# The Long-Time Behavior of 2D Nonautonomousg-Navier-Stokes Equations with Weak Dampness and Time Delay

Authors: Xiaoxia Wang; Jinping Jiang
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we discuss the long-time behavior ofg-Navier-Stokes equations with weak dampnesss and time delay. The uniformly attracting sets of processes are obtained. On the basis of the method with asymptotic compactness, the existence of the uniform attractor for the equation is proved with the restriction of the forcing term belonging to translational compacted function space.


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## Body

## 1. Introduction

The understanding of the behavior with dynamical systems was one of the most important problems of modern mathematical physics (see [1–17]). In the last decades, g-Navier-Stokes equations have received increasing attention due to their importance in the fluid motion. In [2–4], the existence of weak solution and strong solution for the 2D g-Navier-Stokes equation on some bounded domain was studied. The Hausdorff and fractal dimension of the global attractor about the 2D g-Navier-Stokes equation for the periodic and Dirichlet boundary conditions and the global attractor of the 2D g-Navier-Stokes equation on some unbounded domains were researched in [5]. In [6–10], the finite dimensional global attractor and the pullback attractor for g-Navier-Stokes equation were studied. Moreover, Anh et al. studied long-time behavior for 2D nonautonomous g-Navier-Stokes equations and the stability of solutions to stochastic 2D g-Navier-Stokes equation with finite delays in [11, 12]; Quyee researched the stationary solutions to 2D g-Navier-Stokes equation and pullback attractor for 2D g-Navier-Stokes equation with infinite delays in [13]. Recently, the random attractors for the 2D stochastic g-Navier-Stokes equation were researched in [14]. From these researches, we can see that the attractor of 2D g-Navier-Stokes equation is still important. We would like to use the theory of uniform attractors to study it. So, the present research is necessary and has a theoretical basis.In this paper, we study the existence of the uniform attractor of theg-Navier-Stokes equation with weak dampness and time delay which have the following form:
(1)∂u∂t−νΔu+u·∇u+αu+∇p=fx,t+ht,ut,onτ,+∞×Ω,∇·gu=0,onτ,+∞×Ω,ux,t=0,onτ,+∞×∂Ω,uτ,x=u0xx∈Ω,where ut,x∈R2 and pt,x∈R denote the velocity and pressure, respectively. ν>0 is the viscosity coefficient, αu denotes linear dampness, and α>0 is positive constant. f=fx,t∈L2Ω2 is the time-dependent external force term, ht,ut is another external force term with time delay. 0<m0≤g=gx1,x2≤M0 and g=gx1,x2 are suitable real-valued smooth functions; when g=1, Equation (1) becomes the usual 2D Navier-Stokes equations.This paper is organized as follows. In Section2, we first introduce some notations and preliminary results for the g-Navier-Stokes equation. In Section 3, we prove existence of the uniform attractor of 2D g-Navier-Stokes equation with weak dampness and time delay on the bounded domains.


## 2. Preliminaries

We assume that the Poincare inequality holds onΩ, i.e., there exists λ1>0, such that
(2)∫Ωϕ2gdx≤1λ1∫Ω∇ϕ2gdx,∀ϕ∈H01Ω.LetL2g=L2Ω2 with the inner products u,v=∫Ωu·νgdx and the norms ·=·,·1/2,u,v∈L2g. Let H01g=H01Ω2, which is endowed with the inner products u,v=∫ΩΣj=12∇uj·∇vjgdx and the norms ·=·,·1/2, where u=u1,u2,v=v1,v2∈H01g.LetDΩ be the space of C∞ function with the compact support contained in Ω, and let ℵ=v∈DΩ2:∇·gv=0onΩ; the closure of ℵ in L2g is Hg; the closure of ℵ in H01g is Vg. Hg has the inner product and norm of L2g, And Vg has the inner product and norm of H01g.It follows from (2) that
(3)u2≤1λ1u2,∀u∈Vg.We define ag-Laplacian operator as follows: −Δgu=−1/g∇·g∇u=−Δu−1/g∇g·∇u.Using theg-Laplacian operator, we rewrite the first Equation (1) as follows:
(4)∂u∂t−νΔgu+ν∇gg·∇u+u,∇u+αu+∇p=f+ht,ut.From [2], we can define a g-orthogonal projection Pg:L2g⟶Hg and a g-Stokes operator Agu=−Pg1/g∇·g∇u.Applying the projectionPg into (4), we can obtain the following weak formulation of (1): let f∈Vg and u0∈Hg, we find that
(5)u∈L∞0,T;Hg∩L20,T;Vg,T>0,such that∀v∈Vg,∀t>0.
