# The Existence of WeakD-Pullback Exponential Attractor for Nonautonomous Dynamical System

Authors: Yongjun Li; Xiaona Wei; Yanhong Zhang
Journal: The Scientific World Journal
 (2016)

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## Abstract

First, for a processU(t,τ)∣t≥τ, we introduce a new concept, called the weak D-pullback exponential attractor, which is a family of sets M(t)∣t≤T, for any T∈R, satisfying the following: (i) M(t) is compact, (ii) M(t) is positively invariant, that is, U(t,τ)M(τ)⊂M(t), and (iii) there exist k,l>0 such that dist(U(t,τ)B(τ),M(t))≤ke-(t-τ); that is, M(t) pullback exponential attracts B(τ). Then we give a method to obtain the existence of weak D-pullback exponential attractors for a process. As an application, we obtain the existence of weak D-pullback exponential attractor for reaction diffusion equation in H01 with exponential growth of the external force.


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## Body

## 1. Introduction

Pullback attractor is a suitable concept to describe the long time behavior of infinite dimensional nonautonomous dynamical systems or process generated by nonautonomous partial differential equations. There are many references concerned with the existence of pullback attractors for nonautonomous PDEs (see [1–5]). In [2], Caraballo introduced the notion of D-pullback attractor for nonautonomous dynamical systems and gave a general method to prove the existence of D-pullback attractor. However, pullback attractors or D-pullback attractors attract any bounded set of phase space, but the attraction to it may be arbitrarily slow. In order to describe the attracting speed, the concept of pullback exponential attractor is put forward (see [6]), which is a positively invariant family of compact subsets with finite fractal dimension (see [7, 8]) and exponentially attracts each bounded subset. In [6], a new method is given to prove the existence of pullback exponential attractor and it is applied to reaction diffusion equation when the external force is normal; in [9], the same result is obtained when the nonlinear term f(t,u) satisfies ft,u-ft,v≤ξtu-v. In fact, these conditions are relatively strict; for general conditions, we can not get the result.Motivated by these problems and some ideas in [3–6], we introduce a new attractor, called the weak D-pullback exponential attractors that is for a process Ut,τ, for any T∈R, there exists a family of sets M(t)∣t≤T satisfying the following:(i)
M(t) is compact.(ii)
M(t) is positively invariant; that is, U(t,τ)M(τ)⊂M(t).(iii)
∀t∈(-∞,T], there exist k,l>0 such that dist⁡(U(t,τ)B(τ),M(t))≤ke-(t-τ), that is, M(t) pullback exponential attracts B(τ) for all Bt∈D.Compared with the pullback exponential attractor, the fractal dimension of the weakD-pullback exponential attractor is not necessarily uniformly bounded or even unbounded, and the positively invariant only holds for any t∈(-∞,T], compared with the D-pullback attractor, which pullback attracts bounded set with exponential speed and contains D-pullback attractor.The paper is organized as follows. In Section2, we recall some basic concepts about pullback attractor. In Section 3, we construct a weak D-pullback exponential attractor for nonautonomous dynamical systems and we provided a method to verify the existence of weak D-pullback exponential attractor. In Section 4, we apply our result to prove the existence of weak D-pullback exponential attractor for nonautonomous reaction diffusion system in H01 with exponential growth of the external force.


