# Pullback Attractors for Nonautonomous Degenerate Kirchhoff Equations with Strong Damping

Authors: Honglv Ma; Jing Wang; Jun Xie
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we obtain the existence of pullback attractors for nonautonomous Kirchhoff equations with strong damping, which covers the case of possible generation of the stiffness coefficient. For this purpose, a necessary method via “the measure of noncompactness” is established.


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## Body

## 1. Introduction

LetΩ⊂ℝn be a bounded domain with smooth boundary ∂Ω. We consider the following Kirchhoff wave model with strong damping:
(1)utt−Δut−ϕ∇u2Δu+fu=hx,t,inΩ×τ,∞,u∂Ω=0,ux,τ=uτ0x,utx,τ=uτ1x,x∈Ω,τ∈ℝ,wherehx,t is a time-dependent external force term, uτ0 and uτ1 are initial data, and ϕ and f are nonlinear functions specified later.To describe small vibrations of an elastic stretched string, Kirchhoff [1] introduced the equation
(2)ρh∂2u∂t2=p0+Eh2L∫0L∂u∂x2dx∂2u∂x2+g,whereu=ux,t is the lateral deflection, 0<x<L the space coordinate, t≥0 the time, E the Young’s modulus, ρ the mass density, h the cross-section area, L the length, p0 the initial axial tension, and g the external force. It has been called the Kirchhoff equation since then. In general, we call the Kirchhoff equation nondegenerate if the stiffness ϕ satisfies the strict hyperbolicity condition ϕs≥c>0 and degenerate if ϕs≥0 on ℝ+. Obviously, the degenerate stiffness coefficient ϕs in (1) corresponds to the case that the initial axial tension equals zero.From the mathematical point of view, global existence of the model like (2) has been proven in a multitude of special situations in Ω⊂ℝn. We refer to [2–5] for the analytic data, [6–9] for the dispersive equations and small data, and [10–15] for the weak damped equations.Introducing the strong damping term−Δut provides an additional a priori estimate. Certainly, from the physical point of view, the dissipative plays an important spreading role for the energy gathered arising from the nonlinearity in a real process. Concerning Kirchhoff equations with strong dissipation, the first result on the well-posedness we are aware of was obtained by Nishihara [16]. He proved the global existence of the solution for the model utt−Δut−m∇uΔu=0. In recent years, many mathematicians and physicists paid their attentions to this type of problem and obtained the well-posedness under different types of hypotheses, such as the absent source term [17] and the subcritical source term [18–23]. In general, the exponent p∗=n+2/n−2+ is called to be critical when someone studies the problem in H01Ω×L2Ω. Assuming the stiffness factor is nondegenerate (ϕs≥ϕ0>0), References [18–24] also proved the existence of the attractor. In the case of possible degeneration of the stiffness coefficient and the case of supercritical source term (p∗<p<n+4/n−4+), the first result on the well-posedness we are aware of is given by Chueshov [25]. However, when he proved the existence of a global attractor for problem (1) in the natural energy space H01Ω∩Lp+1Ω×L2Ω endowed with a partially strong topology (in the sense, if u0n,u1n⟶u0,u1 with a partially strong topology, then u0n,u1n⟶u0,u1 strongly in H01×L2 and u0n⇀u0 weakly in Lp+1), he assumed that
(3)ϕs>0,∀s≥0,ϕ∈C1ℝ+.Under this condition, one can conclude thatϕ∇ut2≥c0>0 if ∥∇ut∥ is bounded for t∈ℝ+. Recently, Ma et al. [26] proved the existence of the global attractor in the case of degeneration for the autonomous Kirchhoff system.The pullback attractor is a basic concept to study the longtime dynamics of nonautonomous evolution equations (see [27–32] and references therein). It is worth mentioning that there are only a few recent results devoted to the pullback attractor for nonautonomous systems like (1). In 2013, Wang and Zhong [33] investigated the upper semicontinuity of pullback attractors for problem (1) with ϕs=1+εs (ε>0) and ∣f′u∣≤Cu2/n−4++1n≥3. Recently, Li and Yang [34] studied the robustness of pullback attractors with ϕ′s≥0,ϕ0=ϕ0>0. We notice that all these publications assume that the stiffness factor is nondegenerate, or more precisely, ϕ0>0 and ϕ is nondecreasing.In this paper, we consider the problem (1) under the degenerate hyperbolicity condition ϕs≥0. We do not assume that ϕ is monotone and allow ϕ0=0, such as ϕs=bsγ (degenerate and monotone) or ϕs=1+sin2ssγ (degenerate and nonmonotone) with γ≥1. Based on the result in [25, 26], we prove the existence of pullback attractors in H01Ω×L2Ω if ϕ is really degenerate. To overcome the difficulties caused by the degeneration, we first established a method (condition (D-PC)) via “the measure of noncompactness” (some ideas come from [35, 36]) to prove that the process is pullback D-asymptotically compact.The paper is organized as follows. In Section2, we introduce some preliminaries and establish a necessary abstract result (see Theorem 5). In Section 3, we discuss the existence of pullback attractors for the equation (1) (see Theorem 12).


