# Long-Term Dynamic Behavior of a Higher-Order Coupled Kirchhoff Model with Nonlinear Strong Damping

Authors: Penghui Lv; Yan Liu; Shasha Yu
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study the long-time dynamics problem of a class of higher-order Kirchhoff coupled systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and Faedo–Galerkin method; secondly, the family of global attractors of these problems is proved by using the compactness theorem. The results of the Kirchhoff coupled group are promoted through research.


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## Body

## 1. Introduction

In this paper, we mainly consider the dynamic behavior of the higher-order Kirchhoff-type coupled equations on the bounded smooth domainΩ⊂Rℕ:(1)utt+N1∇mu2−Δmut+M∇mu2+∇2mv2−Δmu+g1u,v=f1x,vtt+N2∇2mv2−Δ2mvt+M∇mu2+∇2mv2−Δ2mv+g2u,v=f2x,with boundary conditions:(2)ux=0,∂iu∂ni=0,i=1,…,m−1,vx=0,∂jv∂nj=0,j=1,…,2m−1,m>1,and initial conditions:(3)ux,0=u0x,utx,0=u1x,vx,0=v0x,vtx,0=v1x,x∈Ω.This is a kind of important generalized quasilinear wave equations of higher order. The proposed equation in this paper originated from Kirchhoff’s vibration problem of stretchable strings in 1883:(4)ρh∂2u∂t2=p0+Eh2L∫0L∂u∂x2dx∂2u∂t2,where 0<x<L,t≥0,u=ux,t is the lateral displacement in space coordinate x and time coordinate t, E represents Young’s modulus, ρ represents the mass density, h represents the cross-sectional area, L represents the length, and p0 represents the axial tension of the accident. The research studies on the long-time behavior of various forms of Kirchhoff-type equations have attracted the attention of many scholars in recent decades and have also achieved rich results [1–8]. Igor Chueshov [1] studied the well-posedness and long-time dynamical behavior of the following Kirchhoff equation with a nonlinear strong damping term:(5)utt+σ∇u2Δut−ϕ∇u2Δu+fu=hx.Guoguang Lin, Penghui Lv, and Ruijin Lou [2] studied the global dynamics of the following generalized nonlinear Kirchhoff–Boussinesq equations with damping terms:(6)utt+αut−βΔut+Δ2u=divg∇u2∇u+Δhu+fx.This paper proved that the semigroup conformed to the squeezing property and then obtained the existence of the exponential attractor of the system; then, it used the spectral interval theory to prove that the system had an inertial manifold.Marina Ghisi and Massimo Gobbino [3] studied the global existence and local existence of solutions to the following Kirchhoff model with strong damping:(7)uttt+2δAσutt+MA1/2ut2Aut=0.Mitsuhiro Nakao [4] proved the initial-boundary value problem of the quasilinear Kirchhoff-type wave equation with standard dissipation ut:(8)utt−1+∇ut22Δu+ut+gx,u=fx.With the deepening of research, scholars began to turn their research directions to the dynamics of the higher-order Kirchhoff equations. Ye Yaojun and Tao Xiangxing [9] studied the initial-boundary value problem of the following kind of higher-order Kirchhoff-type equation with nonlinear dissipation term:(9)utt+ΦDmu2−Δmu+autq−2ut=bur−2u.Lin Guoguang and Zhu Changqing [10] studied the initial and boundary value problems of the following nonlinear nonlocal higher-order Kirchhoff-type equations:(10)utt+MDmu2−Δmu+β−Δmut+gx,ut=fx.The paper obtained the existence and uniqueness of the solution and obtained the existence of the family of global attractors of the problem through the compact method and obtained the finite Hausdorff and Fractal