Calderón–Zygmund estimates for a class of quasilinear parabolic equations

Nonlinear Analysis 176 (2018) 84–99

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Nonlinear Analysis www.elsevier.com/locate/na

Calderón–Zygmund estimates for a class of quasilinear parabolic equations Fengping Yao Department of Mathematics, Shanghai University, Shanghai 200444, China

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abstract

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Article history: Received 4 September 2017 Accepted 11 June 2018 Communicated by Enzo Mitidieri

In this paper we obtain the following local Calderón–Zygmund estimates G (|f |) ∈ Lqloc (ΩT ) ⇒ G (|∇u|) ∈ Lqloc (ΩT )

for any q ≥ 1

of weak solutions for a class of quasilinear parabolic equations

MSC: 35B45 35K55

ut − div (a (|∇u|) ∇u) = div (a (|f |) f ) in ΩT ,

Keywords: Calderón–Zygmund Gradient Regularity Divergence Parabolic Quasilinear p-Laplace

where G(t) =

∫t 0

τ a(τ ) dτ for t ≥ 0. We remark that

(

G(t) = |t|p log 1 + |t|

)

for p > 2

satisfies the given conditions in this work. Moreover, we would like to point out that our results improve the known results for such equations. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In this paper we are concerned with the local Lp -type regularity estimates of weak solutions for the following quasilinear parabolic equations ut − div (a (|∇u|) ∇u) = div (a (|f |) f )

in ΩT = Ω × (0, T ),

n

(1.1) 1

where Ω is an open bounded domain in R and the function a : (0, ∞) → (0, ∞) ∈ C (0, ∞) satisfies ta′ (t) ta′ (t) ≤ sup =: sa < ∞. t>0 a(t) t>0 a(t)

0 ≤ ia =: inf

(1.2)

Especially when a(t) = tp−2 and then p = sa + 2 = ia + 2, (1.1) is reduced to the parabolic p-Laplace equation ( ) ( ) p−2 p−2 ut − div |∇u| ∇u = div |f | f in ΩT . (1.3) E-mail address: [email protected]. https://doi.org/10.1016/j.na.2018.06.008 0362-546X/© 2018 Elsevier Ltd. All rights reserved.

F. Yao / Nonlinear Analysis 176 (2018) 84–99

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Now we denote g(t) = ta(t)

(1.4)

and ∫

t

G(t) =

∫ τ a(τ )dτ =

0

t

for t ≥ 0.

g(τ )dτ

(1.5)

0

Then from (1.2) it is easy to check that g(t) is strictly increasing and continuous over [0, +∞)

(1.6)

G(t) is convex, G(0) = 0 and increasing over [0, +∞).

(1.7)

and

Lp -type regularity is the fundamental theory of partial differential equations, which plays an important role in the theory of elliptic and parabolic equations, and is the basis for the existence and uniqueness of solutions. Lp estimates for the second order elliptic and parabolic problems have been obtained by different techniques. Many authors (see [9,11,17–19,22,24,25,27,29,30]) have studied Lq (q ≥ p) estimates of the gradient of weak solutions for ( ) ( ) p−2 p−2 div |∇u| ∇u = div |f | f in Ω (1.8) and the general case with different assumptions on the coefficients and domains. Recently, Cianchi and Maz’ya [14,15] proved global Lipschitz regularity for the Dirichlet and Neumann elliptic boundary value problems of the form div (a (|∇u|) ∇u) = f

in Ω .