## 2. Preliminaries

In this section, we will give some notations and results. As usual, we denote by∥·∥ and ·,· the norm and the inner product in L2Ω, respectively. Let H=H01Ω×L2Ω. We define the norms in H by u0,u1H2=∇u02+u12.LetX be a Banach space and Ut,τ be a process acting on X. In the following, we recall some definitions and results related to the pullback attractors; more details can be found in [27, 29, 33].Definition 1.
A family of compact setsA=Att∈ℝ is said to be a pullback attractor for process U·,· if
(i)A is invariant, that is, Ut,τAτ=At, for all t≥τ
(ii)A is pullback attracting, i.e., dUt,t−τB,At⟶0,asτ⟶+∞, for all bounded subset B of X, where dB,A is the Hausdorff semidistanceDefinition 2.
A family of setsD=Dtt∈ℝ is said to be a pullback absorbing family for process U·,·, if for all t∈ℝ and all bounded B⊂X, there exists T=Tt,B>0, such that Ut,t−τB⊂Dt, for all τ≥T. In addition, the family D is said to be pullback D-absorbing, if for any t∈ℝ, there exists Tt>0 such that Ut,t−τDt−τ⊂Dt for τ≥Tt.Definition 3.
A processU·,· is said to be pullback D-asymptotically compact in X, if for any t∈ℝ, any sequences τn⟶∞ and xn∈Dt−τn; the sequence Ut,t−τnxnn∈ℕ is relatively compact in X.Lemma 4 (see [29]).
Let the familyD=Dtt∈ℝ be pullback absorbing and U·,· be continuous and pullback D-asymptotically compact in X. Then, the family A=Att∈ℝ defined by
(4)At=⋂s≥0⋃τ≥sUt,t−τDt−τ¯,
is a pullback attractor forU·,·.To verify the pullbackD-asymptotically compact property in X, it suffices to check the following condition.


### 2.1.D-Pullback Condition (D-PC)

For anyδ>0 and t∈ℝ, there exist τ0=τ0t,D,δ>0 and a finite dimensional space X1 of X such that
(5)PUt,t−τ0Dt−τ0isbounded,I−PUt,t−τ0xX<δ,∀x∈Dt−τ0,whereP:X⟶X1 is a bounded projector.Theorem 5.
Let the familyD=Dtt∈ℝ be a pullback D-absorbing family of the process Ut,τ. If the D-pullback condition (D-PC) holds, then U·,· is pullback D-asymptotically compact in X.Proof.
By Definition3, the result will be proven if we can show that for any t∈ℝ, any sequences τn⟶∞ and xn∈Dt−τn, Ut,t−τnxnn∈ℕ is relatively compact in X.
For everyδ>0, condition (D-PC) implies that there exist τ0=τ0t,D,δ>0 and the finite dimensional space X1, such that (5) holds. Then, we have
(6)γUt,t−τ0Dt−τ0≤γPUt,t−τ0Dt−τ0+γI−PUt,t−τ0Dt−τ0≤γN0,δ≤2δ,
whereγ is the measure of noncompactness defined as
(7)γB=infδ>0∣Badmitsafinitecoverbysetswhosediameter≤δ.
On the other hand, the properties ofD give that there exists Tt−τ0>0, such that for τ≥Tt−τ0, Ut−τ0,t−τ0−τDt−τ0−τ⊂Dt−τ0 and
(8)Ut,t−τ0−τDt−τ0−τ=Ut,t−τ0Ut−τ0,t−τ0−τDt−τ0−τ⊂Ut,t−τ0Dt−τ0.
Then, we can findN0, such that γ⋃n>N0Ut,t−τnxn≤2δ, which means that Ut,t−τnxnn∈ℕ has a finite 4δ-net for any δ>0. The proof is complete.


