New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math.24(2018) 53–81.

Regularity for weak solutions to nondiagonal quasilinear degenerate parabolic systems with controllable

growth conditions

Yan Dong and Dongyan Li

Abstract. The aim of this paper is to study regularity for weak so- lutions to the nondiagonal quasilinear degenerate parabolic systems re- lated to H¨ormander’s vector fields, where the lower order items satisfy controllable growth conditions. Higher Morrey regularity is proved by establishing a reverse H¨older inequality for weak solutions, and then H¨older regularity is obtained by the isomorphic relationship.

Contents

1. Introduction 53

2. Preliminaries 56

3. Higher integrability 59

4. Proofs of main theorems 71

References 80

1. Introduction

Regularity for weak solutions to divergence elliptic and parabolic systems in Euclidean spaces has been studied by many authors (see [1]–[3], [8], [15], [21], [23]–[24], [27] and the references therein). For diagonal elliptic systems, Giaquinta in [14] proved gradient estimates in Morrey spaces for weak so- lutions to linear elliptic systems with H¨older continuous coefficients. For nondiagonal elliptic systems, Wiegner in [25] considered H¨older estimates for weak solutions when the lower order items satisfy the natural growth

Received November 22, 2017.

2010Mathematics Subject Classification. 35K40, 35D30.

Key words and phrases. Nondiagonal parabolic system; H¨ormander’s vector fields; con- trollable growth condition; regularity.

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11701162); Natural Science Foundation Research Project of Shaanxi Province (Grant No. 2016JQ1029); Natural Science Foundation of Education Department of Shaanxi Provincial Government(Grant No. 16JK1320); Research Fund for the Doctoral Program of Hubei University of Economics (Grant No. XJ16BS28).

ISSN 1076-9803/2018

53

YAN DONG AND DONGYAN LI

conditions. Giaquinta and Struwe in [16] treated partial regularity for weak solutions to diagonal quasilinear parabolic systems with the natural growth conditions and got H¨older regularity.

Many scholars have studied degenerate elliptic and parabolic systems formed by H¨ormander’s (see [18]) vector fields. For linear diagonal ellip- tic systems, Di Fazio and Fanciullo in [5] obtained Morrey estimates for weak solutions, and then got H¨older estimates by Poincar´e inequality and isomorphic relationship. For nonlinear nondiagonal elliptic systems, Dong and Niu in [10] considered nondiagonal quasilinear degenerate elliptic sys- tems with the low order terms satisfy special growth conditions, they got higher Morrey estimates for weak solutions by the reverse H¨older inequality, and then obtained H¨older estimates by Morrey lemma. For diagonal para- bolic systems the reader can refer to [6, 7]. And for some other studies, we quote [9, 11, 26] and references therein. Then the nondiagonal parabolic sys- tems whether have a corresponding regular conclusion? This is the content of this paper.

In this paper, we consider quasilinear nondiagonal parabolic systems with the low order terms satisfying the controllable growth conditions. We gen- eralized the results of [11, 16, 25]. As far as we know, when the low order terms satisfying the natural growth conditions, the weak solutions must be bounded, this condition can be abandoned when the lower order satisfy- ing the controllable growth conditions. And due to the lack of a parabolic Poincar´e inequality, it is more difficult to study. In order to solve these problems, we first introduce the average on the ball of weak solutions, and then get higher integrability for weak solutions and a parabolic Poincar´e inequality. Concretely, we consider the following nondiagonal quasilinear degenerate parabolic system

(1.1) uⁱ_t+X_α^∗(a^αβ_ij (z, u)X_βu^j) =gi(z, u, Xu) +X_α^∗f_i^α(z, u), where

X_α=

n

X

k=1

b_αk(x) ∂

∂x_k

(b_αk(x)∈C^∞(Ω)) is a family of real smooth vector fields in a neighborhood Ω of some bounded domain Ω˜ ⊂Rⁿ (q≤n) and satisfy H¨ormander’s condi- tion (see Section 2) free up to the orders,i, j= 1,2, . . . , N;α, β= 1,2, . . . , q;

X_α^∗ =−X_α+cα

(cα =−Pn k=1

∂bαk

∂x_k ∈C^∞(Ω)) is the transposed vector field ofXα. In this paper, we first establish higher integrability for weak solutions, and get a reverse H¨older inequality for weak solutions by the reverse H¨older inequality on the homogeneous space, and then obtain higher Morrey estimates, finally get H¨older estimates by isomorphic relationship.

Before stating our main results, we need several assumptions of (1.1).