(1.9)

Moreover, Cianchi and Maz’ya [16] obtained a sharp estimate for the decreasing rearrangement of the length of the gradient for the Dirichlet and Neumann elliptic boundary value problems of (1.9). Different from the elliptic case (1.8), (1.3) is not homogeneous even if f ≡ 0, which is one of the most difficulties (see [6]). Kinnunen and Lewis [23] obtained a reverse H¨older inequality of the gradient for weak solutions of (1.3) and the general case. Furthermore, Acerbi and Mingione [1] obtained the following estimates in Sobolev spaces p

p

|f | ∈ Lqloc (Ω ) ⇒ |∇u| ∈ Lqloc (Ω )

for any q ≥ 1

(1.10)

with ∫

pq

[(∫

|∇u| dz ≤ C Qr

p

|∇u| dz Q2r

)q

∫ |f |

+

pq

]p/2 + 1dz

,

(1.11)

Q2r

where Q2r = B2r × (−4r2 , 4r2 ] ⊂ ΩT , for weak solutions of (1.3) and the general case. Moreover, many authors [4,7,20,28] proved Lipschitz regularity, Caccioppoli-type estimate and existence of weak solutions for (1.1), respectively. The purpose of this paper is to extend (1.10) and (1.11) for weak solutions of (1.1). In particular, we are interested in the following local Calder´on–Zygmund estimates like G (|f |) ∈ Lqloc (ΩT ) ⇒ G (|∇u|) ∈ Lqloc (ΩT )

for any q ≥ 1

(1.12)

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with the estimate ∫

q

[∫

q

(G (|∇u|)) dz ≤ C QR (z0 )

(∫

(

(G (|f |)) + 1dz +

G

Q4R (z0 )

Q4R (z0 )

) )q ] sa2+2 1 , |u| dz R

(1.13)

where z0 = (x0 , t0 ), Q4R (z0 ) ⊂ ΩT and C is independent of u and f . Actually, if a(t) = tp−2 , (1.12) is reduced to (1.10) in [1]. Moreover, we remark that this paper is a natural follow-up to the papers [8,35]. In [35] we proved that G (|f |) ∈ Lqloc (Ω ) ⇒ G (|∇u|) ∈ Lqloc (Ω )

for any q ≥ 1

with the estimate ∫

(

)q G (|∇u|) dx ≤ C

{∫

BR

(

)q G (|f |) dx +

(

(∫

B2R

G B2R

) )q } 1 |u| dx R

for weak solutions of the following quasilinear elliptic equations div (a (|∇u|) ∇u) = div (a (|f |) f )

in Ω .

(1.14)

Furthermore, Byun and Cho [8] obtained the global gradient estimates in the frame of the general Orlicz spaces for weak solutions of (1.14) in a Reifenberg domain. As one of natural generalizations of Sobolev spaces, Orlicz–Sobolev spaces have been extensively studied since it was introduced by Orlicz [31] (see [2,5,10,13,26,33,34]). The theory of Orlicz spaces plays a crucial role in a very wide spectrum (see [32]). Here for the reader’s convenience, we will give some definitions and preliminary lemmas on the general Orlicz–Sobolev spaces. We say that a function G : [0, +∞) → [0, +∞) is a Young function if it is convex and G(0) = 0. Definition 1.1. A Young function G is called an N -function if 0 < G(t) < ∞ for t > 0 and lim

t→+∞

G(t) t = lim = +∞. t→0+ G(t) t

(1.15)

˜ of a Young function G is defined as Moreover, the Young conjugate G ˜ = sup {st − G(s)} G(t)

for t ≥ 0.

s≥0

˜ is also an N -function. Actually, if G is an N -function, then G Definition 1.2. A Young function G is said to satisfy the global ∆2 condition, denoted by G ∈ ∆2 , if there exists a positive constant K such that for every t > 0, G(2t) ≤ KG(t).

(1.16)

Moreover, a Young function G is said to satisfy the global ∇2 condition, denoted by G ∈ ∇2 , if there exists a number θ > 1 such that G(t) ≤

G(θt) 2θ

for every t > 0.

Examples 1.3. (1) G1 (t) = (1 + t) log(1 + t) − t ∈ ∆2 , but G1 (t) ̸∈ ∇2 . (2) G2 (t) = et − t − 1 ∈ ∇2 , but G2 (t) ̸∈ ∆2 . ( ) (3) G3 (t) = tp log 1 + t ∈ ∆2 ∩ ∇2 , p > 1.