(6)ddtu,v+νu,v+bgu,u,v+αu,v+νRu,v=f,v+ht,ut,v,(7)u0=u0,where bg:Vg×Vg×Vg⟶R is given by
(8)bgu,v,w=∑i,j=12∫ui∂vj∂xwjgdx,and Ru=Pg1/g∇g·∇u, such that Ru,v=b∇g/g,u,v,∀u,v∈Vg. Then, the weak formulation of (6) and (7) is equivalent to the functional equations
(9)dudt+νAgu+Bu+αu+νRu=f+h,(10)u0=u0,where Ag:Vg⟶Vg′ is the g-Stokes operator defined by Agu,v=u,v,∀u,v∈Vg.Bu=Bu,u=Pgu·∇u is bilinear operator and B:Vg×Vg⟶Vg′,Bu,v,w=bgu,v,w,∀u,v,w∈Vg, where B and R satisfy the following inequalities [2, 4]:
(11)BuVg′≤cuu,RuVg′≤∇g∞m0λ11/2u,∀u∈Vg.LetT>τ, u:τ−r,T⟶L2Ω2. For every t∈τ,T, we define uts=ut+s,s∈−h,0. For convenience, we denote CHg=C0−h,0;Hg,CVg=C0−h,0;Vg,LHg2=L2−h,0;Hg,LVg2=L2−h,0;Vg.Leth:R×CHg⟶L2Ω2 satisfy the following assumptions:
(I)
∀ξ∈CHg,t∈R⟶ht,ξ∈L2Ω2 is measureable,(II)
∀t∈R,ht,0=0,(III)
∃Lg>0, such that ∀t∈R,∀ξ,η∈CHg, there is ht,ξ−ht,η≤Lgξ−ηCHg.(IV)
∃m0≥0,Cg>0,∀m∈0,m0,τ≤t,u,v∈C0τ−r,t;Hg, such that(12)∫τtemshs,us−hs,vs2ds≤Cg2∫τ−rtemsus−vs2ds.∀t∈τ,T, ∀u,v∈L2τ−r,T;Hg, from (IV), we have
(13)∫τths,us−hs,vsL2Ω22ds≤Cg2∫τ−rtus−vs2ds.Definition 1.
Letu0∈Hg,ϕ∈LHg2,f∈LLoc2R;Vg′ and h:R×CHg⟶L2Ω2 satisfy the hypotheses (I)-(IV). For every τ∈R, a function u∈L2τ,T;Vg∩L∞τ,T;Hg,∀T>τ is called a weak solution of problem (1) if it fulfils
(14)ddtut+νAgut+But+αut+νRut=ft+ht,utonD′τ,+∞;Vg′uτ=u0.We can obtain the following theorem by the standard Faedo-Galerkin methods, where we letT>τ>0. Other cases can be similarly proved.Theorem 2.
Letf∈LLoc2R;Vg′,u0x∈Hg,h∈L2Ω2 satisfies the assumptions (I)-(IV), there exists a unique solution
(15)ux,t∈L∞0,T;Hg∩L20,T;Vg,∀T>0,
such that (6) and (7) holds.Proof.
We apply the Faedo-Galerkin methods. SinceVg is separable and ℵ is dense in Vg, there exists a sequence wii∈N∈ℵ, which forms a complete orthonormal system in Hg and a basic for Vg. Let m be a positive integer, for each m, we define an approximate solution um of (6) as um=∑i=1mϕimtwi, which satisfies
(16)um′t,wj+νumt,wj+bumt,umt,wj+αumt,wj+b∇gg,umt,wj=ft,wj+Ht,ut,wj,for t∈0,T,j=1,⋯,m and um0=u0m, where u0m is the orthegonal projection in Hg of u0 onto the space spanned by w1,⋯,wm. Then, we can obtain
(17)∑i=1mwi.wjϕim′t+ν∑i=1mwi,wjϕimt+∑i,l=1mbwi,wl,wjϕimtϕlmt+α∑i=1mwi,wjϕimt+∑i=1mb∇gg,wi,wjϕimt=ft,wj+ht,ut,wj.