(H1) Let coefficients

a^αβ_ij (z, u) =A^αβ(z)δ_ij+B_ij^αβ(z, u),

whereA^αβ(z) ∈V M O∩L^∞, A^αβ(z)=A^βα(z) satisfy the ellipticity condition,B_ij^αβ(z, u) are bounded and measurable, that is, there exist positive constantsλ₀, µ₀ andδ,0< λ₀≤µ₀,0< δ <1, such that for anyz∈Q_T,Q_T = Ω×(0, T),ξ ∈R^(q+1)N,

λ0|ξ|² ≤A^αβ(x)ξαξβ ≤µ0|ξ|², lim

R→0ηR

A^αβ(z)

= 0,

B_ij^αβ(z, u)

≤δλ₀.

(H2) Letu∈W₂^1,1(QT,R^N), f_i^α(z, u), gi(z, u, Xu) satisfy

|f_i^α(z, u)| ≤µ1

|u|^γ2 +fⁱ(z)

,

|g_i(z, u, Xu)| ≤µ2

|Xu|^2(1−γ1)+|u|^γ−1+gⁱ(z)

,

where fⁱ(z) ∈ L^σ(Q_T) (σ > Q+ 2), gⁱ(z) ∈ L^τ(Q_T) (τ > Q+ 2), γ = ^2(Q+2)_Q . Let g = gⁱ

, f = fⁱ

, ˜q = ^2(Q+2)_Q+4 , the definitions ofV M O(QT),ηR A^αβ(z)

,W₂^1,1(QT,R^N) andQ see Section 2.

Ifu∈W₂^1,1(QT,R^N) and for anyψ∈C₀^∞(Ω,R^N), Z Z

QT

h

uⁱ_tψⁱ+a^αβ_ij X_αψⁱX_βu^ji dz=

Z Z

QT

g_iψⁱ+f_i^αX_αψⁱ dz, we say that u is a weak solution to (1.1).

Now the main results of the paper are the following.

Theorem 1.1. Let u ∈ W₂^1,1(Q_T,R^N) be a weak solution to (1.1). Sup- pose that assumptions (H1)–(H2) are satisfied. Then there exists a positive constant ε0 such that for any p∈[2,2 + ˜qε0), we have

u∈L

pγ

2,Q+2−p+pκ

loc QT,R^N

, Xu∈L^{p,Q+2−p+pκ}_loc QT,R^N , where κ= min

n

1−^q(Q+2)˜ _2τ ,1− ^Q+2_σ o .

Theorem 1.2. Under the assumptions in Theorem 1.1, we have u∈C_loc^κ QT,R^N

, where κ= minn

1−^q(Q+2)˜ _2τ ,1− ^Q+2_σ o .

This paper is organized as follows. In Section 2, we introduce H¨ormander’s vector fields and some related function spaces, and then recall several tech- nical lemmas. Section 3 is devoted to establishing higher L^p regularity for gradient of weak solutions to (1.1). The proofs of Theorem 1.1 and Theorem 1.2 are given in Section 4.

YAN DONG AND DONGYAN LI

2. Preliminaries

Denote the commutator of vector fieldsX₁, . . . , X_q by X_β = [X_βd,[X_βd−1, . . .[X_β2, X_β1]. . .]], for|β|=d.

We recall thatdis the length of Xβ.

Definition 2.1. If for every x0 ∈ Ω ⊂ Rⁿ, {X_β(x0)}_|β|≤s spans Rⁿ, then we say that the system X = (X₁, . . . , X_q) satisfies H¨ormander’s condition of step s.

Following [26], we assume that H¨ormander type vector fields X₁, . . . , X_q are free up to the order s. For every multi-index I = (i1, i2, . . . , i_k), the length of I is defined by|I|=k. Ifi_k ≤q, then we set

X_I =XiiXi2. . . Xi_k.

Definition 2.2 (Carnot–Carath´eodory distance). Let Ω be a bounded do- main in Rⁿ. An absolutely continuous curve γ : [0, T] → Ω is called a sub-unit curve with respect to X = (X1,· · ·, Xq), if γ⁰(t) exists for a.e.

t∈[0, T] and satisfies hγ⁰(t), ξi² ≤

q

X

j=1

hX_j(γ(t)), ξi², for any ξ ∈Rⁿ.

We denote the length of this curve byl_S(γ) =T. Given anyx, y∈Ω, let Φ(x, y) be the collection of all sub-unit curves connecting x and y, define the Carnot–Carath´eodory distance which induced by X by

dX(x, y) = inf{l_S(γ) :γ ∈Φ(x, y)}.

With this distance, we denote a metric ball of radiusR centered atx₀ by BR(x0) =B(x0, R) ={x∈Ω :d(x0, x)< R}.

If one does not need to consider the center of the ball, then we also write BR instead ofB(x0, R).

It is well known that the doubling property (see [22]) for metric balls holds true, i.e., there exist positive constants cD and RD, such that for any x₀ ∈Ω, 0<2R < R_D,

|B(x₀,2R)| ≤cD|B(x₀, R)|.

