,

b 0 M ( m u 2 ) d d t ( a ( x ) m + k u , m + k u ) + 2 ε ( a ( x ) m + k u , m + k u ) d d t ( b 0 M 0 ( a ( x ) m + k u , m + k u ) ) + 2 ε b 0 M 0 ( a ( x ) m + k u , m + k u ) ;

else

b 0 M ( m u 2 ) d d t ( a ( x ) m + k u , m + k u ) + 2 ε ( a ( x ) m + k u , m + k u ) d d t ( b 0 C ( m u 2 ) ( a ( x ) m + k u , m + k u ) ) + 2 ε b 0 C ( m u 2 ) ( a ( x ) m + k u , m + k u ) ,

then (4.41) is transformed into

(4.42) d d t [ k v 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u , m + k u ) + ε 2 k u 2 ] + 2 ( a ( x ) m + k v , m + k v ) 6 + ε 4 a 00 m + k v 2 3 ε k v 2 + 2 ε b 0 M 0 ( C ( m u 2 ) ) ( a ( x ) m + k u , m + k u ) 3 ε a 00 b 0 M 0 4 m + k u 2 2 ε 2 ( a ( x ) m + k u , m + k u ) ε 2 a 00 m + k u 2 + ε 3 k u 2 i = 1 m k ( 1 α i ) a 00 4 ( 1 α i ) ( m k ) 1 + α i 1 α i ( C m k i a i C i ) 2 1 α i v 2 + 2 ε i = 1 m k ( 1 α i ) a 00 8 α i ( m k ) α i α i 1 2 ( m k ) a 00 b 0 M 0 ( C m k i a i C i ) 2 1 1 α i v 2 + 2 i = 1 m k ( 1 α i ) ( ε 1 a 00 1 b 0 M 0 ) 1 1 α i a 00 8 α i ( m k ) α i α i 1 ( C m k i a i C i ( 1 + m u 2 q ) ) 2 1 α i v 2 + 4 ε 1 a 00 1 b 0 a 0 2 M 0 ( 1 + m u 2 q ) 2 w ( t ) 2 m + k h 2 + 2 λ 1 m a 00 k f ( x , t ) 2 + 2 C k 1 ( r 01 p ( τ , w ) + ϕ 5 ( x ) 2 ) + 5 a 00 1 a 0 2 w ( t ) 2 m + k 2 + 2 ε w ( t ) 2 k h 2 + 4 a 00 1 w ( t ) 2 i = 1 m k ( C m k i a i ) 2 λ 1 i m + k h 2 .

By 2 ( a ( x ) m + k v , m + k v ) 2 a 00 m + k v 2 , ( a ( x ) m + k u , m + k u ) a 00 m + k u 2 , (4.1), we have

(4.43) d d t [ k v 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u , m + k u ) + ε 2 k u 2 ] + σ 1 [ k v 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u , m + k u ) + ε 2 k u 2 ] C k 2 ( 1 + w ( t ) 2 ) + 2 λ 1 m a 00 k f ( x , t ) 2 .

When k = m , (4.43) is also true, which will not be detailed here.

Using the Gronwall inequality to integrate (4.43) over [ τ t , τ ] and replacing w by θ τ w we obtain

(4.44) e σ 1 τ [ k v 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u , m + k u ) + ε 2 k u 2 ] e σ 1 ( τ t ) [ k v ( τ t ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u ( τ t ) , m + k u ( τ t ) ) + ε 2 k u ( τ t ) 2 ] + τ t τ e σ 1 ξ C k 2 ( 1 + w ( ξ ) 2 ) d ξ + 2 λ 1 m a 00 τ t τ e σ 1 ξ k f ( x , ξ ) 2 d ξ ,

moreover,

(4.45) k v 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u , m + k u ) + ε 2 k u 2 e σ 1 t [ k v ( τ t ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u ( τ t ) , m + k u ( τ t ) ) + ε 2 k u ( τ t ) 2 ] + e σ 1 τ τ t τ e σ 1 ξ C k 2 ( 1 + w ( ξ ) 2 ) d ξ + 2 λ 1 m a 00 e σ 1 τ τ t τ e σ 1 ξ k f ( x , ξ ) 2 d ξ .

Since ( u ( τ t ) , v ( τ t ) ) B k ( τ t , θ τ w ) , when t + ,

(4.46) e σ 1 t [ k v ( τ t ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u ( τ t ) , m + k u ( τ t ) ) + ε 2 k u ( τ t ) 2 ] e σ 1 t [ k v ( τ t ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) a 0 m + k u ( τ t ) 2 + ε 2 k u ( τ t ) 2 ] 0 ,

then there exists T k = T k ( τ , w , B k ) such that for all t T k

(4.47) e σ 1 t [ k v ( τ t ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) a 0 m + k u ( τ t ) 2 + ε 2 k u ( τ t ) 2 ] 1 .

