The main result of a global attractor reads as follows.

Proof.
We are going to apply Lemmas 11 and 12 to prove the asymptotic smooth. Given initial data (u0,u1,θ0) and (v0,v1,θ~0) in a bounded invariant set B⊂H0, let (u,θ),(v,θ~) be the corresponding weak solutions of problem (8)–(11). Then the differences w=u-v, ϑ=θ-θ~ are the weak solutions of (76)wtt+Δ2w+Δ2wt+νΔϑ+Δg=σ∇u2Δu-σ∇v2Δv+Δϕ-Δf1,ϑt-Δϑ+Δf2-νΔwt=0,w=∂w∂v=0,ϑ=0,w0=u0-v0,wt0=u1-v1,ϑ0=θ0-θ~0,where (77)Δϕ=ϕ∫Ω∇u∇utdxΔu-ϕ∫Ω∇v∇vtdxΔv,Δg=gut-gvt,Δf1=f1u-f1v,Δf2=f2θ-f2θ~.Let us define (78)Ewt=wt2+Δw2+σ∇u2∇w2+ϑ2.As before, by density, we can assume formally that w is sufficiently regular. Then, multiplying the first equation in (76) by wt and integrating over Ω and multiplying the second equation in (76) by ϑ and integrating over Ω and then taking the sum, we get (79)12ddtEwt+Δwt2+∇ϑ2+∫ΩΔgwtdx+∫ΩΔf2ϑ dx=-σ′∇u2∇w2∫ΩΔuutdx+Δσ∫ΩΔvwtdx+∫ΩΔϕwtdx-∫ΩΔf1wtdx,where(80)Δσ=σ∇u2-σ∇v2.Let us estimate the right hand side of (79).

Considering the continuity of σ′(·) and estimate (39)(81)-σ′∇u2∇w2≤C∇w2,Applying the Mean Value Theorem combined with estimate (39), by Young inequality, we get (82)Δσ∫ΩΔvwtdx≤C∇wr+2/r+1+k44wtr+2r+2.Also use the Mean Value Theorem combined with estimate (39) and Young inequality to get(83)∫ΩΔϕwtdx=∫Ωϕ∫Ω∇u∇utdxΔwwtdx-∫Ωϕ∫Ω∇u∇utdx-ϕ∫Ω∇v∇vtdxΔvwtdx=∫Ωϕ′ξ2∫Ω∇u∇utdxΔwwtdx-∫Ωϕ′ξ3∫Ω∇w∇wtdxΔvwtdx≤CwΔwt+CwΔwtwt≤14Δwt2+C2w2+14Δwt2+C2w2wt2≤14Δwt2+C2∇w2+C2w2wtr+22≤14Δwt2+C2∇w2+C4k3w2r+2/r+k34wtr+2r+2,where ξ2 is among 0 and ∫Ω∇u∇utdx, and ξ3 is among ∫Ω∇u∇utdx and ∫Ω∇v∇vtdx.