We can write the differential equations in the usual form(18)∑i=1mϕim′t+∑j=1mαijϕjmt+∑j,k=1mαijkϕjmtϕkmt=∑j=1mβijft,wj+∑j=1mγijht,ut,wj,where αij,αijk,βij∈R.
Letϕim0 be the ith component of u0m. The nonlinear ordinary differential system (18) has a maximal solution defined on some interval 0,tm. If tm<T, then umt⟶∞ as t⟶tm. The following we will prove tm=T. We need several estimates to do.
We multiply (16) by ϕjmt and add these equations for j=1,⋯,m to obtain
(19)um′t,umt+νumt2=ft,umt+ht,ut,umt−αumt2−b∇gg·∇umt,umt.
Then, we have(20)ddtumt2+2νumt2=2ft,umt+2ht,ut,umt−2b∇gg·∇umt,umt−2αumt2≤2ftVg′umtVg+2ht,utVg′umtVg+2m∇g∞umtumt+umt2+2αumt2≤νumt2+8νft2+νumt2+8νht,ut2+2νm2∇g∞2umt2+νumt2+2αumt2,
so that(21)ddtumt2≤8νft2+ht,ut2+2νm2∇g∞2+ν+2αumt2.
LetK=2/νm2∇g∞2+ν+2α, then
(22)ddtumt2≤Kumt2+8νft2+ht,ut2.
By the Gronwall inequality, we have(23)umt2≤eKtum02+8ν∫0tfsVg′2+ht,us2ds.
Hence,(24)sups∈0,Tums2≤eKTum02+8ν∫0tfsVg′2+ht,us2ds,which implies that the sequence um remains in bounded set of L∞0,T;Hg. From (22), we have
(25)ddtumt2≤Kλ1umt2+8νft2+ht,ut2.
Then,(26)ddtumt2−Kλ1umt2≤8νft2+ht,ut2.
We intergrate (26) from 0 to T; we have
(27)umt2−Kλ1∫0Tumt2dt≤u0m2+8ν∫0Tft2+ht,ut2dt.
So, theum remains in a bounded set of L20,T;Vg.
Letu~m denotes the function from R into Vg, which is equal to um on 0,T and to 0 on the complement of this interval. The Fourier transform of u~m is denoted by u^m. Then, we will show that there exist a positive constant C and γ such that
(28)∫−∞+∞τ2γu∧mτ2dτ<C.
Since theum remains in a bounded set of L20,T;Vg, the u~m remains in a bounded set of HγR;Vg,Hg. Since u~m has two discontinuities at 0 and T, the distribute derivative of u~m is given by
(29)ddtu~m=ϕ~m+um0δ0−umTδT,where δ0 and δT are the dirac distributions at 0 and T, and ϕm=um′ is the derivative of um on 0,T. We obtain that
(30)ddtu^m,wj=f^m,wj+u0m,wjδ0−umT,wjδT,for j=1,⋯,m, where δ0 and δT are distributions at 0 and T, fm=f+h−νAum−bum−νRum−αum and f~m=fm on 0,T. By the Fourier transform, we have
(31)2iπτu^m,wj=f^m,wj+u0m,wj−umT,wjexp−2iπTτ,where u^m and f^m denoting the Fourier transforms of u~m and f~m, respectively. We multiply (31) by ϕ^jmτ and add the resulting equations for j=1,⋯,m; we get
(32)2iπτu∧mτ2=f^mτ,u^mτ+u0m,u^mτ−umT,u^mτexp−2iπTτ.
We obtain(33)∫0Tfmtdt≤∫0Tft+ht,ut+νumt+c∇g∞um+αλ1um2+cumt2dt.
So,fmt belongs to a bounded set in the space L10,T;Vg′. For ∀m, we have supτ∈Rf^mτ≤C. Since um0 and umT are bounded, from (31), we obtain
(34)τu∧mτ2≤C1u^mτ+C2u^mτ≤C3u^mτ.