When ξ w ( ξ ) at most polynomial growth,

e σ 1 τ τ e σ 1 ξ C k 2 ( 1 + w ( ξ ) 2 ) d ξ r 0 k ( τ , w ) .

We get from (4.45)–(4.47), ( a ( x ) m + k u , m + k u ) a 00 m + k u 2 that

(4.48) k v ( τ , τ t , θ τ w , v τ t ) 2 + m + k u ( τ , τ t , θ τ w , v τ t ) 2 C k 3 1 + 2 λ 1 m a 00 e σ 1 τ τ e σ 1 ξ k f ( x , ξ ) 2 d ξ + r 0 k ( τ , w ) r 1 k ( τ , w ) ,

and r 1 k ( τ , w ) are bounded.

Lemma 4.1 is derived from (4.23) and (4.48).

Lemma 4.1 is proved.

Considering the eigenvalue problem

( Δ ) m + k u = λ m + k u , u Γ = 0 ,

the problem (3.8) has a family of eigenfunctions { e j } j = 1 with the eigenvalues { λ j } j = 1 : λ 1 λ 2 λ j ( j ) , such that { e j } j = 1 is an orthonormal basis of L μ 2 ( Ω ) . Given n let Q n = span { e 1 , , e n } and P n : V k ( Ω ) Q n be the projection operator.

Lemma 4.2

Suppose M satisfies ( M ) , h ( x ) V m + k ( Ω ) , k = 0 , 1 , , m (3.2)–(3.6) hold f ( x , t ) satisfies ( F 1 ) and for η > 0 , τ R , w Ω 1 , B k = { B k ( τ , w ) : τ R , w Ω 1 } D k , there exists T k = T k ( τ , w , B k , η k ) > 0 , N k = N k ( τ , w , η k ) 0 such that the solution of (3.8) satisfies for t T k , n N k

( I P n ) v ( τ , τ t , θ τ w ) V k 2 + ( I P n ) u ( τ , τ t , θ τ w ) V m + k 2 η k .

Proof

Let u n , 1 = P n u , u n , 2 = u u n , 1 , v n , 1 = P n v , v n , 2 = v v n , 1 . Applying ( I P n ) to the second equation of (3.8), we obtain

(4.49) d v n , 2 d t = ε v n , 2 a ( x ) ( Δ ) m v n , 2 + ε a ( x ) ( Δ ) m u n , 2 ε 2 u n , 2 b ( x ) ( I P n ) M ( m 2 ) ( Δ ) m u ( I P n ) ( g ( x , u ) + f ( x , t ) ) + ε h ( x ) w ( t ) a ( x ) ( Δ ) m h ( x ) w ( t ) .

Taking the inner product of the resulting equation (4.49) with v n , 2 in L μ 2 ( Ω ) , we have

(4.50) 1 2 d d t v n , 2 L μ 2 2 = ε v n , 2 L μ 2 2 m v n , 2 2 + ε ( ( Δ ) m u n , 2 , v n , 2 ) ε 2 ( u n , 2 , v n , 2 ) L μ 2 b 0 ( ( I P n ) M ( m u 2 ) ( Δ ) m u , v n , 2 ) ( ( I P n ) g ( x , u ) , v n , 2 ) L μ 2 + ( ( I P n ) f ( x , t ) , v n , 2 ) L μ 2 + ε w ( t ) ( h ( x ) , v n , 2 ) L μ 2 w ( t ) ( ( Δ ) m h , v n , 2 ) .

Applying I P n to the first equation of (3.8), we get

(4.51) v n , 2 = d u n , 2 d t + ε u n , 2 h w ( t ) .

For the third and fourth terms on the right-hand side of (4.49), we obtain

(4.52) ε ( ( Δ ) m u n , 2 , v n , 2 ) ε 2 ( u n , 2 , v n , 2 ) L μ 2 = ε ( ( Δ ) m u n , 2 , d u n , 2 d t + ε u n , 2 h w ( t ) ) ε 2 ( u n , 2 , d u n , 2 d t + ε u n , 2 h w ( t ) ) L μ 2 = ε 2 d d t m u n , 2 2 + ε 2 m u n , 2 2 ε w ( t ) ( ( Δ ) m u n , 2 , h ) ε 2 2 d d t u n , 2 L μ 2 2 ε 3 u n , 2 L μ 2 2 + ε 2 w ( t ) ( u n , 2 , h ) L μ 2 ,

for the fifth term on the right-hand side of (4.49), we have

(4.53) b 0 ( ( I P n ) M ( m u 2 ) ( Δ ) m u , v n , 2 ) = b 0 ( M ( m u 2 ) ( Δ ) m u n , 2 , d u n , 2 d t + ε u n , 2 h w ( t ) ) = 1 2 b 0 M ( m u 2 ) d d t m u n , 2 2 + ε b 0 M ( m u 2 ) m u n , 2 2 b 0 M ( m u 2 ) w ( t ) ( ( Δ ) m u n , 2 , h ) ,

for the sixth term on the right-hand side of (4.49), we have

(4.54) ( ( I P n ) g ( x , u ) , v n , 2 ) L μ 2 = ( ( I P n ) g ( x , u ) , d u n , 2 d t + ε u n , 2 h w ( t ) ) L μ 2 = d d t ( ( I P n ) g ( x , u ) , u n , 2 ) L μ 2 ( ( I P n ) g u ( x , u ) u t , u n , 2 ) L μ 2 + ε ( ( I P n ) g ( x , u ) , u n , 2 ) L μ 2 ( ( I P n ) g ( x , u ) , h w ( t ) ) L μ 2 ,

for the seventh, eighth, and ninth terms on the right-hand side of (4.49), we have

