Global Attractor for Nonlinear Wave Equations with Critical Exponent on Unbounded Domain

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Abstract

Asymptotic and global dynamics of weak solutions for a damped nonlinear wave equation with a critical growth exponent on the unbounded domain ℝn(n ≥ 3) is investigated. The existence of a global attractor is proved under typical dissipative condition, which features the proof of asymptotic compactness of the solution semiflow in the energy space with critical nonlinear exponent by means of Vitali-type convergence theorem.

Abstract

Asymptotic and global dynamics of weak solutions for a damped nonlinear wave equation with a critical growth exponent on the unbounded domain ℝn(n ≥ 3) is investigated. The existence of a global attractor is proved under typical dissipative condition, which features the proof of asymptotic compactness of the solution semiflow in the energy space with critical nonlinear exponent by means of Vitali-type convergence theorem.

1 Introduction

In this paper, we study the asymptotic behavior of solutions of a damped semilinear wave equation with nonlinearity of a critical growth exponent over the Euclidean space ℝn of arbitrary dimension n ≥ 3,

uttΔu+βut+f(x,u)+αu=g(x)

for t ≥ 0, with the initial condition

u(x,0)=u0(x),ut(x,0)=u1(x),

where α and β are arbitrary positive constants, g is a given functions defined on ℝn and f (x, u) is a nonlinear function satisfying some typical dissipative conditions to be specified.

The asymptotic dynamics of global weak solutions for deterministic nonlinear wave equations and for more general nonlinear hyperbolic evolutionary equations with linear or nonlinear damping have been studied in last three decades by many authors, e.g. [1]- [4], [6]- [8], [12]- [14], [16], [19]- [22], [26]. The obtained results focused on the existence of global attractors under certain assumptions.

In the arena of stochastic wave equations driven by additive or multiplicative noise, the solution mapping defines a random dynamical system or called a cocycle on a state space with a parametric base space. The existence of random attractors for stochastic damped wave equations has been studied in [9], [11], [15], [17],[18], [23]- [25].

However, the existence problem of global attractors remains open for damped nonlinear wave equations with nonlinearity of a critical growth exponent and on the unbounded domain Rn with arbitrary dimension. This is the topic of this work.

In case of nonlinearity with higher or critical growth exponents and on the unbounded domain, the issue of asymptotic compactness for the weak or mild solutions of nonlinear damped wave equations becomes difficult to handle due to not only the lack of compactness of the Sobolev embeddings but also the necessarily involved high-order integrable function spaces, in addition to the local existence and regularity of solutions in such spaces. In this work we shall tackle this challenging problem and prove the existence of a global attractor by means of

1) the uniform estimates for absorbing property and norm-smallness of solutions outside a large ball,

2) the esimates of the extended energy functional for the compactness in the space H1(ℝn) × L2(ℝn) and

3) the Vitali-type convergence criterion (Theorem 8) for the function space Lp(ℝn) shown in the paper. This new approach has potential applications to many other nonlinear and stochastic PDEs and to longtime and asymptotic dynamics of various problems with complex and nonlinear interactions.

In Section 2, we briefly recall basic concepts and results related to semiflow and global attractors. In Section 3, we shall conduct uniform estimates of the weak solutions for absorbing sets and for tail parts. In Section 4, we shall establish the intricate asymptotic compactness of the solution semiflow with respect to the Hilbert energy space H1(ℝn) × L2(ℝn). In Section 5, we prove the crucial asymptotic compactness of the first component of solutions in Lp(ℝn). Then the existence of a global attractor for this nonlinear damped wave equation is finally proved.

In this paper, we shall use || · || and 〈·,·〉 to denote the norm and inner product of L2(ℝn), respectively. The norm of Lr(ℝn) with r ≠ 2 or a Banach space X will be denoted by || · ||r or || · ||X . We use c, C or Ci to denote generic or specific positive constants.

2 Preliminaries and Assumptions

Let (X,|| · ||X ) be a real Banach space. The following are the basic concepts and result on the topic of global attractor for infinite dimensional dynamical systems, cf. [2], [8], [20] and [22].

Definition 1

A mapping Φ : ℝ+ × XX is called a semiflow on X, if the following conditions are satisfied:

  • (i) Φ(0, ·) is the identity on X.
  • (ii) Φ(t + s, ·) = Φ(t, Φ(s, ·)), for any t, s ≥ 0.
  • (iii) Φ : ℝ+ × XX is a continuous mapping.

Definition 2

Let Φ be a semiflow on X. A bounded set KX is called an absorbing set for Φ if for any bounded subset BX there exists a finite time TB > 0 such that

Φ(t,B)={Φ(t,x):xB}K, for all  t>TB.

Φ is called asymptotically compact in X if for any given bounded set BX it holds that

{Φ(tm,xm)}m=1 has a convergent subsequence in X,

whenever tm → ∞ and {xm}m=1B.

Definition 3

Let F be a semiflow on X. A set 𝒜X is called a global attractor for Φ, if the following conditions are satisfied:

  • (i) 𝒜 is a compact and invariant set in the sense that Φ(t, 𝒜 ) = 𝒜, for all t ≥ 0.
  • (iii) 𝒜 attracts every bounded set B in X,

limtdistX(Φ(t,B),A)=0,

where distX (·,·) is the Hausdorff semi-distance with respect to the X-norm.

Theorem 1

Let Φ be a semiflow on a Banach space X. If the following two conditions are satisfied:

  • (1) there is a bounded absorbing set K for the semiflow Φ in X, and
  • (2) Φ is asymptotically compact in X,

then there exists a global attractor 𝒜 in X for the semiflow Φ, which is given by

A=ω(K)=τ0tτΦ(t,K)¯.

Now we formulate the original initial value problem of the nonlinear damped wave equation (1)-(2). Let ξ = ut + δ u, where δ is a positive number to be specified later. Then (1)-(2) can be rewritten as

ut+ δu=v,vtδv+(δ2+α+A)u +β(vδu)+f(x,u)=g(x)u(x,τ)=u0(x),v(x,τ )=v0(x)=u1(x)+δu0(x),

where the linear operator A = Δ : H2(ℝn) → L2(ℝn).

Standing Assumption. Throughout the paper, assume that the nonlinear term fC1(ℝn ×, ℝ), n ≥ 3, and its antiderivative F(x,u)=0uf(x,s)ds satisfy the following conditions:

|f(x,u)|C1|u|p1+ϕ1(x),ϕ1(x)H1(n),

f(x,u)uC2F(x,u)ϕ2(x),ϕ2(x)L1(n),

F(x,u)C3|u|pϕ3(x),ϕ3(x)L1(n),

where C1, C2 and C3 are positive constants and 1pn+2n2 is arbitrarily given. Assume that gH1(ℝn).

Define the phase space

E=(H1(n)Lp(n))×L2(n)

endowed with the norm

(u,v)(H1Lp)×L2=(u2+u2+v2)12+uLp,for (u,v)E.

Lemma 2

For any given g0 = (u0, v0) ∈ E, the initial value problem (4) has a unique global weak solution

(u(,u0),v(,v0))C([0,),E).

Moreover, for any t ≥ 0, the solution (u(t, u0), v(t, v0)) is weakly continuous with respect to g0 = (u0, v0) ∈ E in the sense that

(u(t,u0,m),v(t,v0,m))(u(t,u0),v(t,v0))

weakly in E, provided that g0, m = (u0,m, v0, m) ⇀ g0 = (u0, v0) weakly in E.

Proof. The local existence and uniqueness of a weak solution for this problem (4) in the phase space E = (H1(ℝn) ⋂ Lp(ℝn)) × L2(ℝn) and its weakly continuous dependence on the initial data can be established by the Galerkin approximation method as in [8, Chapter XV] and [3]. Also see [20], [22] and [25]. Here the detail is omitted. The proof of the global existence of weak solutions will be included in the proof of Lemma 3 below.