Furthermore, it follows that for any R≤R_D and t∈(0,1),

|B_tR| ≥c⁻¹_D t^Q|B_R|.

The numberQ= log₂cD is called a locally homogeneous dimension rela- tive to Ω. Clearly, Q≥n.

As in [26], we assume that there exist two positive constants c1 and c2, such that

c1R^Q ≤ |B_R| ≤c2R^Q.

Throughout this paper, we denote z₀ = (x₀, t₀) ∈ Q_T ⊂Rⁿ⁺¹. A para- bolic cylinder with vertex at z0 is defined by

QR(z0) =BR(x0)×

t0−R²

2 , t0+R² 2

.

Let us denote I_R(t₀) =

t₀−^R₂², t₀+^R₂²i

, and the parabolic boundary of QRby

∂_pQ_R(z₀) =

∂B_R(x₀)×

t₀−R²

2 , t₀+R² 2

∪

B_R(x₀)×

t₀−R² 2

. We denote the Lebesgue measure of B(x, R) in the n-dimensional space by|B(x, R)|, and the Lebesgue measure ofQR(z0) in the (n+1)-dimensional space by|Q_R(z0)|. To simplify the notations, in the sequel,Q_R(z0),B_R(x0), I_R(t0),

s q

P

i=1

|X_iu|² and (x, t) are written as Q_R, B_R, I_R, |Xu| and z, re- spectively.

For any (x, t),(y, s)∈QT, we denote d_p((x, t),(y, s)) =p

d_X(x, y)²+|t−s|

as the parabolic distance in Q_T.

Definition 2.3 (Sobolev space). Letm, k be 0 or 1, 1≤p <+∞. The set W_p^m,k QT,R^N

={u;Xαu, ∂_t^ru∈L^p(QT),0≤ |α| ≤m,0≤r ≤k}

is called a parabolic Sobolev space related to H¨ormander’s vector fields with the norm

kuk_Wm,k

p = X

|α|≤m

kX_αuk_Lp+X

r≤k

kX_t^ruk_Lp.

Definition 2.4 (Morrey space). Let 1 ≤ p < +∞, λ ≥ 0. We say that f ∈L^p(Q_T) belongs to the Morrey space L^p,λ(Q_T,R^N) if

kfk_Lp,λ= sup

z0∈Q_T,0≤ρ≤d₀

1 ρ^λ

Z Z

QT∩Qρ(z0)

|f|^pdz

!¹_p

<∞, whered0 is the diameter of Q_T.

Definition 2.5 (Campanato space). Let 1 ≤p < +∞, λ≥0. A function f ∈L^p_loc(QT) is said to belong to the Campanato spaceL^p,λ(QT,R^N) if

kfk_Lp,λ= [f]_p,λ+kfk_Lp <+∞, where [f]_p,λ = sup

z0∈QT,0<ρ<d0

1 ρ^λ

RR

QT∩Qρ(z0)

f−f_QT∩Qρ(z0)

pdz¹_p

< +∞, d0 is the diameter of QT, and f_QT∩Qρ(z0)= _|Q ¹

T∩Qρ(z0)|

RR

QT∩Qρ(z0)f(z)dz.

YAN DONG AND DONGYAN LI

Definition 2.6 (H¨older space). For any 0< k ≤1, the space C^k(Q_T,R^N) is the set of functions satisfying

[f]_k,Q

T , sup

z1,z2∈Q_T,z16=z₂

|f(z1)−f(z2)|

dp(z1, z2)^k <∞.

We also define a norm by

|f|_k,Q

T ,sup

QT

|f|+ [f]_k,Q

T.

Definition 2.7 (BM O and V M O spaces). For anyf ∈L¹_loc(Q_T), we set ηR(f)

= sup

z0∈Q,0≤ρ≤R

1

|Q_T ∩Q_ρ(z₀)|

Z Z

QT∩Q_ρ(z0)

f(z)−f_QT∩Qρ(z0)(z) dz

! ,

where f_QT∩Q_ρ(z0) = _|Q ¹

T∩Qρ(z0)|

RR

QT∩Qρ(z0)f(z)dz. If sup

R>0

ηR(f)<+∞, we say that f belongs to BM O(QT) (Bounded Mean Oscillation). Moreover, if η_R(f) → 0 as R → 0, we say that f belongs to V M O(Q_T) (Vanishing Mean Oscillation).

Lemma 2.8 (See [17]). LetH(ρ) be a nonnegative increasing function, and for any 0< ρ < R≤R0=dist(x0, ∂Ω),

H(ρ)≤A ρ

R ˜a

+ε

H(R) +BR^˜b,

where A,˜aand˜b are positive constants with˜a >˜b. Then there exist positive constants ε1 = ε1(A,˜a,˜b) and c = c(A,˜a,˜b), such that for any ε < ε1, it follows

H(ρ)≤c ρ

R ^˜b

H(R) +Bρ^˜b

.