## 2.1.D-Pullback Condition (D-PC)

For anyδ>0 and t∈ℝ, there exist τ0=τ0t,D,δ>0 and a finite dimensional space X1 of X such that
(5)PUt,t−τ0Dt−τ0isbounded,I−PUt,t−τ0xX<δ,∀x∈Dt−τ0,whereP:X⟶X1 is a bounded projector.Theorem 5.
Let the familyD=Dtt∈ℝ be a pullback D-absorbing family of the process Ut,τ. If the D-pullback condition (D-PC) holds, then U·,· is pullback D-asymptotically compact in X.Proof.
By Definition3, the result will be proven if we can show that for any t∈ℝ, any sequences τn⟶∞ and xn∈Dt−τn, Ut,t−τnxnn∈ℕ is relatively compact in X.
For everyδ>0, condition (D-PC) implies that there exist τ0=τ0t,D,δ>0 and the finite dimensional space X1, such that (5) holds. Then, we have
(6)γUt,t−τ0Dt−τ0≤γPUt,t−τ0Dt−τ0+γI−PUt,t−τ0Dt−τ0≤γN0,δ≤2δ,
whereγ is the measure of noncompactness defined as
(7)γB=infδ>0∣Badmitsafinitecoverbysetswhosediameter≤δ.
On the other hand, the properties ofD give that there exists Tt−τ0>0, such that for τ≥Tt−τ0, Ut−τ0,t−τ0−τDt−τ0−τ⊂Dt−τ0 and
(8)Ut,t−τ0−τDt−τ0−τ=Ut,t−τ0Ut−τ0,t−τ0−τDt−τ0−τ⊂Ut,t−τ0Dt−τ0.
Then, we can findN0, such that γ⋃n>N0Ut,t−τnxn≤2δ, which means that Ut,t−τnxnn∈ℕ has a finite 4δ-net for any δ>0. The proof is complete.