3 Uniform Estimates of Solution Trajectories

In this section, we shall derive uniform estimates on the solutions of the nonlinear damped wave equation(4) defined on ℝn in a long run. These a priori estimates pave the way to proving the existence of absorbing set and the asymptotic compactness of the semiflow Φ. In particular, we will show that tails of the solutions for large spatial variables are uniformly small when time is sufficiently large.

Define a new norm of E by

(u,v)E=(v2+(α+δ2βδ)u2+u2)12+uLp,

in which and hereafter let δ be a fixed positive constant satisfying

α+δ2βδ>0  and  β3δ>0.

Obviously the norm || · ||E in (9) and the Sobolev norm || · ||(H1Lp)×L2 in (8) are equivalent.

3.1 Absorbing Set

The next lemma shows that there exists an absorbing set in the Banach space E for the semiflow Φ generated by the weak solutions (u(t, u0), (v(t, v0)) to the problem (4),

Φ(t,g0)=(u(t,u0),v(t,v0)),t0,g0=(u0,v0).

Lemma 3

There exists an absorbing set K ⊂ E for the solution semiflow Φ of the problem (4). For any bounded set B ⊂ E, there exists a finite TB > 0, such that

Φ(t,B)K, for all t>TB.

Proof. Take the inner product of the second equation of (4) with v in L2(ℝn) to get

12ddtv2δv2+(α+δ2) u,v+Au,v+f(x,u),v=β(vδu,v+g(x),v.

Then we find that

u,v= u,ut+δu=12ddtu2+δu2 and Au,v=12ddtu2+δu2.

For the last term on the left-hand side of (11), we have

f(x,u),v=ddtnF(x,u)dx+δf(x,u),u.

By (6), we get

δf(x,u),uδC2nF(x,u)dx+δnϕ2dx.

For the last term on the right-hand side of (11),

g,vgvg22(βδ)+βδ2v2.

Substitute the above inequalities into (11) to obtain

12ddt[v2+(α+δ2βδ)u2+u2+2nF(x,u)dx]+δ2[v2+(α+δ2βδ)u2+u2]+δC2nF(x,u)dx 3δβ2v2+g22(βδ)+δϕ2L1g22(βδ)+δϕ2L1,t0.

where the term (3δ − β)||v||2/2 ≤ 0 due to (10). Let σ be a fixed positive constant:

σ=min{δ,δC2}>0.

Note that ∫n (F(x, u) + ϕ3(x))dx ≥ 0 due to (7). It follows from (12) and (13) that

ddt[v2+(α+δ2βδ)u2+u2+2n(F(x,u)+ϕ3(x))dx]+σ[v2+(α+δ2βδ)u2+u2+2n(F(x,u)+ϕ3(x))dx] g2βδ+2δ(C2ϕ3L1+ϕ2L1),t0.

Apply Gronwall inequality to (14) and see that every weak solution of (4) satisfies

v(t)2+(α+δ2βδ)u(t)2+u(t)2+2n(F(x,u(t))+ϕ3(x))dx eσ(tτ)[v02+(α+δ2βδ)u02+u02+2nF(x,u0)dx]+2eσ(tτ)ϕ3L1+1σ(2δ(C2ϕ3L1+ϕ2L1)+g2βδ).

Thus for any given bounded set BE and any (u0, v0) ∈ B, we have

v(t)2+(α+δ2βδ)u(t2+u(t)2+2n(F(x,u(t))+ϕ3(x))dx eσt[v02+(α+δ2βδ)u02+u02+2nF(x,u0)dx+2ϕ3L1]+1σ(2δ(C2ϕ3L1+ϕ2L1)+g2βδ)t0.

According to the assumption (5) and (6), there exists a constant c = c(C1, C2, ϕ1, ϕ2) > 0 such that

nF(x,u0)dxc(1+u02+u0Lpp).

It follows that, for any given bounded set BE and (u0, v0) ∈ B, there exist a constant C > 0 and a finite TB > 0 such that

eσt [v02+(α+δ2βδ)u02+u02+2nF(x,u0)dx+2ϕ3L1]Ceσt(1+v02+u0H12+u0Lpp)1, for all t>TB.

Substitute (17) into the right-hand side of the last equality in (16) and note that (7) implies

2n(F(x,u(t,u0)+ϕ3(x))dx2C3u(t,u0)Lpp.

Then it results in

v(t,v0)2+(α+δ2βδ)u(t,u0)2+u(t,u0)2+2C3u(t,u0)Lpp1+1σ(2δ(C2ϕ3L1+ϕ2L1)+g2βδ), for  t>TB.

The inequality (18) show that Φ(t, B) ⊂ K = BE(0, R) for t > TB, where the radius of the ball BE(0, R) in E is

R= (1min{1,(α+δ2βδ)}[1+1σ(2δ(C2ϕ3L1+ϕ2L1)+g2βδ)])12+(12C3[1+1σ(2δ(C2ϕ3L1+ϕ2L1)+g2βδ)])1p.

Therefore, this set K = BE(0, R) is an absorbing set in the phase space E for the solution semiflow Φ. The proof is completed.

3.2 Tail Estimate

Next we conduct uniform estimates on the tail parts of the weak solutions for large spatial and time variables. These estimates play key roles in proving the asymptotic compactness in the space E of the dynamical systems F generated by the nonlinear wave equation (4) on the unbounded domain ℝn.

Lemma 4

For every bounded set B ⊂ E and 0 < η ≤ 1, there exists T = T (B, η) > 0 and V = V (η) ≥ 1 such that the semiflow Φ generated by the nonlinear damped wave equation (4) satisfies

Φ(t,B)E(RnBr)=maxg0BΦ(t,g0)ζBrcE<η,

for all t > T and every r > V , where ζBrc(x) is the characteristic function of the set {x ∈ ℝn : |x| > r}.

Proof. Choose a smooth and nondecreasing function ρ such that 0 ≤ ρ(s) ≤ 1 for all s ∈ [0, ∞) and

ρ(s)={0,  if0s<1,1,  ifs>2,

with 0 ≤ ρ(s) ≤ 2 for s ≥ 0. Taking the inner product of the second equation of (4) with ρ(|x|2/r2)v in L2(ℝn), we get

12ddtnρ(|x|2r2)|v|2dxδnρ(|x|2r2)|v|2dx+(α+δ2)nρ(|x|2r2)uvdx+n(Au)ρ(|x|2r2)vdx+nρ(|x|2r2)f(x,u)vdx=nρ(|x|2r2)gvdxnρ(|x|2r2)β(vδu)vdx.

Hence we have

12ddtnρ(|x|2r2)|v|2dx+(α+δ2βδ)nρ(|x|2r2)uvdx+ (βδ)nρ(|x|2r2)|v|2dx+n(Au)ρ(|x|2r2)vdx+nρ(|x|2r2)f(x,u)vdx δ2nρ(|x|2r2)|v|2dx+nρ(|x|2r2)gvdx.

For the second term on the left-hand side of (23), by (4) we have

(α+δ2βδ)nρ(|x|2r2)uvdx=(α+δ2βδ)nρ(|x|2r2)u(ut+δu)dx (α+δ2βδ)(12ddtnρ(|x|2r2)|u|2dx+δnρ(|x|2r2)|u|2dx)δ2(α+δ2βδ)nρ(|x|2r2)|u|2dx.

