Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on RN

J. Math. Anal. Appl. 434 (2016) 1826–1851

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Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on RN ✩ Zhijian Yang ∗ , Pengyan Ding School of Mathematics and Statistics, Zhengzhou University, No. 100, Science Road, Zhengzhou 450001, PR China

a r t i c l e

i n f o

Article history: Received 9 March 2015 Available online 21 October 2015 Submitted by X. Zhang Keywords: Kirchhoff equation Cauchy problem Well-posedness Global attractor Exponential attractor

a b s t r a c t The paper studies the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on RN : utt −Δut −M (∇u2 )Δu +ut +g(x, u) = f (x). We establish the well-posedness, the existence of the global and exponential attractors in natural energy space H = H 1 (RN ) × L2 (RN ) in critical nonlinearity case. These results improve not only the recent ones achieved by Yang (2007) [32], but also ones achieved by Conti, Pata and Squassina (2005) [9] to some extent. © 2015 Published by Elsevier Inc.

1. Introduction In this paper, we are concerned with the well-posedness and the existence of global and exponential attractors to the Cauchy problem of the Kirchhoff equation with strong dissipation and critical nonlinearity utt − Δut − M (∇u2 )Δu + ut + g(x, u) = f (x) in RN × R+ , u(x, 0) = u0 (x),

ut (x, 0) = u1 (x),

x∈R , N

(1.1) (1.2)

where N ≥ 3, M (s), g(x, u) are nonlinear functions specified later, and f (x) is an external force term. When the space dimension N = 1, Eq. (1.1), with M (s) = a + bs (a ≥ 0, b > 0) and without dissipative terms −Δut and ut , was firstly introduced by Kirchhoff [18] to describe small vibrations of an elastic stretched string. The term −Δut occurs in the study of the motion of viscoelastic materials, for instance, the string is made up of the viscoelastic material of rate-type [10], and it indicates that the stress is proportional not only to the strain, as with the Hooke law, but also to the strain rate as in a linearized Kelvin–Voigt material. ✩

The work is supported by National Natural Science Foundation of China (No. 11271336).

* Corresponding author. E-mail addresses: [email protected] (Z. Yang), [email protected] (P. Ding). http://dx.doi.org/10.1016/j.jmaa.2015.10.013 0022-247X/© 2015 Published by Elsevier Inc.

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There have been many researches on the global existence and decay properties of solutions to the Kirchhoff equation with dissipation −Δut or ut or h(ut ), with h(s)s ≥ 0, s ∈ R. For the IBVP of the type of Eq. (1.1) on a bounded domain Ω ⊂ RN , there have been a lot of well-posedness results in the literature (see [4,19, 23–25] and references therein). There are also a few recent results on the global attractor (see [12,22,36,35]), but all these results require the sub-criticality of the source term g(x, u). Chueshov [7] first studied the well-posedness and the global attractor for the IBVP of the Kirchhoff equation with strong nonlinear damping utt − σ(∇u2 )Δut − φ(∇u2 )Δu + g(u) = h(x)

(1.3)

in natural energy space H(Ω) = H01 (Ω) ∩ Lp+1 (Ω) × L2 (Ω). His results allow that the growth exponent p +2 +4 ∗∗ of the nonlinearity g(u) is supercritical, that is, p∗ < p < p∗∗ , with p∗ ≡ (NN−2) ≡ (NN−4) +,p + . Here the ∗ 1 p+1 ∗ growth exponent p is called critical for H (Ω) → L (Ω) as p ≤ p . He established a finite-dimensional global attractor in the sense of partially strong topology in H(Ω). In particular, in nonsupercritical case: (i) the partially strong topology becomes strong; (ii) an exponential attractor is obtained in H(Ω) by virtue of the strong quasi-stability estimates. Moreover, Chueshov [6] also studied the global well-posedness and the longtime dynamics for the Kirchhoff equations with a structural damping of the form σ(∇u2 )(−Δ)θ ut , with 1/2 ≤ θ < 1, at an abstract level. For the related works on the quasilinear wave equations (rather than the semilinear ones) with strong damping, one can see [5,15,22]. Recently, Yang, Ding and Liu [34] put forward a functional analysis method and used it to construct a bounded absorbing set in H(Ω), which is of higher global regularity. They removed the restriction of “partially strong topology” for H(Ω) in [7] and established a strong global attractor in supercritical nonlinearity case. But for the Kirchhoff equation (1.1) on an unbounded domain, there are only a few recent results (see [32]) on the existence of global attractor, and it seems that little is known on that of exponential attractor. The reason is that when Ω is unbounded, the compactness of the Sobolev embedding which is indispensable for constructing the global attractor is lost. Several remedies for the evolution equation on an unbounded domain have been found to overcome this difficulty. One of them consists in working in weighted Sobolev spaces as some authors done (see [1,2,11,16,21,37]). But unfortunately, this attempt is no use for Eq. (1.1) because of the appearance of the Kirchhoff nonlinearity M (∇u2 )Δu. Other approaches developed for unbounded domain are the usual Sobolev spaces (see [29]). One of them consists in using a suitable semigroup decomposition as done by Feireisl in [13,14] for the damped wave equation, but the method ceases to be effective for the strongly damped wave equation because the finite propagation speed of initial disturbances, which plays a key role in [13,14], is lost. Wang [30] presented a new method of investigating the existence of global attractor for reaction–diffusion equations in unbounded domains. By approaching RN by a bounded domain Ωk , and combining the compactness of Sobolev embedding in bounded domain Ωk with the tail estimates with respect to spatial variables, he proved the asymptotic compactness of the solution semigroup and then established the global attractor in L2 (RN ). This method has been also used in other reaction–diffusion equations and systems (see [17,28]). For the strongly damped semilinear wave equation on R3 : utt − Δut − Δu + g(x, u) + φ(x)ut = f (x, t),

(1.4)

Belleri and Pata [3] presented a technique based on a decomposition of the solution by means of suitable cut-off functions to investigate its longtime dynamics. In particular, when f (x, t) ≡ f (x) and the nonlinearity g(x, u) is of subcritical growth 3, they showed that Eq. (1.4) possesses a global attractor.

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Conti, Pata and Squassina [9] further studied the existence of global attractor for the strongly damped semilinear wave equation on R3 with critical nonlinearity: utt − Δut − Δu + g(x, u) + φ(x, ut ) = f (x).

(1.5)

Based on the technique introduced in [3], they improved the main result of [3], namely, they allowed the nonlinear term g(x, u) to reach the critical power 5. But what about the fractal dimension of the global attractor? What about the existence of the exponential attractor? These questions remain unanswered. For the investigation of the longtime dynamics of the Kirchhoff equation on RN , recently, by combining the decomposition of the solution semigroup with the tail estimates, both on time and spatial variables, Yang [32] has proved the existence of global attractor of Eq. (1.1) in phase space X2 = H 2 (RN ) × H 1 (RN ) provided that M (s) = 1 + sm/2 , m ≥ 1, g ∈ C 1 (RN × R) and |

∂g | ≤ β(|s|p−1 + L1 (x)), ∂s

|∇x g| ≤ γ(|s|p + L2 (x)),

+2 4 with 1 ≤ p < p∗ and p ≤ p˜, where p∗ = (NN−2) ˜ = (N −4) +, p + , L1 ∈ L Obviously, there are still two unsolved questions in [32]:

2(p+1) p−1

(RN ), L2 ∈ L2 (RN ).

