# L p Estimates for Weak Solutions to Nonlinear Degenerate Parabolic Systems

Authors: Na Wei; Xiangyu Ge; Yonghong Wu; Leina Zhao
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This paper is devoted to theLp estimates for weak solutions to nonlinear degenerate parabolic systems related to Hörmander’s vector fields. The reverse Hölder inequalities for degenerate parabolic system under the controllable growth conditions and natural growth conditions are established, respectively, and an important multiplicative inequality is proved; finally, we obtain the Lp estimates for the weak solutions by combining the results of Gianazza and the Caccioppoli inequality.


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## Body

## 1. Introduction

TheLp estimates of the weak solutions to elliptic and parabolic systems under the controllable growth conditions and natural growth conditions in the Euclidean space Rn were well studied (see [1–3]). Roughly speaking, if the weak solution is u∈W1,2(Ω,RN) with Ω⊂Rn being an open set, then there exists an index p>2 such that u∈Wloc1,p(Ω,RN). This result plays an important role in the study of the Hölder partial regularities. However, the index p cannot be arbitrarily large. Folland established the local Lp regularity and local Schauder estimates of the sub-Laplace operator structured on the vector fields of homogeneous group (satisfying Hömander’s rank condition) in [4]. Rothschild and Stein [5] proved the local Lp estimates of the square sum operators structured on the smooth Hörmander’s vector fields, through establishing the lifting and approximating theory of Hörmander’s vector fields and the local theory of the second-order invariant differential operator on the homogeneous group. Then, the estimates of weak solutions to the linear and nonlinear subelliptic equations and degenerate parabolic equations stemmed from the noncommutative vector fields that caused extensive concern; see [6–13]. In particular, Gianazza obtained in [11] the Lp estimates of the derivatives of the weak solutions to nonlinear diagonal equations related to the Hörmander vector fields. Recently, the Lp estimates of weak solutions to nonlinear subelliptic systems related to Hörmander’s vector fields under the superquadratic controllable growth condition and natural growth condition were studied in [14] and optimal partial regularity for subelliptic systems related to Hörmander’s vector fields was obtained in [15]. There are also some papers related on regularity for weak solutions to the quasilinear degenerate elliptic systems and degenerate elliptic systems with VMO coefficients were studied in [16, 17]. Those works motivated us to study the interesting question: can we get the Lp estimates of weak solutions to the nonlinear degenerate parabolic systems structured on Hörmander’s vector fields?LetΩ⊂Rn be a bounded and open set. We consider the nonlinear degenerate parabolic systems related to Hömander’s condition(1)∂tuα-∑i=1kXi∗Aiαt,ξ,u,Xu=Bαt,ξ,u,Xu,α=1,…,N,under controllable growth conditions and natural growth conditions, where the controllable growth conditions are(2)Aiαt,ξ,u,PPiα≥λP2-Λur-f2,f∈Ls1Ω,Aiαt,ξ,u,P≤ΛP+ur/2+fiα,0≤fiα∈Ls2Ω,Bαt,ξ,u,P≤ΛP21-1/r+ur-1+fα,0≤fα∈Ls3Ω,with s1>2,s2>2,s3>r/r-1, and r=2∗=2Q+2/Q being a Sobolev critical exponent; and the natural growth conditions are(3)Aiαt,ξ,u,PPiα≥λP2-f2,f∈Ls1Ω,(4)Aiαt,ξ,u,P≤ΛP+fiα,0≤fiα∈Ls2Ω,(5)Bαt,ξ,u,P≤ΛP2+fα,0≤fα∈Ls3′Ω,with s3′>1.This paper is organized as follows: in Section2, we introduce some notation, definitions, and basic facts. In Section 3, we state our main results. Section 4 is devoted to some important lemmas, including the Sobolev embedding theorem, the Poincaré inequalities, and reverse Hölder’s inequalities for the parabolic case. In particular, we establish an important multiplicative inequality by using the Fefferman-Phong inequality. We prove the main results in Section 5.


## 2. Preliminary

LetΩ⊂Rn be a bounded, open, and path-connected set and Xii=1k a family of C∞ real-valued vector fields defined in a neighborhood of the closure Ω¯ of Ω. For a multi-index α=i1,…,ik, we denote by Xα the commutator(6)Xi1,Xi2,…,Xik-1,Xik,of length l=α. Throughout this paper, we suppose that the vector fields X=X1,…,Xk satisfy Hömander’s condition: there exists some positive integer s such that Xαα≤s span the tangent space of Rn at each point of Ω; that is, (7)rank LieX1,…,Xk≡n.Assume that an admissible pathγ is a Lipschitz curve γ:[a,b]↦Ω such that there exist cit,a≤t≤b satisfying ∑i=1kcit2≤1 and γ′t=∑i=1kcitXiγt. Then we define a metric associated to X1,…,Xk on Ω(8)ϱccξ,η≕min⁡b≥0:∃γ:0,b↦Ω,s.t.γ0=ξ,γb=η.We refer to it as Carnot-Carathéodory metric, and we call it C-C metric for short. It is showed in [18] that ϱcc is a distance.We can define the C-C ball and the C-C sphere as(9)BRξ=Bξ,R=η:ϱccξ,η<R,(10)∂BRξ=∂Bξ,R=η:ϱccξ,η=R,respectively, from (8). In [18], the authors proved that the Lebesgue measure satisfies the doubling property related to the C-C ball: given a bounded set Ω⊂Rn, there are positive constants C1 and R0>0 such that(11)B2Rξ≤C1BRξ,for ξ∈Ω and 0<R<R0. Let Q=log2⁡C1; the number Q acts as a dimension and is called the local homogeneous dimension related to Ω and the system {Xi}i=1k.We useξ,t to denote a point in Ω×R and ξ∈Ω⊂Rn. We define the parabolic Carnot- Carathéodory distance between the two points in Ω×R as (12)dPξ,t,ξ′,t′=ϱccξ,ξ′2+t-t′.We use the notationQRξ,t=BR(ξ)×(t-R2,t) for parabolic cylinders in Ω×R, where BR(ξ) is defined in (9).For an open setΩ⊂Rn, we set ΩR(ξ)=BR(ξ)∩Ω. For an open set U⊂Rn+1, we let UR(ξ,t)=U∩QR(ξ,t). We write U(t0) for the set of all points (ξ,t0) in U and I(U) for the set of all t such that U(t) is nonempty. We also write U≔UTS for the cylinder (S,T)×Ω, where -∞<S<T<∞ and UT for U when S=0. We denote by W20,1(U) the Hilbert space with the inner product (13)u,vW20,1U≔∫Uuv+∑k=1dXkuXkv.Let W21,1(U) be subspace of W20,1(U) such that ∂tu∈L2(U) and V2(U) are to be the Banach space consisting of all elements of W20,1(U) having a finite norm uV2(U)≔uU, with uU2=XuL2(U)2+supt∈I(U)ut,·L2Ut2. Set V0,2U to be the set of all functions u in V2U vanishes on the lateral boundary ∂lU≔(S,T)×∂Ω of U.From the well-known Sobolev-like embedding theorem (see [19], II,3), we have(14)uLrU≤NQuU,∀u∈V0,2U,r=2Q+2Q.We denoteHp-1(U) to be the space consisting of all functions u satisfying (15)inf⁡FLpU+hLpU∣u=divXF+h<∞,with divXF=∑i=1kXiFi. It is easy to see that Hp-1(U) is a Banach space. For any u∈Hp-1(U), we can define the norm, (16)uHp-1U=inf⁡FLpU+hLpU∣u=divXF+h,and define Hp1(U)=u:u,Xu∈LpU,∂tu∈Hp-1U. Note that H21(U)⊂V2(U).