Lemma 2.9 (Iterative lemma, see [4]). Let ϕ(t) be a nonnegative bounded function on[T0, T1], whereT1 > T0 ≥0. Suppose that for any s, t:T0 ≤t <

s≤T₁, ϕ satisfies

ϕ(t)≤θϕ(s) + A

(s−t)^α +B,

where θ, A, B and α are nonnegative constants, and θ < 1. Then for any T0≤ρ < R≤T1, one has

ϕ(ρ)≤c A

(R−ρ)^α +B

, where c depends only on α andθ.

Lemma 2.10 (Sobolev–Poincar´e inequality, see [19, 20]). For any open domain Ω⁰, Ω¯⁰ ⊂⊂Ω, there exist positive constants R0 and c, such that for any0< R≤R₀, B_R⊂Ωand u∈C^∞(B_R), we have

(2.1)

1

|B_R| Z

B_R

|u−uR|^p0dx ¹

p0

≤cR 1

|B_R| Z

B_R

|Xu|^pdx ¹_p

, where 1< p < Q,1≤p⁰≤ _Q−p^pQ ,uR= _|B¹

R|

R

BRu(x)dx,R0 and cdepend on Ω⁰ and Ω. In particular, if u∈C₀^∞(BR), then

(2.2)

1

|B_R| Z

BR

|u|^p0dx ¹

p0

≤cR 1

|B_R| Z

BR

|Xu|^pdx ¹_p

.

Lemma 2.11 (Reverse H¨older inequality, see [13]). Let g,ˆ fˆbe nonnegative onQ_T and satisfy

ˆ

g∈L^qˆ(Q_T) and ˆf ∈L^q0(Q_T), 1<q < qˆ ⁰.

Assume that there exist constantsˆb >1andθˆsuch that for anyQ2R⊂⊂QT

the following inequality holds 1

|Q_R| Z Z

Q_R

ˆ

g^qˆdz≤ˆb



 1 Q_4R/3

Z Z

Q_4R/3

ˆ gdz

!qˆ

+ 1

Q_4R/3

Z Z

Q_4R/3

fˆ^qˆdz





+ˆθ 1 Q_4R/3

Z Z

Q_4R/3

ˆ g^qˆdz,

Then there exist positive constantsε0 andθ0 =θ0(ˆq, QT) such that ifθ < θˆ 0, then for any pˆ∈[ˆq,qˆ+ε0), ˆg∈L^p_loc^ˆ (QT), and

1

|Q_R| Z Z

QR

ˆ g^pˆdz

_p¹_ˆ

≤c

"

1

|Q_2R| Z Z

Q_2R

ˆ g^qˆdz

¹_qˆ +

1

|Q_2R| Z Z

Q_2R

fˆ^pˆdz ¹_pˆ#

,

where c and ε0 depend onˆb,θ,ˆ qˆand Q.

Lemma 2.12 (See [12]). The spaces

L^2,Q+2+2κ(QT,R^N) and C^κ( ¯QT,R^N) (0< κ <1)are topologically and algebraically isomorphic.

3. Higher integrability

We first introduce two cutoff functionsξ(x) andη(t)(see [8]) such that for any 0< ρ < R,B_R⊂Ω,

ξ(x)∈C₀^∞(BR), 0≤ξ ≤1, |Xξ| ≤ C

R−ρ and ξ = 1 inBρ;

YAN DONG AND DONGYAN LI

η(t) =







2t−2 t0−^R₂² R²−ρ² , t∈

t₀− ^R₂², t₀−^ρ₂² , 1, t∈h

t0− ^ρ₂², t0+^R₂² i

. Setting _|B¹

R|

R

BRξ²dx=N1, we denote the average ofu(x, t) onB_R by

¯ u(t) =

Z

BR

ξ²dx −1Z

BR

uξ²dx= 1 N1|B_R|

Z

BR

uξ²dx.

Lemma 3.1. Let u ∈ W₂^1,1(Q_T,R^N) be a weak solution to (1.1). Then u∈L^γ_loc(Q_T), and for any Q_R⊂⊂Q_T, we have

(3.1)

Z Z

Q_R

|u|^γdz≤csup

IR

Z

B_R

|u|²dx _Q² Z Z

Q_R

|Xu|²dz+c|Q_R|. Proof. By Young’s inequality and H¨older’s inequality,

|¯u(t)|=

1 N1|B_R|

Z

B_R

uξ²dx (3.2)

≤ 1

N1|B_R|

ε Z

BR

|u|²ξ²dx+cε

Z

BR

ξ²dx

≤ ε

N₁|B_R| Z

BR

|u|²ξ²dx+cε, Z

BR

|u|²ξ²dx≤ Z

BR

|u|^γdx ²_γZ

BR

ξ^Q+2dx ^γ−2_γ (3.3)