Letγ<1/4, we have
(35)τ2γ≤C4γ1+τ1+τ1−2γ,∀τ∈R,then
(36)∫−∞+∞τ2γu∧mτ2dτ≤C4γ∫−∞+∞1+τ1+τ1−2γu∧mτ2dτ.≤C5∫−∞+∞u^mτVg1+τ1−2γdτ+C6∫−∞+∞u∧mτ2dτ.
Sinceum∈L20,T;Vg, by the Parseval equality ∫−∞+∞u^mτVg2dτ<C and by the Schwarz inequality and the Parseval equality, we obtain
(37)∫−∞+∞u^mτVg1+τ1−2γdτ≤∫−∞+∞11+τ1−2γ2dτ1/2∫−∞+∞umτ2dτ1/2<C.
So,um∈HγR;Vg,Hg, and um remains in a bounded set of L∞0,T;Hg,L20,T;Vg and HγR;Vg,Hg. There exists an element u∈L20,T;Vg∩L∞0,T;Hg and a subsequence um′ such that um′⟶u in L20,T;Vg weakly and um′⟶u in L∞0,T;Vg weak-star as m′⟶∞. For any B∈Rn, we have um′B⟶uB strongly in L20,T;HgB.
For any supportBj of wj, we have um′Bj⟶uBj strongly in L20,T;HgBj. Let ψ be a continuously differentiable function on 0,T with ψT=0, we multiply (16) by ψt, then integrate by parts,
(38)−∫0Tumt,ψ′twjdt+ν∫0Tumt,wjψtdt+∫0Tbumt,umt,wjψtdt+∫0Tαumt,wjψtdt+∫0Tb∇gg,umt,wjψtdt=u0m,wjψ0+∫0Tft+ht,ut,wjψtdt.
We have(39)−∫0Tut,vψ′tdt+ν∫0Tut,vψtdt+∫0Tbut,ut,vψtdt+∫0Tαut,vψtdt+∫0Tb∇gg,ut,vψtdt=u0,vψ0+∫0Tft+ht,ut,vψtdt.where ∀v∈Vg.
Finally, we prove thatu satisfies (7). We multiply (6) by ψ and integrate
(40)−∫0Tut,vψ′tdt+ν∫0Tut,vψtdt+∫0Tbut,ut,vψtdt+∫0Tαut,vψtdt+∫0Tb∇gg,ut,vψtdt=u0,vψ0+∫0Tft+ht,ut,vψtdt.
We compare (39) with (40) to obtain u0−u0,vψ0=0. Let ψ0=1, then we have u0−u0,v=0,∀v∈Vg. So, u0=u0.
Now, we will prove the solution of (6) and (7) is unique. We let u1 and u2 be the solutions of (9) and u=u1−u2. We have
(41)∂u∂t+νAu+Bu+νRu+αu=−Bu1+Bu2,(42)u0=0.
We take the scalar product of (41) with ut, then
(43)ddtut2+2νut2+2b∇gg,ut,ut+2αut2=2bu2t,u2t,ut−2bu1t,u1t,ut=−2but,u2t,u,2but,u2t,u≤Cututu2t≤νut2+C2νut2u2t2,2b∇gg,ut,ut≤2C∇g∞uu≤νut2+C2ν∇g∞2ut2.
Therefore,(44)ddtut2+2νut2≤νut2+C2ν∇g∞2ut2+νut2+C2νut2u2t2+αut2.
Then,(45)ddtut2≤C2ν∇g∞2+C2νu2t2+αut2.
We have(46)ut2≤u02exp∫0tC2ν∇g∞2+C2νu2t2+αds.
Hence,ut2=0,∀t∈0,T. So, u1=u2.
From [15], we can define a family of two parametric maps Uft,τ=Uft,τ∣t≥τ,τ∈R in Hg,
(47)Uft,τ:E⟶E,t≥τ,τ∈R.
Here,f∈L∞R+;Vg′ is called the time symbol of the system. We have the following concepts and conclusions from [15].Definition 3.
For the given time symbolf∈L∞R+;Vg′, a family of two-parametric maps Ut,τ with t≥τ≥0 is called a process in Hg, if
(48)Uft,sUs,τ=Uft,τ,∀t≥s≥τ,τ∈R,Ufτ,τ=Id,τ∈R.