## 2. Preliminaries

LetX be a complete metric space; let B(X) be the set of all bounded subsets of X; D is a nonempty class of parameterised sets D^=D(t)∣t∈R,D(t)∈B(X) or D^=D(n)∣n∈Z,D(n)∈B(X); and a two-parameter family of mappings U(t,τ)∣t≥τ=U(t,τ)∣t≥τ,t,τ∈R act on X, that is, U(t,τ):X→X, ∀t≥τ.Definition 1.
A two-parameter family of mappingsUt,τ is said to be a process in X, if(1)
U(t,s)U(s,τ)=U(t,τ), ∀t≥s≥τ,(2)
U(τ,τ)=Id is the identity operator, τ∈R.The pair (U(t,τ),X) is generally referred to as a nonautonomous dynamical system, and (U(n,m),X) (n,m∈Z) is called a nonautonomous discrete dynamical system generated by (U(t,τ),X). If x→U(t,τ)x is continuous in X, we say that the process is continuous process; if U(t,τ)xn⇀U(t,τ)x as xn→x, we say that the process is the norm-to-weak continuous process. Obviously, continuous process is also a norm-to-weak continuous process.Definition 2.
A family of setsB(t)∣t∈R∈D is called D-pullback bounded absorbing sets for the process Ut,τ if, for any t∈R and any bounded sets D(t)∣t∈R∈D, there exists τ0(t,D(t))≤t such that U(t,τ)D(τ)⊂B(t) for all τ≤τ0.Definition 3.
The familyA=A(t)∣t∈R⊂B(X) is said to be a D-pullback attractor for U(t,τ) if the following hold: (1)
A(t) is compact for all t∈R;(2)
A is invariant; that is, U(t,τ)A(τ)=A(t)  ∀t≥τ;(3)
A is D-pullback attracting; that is, limτ→-∞dist⁡(U(t,τ)B(τ),A(t))=0, ∀Bt∈D, and t∈R;(4)
ifCtt∈R is another family of closed attracting sets, then A(t)⊂C(t)  ∀t∈R.Here dist⁡(·,·) denotes the nonsymmetric Hausdorff distance between sets in X; that is, dist⁡(A,B)=supa∈A⁡infb∈B⁡a-b.Definition 4.
The Kuratowski measure of noncompactnessα(B) of B⊂X is defined by (1)αB=inf⁡δ>0∣Badmits  a  finite  cover  by  sets  of  diameter≤δ.The following summarizes some of the basic properties of the measure of noncompactness.Lemma 5 (see [10]).
LetB,B1,B2⊂X. Then (1)
α(B)=0 if, and only if, B¯ is compact;(2)
α(B1+B2)≤α(B1)+α(B2);(3)
α(B1)≤α(B2) for B1⊂B2;(4)
α(B1∪B2)≤maxαB1,αB2;(5)
ifF1⊃F2… are nonempty closed sets in X such that α(Fn)→0 as n→∞, then F=⋂n=1∞‍Fn is nonempty and compact.In addition, let X be an infinite dimensional Banach space with a decomposition X=X1⊕X2 and let P:X→X1, Q:X→X2 be projectors with dim⁡X1<∞. Then (6)
α(B(ε))=2ε, where B(ε) is a ball of radius ε;(7)
α(B)<ε for any bounded subset B of X for which the diameter of QB is less than ε.Definition 6 (see [3–5]).
A processUt,τ is called D-pullback ω-limit compact for B(t)∣t∈R if, for any ε>0, there exists a τ0(t,B(t),ε)≤t such that α(⋃τ≤τ0‍U(t,τ)B(τ))≤ε.Lemma 7 (see [3–5]).
Assume that the processU(t,τ)∣t≥τ is D-pullback ω-limit compact for B^=B(t)∣t∈R; then, for any sequence τn⊂(-∞,t], τn→-∞ as n→+∞ and for any sequence xn∈B(τn), there exists a convergence subsequence of Ut,τnxn whose limit lies in ω(B^,t); here ω(B^,t) is defined by (2)ωB^,t=⋂s≤t⋃τ≤sUt,τBτ¯.Theorem 8 (see [3, 5]).
LetU(t,τ)∣t≥τ be a continuous or norm-to-weak continuous process and U(t,τ)∣t≥τ is D-pullback ω-limit compact; let B(t)∣t∈R⊂B(X) be a family of D-pullback bounded absorbing sets for the process. Then the process Ut,τ∣t≥τ has a D-pullback attractor A=A(t)∣t∈R, and (3)At=⋂s≤t⋃τ≤sUt,τBτ¯=⋂s≤t⋃τ≤sUt,τB∣B∈BX¯.For a discrete processU(n,m)∣n,m∈Z,n≥m, the above conclusions also hold true.