## 3. Main Results

We callu∈V0,2(UT) the weak solution of the systems (1) if(17)∫Ωuαξ,tφαξ,tdξ0t+∫Ω-uα∂tφα+Aiαt,ξ,u,XuXiφαdξdt=∫0t∫ΩBαt,ξ,u,Xuφαdξdt,for any test functions φ∈W0,21,1(UT) and t∈[0,T].The main results in this paper are as follows.Theorem 1.
(A) Assume thatAiα,Bα satisfy (2), u∈V0,2(UT) is a weak solution to systems (1), and f∈L2(UT), fiα∈L2UT, and fα∈Lr/r-1(UT). Then for any (ξ0,t0)∈Rn+1 and 0<R≤1 with BR/2(ξ0)⊂BR(ξ0)⊂Ω and t0≥R2, one has (18)⨏QR/2ξ0,t0Xu2+urdξdt≤C⨏QRξ0,t0Xuq+urq/2dξdt2/q+C⨏QRξ0,t0f2+f¯2+F2dξdt+Csupt∈t0-R2,t0⁡∫BRξ0u2dξ2/Q⨏QRξ0,t0Xu2dξdt,where f¯=fiα,F=fα1/2r/r-1, C=Cn,λ,Λ,uV2,fL2,fαLr/r-1,fiαL2, q=2(Q+2)/Q+4∈1,2, and 1/r+1/q=1.
(B) Letu∈V0,2(UT) be a weak solution of (1), f,fiα∈Lσ(UT),σ∈(2,∞); fα∈Lτ(UT),τ∈r/r-1,∞. Then there exists p>2 such that u∈Hp1(UT). Moreover, the inequality, (19)uLrp/2ε,T×Ω+XuLpε,T×Ω≤C,is valid for any 0<ε<T, where C=C(n,λ,Λ,σ,τ,u,fLσUT,  fiαLσUT,fαLτUT,ε,T,Ω), p=p(n,λ,Λ,σ,τ,u,fLσUT,  fiαLσUT,fαLτUT).Theorem 2.
(C) Suppose thatAiα,Bα satisfy the natural growth conditions (3)–(5); u∈V0,2(UT)∩L∞(UT) is the weak solution of systems (1), supΩ×(t-R2,t)⁡u=M, 2ΛM<λ (λ is the same as in (3)), and f∈L2(UT), fiα∈L2(UT), fα∈Lr/r-1(UT). Then for any ξ0,t0∈Rn+1 and 0<R≤1 with BR/2(ξ0)⊂BR(ξ0)⊂Ω and t0≥R2, one has (20)⨏QR/2ξ0,t0Xu2dξdt≤C⨏QRξ0,t0Xuqdξdt2/q+C⨏QRξ0,t0f2+f¯2+F2dξdt+Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q⨏QRξ0,t0Xu2dξdt,for R<R0, where f¯=(fiα),F=fαr/2(r-1),C=C(n,λ,Λ,uV2,fL2,fαLr/r-1,fiαL2). R0 depend on M and the L2 norm for Xu.
(D) Letu∈V0,2(UT) be the weak solution of (1) and supΩ×(t-R2,t)u=M, 2ΛM<λ, f,fiα∈Lσ(UT),σ∈(2,∞); fα∈Lτ(UT),τ∈(r/r-1,∞). Then there exists p>2 such that u∈Hp1(UT). Moreover, the inequality, (21)XuLpε,T×Ω≤C,is valid for any 0<ε<T, where C=C(n,λ,Λ,σ,τ,u,fLσUT, fiαLσUT, fαLτUT,ε,T, Ω,M), and p=p(n,λ,Λ,σ,τ,u,fLσUT,  fiαLσUT,fαLτUT,M).