By the Holder inequality, Minkowski inequality combined with the estimate of (39), and Young inequality, we obtain (84)∫ΩΔf1wtdx≤C∇wr+2/r+1+k44wtr+2r+2.On the other hand, considering assumptions (17) and (18) of f2(·) and g(·), (85)∫ΩΔf2ϑ dx≥k3ϑϱ+2ϱ+2,∫ΩΔgwtdx≥k4wtr+2r+2.Thus by inserting (81)–(85) into (79), we get that (86)12ddtEwt+14Δwt2+∇ϑ2+k44wtr+2r+2+k3ϑϱ+2ϱ+2≤C∇w2+∇wr+2/r+1+∇w2r+2/r.Then integrating from t to t+1 and defining an auxiliary function F2(t), we get (87)14∫tt+1Δwt2ds+∫tt+1∇ϑ2ds+k44∫tt+1wtr+2r+2ds+k3∫tt+1ϑϱ+2ϱ+2ds≤Ewt-Ewt+1+C∫tt+1∇w2+∇wr+2/r+1+∇w22r+2/rds=Ft2.It is obvious that(88)Ewt+1≥Ett,(89)14∫tt+1Δwt2ds≤Ft2,∫tt+1∇ϑ2ds≤F2t,(90)k44∫tt+1wtr+2r+2ds≤Ft2,k3∫tt+1ϑϱ+2ϱ+2ds≤Ft2.Then by multiplying first equation in (76) by w and integrating over Ω again, we obtain that (91)Δw2+σ∇u2∇w2=-ddt∫Ωwtw dx+wt2-∫ΩΔ2wtw dx+Δσ∫ΩΔvw dx-∫ΩΔf1w dx-∫ΩΔgw dx+∫Ωϕ∫Ω∇u∇utdx-ϕ∫Ω∇v∇vtdxΔvw dx-ϕ∫Ω∇u∇utdx∇w2-ν∫ΩϑΔw dx.Integrating from t1 to t2, we get (92)∫t1t2Δw2+σ∇u2∇w2ds=∫Ωwtt2wt2dx-∫Ωwtt1wt1dx+∫t1t2wt2dt-∫t1t2∫ΩΔwtΔw dx ds+∫t1t2Δσ∫ΩΔvw dx ds-∫t1t2∫ΩΔf1w dx ds-∫t1t2∫ΩΔgw dx ds+∫t1t2∫Ωϕ∫Ω∇u∇utdx-ϕ∫Ω∇v∇vtdxΔvw dx ds-∫t1t2ϕ∫Ω∇u∇utdx∇w2ds+ν∫t1t2∫Ω∇ϑ∇w dx ds.Now let us estimate the right hand side of (92). Firstly, from the first inequality of (90), by holder inequality we infer that (93)∫tt+1wt2ds=∫tt+1∫Ωwt2dx ds≤Ωr/r+2∫tt+1∫Ωwt2r+2/2dx ds2/r+2≤CFt4/r+2;thus there exists t1∈[t,t+1/4] and t2∈[t+3/4,t+1] such that (94)wtt12≤CF4/r+2t,wtt22≤CF4/r+2t;then we can deduce that(95)∫Ωwtt2wt2dx-∫Ωwtt1wt1dx≤CFt4/r+2+18supt≤σ≤t+1Ewσ.Use Schwarz inequality combined with the estimate of (39) and Holder inequality to obtain (96)∫t1t2∫ΩΔwtΔw dx ds≤C∫t1t2Δwtds≤C∫t1t21 ds1/2∫t1t2Δwt2ds1/2≤CFt.Apply the Mean Value Theorem combined with estimate (39) to get (97)∫t1t2Δσ∫ΩΔvw dx ds≤C∫t1t2∇w2ds.Assumption (14) of f(·) and the estimate of (39) imply that(98)∫t1t2∫ΩΔf1w dx ds≤C∫t1t2∇w2ds.Also from assumption (19) of g(·) and the estimate of (39) combined with (94), we have (99)∫t1t2∫ΩΔgw dx ds≤C∫t1t2wtΔwds≤CFt4/r+2+18supt≤σ≤t+1Ewσ.Using the Mean Value Theorem and considering the assumption of ϕ(·) and the estimate of (39), we have (100)∫t1t2∫Ωϕ∫Ω∇u∇utdx-ϕ∫Ω∇v∇vtdxΔvw dx ds=∫t1t2ϕ′ξ4∫Ω∇w∇wtdx∫ΩΔvw dx ds≤C∫t1t2wwtds≤C∫t1t2∇w2ds+C∫t1t2wt2ds,(101)∫t1t2ϕ∫Ω∇u∇utdx∇w2ds≤C∫t1t2∇w2ds,where ξ4 is among ∫Ω∇u∇utdx and ∫Ω∇v∇vtdx.