(4.55) ( ( I P n ) f ( x , t ) , v n , 2 ) L μ 2 1 4 m v n , 2 2 + 1 α n + 1 λ n + 1 m 1 ( I P n ) f ( x , t ) L μ 2 2 ,

(4.56) ε w ( t ) ( h ( x ) , v n , 2 ) L μ 2 1 4 m v n , 2 2 + ε w ( t ) 2 α n + 1 λ n + 1 m 1 h ( x ) L μ 2 2 ,

(4.57) w ( t ) ( ( Δ ) m h , v n , 2 ) 1 4 m v n , 2 2 + ε w ( t ) 2 m h ( x ) 2 .

When d d t m u n , 2 2 + 2 ε m v n , 2 2 0

(4.58) b 0 M ( m u 2 ) d d t m u n , 2 2 + 2 ε m v n , 2 2 d d t ( b 0 M 0 m u n , 2 2 ) + 2 ε b 0 M 0 m v n , 2 2 ,

else

(4.59) b 0 M ( m u 2 ) d d t m u n , 2 2 + 2 ε m v n , 2 2 d d t ( b 0 C ( m u 2 ) m u n , 2 2 ) + 2 ε b 0 C ( m u 2 ) m v n , 2 2 .

By using Young’s inequality and Holder’s inequality, we can get

(4.60) 2 b 0 M ( m u 2 ) w ( t ) ( ( Δ ) m u n , 2 , h ) 2 b 0 M ( m u 2 ) w ( t ) m u n , 2 w ( t ) m h ε b 0 M 0 2 m u n , 2 2 + 2 b 0 C ( m u 2 ) ε M 0 w ( t ) 2 m h 2 ,

(4.61) 2 ε 2 w ( t ) ( h ( x ) , u n , 2 ) L μ 2 2 ε w ( t ) ( ( Δ ) m u n , 2 , h ( x ) ) ε 3 u n , 2 L μ 2 2 + ε 2 m u n , 2 2 + ε w ( t ) 2 h L μ 2 2 + w ( t ) 2 m h 2 .

By (3.5), we have

(4.62) 2 ( ( I P n ) g u ( x , u ) u t , u n , 2 ) L μ 2 2 ϕ 4 L μ 4 u t L μ 2 u n , 2 L μ 4 + 2 c 4 u t L μ 2 u L μ 2 p p 1 u n , 2 L μ 2 p ε b 0 M 0 2 m u n , 2 2 + C 05 ε b 0 M 0 λ n + 1 1 m u t L μ 2 2 + C 06 λ n + 1 1 m u t L μ 2 2 m u 2 p 2 ,

(4.63) 2 w ( t ) ( ( I P n ) g ( x , u ) , h ) L μ 2 2 w ( t ) ϕ 1 L μ 2 h L μ 2 + 2 c 1 w ( t ) u L μ 2 p p h L μ 2 2 w ( t ) ϕ 1 L μ 2 h L μ 2 + 2 C 07 w ( t ) m u p h L μ 2 .

By substituting (4.52)–(4.63) into (4.50), we have

(4.64) d d t [ v n , 2 L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 2 + ε 2 u n , 2 L μ 2 2 + 2 ( ( I P n ) g ( x , u ) , u n , 2 ) L μ 2 ] + 1 2 m v n , 2 2 2 ε v n , 2 L μ 2 2 + ( 2 ε M ( m u 2 ) ε M 0 ) b 0 m u 2 3 ε 2 m u 2 + ε 3 u L μ 2 2 + 2 ε ( ( I P n ) g ( x , u ) , u n , 2 ) L μ 2 1 + 2 b 0 C ( m u 2 ) ε M 0 + 2 ε w ( t ) 2 m h 2 + ε + 2 ε α n + 1 λ n + 1 m 1 w ( t ) 2 h ( x ) L μ 2 2 + C 05 ε b 0 M 0 λ n + 1 1 m u t L μ 2 2 + C 06 λ n + 1 1 m u t L μ 2 2 m u 2 p 2 + ( 2 ϕ 1 L μ 2 h L μ 2 + 2 C 08 m u p ) w ( t ) h L μ 2 + 2 α n + 1 λ n + 1 m 1 ( I p n ) f ( x , t ) L μ 2 2 .

