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In this paper, we mainly use operator decomposition technique to prove the global attractors which in

In this paper, we discuss the regularity of global attractors for the following Kirchhoff wave equation

u t t − ( 1 + ϵ ‖ ∇ u ‖ 2 ) Δ u − Δ u t + f ( u t ) + h ( u ) = g ( x ) in Ω × ℝ + , (1.1)

u | ∂ Ω = 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , (1.2)

where Ω is a bounded domain in ℝ 3 with the smooth boundary ∂ Ω , ϵ ∈ [ 0,1 ] , f ( m ) and h ( m ) are nonlinear functions and g ( x ) is an external force term which is independent of time.

G. Kirchhoff [

u t t − Δ u − Δ u t + f ( u t ) + h ( u ) = g ( x ) , (1.3)

which described the thermal evolution and h ( u ) denoted a source term depending nonlinearly on displacement, f ( u t ) denoted a nonlinearly temperature-dependent internal source term [

Since ϵ > 0 , the following quasi-linear wave equation of Kirchhoff type

u t t − ( 1 + ‖ ∇ u ( t ) ‖ 2 ) Δ u + u t + g ( u ) = f ( x ) (1.4)

was studied by M. Nakao, and the author proved the existence and absorbing properties of attractors in a local sense [

Based on these, the purpose of this paper is to prove the global attractor of problem (1.1)-(1.2), which attracts every H 0 1 ( Ω ) × L 2 ( Ω ) -bounded set that is compacted in H 2 ( Ω ) × H 0 1 ( Ω ) by the way in ( [

The paper is arranged as follows. In Section 2, we verify some preliminaries. In Section 3, we prove the existence of the global attractor. In Section 4, we prove the regularity of the global attractor.

Let A = − Δ on L 2 ( Ω ) with D ( A ) = H 2 ∩ H 0 1 , and A strictly positive on H 0 1 . We define the spaces H m = D ( A m 2 ) , ( m ∈ ℝ ) are Hilbert spaces with the following scalar products and the norms

〈 u , v 〉 m = 〈 A m 2 u , A m 2 v 〉 , ‖ u ‖ H m = ‖ A m 2 u ‖ . (2.1)

Let λ 1 ( > 0 , λ < λ 1 ) be the first eigenvalue of A, then B = A − λ I with D ( B ) = D ( A ) .

We define the phase space X = H 0 1 × L 2 with usual graph norm. Let φ ( ξ ) = f ( ξ ) + λ ξ , then problem (1.1)-(1.2) becomes

u t t + ( 1 + ϵ ‖ A 1 2 u ‖ 2 ) A u + B u t + φ ( u t ) + h ( u ) = g , (2.2)

u ( 0 ) = u 0 , u t ( 0 ) = u 1 . (2.3)

For any s > r , we have the continuous embeddings ,

(2.4)

and the following inequalities hold true:

Interpolation inequality: if r = θ s + ( 1 − θ ) q , where r , s , q ∈ ℝ , s ≥ q and θ ∈ [ 0,1 ] , then there exists a constant C > 0 such that

‖ u ‖ r ≤ C ‖ u ‖ s θ ‖ u ‖ q 1 − θ , ∀ u ∈ H s . (2.5)

The Generalized Poincare inequality:

λ 1 ‖ u ‖ α 2 ≤ ‖ u ‖ α + 1 2 , ∀ u ∈ H α + 1 , (2.6)

where λ 1 > 0 is the first eigenvalue of A.

The Young’s inequality with ε : Let a > 0 , b > 0 , ε > 0 , p > 1 , q > 1 , and 1 p + 1 q = 1 , then

a b ≤ ε a p p + ε − q p b q q , (2.7)

especially, when p = q = 2 , then

a b ≤ ε a 2 + b 2 4 ε . (2.8)

The Gronwall inequality (differential form): let η ( ⋅ ) is nonnegative continuous differentiable function (or nonnegative absolutely continuous function), and satisfy

η ′ ( t ) ≤ ϕ ( t ) η ( t ) + φ ( t ) , t ∈ [ 0, T ] , (2.9)

here ϕ ( t ) , φ ( t ) are nonnegative integrable functions, then

η ( t ) ≤ e ∫ 0 t ϕ ( s ) d s [ η ( 0 ) + ∫ 0 t φ ( s ) d s ] , ∀ t ∈ [ 0, T ] . (2.10)

Throughout this paper, we will denote by C a positive constant which is various in different line or even in the same line and use the following abbreviations:

L p = L p ( Ω ) , ‖ ⋅ ‖ = ‖ ⋅ ‖ L 2 , ‖ u ‖ m = ‖ u ‖ H m , ‖ u ‖ 1 = ‖ u ‖ H 0 1

with p ≥ 1 .