Finally, use Young inequality to get (102)ν∫t1t2∫Ω∇ϑ∇w dx ds≤ν22∫t1t2∇ϑ2+12∫t1t2∇w2ds.By inserting (93) and (95)–(102) into (92), we obtain that (103)∫t1t2Δw2+σ∇u2∇w2ds≤3C∫t1t2∇w2ds+2C∫t1t2wt2ds+C∫t1t2∇ϑ2ds+2CFt4/r+2+14supt≤σ≤t+1Ewσ+CFt.Considering (89) and (93), from (103), we have (104)∫t1t2Δw2+σ∇u2∇w2ds≤3C∫tt+1∇wds+CFt+CF2t+4CFt4/r+2+14supt≤σ≤t+1Ewσ.Using Holder inequality with 1/ϱ+2+ϱ+1/ϱ+2=1, (105)∫t1t2ϑ2ds≤C∫t1t2ϑϱ+2ds≤C∫t1t21ϱ+2/ϱ+2dsϱ+1/ϱ+2∫t1t2ϑϱ+2ϱ+2ds≤C∫t1t2ϑϱ+2ϱ+2ds≤CF2t.Then from the definition of Ew(t) and (93), (104), and (105), we obtain that (106)∫t1t2Ewsds≤5CFt4/r+2+2CF2t+CFt+14supt≤σ≤t+1Ewσ+3C∫tt+1∇wds.For (106), by using Mean Value theorem, there exists t∗∈[t1,t2] such that(107)Ewt∗≤10CFt4/r+2+2CFt+4CF2t+12supt≤σ≤t+1Ewσ+6C∫tt+1∇wds.From (87), we see that (108)Ewt≤Ewt+1+F2t.Let Ew=supt≤σ≤t+1Ew(σ) with τ∈[t,t+1]; then integrate (86) over [t,τ] and over [t∗,t+1] to have (109)supt≤σ≤t+1Ewσ≤Ewτ≤Ewt+1+F2t+C∫tt+1∇w2+∇w2r+2/rds≤Ewt∗+F2t+C∫tt+1∇w2+∇w2r+2/rds.Inserting (107) into (109), we obtain (110)supt≤σ≤t+1Ewσ≤10CFt4/r+2+2CFt+4CF2t+12supt≤σ≤t+1Ewσ+6C∫tt+1∇wds+F2t+C∫tt+1∇w2+∇w2r+2/rds.Therefore from the boundary of 1+F(t)(1-4/r+2)+F(t)2-4/r+2, we have (111)supt≤σ≤t+1Ewσ≤CFt4/r+2+C∫tt+1∇w+∇w2+∇w2r+2/rds.Therefore (112)supt≤σ≤t+1Ewσ1+r/2≤CEwt-Ewt+1+Csup0≤σ≤T∫σσ+1∇w+∇w2+∇w2r+2/rds.From Nakao’s Lemma 11, there exists CB>0 and CT>0 such that (113)Ewt≤CBt-1+-2/r+CTsup0≤σ≤T∫σσ+1∇w+∇w2+∇w2r+2/rds2/r+2, 0≤t≤T.From the definition of Ew(t), we have (114)w,wt,ϑH0≤CBt-1+-2/r+CTsup0≤σ≤T∫σσ+1∇w+∇w2+∇w2r+2/rds2/r+2.Given ε>0, we choose T large such that(115)CBt-1+-2/r≤ε,and define ϖT:H0×H0→R as (116)ϖTu0,u1,θ0,v0,v1,θ~0=CTsup∫σσ+1∇w+∇w2+∇w2r+2/rds2/r+2.Then from (114)–(116), we get (117)STu0,u1,θ0-STv0,v1,θ~0H0≤ε+ϖTu0,u1,θ0,v0,v1,θ~0for all (u0,u1,θ0),(v0,v1,θ~0)∈B.

Let (un0,un1,θn0) be a given sequence of initial data in B. Then the corresponding sequence (un,utn,θn) of solutions of the problem (8)–(11) is uniformly bounded in H0, because B is bounded and positively invariant. So {un} is bounded in C([0,∞),H02(Ω))∩C1([0,∞),L2(Ω)). Since H02(Ω)↪H01(Ω) compactly, there exists a subsequence unk which converges strongly in C([0,T],H01(Ω)). Therefore (118)limk→∞ liml→∞∫0T∇unks-∇unls+∇unks-∇unls2+∇unks-∇unls2r+2/rds=0,limk→∞ liml→∞ϖTunk0,unk1,θnk0,unl0,unl1,θnl0=0.So S(t) is asymptotically smooth in H0. That is, Lemma 12 holds. Thus Theorem 14 is proved.