## 3. Existence of Pullback Attractors

In this section, we will prove the existence of the pullback attractor whenϕs is really degenerate and fu is subcritical. We assume that f, ϕ, and h satisfy the following conditions.Assumption 6.
The functionϕ∈C1ℝ+, ϕs≥minL1sα,L2 for s∈ℝ+, and some constants α≥0, L1,L2>0. Moreover, there exists δ0>0 such that
(9)liminfs⟶+∞sϕs−δ0Φs>−∞,
whereΦs=∫0sϕtdt.Assumption 7.
fu is a C1 function, f0=0, f′s≥−c1, and s∈ℝ,
(10)μf=liminf∣s∣⟶+∞fss>−λ1ϕ∞withϕ∞=liminfs⟶+∞ϕs,and the following properties hold:
(i) ifn=1, then f is arbitrary
(ii) ifn=2, then
(11)∣f′u∣≤C1+∣up−1with1≤p<∞,
(iii) ifn≥3, then
(12)f′u≤C1+∣up−1with1≤p<p∗=n+2n−2,
wherec1 and C are positive constants and λ1 is the first eigenvalue of −Δ.Assumption 8.
h,∂th∈Lloc2ℝ,L2Ω and
(13)∫−∞th·,s2ds<+∞,∀t∈ℝ.Remark 9.
(1)ϕs=L1sα or ϕ~s=1+sin2ssαα≥1 satisfies Assumption 6. It indicates that we include into the consideration the case of possibly degenerate ϕ since ϕ0=0. Moreover, because ϕ∞=+∞ in this case, μf>−λ1ϕ∞ becomes μf>−∞. If α=0, then ϕs is a constant, and equation (1) is the nonlinear wave equation with strong damping.
(2) Assumptions6 and 7 imply that there exist constants c0>0, θ1>0 with 0<ϕ1<ϕ∞, 0<ϕ1λ1−θ1<<1 such that
(14)Φs≥ϕ1·s−c0ϕ1∀s∈ℝ+,(15)Fs≥−θ12s2−c2,fss≥−θ1s2−c2,fss−Fs≥−c12s2,∀s∈ℝ,
whereFs=∫0sftdt.
The well-posedness of the problem(16)∂ttu−σ∇u2Δ∂tu−ϕ∥∇u∥2Δu+fu=hx,inΩ×0,∞,u∂Ω=0,ux,0=u0x,utx,0=u1x,x∈Ω,
has been established by Chueshov [25] in the autonomous case. Noticing that the conditions of ϕ,f are more general than the above Assumptions 6–8, we can obtain the following Proposition 10 by a similar argument as in [25], except for the treatment of hx,t. The reader is referred to the Appendix for a detailed proof of these facts.Proposition 10.
Let Assumptions6–8 be in force. Then, for τ,T∈ℝτ<T and uτ0,uτ1∈H, problem (1) has a unique weak solution u with u,ut∈Cτ,T;H and
(1) for everyt∈τ,T, there exists C=CR,τ,T>0 such that
(17)utt2+∇ut2+∫τt∇uts2ds≤C,(18)Eut,utt+2∫st∇utr2−h,utdr=Eus,uts,τ≤s<t,
whereEu0,u1=u12+Φ∇u02+2∫ΩFu0dx, uτ0,uτ1H≤R
(2) for everyt∈τ,T, there exists K=KR,τ,T>0 such that
(19)utttH−12+∇utt2≤K1+1t−τ21+∫τT∥h·,s∥2+∥ht·,s∥2ds.
(3) the Lipschitz stability(20)zt,zttH2≤Kzτ,ztτH2,
holds forzt=u1t−u2t, where u1,u2 are two weak solutions of problem (1) with initial data ui,τ0,ui,τ1, ui,τ0,ui,τ1H≤R,i=1,2.We define the solution operatorUt,τ:H⟶H associated to problem (1) as