## 3. The Existence of WeakD-Pullback Exponential Attractor

LetX be a Banach space; · denotes the norm of X, D is a nonempty class of parameterised sets D^=D(t)∣t∈R⊂B(X) or D^=D(n)∣n∈Z⊂B(X), and Ut,τ is a continuous process on X.Now, we give our main theorems which describe the relationship between the measure of noncompactness and the weakD-pullback exponential attractor.Theorem 9.
Assume thatBn∈D is positively invariant D-pullback bounded absorbing sets of Un,m; that is, for any Dn∈D, N∈Z, there exists T∈N, such that U(n,m)D(m)⊂B(n) for any n-m≥T, and U(n,m)B(m)⊂B(n) for any m≤n≤N; then the following are equivalent: (I)
The measure of noncompactnessD-pullback decays exponentially for the discrete process {U(n,m)}; that is, there exist k,l>0 such that(4)α⋃k≤mUn,kBk≤ke-ln-m,foranym≤n≤N.(II)
The processUn,m has a weak D-pullback exponential attractor; that is, there exists a family of sets M(n)∣n≤N satisfying the following:
(1)
M(n) is compact;(2)
M(n) is positively invariant; that is, U(n,m)M(m)⊂M(n);(3)
M(n)∣n∈Z attracts Dn exponentially in a D-pullback sense; more precisely, (5)dist⁡Un,mDm,Mn≤ke-ln-m,foranyDn∈D.Proof.
(
(
I
)
⇒
(
I
I
)
) Since the measure of noncompactness D-pullback decays exponentially for {U(n,m)}, from Definition 6, we find that Un,m is D-pullback ω-limit compact. By Theorem 8, we get that (6)An=⋂k≤n⋃m≤kUn,mBm¯is a D-pullback attractor of Un,m. Using (3) of Lemma 5, we find that (7)αUn,mBm≤α⋃k≤mUn,kBk≤ke-ln-m,and by the definition of the measure of noncompactness, for any n≥m, there exist finite points xn,im∈B(n) such that Un,mBm⊂⋃i=1nmBxn,im,ke-ln-m. Letting Wnm=xn,im∣i=1,2,…,nm and Mk=⋃n=0+∞⋃i=0nUk,k-iWk-in-i, we get(8)Mk+1=⋃n=0+∞⋃i=0nUk+1,k+1-iWk+1-in-i=⋃n=0+∞⋃i=0nUk+1,kUk,k+1-iWk+1-in-i⊃⋃n=1+∞⋃i=0nUk+1,kUk,k-i-1Wk-i-1n-1-i-1⊃Uk+1,k⋃n=1+∞⋃i=1nUk,k-i-1Wk-i-1n-1-i-1=Uk+1,k⋃n=1+∞⋃i=0n-1Uk,k-iWk-in-1-i=Uk+1,k⋃n=0+∞⋃i=0nUk,k-iWk-in-i=Uk+1,kMk.Consequently, for all n∈Z, the family M(n)∣n≤N is positively invariant.
LetM(n)=M(n)∪A(n); we claim that M(n)∣n≤N satisfies (II).
(Compactness) for any sequencexk∈M(n), there exist mk and yk such that xk=U(n,mk)yk. By (I), we get that the process Un,m is pullback D-ω-limit compact; we deduce from Lemma 7 that xk has subsequence convergent in M(n). We get that M(n) is compact.
(Positively invariant) sinceU(n+1,n)M(n)⊂M(n+1), U(n+1,n)A(n)=A(n+1), we get (9)Un+1,nMn=Un+1,nMn∪An⊂Mn+1.
(Exponential attracting) for anyDm∈D, there exists T∈N, such that (10)Um,m-TDm-T⊂Bm.Since Bn is positively invariant, we get(11)Un,m-TDm-T=Un,mUm,m-TDm-T⊂Un,mBm⊂⋃i=1nmBxn,im,ke-ln-m,so we obtain (12)dist⁡Un,m-TDm-T,Mn≤dist⁡Un,mBm,Mn.Since Wnm⊂M(n), we get (13)dist⁡Un,m-TDm-T,Mn≤dist⁡Un,mBm,Wnm≤ke-ln-mfor any n≥m.