Now, we define translation operator inL∞R+;Vg′. ∀f∈L∞R+;Vg′.
(49)Thfs=fs+h,∀h≥0,s∈R.
We have(50)ThfL∞R+;Vg′≤fL∞R+;Vg′,∀h≥0,f∈L∞R+;Vg′.
DenoteΣ=Thfx,s=fx,s+h,∀h∈R, where T· is the positive invariant semigroups acting on Σ and satifying ThΣ⊂Σ,∀h≥0 and
(51)UThft,τ=Uft+h,τ+h,∀h≥0,t≥τ≥0.
LetρF>0 be a constant, obviously
(52)Σ⊂f∈L∞R+;Vg′:fL∞R+;Vg′≤ρF.
LetE be the Banach space; we use BE to denote the set of all bounded sets on E and consider a family of processes Uft,τ with f∈Σ, the parameter f is called the symbols of the process family Uft,τ, Σ is called the symbol space, and we assume that Σ is a complete metric space.Definition 4.
A family of processesUft,τ,f∈Σ is called uniformly bounded (w.r.t.f∈Σ), if any B∈BE, both
(53)⋃f∈Σ⋃τ∈R⋃t≥τUft,τB∈BE.Definition 5.
A setB0⊂E is said to be uniformly absorbing for the family of processes Uft,τ,f∈Σ, if for any τ∈R and each B∈BE, there exists t0=t0τ,B≥τ, such that for all t≥t0,
(54)⋃f∈ΣUft,τB⊆B0.Definition 6.
A setP⊂E is said uniformly atttracting set of Uft,τ,f∈Σ, if for any τ∈R, there is
(55)limt⟶+∞supf∈ΣdistEUft,τB,P=0.
A family of processesUft,τ,f∈Σ is said to uniformly compact, if there exists a compacted uniformly absorbed set in Uft,τ,f∈Σ. A family of processes Uft,τ,f∈Σ is said to uniformly asymptotic compact, if there exists a compacted uniformly atttracting set in Uft,τ,f∈Σ.Definition 7.
A closed setAΣ⊂E is said to be the uniform attractor of the family of processes Uft,τ,f∈Σ, if
(1)
AΣ⊂E is uniformly attractive(2)
AΣ⊂Eis included in any uniformly attracting set of Uft,τ,f∈Σ, that is AΣ⊂A′.Theorem 8.
Letfγθ:γ∈Γ⊂C=C−r,0;X be equicontinuous and for any ∀θ∈−r,0,fγθ:γ∈Γ is quasicompact in X, then fγθ:γ∈Γ is relatively compact in C−r,0;X.Lemma 9 (Uniform Gronwall lemma).
Letg,h,y be local integrable function on t0,∞, y′ is also local integrable on t0,∞, and y′t≤gtyt+ht,∀t≥t0.∫tt+rgsds≤a1,∫tt+rhsds≤a2,∫tt+rysds≤a3, where r,a1,a2,a3 is positive constant. Then,
(56)yt≤a3r+a2expa1,∀t≥t0.


## 3. The Existence of Uniform Attractor for 2Dg-Navier-Stokes Equations in Bounded Domain

First, we prove the existence of uniformly absorbing set inCHg and CVg; we define u·=u·;τ,u0,ϕ,f, where f is translation compact function. That is,
(57)fb2=fLb2R,Hg2=supt∈R∫tt+1fs2ds<∞.The following we useLc2R,Hg to represent the translation compact function class.Lemma 10.
Let for anyτ≤t, m0>0,f∈Lc2R,Hg, assume that (I)-(IV) hold, then there exist bounded absorbing sets Btt∈R of process family Uft,τ:t≥τ in CHg.Proof.
SinceD~⊂MHg2=Hg×CHg is bounded, then there exists d~≥0, such that
(58)u02+ϕLHg22≤d2~,∀u0,ϕ∈D~.