(21)Ut,τuτ0,uτ1=ut,utt,∀t≥τ,τ∈ℝ,whereu is the weak solution of problem (1) corresponding to initial data uτ0,uτ1∈H. Then, we know from Proposition 10 that Ut,τ:H⟶H is a continuous evolution process. For convenience, we denote by ξut=ut,utt for any function ut. As uτ,utτ=uτ0,uτ1, we also denote uτ0,uτ1 by ξuτ.Lemma 11.
Let Assumptions6–8 be valid. Then, the process U·,· defined in (21) has a pullback D-absorbing family D=Dtt∈ℝ. Moreover, Dt is bounded in H1=H01Ω×H01Ω for every t∈ℝ.Proof.
As usual, the argument below can be justified by considering Galerkin approximations. Using the multiplierut+ηu in Equation (1), we have that
(22)ddtWηξut+Kξut=0,t≥τ,where
(23)Wηξut=ut2+Φ∇u2+2Fu,1+η∇u2+2ut,u≥1−ηut2+ϕ1·∇u2−c0ϕ1−θ1u2−2c2·mesΩ+η∇u2−ηu2≥κξutH2−C3,(24)Kξut=2∇ut2−2ηut2+2ηϕ∇u2∇u2+fu,u−2h,ut+ηu,
forη>0 which is small enough, κ>0 is a positive constant, and κ,C3 are independent of ξut.
Since Assumption6 implies that there exists L3>0 such that
(25)δ0Φ∇u2≤ϕ∇u2∇u2+L3,combining with (15), we have that
(26)Wηξut≤1+ηut2+Φ∇u2+2fu,u+c1u2+η∇u2+u2≤κ1ξutH2+Φ∇u2+2fu,u≤κ1ξutH2+Φ∇u2+C3uLp+1p+1+C4,(27)Kξut≥∇ut2+λ1−2ηut2+2η−εϕ∇u2∇u2+εδ0Φ∇u2−L3+2ηfu,u−δut2−δη2u2−2δh·,t2.
Then, we can findη>0,ε>0,δ>0 small enough such that
(28)Kξut−δWηξut≥∇ut2+λ1−2η−δ1+η−δut2+2η−εϕ∇u2∇u2+εδ0Φ∇u2−εL3+2η−δfu,u−c1δu2−δΦ∇u2−δη∇u2−δη2+ηu2−2δh·,t2≥∇ut2+2η−εϕ1·∇u2−c0ϕ1+εδ0−δΦ∇u2−2ηθ1λ1∇u2−δη∇u2−δc1+η2+ηu2−2δh·,t2−C4≥∇ut2−C1+h·,t2.
By (22) and (28), we get that
(29)ddtWηξut+δWηξut+∇ut2≤C1+h·,t2.
According to the Gronwall inequality, we have(30)Wηξut≤Wηξuτe−δt−τ+C1+∫τth·,s2ds.
Then, (23), (26), and H01Ω°Lp+1Ω yield that
(31)ξutH2≤1κWηξuτe−δt−τ+C1+∫τth·,s2ds≤κ1κξuτH2+1κΦ∇u2+C3κuLp+1p+1+C4κe−δt−τ+C1+∫τth·,s2ds≤CξuτH2+Φ∇u2+∇up+1e−δt−τ+C1+∫τth·,s2ds≤CξuτH2+ΦξuτH2+ξuτH2p+1/2e−δt−τ+C1+∫τth·,s2ds≜QξuτH2e−δt−τ+C1+∫τth·,s2ds,
whereQx=Cx+Φx+xp+1/2>0 is a monotone positive function on ℝ+. Let
(32)D0t=ξ∈H∣ξH≤Rt,withR2t=2C1+hL2−∞,t;L22,t∈ℝ.
Obviously,D0=D0tt∈ℝ is a pullback absorbing family of the process Ut,τ in H. Moreover, for every t∈ℝ, there exists a Tt>0 such that
(33)Ut,t−τD0t−τ⊂D0t,Ut−1,t−τD0t−τ⊂D0t−1,forτ≥Tt.
LetDt=⋃τ≥TtUt,t−τD0t−τ¯H. By a standard procedure (see, e.g., Theorem 3.1 of [34]), we know that D=Dtt∈ℝ is a pullback absorbing family. Moreover, Dt is bounded in H1 for every t∈ℝ, and there exists a Tt>0 such that Ut,t−τDt−τ⊂Dt for τ≥Tt.For simplicity, we assume thatα>0 and L1=L2=1 in the following.Theorem 12.