dimensions.System coupling originates from physics and is a metric used to refer to the mutual dependence of two entities on each other. Coupled systems refer to the system coupling occurring when some conditions or parameters are appropriate, and the potential energy of the system can make different systems realize the combination of structure and function and then produce new functions. The Kirchhoff models are mathematical equations derived from a physical background, and it will be natural to consider their coupled systems. Later, scholars gradually consider the dynamics of the Kirchhoff equations under the coupled effect. For example, Wang Yu and Zhang Jianwen [11] studied the long-term dynamics problem of a class of coupled beam equations with strong damping under nonlinear boundary conditions. Guoguang Lin and Ming Zhang [12] have studied the initial-boundary value problem of the following Kirchhoff coupling group with strong damping and source term:(11)utt−βΔut−M∇u2+∇v2Δu+g1u,v=f1x,vtt−βΔvt−M∇u2+∇v2Δv+g2u,v=f2x.In this paper, the finite Hausdorff dimension of the global attractor is obtained.In recent years, Guoguang Lin et al. [13–15] focused on the dynamics of a class of higher-order Kirchhoff coupled equations and obtained a series of ideal results. More conclusions about higher-order Kirchhoff-type systems can also be found in [16–19].For the higher-order Kirchhoff coupled problems, there are few articles at present, and the problem of higher-order beam-plate coupled with nonlinear strong damping has not been studied. The main difficulty lies in the estimation and processing of the harmonic term and the nonlinear damping term, and when proving the uniqueness, the nonlinear damping brings some difficulties. In this regard, this paper overcomes these difficulties and obtains the global solution of the problem and the family of global attractors.


## 2. Preparatory Knowledge

In this paper, we use⋅ and ⋅,⋅ to represent the norm and inner product in H=L2Ω, respectively. In order to get a more ideal conclusion, given the following assumptions:(i)
A1.Ms is a continuous function on the interval 0,+∞, Ms∈C1R+, and 1M′s≥0, 2M0≡M0>0.(ii)
A2. For any u,v∈H, let Ju,v=∫ΩG1u,v+G2u,vdx, where G1u,v=∫0ug1s,vds,G2u,v=∫0vg2u,sds, then for any μ≥0, there exists C1≥0,Cμ≥0,Cμ′≥0, such that G1u,v+G2u,v−C1Ju,v+μ∇mu2+∇2mv2≥−Cμ, Ju,v+μ∇mu2+∇2mv2≥−Cμ′.(iii)
A3.gju,vj=1,2∈C1R, and gju,v≤C21+upj+vqj; gjuu,v≤C31+upj−1+vqj; gjvu,v≤C41+upj+vqj−1. Among them, when ℕ=1,2, 1≤pjqj; when 3≤ℕ≤4m, 1≤pjqj≤2m/ℕ−2m.(iv)
A4.Njsj≥Nj0, Nj0j=1,2 are positive constants, and there exists ρ>0, so that Ms1+s2−ρN1s1−ρN2s2>0.Next, the research phase space of this paper is given as follows:(12)V0=H,Vm=HmΩ∩H01Ω,X0=VmΩ×HΩ×V2m×HΩ,Xk=Vm+kΩ×VkΩ×V2m+2k×V2kΩ,k=1,2,…,m,and the norms of the corresponding spaces are as follows:(13)u,y1,v,y2Xk2=∇m+ku2+∇ky12+∇2m+2kv2+∇2ky22,k=0,1,…,m.Meanwhile, there exists the general form of Poincare’s inequality:λ1∇ru2≤∇r+1u2, where λ1 is the first eigenvalue of −Δ with a homogeneous Dirichlet boundary on Ω. In this paper, Ci is a constant, and C⋅ is a constant that depends on the parameters in parentheses.Lemma 1 (see [20]).
Lety:R+⟶R+ be an absolutely continuous positive function on 0,+∞, which satisfies for some δ>0 the differential inequality(14)ddtyt+2δyt≤gtyt+K,t>0,where K≥0; for t≥s≥0, there exists a≥0, such that ∫stgτdτ≤δt−s+a, then(15)yt≤eay0e−δt+Keaδ,t≥0.Lemma 2 (see [10]).