((II)⇒(I)) By the definition of dist⁡(·,·), we get (14)dist⁡Un,mBm,Mn=supx∈Bm⁡infy∈Mn⁡dUn,mx,y≤ke-ln-m,and, for any x∈B(m), we have (15)infy∈Mn⁡dUn,mx,y≤ke-ln-m.Therefore, for any x∈B(m), there exists yx∈M(n), such that (16)dUn,mx,yx<2ke-ln-m.We get (17)Un,mx∈Byx,2ke-ln-m.Since M(n) is a compact set, we get that there exist y1,y2,…,yl∈M(n) such that (18)Mn⊂⋃i=1lByi,2ke-ln-m.Therefore, for any yx, there exists yix∈y1,y2,…,yl such that (19)dyx,yix≤2ke-ln-m,dUn,mx,yix≤dUn,mx,yx+dyx,yix≤4ke-ln-m.We get (20)Un,mBm⊂⋃i=1lByi,4ke-ln-m,and, by Definition 4, we obtain (21)αUn,mBm≤Ke-ln-mK=4k,and, by (4) of Lemma 5, we get (22)α⋃k≤mUn,mBm≤Ke-ln-m,which say that the measure of noncompactness D-pullback decays exponentially.Theorem 10.
Assume thatBt∈D is positively invariant D-pullback bounded absorbing sets of Ut,τ; that is, for any Dt∈D, R∈R, there exists T≥0, such that U(t,τ)D(τ)⊂B(t) for any t-τ≥T, and U(t,τ)B(τ)⊂B(t) for any t≤R, and there exists a continuous function r(t) that satisfies U(t,τ)x-U(t,τ)y≤r(t)x-y for any x,y∈B(τ), t-τ≤1; then the following are equivalent:(I)
The measure of noncompactnessD-pullback decays exponentially for the process Ut,τ; that is, there exist k,l>0 such that(23)α⋃s≤τUt,sBs≤ke-lt-τ,foranyτ≤t≤R.(II)
The processUt,τ has a weak D-pullback attractor; that is, there exists a family of sets M(t)∣t≤R satisfying the following:
(1)
M(t) is compact;(2)
M(t) is positively invariant; that is, U(t,τ)M(τ)⊂A(t);(3)
M(t)∣t≤R attracts Dt exponentially in D-pullback sense; more precisely,(24)dist⁡Ut,τDτ,Mt≤kte-lt-τ,foranyDt∈D.Proof.
((I)⇒(II)) By Theorem 9, we know that the discrete process Un,m generated by Ut,τ has a weak D-pullback exponential attractor Mn, that is, M(n) is compact and positively invariant and D-pullback exponentially attracts Dn∈D. We set M(t)=U(t,k)M(k), t∈[k,k+1), for all k≤R. As proof of Theorem 9, it is easy to prove that M(t) is compact and positively invariant. Next, we will prove that M(t) attracts Dn∈D exponentially in D-pullback sense.
For anyDt∈D, there exists T∈N such that U(t,τ)D(τ)⊂B(t) for any t-τ≥T. For discrete process Un,m, by Theorem 9, there exist k0,l0>0 such that (25)dist⁡Un,mBm,Mn≤k0e-l0n-m.For any t,τ∈R, there exist t0,τ0∈[0,1) such that t=n+t0, τ=m+τ0; therefore(26)dist⁡Ut,τDτ,Mt=dist⁡Un+t0,m+τ0Dm+τ0,Mn+t0≤dist⁡Un+t0,nUn,m+τ0Dm+τ0,Un+t0Mn≤rtdist⁡Un,m+τ0Dm+τ0,Mn≤rtdist⁡Un,m+T+1Um+T+1,m+τ0Dm+τ0,Mn≤rtdist⁡Un,m+T+1Bm+T+1,Mn≤rtk0e-l0n-m-T-1,dist⁡Ut,τDτ,Mt≤rtk0e-l0t-t0-τ+τ0-T-1≤rtk0e-l0-t0+τ0-T-1e-l0t-τ≤rtk0el0T+2e-l0t-τ.We obtain that M(t)∣t≤R attracts Dt exponentially in a D-pullback sense.
((II)⇒(I)) The proof is the same as that of Theorem 9, so we omit it.We now present a method to verify that the measure of noncompactnessD-pullback decays exponentially for the process Ut,τ.LetX be a uniformly convex Banach space; that is, for all ε>0, there exists δ>0 such that, given x,y∈X, x≤1, y≤1, x-y>ε; then x+y/2<1-δ. Requiring a space to be uniformly convex is not a severe restriction in application, since this property is satisfied by all Hilbert spaces, the Lp space with 1<p<∞, and most Sobolev spaces Wk,p with 1<p<∞.Definition 11 (enhanced flattening property).