≤(N₁|B_R|)^γ−2γ Z

BR

|u|^γdx ²_γ

. Then by (3.2) and (3.3), we get

Z Z

QR

|¯u(t)|^γdz≤ |B_R| Z

IR

|¯u(t)|^γdt (3.4)

≤ |B_R| Z

IR

ε N₁|B_R|

Z

BR

|u|²ξ²dx+c_ε γ

dt

≤ ε

N1γ|B_R|^γ−1 Z

IR

Z

BR

|u|²ξ²dx γ

dt+c_ε|Q_R|

≤ ε

N₁^γ|B_R|^γ−1 sup

IR

Z

BR

|u|²dx ^γ₂ Z

IR

Z

BR

|u|²ξ²dx ^γ₂

dt+cε|Q_R|

≤ ε

N1γ|B_R|^γ−1 sup

IR

Z

BR

|u|²dx ^γ₂

· Z

IR

(N₁|B_R|)

γ−2 γ

Z

BR

|u|^γdx ²_γ!^γ₂

dt+c_ε|Q_R|

≤ ε N1

γ+2

2 |B_R|^γ2 sup

IR

Z

B_R

|u|²dx ^γ₂ Z Z

Q_R

|u|^γdz+cε|Q_R|. By (2.1) and H¨older’s inequality, one has

Z

BR

|u−u(t)|¯ ^2(Q+1)Q−1 dx= Z

BR

|u−u(t)|¯ ^2(Q+1)Q _Q−1^Q

dx (3.5)

≤ Z

BR

X

|u−u(t)|¯ ^2(Q+1)Q

dx _Q−1^Q

≤

2(Q+ 1) Q

Z

BR

|u−u(t)|¯ ^Q+2Q Xu dx

_Q−1^Q

≤c Z

BR

|u−u(t)|¯ ^γdx ¹

2Z

BR

|Xu|²dx ¹

2

!_Q−1^Q .

By H¨older’s inequality and (3.5), Z Z

QR

|u−u(t)|¯ ^γdz

≤ Z

I_R

Z

B_R

|u−u(t)|¯ ² _Q¹Z

B_R

|u−u(t)|¯

2(Q+1) Q−1

^Q−1_Q dt

≤c Z

IR

Z

BR

|u−u(t)|¯ ² _Q¹Z

BR

|u−u(t)|¯ ^γdx ¹₂Z

BR

|Xu|²dx ¹₂

dt

≤csup

I_R

Z

BR

|u|²dx _Q¹ Z

IR

Z

BR

|u−u(t)|¯ ^γdx ¹₂Z

BR

|Xu|²dx ¹₂

dt

≤csup

IR

Z

BR

|u|²dx

_Q¹Z Z

QR

|u−u(t)|¯ ^γdz

¹₂Z Z

QR

|Xu|²dz ¹₂

. So we have

(3.6)

Z Z

QR

|u−u(t)|¯ ^γdz ≤csup

I_R

Z

BR

|u|²dx _Q² Z Z

QR

|Xu|²dz.

And by (3.4) and (3.6), Z Z

QR

|u|^γdz

≤c Z Z

QR

|u−u(t)|¯ ^γdz+c Z Z

QR

|¯u(t)|^γdz

≤csup

IR

Z

BR

|u|²dx _Q² Z Z

QR

|Xu|²dz

YAN DONG AND DONGYAN LI

+ ε

N1

γ+2

2 |B_R|^γ2 sup

IR

Z

B_R

|u|²dx ^γ₂ Z Z

Q_R

|u|^γdz+cε|Q_R|. Choosingε small enough, then we get

Z Z

QR

|u|^γdz≤csup

IR

Z

BR

|u|²dx _Q² Z Z

QR

|Xu|²dz+c|Q_R|. Lemma 3.2. Let u ∈W₂^1,1(QT,R^N) be a weak solution to (1.1). Then for any0< ρ < R, Q_R⊂⊂Q_T, we have

sup

Iρ

Z

Bρ

|u−u(t)|¯ ²dx+ Z Z

Qρ

|Xu|²dz (3.7)

≤ c

(R−ρ)² Z Z

QR

|u−u(t)|¯ ²dz+c Z Z

QR

|u|^γ+|f|²+|g|^q˜ dz.

Proof. Let B_ρ ⊂ B_R ⊂ Ω, multiplying both sides of (1.1) by the test function (u−u(t))¯ ξ²(x)η(t), and integrating on

Q⁰_R=B_R(x0)×

t0−R² 2 , s

(s≤t0+^R₂²), we get Z Z

Q⁰_R

h

uⁱ_t+X_α^∗

a^αβ_ij X_βu^j i

uⁱ−u(t)¯ ξ²ηdz (3.8)

= Z Z

Q⁰_R

[g_i+X_α^∗f_i^α] uⁱ−u(t)¯ ξ²ηdz.