LetX be a Banach space, and the continuous operator semigroup Stt≥0 satisfies the following:(1)
The semigroupStt≥0 is uniformly bounded in X, that is, ∀R0>0; there exists a positive constant C0R0 such that when uX≤R0,(16)StuX≤C0R0,∀t∈0,+∞.(2)
There exists a bounded absorbing setB0 in X, and for any bounded set B⊂X, there exists a moment t0 such that(17)StB⊂B0t≥t0.(3)
Fort>0,St is a fully continuous operator.
Then, the semigroupStt≥0 has a global attractor A in X, and(18)A=ωB0=∩τ≥0∪t≥τStB0¯.


## 3. Main Conclusion

Letε>0 be small enough and λ1mN10−2−4ε−2ε2≥0,λ12mN20−2−4ε−2ε2≥0.Lemma 3.
Assume that assumptionsA1−A4 hold if fj∈HΩ and initial data u0,u1,v0,v1∈X0; then, for R0>0, there exist positive constants CR0 and t0, so that when t≥t0, u,y1,v,y2 determined by problems (1)–(3) satisfies(19)u,y1,v,y2X02=∇mu2+y12+∇2mv2+y22≤CR0,where y1=ut+εuandy2=vt+εv.Proof.
Multiplying the first equation of1 by y1 in HΩ and the second one by y2 in HΩ, we infer that(20)12ddty12+y22+∫0∇mu2+∇2mv2Mτdτ+2Ju,v+εM∇mu2+∇2mv2∇mu2+∇2mv2−εy12+y22+ε2u,y1+v,y2+N1∇mu2∇my12+N2∇2mv2∇2my22−εN1∇mu2∇my1,∇mu−εN2∇2mv2∇2my2,∇2mv+εg1u,v,u+εg2u,v,v=f1,y1+f2,y2.
Using Holder’s inequality, Young’s inequality, Poincare’s inequality, and so on to process the items in (20),(21)−εy12+y22+ε2u,y1+v,y2≥−ε−ε22y12+y22−ε22u2+v2≥−ε−ε22y12+y22−ε22λ1−m∇mu2−ε22λ1−2m∇2mv2,(22)N1∇mu2∇my12+N2∇2mv2∇2my22−εN1∇mu2∇my1,∇mu−εN2∇2mv2∇2my2,∇2mv≥12N1∇mu2∇my12+12N2∇2mv2∇2my22−ε22N1∇mu2∇mu2−ε22N2∇2mv2∇2mv2≥12λ1mN1∇mu2y12+12λ12mN2∇2mv2y22−ε22N1∇mu2∇mu2−ε22N2∇2mv2∇2mv2,(23)f1,y1+f2,y2≤f1y1+f2y2≤12y12+12y22+12f12+12f22.
Substituting (21)–(23) into (20), we can obtain(24)12ddty12+y22+∫0∇mu2+‖∇2mv‖2Mτdτ+2Ju,v++12λ1mN1∇mu2−12−ε−ε22y12+12λ12mN2∇2mv2−12−ε−ε22y22+εM∇mu2+∇2mv2∇mu2+∇2mv2−ε22N1∇mu2+ε22λ1−m∇mu2−ε22N2∇2mv2+ε22λ1−2m∇2mv2≤−εg1u,v,u−εg2u,v,v+12f12+12f22.
ByA1, we get(25)εM∇mu2+∇2mv2∇mu2+∇2mv2≥ε4∫0∇mu2+∇2mv2Mτdτ+3ε4M∇mu2+∇2mv2∇mu2+∇2mv2,and according to A2, we get(26)−εg1u,v,u−εg2u,v,v≤−εC1Ju,v+εμ∇mu2+∇2mv2+εCμ.