Let Assumptions6–8 be in force. Then, the process U·,· possesses a pullback attractor A=Att∈ℝ as shown in (4). Moreover, At is bounded in H1 for every t∈ℝ.Proof.
According to Lemma4, Theorem 5, Lemma 11, and the continuity of Ut,τ:H⟶H, it suffices to show that Ut,τ satisfies the condition (D-PC). Let ejj=1∞ be an orthonormal basis and λjj=1∞ be the corresponding eigenvalues of L2Ω which consists of eigenvectors of −Δ, i.e., −Δej=λjej,j∈ℕ. Let Vm×Wm=spane1,⋯,em×spane1,⋯,em in H and Pm=Pm1,Pm2:H⟶Vm×Wm be an orthogonal projector. Denote Qm=I−Pm, u=Pm1u+Qm1u=u1+u2, and ξuτt=ut,utt=Ut,τuτ0,uτ1 with uτ0,uτ1∈Dτ, t≥τ.
Letε>0 and t0∈ℝ be given. Without loss of generality, we assume ε<1/4.
For everyτ≥1 and every ut0−τ0,ut0−τ1∈Dt0−τ, let
(34)u,utt=ξut0−τt=Ut,t0−τut0−τ0,ut0−τ1∈Ut,t0−τDt0−τ⊂Ut,t0−τD0t0−τ.
DenoteZt=1/2ut2+u2+∇u2. It is easy to see that
(35)Zt0−τ+1≤121+1λ1ξut0−τt0−τ+1H2≤121+1λ1Qξut0−τt0−τH2·e−δ+C1+∫−∞t0−τ+1h·,s2ds.
Sinceξut0−τt0−τ=ut0−τ0,ut0−τ1∈Dt0−τ⊂D0t0−τ, we find
(36)ξut0−τt0−τH≤Rt0−τ≤Rt0,∀τ≥0.
Thus,(37)Zt0−τ+1≤121+1λ1QR2t0e−δ+C1+∫−∞t0h·,s2ds≜C5t0,∀τ≥1,
whereC5t0 is independent of τ. Then, there exists τ0>1 such that
(38)Zt0−τ0−1e−2ε2ατ0−1<ε22.
On the other hand, for everyut0−τ00,ut0−τ01∈Dt0−τ0, using (16) and (18), we get that
(39)∇ut2+∇utt2+utttH−12≤K0,fort∈t0−τ0+1,t0,
whereut,utt=Ut,t0−τ0ut0−τ00,ut0−τ01. Using H01Ω°LqΩ (2≤q≤p∗=2n/n−2), one can find M≥1,L0<t0 (without loss of generality, we assume L0<0), such that for every t∈t0−τ0+1,t0,
(40)futLp+1/p+∫tt+1uttsH−12ds1/2+∫−∞t0h·,s2ds1/2<M,∫−∞L0h·,s2ds<ε24.
By the Sobolev embedding theorem, we know that the embeddingH01Ω°L2Ω is compact. Then, the boundedness of ut,uttt∈t0−τ0+1,t0 in H01Ω×H01Ω implies that ut,uttt∈t0−τ0+1,t0 is compact in L2Ω×L2Ω. Therefore, for ε1=ε2+2α/4M1+t0−L0, there exists m0∈ℤ+, such that for every t∈t0−τ0+1,t0,
(41)u2t,ut2tL2×L2<ε1,(42)u2Lp+1≤u2θ·u2Lp∗1−θ≤Cu2θ·∇u21−θ≤C6u2θ<ε12M,
whereu=Pm01u+I−Pm01u≜u1+u2, ut=Pm02ut+I−Pm02ut≜ut1+ut2, and 1/p+1=θ/2+1−θ/p∗.Now, we will consider two situations. Without loss of generality, we assume0<ε<1/3.Case 1.
For everyut0−τ00,ut0−τ01∈Dt0−τ0, the inequality
(43)∇ut>ε,holds for any t∈t0−τ0+1,t0, where u,utt=ξut0−τ0t.Multiplying (1) by u2, we have that
(44)ddtut2,u2+12∇u22+ϕ∇ut2∇u22≤ut22+fu,u2+h·,t,u2.LetYt=ut2,u2+1/2∇u22. Since ϕ∇u2≥min∇u2α,1≥minε2α,1=ε2α in this case, the above inequality implies that