By (H1), one has Z Z

Q⁰_R

h

uⁱ_t+X_α^∗

a^αβ_ij X_βu^ji

uⁱ−u(t)¯ ξ²ηdz

= Z Z

Q⁰_R

h

uⁱ_t uⁱ−u(t)¯

ξ²η+a^αβ_ij Xβu^jXα uⁱ−u(t)¯ ξ²ηi

dz

= Z Z

Q⁰_R

h

uⁱ_t uⁱ−u(t)¯

ξ²η+a^αβ_ij ξ²ηX_αuⁱX_βu^j + 2a^αβ_ij uⁱ−u(t)¯

ξηX_αξX_βu^ji dz

= Z Z

Q⁰_R

1 2

uⁱ−u(t)¯

2η

t

ξ²−1 2

uⁱ−u(t)¯

2ξ²ηt

+A^αβδijξ²ηXαuⁱX_βu^j

dz

+ Z Z

Q⁰_R

B_ij^αβξ²ηX_αuⁱX_βu^j + 2A^αβδ_ij uⁱ−u(t)¯

ξηX_αξX_βu^jdz

+ Z Z

Q⁰_R

2B_ij^αβ uⁱ−u(t)¯

ξηX_αξX_βu^jdz,

and Z Z

Q⁰_R

[gi+X_α^∗f_i^α] uⁱ−u(t)¯ ξ²ηdz

= Z Z

Q⁰_R

g_i uⁱ−u(t)¯

ξ²η+f_i^αX_α uⁱ−u(t)¯ ξ²η

dz

= Z Z

Q⁰_R

gi uⁱ−u(t)¯

ξ²η+f_i^αξ²ηXαuⁱ+ 2ξη uⁱ−u(t)¯

f_i^αXαξ dz.

By the above, (3.8) can be written as

Z Z

Q⁰_R

1 2

uⁱ−u(t)¯

2η

t

ξ²+A^αβδijξ²ηXαuⁱX_βu^j

dz (3.9)

= Z Z

Q⁰_R

1 2

uⁱ−u(t)¯

2ξ²ηt−B_ij^αβξ²ηXαuⁱX_βu^j

−2 Z Z

Q⁰_R

A^αβδ_ij uⁱ−u(t)¯

ξηX_αξX_βu^jdz

−2 Z Z

Q⁰_R

B_ij^αβ uⁱ−u(t)¯

ξηX_αξX_βu^jdz

+ Z Z

Q⁰_R

gi uⁱ−u(t)¯

ξ²η+f_i^αξ²ηXαuⁱ+ 2ξη uⁱ−u(t)¯

f_i^αXαξ dz.

By (H2) and Young’s inequality, Z Z

Q⁰_R

gi uⁱ−u(t)¯ ξ²ηdz (3.10)

≤µ₂ Z Z

Q⁰_R

|Xu|^2(1−1γ)+|u|^γ−1+gⁱ(z)

uⁱ−u(t)¯ ξ²ηdz

≤ε Z Z

Q⁰_R

|Xu|²ξ²ηdz+cε

Z Z

Q⁰_R

|u−u(t)|¯ ^γξ²ηdz +c_ε

Z Z

Q⁰_R

|u|^γξ²ηdz+c_ε Z Z

Q⁰_R

|g|^q˜ξ²ηdz, Z Z

Q⁰_R

f_i^αξ²ηXαuⁱ+ 2ξη uⁱ−u(t)¯

f_i^αXαξ dz (3.11)

≤µ₁ Z Z

Q⁰_R

|u|^γ2 +fⁱ(z)

ξ²ηX_αuⁱdz

+ 2µ₁ Z Z

Q⁰_R

ξη(u−u(t))¯

|u|^γ2 +fⁱ(z) X_αξdz

YAN DONG AND DONGYAN LI

≤2ε Z Z

Q⁰_R

|Xu|²ξ²ηdz+c_ε Z Z

Q⁰_R

|u|^γξ²ηdz+c_ε Z Z

Q⁰_R

|f|²ξ²ηdz

+ 2ε Z Z

Q⁰_R

|u−u(t)|¯ ²|Xξ|²ηdz.