(i) g(x, u) in [32] excludes g(u) as its special case because of the assumptions on L1 , L2 ; (ii) the growth exponent p∗ of the nonlinearity is critical for the natural energy space H = H 1 (RN )×L2 (RN ), but is not critical for the phase space X2 . Does Eq. (1.1) possess global and exponential attractors in natural energy space H in critical nonlinearity case: p = p∗ ? To the best of our knowledge, the questions are still open. The purpose of the present paper is to solve these questions. First, we use the energy method combined with the monotone technology and a new limiting process to get the well-posedness of solutions in natural energy space H. Second, we use the direct tail cut-off method in H, which is different form those used in [3,9,32], and the recently developed quasi-stable estimate to establish the existence of the global and exponential attractors in H for 1 ≤ p ≤ p∗ . These results improve the recent ones archived by the first author in [32]. The novelty of this paper is that it overcomes the essential difficulties: “both the Sobolev embedding on RN and the critical growth of g cause the lack of compactness” and establishes the well-posedness, the existence of the global and exponential attractors for the Kirchhoff equation with critical nonlinearity. Especially, our results cover the case M (s) ≡ 1, and at this time Eq. (1.1) becomes Eq. (1.5), with φ(x, ut ) = ut , so the results of the present paper also improve the results in [9] to some extent. The paper is organized as follows. In Section 2, we discuss the well-posedness of the Cauchy problem (1.1)–(1.2). In Section 3, we establish the existence of the global attractor. In Section 4, we study the exponential attractor. 2. Well-posedness We first introduce the following abbreviations:  Lp = Lp (RN ),

W s,p = W s,p (RN ),

H s = W s,2 ,

 =

,

 ·  =  · L2 ,

 · p =  · Lp ,

RN

with p ≥ 1. The notation (·, ·) for L2 -inner product will also be used for the notation of duality pairing between the dual spaces. We use the same letter C to denote different positive constants, C(·, ·) to denote

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positive constants depending on the quantities appearing in parenthesis. The sign H1 → H2 denotes that the functional space H1 continuously embeds into H2 and H1 →→ H2 denotes that H1 compactly embeds into H2 . Let H = H 1 × L2 , which is equipped with the usual graph norm. Assumption 1. (i) g = g(x, s) ∈ C(RN × R), with g(x, ·) ∈ C 2 (R) for almost every x ∈ RN , and (G1 ) g(·, 0) ∈ L2 ; (G2 ) |g  (x, 0)| ≤ C, (G3 )

lim inf |s|→∞

|g  (x, s)| ≤ c1 (1 + |s|p−2 ), x ∈ RN , s ∈ R, 2 ≤ p ≤ p∗ ;

g(x, s) ≥ 0 uniformly as |x| ≤ r0 ; s

(G4 ) (g(x, s) − g(x, 0))s ≥ c2 s2 , ∀s ∈ R, |x| > r0 , where and in the following g  (x, s) stands for the derivative for the second variable, p∗ = ci and r0 are positive constants. (ii) M ∈ C 1 (R+ ), M  (s) ≥ 0, M (0)  M0 > 0. (iii) f ∈ L2 , ξu (0) = (u0 , u1 ) ∈ H, (u0 , u1 )H ≤ R0 .

N +2 N −2

(N ≥ 3),

Without loss of generality, we assume g(x, 0) = 0. Or else, let g1 (x, s) = g(x, s) − g(x, 0), f1 (x) = f (x) − g(x, 0), and replace g and f in Eq. (1.1) by g1 and f1 , respectively. Obviously, g1 (x, 0) = 0, f1 ∈ L2 , and g1 (x, s) satisfies Assumption 1: (i). Example. We now give some examples for the functions g, M , f satisfying Assumption 1. Let g(x, s) = a0 (x)|s|p−1 s + a1 (x)s, where a0 , a1 ∈ C(RN ), 0 ≤ a0 (x) ≤ C, 0 < c2 ≤ a1 (x) ≤ C. Obviously, assumptions (G1 ) and (G3 ) hold, and g  (x, s) = pa0 (x)|s|p−1 + a1 (x), g(x, s)s ≥ c2 s2 ,

|g  (x, s)| = |p(p − 1)a0 (x)|s|p−3 s| ≤ c1 (1 + |s|p−2 ),

|g  (x, 0)| = |a1 (x)| ≤ C.

That is, g satisfies Assumption 1: (i). Let M (s) = a + bsα ,

f (x) =

1 , 1 + |x|β

where a > 0, b ≥ 0, α ≥ 1, β > N/2. Obviously, M and f satisfy Assumption 1: (ii) and (iii), respectively. Theorem 2.1. Let Assumption 1 be in force. Then problem (1.1)–(1.2) admits a unique solution u ∈ C(R+ ; H 1 ), with ut ∈ C(R+ ; L2 ) ∩ L2 (R+ ; H 1 ). This solution possesses the following properties:

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(1) t u(t)2H 1

+ ut (t) +

ut (τ )2H 1 dτ ≤ C(R0 , f ),

2

t > 0.

(2.1)

t > 0.

(2.2)

0

(2) For any a > 0, ut ∈ L∞ (a, T ; H 1 ),

utt ∈ L∞ (a, T ; L2 ),

and there exists a small constant κ > 0 such that ut (t)2H 1 + utt (t)2 ≤

t2 + 1 C(R0 )e−κt + C(f ), t2

(3) The following Lipschitz continuity holds: t z(t)2H 1

+ zt (t) +

zt (τ )2H 1 dτ ≤ Cekt (z(0)2H 1 + zt (0)2 ),

2

t ≥ 0,

(2.3)

0

for some k > 0, where z = u − v, u, v are the solutions of problem (1.1)–(1.2) corresponding to initial data (u0 , u1 ) and (v0 , v1 ), respectively. We first state some lemmas, which are indispensable for our proof. Lemma 2.1. (See [27].) Let X, B and Y be the Banach spaces, X →→ B → Y , W = {u ∈ Lp (0, T ; X)|ut ∈ L1 (0, T ; Y )}, ∞

W1 = {u ∈ L (0, T ; X)|ut ∈ L (0, T ; Y )}, r

with 1 ≤ p < ∞, with r > 1.

Then, W →→ Lp (0, T ; B),

W1 →→ C([0, T ]; B).

Lemma 2.2. (See [29].) Let X, Y be two Banach spaces such that X → Y . If φ ∈ L∞ (0, T ; X) ∩Cw ([0, T ]; Y ), then φ ∈ Cw ([0, T ]; X). Lemma 2.3. (See [20,31].) Let Ω be an open set in RN . Then H01 (Ω) → Lq (Ω), with q = is, uLq (Ω) ≤ c0 ∇uL2 (Ω) ,

∀u ∈ H01 (Ω),

where c0 = c0 (N ) is a positive constant independent of Ω. Lemma 2.4. Let Ω be an open set in RN . Then uLp+1 (Ω) ≤ CuH 1 (Ω) , for 1 ≤ p ≤ p∗ ≡

N +2 N −2

∀u ∈ H01 (Ω)

(N ≥ 3), where C is a positive constant independent of Ω.

2N N −2

(N ≥ 3), that

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Proof. Lemma 2.3 implies that the conclusion of Lemma 2.4 holds for p = p∗ . When 1 ≤ p < p∗ , by the interpolation theorem and Lemma 2.3, 1−θ uLp+1 (Ω) ≤ C(θ)uθLp∗ +1 (Ω) uL 2 (Ω) ≤ C(θ)uH 1 (Ω) ,

with θ =

N (p−1) 2(p+1) .

Lemma 2.5. Let y : R+ → R+ be an absolutely continuous function satisfying d y(t) + 2 y(t) ≤ h(t)y(t) + z(t), dt where > 0, z ∈ L1loc (R+ ),

t s

t > 0,

h(τ )dτ ≤ (t − s) + m for t ≥ s ≥ 0 and some m > 0. Then

t   −t y(t) ≤ e y(0)e + |z(τ )|e−(t−τ ) dτ , m

t > 0.

0

Lemma 2.6 (Gronwall-type lemma). (See [3].) Let X be a Banach space, and let Z ⊂ C(R+ , X). Let Φ : X → R be a continuous function such that sup Φ(z(t)) ≥ −η,

t∈R+

Φ(z(0)) ≤ K

for some η, K ≥ 0 and every z ∈ Z. In addition, assume that for every z ∈ Z the function t → Φ(z(t)) is continuously differentiable, and satisfies the differential inequality d Φ(z(t)) + δz(t)2X ≤ k dt for some δ > 0 and k ≥ 0 independent of z ∈ Z. Then, for every γ > 0 there is t0 = Φ(z(t)) ≤ sup {Φ(ζ) : δζ2X ≤ k + γ},

η+K γ

> 0 such that

t ≥ t0 .

ζ∈X

Proof of Theorem 2.1. Let Ω = ΩR be a ball in RN with radius R. We first consider the auxiliary IBVP of Eq. (1.1) on Ω: ⎧ 2 ⎪ ⎨ utt − Δut − M (∇uL2 (Ω) )Δu + ut + g(x, u) = f (x), u|∂Ω = 0, ⎪ ⎩ u(x, 0) = u ˜R ut (x, 0) = u ˜R x ∈ Ω, 0 (x), 1 (x),

(x, t) ∈ Ω × R+ , (2.4)

where the functions u ˜R ˜R i (i = 0, 1) are of the forms: u i (x) = θ(|x|)ui (x). Here θ(x) is a smooth function, θ(x) = 1

as |x| ≤ R − 1;

θ(x) = 0 as |x| ≥ R and 0 ≤ θ(x) ≤ 1,

|∇θ(x)| ≤ C,

x ∈ RN .

Let H(Ω) = H01 (Ω) × L2 (Ω). For brevity, for problem (2.4), we let the notation (·, ·) stand for either the L2 (Ω)-inner product or the duality pairing between dual spaces.