Combining (25) and (26), (24) becomes(27)ddty12+y22+∫0∇mu2+∇2mv2Mτdτ+2Ju,v++λ1mN1∇mu2−1−2ε−ε2y12+λ12mN2∇2mv2−1−2ε−ε2y22+ε2∫0∇mu2+∇2mv2Mτdτ+2εC1Ju,v+3ε2M∇mu2+∇2mv2−2εμ−ε2N1∇mu2−ε2λ1−m∇mu2+3ε2M∇mu2+∇2mv2−2εμ−ε2N2∇2mv2−ε2λ1−2m∇2mv2≤2εCμ+f12+f22.
LetH1t=y12+y22+∫0∇mu2+∇2mv2Mτdτ+2Ju,v and σ1=minλ1mNj0−1−2ε−ε2,ε/2,εC1, we infer from (27) that(28)ddtH1t+σ1H1t≤2εCμ+f12+f22.
Using Gronwall’s inequality, we get(29)H1t≤H10e−σ1t+2εCμ+f12+f22σ1.
According toA1A2, taking μ=M0/4, we have(30)H1t≥y12+y22+M0∇mu2+∇2mv2+2Ju,v≥y12+y22+M02∇mu2+∇2mv2−2Cμ′≥C5y12+y22+∇mu2+∇2mv2−2Cμ′,where C5=min1,M0/2.
Then,(31)u,y1,v,y2X02=∇mu2+y12+∇2mv2+y22≤H1t+2Cμ′C5≤H10e−σ1t+2Cμ′C5+2εCμ+f12+f22σ1C5,and thus(32)limt⟶∞¯u,y1,v,y2X02≤2Cμ′C5+2εCμ+f12+f22σ1C5=R0.
Therefore, there exist positive constantsCR0 and t0, such that when t≥t0, we get(33)u,y1,v,y2X02=∇mu2+y12+∇2mv2+y22≤CR0.
The proof of Lemma3 is completed.Lemma 4.
Assume that assumptionsA1−A4 hold, if f1∈Vk,f2∈V2k and initial data u0,u1,v0,v1∈Xk,k=1,2,…,m; then, for Rk>0, there exist positive constants CRk and tk, so that when t≥tk, u,y1,v,y2 determined by problems (1)–(3) satisfies(34)u,y1,v,y2Xk2=∇m+ku2+∇ky12+∇2m+2kv2+∇2ky22≤CRk,where y1=ut+εuandy2=vt+εv.Proof.
Multiplying the first equation of1 by −Δky1 in HΩ and the second one by −Δ2ky2,k=1,2,…,m in HΩ, we infer that(35)12ddt∇ky12+∇2ky22+M∇mu2+∇2mv2∇m+ku2+∇2m+2kv2+εM∇mu2+∇2mv2∇m+ku2+∇2m+2kv2−ε∇ky12+∇2ky22+ε2∇ku,∇ky1+∇2kv,∇2ky2+N1∇mu2∇m+ky12+N2∇2mv2∇2m+2ky22−εN1∇mu2∇m+ky1,∇m+ku−εN2∇2mv2∇2m+2ky2,∇2m+2kv+g1u,v,−Δky1+g2u,v,−Δ2ky2=∇m+ku2+∇2m+2kv22ddtM∇mu2+∇2mv2+f1,−Δky1+f2,−Δ2ky2.