Inserting (3.10) and (3.11) into (3.9), and by (H1) and Young’s inequality, we get

Z Z

Q⁰_R

1

2|u−u(t)|¯ ²η

t

ξ²dz+λ₀ Z Z

Q⁰_R

|Xu|²ξ²ηdz

≤ Z Z

Q⁰_R

1

2|u−u(t)|¯ ²ξ²ηtdz+δλ0

Z Z

Q⁰_R

|Xu|²ξ²ηdz

+c_ε Z Z

Q⁰_R

|u−u(t)|¯ ²|Xξ|²ηdz+ 5ε Z Z

Q⁰_R

|Xu|²ξ²ηdz

+cε

Z Z

Q⁰_R

|u−u(t)|¯ ^γξ²ηdz+cε

Z Z

Q⁰_R

|u|^γξ²ηdz+cε

Z Z

Q⁰_R

|g|^q˜ξ²ηdz

+cε

Z Z

Q⁰_R

|f|²ξ²ηdz+ 2ε Z Z

Q⁰_R

|u−u(t)|¯ ²|Xξ|²ηdz

≤ Z Z

Q⁰_R

|u−u(t)|¯ ² 1

2ξ²η_t+c_ε|Xξ|²η+ 2ε|Xξ|²η

dz

+ (δλ₀+ 5ε) Z Z

Q⁰_R

|Xu|²ξ²ηdz+c_ε Z Z

Q⁰_R

|u−u(t)|¯ ^γξ²ηdz +cε

Z Z

Q⁰_R

|u|^γξ²ηdz+cε

Z Z

Q⁰_R

|f|²+|g|^q˜ ξ²ηdz.

Then Z

BR

1

2|u−u(t)|¯ ²ξ²ηdx+ (λ0−δλ0−5ε) Z Z

Q⁰_R

|Xu|²ξ²ηdz

≤ Z Z

Q⁰_R

|u−u(t)|¯ ² 1

2ξ²ηt+cε|Xξ|²η+ 2ε|Xξ|²η

dz

+c_ε Z Z

Q⁰_R

|u−u(t)|¯ ^γξ²ηdz+c_ε Z Z

Q⁰_R

|u|^γξ²ηdz

+c_ε Z Z

Q⁰_R

|f|²+|g|^q˜ ξ²ηdz.

Choosingεsmall enough such thatλ0−δλ0−5ε >0, then by properties of ξ, η, we get

sup

Iρ

Z

Bρ

|u−u(t)|¯ ²dx+ Z Z

Qρ

|Xu|²dz

≤c Z Z

Q⁰_R

|u−u(t)|¯ ²

ξ²

R²−ρ² + cη

(R−ρ)² + cη (R−ρ)²

dz

+c Z Z

Q⁰_R

|u−u(t)|¯ ^γ+|u|^γ+|f|²+|g|^q˜ ξ²ηdz

≤ c

(R−ρ)² Z Z

Q⁰_R

|u−u(t)|¯ ²dz+c Z Z

Q⁰_R

|u|^γ+|f|²+|g|^q˜

ξ²ηdz.

Lemma 3.3. Let u ∈W₂^1,1(QT,R^N) be a weak solution to (1.1). Then for any0< ρ < R, Q_R⊂⊂Q_T, we have

Z Z

Qρ

|u|²dz (3.12)

≤ cR⁴ (R−ρ)²

Z Z

QR

|Xu|²dz+cR² Z Z

QR

|u|^γ+|f|²+|g|^q˜ dz.

Proof. Let Bρ ⊂ BR ⊂ Ω, multiplying both sides of (1.1) by the test functionuξ²(x)η(t) and integrating on

Q⁰_R=B_R(x₀)×

t₀−R² 2 , s

(s≤t0+^R₂²), one has Z Z

Q⁰_R

1 2

uⁱ

2η

t

ξ²dz (3.13)

= Z Z

Q⁰_R

1 2

uⁱ

2ξ²ηtdz− Z Z

Q⁰_R

A^αβδijξ²ηXαuⁱX_βu^jdz

− Z Z

Q⁰_R

B_ij^αβξ²ηX_αuⁱX_βu^jdz

−2 Z Z

Q⁰_R

A^αβδ_ijuⁱξηX_αξX_βu^jdz−2 Z Z

Q⁰_R

B_ij^αβuⁱξηX_αξX_βu^jdz

+ Z Z

Q⁰_R

giuⁱξ²η+f_i^αξ²ηXαuⁱ+ 2ξηuⁱf_i^αXαξ dz.

By (H2) and Young’s inequality, Z Z

Q⁰_R

giuⁱξ²ηdz

≤c_ε Z Z

Q⁰_R

|Xu|²ξ²ηdz+ (2ε+µ₂) Z Z

Q⁰_R

|u|^γξ²ηdz+c_ε Z Z

Q⁰_R

|g|^q˜ξ²ηdz,

Z Z

Q⁰_R

f_i^αξ²ηX_αuⁱ+ 2ξηuⁱf_i^αX_αξ dz

≤µ₁ Z Z

Q⁰_R

|u|^γ2 +fⁱ(z)

ξ²ηX_αuⁱdz

YAN DONG AND DONGYAN LI

+ 2µ₁ Z Z

Q⁰_R

ξηuⁱ

|u|^γ2 +fⁱ(z) X_αξdz

≤2ε Z Z

Q⁰_R

|Xu|²ξ²ηdz+c_ε Z Z

Q⁰_R

|u|^γξ²ηdz+c_ε Z Z

Q⁰_R

|f|²ξ²ηdz

+ 2ε Z Z

Q⁰_R

|u|²|Xξ|²ηdz.