For the fourth term on the left-hand side of (23),

n(Au)ρ(|x|2r2)vdx=n(Au)ρ(|x|2r2)(ut+δu)dx=n(u)(ρ(|x|2r2)(ut+δu))dx=n(u)2xr2ρ(|x|2r2)vdx+n(u)ρ(|x|2r2)(ut+δu)dx=n(u)2xr2ρ(|x|2r2)vdx+12ddtnρ(|x|2r2)|u|2dx+δnρ(|x|2r2)|u|2dx.

Since 0 ≤ ρ(s) ≤ 2, it follows that

n(Au)ρ(|x|2r2)vdxr|x|2r4|x|r2|(u)v|dx+12ddtnρ(|x|2r2)|u|2dx+δnρ(|x|2r2)|u|2dx22rr|x|2r(|u|2+|v|2)dx+12ddtnρ(|x|2r2)|u|2dx+δ2nρ(|x|2r2)|u|2dx.

For the fifth term on the left-hand side of (23), by (5)-(7), we have

nρ(|x|2r2)f(x,u)vdx=nρ(|x|2r2)f(x,u)(ut+δu)dx ddtnρ(|x|2r2)F(x,u)dx+δnρ(|x|2r2)(C2F(x,u)+ϕ2(x))dx.

For the last term on the right-hand side of (23), we see

nρ(|x|2r2)gvdx12(βδ)nρ(|x|2r2)|g|2dx+βδ2nρ(|x|2r2)|v|2dx.

Now substitute (24)-(27) into (23), we obtain

12ddtnρ(|x|2r2)(|v|2+(α+δ2βδ)|u|2+|u|2+2F(x,u))dx+δ2nρ(|x|2r2)|v|2dx+δ2nρ(|x|2r2)((α+δ2βδ)|u|2+|u|2)dx+δC2nρ(|x|2r2)F(x,u)dx 22rr|x|2r(|u|2+|v|2)dx+nρ(|x|2r2)(|g|2βδ+δ|ϕ2|)dx.

Since gL2(ℝn) and ϕ2, ϕ3L1(ℝn), for any η > 0, there exists K0 = K0(η) 1 such that for all rK0,

nρ(|x|2r2)(|g|2βδ+δ|ϕ2|+2σ|ϕ3|)dx|x|r(|g|2βδ+δ|ϕ2|+2σ|ϕ3|)dx<η.

By (13) and (28)-(29), there exists K1 = K1(η) ≥ 1 such that for all r > K1,

ddtnρ(|x|2r2)(|v|2+(α+δ2βδ)|u|2+|u|2+2(F(x,u)+ϕ3))dx+σnρ(|x|2r2)(|v|2+(α+δ2βδ)|u|2+|u|2+2(F(x,u)+ϕ3))dx η[1+r|x|2r(|u|2+|v|2)dx].

Therefore, for any t > 0 and r > K1, it holds that

nρ(|x|2r2)[|v(t)|2+(α+δ2βδ)|u(t)|2+|u(t)|2+2(F(x,u(t))+ϕ3(x))]dx eσtnρ(|v0|2+(α+δ2βδ)|u0|2+|u0|2+2(F(x,u0)+ϕ3(x)))dx+ησ+η0teσ(ts)r|x|2r(|u(s)|2+|v(s)|2)dxds.

Next we conduct estimates of the terms on the right-hand side of in (30). For the first term, there exists T1 = T1(B, η) > 0 and a constant C4 > 0 such that

eσt nρ(|x|2r2)(|v0|2+(α+δ2βδ)|u0)|2+|u0|2+2(F(x,u0)+ϕ3))dx  eσtn(|v0|2+(α+δ2βδ)|u0|2+|u0|2)dx+2eσtn[1C2(C1|u0|p+|u0||ϕ1(x)|+|ϕ2(x)|)+|ϕ3(x)|]dx C4eσt((v0,u0)2+u02+u0Lpp+ϕ12+ϕ2L1+ϕ3L1)<η

for all t > T1. For the second integral term on the right-hand side of (30), applying the Gronwall inequality to(14) while taking the spatial integral over the region r|x|2r, with (13) in mind, we get

t0eσsr|x|2r(|u(s)|2+|v(s)|2)dxds eσ(s+t)(v02+(α+δ2βδ)u02+u02)+2eσ(s+t)n(F(x,u0)+ϕ3(x))dx+1σ(2δ(C2ϕ3L1+ϕ2L1)+1βδg2).

Based on (31) and (32), there exists T2 = T2(B, η) > 0 such that

t0eσsr|x|2r(|u(s)|2+|v(s)|2)dxdsCteσt[(u0,v0)2+u02+n(F(x,u0)+ϕ3(x))dx]+1σ(C6+1βδg2)Cteσt[(u0,v0)2+u02+ϕ3L1+1C2(C1u0Lpp+u02+ϕ12+ϕ2L1)]+1σ(2δ(C2ϕ3L1+ϕ2L1)+1βδg2)M, for all tT2,

where the constant

M=1+1σ(2δ(C2ϕ3L1+ϕ2L1)+1βδg2).

Now assemble all these estimates and substitute (31) and (33) into (30). It shows that for any bounded set B and any 0 < η ≤ 1, as long as r > V = max {K0, K1} and t > max {T1, T2}, one has

|x|2r(|v(t)|2+(α+δ2βδ)|u(t)|2+|u(t)|2+|u(t)|p)dxnρ(|x|2r2)(|v(t)|2+(α+δ2βδ)|u(t)|2+|u(t)|2)dx+2C3nρ(F(x,u(t)+ϕ3(x))dx(1+1C3)(2+M)η.

By (9), the above inequality (34) demonstrates that for any bounded BE it holds that

Φ(t,B)E(RnBR)=maxg0BΦ(t,g0)E(RnBR)1+1C3(2+M)η1/2+1+1C3(2+M)η1/p,

where R=2r. (35) implies that (20) is satisfied as stated in this theorem just by renaming r to be R and η to be ((1 + 1/C3)(2 + M)η)1/2 + ((1 + 1/C3)(2 + M)η)1/p. The proof is completed.

4 Asymptotic Compactness in H1(ℝn) × L2(ℝn)

In this section, we shall prove the asymptotic compactness in the space H1(ℝn) × L2(ℝn) of the solution semiflow Φ associated with the nonlinear damped wave equation (4).

Lemma 5

The following statements hold for Lp(ℝn).

  • 1) For 1 ≤ p <, let {ψm} be a sequence and ψ be a function in Lp(ℝn) such that ||ψm − ψ||Lp → 0 as m →. Then there exists a subsequence {ψmk} such that
    limkψmk(x)=ψ(x), a.e. on n.
  • 2) For 1 < p <, if a sequence {ψm} and a function ψ in Lp(ℝn) satisfy the following two conditions:
    limmψm(x)=ψ(x), a.e. on nand {ψm}is bounded in  Lp(n),
    then ψm → ψ weakly in Lp(ℝn), as m →.
  • 3) For 1 < p <, if a sequence {ψm} and a function ψ in Lp(ℝn) satisfy the following two conditions:
    limmψm(x)=ψ(x), a.e. on nandlimmψmLp=ψLp,
    then limm→ ||ψm − ψ||Lp = 0.

Proof. Since ℝn with the Lebesgue measure is a σ-finite measure space, the first item is a standard result in Real and Functional Analysis.

For the second item, since Lp(ℝn) is a reflexive Banach space for 1 < p < ∞, the boundedness of {ψm} in Lp(ℝn) implies that there is φLp(ℝn) such that ψm → φ weakly as m → ∞. By Mazur’s lemma, this weak convergence implies there exists a sequence {ζm} ⊂ Lp(ℝn) such that

ζmcovn{ψm,ψm+1,} and ζmφ  strongly in  Lp(n).

From the condition ψm → ψ a.e. and ζmconv(i=mψi), it follows that

ζmψa.e. in n.