Using Holder’s inequality, Young’s inequality, Poincare’s inequality, and so on to process the items in (35),(36)−ε∇ky12+∇2ky22+ε2∇ku,∇ky1+∇2kv,∇2ky2≥−ε−ε22∇ky12+∇2ky22−ε22∇ku2+∇2kv2≥−ε−ε22∇ky12+∇2ky22−ε22λ1−m∇m+ku2−ε22λ1−2m∇2m+2kv2,N1∇mu2∇m+ku2+N2∇2mv2∇2m+2kv2−εN1∇mu2∇m+ky1,∇m+ku−εN2∇2mv2∇2m+2ky2,∇2m+2kv≥12N1∇mu2∇m+ky12+12N2∇2mv2⋅∇2m+2ky22−ε22N1∇mu2∇m+ku2−ε22N2∇2mv2∇2m+2kv2,g1u,v,−Δky1+g2u,v,−Δ2ky2≤g1u,v∇2ky1+g2u,v∇4ky2≤N104∇m+ky12+λ1k−mN10g1u,v2+N204∇2m+2ky22+λ12k−2mN20g2u,v2,f1,−Δky1+f2,−Δ2ky2≤∇kf1∇ky1+∇2kf2∇2ky2≤12∇ky12+12∇2ky22+12∇kf12+12∇2kf22,
According toA3, we get(37)g1u,v2=∫Ωg1u,v2dx≤∫ΩC21+up1+vq12dx≤C6∫Ω1+u2p1+v2q1dx≤C71+u2p12p1+v2q12q1,and g2u,v2≤C81+u2p22p2+v2q22q2. Furthermore, using the Gagliardo–Nirenberg inequality, we conclude(38)u2pj2pj≤C9j∇muℕpj−1/mu2mpj−ℕpj−1/m,v2qj2qj≤C10j∇2mvℕqj−1/2mv4mqj−ℕqj−1/2m,then(39)g1u,v2+g2u,v2≤CR0.
Combining (36)–(38) and (39), (28) becomes(40)12ddt∇ky12+∇2ky22+M∇mu2+∇2mv2∇m+ku2+∇2m+2kv2+2N1∇mu2−N10λ1m−2−4ε−2ε24∇ky12+2N2∇2mv2−N20λ12m−2−4ε−2ε24⋅∇2ky22+εM∇mu2+∇2mv2−ε22N1∇mu2−ε22λ1−m∇m+ku2+εM∇mu2+∇2mv2−ε22N2∇2mv2−ε22λ1−2m∇2m+2kv2≤∇m+ku2+∇2m+2kv22ddtM∇mu2+∇2mv2+12∇kf12+12∇2kf22+CR0,λ1≤∇m+ku2+∇2m+2kv2M′∇mu2+∇2mv2⋅∇mu,∇mut+∇2mv,∇2mvt+12∇kf12+12∇2kf22+CR0,λ1≤C8∇mut+∇2mvt∇m+ku2+∇2m+2kv2+12∇kf12+12∇2kf22+CR0,λ1.
LetH2t=∇ky12+∇2ky22+M∇mu2+∇2mv2∇m+ku2+∇2m+2kv2 and σ2=minλ1mNj0−2−4ε−2ε2/2,ε/2, we have(41)ddtH2t+σ2H2t≤C9∇mut+∇2mvtH2t+∇kf12+∇2kf22+CR0,λ1.
Taking the scalar product inHΩ of (1) with ut,vt, we obtain(42)12ddtut2+vt2+∫0∇mu2+∇2mv2Mτdτ+2Ju,v−2f1,u−2f2,v+N1∇mu2∇mut2+N2∇2mv2∇2mvt2=0,and integrating (42) in dt on 0,t, we get(43)∫0t∇mutτ2+∇2mvtτ2dτ≤1minN10,N20∫0tN1∇muτ2∇mutτ2+N2∇2mvτ2∇2mvtτ2dτ≤1minN10,N20u12+v12+∫0∇mu02+∇2mv02Mτdτ+2Ju0,v0−2f1,u0−2f2,v0≤C10,and then we have(44)C9∫st∇mutτ+∇2mvtτdτ≤σ22t−s+a,fort>s≥0and somea>0.
Combining (41) and (44), according to Lemma 1, we know(45)H2t≤C11H20e−σ2/2t+C12.