Putting the above into (3.13) and by (H1), we get Z

BR

1

2|u|²ξ²ηdx

≤ Z Z

Q⁰_R

1

2|u|²ξ²ηtdz+µ0

Z Z

Q⁰_R

|Xu|²ξ²ηdz+δλ0

Z Z

Q⁰_R

|Xu|²ξ²ηdz

+ 2ε Z Z

Q⁰_R

|u|²|Xξ|²ηdz+ 3c_ε Z Z

Q⁰_R

|Xu|²ξ²ηdz

+ (2ε+µ2+cε) Z Z

Q⁰_R

|u|^γξ²ηdz+cε

Z Z

Q⁰_R

|g|^q˜ξ²ηdz

+ 2ε Z Z

Q⁰_R

|Xu|²ξ²ηdz+cε

Z Z

Q⁰_R

|f|²ξ²ηdz+ 2ε Z Z

Q⁰_R

|u|²|Xξ|²ηdz

≤ Z Z

Q⁰_R

|u|² 1

2ξ²η_t+ 4ε|Xξ|²η

dz

+ Z Z

Q⁰_R

|Xu|²(µ₀+δλ₀+ 3c_ε+ 2ε)ξ²ηdz +

Z Z

Q⁰_R

|u|^γ(2ε+µ2+cε)ξ²ηdz+cε

Z Z

Q⁰_R

|f|²+|g|^˜q ξ²ηdz.

By properties ofξ, η, Z Z

Qρ

|u|²dz≤ |I_ρ|sup

Iρ

Z

Bρ

|u|²dx

≤ρ² Z Z

QR

|u|²

ξ²η_t+ 8ε|Xξ|²η dz

+ 2ρ² Z Z

QR

|Xu|²(µ₀+δλ₀+ 3c_ε+ 2ε)ξ²ηdz + 2ρ²

Z Z

QR

|u|^γ(2ε+µ2+cε)ξ²ηdz+ 2cερ² Z Z

QR

|f|²+|g|^q˜ ξ²ηdz

≤ Z Z

QR

|u|²

2ρ²ξ²

R²−ρ² + 8cερ²η (R−ρ)²

dz+cε

ρ²(R−ρ)² (R−ρ)²

Z Z

QR

|Xu|²ξ²ηdz

+ 2ρ² Z Z

QR

|u|^γ(2ε+µ2+cε)ξ²ηdz+ 2cερ² Z Z

QR

|f|²+|g|^q˜ ξ²ηdz

≤θ Z Z

QR

|u|²dz+ cεR⁴ (R−ρ)²

Z Z

QR

|Xu|²dz

+c_εR² Z Z

QR

|u|^γ+|f|²+|g|^q˜ dz,

whereθ= _R^2ρ2²−ρ^ξ22 + ^8cερ2η

(R−ρ)². By choosingεsmall enough such thatθ∈(0,1),

then by Lemma 2.9 we obtain (3.12).

Theorem 3.4. Let u ∈ W₂^1,1(Q_T,R^N) be a weak solution to (1.1). Then there exists a positive constantε0 such that for anyp∈[2,2 + ˜qε0), we have u∈L

pγ 2

loc(Q_T), Xu∈L^p_loc(Q_T), and for any Q_2R⊂⊂Q_T, 1

|Q_R| Z Z

QR

|Xu|²+|u|^γp₂ dz

≤c

"

1

|Q_2R| Z Z

Q2R

|Xu|²+|u|^γ dz

^p₂

+ 1

|Q_2R| Z Z

Q2R

|f|²+|g|^q˜+ 1

p 2dz

# .

Proof. By (3.7) and (2.1), sup

I_4R/5

Z

B_4R/5

|u−u(t)|¯ ²dx

!¹₂ (3.14)

≤ c

R² Z Z

QR

|u−u(t)|¯ ²dz+c Z Z

QR

|u|^γdz ¹₂

+c Z Z

QR

|f|²+|g|^q˜ dz

¹

2

≤c Z Z

QR

|Xu|²+|u|^γ dz

¹₂ +c

Z Z

QR

|f|²+|g|^q˜ dz

¹₂ .

By H¨older’s inequality, (2.1) and (2.2), Z

I4R/5

Z

B4R/5

|u−u(t)|¯ ²dx

!¹

2

dt (3.15)

≤ Z

IR

Z

BR

|u−u(t)|¯ ^q˜dx _{2 ˜}¹_qZ

BR

|u−u(t)|¯ ^γdx _2γ¹

dt

≤cR^1q˜ Z

IR

Z

BR

|Xu|^q˜dx _{2 ˜}¹_qZ

BR

|Xu|²dx ¹₄

dt