LetX be a uniformly convex Banach space; for a family of bounded sets Bt⊂X, there exist k,l,T>0, and for any finite dimension subspace X1 of X, such that(i)
Pm(⋃t-τ≥s‍U(t,τ)B(τ)) is bounded;(ii)
(I-Pm)⋃t-τ≥s‍U(t,τ)x≤ke-l(t-τ)+k(t,m), ∀x∈B(τ),for all s≥T. Here · denote the norm in X and k(t,s) is real-valued function satisfying(27)lims→+∞⁡kt,s=0.Theorem 12.
Assume that the processUt,τ satisfies the enhanced flattening property; then the measure of noncompactness D-pullback decays exponentially for Ut,τ.Proof.
For anyBt∈D, from (2) and (7) of Lemma 5, and the enhanced flattening property, we get (28)α⋃t-τ≥TUt,τBτ≤αPm⋃t-τ≥TUt,τBτ+αI-Pm⋃t-τ≥TUt,τBτ=αI-Pm⋃t-τ≥TUt,τBτ≤ke-lt-τ+kt,m.Since k(t,m)→0, for ε0=ke-l(t-τ), there exists M>0, for any m>M; we have(29)kt,m<ke-lt-τ.Hence, α(⋃t-τ≥TU(t,τ)B(t))≤2ke-l(t-τ); that is, the measure of noncompactness of Ut,τD-pullback decays exponentially.
LetR be the set of all functions r(t):R→(0,+∞) such that limt→-∞tβeλtr2(t)=0 for some β≥0, λ>0, and denote by D the class of all families D=D(t)∣t∈R⊂B(X) such that D(t)⊂B¯(r(t)) for some r(t)∈R, B¯(r(t)) denote the closed ball in X with radius r(t).Theorem 13.
Assume that the processUt,τ satisfies (30)Ut,τuτ2≤K0t-τβe-λt-τuτ2+K1+K2eαtfor some β≥0, 0<α<λ, and t-τ≥T′ and for any t≤R; then the process Ut,τ has a family of positively invariant D-pullback bounded absorbing sets B(t)∣t≤R; that is, for any D∈D, there exists T>0 such that U(t,τ)D(τ)⊂B(t) for any t-τ≥T and U(t,τ)B(τ)⊂B(t).Proof.
Let us define(31)Dt=x∈X∣x≤2K1+K2eαt,t≤R.For every D0t∈D, there exists T0>0 such that (32)Ut,t-sD0t-s⊂Dt,s≥T0,t≤R.Obviously, Dt is a family of D-pullback bounded absorbing sets. Moreover, there exists T>0 such that (33)Ut,t-sDt-s⊂Dt,s≥T,t≤R.Note that these can not hold for any t∈R. Let (34)Bt=⋃s≥TUt,t-sDt-s¯,t≤R.We know that B(t)⊂D(t) and Bt is also a family of D-pullback bounded absorbing sets. We also have(35)Ut,τBτ=⋃s≥TUt,t-t-τ+sDt-t-τ+s¯⊂⋃s≥TUt,t-sDt-s¯=Bt.By Theorems10–13, we get the following theorem.Theorem 14.
LetX be a uniformly convex Banach space; Ut,τ is a process on X, and the process Ut,τ satisfies the following:(I)
U(t,τ)uτ2≤K0(t-τ)βe-λ(t-τ)uτ2+K1+K2eα|t| for some β≥0, 0<α<λ, and t-τ≥T and any t≤R.(II)
(I-Pm)(⋃t-τ≥sU(t,τ)x)≤ke-l(t-τ)+k(t,m), ∀x∈Bτ=x:x≤2K1+K2eατ}, for all s≥T. Here m is the dimension of subspace X1 of X, and k(t,s) is real-valued function that satisfies(36)lims→+∞⁡kt,s=0.(III)
U(t,τ)x-U(t,τ)y≤r(t)x-y, for any t-τ<1, x,y∈B(τ); then the process Ut,τ has a weak D-pullback exponential attractor; that is, for any R∈R, there exists a family of sets M(t)∣t≤R satisfying the following:
(1)
M(t) is compact.(2)
M(t) is positively invariant; that is, U(t,τ)M(τ)⊂M(t).(3)
M(t) attracts Dt exponentially in a D-pullback sense; more precisely,(37)dist⁡Ut,τDτ,Mt≤βte-lt-τ,foranyDt∈D.