Because, when N = 1 , 2 , then 1 p < + ; when N = 3 , 4 , then 1 p < N N 2 . Moreover, when n , λ n so given η 0 > 0 , there exists N 01 = N 01 ( η 0 ) 1 for n N 01

(4.65) 1 + 2 b 0 C ( m u 2 ) ε M 0 + 2 ε w ( t ) 2 m h 2 + ε + 2 ε α n + 1 λ n + 1 m 1 w ( t ) 2 h ( x ) L μ 2 2 + C 05 ε b 0 M 0 λ n + 1 1 m u t L μ 2 2 + C 06 λ n + 1 1 m u t L μ 2 2 m u 2 p 2 + ( 2 ϕ 1 L μ 2 h L μ 2 + 2 C 08 m u p ) w ( t ) h L μ 2 + 2 α n + 1 λ n + 1 m 1 ( I p n ) f ( x , t ) L μ 2 2 C 09 η 0 ( 1 + w ( t ) 2 + u t L μ 2 6 + m u 6 ) + 2 α n + 1 λ n + 1 m 1 ( I p n ) f ( x , t ) L μ 2 2 .

Then, there is an appropriate positive constant σ 2 so that (4.64) can be reduced to

(4.66) d d t [ v n , 2 L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 2 + ε 2 u n , 2 L μ 2 2 + 2 ( ( I P n ) g ( x , u ) , u n , 2 ) L μ 2 ] + σ 2 [ v n , 2 L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 2 + ε 2 u n , 2 L μ 2 2 + 2 ( ( I P n ) g ( x , u ) , u n , 2 ) L μ 2 ] C 09 η 0 ( 1 + w ( t ) 2 + u t L μ 2 6 + m u 6 ) + 2 α n + 1 λ n + 1 m 1 ( I p n ) f ( x , t ) L μ 2 2 ,

integrating (4.66) over ( τ t , τ ) with t 0 we get for all n N 01

(4.67) v n , 2 ( τ , τ t , w ) L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 ( τ , τ t , w ) 2 + ε 2 u n , 2 ( τ , τ t , w ) L μ 2 2 + 2 ( ( I P n ) g ( x , u ) , u n , 2 ( τ , τ t , w ) ) L μ 2

e σ 2 t [ v n , 2 ( τ t ) L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 ( τ t ) 2 + ε 2 u n , 2 ( τ t ) L μ 2 2 + 2 ( ( I P n ) g ( x , u ( τ t ) ) , u n , 2 ( τ t ) ) L μ 2 ] + C 09 η 0 τ t τ e σ ( s t ) ( 1 + w ( s ) 2 + u t ( s , τ t , w , u 1 τ ) L μ 2 6 + m u ( s , τ t , w , u 1 τ ) 6 ) d s + 2 α n + 1 λ n + 1 m 1 τ t τ e σ ( s t ) ( I p n ) f ( x , s ) L μ 2 2 d s .

Replacing w by θ τ w in (4.67) for every t R + , τ R , w Ω 1 , n N 01 , we obtain

(4.68) v n , 2 ( τ , τ t , θ τ w ) L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 ( τ , τ t , θ τ w ) 2 + ε 2 u n , 2 ( τ , τ t , θ τ w ) L μ 2 2 + 2 ( ( I P n ) g ( x , u ) , u n , 2 ( τ , τ t , θ τ w ) ) L μ 2 e σ 2 t [ v n , 2 ( τ t ) L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 ( τ t ) 2 + ε 2 u n , 2 ( τ t ) L μ 2 2 + 2 ( ( I P n ) g ( x , u ( τ t ) ) , u n , 2 ( τ t ) ) L μ 2 ] + C 09 η 0 τ t τ e σ ( s t ) ( 1 + w ( s ) 2 + u t ( s , τ t , θ τ w , u 1 τ ) L μ 2 6 + m u ( s , τ t , θ τ w , u 1 τ ) 6 ) d s + 2 α n + 1 λ n + 1 m 1 τ t τ e σ ( s t ) ( I p n ) f ( x , s ) L μ 2 2 d s .

From (3.8), f ( x , t ) satisfies ( F 1 ) , h V m and Lemma 4.1, we have

(4.69) v n , 2 ( τ , τ t , θ τ w ) L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 ( τ , τ t , θ τ w ) 2 + ε 2 u n , 2 ( τ , τ t , θ τ w ) L μ 2 2 + 2 ( ( I P n ) g ( x , u ) , u n , 2 ( τ , τ t , θ τ w ) ) L μ 2 e σ 2 t [ v n , 2 ( τ t ) L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 ( τ t ) 2 + ε 2 u n , 2 ( τ t ) L μ 2 2 + 2 ( ( I P n ) g ( x , u ( τ t ) ) , u n , 2 ( τ t ) ) L μ 2 ] + C 09 η 0 r 00 3 τ t τ e σ ( s t ) ( 1 + w ( s ) 2 + w ( s ) 6 ) d s + 2 α n + 1 λ n + 1 m 1 τ t τ e σ ( s t ) ( I p n ) f ( x , s ) L μ 2 2 d s ,

by ( u τ t , v τ t ) B 0 ( τ t , θ τ w ) , then

(4.70) e σ 2 t [ v n , 2 ( τ t ) L μ 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) m u n , 2 ( τ t ) 2 + ε 2 u n , 2 ( τ t ) L μ 2 2 + 2 ( ( I P n ) g ( x , u ( τ t ) ) , u n , 2 ( τ t ) ) L μ 2 ] 0 , t .