Assumption 2.1.

1) φ ∈ C 1 ( ℝ ) , φ ( 0 ) = 0 , and

0 ≤ φ ′ ( s ) ≤ C ( 1 + | s | q − 1 ) , s ∈ ℝ , (2.11)

where 1 ≤ q ≤ p * ≡ N + 2 N − 2 = 5 if N = 3

2) h ∈ C 1 ( ℝ ) , h ( 0 ) = 0 ,

lim inf | s | → ∞ h ′ ( s ) > − λ 1 , | h ′ ( s ) | ≤ C ( 1 + | s | p − 1 ) , s ∈ ℝ , (2.12)

where 1 ≤ p ≤ p * = 5 if N = 3 .

3)

g ∈ L 2 , ( u 0 , u 1 ) ∈ X with ‖ ( u 0 , u 1 ) ‖ X ≤ R (2.13)

Definition 2.2. Let S ( t ) t ≥ 0 be a semigroup on a metric space ( E , d ) . A subset A of E is called a global attractor for the semigroup, if A is compact and enjoys the following properties:

1) A is invariant, i.e. S ( t ) A = A , ∀ t ≥ 0 ;

2) A attracts all bounded set of E. That is, for any bounded subset B of E,

d i s t ( S ( t ) B , A ) → 0, as t → 0.

Next we only formulate the following results, which is proved in [

Lemma 2.3. Let (2.11)-(2.13) be valid. Then problem (2.2)-(2.3) admits a unique weak solution u, with ( u , u t ) ∈ L ∞ ( ℝ + ; X ) ∩ C ( ℝ + ; X ) , u t ∈ L 2 ( ℝ + ; H 0 1 ) . Moreover, this solution possesses the following properties:

(Dissipativity)

‖ ( u , u t ) ( t ) ‖ X 2 + ∫ t ∞ ( ‖ u t ( τ ) ‖ H 0 1 2 + ( φ ( u t ) , u t ) ) d τ ≤ C ( R ) e − k t + C 0 , t ≥ 0, (2.14)

where k denotes a small positive constant, C ( R ) and C 0 = C ( ‖ f ‖ H − 1 ) are positive constants.

Lemma 2.4. Let (2.11)-(2.13) be valid and when p = 5 , h ∈ C 2 ( ℝ ) . Then

‖ u t ( t ) ‖ H 0 1 2 + ‖ u t t ( t ) ‖ L 2 2 ≤ R 0 2 , t > 0. (2.15)

Actually, by exploiting (2.11) and (2.14), we can get u , u t are respectively bounded in H 0 1 , L 2 .

For every fixed x ∈ B 0 , we split the solution S ( t ) x = ( u ( t ) , u t ( t ) ) into the sum η ^ ( t ) + ζ ^ ( t ) , where η ^ ( t ) = ( v ^ ( t ) , v ^ t ( t ) ) and ζ ^ ( t ) = ( w ^ ( t ) , w ^ t ( t ) ) solve the Cauchy problems

{ v ^ t t + ( 1 + ϵ ‖ A 1 2 u ^ ‖ 2 ) A v ^ + B v ^ t + φ 0 ( v ^ t ) + h 0 ( v ^ ) = 0 , η ^ ( 0 ) = x , (3.1)

{ w ^ t t + ( 1 + ϵ ‖ A 1 2 u ^ ‖ 2 ) A w ^ + B w ^ t = ρ ^ , ζ ^ ( 0 ) = 0 , (3.2)

here

ρ ^ = g − [ φ 0 ( u t ) + h 0 ( u ) ] + [ φ 0 ( v ^ t ) + h 0 ( v ^ ) ] + [ φ 1 ( u t ) + h 1 ( u ) ] .