On the other hand, by the first statement in this lemma, the strong convergence in (38) implies that there exists a subsequence {ζmk} such that ζmk → σ a.e. as k → ∞. Therefore, (39) leads to ψ = φ a.e. on ℝn so that ψm → ψ weakly as m → ∞. The third item is a known result in Functional Analysis, cf. [5, Chapter 4]. Thus the proof is completed.

Let us define the following energy functional on E: for (u, v) ∈ E,

Γ(u,v)=v2+(α+δ2βδ)u2+u2+2n(F(x,u)+ϕ3(x))dx.

Compare (9) and (40), we see that

Γ(u,v)(u,v)E2+2n(F(x,u)+ϕ3(x))dx.

Lemma 6

For every bounded set BE and any integer k > 0, there exists a constant M1 = M1(B, k) > 0 such that for all m > M1 one has tm > k with the property that

Γ(u(tmt,u0,m),v(tmt,v0,m))R+1+1σ[2δ(C2ϕ3L1+ϕ2L1)+g2βδ]

for all t ∈ [0, k] and (u0, m, v0,m) ∈ B, where the constant R is the same as in (19).

Proof. Integrate the inequality (14) over the time interval [0, t] [0, k], where δσ by (13). Similar to (18), there exists M1 = M1(B, k) > 0 such that for all m > M1 one has tm > k and

Γ(u(tmt,u0,m),v(tmt,v0,m))eσ(kt)Γ(u(tmk,u0,m),v(tmk,v0,m))+tkeσ(st)(2δ(C2ϕ3L1+ϕ2L1)+g2βδ)ds R+1+1σ[2δ(C2ϕ3L1+ϕ2L1)+g2βδ],t[0,k].

Therefore, (42) is proved.

Theorem 7

For every bounded set B and for any sequences tmand g0,m = (u0,m, v0,m) ∈ B, the sequence {Φ(tm,g0,m)}m=1 has a strongly convergent subsequence in H1(ℝn) × L2(ℝn), where Φ is the solution semiflow generated by the nonlinear damped wave equation (4).

Proof. The proof goes through the following steps.

Step 1

By Lemma 3, there is a constant M2 = M2(B) > 0 such that for all mM2 and g0,mB, we have

Φ(tm,g0,m)ER+1

where R > 0 is given by (19). Then there is (u˜,v˜)E such that, up to a subsequence and relabeled as the same,

Φ(tm,g0,m)(u˜,v˜)weakly in  EandΦ(tm,g0,m)(u˜,v˜)weakly in   H1(n)×L2(n).

Since E is a reflexive and separable Banach space, the weak lower-semicontinuity of the E-norm and of the norm of H1(ℝn) × L2(ℝn) as well implies that

liminfmΦ(tm,g0,m)E(u˜,v˜)E,liminfmΦ(tm,g0,m)H1×L2(u˜,v˜)H1×L2.

Next we want to prove that in the Hilbert space H1(ℝn) × L2(ℝn),

Φ(tm,g0,m)(u˜,v˜)strongly.

It suffices to show that

limsupmΦ(tm,g0,m)H1×L2(u˜,v˜)H1×L2.

If so, then (46) and (48) will lead to limmΦ(tm,g0,m)H1×L2=(u~,v~)H1×L2. By the item 3 of Lemma 5, we shall obtain (47).

Step 2

By Lemma 6 and (7), there exists a constant C > 0 such that, for any given integer k > 0 and all mM1(B, k), one has tm > k and

(u(tmt,u0,m),v(tmt,v0,m))EC[R+1+1σ(2δ(C2ϕ3L1+ϕ2L1)+g2βδ)]1/2+C[R+1+1σ(2δ(C2ϕ3L1+ϕ2L1)+g2βδ)]1/p,t[0,k],

for any (u0,m, v0,m) ∈ B. In particular, (49) is satisfied for t = k.

According to Banach-Alaoglu theorem, there exists a sequence {u˜k,v˜k}k=1 in the space E and subsequences of {tm}m=1 and {(u0,m,v0,m)}m=1 again relabeled as the same, such that for every integer k ≥ 1,

(u(tmk,u0,m),v(tmk,v0,m))(u˜k,v˜k) Weakly inE,

as m → ∞, which can be extracted through a diagonal selection procedure as in Real Analysis.

By the weakly continuous dependence on the initial data of the weak solutions stated in Lemma 2, here the weak convergence (50) together with the concatenation

(u(tm,u0,m),v(tm,v0,m))=(u(k,u(tmk,u0,m)),v(k,v(tmk,v0,m)))

implies that for all integers k ≥ 1, when m → ∞,

(u(tm,u0,m),v(tm,v0,m))(u(k,u˜k),v(k,v˜k))weakly in  E.

Thus (45) and (52) validate the following equality that for all positive integers k,

(u˜,v˜)=(u(k,u˜k),v(k,v˜k)).

By the similar argument from (11) to(12), the weak solutions (u, v) of (4) satisfy

ddtΓ(u(t,u0),v(t,v0))+2σΓ(u(t,u0),v(t,v0))G(u(t,u0),v(t,v0)),

where

G(u,v)=2(βδσ)v22(δσ)(α+δ2βδ)u22(δ &σ)u2+4σn(F(x,u)+ϕ3(x))dx2δf(x,u),u+2g,v.

From (53) and (54), for any integer k ≥ 1 we have

Γ(u˜,v˜)e2σkΓ(u˜k,v˜k)+0ke2σξG(u(ξ,u˜k),v(ξ,v˜k))dξ.

Step 3

On the other hand, from the concatenation (51) and the inequality (54), we obtain

Γ (u(tm,u0,m),v(tm,v0,m))e2σkΓ(u(tmk,u0,m),v(tmk,v0,m))2(βδσ)0ke2σξv(ξ,v(tmk,v0,m))2dξ2(δσ)(α+δ2βδ)0ke2σξu(ξ,u(tmk,u0,m))2dξ2(δσ)0ke2σξu(ξ,u(tmk,u0,m))2dξ+4σ0ke2σξn(F(x,u(ξ,u(tmk,u0,m)))+ϕ3(x))dxdξ2δ0ke2σξnf(x,u(ξ,u(tmk,u0,m)))u(ξ,u(tmk,u0,m))dxdξ+20ke2σξng(x)v(ξ,v(tmk,v0,m))dxdξ.

Below we treat all these terms on the right-hand side of the inequality (57).

1) For the first term on the right-hand side of (57), by (42) in Lemma 6, for all mM1(B, k) we have

e2σkΓ(u(tmk,u0,m),v(tmk,v0,m)) e2σk(R+1+1σe12σk[2δ(C2ϕ3L1+ϕ2L1)+g2βδ])eσk(R+1+1σ[2δ(C2ϕ3L1+ϕ2L1)+g2βδ]).

2) For the second term on the right-hand side of (57), by (50) and the weakly continuous dependence of solutions on the initial data stated in Lemma 2, we find that for any ξ ∈ [0, k], when m → ∞,

v(ξ,v(tmk,v0,m))v(ξ,v˜k) weakly in  L2(n),

which implies that for all ξ ∈ [0, k],

liminfmv(ξ,v(tmk,v0,m))2v(ξ,v˜k)2.

By (59) and Fatou’s lemma we obtain

liminfm0ke2σξv(ξ,v(tmk,v0,m))2dξ 0ke2σξliminfmv(ξ,v(tmk,v0,m))2dξ0ke2σξv(ξ,v˜k)2dξ.

Since (10) and (13) implies β − δ − σβ − 2δ > 0, (60) leads to

limsupm2(βδσ)0ke2σξv(ξ,v(tmk,v0,m))2dξ=2(βδσ)liminfm0ke2σξv(ξ,v(tmk,v0,m))2dξ 2(βδσ)0ke2σξv(ξ,v˜k)2dξ.