Taking the inner product of (4.49) with ( Δ ) k v n , 2 , k = 1 , 2 , , m 1 in L 2 ( Ω ) , we have

(4.71) 1 2 d d t k v n , 2 2 = ε k v n , 2 2 ( a ( x ) ( Δ ) m v n , 2 , ( Δ ) k v n , 2 ) + ε ( a ( x ) ( Δ ) m u n , 2 , ( Δ ) k v n , 2 ) ε 2 ( u n , 2 , ( Δ ) k v n , 2 ) ( b ( x ) M ( m u 2 ) ( Δ ) m u n , 2 , ( Δ ) k v n , 2 ) ( ( I P n ) g ( x , u ) , ( Δ ) k v n , 2 ) + ( f ( x , t ) , ( Δ ) k v n , 2 ) + ε w ( t ) ( h , ( Δ ) k v n , 2 ) w ( t ) ( a ( x ) ( Δ ) m h , ( Δ ) k v n , 2 ) .

Then applying I P n to the first equation of (3.8), we obtain

(4.72) v n , 2 = d u n , 2 d t + ε u n , 2 h w ( t ) .

Combining the processing method of Lemma 4.1,

(4.73) d d t [ k v n , 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u n , 2 , m + k u n , 2 ) + ε 2 k u n , 2 2 ] + 2 ( a ( x ) m + k v n , 2 , m + k v n , 2 ) 6 + ε 4 a 00 m + k v n , 2 2 3 ε k v n , 2 2 + 2 ε b 0 M 0 ( C ( m u 2 ) ) ( a ( x ) m + k u n , 2 , m + k u n , 2 ) 3 ε a 00 b 0 M 0 4 m + k u n , 2 2 2 ε 2 ( a ( x ) m + k u n , 2 , m + k u n , 2 ) ε 2 a 00 m + k u n , 2 2 + ε 3 k u n , 2 2 i = 1 m k ( 1 α i ) a 00 4 ( 1 α i ) ( m k ) 1 + α i 1 α i ( C m k i a i C i ) 2 1 α i v n , 2 2 + 2 ε i = 1 m k ( 1 α i ) a 00 8 α i ( m k ) α i α i 1 2 ( m k ) a 00 b 0 M 0 ( C m k i a i C i ) 2 1 1 α i v n , 2 2 + 2 ( ε 1 a 00 1 b 0 M 0 ) 1 1 α i i = 1 m k ( 1 α i ) a 00 8 α i ( m k ) α i α i 1 ( C m k i a i C i ( 1 + m u 2 q ) ) 2 1 α i v n , 2 2 + 4 ε 1 a 00 1 b 0 a 0 2 M 0 ( 1 + m u 2 q ) 2 w ( t ) 2 m + k h 2 + 2 λ n + 1 m a 00 k f ( x , t ) 2 + 2 C k 1 ( r 01 p ( τ , w ) + ϕ 5 ( x ) 2 ) + 5 a 00 1 a 0 2 w ( t ) 2 m + k h 2 + 2 ε w ( t ) 2 k h 2 + 4 a 00 1 w ( t ) 2 i = 1 m k ( C m k i a i ) 2 λ 1 i m + k h 2 .

Then there is a positive constant σ 1 (4.73) can be reduced to

(4.74) d d t [ k v n , 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u n , 2 , m + k u n , 2 ) + ε 2 k u n , 2 2 ] + σ 1 [ k v n , 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u n , 2 , m + k u n , 2 ) + ε 2 k u n , 2 2 ] C k 4 ( 1 + w ( t ) 2 ) + 2 λ n + 1 m a 00 k f ( x , t ) 2 .

when k = m , (4.74) also holds.□

Integrating (4.74) over ( τ t , τ ) with t 0 , we get for all n N k 1

(4.75) k v n , 2 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u n , 2 , m + k u n , 2 ) + ε 2 k u n , 2 2 e σ 1 t [ k v n , 2 ( τ t ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u n , 2 ( τ t ) , m + k u n , 2 ( τ t ) ) + ε 2 k u n , 2 ( τ t ) 2 ] + e σ 1 τ τ t τ e σ 1 ξ C k 4 ( 1 + w ( ξ ) 2 ) d ξ + 2 λ 1 m a 00 e σ 1 τ τ t τ e σ 1 ξ k f ( x , ξ ) 2 d ξ .