(45)ddtYt+2ε2αYt≤2ε2αut2,u2+ut22+fu,u2+h·,t,u2≤2ε2α·ut2·u2+ut22+fuLp+1/p·u2Lp+1+h·,t·u2≜Wεt.By Gronwall’s inequality, we obtain that(46)Yt≤Yt−τ−1e−2ε2ατ−1+e−2ε2αt∫t−τ−1te2ε2αsWεsds.Since(47)Yt0−τ+1≤Zt0−τ+1≤C5t0,(37) yields that
(48)Yt0−τ0−1e−2ε2ατ0−1<ε22.Combining (45), we have
(49)Yt0≤Yt0−τ0+1e−2ε2ατ0−1+e−2ε2αt0∫t0−τ0+1t0e2ε2αsWεsds≤ε22+e−2ε2αt0∫t0−τ0+1t0e2ε2αs2ε12ε2α+1ds+e−2ε2αt0∫t0−τ0+1t0e2ε2αs·h·,s·u2sds≤ε22+1+1ε2αε12+e−2ε2αt0∫t0−τ0+1t0e2ε2αs·h·,s·u2sds.IfL0≤t0−τ0+1, by the Hölder inequality, we have that
(50)I1=e−2ε2αt0∫t0−τ0+1t0e2ε2αs·h·,s·u2sds≤e−2ε2αt0·e2ε2αt0ε1·t0−L0·M<ε24.On the other hand, ifL0>t0−τ0+1, we get that
(51)I1≤e−2ε2αt0∫L0t0e2ε2αt0·h·,s·ε1ds+12∫t0−τ0+1L0e2ε2αsu2s2+h·,s2ds<ε1·t0−L0·M+ε124ε2α+ε28<ε22.The above inequalities guarantee thatYt0<9ε2/8. And because
(52)Yt0=ut2t0,u2t0+12∇u2t02≥12∇u2t02−ut2t0·u2t0≥12∇u2t02−ε12,we get that
(53)∇u2t02≤2Yt0+ε12<29ε28+ε28<4ε2,i.e.,∇u2t0<2ε.Case 2.
There existut0−τ00,ut0−τ01∈Dt0−τ0 and t1∈t0−τ0+1,t0 such that
(54)∇ut1≤εwithu,utt=ξut0−τt.
In this case, we claim that the following inequality is true, i.e., for everyt1≤t≤t0,
(55)∇u2t<2ε,foru2=Qm01u.
In fact, if this claim is not true, the continuity of∇u2t gives that
(56)E=t∣t∈t1,t0,∇u2t=2ε,is not an empty set. Let t3=infE. It is easy to prove that ∇u2t3=2ε. Moreover, by the definition of t3, we have that
(57)∇u2t<2ε,∀t∈t1,t3.According to the intermediate value theorem, we know that the set(58)E1=t∣t∈t1,t3,∇u2t=32ε,is not empty. Denoting t2=supE1, we can conclude from the definition of supremum that
(59)∇u2t2=32ε.Thus,(60)32ε<∇u2t<3ε,∀t∈t2,t3,∇u2t2=32ε,∇u2t3=2ε.Notice that∇u≥∇u2 and ∇u2≤1 for t∈t2,t3; we have that ϕ∇u2≥∇u22α. Then, integrating (43) on t2,t3, we have that
(61)ut2t3,u2t3+12∇u2t32−ut2t2,u2t2+12∇u2t22+∫t2t3ϕ∇us2∇u2s2ds≤∫t2t3uts22+fus,u2s+h·,s,u2sds.It implies that(62)∇u2t32+2∫t2t3∇u2s2+2αds≤∇u2t22−2ut2t3,u2t3+2ut2t2,u2t2+2∫t2t3uts22+fus,u2s+h·,s,u2sds.Combing (40), (41), and (59),we get
(63)∇u2t32+232ε2α+2t3−t2≤∇u2t32+2∫t2t3∇u2s2α+2ds≤∇u2t22+2ut2t2·u2t2+2ut2t3·u2t3+2∫t2t3ut2s2+fu,u2+h·,s,u2ds≤32ε2+4ε12+2ε12t3−t2+2∫t2t3fuLp+1/p∥u2Lp+1ds+2ε1∫t2t3h·,s2ds1/2t3−t2≤32ε2+4ε12+2ε12t3−t2+2∫t2t3M·ε12Mds+2ε1Mt3−t2≤32ε2+4ε12+4ε12t3−t2+ε1Mt3−t2+ε1M≤94ε2+ε4+4α4+ε4+4α4t3−t2+ε2+2α4t3−t2+ε2+2α4<114ε2+12ε2+2αt3−t2.Thus,∇u2t32<11/4ε2, which is in contradiction with (59), and condition (D-PC) holds. This completes the proof.



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*Source: 7575078-2021-12-20.xml*