Similarly for the third and fourth terms, by (50) and Fatou’s lemma we obtain

limsupm 2(δσ)(α+δ2βδ)0ke2σξu(ξ,u(tmk,u0,m))2dξ2(δσ)(α+δ2βδ)0ke2σξu(ξ,u˜k)2dξ,limsupm 2(δσ)0ke2σξu(ξ,u(tmk,u0,m))2dξ2(δσ)0ke2σξu(ξ,u˜k)2dξ.

3) For the fifth term on the right-hand side of (57), we have

|0ke2σξn(F(x,u(ξ,u(tmk,u0,m)))F(x,u(ξ,u˜k)))dxdξ|0ke2σξ|x|>r|F(x,u(ξ,u(tmk,u0,m)))F(x,u(ξ,u˜k))|dxdξ+0ke2σξ|x|r|F(x,u(ξ,u(tmk,u0,m)))F(x,u(ξ,u˜k))|dxdξ.

A) For any given η > 0, by the proof of Lemma 4 adapted to the time interval [k, ∞), there exist M3 = M3(B, η) > M2 and K = K(B, η) ≥ 1 such that for ξ ∈ [0, k], whenever r > K and m > M3, one has

|x|>r(|u(tmξ,u0,m)|2+|u(tmξ,u0,m)|p+|ϕ1|2+|ϕ2|+|ϕ3|)dx<η.

In view of (5) and (6), there is a constant L1 > 0 such that for any ξ ∈ [0, k] one has

|x|>r|F(x,u(tmξ,u0,m))|dx|x|>rL1(|u(tmξ, u0,m)|2+|u(tmξ,u0,m)|p+|ϕ1|2+|ϕ2|+|ϕ3|)dx<L1η,

for all r > K and m > M3.

B) Since (50) shows that

u˜k=(weak)limmu(tmk,u0,m) in L2(n)Lp(n),

by the weakly continuous dependence of solutions on intial data stated in Lemma 2, by the weak lower-semicontinuity of the L2 and LP norms, it yields from (64) that

0ke2σξ|x|>r|F(x,u(kξ,u˜k))|dxdξ0ke2σξ|x|>rL1(|u(kξ,u˜k)|2+|u(kξ,u˜k)|p+|ϕ1|2+|ϕ2|+|ϕ3|)dxdξ=0ke2σξL1(u(kξ,u˜k)L2(n\Br)2+u(kξ,u˜k)Lp(n\Br)p)dξ+0ke2σξL1|x|>r(|ϕ1|2+|ϕ2|+|ϕ3|)dxdξ0ke2σξL1[liminfmu(kξ,u˜k)L2(n\Br)2+liminfmu(kξ,u˜k)Lp(n\Br)p]dξ+0ke2σξL1|x|>r(|ϕ1|2+|ϕ2|+|ϕ3|)dxdξL12ση,forr>K,m>M3.

The above two inequalities show that there exists a constant L2 = L1 (1 +1/(2σ)) > 0 such that the first term on the right-hand side of (63) satisfies

0ke2σξ|x|>r|F(x,u(kξ,u(tmk,u0,m)))F(x,u(kξ,u˜k))|dxdξ 0ke2σξ|x|>r(|F(x,u(tmξ,u0,m))|+|F(x,u(kξ,u˜k))|)dxdξL2η,

for all r > K and m > M3.

C) For the second term on the right-hand side of (63), by (50) we have

u(kξ,u(tmk,u0,m))u(kξ,u˜k)weaklyinH1(Br)Lp(Br).

Since H1(Br) is compactly embedded in L2(Br), it follows that for any ξ ∈ [0, k],

u(kξ,u(tmk,u0,m))u(kξ,u˜k)stronglyinL2(Br).

Then by the first item of Lemma 5 and the continuity of F (x, u),

F(x,u(kξ,u(tmk,u0,m)))F(x,u(kξ,u˜k)) in  Br, as m.

On the other hand, by the Standing Assumption and Lemma 6, we have the following uniform bound that there is a constant L3 > 0 such that

|x|<r|F(x,u(kξ,u(tmk,u0,m)))|dxL1(u(kξ,u(tmk,u0,m))L2(Br)2+u(kξ,u(tmk,u0,m))Lp(Br)p+ϕ12+ϕ2L1(n)+ϕ3L1(n)) L3[R+1+1σe12σk(2δ(C2ϕ3L1+ϕ2L1)+g2βδ)+ϕ12+ϕ2L1+ϕ3L1]

for any ξ ∈ [0, k] and m > M1. By the second item of Lemma 6, it follows from (67) and (68) that

F(x,u(kξ,u(tmk,u0,m)))F(x,u(kξ,u˜k))weakly in L1(Br),

as m → ∞. Consequently, when m → ∞,

|x|<rF(x,u(kξ,u(tmk,u0,m)))dx|x|<rF(x,u(kξ,u˜k))dx.

Furthermore, (68) shows that

||x|<r[F(x,u(kξ,u(tmk,u0,m)))F(x,u(kξ,u˜k))]dx| L3[R+1+1σe12σk(2δ(C2ϕ3L1+ϕ2L1)+g2βδ)+ϕ12+ϕ2L1+ϕ3L1]+F(,u(kξ,u˜k))L1(n).

According to Lebesgue dominated convergence theorem, (69) and (70) imply that for every integer k ≥ 1 and any rK,

limm0ke2σξ|x|<rF(x,u(kξ,u(tmk,u0,m)))dxdξ=0ke2σξ|x|<rF(x,u(kξ,u˜k))dxdξ.

Put together (63), (65) and (71). Then we obtain

limm0ke2σξn(F(x,u(kξ,u(tmk,u0,m)))+ϕ3(x))dxdξ=0ke2σξn(F(x,u(kξ,u˜k))+ϕ3(x))dxdξ.

4) By an argument similar to the proof of (72), we can also prove the convergence of the sixth term on the right-hand side of (57),

limm0ke2σξnf(x,u(kξ,u(tmk,u0,m)))u(kξ,u(tmk,u0,m))dxdξ=0ke2σξnf(x,u(kξ,u˜k))u(kξ,u˜k)dxdξ,limm0ke2σξng(x)v(kξ,v(tmk,v0,m))dxdξ=0ke2σξng(x)v(kξ,v˜k)dxdξ.

Step 4

Take the limit of (57) as m → ∞ and assemble together the results shown above in the items 1) through 5) of Step 3. Then we get

limsupmΓ(u(tm,u0,m),v(tm,v0,m))eσk(R+1+1σ[2δ(C2ϕ3L1+ϕ2L1)+g2βδ])2(βδσ)0ke2σξv(kξ,v˜k)2dξ2(δσ)(α+δ2βδ)0ke2σξu(kξ,u˜k)2dξ2(δσ)0ke2σξu(kξ,u˜k)2dξ+4σ0ke2σξn(F(x,u(kξ,u˜k))+ϕ3(x))dxdξ+20ke2σξn[g(x)v(kξ,v˜k)δf(x,u(kξ,u˜k))u(kξ,u˜k)]dxdξ.

From (56) and (73) it follows that

limsupmΓ(u(tm,u0,m),v(tm,v0,m))eσk(R+1+1σ[2δ(C2ϕ3L1+ϕ2L1)+g2βδ])+0ke2σξG(u(kξ,u˜k),v(kξ,v˜k))dξ=eσk(R+1+1σ[2δ(C2ϕ3L1+ϕ2L1)+g2βδ])+Γ(u˜,v˜)e2σkΓ(u˜k,v˜k)eσk(R+1+1σ[2δ(C2ϕ3L1+ϕ2L1)+g2βδ])+Γ(u˜,v˜).