## 4. Some Lemmas

Lu established Pioncaré’s inequality in [20]; we can similarly derive the following lemma by replacing the C-C ball BR with the parabolic cylinder QR.Lemma 3.
There existC and R such that for any u∈Wq0,1(QR(ξ0,t0)), 1<q<Q,1≤p≤qQ/Q-q(22)⨏QRξ0,t0u-u¯tpdξdt1/p≤CR⨏QRξ0,t0Xuqdξdt1/q,where u¯(t)≔1/C1⨏QRξ0,t0uη4dξdt; η is the cut-off function related to ξ.From (22), we can get the following Sobolev-Poincaré type inequality immediately.Corollary 4.
Letu∈Wq0,1QRξ0,t0, 1<q<Q, then u∈LpQRξ0,t0, p=Qq/Q-q, and one has(23)∫QRξ0,t0u-u¯tpdξdt1/p≤C∫QRξ0,t0Xuqdξdt1/q.Next, we recall Fefferman-Phong’s inequality (see [21]) to get the multiplicative inequality which plays an important role in this paper.Theorem 5 (Fefferman-Phong’s inequality).
LetΩ⊂Rn be an open set and Q a homogeneous dimension. Set R0>0; suppose that 1<p<Q,1<s≤Q/p,β≥1, and V∈Ls,ps(βB), where B=BR(ξ0),ξ0∈Ω,0<R<R0, and βB⊂Ω. Then there exists C=C(Ω,X)>0 such that for any u∈L1,p(βB)(24)∫Bu-uBpVdξ≤CVLs,psβB∫βBXudξ,where uB=1/B∫Budξ and u∈Lp,μΩμ>0 implies u∈Llocp(Ω) and (25)supξ∈Ω,0<r<dimΩrμBrξ∩Ω∫Brξ∩Ωupdξ1/p<∞.Lemma 6 (multiplicative inequality forp).
LetΩ⊂Rn be an open set and Q a homogeneous dimension. Let R0>0, 1<p<Q, β≥1, ξ0∈Ω, 0<R<R0, and βB⊂Ω, where B=BR(ξ0). Then there exists C=C(Ω,X)>0 such that for any u∈W1,p(βB)(26)uLrB≤CuLpβBp/Q+pXuLpBR+R-1uLpBRQ/Q+p, where r=p(Q+p)/Q.Proof.
Sinceu∈W1,p(βB), we have u∈LpQ/Q-p by Sobolev’s embedding theorem; then (27)∫Bu-uBrdξ=∫Bu-uBpu-uBp2/Qdξ.Let u-uBp2/Q=V; then V∈LQ2/Q-p1/p(βB). Let s=Q/p; one has V∈LQ/Q-pQ/p(βB)⊂LQ/p(βB)=LQ/p,Q(βB).
By Theorem5, we obtain (28)∫Bu-uBpVdξ≤CVLQ/pβB∫βBXudξ,VLQ/pβB=u-uBp2/QLQ/pβB=u-uBLpβBp2/Q.Therefore (29)u-uBLrB≤Cu-uBLpβBp2/Q∫βBXudξQ/pQ+p=u-uBLpβBp/Q+pXuLpβBQ/Q+p,and then (30)uLrB≤Cu-uBLpβBp2/Q∫βBXudξQ/pQ+p≤Cu-uBLpβBp/Q+pXuLpβBQ/Q+p+uBLrB≤CuBLpβBp/Q+p+uLpβBp/Q+pXuLpβBQ/Q+p+uBLrB,uBLrB=1B∫BudξB1/r≤1B∫Budξ1/pBp-1/pB1/r≤uLpRp-1Q/p+Q2/pQ+pRQ=uLpR-1Q/p+Q,uBLpβBp/Q+p=1B∫Budξp/Q+pB1/Q+p≤1Bp/Q+p∫Budξ1/pBp-1/pp/Q+pB1/Q+p=uLpβBp/p+Q.Whenp=2, we get the multiplicative inequality that we will use in this paper.Lemma 7.
For anyR>0 and u∈V0,2(QR), it holds that (31)uLrBR≤CduL2BR2/Q+2XuL2BR+R-1uL2BRQ/Q+2.Gianazza [11] showed a reverse Hölder inequality on homogeneous spaces. Since the ball BR(ξ0) which is induced by the vector fields is a homogeneous space, it is obvious that the cylinder QR(ξ0,t0)=BR(ξ0)×(t-R2,t) is also a homogeneous space. Therefore, we just need to replace the ball BR(ξ0) in [11] with the parabolic cylinder QR(ξ0,t0). Similarly, we can getLemma 8.
Letg and h be nonnegative functions and satisfy the following.(1)
g∈Lγ(Q), γ>1; h∈Lβ(Q), β>γ.(2)
Assume that∀(ξ0,t0)∈Q,QR(ξ0,t0)⊂Q and ∀R:0<R<1/2dist((ξ0,t0),∂Q)∧R0; one has(32)⨏QRξ0,t0gγdξdt≤C0⨏Q8Rξ0,t0gdξdtγ+⨏Q8Rξ0,t0hγdξdt+θ⨏Q8Rξ0,t0gγdξdt,where the constantsR0,θ, and C0 are positive, θ∈(0,1). Then there exist ε>0 and C>0, such that g∈Llocp(Q),∀p∈[γ,γ+ε). Moreover, for QR(ξ0,t0)⊂Q,R<R0(33)⨏QRξ0,t0gpdξdt1/p≤C⨏Q8Rξ0,t0gγdξdt1/γ+C⨏Q8Rξ0,t0hpdξdt1/p,is valid, whereC and ε depend on C0,θ,γ,β,K(≥1) and n.We will use the following cut-off function. Letη(ξ)∈C0∞(BR(ξ0)),ζ(t)∈C0∞(-1,1) be cut-off functions and satisfy 0≤η,ζ≤1 (on BR(ξ0)), η≡1 (on BR/2(ξ0)), and ζ≡1 (on (-1/2,1/2)), Xη≤C/R. Note that ∫BRξ0η4dξ=C1BRξ0, where C1 is independent of R.Lemma 9 (Caccioppoli inequality).
Letu∈V0,2(UT) be the weak solutions of systems (1) and f∈L2(UT), fiα∈L2(UT), and fα∈Lr/r-1(UT). Then for any ξ0∈Rd, 0<R≤R0, and R2≤t0≤T(34)supt∈t0-R2,t0∫ΩRξ0u2ξ·tη4ζ2dξ+λ∫URξ0,t0Xu2η4ζ2≤CR-2∫URξ0,t0u2η2ζ+C∫URξ0,t0ur+C∫URξ0,t0f2+fiα2+fαr/r-1,where C=C(n,λ,Λ).Proof.
Take the test functionϕ=u(ξ,t)η4ζ2. Substituting ϕ in (17), we have (35)12∫ΩRξ0u2η4ζ2dξ+∫t0-R2t∫ΩRξ0Aiαt,ξ,u,XuXi∗uη4ζ2dξdt=∫t0-R2t∫ΩRξ0Bαt,ξ,u,Xuuη4ζ2dξdt+∫t0-R2t∫ΩRξ0u2η4ζ∂tζdξdt.By using the controllable growth conditions (2)(36)14supt∈t0-R2,t0∫ΩRξ0u2η4ζ2dξ+λ2∫QRξ0,t0Xu2η4ζ2dξdt≤Λ2∫URξ0,t0urη4ζ2dξdt+12∫URξ0,t0f2η4ζ2dξdt+4Λ∫URξ0,t0Xuuη3Xηζ2dξdt+4Λ∫URξ0,t0ur/2uη3Xηζ2dξdt+4Λ∫URξ0,t0fiαuη3Xηζ2dξdt+∫URξ0,t0Bαt,ξ,u,Xuuη4ζ2dξdt+∫URξ0,t0u2η4ζ∂tζdξdt≔G1+G2+G3+G4+G5+G6+G7. We estimate the following terms by Young’s inequality:(37)G3=4Λ∫URξ0,t0Xuuη3Xηζ2≤λ16∫URξ0,t0Xu2η4ζ2+C∫URξ0,t0u2Xη2η2ζ2≤λ16∫URξ0,t0Xu2η4ζ2+CR-2∫URξ0,t0u2η2ζ2,G4=4Λ∫URξ0,t0ur/2uη3Xηζ2≤λ16∫URξ0,t0urη4ζ2+C∫URξ0,t0u2Xη2η2ζ2≤λ16∫URξ0,t0urη4ζ2+CR-2∫URξ0,t0u2η2ζ2,G5=4Λ∫URξ0,t0fiαuη3Xηζ2≤λ16∫URξ0,t0fiα2η4ζ2+C∫QRξ0,t0u2Xη2η2ζ2≤λ16∫URξ0,t0fiα2η4ζ2+CR-2∫URξ0,t0u2η2ζ2,G6=∫URξ0,t0Bαt,ξ,u,Xuuη4ζ2≤C∫URξ0,t0Xu21-1/ruη4ζ2+C∫URξ0,t0ur-1uη4ζ2+C∫QRξ0,t0fαuη4ζ2≤λ16∫URξ0,t0Xu2η4ζ2+C∫URξ0,t0ur+C∫QRξ0,t0fαr/r-1.Combining the above estimations and ζ2≤ζ, then we get (34).Corollary 10.
Under the same assumptions as in Lemma9, we have(38)supt∈t0-R2/4,t0∫ΩR/2ξ0u2dξ⟶0,as R→0, uniformly in ξ0 and t0.Proof.
By Hölder’s inequality(39)R-2∫URξ0,t0u2≤C∫URξ0,t0ur2/r. From this inequality and (34) we have (40)supt∈t0-R2/4,t0∫ΩR/2ξ0u2dξ≤C∫URξ0,t0ur+f2+fiα2+fαr/r-1+C∫URξ0,t0ur2/r.Then f,fα,fiα,u, the assumption of g, and (14), as well as the absolute continuity of Lebesgue, integrals imply (38).