For anyu0,ϕ∈D~,τ∈R, we define u·=u·;τ,u0,ϕ, then taking the inner product of (9) with ut, we have
(59)12ddtu2+νu2+αu,u+νRu,u=f,u+ht,ut,uddtu2+2νu2+2αu2=2f,u+2ht,ut,u−2νRu,u≤f2σ+σu2+1Cght,ut2+Cgu2+2ν∇g∞m0λ11/2u2ddtu2+2ν1−∇g∞m0λ11/2u2+2αu2≤f2σ+σu2+1Cght,ut2+Cgu2ddtu2+2νλ11−∇g∞m0λ11/2+2αu2≤f2σ+σu2+1Cght,ut2+Cgu2ddtemtut2=memtut2+emtddtut2≤memtut2+emtf2σ+1Cght,ut2+σ+Cgu2−2νλ11−∇g∞m0λ11/2+2αu2.
Letβ=1−∇g∞/m0λ11/2+2α, then
(60)ddtemtut2≤memtut2+emtf2σ+1Cght,ut2+σ+Cg−2νλ1βu2.
Integrating both sides fromτ to t, then
(61)emtut2−emτuτ2≤∫τtemsf2σds+∫τtemsCghs,us2ds+∫τtemsm+σ+Cg−2νλ1βus2ds.
Then,(62)emtut2≤∫τtemsf2σds+∫τtemsCght,ut2ds+∫τtemsm+σ+Cg−2νλ1βus2dsut2≤e−mt∫τtemsf2σds+e−mt∫τtemsCght,ut2ds+e−mt∫τtemsm+σ+Cg−2νλ1βus2ds,for
(63)e−mt∫τtemsCght,ut2ds≤e−mtCg∫τ−hτemsϕs−τ2ds.
Lets−τ=θ, then
(64)e−mt∫τtemsCght,ut2ds≤e−mtCgemτ∫−h0ϕθ2dθe−mt∫τtemsf22ds=1σ∫τte−mt−sf2ds≤1σ∫t−1te−mt−sfs2ds+∫t−2t−1e−mt−sfs2ds+⋯≤1σ1+e−m+e−2m+⋯supt∈R∫tt+1f2ds=1σ1−e−mfb2.
Takingm∈0,m0, such that m+σ+Cg−2νλ1β<1, then ∇g∞<m02νλ11+2α−σ−Cg+1/2νλ11/2, so
(65)ut2≤1σ1−e−mfb2+d~21+Cge−mt·emτt≥τ.
Lett≥τ+h,∀θ∈−h,0, then
(66)ut+θ2≤fb2σ1−e−m+d~21+Cge−mt+θ·emτ≤fb2σ1−e−m+d~2emh1+Cge−mt·emτ.
Then,(67)utCHg2≤fb2σ1−e−m+d~2emh1+Cge−mt·emτt≥τ+h.
LetB1=ut∣utCHg2≤fb2/σ1−e−m, we will prove the existence of the uniformly absorbing bounded set in CVg. First, we must prove the boundedness of ∫tt+1us2ds.Lemma 11.
Given thatD∈BMHg2, then there exist THgD and constant IVg, such that
(68)∫tt+1us2ds≤IVg.∀t≥THgD+r+1,where BMHg2 denotes any bounded set on the MHg2.Proof.
Taking the inner product of (9) with ut,
(69)ddtu2+2νu2+2αu2=2f,u+2ht,ut,u−2νRu,u≤f2σ+σu2+1Cght,ut2+Cgu2+2ν∇g∞m0λ11/2u2≤f2σ+σλ1u2+1Cght,ut2+Cgλ1u2+2ν∇g∞m0λ11/2u2.
Then,(70)ddtu2+2ν−σλ1−Cgλ1−2ν∇g∞m0λ11/2u2+2αu2≤f2σ+1Cght,ut2.
Integrating on both sides int,t+1, we have
(71)ut+12−ut2+2ν−σλ1−Cgλ1−2ν∇g∞m0λ11/2∫tt+1u2ds≤∫tt+1f2σds+1Cg∫tt+1ht,ut2ds≤fb2+Cg∫t−ht+1us2ds≤fb2+Cg∫t−htus2ds+Cg∫tt+1us2ds≤fb2+Cg∫t−htus2ds+Cgλ1∫tt+1us2ds.
Then,(72)2ν−σλ1−2Cgλ1−2ν∇g∞m0λ11/2∫tt+1u2ds≤fb2+Cg∫t−htus2ds+ut2≤fb2+Cg∫t−htusCHg2ds+ρHg2≤fb2+1+hCgρHg2.