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Lemma 2.7. Let Assumption 1: (i) be in force, Ω = ΩR , with R > r0 . Then for any η > 0, there exist ρ(η) > 0 and c(η) ≥ 0 such that the following inequalities hold for every u ∈ H01 (Ω): G(u) ≥ −ηuL2 (Ω) − c(η), (g(·, u), u) + η∇u2L2 (Ω) ≥ ρ(η)u2L2 (Ω) − c(η), where G(u) =

 Ω

G(x, u)dx, G(x, u) =

u 0

(2.5)

g(x, τ )dτ .

Proof. For any θ ∈ H01 (ΩR ), by virtue of Lemma 2.3 and the fact Lq (Ωr0 ) → L2 (Ωr0 ), with q =  c3 θ2L2 (ΩR )

−

∇θ2L2 (ΩR )

≤ c3

|θ|2 dx + c3 θ2L2 (Ωr

0)

2N N −2 ,

2 − c−2 0 θLq (ΩR )

ΩR \Ωr0



2 |θ|2 dx + (c3 |Ωr0 | N − c−2 0 )θLq (Ωr 2

≤ c3

0)

ΩR \Ωr0



≤

|θ|2 dx,

(2.6)

ΩR \Ωr0 −N with 0 < c3 ≤ min{1, c−2 }. By virtue of (2.6), similar to the proof in [26] for Ω = RN , one can 0 |Ωr0 | easily get the conclusion of Lemma 2.7. We omit the process here. The proof is complete. 2

Now, we formally give some a priori estimates to the solutions of problem (2.4). Using the multiplier ut + u in (2.4), we get d Φ(ξu (t)) + K(ξu (t)) = 0, dt

(2.7)

where ξu (t) = (u(t), ut (t)), 1 ut 2L2 (Ω) + Φ(ξu (t)) = 2

∇u2L2 (Ω)



M (s)ds + 2G(u) − 2(u, f )



0

  1 + (u, ut ) + u2H 1 (Ω) , 2 2 K(ξu (t)) = ut H 1 (Ω) − ut 2L2 (Ω) + M (∇u2L2 (Ω) )∇u2L2 (Ω) + (g(·, u), u) − (u, f ).

(2.8)

Obviously, Φ : H(Ω) → R is a continuous function. Making use of (2.5) and the fact: M (s) ≥ M0 > 0, we infer from (2.8) that Φ(ξu (t)) ≥ κ(ut 2L2 (Ω) + u2H 1 (Ω) ) − C(f L2 (Ω) ), K(ξu (t)) ≥

1 ut 2H 1 (Ω) + κ(ut 2L2 (Ω) + u2H 1 (Ω) ) − C(f L2 (Ω) ) 2

(2.9)

for > 0 suitably small, where and in the following κ stands for a small positive constant. Obviously,   p+1 2 2 2 2 Φ(ξu (0)) ≤ C ˜ uR uR uR uR uR 1 L2 (Ω) + ˜ 0 H 1 (Ω) + M (∇˜ 0 L2 (Ω) )∇˜ 0 L2 (Ω) + ˜ 0 Lp+1 (Ω) + C(f L2 (Ω) ) ≤ C(R0 , f L2 (Ω) ).

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Inserting (2.9) into (2.7), we have d Φ(ξu (t)) + κξu (t)2H(Ω) ≤ C(f L2 (Ω) ). dt

(2.10)

Applying Lemma 2.6 to (2.10) (with η = C(f L2 (Ω) ), k = C(f L2 (Ω) ), K = C(R0 , f L2 (Ω) ), δ = κ, γ = 1) we get Φ(ξu (t)) ≤ sup {Φ(ζ)|ζ2H(Ω) ≤ ζ∈H(Ω)

C(f L2 (Ω) ) + 1 }, κ

t ≥ t0 = C(R0 , f L2 (Ω) ).

Therefore, u(t)2H 1 (Ω) + ut (t)2L2 (Ω) ≤ C(R0 , f L2 (Ω) ),

t ≥ t0 = C(R0 , f L2 (Ω) ).

(2.11)

Integrating (2.10) on (0, t), with t ≤ t0 , we get u(t)2H 1 (Ω) + ut (t)2L2 (Ω) ≤ C(R0 , f L2 (Ω) ),

∀t ∈ [0, t0 ].

(2.12)

Letting = 0 in (2.7), integrating the resulting expression over (0, t) and making use of (2.11)–(2.12) we get t ut (τ )2H 1 (Ω) dτ ≤ C(R0 , f L2 (Ω) ).

(2.13)

0

The combination of (2.11)–(2.12) with (2.13) means that (2.1) holds on Ω. On account of H01 (Ω) → Lp+1 (Ω), 1 L1+ p (Ω) → H −1 (Ω), we infer from Eq. (2.4) and (2.11)–(2.13) that T utt (t)2H −1 (Ω) dt ≤ C(R0 , f L2 (Ω) , T ).

(2.14)

0

Differentiating Eq. (2.4) with respect to t and letting v = ut , we get vtt − Δvt − M (∇u2L2 (Ω) )Δv − 2M  (∇u2L2 (Ω) )(∇u, ∇ut )Δu + vt + g  (·, u)v = 0.

(2.15)

Using the multiplier vt + v in (2.15), we have d Ψ(t) + vt 2H 1 (Ω) + κv2L2 (Ω) + (g  (·, u)v, vt + v) dt = vt 2L2 (Ω) + κv2L2 (Ω) + M  (∇u2L2 (Ω) )(∇u, ∇ut )∇v2L2 (Ω) − 2M  (∇u2L2 (Ω) )(∇u, ∇ut )(∇u, ∇vt )   − M (∇u2L2 (Ω) )∇v2L2 (Ω) + 2M  (∇u2L2 (Ω) )(∇u, ∇v)2 ≤ vt 2L2 (Ω) + κv2L2 (Ω) + C∇ut L2 (Ω) (∇v2L2 (Ω) + ∇vt L2 (Ω) ) + C∇v2L2 (Ω) ≤ vt 2L2 (Ω) + κv2L2 (Ω) + C∇ut L2 (Ω) ∇v2L2 (Ω) + vt 2H 1 (Ω) + Cv2H 1 (Ω) , where

(2.16)

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Ψ(t) =

   1 1 vt 2L2 (Ω) + M (∇u2L2 (Ω) )∇v2L2 (Ω) + (v, vt ) + v2H 1 (Ω) 2 2

∼ v2H 1 (Ω) + vt 2L2 (Ω)

(2.17)

for > 0 suitably small, and where we have used (2.11)–(2.12). By Lemma 2.4, |(g  (·, u)v, vt + v)|   2 p+1 p+1 ( v + v v  ) ≤ C vL2 (Ω) vt L2 (Ω) + v2L2 (Ω) + up−1 p+1 p+1 L (Ω) t L (Ω) L (Ω) (Ω) L 2 1 1 ≤ vt 2L2 (Ω) + Cv2L2 (Ω) + Cup−1 H 1 (Ω) ( vH 1 (Ω) + vH (Ω) vt H (Ω) )

≤ 2 vt 2H 1 (Ω) + Cv2H 1 (Ω) ,

(2.18)

where and in the context C is a positive constant independent of Ω. Inserting (2.18) into (2.16), we get 1 d Ψ(t) + κΨ(t) + vt 2H 1 (Ω) ≤ C∇ut L2 (Ω) Ψ(t) + Cv2H 1 (Ω) . dt 2

(2.19)

When 0 < t ≤ 1, multiplying (2.19) by t2 , we have  1 d2 t Ψ(t) + κt2 Ψ(t) + t2 vt 2H 1 (Ω) dt 2 ≤ C∇ut L2 (Ω) t2 Ψ(t) + Cv2H 1 (Ω) + Ct(vt 2L2 (Ω) + v2H 1 (Ω) ) ≤

1 2 t vt 2H 1 (Ω) + C∇ut L2 (Ω) t2 Ψ(t) + C(v2H 1 (Ω) + vt 2H −1 (Ω) ), 2

(2.20)

where we have used the interpolation inequality Ctvt 2L2 (Ω) ≤ Ctvt H 1 (Ω) vt H −1 (Ω) ≤

1 2 t vt 2H 1 (Ω) + Cvt 2H −1 (Ω) . 2

On account of t ∇ut (τ )L2 (Ω) dτ ≤ C

C s

 t

∇ut (τ )2L2 (Ω) dτ

1/2

(t − s)1/2 ≤

κ (t − s) + m 2

s

for t > s ≥ 0 and some m > 0, applying Lemma 2.5 to (2.20), we have t2 Ψ(t) ≤ C1 ,

ut (t)2H 1 (Ω) + utt (t)2L2 (Ω) ≤

C1 , t2

0 < t ≤ 1,

(2.21)

where and in the following C1 = C(R0 , f L2 (Ω) ). When t ≥ 1, applying Lemma 2.5 to (2.19) on (1, t), we get ut (t)2H 1 (Ω) + utt (t)2L2 (Ω) ≤ C1 e−κt + C1 < C1 .