ByA1, we have(46)H2t≥∇ky12+∇2ky22+M∇mu2+∇2mv2∇m+ku2+∇2m+2kv2≥C13∇ky12+∇2ky22+‖∇m+ku‖2+‖∇2m+2kv‖2,then(47)u,y1,v,y2Xk2=∇ky12+∇2ky22+∇m+ku2+∇2m+2kv2≤C11H20e−σ2/2t+C12C13,that is,(48)limt⟶∞¯u,y1,v,y2Xk2≤Rk.
Therefore, there exist positive constantsCRk and tk, such that when t≥tk, u,y1,v,y2 satisfies(49)u,y1,v,y2Xk2=∇ky12+∇2ky22+∇m+ku2+∇2m+2kv2≤CRk,k=1,2,…,m.
The proof of Lemma4 is completed.Theorem 1.
Assume that assumptionsA1−A4 hold if f1∈Vk,f2∈V2k and initial data u0,u1,v0,v1∈Xk,k=0,1,…,m, then problems (1)– (3) have a unique solution u,v satisfying(50)u∈L∞0,∞;Vm+k;ut∈L∞0,∞;H∩L20,T;Vk;v∈L∞0,∞;V2m+2k;vt∈L∞0,∞;H∩L20,T;V2k.Proof.
According to [10] and the Faedo–Galerkin method, it is easy to obtain that (1)– (3) have global solutions combining with Lemma 3 and Lemma 4.
The uniqueness of the solution is as follows:
Letu1,v1 and u2,v2 be two solutions of problems (1)–(3) with the same initial value. Write w=u1−u2,z=v1−v2, then w,z satisfies(51)wtt+12σ12t−Δmwt+12Φ12t−Δmw+G1u1,u2,v1,v2;t=0,ztt+12σ34t−Δ2mzt+12Φ12t−Δ2mz+G2u1,u2,v1,v2;t=0,where(52)σ12=σ1t+σ2t,Φ12t=Φ1t+Φ2t,σit=N1∇mui2,Φit=M∇mui2+∇2mvi2,i=1,2,σ34=σ3t+σ4t,σjt=N2∇2mvj2,j=3,4,G1u1,u2,v1,v2;t=12σ1t−σ2t−Δmut1+ut2+Φ1t−Φ2t−Δmu1+u2+g1u1,v1−g1u2,v2,G2u1,u2,v1,v2;t=12σ3t−σ4t−Δ2mvt1+vt2+Φ1t−Φ2t−Δ2mv1+v2+g2u1,v1−g2u2,v2.
From Lemma3, it can be known that σ12′≤CR0∇mut1+∇mut2 and σ34′≤CR0∇2mvt1+∇2mvt2.
Taking the scalar product inHΩ of (51) with wt,zt, we can obtain(53)12ddtwt2+zt2+14Φ0⋅∇mw2+∇2mz2+12σ12t∇mwt2+12σ34t∇2mzt2+G1u1,u2,v1,v2;t,wt+G2u1,u2,v1,v2;t,zt=0.
From Lemma3 and A1, we know M0≤M≤CR0,H10≡M1; when d/dt∇mw2+∇2mz2≥0, we write Φ0=2M0; otherwise, Φ0=2M1.
LetG1u1,u2,v1,v2;t,wt=G11+G12+G13,G2u1,u2,v1,v2;t,zt=G21+G22+G23, we have(54)G11=12σ1t−σ2t∇mut1+ut2,∇mwt≤CR0∇mut1+∇mut2∇mw∇mwt≤σ1208∇mwt2+2CR0σ120∇mut12+∇mut22∇mw2,(55)G12=12Φ1t−Φ2t∇mu1+u2,∇mwt≤CR0∇mw+∇2mz∇mwt≤σ1208∇mwt2+2CR0σ120∇mw2+∇2mz2,(56)G13=g1u1,v1−g1u2,v2,wt≤CR0wt2+∇mw2+∇2mz2,(57)G21=12σ3t−σ4t∇2mvt1+vt2,∇2mzt≤CR0∇2mvt1+∇2mvt2∇2mz∇2mzt≤σ3408∇2mzt2+2CR0σ340∇2mvt12+∇2mvt22∇2mz2,(58)G22=12Φ1t−Φ2t∇2mv1+v2,∇2mzt≤CR0∇mw+∇2mz∇2mzt≤σ3408∇2mzt2+2CR0σ340∇mw2+∇2mz2,(59)G23=g2u1,v1−g2u2,v2,zt≤CR0zt2+∇mw2+∇2mz2,where σ120=2N10andσ340=2N20.