(1.17)

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Remark 1.4. A Young function G ∈ ∆2 ∩ ∇2 if and only if there exist constants A2 ≥ A1 > 0 and α1 ≥ α2 > 1 such that ( s )α 1 ( s )α2 G(s) ≤ ≤ A2 for 0 < t ≤ s. (1.18) A1 t G(t) t Actually, (1.18) implies that (1.15), α2 G(θ1 t) ≤ A−1 1 θ1 G(t)

and

G(θ2 t) ≤ A2 θ2α1 G(t)

(1.19)

for 0 < θ1 ≤ 1 ≤ θ2 < ∞. Lemma 1.5. If G is an N -function, then G satisfies the following Young’s inequality ˜ st ≤ G(s) + G(t) for any s, t ≥ 0. Furthermore, if G ∈ ∇2 ∩ ∆2 is an N -function, then G satisfies the following Young’s inequality with ϵ > 0 ˜ st ≤ ϵG(s) + C(ϵ)G(t) for any s, t ≥ 0. Definition 1.6. Assume that G is an N -function. Then the Orlicz class K G (Ω ) is the set of all measurable functions h : Ω → R satisfying ∫ G(|h|)dx < ∞. Ω G

The Orlicz space L (Ω ) is the linear hull of K G (Ω ) endowed with the Luxemburg norm ) } { ( ∫ |h(x)| dx ≤ 1 . ∥h∥LG (Ω) = inf k > 0 : G k Ω ⏐ { } Furthermore, the Orlicz–Sobolev space W 1,G (Ω ) = h ∈ LG (Ω ) ⏐ ∇h ∈ LG (Ω ) , endowed with the norm ∥h∥W 1,G (Ω) = ∥h∥LG (Ω) + ∥∇h∥LG (Ω) . The subspace W01,G (Ω ) is the closure of C0∞ (Ω ) in W 1,G (Ω ). Actually, K G (Ω ) = LG (Ω ) for G ∈ ∆2 (see [2]). For weak solutions of (1.1) are usually defined as below. 1,G ∞ 2 G Definition 1.7. Assume that f ∈ LG loc (ΩT ). A function u ∈ Lloc (0, T ; L (Ω )) ∩ Lloc (0, T ; Wloc (Ω )) is a local weak solution of (1.1) if for any compact subset K of Ω and for any subinterval [t1 , t2 ] of (0, T ) we have ∫ ∫ t2 ∫ ⏐ t2 ∫ t2 ∫ { } ⏐ a (|f |) f · ∇φdxdt uφdx⏐ + −uφt + a (|∇u|) ∇u · ∇φ dxdt = − K

t1

t1

2 for any φ ∈ L∞ loc (0, T ; L (K)) ∩

K

t1

K

1,G LG loc (0, T ; W0 (K)).

Now we state the main result of this work. Theorem 1.8. If u is a local weak solution of (1.1) in ΩT with (1.2) and G (|f |) ∈ Lqloc (ΩT ) for any q ≥ 1, then we have G (|∇u|) ∈ Lqloc (ΩT ) with the estimate (1.13).

2. Proof of the main result In this section we will finish the proof of the main result, Theorem 1.8. We first recall the following result.

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Lemma 2.1 (see [14], Proposition 2.9). If a(t) satisfies (1.2) and G is defined in (1.5), then we have (1) G(t) is strictly convex N -function and ˜ (g(t)) ≤ C0 G(t) for t ≥ 0 and some C0 > 0. G

(2.1)

(2) G(t) ∈ ∆2 . Lemma 2.2. Assume that a(t) satisfies (1.2) and G(t) is defined in (1.5). (1) For any t > 0 we find that θ ia ≤

a(θt) ≤ θsa for θ ≥ 1. a(t)

(2.2)

(2) For any t > 0 we have 2 + ia ≤

tG′ (t) tg(t) = ≤ 2 + sa < ∞ G(t) G(t)

(2.3)

G(θt) ≤ θsa +2 for θ ≥ 1. G(t)

(2.4)

and θia +2 ≤ (3) G(t) ∈ ∇2 . Proof . (1) From (1.2) we find that ia a′ (θt) sa ≤ ≤ θt a(θt) θt

for any

θ, t > 0.