Replacing w by θ τ w in (4.75), we obtain for every t R + , τ R , w Ω 1 , n N k 1

(4.76) k v n , 2 ( τ , τ t , θ τ w ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) × ( a ( x ) m + k u n , 2 ( τ , τ t , θ τ w ) , m + k u n , 2 ( τ , τ t , θ τ w ) ) + ε 2 k u n , 2 ( τ , τ t , θ τ w ) 2 e σ 1 t [ k v n , 2 ( τ t ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u n , 2 ( τ t ) , m + k u n , 2 ( τ t ) ) + ε 2 k u n , 2 ( τ t ) 2 ] + e σ 1 τ τ t τ e σ 1 ξ C k 4 ( 1 + w ( ξ ) 2 ) d ξ + 2 λ 1 m a 00 e σ 1 τ τ t τ e σ 1 ξ k f ( x , ξ ) 2 d ξ .

From (3.8), h V m + k , f ( x , t ) satisfies ( F 1 ) Lemma 4.1 and ( u τ t , v τ t ) B k ( τ t , θ τ w ) for t +

(4.77) e σ 1 t [ k v n , 2 ( τ t ) 2 + ( b 0 M 0 ( C ( m u 2 ) ) ε ) ( a ( x ) m + k u n , 2 ( τ t ) , m + k u n , 2 ( τ t ) ) + ε 2 k u n , 2 ( τ t ) 2 ] 0 .

Combining (4.69), (4.70), (4.76), (4.77), and Lemma 4.1, we can get the conclusion of Lemma 4.2.

Lemma 4.2 is proved.

5 The existence of the family of random attractors

In this section, we shall prove the existence of the family of random pullback attractors for system (3.8). From Lemma 4.1, we know that for P a.e. D k = { D k ( τ , w ) : τ R , w Ω 1 } D k and w Ω 1 , there exists T k = T k ( D k , w ) such that for all t T k

(5.1) v ( τ , τ t , θ τ w , v τ t ) V k 2 + m + k u ( τ , τ t , θ τ w , u τ t ) V m + k 2 r 1 k ( τ , w ) .

Let

(5.2) B k ( τ , w ) = { ( u , v ) V m + k × V k : v V k 2 + u V m + k 2 r 1 k ( τ , w ) } .

Then, by (5.2), B k = { B k ( τ , w ) } w Ω 1 are the closed absorption sets of Φ k in X k . We are now ready to prove the asymptotic compactness of Φ k in X k .

Lemma 5.1

Suppose M satisfies ( M ) , h ( x ) V m + k ( Ω ) , (3.2)–(3.6) hold f ( x , t ) satisfies, ( F 1 ) , then Φ k is asymptotically compact in X k , that is, for every τ R , w Ω 1 , the sequence { Φ k ( t i , τ t i , θ t i w , ( u τ , i , v τ , i ) ) } has a convergent subsequence in X k provided t i and

( u τ , i , v τ , i ) D k ( τ t i , θ t i w ) ; D k = { D k ( τ , w ) : τ R , w Ω 1 } D k .

Proof

We first let t i , it follows from Lemma 4.1 that there exist i 1 = i 1 ( τ , w , D k ) > 0 such that for every i i 1

(5.3) v ( τ , τ t i , θ τ w , v τ ) V k 2 + u ( τ , τ t i , θ τ w , u τ ) V m + k 2 r 1 k ( τ , w ) ,

next by using Lemma 4.2 for η k > 0 , there are i k 2 = i k 2 ( η k , w , B k ) and N k = N k ( η k , w ) > 0 such that for every i k i k 2

(5.4) ( I P n ) v ( τ , τ t k i , θ τ w , v τ ) V k 2 + ( I P n ) u ( τ , τ t k i , θ τ w , u τ ) V m + k 2 η k .

By using (5.3), we find that { P N k ( u ( τ , τ t i , w ) , v ( τ , τ t i , w ) ) } is bounded in P N k X k and P N k X k is finite dimensional, which associates with (5.4) implies that { ( u ( τ , τ t i , w ) , v ( τ , τ t i , w ) ) } is precompact in X k .□

Theorem 5.2

Suppose M satisfies ( M ) , h ( x ) V m + k ( Ω ) , (3.2)–(3.6) hold f ( x , t ) satisfies ( F 1 ) , then the family of cocycles Φ k generated by (3.8) has a family of pullback D k attractors { A k } = { { A k ( τ , w ) } D k ( k = 1 , 2 , , m ) } in X k and can be expressed as follows:

A k ( τ , w ) = τ 0 t τ Φ k ( t , τ t , θ t w , B k ( τ t , θ t w ) ) ¯ , τ R , w Ω 1 .

Proof

From (5.2), Lemmas 5.1 and 2.10, the conclusion of Theorem 5.2 can be obtained.