Having set φ ( u t ) + h ( u ) = [ φ 0 ( u t ) + h 0 ( u ) ] + [ φ 1 ( u t ) + h 1 ( u ) ] , and satisfying

φ 0 ( u t ) u t ≥ 0, φ ′ 0 ( u t ) ≥ C , | φ 0 ( u t ) − φ 0 ( v t ) | ≤ C | u t − v t | ( | u t | 4 + | v t | 4 ) , | φ 1 ( u t ) | ≤ C ( 1 + | u t | ) . (3.3)

h 0 ( u ) u ≥ 0, | h 0 ( u ) − h 0 ( v ) | ≤ C | u − v | ( | u | 4 + | v | 4 ) , | h 1 ( u ) | ≤ C ( 1 + | u | ) . (3.4)

From now on, c 0 , υ 0 > 0 and J 0 will denote generic constants and a generic function, respectively, depending only on B 0 .

Theorem 3.1. Let (2.11)-(2.13) be valid, then the solution semigroup S ( t ) possesses a global attractor B in X.

Proof. Estimate (2.14) shows

‖ ( u , u t ) ( t ) ‖ X 2 ≤ C ( R ) e − k t + C 0 , t ≥ 0,

such that the ball B 0 = { ( u , u t ) ∈ X | ‖ ( u , u t ) ‖ X ≤ R 0 } is an absorbing set of the semigroup S ( t ) in X for R 0 > C 0 ( ‖ g ‖ H − 1 ) .

In order to prove the existence of the global attractors, now we need to prove the asymptotic compactness.

Multiplying the first equation of (3.1) by v ^ t + γ v ^ and integrating over Ω , we get

〈 v ^ t t + ( 1 + ϵ ‖ A 1 2 u ^ ‖ 2 ) A v ^ + B v ^ t + φ 0 ( v ^ t ) + h 0 ( v ^ ) , v ^ t + γ v ^ 〉 = 0.

By using ϵ ≥ 0, φ 0 ( u t ) u t ≥ 0, h 0 ( u ) u ≥ 0 and the generalized Poincare inequality, then

1 2 d d t [ ‖ v ^ t ‖ 2 + ‖ v ^ ‖ 1 2 + γ ( ‖ v ^ ‖ 1 2 + 〈 v ^ t , v ^ 〉 − λ ‖ v ^ ‖ 2 ) ] ≤ ( γ − λ 1 + λ ) ‖ v ^ t ‖ 2 − γ ‖ v ^ ‖ 1 2 − γ ∫ Ω φ 0 ( v ^ t ) v ^ d x − ∫ Ω h 0 ( v ^ ) v ^ t d x ,

By λ < λ 1 , we know

1 2 d d t [ ‖ v ^ t ‖ 2 + ‖ v ^ ‖ 1 2 + γ ( ‖ v ^ ‖ 1 2 + 〈 v ^ t , v ^ 〉 − λ ‖ v ^ ‖ 2 ) ] ≥ 1 2 d d t [ ‖ v ^ t ‖ 2 + ‖ v ^ ‖ 1 2 + γ ( 〈 v ^ t , v ^ 〉 − λ 1 ‖ v ^ ‖ 2 ) ] ,

where γ > 0 is small enough such that

E ( t ) = 1 2 [ ‖ v ^ t ‖ 2 + ‖ v ^ ‖ 1 2 + γ ( 〈 v ^ t , v ^ 〉 − λ 1 ‖ v ^ ‖ 2 ) ] ∼ 1 2 [ ‖ v ^ t ‖ 2 + ‖ v ^ ‖ 1 2 ] . (3.5)

Actually, noting that φ ′ 0 ( v ^ t ) ≥ C , and by exploiting (2.8) and (2.12), we deduce that

− ∫ Ω φ 0 ( v ^ t ) v ^ d x = − ∫ Ω φ 0 ( v ^ t ) − φ ( 0 ) v ^ t − 0 v ^ t v ^ d x = − ∫ Ω φ ′ 0 ( v ^ t ) v ^ t v ^ d x ≤ − C 〈 v ^ t , v ^ 〉 , (3.6)

and

− ∫ Ω h 0 ( v ^ ) v ^ t d x ≤ ∫ Ω λ 1 v ^ t v ^ d x ≤ 1 2 γ ‖ v ^ ‖ 2 + γ λ 1 2 ‖ v ^ t ‖ 2 . (3.7)

From (3.5)-(3.7), we get

d d t E ( t ) ≤ ( γ + λ − λ 1 + γ λ 1 2 ) ‖ v ^ t ‖ 2 − γ ‖ v ^ ‖ 1 2 + 1 2 γ ‖ v ^ ‖ 2 − C 〈 v t , v 〉 ≤ − C E ( t ) ,

where γ > 0 is small enough such that ( γ + λ − λ 1 + γ λ 1 2 ) is negative. Furthermore, by the Gronwall inequality, we can get