(2.22)

The combination of (2.21) with (2.22) gives ut (t)2H 1 (Ω) + utt (t)2L2 (Ω) ≤

t2 + 1 C1 e−κt + C(f L2 (Ω) ), t2

t > 0.

(2.23)

Z. Yang, P. Ding / J. Math. Anal. Appl. 434 (2016) 1826–1851

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Now, we look for the approximate solutions of Eq. (2.4) of the form n

n

u (t) =

Tjn (t)wj ,

j=1

where −Δwj = λj wj , j = 1, 2, . . . , wj |∂Ω = 0, with (untt , wj ) + (∇unt , ∇wj ) + M (∇un 2L2 (Ω) )(∇un , ∇wj ) + (unt , wj ) + (g(·, un ), wj ) = (f, wj ), n

u (0) = u ˜0n ,

unt (0)

t > 0,

j = 1, 2, . . . , n,

(2.24)

=u ˜1n ,

and where (˜ u0n , u ˜1n ) → (˜ uR ˜R 0 ,u 1 ) in H(Ω). Obviously, the estimates (2.1) (on Ω), (2.14) and (2.23) hold for un . So we can extract a subsequence, still denoted by {un }, such that un → u unt → ut u ˜ntt → utt

weakly∗ in L∞ (0, T ; H01 (Ω)); weakly∗ in L∞ (0, T ; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)); weakly in L2 (0, T ; H −1 (Ω));

−M (∇un 2L2 (Ω) )Δun → ξ

weakly∗ in L∞ (0, T ; H −1 (Ω)).

(2.25)

It follows from (2.25) and Lemma 2.1 that unt → ut

in L2 (0, T ; L2 (Ω));

un → u in C([0, T ]; L2 (Ω)) and a.e. in Ω for any t ∈ [0, T ]; g(·, un ) → g(·, u)

weakly∗ in L∞ (0, T ; L1+ p (Ω)). 1

Letting n → ∞ in (2.24), we get that the limiting function u ∈ L∞ (0, T ; H01 (Ω)) solves utt − Δut + ξ + ut + g(·, u) = f, u(0) = u ˜R 0,

ut (0) = u ˜R 1.

Now we show ξ = −M (∇u2L2 (Ω) )Δu. Define the operator B : H01 (Ω) → H −1 (Ω), (Bu, v) = M (∇u2L2 (Ω) )(∇u, ∇v),

∀u, v ∈ H01 (Ω).

Lemma 2.8. (i) The operator B : H01 (Ω) → H −1 (Ω) is monotone, that is, for any u, v ∈ H01 (Ω), (Bu − Bv, u − v) ≥ 0. (ii) The operator B is semicontinuous, that is, for any u, v, w ∈ H01 (Ω), (B(u + λv) − Bu, w) → 0

as λ → 0.

(2.26)

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Proof. (i) For any u, v ∈ H01 (Ω),  2 ˜ 12 (t) ∇(u + v), ∇(u − v) , (Bu − Bv, u − v) = M12 (t)∇(u − v)2L2 (Ω) + M where  1 M (∇u2L2 (Ω) ) + M (∇v2L2 (Ω) ) > 0, 2 1 1 ˜ M  (λ∇u2L2 (Ω) + (1 − λ)∇v2L2 (Ω) )dλ > 0. M12 (t) = 2

M12 (t) =

0

So the operator B is monotone. (ii) When λ → 0, for any u, v, w ∈ H01 (Ω), ∇u + λ∇v2L2 (Ω) = ∇u2L2 (Ω) + 2λ(∇u, ∇v) + λ2 ∇v2L2 (Ω) → ∇u2L2 (Ω) , M (∇u + λ∇v2L2 (Ω) ) → M (∇u2L2 (Ω) ), (∇(u + λv), ∇w) → (∇u, ∇w),

(B(u + λv), w) → (Bu, w),

that is, the operator B is semicontinuous. Let t (Bun − Bv, un − v)dτ (≥ 0),

ξn (t) = 0

where un as shown in (2.24) and v ∈ H01 (Ω). Obviously, t

t n

−(untt − Δunt + unt + g(·, un ) − f, un )dτ

n

(Bu , u )dτ = 0

0

t =

−(unt , un )

unt (τ )2L2 (Ω) dτ −

+ (˜ u1n , u ˜0n ) +

 1 n u0n 2H 1 (Ω) u (t)2H 1 (Ω) − ˜ 2

0

t

t (f, un )dτ −

+ 0

(g(·, un ), un )dτ. 0

It follows from (2.26) that g(x, un )un → g(x, u)u a.e. in Qt = Ω × [0, t]. We infer from assumptions (G3 ) and (G4 ) that g(x, s)s + s2 ≥ −C(r0 ),

|x| ≤ r0 ;

g(x, s)s ≥ c2 s2 ≥ 0,

So g(x, s)s + s2 ≥ −C(r0 ),

x ∈ Ω, s ∈ R.

|x| > r0 .

(2.27)

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By the Fatou lemma and (2.26), t 

t  (g(x, u)u + u )dxdτ ≤ lim inf 2

(g(x, un )un + (un )2 )dxdτ

n→∞

0 Ω

0 Ω

t  ≤ lim inf

t  n

n

0 Ω

t

u2 dxdτ,

(g(x, u )u dxdt +

n→∞

0 Ω

t (g(·, u), u)dτ ≤ lim inf

(g(·, un ), un )dτ.

n→∞

0

(2.28)

0

By virtue of (2.25)–(2.26) and (2.28), t

t (Bu , u )dτ ≤ −(ut , u) + n

lim sup n→∞

n

(˜ uR ˜R 0 ,u 1)

+

0

1 ut (τ )2L2 (Ω) dτ − u(t)2H 1 (Ω) 2

0

1 R 2 u  1 + ˜ + 2 0 H (Ω)

t

t (f, u)dτ −

0

(g(·, u), u)dτ. 0

Obviously, t

t (Bu , v)dτ → n

0

t

t (Bv, u )dτ → n

(ξ, v)dτ, 0

0

(Bv, u)dτ 0

as n → ∞. Therefore, t 0 ≤ lim sup ξn (t) ≤

(ξ − Bv, u − v)dτ.

n→∞

0

Taking v = u − λw, with λ > 0, w ∈ H01 (Ω), we get t 0≤

(ξ − B(u − λw), w)dτ. 0

Letting λ → 0, we infer from Lemma 2.8 that t 0≤

(ξ − Bu, w)dτ. 0

By the arbitrariness of w, ξ = Bu, u is a solution of the problem (2.4). By the lower semi-continuity of the norm of the weak∗ limit, the estimates (2.1)–(2.2) (on Ω) hold for u. Now, we show the existence of solutions of the Cauchy problem (1.1)–(1.2). For brevity, in the following, we use the abbreviations as is shown in the beginning of this section. Let uR ∈ L∞ (0, T ; H01 (ΩR )) be the solution of the auxiliary problem (2.4). Define the natural extension of uR on RN :

Z. Yang, P. Ding / J. Math. Anal. Appl. 434 (2016) 1826–1851

1838

 R

u ˜ =

uR ,

|x| ≤ R,

0,

|x| > R,

fR =

f, |x| ≤ R, 0,

|x| > R.

A simple calculation shows that  ∇˜ u = R

∇uR ,

|x| ≤ R,

0,

|x| > R.

Indeed, for any φ ∈ C0∞ (RN ), noticing that uR |∂ΩR = 0, we have 





u ˜ ∇φdx =

u ∇φdx = −

R

R

ΩR

 ∇u φdx = − R

∇˜ uR φdx.