Take limit k → ∞ of the above inequality to obtain limsup

limsupmΓ(u(tm,u0,m),v(tm,v0,m))Γ(u˜,v˜).

On the other hand, from (53), (67) and (68) we have

limmnF(x,u(tm,u0,m))dx=nF(x,u˜)dx,

which along with (74) shows that

limsupm(v(tm,v0,m)2+(α+δ2βδ)u(tm,u0,m)2+u(tm,u0,m)2)v˜2+(α+δ2βδ)u˜2+u˜2.

Step 5

Note that the norm of H1(ℝn) × L2(ℝn) is equivalent to

(u,v)Π=defΓ(u,v)2Rn(F(x,u)+ϕ3(x))dx=v2+(α+δ2βδ)u2+u2.

Same as the second inequality in (46), from the weak convergence shown by (45), for any sequence {g0,m=(u0,m,v0,m)}m=1B, we have

liminfmΦ(tm,g0,m)Π(u˜,v˜)Π.

Meanwhile, (76) implies that limsup

limsupmΦ(tm,g0,m)Π(u˜,v˜)Π.

Thus we have proved lim

limmΦ(tm,g0,m)Π=(u˜,v˜)Π.

Finally, for the Hilbert space H1(ℝn) × L2(ℝn), the weak convergence (45) and the norm convergence (77) imply the strong convergence. Therefore, up to finite steps of subsequence selections (always relabeled as the same), we reach the conclusion that

limmΦ(tm,g0,m)=(u~,v~)strongly H1(Rn)×L2(Rn).

Thus the proof is completed.

5 The Existence of Random Attractor

In this section we shall first prove an instrumental convergence theorem in the space Lp(X,, μ) of Vitalitype. It will pave the way to prove asymptotic compactness of the first component of the semiflow Φ in the space Lp(ℝn) for any exponent 1pn+2n2. This is the crucial and final step to accomplish the proof of the existence of a global attractor for this dynamical system F for the nonlinear damped wave equation (1).

Theorem 8

Let (X, , μ) be a σ-finite measure space and assume that a sequence {fm}m=1Lp(X,M,μ) with 1 ≤ p < ∞ satisfies

limmfm(x)=f(x), a.e.

Then fLp(X,, μ) and

limmfmfLp(X,M,μ)=0

if and only if the following two conditions are satisfied:

  • (a) For any given ε > 0, there exists a set Aε ∈ ℳ such that μ(Aε) < ∞ and
    X\Aε|fm(x)|pdμ<ε,for allm1.
  • (b) The absolutely continuous property of the Lp integrals is satisfied uniformly, i.e.
    limμ(Y)0Y|fm(x)|pdμ=0,uniformly for all m1.

Proof. First we prove the necessity.

Statement (a): Under the condition (79), for an arbitrarily given ε > 0 there exists an integer N = N(ε) ≥ 1 such that

fmfLp(X,M,μ)p<ε2p,for all m>N.

Since fLp(X,, μ), there exist measurable sets Bε and Sε both of finite measure, such that

X\Bε|f(x)|pdμ<ε2p andX\Sε|fm(x)|pdμ<ε, for m=1,,N.

Put Aε = BεSe. Then μ(Aε) < ∞ and we have

XAε|fm(x)|pdμ=XAε(|fm(x)f(x)|+|f(x)|)pdμ2p1X|fm(x)f(x)|pdμ+XBε|f(x)|pdμ<ε2+ε2=ε,form>N.

Besides, from the second inequality in (83) it follows that

X\Aε|fm(x)|pdμX\Sε|fm(x)|pdμ<ε, for m=1,,N.

for m = 1,··· , N.

Therefore, the statement (a) is valid.

Statement (b): By the absolutely continuous property of Lebesgue integral on a σ-finite measure space, for any given ε > 0, there exists δ0 = δ0(ε) > 0 such that whenever μ(Y ) < δ0 one has

Y|f(x)|pdμ<ε2p andY|fm(x)|pdμ<ε, for  m=1,,N,

where N = N(ε) is the same integer in (82). Then for any measurable set YX with μ(Y) < δ0 one also has

Y|fm(x)|pdμ2p1(X|fm(x)f(x)|pdμ+Y|f(x)|pdμ)<ε, for  m>N.

Thus the statement (b) is valid.

Next we prove the sufficiency. Suppose the two conditions (a) and (b) are satisfied. First of all, by the condition (a) and Fatou’s Lemma, for an arbitrarily given ε > 0 there exists a set Aε of finite measure with

supm1X\Aε|fm(x)|pdμ<ε,

which implies that the limit function f in the assumption (78) satisfies

X\Aε|f(x)|pdμliminfmX\Aε|fm(x)|pdμ<ε.

Hence it follows that

fLp(X\Aε) and fmfLp(X\Aε)<2ε1/p,  for all  m1.

Therefore, the proof of fLP(X, , μ) and (79) is reduced to proving that

fLp(Y)andlimmfmfLp(Y)=0,

for any given measurable set YX with μ(Y) < ∞.

By the condition (b), for any given ε > 0, there exists δ1 = δ1(ε) > 0 such that for any SX with μ(S) < δ1 one has

S|fm(x)|pdμ<εp, uniformly in  m1.

Consequently, by Fatou’s lemma,

S|f(x)|pdμliminfmS|fm(x)|pdμ<εp.

By Egorov’s theorem on Lebesgue integral over such a set Y of finite measure in the space (X,, μ), there exists a measurable subset BY with μ(Y\B) < δ1 such that

limmfm(x)=f(x), uniformly a.e. on  B,

so that there exists an integer m0 = m0(ε) ≥ 1 such that

fmfLp(B)<ε, for all  m>m0.

Combining (88), (89) and (90), we get

fmfLp(Y)fmLp(Y\B)+fLp(Y\B)+fmfLp(B)<3ε, for m>m0.

Therefore, (87) is proved. The proof is completed.

Finally we present and prove the main result of this work on the existence of a global attractor for this semiflow Φ generated by the nonlinear damped wave equation (1) on the product Banach space with critical exponent and arbitrary space dimension.

Theorem 9

Under the Standing Assumption, the semiflow Φ generated by the nonlinear damped wave equation (1) in the converted problem (4) on the space E = (H1(ℝn) ⋂ Lp(ℝn)) × L2(ℝn) has a global attractor A in E.

Proof. Lemma 3 shows that there exists an absorbing set, the K = BE(0, R) in the space E for the semiflow Φ. It suffices to prove that the semiflow F is asymptotically compact in E.

(1) Theorem 7 shows that for any given bounded set BE and any sequences tm → ∞ and {g0,m = (u0,m, v0,m)} ⊂ B, the sequence {Φ(tm,g0,m)}m=1 has a convergent subsequence, which is relabeled by the same, such that

Φ(tm,g0,m)(u˜,v˜) strongly in  H1(n)×L2(n),

and consequently

uΦ(tm,g0,m)u˜  strongly in  L2(n).

Here ℙu : (u, v) ↦ u is the component projection.

(2) Applying the first item in Lemma 5 to the space L2(ℝn), it follows from (91) that there exists a subsequence {Φ(tmk,g0,mk)}k=1 of {Φ(tm,g0,m)}m=1 such that

limkΦ(tmk,g0,mk)(x)=(u˜(x),v˜(x)), a.e. in n.

Hence we have

limkuΦ(tmk,g0,mk)(x)=u˜(x), a.e. in  n.

Therefore, the assumption (78) in Theorem 8 is satisfied by the sequence of functions {uΦ(tmk,g0,mk)(x)}k=1 in Lp(ℝn).