When2ν−σ/λ1−2Cg/λ1−2ν∇g∞/m0λ11/2>0, that is ∇g∞<2νλ1−σ−2Cg/2νλ11/2, we have
(73)∫tt+1us2ds≤IVg.∀t≥THgD+r+1,where
(74)IVg=12ν−σ/λ1−2Cg/λ1−2ν∇g∞/m0λ11/2fb2+1+hCgρHg2.ρHg=fb2σ1−e−m.Lemma 12.
For anyτ≤t, m0>0,f∈Lc2R,Hg. Assume that (I)-(IV) hold, then there exists uniformly bounded absorbing set B2⊂CVg of process family Uft,τ:t≥τ in CVg.Proof.
LetD∈BMHg2, taking the inner product of (9) with Agu, we obtain
(75)12ddtu2+νAgu2+αu2+Bu,u,Agu+νRu,Agu=f,Agu+ht,ut,Agu,ddtu2+2νAgu2+2αu2≤2f,Agu+2ht,ut,Agu−2Bu,u,Agu−2νRu,Agu,for
(76)2f,Agu+2ht,ut,Agu≤2Aguf+ht,ut≤ν2Agu2+4νf2+ht,ut2,2Bu,u,Agu≤2c1u1/2uAgu3/2≤ν2Agu2+2c1′ν3u2u4,2νRu,Agu≤2νRu·Agu≤2ν∇g∞m0u·Agu≤2ν∇g∞m01λ1Agu2,∀u∈DAgu.
Then,(77)ddtu2+2νAgu2≤ν2Agu2+4νf2+ht,ut2+ν2Agu2+2c1′ν3u2u4+2ν∇g∞m0λ11/2Agu2−2αu2,ddtu2+ν−2ν∇g∞m0λ11/2Agu2≤4νf2+ht,ut2+2c1′ν3u2u4−2αu2,ddtu2+ν1−2∇g∞m0λ11/2λ1u2≤4νf2+Lg2usCHg2+2c1′ν3u2u4−2αu2≤4νf2+Lg2ρHg2+2c1′ν3u2u4.
Applying Lemma9,
(78)ur2≤a3+a2ea1∀r≥t0+1,s≥T~D~,where a3=IVg,a2=4/vf2+Lg2ρHg2,a1=2c1′/ν3ρHg2IVg. If taking s≥T~D~+1+h, then
(79)supθ∈−h,0ut0+θ2≤a3+a2ea1=ρVg2.
Letu·=u·;t−s,u0,ϕ, so ut·∈CVg,∀s>h. Then,
(80)B2=ut∣utCVg≤ρVg,∀t∈R,s≥T~D~+1+h.From [16], we have the following definition.Definition 13.
LetE be Banach space, if ∀ε>0, there exists η>0, such that
(81)sup∫tt+ηfE2ds<ε.
Then,f∈Lloc2R,E is called normal function.We will take the sets of all normal function classes inLloc2R,E as Ln2R;E. From [17], we can see that Ln2R;E is the true subspace of Lc2R;E. Therefore, the translation compact function must be a normal function.Theorem 14.
Suppose that nonlinear termh satisfies (I)-(III), f is translation compact function in Lloc2R,Hg, then process family Uf·,·∣f∈Σ exist uniform attractor AΣ, and AΣ⊂Hg×CHg.Proof.
SinceB2 is bounded set in CVg and uniform absorbed set of Uf·,·∣f∈Σ. For each τ∈R, we take a set
(82)B3=⋃f∈ΣUfτ+r,τjB2,where j denotes any compact self-adjoint operator, then B3⊂B2⊂B1, and B3 is another uniform absorbing bounded set of Uf·,·∣f∈Σ in CVg. Now, we will prove B3 is relatively compact in CHg. From Theorem 8, we only need to prove B3 is equicontinuous and uniform bounded in CHg. From the definition of B3, we can obtain it is uniformly bounded. Now, we will prove B3 is equicontinuous. For any θ1,θ2∈−r,0,ϕ∈B2,f∈Σ,
(83)Ufτ+r,τjϕθ1−Ufτ+r,τjϕθ2=uτ+r+θ1;τ,jϕ,f−uτ+r+θ2;τ,jϕ,f.