Theorem 5.2 is proved.□

Note 5.3

Theorem 5.2 shows the family of cocycles Φ k generated by (3.8) has a unique pullback attractor A k , respectively, in the space X k ( k = 0 , 1 , , m ) , which together form a family of pullback attractors { A k } . At the same time, according to Lemma 4.1 and (5.2) and the tight embedding of X k X 0 , k = 1 , 2 , , m , get the corresponding a family of pullback attractors { A k } , which is ( X k , X 0 ) the family of random weak attractors, which means that the family of cocycles Φ k has uniformly asymptotically compact absorption sets B k ( τ , w ) X 0 , k = 1 , 2 , , m , where B k ( τ , w ) are the bounded sets in X k , i.e., Φ k are asymptotically compact in X 0 .

Acknowledgements

The authors thank the referees for valuable comments and suggestions, which improved the presentation of this manuscript.

  1. Funding information: This work was partially supported by the fundamental research fund of Yunnan Education Department (2020J0908) and the basic science (NATURAL SCIENCE) research project of colleges and universities in Jiangsu Province (21KJB110013).

  2. Conflict of interest: The authors state no conflict of interest.

  3. Data availability statement: No data, models,or code are generated or used during the study.

References

[1] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations 252 (2011), no. 2, 1229–1262, https://doi.org/10.1016/j.jde.2011.08.022. Search in Google Scholar

[2] Z. Yang and P. Ding, Long-time dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on RN, J. Math. Anal. Appl. 434 (2016), no. 2, 1826–1851, https://doi.org/10.1016/j.jmaa.2015.10.013. Search in Google Scholar

[3] Z. Yang and Z. Liu, Exponential attractor for the Kirchhoff equations with strong nonlinear damping and supercritical nonlinearity, Appl. Math. Lett. 46 (2015), 127–132, https://doi.org/10.1016/j.aml.2015.02.019. Search in Google Scholar

[4] G. Lin, P. Lv, and R. Lou, Exponential attractors and inertial manifolds for a class of nonlinear generalized Kirchhoff-Boussinesq model, Far East J. Math. Sci. 101 (2017), no. 9, 1913–1945, https://doi.org/10.17654/MS101091913. Search in Google Scholar

[5] H. Ma, J. Zhang, and C. Zhong, Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension, J. Math. Anal. Appl. 484 (2020), 123670, https://doi.org/10.1016/j.jmaa.2019.123670.10.1016/j.jmaa.2019.123670Search in Google Scholar

[6] Y. Li, Z. Yang, and P. Ding, Regular solutions and strong attractors for the Kirchhoff wave model with structural nonlinear damping, Appl. Math. Lett. 104 (2020), 106258, https://doi.org/10.1016/j.aml.2020.106258. Search in Google Scholar

[7] K. Ono, Global existence, decay and blow up of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations 137 (1997), no. 2, 273–301, https://doi.org/10.1006/jdeq.1997.3263. Search in Google Scholar

[8] S. A. Messaoudi and B. Said Houari, A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation, Appl. Math. Lett. 20 (2007), no. 8, 866–871, https://doi.org/10.1016/j.aml.2006.08.018. Search in Google Scholar

[9] F. Li, Global existence and blow-up of solutions for a higher-order Kirchhoff-type equation with nonlinear dissipation, Appl. Math. Lett. 17 (2004), no. 12, 1409–1414, https://doi.org/10.1016/j.am1.2003.07.014. Search in Google Scholar

[10] P. G. Papadopoulos and N. M. Stavrakakis, Global existence and blow-up results for an equation of Kirchhoff type on RN, Topol. Methods Nonlinear Anal. 17 (2001), no. 1, 91–109, https://doi.org/10.12775/TMNA.2001.006. Search in Google Scholar

[11] H. Chen and G. Liu, Well-posedness for a class of Kirchhoff equations with damping and memory terms, IMA J. Appl. Math. 80 (2015), no. 6, 1808–1836, https://doi.org/10.1093/imamat/hxv018. Search in Google Scholar

[12] H. Li, Y. You, and J. Tu, Random attractors and averaging for nonautonomous stochastic wave equations with nonlinear damping, J. Differential Equations 258 (2015), no. 1, 148–190, https://doi.org/10.1016/j.jde.2014.09.007. Search in Google Scholar

[13] S. Zhou and M. Zhao, Random attractors for damped nonautonomous wave equations with memory and white noise, Nonlinear Anal. 120 (2015), 202–226, https://doi.org/10.1016/j.na.2015.03.009. Search in Google Scholar

[14] B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations 246 (2008), no. 6, 2506–2537, https://doi.org/10.1016/j.jde.2008.10.012. Search in Google Scholar

[15] Z. Wang, S. Zhou, and A. Gu, Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal. Real World Appl. 12 (2011), no. 6, 3468–3482, https://doi.org/10.1016/j.nonrwa.2011.06.008. 10.1016/j.nonrwa.2011.06.008Search in Google Scholar