‖ η ^ ( t ) ‖ H 0 1 × L 2 ≤ c 0 e − υ 0 t ‖ x ‖ H 0 1 × L 2 . (3.8)

Next multiplying the first equation of (3.2) by A 1 4 ω ^ t + γ A 1 4 ω ^ and integrating over Ω , we get

| 〈 w ^ t t + ( 1 + ϵ ‖ A 1 2 u ‖ 2 ) A w ^ + B w ^ t , A 1 4 w ^ t + γ A 1 4 w ^ 〉 | = d d t [ 1 2 ‖ w ^ t ‖ 1 4 2 + 1 2 ‖ w ^ ‖ 5 4 2 + γ ( 1 2 ‖ w ^ ‖ 5 4 2 − λ 2 ‖ w ^ ‖ 1 4 2 + 〈 w ^ t , A 1 4 w ^ 〉 ) ] + ϵ ‖ A 1 2 u ‖ 2 〈 A w ^ , A 1 4 w ^ t 〉 + ‖ w ^ t ‖ 5 4 2 − λ ‖ w ^ t ‖ 1 4 2 − γ ‖ w ^ t ‖ 1 4 2 + γ ‖ w ^ ‖ 5 4 2 + ϵ γ ‖ A 1 2 u ‖ 2 〈 A w ^ , A 1 4 w ^ 〉 ≥ d d t ( 1 2 ‖ w ^ t ‖ 1 4 2 + 1 2 ‖ w ^ ‖ 5 4 2 ) + ‖ w ^ t ‖ 5 4 2 − λ ‖ w ^ t ‖ 1 4 2 − γ ‖ w ^ t ‖ 1 4 2 + γ ‖ w ^ ‖ 5 4 2 , (3.9)

where γ > 0 is small enough. Then we define the energy functional

E 1 ( t ) = 1 2 ( ‖ w ^ t ‖ 1 4 2 + ‖ w ^ ‖ 5 4 2 ) , (3.10)

At the same time, by the interpolation inequality, we have

‖ ρ ^ ‖ L 4 3 ( Ω ) = ‖ g + ( φ 0 ( v ^ t ) − φ 0 ( u t ) ) + ( h 0 ( v ^ ) − h 0 ( u ) ) + ( φ 1 ( u t ) + h 1 ( u ) ) ‖ ≤ C ‖ g ‖ + C ‖ w ^ ‖ 5 4 ( ‖ v ^ ‖ 1 4 + ‖ u ‖ 1 4 ) + C ‖ w ^ t ‖ 5 4 ( ‖ v ^ t ‖ 1 4 + ‖ u t ‖ 1 4 ) + C ( 1 + ‖ u t ‖ 1 ) + C ( 1 + ‖ u ‖ 1 ) ≤ c 0 e − υ 0 t ‖ x ‖ H 1 × L 2 2 ‖ w ^ ‖ 5 4 + c e − k t ‖ w ^ t ‖ 5 4 + C ( ‖ g ‖ + ‖ u ‖ 1 + ‖ u t ‖ 1 + 1 ) ≤ c 0 e − υ 0 t ‖ w ^ ‖ 5 4 + c e − k t ‖ w ^ t ‖ 5 4 + C ( 1 + e − k t ) ,

and by the embedding , then

| 〈 ρ ^ , A 1 4 w ^ t + γ A 1 4 w ^ 〉 | ≤ ‖ ρ ^ ‖ L 4 3 ( ‖ A 1 4 w ^ t ‖ L 4 + γ ‖ A 1 4 w ^ ‖ L 4 ) ≤ ‖ ρ ^ ‖ ( ‖ A 1 4 w ^ t ‖ 3 4 + γ ‖ A 1 4 w ^ ‖ 3 4 ) ≤ c 0 e − υ 0 t ‖ w ^ ‖ 5 4 2 + c e − k t ‖ w ^ t ‖ 5 4 2 + δ ‖ w ^ t ‖ 5 4 2 + γ δ ‖ w ^ ‖ 5 4 2 + C ( 1 + e − k t ) . (3.11)

