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 [1] introduced the Equation (1.1) in ℝ 1 without dissipation − Δ u t and nonlinear perturbations f ( u t ) and h ( u ) , and described the oscillation of an elastic stretched string. Furthermore, if the string is made up of the viscoelastic material of rate-type, the equation with the strong damping − Δ u t appeared [2]. Since ϵ = 0 , the Equation (1.1) became the following strongly damped semi-linear wave equation

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 [3]. With different conditions about the growth exponents q and p of the nonlinearities f ( u t ) and h ( u ) , some scholars [4] [5] analyzed the longtime behaviour of solutions of (1.3)-(1.2) by the global and exponential attractors in a bounded region of ℝ 3 . When the nonlinearities are of fully supercritical growth, which lead to that the weak solutions of the equation lose their uniqueness. Z. J. Yang and Z. M. Liu [6] established the existence of global attractor for the subclass of limit solutions of (1.3)-(1.2) by using J. Ball’s attractor theory on the generalized semiflow. Recently, I. Chueshov [7] founded that the Kirchhoff wave equation with strong nonlinear damping was still well-posed and the related evolution semigroup had a finite-dimensional global attractor in H = H 0 1 ∩ L p + 1 × L 2 in the sense of “partially strong topology”. Without “partially strong topology”, P. Y. Ding, Z. J. Yang [8] proved the existence of a finite-dimensional global attractor in the natural energy space. And H. L. Ma and C. K. Zhong [9] proved that global attractors for the Kirchhoff equations with strong nonlinear damping attracted H = H 0 1 × L 2 -bounded set with respect to the H 0 1 × H 0 1 norm.

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 [10]. Replacing u t with − Δ u t , Y. H. Wang and C. K. Zhong [11] proved the upper semicontinuity of pullback attractors in non-autonomous case. Then Z. J. Yang and Y. Q. Wang [12] studied the longtime behavior of the Kirchhoff type equation with a strong dissipation and proved that the continuous semigroup S ( t ) possessed global attractors in the phase spaces with low regularity. As for the Kirchhoff wave equation with strong damping and critical nonlinearities, Z. J. Yang and F. Da [13] also studied the stability for the Kirchhoff wave equation with strong damping and critical nonlinearities and proved the existence of global attractors and exponential attractors. Comparing with many researches about the longtime dynamic behavior of solutions for the Kirchhoff wave equation with different types of dissipations [14] - [23], there are few researches about problem of (1.1)-(1.2). And the attractor is a key point for studying these properties, we introduce readers to see the classical book [24].

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 ( [25], Theorem 3.1). And we also establish the asymptotic compactness of the global attractor by operator decomposition technique ( [24], Theorem 1.1). So these jobs provide a way to research the longtime dynamic behaviour of such Kirchhoff wave equations, and also reflect the strong damped properties of Δ u t to some extent.

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 [13] :

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 and integrating over, exploiting (2.8) and the Hölder’s inequality, the right side becomes

where is small enough, we know is bounded by (2.13) and lemma 2.3. At the same time, the left side becomes

then we define the energy functional

where is small enough such that. By combining (4.5)-(4.7) and the embedding, we get

where is small enough and by, we get are negative. From the Gronwall inequality, we get

which provides the estimate

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

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. Then we split the solution into two parts, one part decays exponentially and the other part satisfies asymptotic behaviour in spaces with higher regularity. By the operator decomposition technique, we get the global attractor which is compactly bounded 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