## 5. The Proof of the Main Results

### 5.1. The Proof of Theorem1

Take the test functionφ=η(ξ)4ζ(t)2(u-u¯(t)), where (41)u¯t=∫BRξ0η4dξ-1∫BRξ0uη4dξ=1C1⨏BRξ0uη4dξ, and then (42)∫BRξ0u¯tη4dξ=C1BRξ01C1⨏BRξ0uη4dξ=∫BRξ0uη4dξ, so we have (43)∫BRξ0∂tu¯tu-u¯tη4ζ2dξ=∂tu¯tζ2∫BRξ0u-u¯tη4dξ=∂tu¯tζ2∫BRξ0uη4-u¯tη4dξ=0,and then (44)∫BRξ0∂tuu-u¯tη4ζ2dξ=∫BRξ0∂tu-∂tu¯tu-u¯tη4ζ2dξ.Substituting φ in (17), we have(45)12∫BRξ0u-u¯t2η4ζ2dξ+∫t0-R2t∫BRξ0Aiαt,ξ,u,XuXi∗u-u¯tη4ζ2dξdt=∫t0-R2t∫BRξ0Bαt,ξ,u,Xuu-u¯tη4ζ2dξdt+∫t0-R2t∫BRξ0u-u¯t2η4ζ∂tζdξdt.By the controllable growth conditions (2) (46)14supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+λ2∫QRξ0,t0Xu2η4ζ2dξdt≤Λ2∫QRξ0,t0urη4ζ2dξdt+12∫QRξ0,t0f2η4ζ2dξdt+4Λ∫QRξ0,t0Xuu-u¯tη3Xηζ2dξdt+4Λ∫QRξ0,t0ur/2u-u¯tη3Xηζ2dξdt+4Λ∫QRξ0,t0fiαu-u¯tη3Xηζ2dξdt+∫QRξ0,t0Bαt,ξ,u,Xuu-u¯tη4ζ2dξdt+∫QRξ0,t0u-u¯t2η4ζ∂tζdξdt≔I1+I2+I3+I4+I5+I6+I7. We estimate each term by using Young’s inequality(47)I3=4Λ∫QRξ0,t0Xuu-u¯tη3Xηζ2≤λ16∫QRξ0,t0Xu2η4ζ2+C∫QRξ0,t0u-u¯t2Xη2η2ζ2≤λ16∫QRξ0,t0Xu2η4ζ2+CR-2∫QRξ0,t0u-u¯t2η2ζ2,I4=4Λ∫QRξ0,t0ur/2u-u¯tη3Xηζ2≤λ16∫QRξ0,t0urη4ζ2+C∫QRξ0,t0u-u¯t2Xη2η2ζ2≤λ16∫QRξ0,t0urη4ζ2+CR-2∫QRξ0,t0u-u¯t2η2ζ2,I5=4Λ∫QRξ0,t0fiαu-u¯tη3Xηζ2≤λ16∫QRξ0,t0fiα2η4ζ2+C∫QRξ0,t0u-u¯t2Xη2η2ζ2≤λ16∫QRξ0,t0fiα2η4ζ2+CR-2∫QRξ0,t0u-u¯t2η2ζ2,I6=∫QRξ0,t0Bαt,ξ,u,Xuu-u¯tη4ζ2≤C∫QRξ0,t0Xu21-1/ru-u¯tη4ζ2+C∫QRξ0,t0ur-1u-u¯tη4ζ2+C∫QRξ0,t0fαu-u¯tη4ζ2≤λ16∫QRξ0,t0Xu2η4ζ2+C∫QRξ0,t0u-u¯tr+C∫QRξ0,t0ur+C∫QRξ0,t0fαr/r-1. Therefore(48)supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+λ∫QRξ0,t0Xu2η4ζ2dξdt≤CR-2∫QRξ0,t0u-u¯t2η2ζ2+C∫QRξ0,t0u-u¯tr+ur+C∫QRξ0,t0f2+fiα2+fαr/r-1≔C0J1+J2+J3.Note that (49)∫BRξ0u-u¯t2dξ=∫BRξ0uξ,t-1C1⨏BRξ0uξ′,tη4ξ′dξ′2dξ=∫BRξ01C1⨏BRξ0uξ,tη4ξ′dξ′-1C1⨏BRξ0uξ′,tη4ξ′dξ′2dξ≤C2∫BRξ0⨏BRξ0uξ,t-uξ′,tdξ′dξ. So we can use Young’s inequality and Sobolev-Poincaré inequality (23) to estimate J1:(50)J1≤CR-2R2∫QRξ0,t0u-u¯t2η4ζ2+C∫QRξ0,t0u-u¯t2≤λ16C0supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2+CR-q∫QRξ0,t0Xuq2/q,where (51)q=2Q+2Q+4∈1,2.Next, we will estimateJ2. By the triangle inequality(52)J2≤C∫QRξ0,t0u-u¯trη4ζ2+C∫QRξ0,t0u¯trη4ζ2≔J21+J22.By Poincaré inequality (22) and Lemma 7(53)∫t0-R2t0∫BRξ0u-u¯trdξdt≤C∫t0-R2t0∫BRξ0u-u¯t2dξ2/Q∫BRξ0Xu2+R-2∫BRξ0u-u¯t2dξdt≤C∫t0-R2t0∫BRξ0u-u¯t2dξ2/Q∫BRξ0Xu2dξdt,then we can estimate J21:(54)J21≤Csupt∈t0-R2,t0∫BRξ0u-u¯t2dξ2/Q∫QRξ0,t0Xu2≤Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q∫QRξ0,t0Xu2.For J22, by the triangle inequality(55)J22≤C∫QRξ0,t0u¯tr≤C∫QRξ0,t0u¯t-cr+C∫QRξ0,t0cr≤C∫QRξ0,t0u¯t-cr+CR2BRξ0cr≔J221+J222,where (56)c=⨏t0-R2t0u¯tdt.Byrq/2>1 and Hölder inequality, we have (57)J222=CR2BRξ0⨏t0-R2t0u¯tdtr=CR2BRξ0⨏t0-R2t01C1⨏BRξ0uη4dξdtr=CR2BRξ0⨏QRξ0,t0ur≤CR2BRξ0⨏QRξ0,t0urq/22/q.To estimate J221, we use the Poincaré inequality in t(58)J221≤CR2BRξ0∫t0-R2t0∂tu¯dtr≤CR2BRξ0∫t0-R2t0BRξ0-1∫BRξ0∂tuη4dξdtr=CR2BRξ0∫t0-R2t0BRξ0-1∫BRξ0Xi∗Aiαt,ξ,u,Xu+Bαt,ξ,u,Xuη4dξdtr≔CR2BRξ0Kr,integrating by parts for K, combining with the controllable growth conditions(59)K≤CR2⨏QRξ0,t0Aiαη3Xη+CR2⨏QRξ0,t0Bαη4≤CR⨏QRξ0,t0Xu+ur/2+fiα+CR2⨏QRξ0,t0Xu21-1/r+ur-1+fα≔CK1+K2+K3+K4+K5+K6.It remains to estimate each term by the Hölder inequality(60)K1=R⨏QRξ0,t0Xu≤R⨏QRξ0,t0Xu2/rrq/22/rq⨏QRξ0,t0Xur-2/r2r/r-2r-2/2r=⨏QRξ0,t0Xuq2/rq∫QRξ0,t0Xu2r-2/2r,K2=R⨏QRξ0,t0ur/2≤R⨏QRξ0,t0urq/22/rq⨏QRξ0,t0ur-2/22r/r-2r-2/2r=⨏QRξ0,t0urq/22/rq∫QRξ0,t0urr-2/2r,K3=R⨏QRξ0,t0fiα≤R⨏QRξ0,t0fiα21/2≤R⨏QRξ0,t0fiα4/rr/2⨏QRξ0,t0fiα2r-2/rr/r-21/2=⨏QRξ0,t0fiα21/r∫QRξ0,t0fiα2r-2/2r,K4=R2⨏QRξ0,t0Xu21-1/r≤R2⨏QRξ0,t0Xu2/rrq/22/rq⨏QRξ0,t0Xu2r-2/rr/r-2r-2/r=⨏QRξ0,t0Xuq2/rq∫QRξ0,t0Xu2r-2/r,K5=R2⨏QRξ0,t0ur-1≤R2⨏QRξ0,t0urq/22/rq⨏QRξ0,t0ur-2r/r-2r-2/r=⨏QRξ0,t0urq/22/rq∫QRξ0,t0urr-2/r,K6=R2⨏QRξ0,t0fα≤R2⨏QRξ0,t0fαr/r-1r-1/r≤R2⨏QRξ0,t0fαr/r-12r-11/r-1⨏QRξ0,t0fαrr-2/r-12r-1/r-2r-2/r-1r-1/r=⨏QRξ0,t0fαr/r-11/r∫QRξ0,t0fαr/r-1r-2/r.By f,fα,fiα,u, the estimation of g, and (14), we get (61)∫QRξ0,t0Xu2,∫QRξ0,t0f2,∫QRξ0,t0fiα2,∫QRξ0,t0fαr-1/r,∫QRξ0,t0ur,that are uniformly bounded. Combining all the above inequalities(62)λ⨏QRξ0,t0Xu2η4ζ2≤C⨏QRξ0,t0Xuq+urq/22/q+C⨏QRξ0,t0f2+fiα2+fαr/r-1+Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q⨏QRξ0,t0Xu2,where C=C(n,λ,Λ,uU2,fL2,fαLr/r-1,fiαL2). (A) is obtained by this estimation and the estimate of J2.Next we will prove (B): firstly, we extendu to QT≔(0,T)×Rn such that if ξ∈Rn∖Ω, then u(ξ,t)=0. It is obvious that u∈V2(Q). Similarly, we extend f,fα, and fiα and let R<R0/4.Set(ξ0,t0)∈QT such that Q4R(ξ0,t0)⊂QT. We consider the case B4R(ξ0)⊂Ω. Following the results in (A)(63)⨏QR/2ξ0,t0Xu2+urdξdt≤C⨏Q4Rξ0,t0Xuq+urq/2dξdt2/q+C⨏Q4Rξ0,t0f2+f¯2+F2dξdt+Csupt∈t0-R2,t0∫B4Rξ0u2dξ2/Q⨏Q4Rξ0,t0Xu2dξdt,where f¯=fiα,F=fα1/2r/(r-1), and C=C(n,λ,Λ,uV2,fL2,fαLr/r-1,fiαL2).By Corollary10, there exists R0′=R0′(u,fL2,fαLr/r-1,fiαL2)>0 small enough, such that for all 0<R≤R0′(64)Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q<1.For p∈2,min⁡σ2r-1/rτ and Q4Rξ0′,t0′⊂QT, we take γ=2/q>1 in Lemma 8, g=Xuq+urq/2,h=fq+fiαq+Fq; then (65)⨏QR/2ξ0′,t0′Xup+urp/2dξdt≤C⨏Q4Rξ0′,t0′Xu2+urdξdtp/2+C⨏Q4Rξ0′,t0′fp+f¯p+Fpdξdt,where p and C depend on n,λ,Λ,uV2,fL2,fαLr/(r-1),fiαL2. Take the suitable parabolic cylinder QR/2ξ0′,t0′ to cover ε,T×Ω such that Q4Rξ0′,t0′⊂QT; the result is obtained.