## 4. Application to Nonautonomous Reaction Diffusion Equation

As an application of Theorem14, we prove the existence of the weak D-pullback exponential attractor in H01(Ω) for the process generated by the solution of the following nonautonomous reaction diffusion equation: (38)ut-▵u+fu=gt,x∈Ω,u∂Ω=0,uτ=uτ,where f∈C1(R,R), g(·)∈Lloc⁡2(R,L2(Ω)), Ω is a bounded open subset of Rn, and there exist p≥2,  ci>0,  i=1,…,5, l>0 such that(39)c1up-c2≤fuu≤c3up+c4,(40)f′u≥-l,f′u≤c51+up-2for all u∈R.We setA:=-▵, naming λ the first eigenvalue of A, and denote H=L2(Ω) by scalar product (·,·) and norm ·; let ((·,·)) and · denote the scalar product and norm of H01(Ω) and ((u,v))=∫Ω‍∇u∇vdx for all u,v∈H01(Ω). Moreover, we suppose for any t∈R that there exist M≥0 and 0≤α<λ such that(41)gt2≤Meαt.For this initial boundary value problem, we know from [7, 8] that, for any τ,T∈R,  T>τ, there exists a unique solution u(·)∈C([τ,T];H)∩L2(τ,T;H01(Ω))∩Lp(τ,T;Lp(Ω)).Thanks to the existence theorem, the initial boundary value problem is equivalent to a processUt,τt≥τ defined by(42)Ut,τ:H×τ,+∞⟶H01ut=Ut,τuτ,where u(t) is the solution of (38)–(40) with uτ as initial data at time τ.Theorem 15 (see [3]).
Assume thatf and g satisfy (39)–(41) and u(t) is a weak solution associated with (38). Then the following inequality holds for t>τ:(43)ut2≤c1+t-τ+1t-τe-λt-τuτ2+1+1t-τ+1+1t-τe-λt∫-∞teλsgs2ds+1+1t-τe-λt∫-∞t∫-∞seλrgr2drds.Theorem 16.
Assume thatf and g satisfy (39)–(41), where 2≤p<+∞(n≤2), 2≤p≤2n-2/n-2 (n≥3). Then the process defined by (42) has a weak D-pullback exponential attractor in H01.Next, we will prove that the process defined by (42) satisfy (I)–(III) of Theorem 14.Proof.
By (42), for t≤0, (44)e-λt∫-∞teλsgs2ds≤Me-λt∫-∞teλse-αsds≤Me-αtλ-α,and, for t>0,(45)e-λt∫-∞teλsgs2ds=e-λt∫-∞0eλsgs2ds+∫0teλsgs2ds≤M1λ-α+eαtλ+α.Therefore, for any t∈R, we have (46)e-λt∫-∞teλsgs2ds≤Mλ-α1+eαt.Using the same proof, we can get (47)e-λt∫-∞t∫-∞seλrgr2drds≤Mλ-α21+eαt.By (43) and using (46) and (47) we find that there exists T>0, for any t-τ≥T; we have (48)Ut,τuτ2≤K0t-τe-λt-τuτ2+K1+K2eαt.
By Theorem13, for any fixed R∈R, the process {Ut,τ} generated by (38) is a family of positively invariant D-pullback bounded absorbing sets B(t)∣t≤R and for any x∈B(t), x≤2K1+K2eαt.