Combining (53)–(59), (52) can be changed:(60)12ddtwt2+zt2+14Φ0⋅∇mw2+∇2mz2≤C141+∇2mvt12+∇2mvt22wt2+zt2+14Φ0⋅∇mw2+∇2mz2,then(61)wt2+zt2+14Φ0⋅∇mw2+∇2mz2≤w12+z12+14Φ0⋅∇mw02+∇2mz02exp∫0tC141+∇2mvt12+∇2mvt22ds.
The uniqueness of the solution is proved.
The proof of Theorem1 is completed.
According to Theorem1, we define a nonlinear operator St on space X0:u0,u1,v0,v1⟶u,ut,v,vt,∀t≥0. It can be known that St is a nonlinear C0−semigroup defined on X0.Theorem 2.
Assume that assumptions (A1)–(A4) hold, iff1∈Vk,f2∈V2k and initial data u0,u1,v0,v1∈Xk,k=1,2,…,m, then problems (1)–(3) have a family of global attractors A in X0:(62)A=Ak,Ak=ωB0k=∩τ≥0∪t≥τStB0k¯,k=1,2,…,m,where B0k=u,ut,v,vt∈Xk:u,ut,v,vtXk2=∇m+ku2+∇kut2+∇2m+kv2+∇2kvt2≤CR0+CR0k. Obviously, B0k are bounded absorbing sets in X0 and compact in X0.(1)
StAk=Ak,∀t≥0,(2)
Ak attracts all bounded sets in X0, that is, ∀Bk⊂X0 are bounded sets in X0, and(63)distStBk,Ak=supx∈Bkinfy∈AkStx−yX0⟶0t⟶∞,Stt≥0 is the solution semigroup generated by problems (1)–(3).Proof.
From Lemma3, ∀R0>0, u0,u1,v0,v1X0≤R0, such that(64)Stu0,u1,v0,v1X02=uVm2+utV02+vV2m2+vtV02≤CR0,which shows that Stt≥0 are uniformly bounded in X0.
Further,B0k=u,ut,v,vt∈Xk:u,ut,v,vtXk2=∇m+ku2+∇kut2+∇2m+kv2+∇2kvt2≤CR0+CR0k are bounded absorbing sets of the semigroup Stt≥0 in X0 because  are compactly embedding, that is, the bounded sets in Xk are compact sets in X0, so the solution semigroup Stt≥0 is a fully continuous operator.
To sum up, we get the family of global attractorsA=Ak of the solution semigroup Stt≥0 in X0, and(65)Ak=ωB0k=∩τ≥0∪t≥τStB0k¯,Ak⊂X0,k=1,2,…,m.
The proof of Theorem2 is completed.Note 1.
Lemma4 and Theorem 2 show that the bounded absorbing sets B0k=u,ut,v,vt∈Xk:u,ut,v,vtXk2=∇m+ku2+∇kut2+∇2m+kv2+∇2kvt2≤CR0+CR0k,k=1,2,…,m are compact bounded absorbing sets in X0. Therefore, using condition 3 in Lemma 2, the operator semigroup St only needs to be a continuous operator; from Theorem 1, it can be seen that the semigroup St is already a continuous semigroup; from this angle, the family of global attractors A=Ak of problems (1)–(3) in X0 can also be obtained.



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*Source: 7044906-2022-06-13.xml*