### 5.2. The Proof of Theorem2

In the similar way to the proof of Theorem1, we get (45). By the natural growth conditions (3)–(5), we have (66)14supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+λ2∫QRξ0,t0Xu2η4ζ2dξdt≤12∫QRξ0,t0f2η4ζ2dξdt+Λ2∫QRξ0,t0Xuu-u¯tη3Xηζ2dξdt+12∫QRξ0,t0fiαu-u¯tη3Xηζ2dξdt+Λ2∫QRξ0,t0Xu2u-u¯tη4ζ2dξdt+12∫QRξ0,t0fαu-u¯tη4ζ2dξdt+∫QRξ0,t0u-u¯t2η4ζ∂tζdξdt≔R1+R2+R3+R4+R5+R6.We estimate each term by Young inequality and Sobolev-Poincaré inequality(67)R2≤εΛ2∫QRξ0,t0Xu2η4ζ2+CΛε∫QRξ0,t0u-u¯t2Xη2η2ζ2≤εΛ2∫QRξ0,t0Xu2η4ζ2+CΛεR-2∫QRξ0,t0u-u¯t2η2ζ2≤εΛ2∫QRξ0,t0Xu2η4ζ2+CΛεR-2∫QRξ0,t0Xu4/rr/4,R3≤C∫QRξ0,t0fiα2η4ζ2+C∫QRξ0,t0u-u¯t2Xη2η2ζ2≤C∫QRξ0,t0fiα2η4ζ2+CR-2∫QRξ0,t0u-u¯t2η2ζ2≤C∫QRξ0,t0fiα2η4ζ2+CR-2∫QRξ0,t0Xu4/rr/4,R4=Λ2∫QRξ0,t0Xu2u-u¯tη4ζ2≤ΛM∫QRξ0,t0Xu2η4ζ2,R5≤C∫QRξ0,t0fαr/r-1η4ζ2+C∫QRξ0,t0u-u¯tr. Therefore(68)supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+λ-2ΛM-εΛ∫QRξ0,t0Xu2η4ζ2dξdt≤CεΛR-2∫QRξ0,t0u-u¯t2η2ζ2+C∫QRξ0,t0u-u¯tr+C∫QRξ0,t0f2+fiα2+fαr/r-1.Sinceλ-2ΛM>0, we take ε=(λ-2ΛM)/3Λ; then (69)supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+∫QRξ0,t0Xu2η4ζ2dξdt≤CR-2∫QRξ0,t0u-u¯t2η2ζ2+C∫QRξ0,t0u-u¯tr+C∫QRξ0,t0f2+fiα2+fαr/r-1≔C0S1+S2+S3. Following the estimate (50) in Theorem 1, one has(70)S1≤λ16C0supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2+CR-q∫QRξ0,t0Xuq2/q.Similar to the estimate of J21 in Theorem 1(71)S2≤Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q∫QRξ0,t0Xu2.By f,fα,fiα,u, the assumption of g, and (14), we know that (72)∫QRξ0,t0Xu2,∫QRξ0,t0f2,∫QRξ0,t0fiα2,∫QRξ0,t0fαr-1/r,∫QRξ0,t0ur.They are uniformly bounded. Combining all the estimations above, we have (73)⨏QRξ0,t0Xu2η4ζ2≤C⨏QRξ0,t0Xuq2/q+C⨏QRξ0,t0f2+fiα2+fαr/r-1+Csupt∈t0-R2,t0∫BRξ0u2dξ2/d⨏QRξ0,t0Xu2, where C=Cn,λ,Λ,uU2,fL2,fαLr/r-1,fiαL2. Then (C) is derived.Now we prove (D). Similar to (B), we just need to takeγ=2/q>1,g=Xuq,h=fq+fiαq+Fq in Lemma 8; then (74)⨏QR/2ξ0′,t0′Xupdξdt≤C⨏Q4Rξ0′,t0′Xu2dξdtp/2+C⨏Q4Rξ0′,t0′fp+f¯p+Fpdξdt,where p and C depend on n,λ,Λ,uV2,fL2,fαLr/(r-1),fiαL2,M. The proof is complete by using suitable parabolic cylinder QR/2ξ0′,t0′ to cover ε,T×Ω such that Q4Rξ0′,t0′⊂QT.