(3) By Lemma 4, for any ε > 0, there exists an integer k0 = k0(B, ε) > 0 and V = V (ε) ≥ 1 such that for all k > k0 one has

n\BV|uΦ(tmk,g0,mk)(x)|pdxΦ(tmk,g0,mk)E(n\BV)p<ε,

for any g0,mkB, where BV is the ball centered at the origin with radius V in ℝn. Then there exists V0 = V0(ε) > 0 such that

n\BV0|uΦ(tmk,g0,mk)(x)|pdx<ε, for  k=1,,k0.

Thus (95) and (96) confirm that with Aε = Bmax{V, V0} the condition (a) in Theorem 8 is satisfied by the sequence of functions {uΦ(tmk,g0,mk)(x)}k=1 in Lp(ℝn).

(4) Finally we prove that the uniform absolutely continuous condition (b) of Theorem 8 is also satisfied by the sequence of functions {uΦ(tmk,g0,mk)(x)}k=1 in Lp(ℝn).

According to the Standing Assumption, for any measurable set Y ⊂ ℝn, we have

C3Y|u|pdxY(F(x,u)+ϕ3(x))dxΓY(u,v), for (u,v)E,

where ΓY (u, v) is analogous to (40) and defined by

ΓY(u,v)=vL2(Y)2+(α+δ2βδ)uL2(Y)2+uL2(Y)2+2Y(F(x,u)+ϕ3(x))dx.

We can integrate the inequality (14) over the time interval [0,tm] to get

ΓY(u(tm,u0,m),v(tm,v0,m)) eσtmΓY(u0,m,v0,m)+0tmeσt(2δ(C2ϕ3L1+ϕ2L1)+gL2(Y)2βδ)dt.

Substitute the expression of ΓY (u0,m, v0,m) with (u0,m, v0,m) ∈ B into the inequality (98). Since (5)-(6) yield

Y(F(x,u)+ϕ3(x))dx1C2[C1uLp(Y)p+uL2(Y)2+ϕ1L2(Y)2+ϕ2L1(Y)],

for any g0,m = (u0,m, v0,m) ∈ B, we get

C3Y|u(tm,u0,m)|pdxΓY(u(tm,u0,m),v(tm,v0,m)) eσtm[v0,mL2(Y)2+(α+δ2βδ)u0,mL2(Y)2+u0,mL2(Y)2]+eσtm1C2[C1u0,mLp(Y)p+u0,mL2(Y)2+ϕ1L2(Y)2+ϕ2L1(Y)]+2δC20tmeσt(ϕ1L2(Y)2+ϕ3L1(Y))dt+0tmeσtβδgL2(Y)2dt.

Due to the absolute continuity of the respective Lebesgue integrals of the functions ϕ1(x), ϕ2(x), ϕ3(x) and g involved in the above inequality (99), for an arbitrarily given η > 0, there exists μ0 = μ0(η) > 0 such that for any measurable set Y ⊂ ℝn with μ(Y ) < μ0 one has

eσtm 1C2(ϕ1L2(Y)2+ϕ2L1(Y))+2δC20tmeσt(ϕ1L2(Y)2+ϕ3L1(Y))dt+0tmeσtβδgL2(Y)2dt1C2(ϕ1L2(Y)2+ϕ2L1(Y))+2δσC2(ϕ1L2(Y)2+ϕ3L1(Y))+1σ(βδ)gL2(Y)2<η2.

Moreover, since BE is a bounded set, there exists a constant Ĉ > 0 such that

eσtm[v0,mL2(Y)2+(α+δ2βδ)u0,mL2(Y)2+u0,mL2(Y)2]+1C2 eσtm[C1u0,mLp(Y)p+u0,mL2(Y)2]eσtm C^(BE(Y)2+BE(Y)p),

where ||B||E(Y) = maxg0∈B(θ−tmω) ||g0 ζY||E with ζY being the characteristic function for the set Y . Clearly,

limteσtBE=0.

For the aforementioned arbitrary η > 0, there exists an integer m0 = m0(B, η) ≥ 1 such that

eσtmC^(BE(Y)2+BE(Y)p)eσtmC^(BE2+BEp)<η2

for all m > m0. Then there esists μ1 = μ1(B, m0, η) > 0 such that for any set Y with μ(Y ) < μ1 one has

eσtjC^(BE(Y)2+BE(Y)p)<η2,j=1,,m0.

Put together (100), (101) and (102) with (99). It shows that

C3Y|u(tm,u0,m)|pdxη2+η2=η, for all  m1,

whenever a measurable set Y ⊂ ℝn satisfies μ(Y) < min{μ0, μ1}. Therefore,

limμ(Y)0Y|uΦ(tmk,g0,m)(x)|pdx=0  uniformly for all  k1,

so that the condition (b) of Theorem 8 is also satisfied by the sequence of functions {uΦ(tmk,g0,mk)(x)}k=1 in Lp(ℝn).

As checked by the above steps (2), (3) and (4) in this proof, all the conditions in Theorem 8 are satisfied by the sequence of functions {uΦ(tmk,g0,mk)(x)}k=1 in Lp(ℝn). Then we apply Theorem 8 to obtain

limkuΦ(tmk,g0,mk)=u˜ strongly in  Lp(n).

Finally, combination of (91) and (105) shows that there exists a convergent subsequence {Φ(tmk,g0,mk)}k=1 of the sequence {Φ(tm,g0,m)}m=1 in the space E = (H1(ℝn) ⋂ Lp(ℝn)) × L2(ℝn). Therefore, the semiflow Φ on the Banach space E is asymptotically compact.

According to Theorem 1, we conclude that there exists a global attractor 𝒜 in E for this semiflow Φ generated by the original nonlinear damped wave equation (1). The proof is completed.

Communicated by Tomás Caraballo

References

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    J. M. Arrieta A. N. Carvalho and J. K. Hale. (1992) A damped hyperbolic equation with critical exponent Communications in Partial Differential Equations 17 841-866.

    • Crossref
    • Export Citation
  • [2]

    A. V. Babin and M. I. Vishik. (1992) Attractors of Evolution Equations North-Holland Amsterdam.

  • [3]

    J. M. Ball. (2004) Global attractors for damped semilinear wave equations Discrete and Continuous Dynamical Systems 10 31-52.

    • Crossref
    • Export Citation
  • [4]

    V. Belleri and V. Pata. (2001) Attractors for semilinear strongly damped wave equations on R3 Discrete and Continuous Dynamical Systems Ser. A 7 719-735.

    • Crossref
    • Export Citation
  • [5]

    H. Brezis. (2011) Functional Analysis Sobolev Spaces and Partial Differential Equations Springer-Verlag New York.

  • [6]

    A.N. Carvalho and J.W. Cholewa. (2002) Attrarctors for strongly damped wave equations with critical nonlinearities Pacific Journal of Mathematics. 207 287-310.

    • Crossref
    • Export Citation
  • [7]

    T. Caraballo A. N. Carvalho J. A. Langa and F. Rivero. (2011) A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor Nonlinear Analysis: Theory Methods and Applications 74 2272-2283.

    • Crossref
    • Export Citation
  • [8]

    V. V. Chepyzhov and M. I. Vishik. (2002) Attractors for Equations of Mathematical Physics American Mathematical Society Providence R.I.

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    H. Crauel A. Debussche and F. Flandoli. (1997) Random attractors Journal of Dynamics and Differential Equations 9 307-341.

    • Crossref
    • Export Citation
  • [10]

    J. Duan K. Lu and B. Schmalfuss. (2003) Invariant manifolds for stochastic partial differential equations The Annals of Probability 31 2109-2135.

    • Crossref
    • Export Citation
  • [11]

    X. Fan. (2008) Random attractor for a damped stochastic wave equation with multiplicative noise International Journal of Mathematics 19 421-437.