ΩR

Obviously, u ˜R ∈ L∞ (0, T ; H 1 (RN )) solves the Cauchy problem 

u ˜R uR uR 2 )Δ˜ uR + u ˜R ˜R ) = fR tt − Δ˜ t − M (∇˜ t + g(·, u u ˜R (0) = u ˜R 0,

in RN × R+ ,

u ˜R ˜R t (0) = u 1,

(2.29)

and the estimates (2.1)–(2.2) and (2.14) hold for u ˜R . Since   g(x, u ˜R ) = g  (x, μ˜ uR ) − g  (x, 0) u ˜R + g  (x, 0)˜ uR = g  (x, μθ˜ uR )μ|˜ uR |2 + g  (x, 0)˜ uR ,

(2.30) 1

where 0 < μ, θ < 1, by assumption (G2 ) and the Sobolev embedding: L1+ p → H −1 , L2 → H −1 , |g(x, u ˜R ) − g  (x, 0)˜ uR | = |g  (x, μθ˜ uR )μ|˜ uR |2 | ≤ C(1 + |˜ uR |p−2 )|˜ uR |2 , uR H −1 ≤ Cg  (x, μθ˜ uR )μ|˜ uR |2 1+ p1 ≤ C(˜ uR 22(p+1) + ˜ uR pp+1 ≤ C, g(·, u ˜R ) − g  (·, 0)˜ p



u H −1 + C ≤ C(˜ u  + 1) ≤ C, g(·, u ˜ )H −1 ≤ g (·, 0)˜ R

R

t ≥ 0.

R

(2.31)

So there exists a limiting function defined on RN , still denoted by u, such that (subsequence if necessary) u ˜R → u

weakly∗ in L∞ (0, T ; H 1 );

u ˜R t → ut u ˜R tt → utt

weakly∗ in L∞ (0, T ; L2 ) ∩ L2 (0, T ; H 1 ); weakly in L2 (0, T ; H −1 );

g(·, u ˜R ) → ϑ

weakly∗ in L∞ (0, T ; H −1 );

uR → ζ M (∇˜ uR 2 )Δ˜

weakly∗ in L∞ (0, T ; H −1 ).

Lemma 2.9. ˜ uR uR 0 − u0 H 1 + ˜ 1 − u1  + fR − f  → 0

as R → +∞.

Proof. Obviously, 2 2 2 ˜ uR uR 0 − u0 H 1 + ˜ 1 − u1  + fR − f   ≤ 2(C 2 + 1) (|u0 |2 + |∇u0 |2 )dx + |x|≥R−1



|x|≥R−1

  |u1 |2 + |f |2 dx → 0.

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i i Let u ˜ Ri , u ˜Rj be the solutions of the Cauchy problem (2.29) corresponding to initial data (˜ uR ˜R 0 ,u 1 ) and Ri Rj ∈ H. Then z = u ˜ −u ˜ solves

R R (˜ u0 j , u ˜1 j )

˜ ij (t)(∇(˜ ztt − Δzt − Mij (t)Δz + zt − M uRi + u ˜Rj ), ∇z)Δ(˜ u Ri + u ˜ Rj ) ˜Rj ) + fRi − fRj , = −g(·, u ˜Ri ) + g(·, u R

i ˜0 j ≡ z0 , z(0) = u ˜R 0 −u

R

i zt (0) = u ˜R ˜1 j ≡ z1 , 1 −u

(2.32)

where  1 M (∇˜ uRi 2 ) + M (∇˜ Mij (t) = u Rj  2 ) , 2

˜ ij (t) = 1 M 2

1

M  (λ∇˜ uRi 2 + (1 − λ)∇˜ uRj 2 )dλ.

0

Using the multiplier zt + z in (2.32), we have   d H1 (t) + zt 2H 1 = K1 (t) − g(·, u ˜Ri ) − g(·, u ˜Rj ) − (fRi − fRj ), zt + z , dt

(2.33)

where H1 (t) =

 1 zt 2 + z2H 1 + (z, zt ) ∼ z2H 1 + zt 2 2

for > 0 suitably small, and ˜ ij (t)(∇(˜ K1 (t) = −Mij (t)(∇z, ∇zt ) − M uRi + u ˜Rj ), ∇zt )(∇(˜ uR i + u ˜Rj ), ∇z)   ˜ ij (t)(∇(˜ − Mij (t)∇z2 + M uRi + u ˜Rj ), ∇z)2 + zt 2 ≤

1 zt 2H 1 + Cz2H 1 . 8

(2.34)

Obviously,

 

˜Ri ) − g(·, u ˜Rj ) − (fRi − fRj ), zt + z

− g(·, u  1 1 ≤ C (|˜ uRi |p−1 + |˜ uRj |p−1 + 1)|z|(|zt | + |z|)dx + (zt 2 + z2 ) + fRi − fRj 2 2 2 ≤ Cz(zt  + z) + C(˜ uRi p−1 uRj p−1 p+1 + ˜ p+1 )zp+1 (zt p+1 + zp+1 ) 1 1 + (zt 2 + z2 ) + fRi − fRj 2 2 2 1 1 ≤ zt 2H 1 + C(zt 2 + z2H 1 ) + fRi − fRj 2 . 8 2

(2.35)

Inserting (2.34)–(2.35) into (2.33), we get 1 d H1 (t) + zt 2H 1 ≤ CH1 (t) + fRi − fRj 2 , dt 8 t 2 2 z(t)H 1 + zt (t) + zt (τ )2H 1 dτ ≤ Cekt (z0 2H 1 + z1 2 ) + C(T )fRi − fRj 2 . 0 i Estimate (2.36) and Lemma 2.9 imply that {(˜ u Ri , u ˜R t )} is a Cauchy sequence in H, that is,

(2.36)

Z. Yang, P. Ding / J. Math. Anal. Appl. 434 (2016) 1826–1851

1840

u ˜Ri → u in L∞ (0, T ; H 1 ), u ˜Ri → u,

i u ˜R t → ut

in L∞ (0, T ; L2 );

g(x, u ˜Ri ) → g(x, u) a.e. in RN × [0, T ].

We claim that ϑ = g(·, u). Indeed, it follows from (2.30) that g(x, u ˜Ri ) = g  (x, μi θi u ˜Ri )μi (˜ uRi )2 + g  (x, 0)˜ u Ri , with 0 < μi , θi < 1. By the compactness of the number sequences {μi }, {θi }, we have (subsequence if necessary) μi → μ,

θi → θ,

μi θ i u ˜Ri → μθu

a.e. in RN × [0, T ],

g  (x, μi θi u ˜Ri )μi (˜ uRi )2 → g  (x, μθu)μu2 ,

g  (x, 0)˜ uRi → g  (x, 0)u a.e. in RN × [0, T ].

By the uniqueness of the limit g(x, u) = g  (x, μθu)μu2 + g  (x, 0)u a.e. in RN × [0, T ]. Obviously (see (2.31)), g  (x, μi θi u ˜Ri )μi (˜ uRi )2 1+ p1 + g  (x, 0)˜ uRi  ≤ C. So, g  (x, μi θi u ˜Ri )μi (˜ uRi )2 → g  (x, μθu)μu2

weakly∗ in L∞ (0, T ; L1+ p ); 1

g  (x, 0)˜ uRi → g  (x, 0)u weakly∗ in L∞ (0, T ; L2 ). Therefore, for any φ ∈ H 1 (→ Lp+1 ) (g(·, u ˜Ri ), φ) = (g  (x, μi θi u ˜Ri )μi (˜ uRi )2 + g  (x, 0)˜ uRi , φ) → (g  (x, μθu)μu2 + g  (x, 0)u, φ) = (g(·, u), φ), that is, ϑ = g(·, u). For any v ∈ H 1 , 

 uRi − M (∇u2 )Δu, v M (∇˜ uRi 2 )Δ˜   = − M (∇˜ uRi 2 ) − M (∇u2 ) (∇˜ uRi , ∇v) − M (∇u2 )(∇˜ uRi − ∇u, ∇v) → 0.

By the uniqueness of the limit, M (∇u2 )Δu = ζ. Therefore, u ∈ L∞ (0, T ; H 1 ), with ut ∈ L∞ (0, T ; L2 ) ∩ L2 (0, T ; H 1 ), is a solution of the Cauchy problem (1.1)–(1.2). By the lower semi-continuity of the norm of the weak∗ limit, the estimates (2.1)–(2.2) hold for u. Let u, v be two solutions of the Cauchy problem (1.1)–(1.2) corresponding to initial data (u0 , u1 ) and (v0 , v1 ) ∈ H, respectively, z = u − v. Similar to the proof of (2.36) (replacing fRi − fRj by 0 there), we get (2.3).

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Noticing that g(·, u)ut ∈ L1 ([0, T ] × RN ), using the multiplier ut in (1.1) and integrating the resulting expression over (t0 , t), we get t ut (τ )2H 1 dτ = E(ξu (t0 ))

E(ξu (t)) +

(2.37)

t0

for t ≥ t0 ≥ 0, where 1 E(ξu ) = ut 2 + 2

2 ∇u 

  M (s)ds + G(x, u)dx − (f, u).

0

Letting t → t0 in (2.37), we arrive at E(ξu (t0 )) = lim E(ξu (t)).