Letθ2>θ1, and denote u·=u·;τ,jϕ,f as u·, then
(84)uτ+r+θ1−uτ+r+θ2=∫τ+r+θ1τ+r+θ2dusdtds≤∫τ+r+θ1τ+r+θ2dusdtds≤∫τ+r+θ1τ+r+θ2νAgus+αus+Bus+f+hs,us+νRusds≤∫τ+r+θ1τ+r+θ2νAgus+c1Agusus+αus+f+hs,us+νRusds≤∫τ+r+θ1τ+r+θ2ν+c1usAgus+LgusCHg+ν∇g∞m0uds+∫τ+r+θ1τ+r+θ2fds+∫τ+r+θ1τ+r+θ2αusds.
We estimate the items on the right end of the above formula, letθ1⟶θ2,
(85)i∫τ+r+θ1τ+r+θ2fds≤supt∈R∫τt+θ2−θ1fds⟶0,∀f∈Σ,ii∫τ+r+θ1τ+r+θ2ν+c1usAgus+LgusCHgds≤∫τ+r+θ1τ+r+θ2ν+c1usAgusds+ρHgLgθ1−θ2≤∫τ+r+θ1τ+r+θ2ν+c1ρVgAgusds+ρHgLgθ1−θ2≤ν+c1ρVgθ1−θ21/2∫τ+r+θ1τ+r+θ2Agus2ds1/2+ρHgLgθ1−θ2.
Since(86)ddtu2+ν1−2∇g∞m0λ11/2Agu2≤4νf2+Lg2usLHg2+2c1′ν3u2u4−2αu2,we let α1=4/vf2+Lg2usLHg2,α2=2c1′/ν3,α3=1−2∇g∞/m0λ11/2,α4=2α.
Whens≥T~D~+1+h,θ1,θ2∈−h,0, and θ2>θ1, then
(87)Agu2≤α1α3+α2α3u2u4−1α3ddtu2−α4α3u2,∫t+θ1t+θ2Agu2ds≤α1α3θ2−θ1+α2α3∫t+θ1t+θ2us2us4ds−1α3ut+θ22+1α3ut+θ12−α4α3∫t+θ1t+θ2us2ds≤α1α3+α2α3ρHg2ρVg4−α4α3ρVg2θ2−θ1+1α3ut+θ12.
Let(88)β1=α1α3+α2α3ρHg2ρVg4−α4α3ρVg2,β2=1α3ρVg2.
Then,(89)∫t+θ1t+θ2Agu2ds<β1θ2−θ1+β2.
So,(90)∫t+r+θ1t+r+θ2ν+c1usAgus+LgusCHgds≤ν+c1ρVgθ1−θ21/2β1θ1−θ2+β21/2+ρVgLgθ1−θ2.
Andα4/α3∫t+r+θ1t+r+θ2αusds≤αρVgθ2−θ1. When θ1⟶θ2,∀ϕ∈B2,f∈Σ, we have
(91)ut+r+θ1−ut+r+θ2≤∫t+r+θ1t+r+θ2fds+∫t+r+θ1t+r+θ2ν∇g∞m0uds+∫t+r+θ1t+r+θ2ν+c1usAgus+LgusCHgds+∫t+r+θ1t+r+θ2αusds≤supt∈R∫tt+θ2−θ1fsds+ν∇g∞m0ρVgθ2−θ1+αρVgθ2−θ1+ν+c1ρVgθ1−θ21/2β1θ1−θ2+β21/2+ρVgLgθ1−θ2⟶0.
Then,B3 is equicontinuous, and B3 is relatively compact in CHg, so B3¯ is compacted uniformly absorbing set of Uf·,·∣f∈Σ in CHg, Let B~3=jB¯3, since Hg×CHg=MHg2,Vg⊂Hg and the embedding mapping is continuous, so B~3 is compact in MHg2; B~3 is also compacted uniformly absorbing set of Uf·,·∣f∈Σ in MHg2. Then, process family Uf·,·∣f∈Σ exists uniform attractor AΣ⊂Hg×CHg.



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*Source: 2034264-2022-07-25.xml*