## 5.1. The Proof of Theorem1

Take the test functionφ=η(ξ)4ζ(t)2(u-u¯(t)), where (41)u¯t=∫BRξ0η4dξ-1∫BRξ0uη4dξ=1C1⨏BRξ0uη4dξ, and then (42)∫BRξ0u¯tη4dξ=C1BRξ01C1⨏BRξ0uη4dξ=∫BRξ0uη4dξ, so we have (43)∫BRξ0∂tu¯tu-u¯tη4ζ2dξ=∂tu¯tζ2∫BRξ0u-u¯tη4dξ=∂tu¯tζ2∫BRξ0uη4-u¯tη4dξ=0,and then (44)∫BRξ0∂tuu-u¯tη4ζ2dξ=∫BRξ0∂tu-∂tu¯tu-u¯tη4ζ2dξ.Substituting φ in (17), we have(45)12∫BRξ0u-u¯t2η4ζ2dξ+∫t0-R2t∫BRξ0Aiαt,ξ,u,XuXi∗u-u¯tη4ζ2dξdt=∫t0-R2t∫BRξ0Bαt,ξ,u,Xuu-u¯tη4ζ2dξdt+∫t0-R2t∫BRξ0u-u¯t2η4ζ∂tζdξdt.By the controllable growth conditions (2) (46)14supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+λ2∫QRξ0,t0Xu2η4ζ2dξdt≤Λ2∫QRξ0,t0urη4ζ2dξdt+12∫QRξ0,t0f2η4ζ2dξdt+4Λ∫QRξ0,t0Xuu-u¯tη3Xηζ2dξdt+4Λ∫QRξ0,t0ur/2u-u¯tη3Xηζ2dξdt+4Λ∫QRξ0,t0fiαu-u¯tη3Xηζ2dξdt+∫QRξ0,t0Bαt,ξ,u,Xuu-u¯tη4ζ2dξdt+∫QRξ0,t0u-u¯t2η4ζ∂tζdξdt≔I1+I2+I3+I4+I5+I6+I7. We estimate each term by using Young’s inequality(47)I3=4Λ∫QRξ0,t0Xuu-u¯tη3Xηζ2≤λ16∫QRξ0,t0Xu2η4ζ2+C∫QRξ0,t0u-u¯t2Xη2η2ζ2≤λ16∫QRξ0,t0Xu2η4ζ2+CR-2∫QRξ0,t0u-u¯t2η2ζ2,I4=4Λ∫QRξ0,t0ur/2u-u¯tη3Xηζ2≤λ16∫QRξ0,t0urη4ζ2+C∫QRξ0,t0u-u¯t2Xη2η2ζ2≤λ16∫QRξ0,t0urη4ζ2+CR-2∫QRξ0,t0u-u¯t2η2ζ2,I5=4Λ∫QRξ0,t0fiαu-u¯tη3Xηζ2≤λ16∫QRξ0,t0fiα2η4ζ2+C∫QRξ0,t0u-u¯t2Xη2η2ζ2≤λ16∫QRξ0,t0fiα2η4ζ2+CR-2∫QRξ0,t0u-u¯t2η2ζ2,I6=∫QRξ0,t0Bαt,ξ,u,Xuu-u¯tη4ζ2≤C∫QRξ0,t0Xu21-1/ru-u¯tη4ζ2+C∫QRξ0,t0ur-1u-u¯tη4ζ2+C∫QRξ0,t0fαu-u¯tη4ζ2≤λ16∫QRξ0,t0Xu2η4ζ2+C∫QRξ0,t0u-u¯tr+C∫QRξ0,t0ur+C∫QRξ0,t0fαr/r-1. Therefore(48)supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+λ∫QRξ0,t0Xu2η4ζ2dξdt≤CR-2∫QRξ0,t0u-u¯t2η2ζ2+C∫QRξ0,t0u-u¯tr+ur+C∫QRξ0,t0f2+fiα2+fαr/r-1≔C0J1+J2+J3.Note that (49)∫BRξ0u-u¯t2dξ=∫BRξ0uξ,t-1C1⨏BRξ0uξ′,tη4ξ′dξ′2dξ=∫BRξ01C1⨏BRξ0uξ,tη4ξ′dξ′-1C1⨏BRξ0uξ′,tη4ξ′dξ′2dξ≤C2∫BRξ0⨏BRξ0uξ,t-uξ′,tdξ′dξ. So we can use Young’s inequality and Sobolev-Poincaré inequality (23) to estimate J1:(50)J1≤CR-2R2∫QRξ0,t0u-u¯t2η4ζ2+C∫QRξ0,t0u-u¯t2≤λ16C0supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2+CR-q∫QRξ0,t0Xuq2/q,where (51)q=2Q+2Q+4∈1,2.Next, we will estimateJ2. By the triangle inequality(52)J2≤C∫QRξ0,t0u-u¯trη4ζ2+C∫QRξ0,t0u¯trη4ζ2≔J21+J22.By Poincaré inequality (22) and Lemma 7(53)∫t0-R2t0∫BRξ0u-u¯trdξdt≤C∫t0-R2t0∫BRξ0u-u¯t2dξ2/Q∫BRξ0Xu2+R-2∫BRξ0u-u¯t2dξdt≤C∫t0-R2t0∫BRξ0u-u¯t2dξ2/Q∫BRξ0Xu2dξdt,then we can estimate J21:(54)J21≤Csupt∈t0-R2,t0∫BRξ0u-u¯t2dξ2/Q∫QRξ0,t0Xu2≤Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q∫QRξ0,t0Xu2.For J22, by the triangle inequality(55)J22≤C∫QRξ0,t0u¯tr≤C∫QRξ0,t0u¯t-cr+C∫QRξ0,t0cr≤C∫QRξ0,t0u¯t-cr+CR2BRξ0cr≔J221+J222,where (56)c=⨏t0-R2t0u¯tdt.Byrq/2>1 and Hölder inequality, we have (57)J222=CR2BRξ0⨏t0-R2t0u¯tdtr=CR2BRξ0⨏t0-R2t01C1⨏BRξ0uη4dξdtr=CR2BRξ0⨏QRξ0,t0ur≤CR2BRξ0⨏QRξ0,t0urq/22/q.To estimate J221, we use the Poincaré inequality in t(58)J221≤CR2BRξ0∫t0-R2t0∂tu¯dtr≤CR2BRξ0∫t0-R2t0BRξ0-1∫BRξ0∂tuη4dξdtr=CR2BRξ0∫t0-R2t0BRξ0-1∫BRξ0Xi∗Aiαt,ξ,u,Xu+Bαt,ξ,u,Xuη4dξdtr≔CR2BRξ0Kr,integrating by parts for K, combining with the controllable growth conditions(59)K≤CR2⨏QRξ0,t0Aiαη3Xη+CR2⨏QRξ0,t0Bαη4≤CR⨏QRξ0,t0Xu+ur/2+fiα+CR2⨏QRξ0,t0Xu21-1/r+ur-1+fα≔CK1+K2+K3+K4+K5+K6.It remains to estimate each term by the Hölder inequality(60)K1=R⨏QRξ0,t0Xu≤R⨏QRξ0,t0Xu2/rrq/22/rq⨏QRξ0,t0Xur-2/r2r/r-2r-2/2r=⨏QRξ0,t0Xuq2/rq∫QRξ0,t0Xu2r-2/2r,K2=R⨏QRξ0,t0ur/2≤R⨏QRξ0,t0urq/22/rq⨏QRξ0,t0ur-2/22r/r-2r-2/2r=⨏QRξ0,t0urq/22/rq∫QRξ0,t0urr-2/2r,K3=R⨏QRξ0,t0fiα≤R⨏QRξ0,t0fiα21/2≤R⨏QRξ0,t0fiα4/rr/2⨏QRξ0,t0fiα2r-2/rr/r-21/2=⨏QRξ0,t0fiα21/r∫QRξ0,t0fiα2r-2/2r,K4=R2⨏QRξ0,t0Xu21-1/r≤R2⨏QRξ0,t0Xu2/rrq/22/rq⨏QRξ0,t0Xu2r-2/rr/r-2r-2/r=⨏QRξ0,t0Xuq2/rq∫QRξ0,t0Xu2r-2/r,K5=R2⨏QRξ0,t0ur-1≤R2⨏QRξ0,t0urq/22/rq⨏QRξ0,t0ur-2r/r-2r-2/r=⨏QRξ0,t0urq/22/rq∫QRξ0,t0urr-2/r,K6=R2⨏QRξ0,t0fα≤R2⨏QRξ0,t0fαr/r-1r-1/r≤R2⨏QRξ0,t0fαr/r-12r-11/r-1⨏QRξ0,t0fαrr-2/r-12r-1/r-2r-2/r-1r-1/r=⨏QRξ0,t0fαr/r-11/r∫QRξ0,t0fαr/r-1r-2/r.By f,fα,fiα,u, the estimation of g, and (14), we get (61)∫QRξ0,t0Xu2,∫QRξ0,t0f2,∫QRξ0,t0fiα2,∫QRξ0,t0fαr-1/r,∫QRξ0,t0ur,that are uniformly bounded. Combining all the above inequalities(62)λ⨏QRξ0,t0Xu2η4ζ2≤C⨏QRξ0,t0Xuq+urq/22/q+C⨏QRξ0,t0f2+fiα2+fαr/r-1+Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q⨏QRξ0,t0Xu2,where C=C(n,λ,Λ,uU2,fL2,fαLr/r-1,fiαL2). (A) is obtained by this estimation and the estimate of J2.Next we will prove (B): firstly, we extendu to QT≔(0,T)×Rn such that if ξ∈Rn∖Ω, then u(ξ,t)=0. It is obvious that u∈V2(Q). Similarly, we extend f,fα, and fiα and let R<R0/4.Set(ξ0,t0)∈QT such that Q4R(ξ0,t0)⊂QT. We consider the case B4R(ξ0)⊂Ω. Following the results in (A)(63)⨏QR/2ξ0,t0Xu2+urdξdt≤C⨏Q4Rξ0,t0Xuq+urq/2dξdt2/q+C⨏Q4Rξ0,t0f2+f¯2+F2dξdt+Csupt∈t0-R2,t0∫B4Rξ0u2dξ2/Q⨏Q4Rξ0,t0Xu2dξdt,where f¯=fiα,F=fα1/2r/(r-1), and C=C(n,λ,Λ,uV2,fL2,fαLr/r-1,fiαL2).By Corollary10, there exists R0′=R0′(u,fL2,fαLr/r-1,fiαL2)>0 small enough, such that for all 0<R≤R0′(64)Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q<1.For p∈2,min⁡σ2r-1/rτ and Q4Rξ0′,t0′⊂QT, we take γ=2/q>1 in Lemma 8, g=Xuq+urq/2,h=fq+fiαq+Fq; then (65)⨏QR/2ξ0′,t0′Xup+urp/2dξdt≤C⨏Q4Rξ0′,t0′Xu2+urdξdtp/2+C⨏Q4Rξ0′,t0′fp+f¯p+Fpdξdt,where p and C depend on n,λ,Λ,uV2,fL2,fαLr/(r-1),fiαL2. Take the suitable parabolic cylinder QR/2ξ0′,t0′ to cover ε,T×Ω such that Q4Rξ0′,t0′⊂QT; the result is obtained.