(2.38)

t→t0

Since u ∈ H 1 (0, T ; H 1 ) → C([0, T ]; H 1 ), u ∈ C([0, T ]; H 1 ), u(t) → u(t0 ),

∇u(t) → ∇u(t0 ) in L2 and a.e. in RN as t → t0 ,

2 ∇u(t  0 )

2 ∇u(t) 

M (s)ds = lim

M (s)ds.

t→t0

0

0

By (2.14), ut ∈ L∞ (0, T ; L2 ) ∩ H 1 (0, T ; H −1 ) → C([0, T ]; H −1 ), L2 → H −1 , so ut ∈ Cw ([0, T ]; L2 ) (see Lemma 2.2), ut (t) → ut (t0 )

weakly in L2 ,

ut (t0 ) ≤ lim inf ut (t). t→t0

By assumptions (G3 )–(G4 ), s G(x, s) + s ≥ −C(r0 ),

|x| ≤ r0 ;

2

g(x, τ )dτ ≥ 0,

G(x, s) =

|x| > r0 .

0

By the Fatou lemma, 

 G(x, u(t0 ))dx + u(t0 )2L2 (Ωr

0)

≤ lim inf

(G(x, u(t)) + u2 (t))dx

t→t0

Ωr0

Ωr0



≤ lim inf

G(x, u(t))dx + u(t0 )2L2 (Ωr ) ,

t→t0



 G(x, u(t0 ))dx ≤ lim inf

G(x, u(t))dx,

t→t0

RN \Ωr0



RN \Ωr0



G(x, u(t0 ))dx =

 +

Ωr0

Therefore, it follows from (2.38) that

0

Ωr0

RN \Ωr0

  G(x, u(t0 ))dx ≤ lim inf G(x, u(t))dx. t→t0

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1842

 1 2 lim sup ut (t) + lim inf G(x, u(t))dx t→t0 t→t0 2    1 1 ut (t)2 + G(x, u(t))dx = ut (t0 )2 + G(x, u(t0 ))dx ≤ lim t→t0 2 2  1 ≤ lim inf ut (t)2 + lim inf G(x, u(t))dx, t→t0 2 t→t0 which implies  lim ut (t)2 = ut (t0 )2 ,

t→t0

lim

t→t0

 G(x, u(t))dx =

G(x, u(t0 ))dx.

So ut (t) ∈ C[0, T ]. By the uniform convexity of L2 , ut ∈ C([0, T ]; L2 ). Therefore, problem (1.1)–(1.2) possesses a unique solution u, with (u, ut ) ∈ C([0, T ], H 1 × L2 ), and the estimates (2.1)–(2.3) hold. Theorem 2.1 is proved. 3. Global attractor We denote the solution in Theorem 2.1 by S(t)(u0 , u1 ) = (u(t), ut (t)). Theorem 2.1 shows that the solution operators S(t) compose a continuous semigroup in H. Theorem 3.1. In addition to Assumption 1, if also M ∈ C 2 (R+ ), g  (x, s) > −l for some constant l > 0. Then the dynamical system (S(t), H) possesses a compact global attractor A. Proof. Estimate (2.1) shows that the dynamical system (S(t), H) is dissipative and B0 = {(u, v) ∈ H | u2H 1 + v2 ≤ R02 } is an absorbing set for R0 suitably large, so there exists a t0 > 0 such that S(t)B0 ⊂ B0 for t > t0 . Let B = [∪t≥t0 +1 S(t)B0 ]H , where [·]H denotes the closure in H. Obviously, B is a forward invariant and bounded closed absorbing set, and it is complete with respect to the norm of H. So it is enough to show that the dynamical system (S(t), B) has a global attractor. We construct the functions ⎧ 0, 0 ≤ s ≤ 1, ⎪ ⎨ K0 (s) = s − 1, 1 < s ≤ 2, ⎪ ⎩ 1, s > 2,  Kδ (s) = (ρδ ∗ K0 )(s) = ρδ (s − y)K0 (y)dy, R

where ρδ (s) is the standard mollifier on R with supp ρδ ⊂ [−δ, δ]. Obviously, Kδ ∈ C ∞ (R),

0 ≤ Kδ (s) ≤ 1,

Kδ (s) = 0

as 0 ≤ s < 1;

Kδ (s) = 1

as s > 2,

with 0 < δ  1. Let ϕ(x) = Kδ ( |x| R ), with R > r0 . A simple calculation shows that ϕ(x) = 0

as |x| < R, 0 ≤ ϕ(x) ≤ 1

and |∇ϕ(x)|2 ≤

C ϕ(x), R2

x ∈ RN .

(3.1)

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1843

Lemma 3.1. Let Assumption 1: (i) be in force and g  (x, s) > −l. Then 

c2 ϕu2 , 2    ϕ2 g(x, u)u − ηG(x, u) dx ≥ ηϕu2 ϕ2 G(x, u)dx ≥

for η : 0 < η ≤

2c2 2c2 +l+2

(< 1).

Proof. When |x| > r0 , taking account of g(x, 0) = 0, g  (x, s) > −l and making use of assumption (G4 ), we have G(x, u) ≥

c2 2 u , 2 u

g(x, u)u − ηG(x, u) = η 

(g(x, u) − g(x, ξ))dξ + (1 − η)g(x, u)u 0

≥ c2 (1 − η) −

ηl  2 u ≥ ηu2 . 2

Since ϕ(x) = 0 as |x| ≤ r0 (< R), we have 



ϕ2 G(x, u)dx ≥

ϕ2 G(x, u)dx =  2

ϕ



|x|>r0

  g(x, u)u − ηG(x, u) dx =

c2 ϕu2 , 2

  ϕ2 g(x, u)u − ηG(x, u) dx

|x|>r0



|ϕu|2 dx = ηϕu2 .

≥η |x|>r0

Lemma 3.2 (Tail estimate). Let S(t)(u0 , u1 ) = (u(t), ut (t)), with (u0 , u1 ) ∈ B. Then for any > 0, there exist positive constants R1 = R1 (R0 ) and T0 = T0 (R0 ) such that (u(t), ut (t))H(ΩC2R ) <

as R > R1 , t > T0 ,

N where and in the following Ω2R is the ball in RN with radius 2R, ΩC 2R = R \ Ω2R .

Proof. Using the multiplier ϕ2 (ut + u) in (1.1), taking account of the boundedness of (u, ut ) in H and (3.1), we have d H2 (t) + K2 (t) dt    = −2 ϕ∇ϕ (ut + u)∇ut + M (∇u2 )(ut + u)∇u dx + M  (∇u2 )(∇u, ∇ut )ϕ∇u2 1 (ϕut 2 + ϕ∇ut 2 ) + 2 ϕ∇u2 + C(ut ∇ϕ2 + ∇ϕ∇u2 + u∇ϕ2 ) + C∇ut ϕ∇u2 2 1 C ≤ (ϕut 2 + ϕ∇ut 2 ) + 2 ϕ∇u2 + 2 + C∇ut ϕ∇u2 , (3.2) 2 R ≤

where

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  1 2 2 2 H2 (t) = ϕut  + M (∇u )ϕ∇u + 2 ϕ2 (G(x, u) − f u)dx 2   1 + (ϕ2 u, ut ) + (ϕu2 + ϕ∇u2 ) , 2 2 K2 (t) = (1 − )ϕut  + ϕ∇ut 2 + M (∇u2 )ϕ∇u2 + (ϕ2 g(·, u), u) − (ϕ2 u, f ). Taking account of M (s) ≥ M0 > 0 and exploiting Lemma 3.1, we get H2 (t) ≥ κ(ϕut 2 + ϕu2 + ϕ∇u2 ) − Cϕf 2 , 1 K2 (t) − (ϕut 2 + ϕ∇ut 2 ) − 2 ϕ∇u2 − η H2 (t) 2  1 η 1 η η  = ( − − )ϕut 2 + ϕ∇ut 2 + (1 − )M (∇u2 ) − (1 + ) ϕ∇u2 2 2 2 2 2       1 2 2 2 2 + ϕ g(x, u)u − ηG(x, u) dx − (1 − η) ϕ uf dx − η (ϕ u, ut ) + ϕu2 2 ≥ κ(ϕut 2 + ϕ∇ut 2 + ϕ∇u2 + ϕu2 ) − Cϕf 2

(3.3)

(3.4)

for > 0 suitably small. Inserting (3.4) into (3.2), we have C d H2 (t) + η H2 (t) ≤ C∇ut H2 (t) + Cϕf 2 (∇ut  + 1) + 2 . dt R

(3.5)

Applying Lemma 2.5 to (3.5), we get H2 (t) ≤ CH2 (0)e

−κt

t +C

 1  e−κ(t−τ ) ϕf 2 (∇ut (τ ) + 1) + 2 dτ R

0

 1  2 + ϕf  ≤ CH2 (0)e−κt + C , R2  1  2 (u(t), ut (t))2H(ΩC ) ≤ Ce−κt + C + f  C 2 L (ΩR ) . 2R R2

(3.6)

(3.6) implies the conclusion of Lemma 3.2. Remark 3.2. For any bounded set B ⊂ H, there exists a tB > 0 such that S(t)B ⊂ B as t ≥ tB . So we infer from Lemma 3.2 that for any > 0, there exist positive constants R1 = R1 (R0 ) and T1 = tB + T0 such that (u(t), ut (t))H(ΩC2R ) < as R > R1 , t > T1 , where (u(t), ut (t)) = S(t)(u0 , u1 ), (u0 , u1 ) ∈ B. Lemma 3.3. Let the assumptions of Theorem 3.1 be in force, and u, v be two solutions of the Cauchy problem (1.1)–(1.2) corresponding to initial data (u0 , u1 ), (v0 , v1 ) ∈ B, z = u − v. Then (z, zt )(t)2H

≤

C(z0 , z1 )2H e−κt

t +C

e−κ(t−τ ) z(τ )2 dτ.