LetR be the set of all functions r:R→(0,+∞) such that limt→-∞⁡teλtr2(t)=0 and denote by D the class of all families D=Dt∣t∈R⊂B(H) such that D(t)⊂B¯(r(t)) for some r(t)∈R, B¯(r(t)) denotes the closed ball in H with radius r(t).
SinceA-1 is a continuous compact operator in H, by the classical spectral theorem, there exist a sequence λjj=1∞, (49)0<λ1≤λ2≤⋯≤λj≤⋯,λj⟶+∞,asj⟶∞,and a family of elements ejj=1∞ of H01(Ω) which are orthogonal in H such that(50)Aej=λjej,∀j∈N.Let Hm=span⁡{e1,e2,…,em} in H and P:H→Hm is an orthogonal projector. For any u∈H we write(51)u=Pu+I-Pu≜u1+u2.
We setu1(t)=U(t,τ)u1τ and u2(t)=U(t,τ)u2τ to be solutions associated with (38) with initial data u1τ,u2τ∈B(τ). Let w(t)=u1(t)-u2(t); by (38) we get (52)wt-▵w+fu1t-fu2t=0.
Taking inner product of (52) with -▵w in H, we have(53)12ddtw2+▵w2+fu1-fu2,-▵w=0.
Taking into account (40) and Hölder inequality, it is immediate to see that (54)fu1-fu2,-▵w≤∫Ωfu1-fu2▵wdx≤12▵w2+12∫Ωfu1-fu22dx,∫Ωfu1-fu22dx=∫Ωf′u1+θu2-u12u1-u22dx≤c∫Ω1+u1p-2+u2p-22u1-u22dx≤c∫Ω1+u12p-1+u22p-1dxp-2/p-1∫Ωu1-u22p-11/p-1≤c1+u12p-12p-2+u22p-12p-2w2p-12.Since 2≤p<∞(n≤2), 2≤p≤n/n-2+1(n≥3), using Sobolev embedding theorem, we obtain (55)∫Ωfu1-fu22dx≤c1+u12p-2+u22p-2w2≤cw2.Since u1(t),u2(t)∈B(t), we get (56)uit2≤2K1+K2eαt,i=1,2.From (53)–(56), we have (57)ddtw2≤c1+eαtw2.Therefore (58)wt2≤ec1+∫t-1teαsdswτ2,t-τ<1.We obtain (59)Ut,τu1τ-Ut,τu2τ2≤ec1+∫t-1teαsdsu1τ-u2τ2.
For anyu∈H we write(60)u=Pmu+I-Pmu≜u1+u2.Taking the inner product of (38) with -▵u2, we have(61)12ddtu22+▵u22+fu,-▵u2=gt,-▵u2.Applying the Poincaré inequality and Hölder inequality, we get(62)ddtu22+λmu22≤2∫Ωfu2dx+gt2.By (40), we find(63)fu≤cup-1+1.Hence,(64)ddtu22+λmu22≤c1+∫Ωus2p-2dx+gt2.Thanks to Sobolev embedding theorem, we obtain(65)ddtu22+λmu22≤c1+u2p-2+gt2.Since u(t)∈B(t), hence (66)u2p-2≤c1+eαp-1t.By (41), we get(67)ddtu22+λmu22≤c1+eαp-1t+eαt.Using Gronwall lemma, we have(68)u2t2≤e-λmt-τu2τ2+ce-λmt∫-∞teλms1+eαp-1s+eαsds≤ce-λm-λt-τe-λt-τuτ2+1λm+1λm-αp-1+1λm-α+eαp-1tλm-αp-1+eαtλm-α.Since e-λ(t-τ)uτ2→0 for any uτ∈B(τ) and λm→+∞, which imply that there exists T>0, for any t-τ≥T, we have (69)u2t2≤ce-λm-λt-τ+kt,m.Here k(t,m)=1/λm+1/λm-α(p-1)+1/λm-α+eα(p-1)|t|/λm-α(p-1)+eα|t|/λm-α; obviously k(t,m)→0 as m→+∞.
Therefore, by (48), (59), and (69), the process Ut,τt≥τ generated by (38) satisfies all the conditions of Theorem 14.



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*Source: 1871602-2016-03-22.xml*