By exploiting (2.8) and the generalized Poincare inequality, from (3.9)-(3.11), we get

d d t E 1 ( t ) ≤ ( c 0 e − k t + δ − 1 ) ‖ w ^ t ‖ 5 4 2 + ( λ + γ ) ‖ w ^ t ‖ 1 4 2 + ( c 0 e − υ 0 t + γ δ − γ ) ‖ w ^ ‖ 5 4 2 + C ( 1 + e − k t ) ≤ − C λ 1 ‖ w ^ t ‖ 1 4 2 + ( λ + γ ) ‖ w ^ t ‖ 1 4 2 + ( c 0 e − υ 0 t − C ) ‖ w ^ ‖ 5 4 2 + C ( 1 + e − k t ) ≤ − ( C − c 0 e − υ 0 t ) E 1 ( t ) + C ( 1 + e − k t ) ,

where δ > 0 is small enough and by λ < λ 1 , we get ( − C λ 1 + λ + γ ) , ( γ δ − γ ) are negative. Then from the Gronwall inequality and noting that ζ ^ ( 0 ) = ( ω ^ ( 0 ) , ω ^ t ( 0 ) ) = 0 , we get

E 1 ( t ) ≤ c 0 e − ν t E 1 ( 0 ) + C ( 1 + e − k t ) ≤ C ( 1 + e − k t )

which provides the following estimate

‖ ζ ^ ( t ) ‖ H 5 4 × H 1 4 ≤ C ( 1 + e − k t ) , (3.12)

From (3.8) and (3.12), we obtain that the evolution semigroup S ( t ) is asymptotically compact in X, so the solution semigroup S ( t ) possesses a global attractor B in H 0 1 × L 2 , which

B = ∩ t 0 ≥ 0 ∪ t ≥ t 0 S ( t ) B 0 ¯ ,

where t 0 > 0 is chosen such that S ( t ) B 0 ⊂ B 0 for t ≥ t 0 .

Now we are in a position to state and prove the main result:

Theorem 4.1. The attractor B of the semigroup S ( t ) on X is bounded in H 2 × H 0 1 .

Proof. Having set x = y + z . For y ∈ B 0 , z ∈ H 2 × H 0 1 , we split the solution into the sum

S ( t ) x = Y ( t ) y + Z ( t ) z ,

where η ( t ) = Y ( t ) y = ( v ( t ) , v t ( t ) ) and ζ ( t ) = Z ( t ) z = ( w ( t ) , w t ( t ) ) solve the following equations with initial data η ( 0 ) = y , ζ ( 0 ) = z ,

{ v t t + ( 1 + ϵ ‖ A 1 2 u ‖ 2 ) A v + B v t = 0 , η ( 0 ) = x , (4.1)

and

{ w t t + ( 1 + ϵ ‖ A 1 2 u ‖ 2 ) A w + B w t = ρ , ζ ( 0 ) = 0 , (4.2)

where ρ ( t ) = g + φ ( u t ) + h ( u ) .

Multiplying the first equation of (4.1) by v t + γ v and integrating over Ω , by ϵ > 0 we get

1 2 d d t [ ‖ v t ‖ 2 + ‖ v ‖ 1 2 + γ 2 ( ‖ v ‖ 1 2 − λ ‖ v ‖ 2 + 〈 v t , v 〉 ) ] + γ ‖ v ‖ 1 2 + ‖ v t ‖ 1 2 − λ ‖ v t ‖ 2 ≤ 0 ,

where γ > 0 is small enough such that

E 2 ( t ) = 1 2 [ ‖ v t ‖ 2 + ‖ v ‖ 1 2 + γ 2 ( ‖ v ‖ 1 2 − λ ‖ v ‖ 2 + 〈 v t , v 〉 ) ] ∼ 1 2 ( ‖ v t ‖ 2 + ‖ v ‖ 1 2 ) .

By λ < λ 1 and the generalized Poincare inequality, we deduce that

d d t E 2 ( t ) ≤ ( λ − λ 1 ) ‖ v t ‖ 2 − γ ‖ v ‖ 1 2 ≤ − C 2 ( ‖ v t ‖ 2 + ‖ v ‖ 1 2 ) , (4.3)

then by the Gronwall inequality, we get

Next multiplying the first equation of (4.2) by

where

then we define the energy functional

where

where

which provides the estimate

From (4.4) and (4.8), for every bounded set

so

Then we finish the proof.

In this paper, we first prove that the Kirchhoff wave equation with strong damping and critical nonlinearities possesses a global attractor in

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

Liu, X.Y. and Gao, P. (2019) Regularity of Global Attractors for the Kirchhoff Wave Equation. Journal of Applied Mathematics and Physics, 7, 2481-2491. https://doi.org/10.4236/jamp.2019.710168