(3.7)

0

Proof. Obviously, z = u − v solves ˜ 12 (t)(∇(u + v), ∇z)Δ(u + v) + g(·, u) − g(·, v) = 0, ztt − Δzt − M12 (t)Δz + zt − M z(0) = u0 − v0 ≡ z0 ,

zt (0) = u1 − v1 ≡ z1 ,

(3.8)

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˜ 12 (t) are as shown in (2.27). Using the multiplier zt in (3.8), we get where M12 (t) and M  1 d ˜ 12 (t)(∇(u + v), ∇z)2 + zt 2 1 zt 2 + M12 (t)∇z2 + M H 2 dt   1 ˜ 12 (t)(∇(ut + vt ), ∇z)(∇(u + v), ∇z) = M  (t)(∇u, ∇ut ) + M  (t)(∇v, ∇vt ) ∇z2 + M 2 1   1 M  (λ∇u2 + (1 − λ)∇v2 ) λ(∇u, ∇ut ) + (1 − λ)(∇v, ∇vt ) dλ(∇(u + v), ∇z)2 + 2 0

− (g(·, u) − g(·, v), zt ) ≤ C(∇ut  + ∇vt )∇z2 − (g(·, u) − g(·, v), zt ).

(3.9)

(i) When 1 ≤ p < p∗ , there exists a δ : 0 < δ << 1 such that H 1−δ → Lp+1 . By the interpolation theorem,  |(g(·, u) − g(·, v), zt )| ≤ C

(|u|p−1 + |v|p−1 + 1)|z||zt |dx

p−1 ≤ Czzt  + C(up−1 p+1 + vp+1 )zp+1 zt p+1 1−δ ≤ Czzt  + CzH 1−δ zt H 1 ≤ Czzt  + Czδ zH 1 zt H 1

≤ zt 2H 1 + 2 z2H 1 + Cz2 .

(3.10)

Inserting (3.10) into (3.9), we get  1 d ˜ 12 (t)(∇(u + v), ∇z)2 + (1 − )zt 2 1 zt 2 + M12 (t)∇z2 + M H 2 dt ≤ 2 z2H 1 + C(∇ut  + ∇vt )∇z2 + Cz2 .

(3.11)

Similarly, using the multiplier z in (3.8) and adding z2 to both sides, we have  1 d ˜ 12 (t)(∇(u + v), ∇z)2 (zt , z) + z2H 1 + z2 + M12 (t)∇z2 + M dt 2 ≤ zt 2 + z2 − (g(·, u) − g(·, v), z) ≤ zt 2 + z2H 1 + Cz2 .

(3.12)

(3.11) + × (3.12) gives d H3 (t) + K3 (t) ≤ C(∇ut  + ∇vt )∇z2 + Cz2 , dt

(3.13)

where    1 ˜ 12 (t)(∇(u + v), ∇z)2 + (zt , z) + 1 z2 1 zt 2 + M12 (t)∇z2 + M H 2 2 ˜ 12 (t)(∇(u + v), ∇z)2 , ∼ zt 2 + z2H 1 + M

H3 (t) =

˜ 12 (t)(∇(u + v), ∇z)2 K3 (t) = (1 − 2 )zt 2H 1 + z2 − 2 2 z2H 1 + M12 (t)∇z2 + M ˜ 12 (t)(∇(u + v), ∇z)2 ∼ zt 2H 1 + z2H 1 + M for > 0 suitably small. Therefore, there exists a κ > 0 such that K3 (t) − κH3 (t) ≥ 0.

(3.14)

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Inserting (3.14) into (3.13), we have d H3 (t) + κH3 (t) ≤ C(∇ut  + ∇vt )H3 (t) + Cz2 , dt t 2 2 −κt (z, zt )(t)H ≤ C(z0 , z1 )H e + C e−κ(t−τ ) z(τ )2 dτ.

(3.15)

0

(ii) When p = p∗ , noticing that p∗ ≥ 2 implies N ≤ 6 and H 1 → L3 , we have 1 d (g(·, u) − g(·, v), zt ) = 2 dt

 1   ˜ g  (x, λu + (1 − λ)v) + l z 2 dλdx + G(t), 0

1 

˜ |G(t)| = 2

1

 

g  (x, λu + (1 − λ)v) λut + (1 − λ)vt z 2 dλdx + l(z, zt )

0 ∗

∗

p −2 2 ≤ C(ut 3 + vt 3 )z23 + C(upp∗ −2 +1 + vp∗ +1 )(ut p∗ +1 + vt p∗ +1 )zp∗ +1 + l|(z, zt )|

≤ C(ut H 1 + vt H 1 )z2H 1 + zt 2 + Cz2 , and (g(·, u) − g(·, v), z) ≥ −lz2 . Hence, we infer from (3.9) and (3.12) that d ˜ ˜ 3 (t) ≤ C(ut H 1 + vt H 1 )z2 1 + Cz2 , H3 (t) + K H dt

(3.16)

where  ˜ 12 (t)(∇(u + v), ∇z)2 ˜ 3 (t) = 1 zt 2 + M12 (t)∇z2 + M H 2 1       1 + g  (x, λu + (1 − λ)v) + l z 2 dxdλ + (zt , z) + z2H 1 2 0

˜ 12 (t)(∇(u + v), ∇z)2 , ∼ zt 2 + z2H 1 + M ˜ 3 (t) = zt 2 1 − 2 zt 2 + z2 + M12 (t)∇z2 + M ˜ 12 (t)(∇(u + v), ∇z)2 − 2 z2 1 K H H ˜ 12 (t)(∇(u + v), ∇z)2 ∼ zt 2H 1 + z2H 1 + M for > 0 suitably small. Repeating the proof of (3.14)–(3.15), we still get (3.7). Lemma 3.3 is proved. Asymptotic compactness. Let {(un0 , un1 )} be a bounded sequence in B, S(t)(un0 , un1 ) = (un (t), unt (t)). Applying Lemma 3.3 to wm,n (t) = un (t + tn − T ) − um (t + tm − T ), with tn > tm > T > 0, t ≥ 0, we obtain (wm,n , wtm,n )(t)2H ≤ Ce−κt + C sup un (tn − T + s) − um (tm − T + s)2 . 0≤s≤t

By taking t = T we get

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2 (un (tn ) − um (tm ), unt (tn ) − um t (tm ))H

≤ Ce−κT + C sup un (tn − s) − um (tm − s)2L2 (ΩC

2R )

0≤s≤T

+ C sup un (tn − s) − um (tm − s)2L2 (Ω2R ) .