## 5.2. The Proof of Theorem2

In the similar way to the proof of Theorem1, we get (45). By the natural growth conditions (3)–(5), we have (66)14supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+λ2∫QRξ0,t0Xu2η4ζ2dξdt≤12∫QRξ0,t0f2η4ζ2dξdt+Λ2∫QRξ0,t0Xuu-u¯tη3Xηζ2dξdt+12∫QRξ0,t0fiαu-u¯tη3Xηζ2dξdt+Λ2∫QRξ0,t0Xu2u-u¯tη4ζ2dξdt+12∫QRξ0,t0fαu-u¯tη4ζ2dξdt+∫QRξ0,t0u-u¯t2η4ζ∂tζdξdt≔R1+R2+R3+R4+R5+R6.We estimate each term by Young inequality and Sobolev-Poincaré inequality(67)R2≤εΛ2∫QRξ0,t0Xu2η4ζ2+CΛε∫QRξ0,t0u-u¯t2Xη2η2ζ2≤εΛ2∫QRξ0,t0Xu2η4ζ2+CΛεR-2∫QRξ0,t0u-u¯t2η2ζ2≤εΛ2∫QRξ0,t0Xu2η4ζ2+CΛεR-2∫QRξ0,t0Xu4/rr/4,R3≤C∫QRξ0,t0fiα2η4ζ2+C∫QRξ0,t0u-u¯t2Xη2η2ζ2≤C∫QRξ0,t0fiα2η4ζ2+CR-2∫QRξ0,t0u-u¯t2η2ζ2≤C∫QRξ0,t0fiα2η4ζ2+CR-2∫QRξ0,t0Xu4/rr/4,R4=Λ2∫QRξ0,t0Xu2u-u¯tη4ζ2≤ΛM∫QRξ0,t0Xu2η4ζ2,R5≤C∫QRξ0,t0fαr/r-1η4ζ2+C∫QRξ0,t0u-u¯tr. Therefore(68)supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+λ-2ΛM-εΛ∫QRξ0,t0Xu2η4ζ2dξdt≤CεΛR-2∫QRξ0,t0u-u¯t2η2ζ2+C∫QRξ0,t0u-u¯tr+C∫QRξ0,t0f2+fiα2+fαr/r-1.Sinceλ-2ΛM>0, we take ε=(λ-2ΛM)/3Λ; then (69)supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2dξ+∫QRξ0,t0Xu2η4ζ2dξdt≤CR-2∫QRξ0,t0u-u¯t2η2ζ2+C∫QRξ0,t0u-u¯tr+C∫QRξ0,t0f2+fiα2+fαr/r-1≔C0S1+S2+S3. Following the estimate (50) in Theorem 1, one has(70)S1≤λ16C0supt∈t0-R2,t0∫BRξ0u-u¯t2η4ζ2+CR-q∫QRξ0,t0Xuq2/q.Similar to the estimate of J21 in Theorem 1(71)S2≤Csupt∈t0-R2,t0∫BRξ0u2dξ2/Q∫QRξ0,t0Xu2.By f,fα,fiα,u, the assumption of g, and (14), we know that (72)∫QRξ0,t0Xu2,∫QRξ0,t0f2,∫QRξ0,t0fiα2,∫QRξ0,t0fαr-1/r,∫QRξ0,t0ur.They are uniformly bounded. Combining all the estimations above, we have (73)⨏QRξ0,t0Xu2η4ζ2≤C⨏QRξ0,t0Xuq2/q+C⨏QRξ0,t0f2+fiα2+fαr/r-1+Csupt∈t0-R2,t0∫BRξ0u2dξ2/d⨏QRξ0,t0Xu2, where C=Cn,λ,Λ,uU2,fL2,fαLr/r-1,fiαL2. Then (C) is derived.Now we prove (D). Similar to (B), we just need to takeγ=2/q>1,g=Xuq,h=fq+fiαq+Fq in Lemma 8; then (74)⨏QR/2ξ0′,t0′Xupdξdt≤C⨏Q4Rξ0′,t0′Xu2dξdtp/2+C⨏Q4Rξ0′,t0′fp+f¯p+Fpdξdt,where p and C depend on n,λ,Λ,uV2,fL2,fαLr/(r-1),fiαL2,M. The proof is complete by using suitable parabolic cylinder QR/2ξ0′,t0′ to cover ε,T×Ω such that Q4Rξ0′,t0′⊂QT.



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*Source: 2741326-2017-01-22.xml*