    • Crossref
    • Export Citation
  • [12]

    E. Feireisl. (1994) Attractors for semilinear damped wave equations on R3 Nonlinear Analysis: Theory Methods and Applications 23 187-195.

    • Crossref
    • Export Citation
  • [13]

    E. Feireisl. (1995) Global attractors for damped wave equations with supercritical exponent J. Differential Equations 116 431-447.

    • Crossref
    • Export Citation
  • [14]

    J. M. Ghidaglia and R. Temam. (1987) Attractors of damped nonlinear hyperbolic equations J. Math. Pures Appl. 66 273-319.

  • [15]

    R. Jones and B. Wang. (2013) Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms Nonlinear Analysis: Real World Applications 14 1308-1322.

    • Crossref
    • Export Citation
  • [16]

    A. Kh. Khanmamedov. (2006) Global attractors for wave equations with nonlinear interior damping and critical exponents Journal of Differential Equations 230 702-719.

    • Crossref
    • Export Citation
  • [17]

    H. Li and Y. You. (2015) Random attractor for stochastic wave equation with arbitrary exponent and additive noise on Rn Dynamics of Partial Differential Equations 12 343-378.

    • Crossref
    • Export Citation
  • [18]

    H. Li Y. You and J. Tu. (2015) Random attractors and averaging for nonautonomous stochastic wave equations with nonlinear damping Journal of Differential Equations 258 148-190.

    • Crossref
    • Export Citation
  • [19]

    M. Nakao. (2006) Global attractors for nonlinear wave equations with nonlinear dissipative terms J. Differential Equations 227 204-229.

    • Crossref
    • Export Citation
  • [20]

    R. Sell and Y. You. (2002) Dynamics of Evolutionary Equations Springer-Verlag New York.

  • [21]

    C. Sun M. Yang and C. Zhong. (2006) Global attractors for the wave equation with nonlinear damping Journal of Differential Equations 227 427-443.

    • Crossref
    • Export Citation
  • [22]

    R. Temam. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag New York.

  • [23]

    B. Wang. (2011) Asymptotic behavior of stochastic wave equations with critical exponents on R3 Transactions of the American Mathematical Society 363 3639-3663.

    • Crossref
    • Export Citation
  • [24]

    B. Wang. (2014) Random attractors for non-autonomous stochastic wave equations with multiplicative noise Discrete and Continuous Dynamical Systems 34 269-300.

    • Crossref
    • Export Citation
  • [25]

    M. Yang J. Duan and P. Kloeden. (2011) Asymptotic behavior of solutions for random wave equation with nonlinear damping and white noise Nonlinear Analysis: Real World Applications 12 464-478.

    • Crossref
    • Export Citation
  • [26]

    Y. You. (2004) Global dynamics of nonlinear wave equations with cubic non-monotone damping Dynamics of Partial Differential Equations 1 65-86.

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    J. M. Arrieta A. N. Carvalho and J. K. Hale. (1992) A damped hyperbolic equation with critical exponent Communications in Partial Differential Equations 17 841-866.

    • Crossref
    • Export Citation
  • [2]

    A. V. Babin and M. I. Vishik. (1992) Attractors of Evolution Equations North-Holland Amsterdam.

  • [3]

    J. M. Ball. (2004) Global attractors for damped semilinear wave equations Discrete and Continuous Dynamical Systems 10 31-52.

    • Crossref
    • Export Citation
  • [4]

    V. Belleri and V. Pata. (2001) Attractors for semilinear strongly damped wave equations on R3 Discrete and Continuous Dynamical Systems Ser. A 7 719-735.

    • Crossref
    • Export Citation
  • [5]

    H. Brezis. (2011) Functional Analysis Sobolev Spaces and Partial Differential Equations Springer-Verlag New York.

  • [6]

    A.N. Carvalho and J.W. Cholewa. (2002) Attrarctors for strongly damped wave equations with critical nonlinearities Pacific Journal of Mathematics. 207 287-310.

    • Crossref
    • Export Citation
  • [7]

    T. Caraballo A. N. Carvalho J. A. Langa and F. Rivero. (2011) A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor Nonlinear Analysis: Theory Methods and Applications 74 2272-2283.

    • Crossref
    • Export Citation
  • [8]

    V. V. Chepyzhov and M. I. Vishik. (2002) Attractors for Equations of Mathematical Physics American Mathematical Society Providence R.I.

  • [9]

    H. Crauel A. Debussche and F. Flandoli. (1997) Random attractors Journal of Dynamics and Differential Equations 9 307-341.

    • Crossref
    • Export Citation
  • [10]

    J. Duan K. Lu and B. Schmalfuss. (2003) Invariant manifolds for stochastic partial differential equations The Annals of Probability 31 2109-2135.

    • Crossref
    • Export Citation
  • [11]

    X. Fan. (2008) Random attractor for a damped stochastic wave equation with multiplicative noise International Journal of Mathematics 19 421-437.

    • Crossref
    • Export Citation
  • [12]

    E. Feireisl. (1994) Attractors for semilinear damped wave equations on R3 Nonlinear Analysis: Theory Methods and Applications 23 187-195.

    • Crossref
    • Export Citation
  • [13]

    E. Feireisl. (1995) Global attractors for damped wave equations with supercritical exponent J. Differential Equations 116 431-447.

    • Crossref
    • Export Citation
  • [14]

    J. M. Ghidaglia and R. Temam. (1987) Attractors of damped nonlinear hyperbolic equations J. Math. Pures Appl. 66 273-319.

  • [15]

    R. Jones and B. Wang. (2013) Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms Nonlinear Analysis: Real World Applications 14 1308-1322.

    • Crossref
    • Export Citation
  • [16]

    A. Kh. Khanmamedov. (2006) Global attractors for wave equations with nonlinear interior damping and critical exponents Journal of Differential Equations 230 702-719.

    • Crossref
    • Export Citation
  • [17]

    H. Li and Y. You. (2015) Random attractor for stochastic wave equation with arbitrary exponent and additive noise on Rn Dynamics of Partial Differential Equations 12 343-378.

    • Crossref
    • Export Citation
  • [18]

    H. Li Y. You and J. Tu. (2015) Random attractors and averaging for nonautonomous stochastic wave equations with nonlinear damping Journal of Differential Equations 258 148-190.

    • Crossref
    • Export Citation
  • [19]

    M. Nakao. (2006) Global attractors for nonlinear wave equations with nonlinear dissipative terms J. Differential Equations 227 204-229.

    • Crossref
    • Export Citation
  • [20]

    R. Sell and Y. You. (2002) Dynamics of Evolutionary Equations Springer-Verlag New York.

  • [21]

    C. Sun M. Yang and C. Zhong. (2006) Global attractors for the wave equation with nonlinear damping Journal of Differential Equations 227 427-443.

    • Crossref
    • Export Citation
  • [22]

    R. Temam. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag New York.

  • [23]

    B. Wang. (2011) Asymptotic behavior of stochastic wave equations with critical exponents on R3 Transactions of the American Mathematical Society 363 3639-3663.

    • Crossref
    • Export Citation
  • [24]

    B. Wang. (2014) Random attractors for non-autonomous stochastic wave equations with multiplicative noise Discrete and Continuous Dynamical Systems 34 269-300.

    • Crossref
    • Export Citation
  • [25]

    M. Yang J. Duan and P. Kloeden. (2011) Asymptotic behavior of solutions for random wave equation with nonlinear damping and white noise Nonlinear Analysis: Real World Applications 12 464-478.

    • Crossref
    • Export Citation
  • [26]

    Y. You. (2004) Global dynamics of nonlinear wave equations with cubic non-monotone damping Dynamics of Partial Differential Equations 1 65-86.

    • Crossref
    • Export Citation
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