(3.17)

C([0, T ], H 1 (Ω2R )) ∩ C 1 ([0, T ], L2 (Ω2R )) →→ C([0, T ], L2 (Ω2R )),

(3.18)

0≤s≤T

Since

we can extract a subsequence, still denoted by {un }, such that un → u

in C([0, T ], L2 (Ω2R ))

for any fixed T > 0, R > 0. For any > 0, by fixing R : R > R1 , T : Ce−κT < /4 and taking n, m large enough such that tn − T > T0 , tm − T > T0 , we infer from Lemma 3.2 and (3.17) that Ce−κT + C sup un (tn − s) − um (tm − s)2L2 (ΩC 0≤s≤T

2R )

C sup un (tn − s) − um (tm − s)2L2 (Ω2R ) < 0≤s≤T

<

. 2

, 2

(3.19) (3.20)

The combination of (3.17) with (3.19)–(3.20) yields 2 (un (tn ) − um (tm ), unt (tn ) − um t (tm )H < ,

that is, S(t) is asymptotically compact on B. Therefore, the dynamical system (S(t), B), and hence the dynamical system (S(t), H), has a global attractor A. Theorem 3.1 is proved. 4. Exponential attractor Definition. Let X be a complete metric space. A set Aexp ⊂ X is said to be an exponential attractor of the dynamical system (S(t), X) if (1) (2) (3) (4)

it is a compact set in X; it has finite fractal dimension in X, i.e. dim f {Aexp , X} < +∞; it is a forward invariant set, i.e. S(t)Aexp ⊂ Aexp , t ≥ 0; it attracts exponentially the bounded sets in X, that is, for any bounded set B ⊂ X, there exists a positive constant γ such that dist X {S(t)B, Aexp } ≤ C(BX )e−γt ,

t ≥ 0,

where BX = supζ∈B ζX . Theorem 4.1. Under the assumptions of Theorem 3.1, the dynamical system (S(t), H) possesses an exponential attractor Aexp . In order to prove Theorem 4.1, we need the following lemma. Lemma 4.2. (See [8].) Let X be a Banach space and M be a bounded closed set in X. Assume that the mapping V : M → M possesses the properties:

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(i) V is Lipschitz on M , i.e., there exists an L > 0 such that V v1 − V v2  ≤ Lv1 − v2 ,

∀v1 , v2 ∈ M ;

(ii) there exist the compact seminorms n1 (x), n2 (x) on X such that   V v1 − V v2  ≤ ηv1 − v2  + K n1 (v1 − v2 ) + n2 (V v1 − V v2 ) for any v1 , v2 ∈ M , where 0 < η < 1 and K > 0 are constants. Then for any κ > 0 and δ ∈ (0, 1 − η) there exists a forward invariant compact set Aκ,δ ⊂ M of finite fractal dimension such that dist(V k M, Aκ,δ ) ≤ q k ,

k = 1, 2, . . . ,

where q = η + δ < 1, and  dim f Aκ,δ ≤ ln

 2K(1 + L2 )1/2   1 −1  · ln m0 +κ , δ+η 1−η

where m0 (R) is the maximal number of pairs (xi , yi ) in X × X possessing the properties xi 2 + yi 2 ≤ R2 ,

n1 (xi − xj ) + n2 (yi − yj ) > 1,

i = j.

That is, the discrete dynamical system (V k , M ) possesses an exponential attractor Aκ,δ . Proof of Theorem 4.1. We have known that the dynamical system (S(t), H) has a forward invariant bounded absorbing set B, which is closed in H. Especially, we see from the estimate (2.2) that B is bounded in H 1 ×H 1 , and for any ξu = (u0 , u1 ) ∈ B, ξu (t) = S(t)ξu = (u(t), ut (t)) ∈ B, and u(t)H 1 + ut (t)H 1 + utt (t) ≤ C,

t ≥ 0.

(4.1)

Define the operator V = S(T ) : B → B. Obviously, V B ⊂ B and V is Lipschitz on B. For any ξu , ξv ∈ B, we infer from Lemma 3.3 and (2.3) that V ξu − V

ξv 2H

−κ(T −t0 )

≤ Ce

T ξu (t0 ) −

ξv (t0 )2H

+C

e−κ(T −s) u(s) − v(s)2 ds

t0

≤

ηT2 ξu

−

ξv 2H

+ C max u(s) − v(s)2 , t0 ≤s≤T

that is, V ξu − V ξv H ≤ ηT ξu − ξv H + Cn1 (ξu − ξv ), where ηT2 = Ce(κ+k)t0 e−κT ,

n1 (ξu ) = max u(s). t0 ≤s≤T

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We claim that n1 (ξu ) is a compact seminorm on H. Indeed, for any bounded sequence {ξun } ⊂ H, ξun H ≤ R∗ , on account of n1 (ξun − ξum ) = max un (s) − um (s) t0 ≤s≤T

≤ max un (s) − um (s)L2 (Ω2R ) + max un (s) − um (s)L2 (ΩC2R ) , t0 ≤s≤T

t0 ≤s≤T

for any > 0, taking R > R1 , t0 > T1 (= tR∗ + T0 ), we infer from Remark 3.2 that max un (s) − um (s)L2 (ΩC2R ) < /2.

t0 ≤s≤T

For fixed R and t0 , taking T such that 0 < ηT < 1, by (3.18) (replacing 0 there by t0 ), one can extract a subsequence, still denoted by {un }, such that un → u

in C([t0 , T ]; L2 (Ω2R )).

That is, there exists an N > 0 such that max un (s) − um (s)L2 (Ω2R ) < /2 as n, m > N.

t0 ≤s≤T

Therefore, n1 (ξun − ξum ) → 0 as n, m → ∞. The claim is valid. By Lemma 4.2, the discrete dynamical system (V k , B) has an exponential attractor A, where V k = S(kT ). Let Aexp =



S(t)A.

0≤t≤T

By the standard method (cf. [33,35]), one easily knows that Aexp is an exponential attractor of the dynamical system (S(t), B). So there exists a γ > 0 such that dist H {S(t)B, Aexp } ≤ Ce−γt ,

t ≥ 0.

We claim that Aexp is an exponential attractor of the dynamical system (S(t), H). Indeed, (i) Obviously, Aexp is a forward invariant set. (ii) For every bounded set B ⊂ H, there exists a tB > 0 such that S(t)B ⊂ B as t ≥ tB , so dist H {S(t)B, Aexp } = dist H {S(t − tB )S(tB )B, Aexp } ≤ dist H {S(t − tB )B, Aexp } ≤ Ce−γ(t−tB ) ≤ C(BH )e−γt . When t < tB , dist H {S(t)B, Aexp } ≤ Ceγt e−γt ≤ C(BH )e−γt .

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(iii) Define the operator F : [0, T ] × A → B,

F (t, ξu ) = ξu (t) = S(t)ξu ,

ξu ∈ A.

It follows from (4.1) and (2.3) that

F (t1 , ξu ) − F (t2 , ξu )H = S(t1 )ξu − S(t2 )ξu H

t2

≤ ξu (τ )H dτ ≤ C|t1 − t2 |, t1

F (t, ξu1 ) − F (t, ξu2 )H = S(t)ξu1 − S(t)ξu2 H ≤ Cξu1 − ξu2 H

(4.2)

for all ξu , ξu1 , ξu2 ∈ B, t, t1 , t2 ∈ [0, T ], that is, the mapping F is Lipschitz continuous, so Aexp = F {[0, T ] × A} (the image of the set [0, T ] × A) is a compact set in H, and (iv) dimf (Aexp , H) ≤ 1 + dimf {A, H} < +∞. So, Aexp is the desired exponential attractor. Theorem 4.1 is proved. Remark 4.1. Theorem 4.1 implies that the global attractor A in Theorem 3.1 has finite fractal dimension. Acknowledgments The authors thank the referee for his valuable comments and suggestions which helped improving the original manuscript. References [1] F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990) 85–108. [2] A.V. Babin, M.I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A 116 (1990) 221–243. [3] V. Belleri, V. Pata, Attractors for semilinear strongly damped wave equations on R3 , Discrete Contin. Dyn. Syst. 7 (2001) 719–735. [4] M.M. Cavalcanti, V.N.D. Cavalcanti, J.S.P. Filho, J.A. Soriano, Existence and exponential decay for a Kirchhoff–Carrier model with viscosity, J. Math. Anal. Appl. 226 (1998) 40–60. [5] F.-X. Chen, B.-L. Guo, P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations 147 (1998) 231–241. [6] I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl. 1 (2010) 86–106. [7] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations 252 (2012) 1229–1262. [8] I. Chueshov, I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., vol. 195, Amer. Math. Soc., Providence, RI, 2008, No. 912. [9] M. Conti, V. Pata, M. Squassina, Strongly damped wave equations on R3 with critical nonlinearities, Commun. Appl. Anal. 9 (2005) 161–176. [10] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin/New York, 1976. [11] M.A. Efendiev, S. Zelik, The attractor for a nonlinear reaction–diffusion system in an unbounded domain, Comm. Pure Appl. Math. 54 (2001) 625–688. [12] X. Fan, S. Zhou, Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhoff-type, Appl. Math. Comput. 158 (2004) 253–266. [13] E. Feireisl, Attractors for semilinear damped wave equations on R3 , Nonlinear Anal. 23 (1994) 187–195. [14] E. Feireisl, Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1051–1062. [15] V. Kalantarov, S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations 247 (2009) 1120–1155.

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