[16] B. Wang, Random attractors for nonautonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. 34 (2014), no. 1, 269–300, https://doi.org/10.3934/dcds.2014.34.269. Search in Google Scholar

[17] Z. Wang and S. Zhou, Random attractor for stochastic nonautonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst. 37 (2016), no. 1, 545–573, https://doi.org/10.3934/dcds.2017022. Search in Google Scholar

[18] Z. Wang, S. Zhou, and A. Gu, Random attractor of the stochastic strongly damped wave equation, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 4, 1649–1658, https://doi.org/10.1016/j.cnsns.2011.09.001. Search in Google Scholar

[19] S. Zhou, F. Yin, and Z. Ouyang, random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst. 4 (2005), no. 4, 883–903, https://doi.org/10.1137/050623097. Search in Google Scholar

[20] H. Yao and J. Zhang, Random attractors for nonautonomous stochastic wave equations with nonlinear damping and white noise, Adv. Difference Equ. 2020 (2020), 221, https://doi.org/10.1186/s13662-020-02664-3. Search in Google Scholar

[21] M. A. Ragusa, Parabolic systems with noncontinuous coefficients, Conference Publications 2003 (2003), 727–733. Search in Google Scholar

[22] E. Guariglia, Primality, fractality, and image analysis, Entropy 21 (2019), no. 3, 304, https://doi.org/10.3390/e21030304. 10.3390/e21030304Search in Google Scholar PubMed PubMed Central

[23] S. Gala and M. A. Ragusa, A new regularity criterion for strong solutions to the Ericksen-Leslie system, Appl. Math. 43 (2016), no. 1, 95–103, https://doi.org/10.4064/am2281-1-2016. Search in Google Scholar

[24] E. Guariglia, Harmonic Sierpinski gasket and applications, Entropy 20 (2018), no. 9, 714, https://doi.org/10.3390/E20090714. 10.3390/e20090714Search in Google Scholar PubMed PubMed Central

[25] P. G. Papadopoulos, M. Karamolengos, and A. Pappas, Global existence and energy decay for mildly degenerate Kirchhoffas equations on RN, J. Interdiscip. Math. 12 (2009), no. 6, 767–783, https://doi.org/10.1080/09720502.2009.10700663. 10.1080/09720502.2009.10700663Search in Google Scholar

[26] N. I. Karachalios and N. M. Stavrakakis, Existence of a global attractor for semilinear dissipative wave equations on RN, J. Differential Equations 157 (1999), no. 1, 183–205, https://doi.org/10.1006/jdeq.1999.3618. Search in Google Scholar

[27] Y. Li, A. Gu, and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations 258 (2015), no. 2, 504–534, https://doi.org/10.1016/j.jde.2014.09.021. Search in Google Scholar

[28] Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations 245 (2008), no. 7, 1775–1800, https://doi.org/10.1016/j.jde.2008.06.031. 10.1016/j.jde.2008.06.031Search in Google Scholar

[29] J. Limaco, H. R. Clark, and L. A. Medeiros, On damped Kirchhoff equation with variable coefficients, J. Math. Anal. Appl. 307 (2004), no. 2, 641–655, https://doi.org/10.1016/j.jmaa.2004.12.032. Search in Google Scholar

[30] M. A. Ragusa, A. Razani, and F. Safari, Existence of radial solutions for a p(x)p(x)-Laplacian Dirichlet problem, Adv. Difference Equ. 2021 (2021), 215, https://doi.org/10.1186/s13662-021-03369-x. Search in Google Scholar

[31] S. Zhang, J. Zhang, and H. Wang, Global attractors for Kirchhoff wave equation with nonlinear damping and memory, Bound. Value Probl. 2020 (2020), 116, https://doi.org/10.1186/s13661-020-01413-5. Search in Google Scholar

[32] G. Zhou, Random attractor for the 3D viscous primitive equations driven by fractional noises, J. Differential Equations 266 (2019), no. 11, 7569–7637, https://doi.org/10.1016/j.jde.2018.12.009. Search in Google Scholar

[33] B. Gess, W. Liu, and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differential Equations 269 (2020), no. 4, 3414–3455, https://doi.org/10.1016/j.jde.2020.03.002. Search in Google Scholar

[34] G. Lin and Y. Chen, A family of global attractors for the higher-order Kirchhoff-type equations and its dimension estimation, Amer. J. Appl. Math. 8 (2020), no. 6, 300–310. 10.11648/j.ajam.20200806.12Search in Google Scholar

[35] G. Lin and Y. Jin, Long-time behavior of solutions for a class of nonlinear higher order Kirchhoff equation, Amer. J. Appl. Math. 7 (2019), no. 1, 21–29, https://doi.org/10.11648/j.ajam.20190701.14. Search in Google Scholar

Received: 2021-06-20
Revised: 2021-07-18
Accepted: 2021-07-19
Published Online: 2022-03-10

© 2022 Penghui Lv et al., published by De Gruyter

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

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/math-2022-0003/html
Scroll to top button