By integrating the above inequality over [1, θ] for θ ≥ 1, we have ∫ θ ∫ θ ′ ∫ θ ia a (θt) sa dθ ≤ dθ ≤ dθ, θt a(θt) 1 1 1 θt which implies that a(t)θia ≤ a(θt) ≤ a(t)θsa . (2) (1.2) implies that ia a(t) ≤ ta′ (t) ≤ sa a(t)

for any t > 0.

Then we have t

∫

t

∫

τ 2 a′ (τ ) dτ = t2 a(t) − 2

τ a(τ ) dτ ≤

ia 0

t

∫

τ a(τ ) dτ

0

0

and t

∫

∫ τ a(τ ) dτ ≥

sa 0

t

τ 2 a′ (τ ) dτ = t2 a(t) − 2

0

∫

t

τ a(τ ) dτ, 0

which implies that 2 ≤ 2 + ia ≤

tG′ (t) t2 a(t) = ∫t ≤ 2 + sa < ∞. G(t) τ a(τ ) dτ 0

Moreover, the proof of (2.4) is totally similar to (2.2).

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(3) From (2.3) we obtain (log G(t))′ ≥ (2 + ia )(log t)′ . 1

For every t > 0, integrating the above inequality over [t, θ1 t] where θ1 = 2 1+ia > 1, we conclude that G(θ1 t) ≥ θ12+ia G(t) = 2θ1 G(t), which implies that G ∈ ∇2 . This finishes our proof.

□

Remark 2.3. If a(t) satisfies (1.2), then from Lemma 2.2 we find that G(t) satisfies (1.18) and (1.19), where α1 = 2 + sa ≥ α2 = 2 + ia . Moreover, from Lemma 2.2 we have L2+sa (Ω ) ⊂ LG (Ω ) ⊂ L2+ia (Ω ) ⊂ L2 (Ω ).

(2.5)

Moreover, we shall give the following two important results. Lemma 2.4. Assume that a(t) satisfies (1.2) and G(t) is defined in (1.5). Then there exists C1 = C1 (ia , sa ) > 0 we have a (|ξ|) ξ · ξ ≥ C1 G (|ξ|)

ξ ∈ Rn .

for any

(2.6)

Proof . We first find that ∫ ξa (|ξ|) = 0

1

d [sξa (|sξ|)] =ξ ds

1

∫

∫

1

a (|sξ|) ds + 0

a′ (|sξ|) |ξ| sξds,

0

which implies that 2

∫

1

∫

1

3

a′ (|sξ|) s|ξ| ds 0 0 ∫ 1 ∫ 1 2 2 ≥ |ξ| a (|sξ|) ds + ia |ξ| a (|sξ|) ds 0 0 ∫ 1 2 = (1 + ia )|ξ| a (|sξ|) ds

a (|ξ|) ξ · ξ = |ξ|

a (|sξ|) ds +

(2.7)

0

in view of (1.2). Moreover, from (2.2) we find that ∫ ∫ 1 ∫ 1 a (|sξ|) ds ≥ a (|sξ|) ds ≥ 0

3 4

1 3 4

1 a 2

(

) ( )sa +3 1 1 |ξ| ds ≥ a (|ξ|) , 2 2

(2.8)

since a g (|sξ|) a (|sξ|) |sξ| a (|sξ|) = = ≥ |sξ| |sξ|

(1

)1

2 |ξ|

|ξ|

2 |ξ|

(

1 = a 2

1 |ξ| 2

)

in view of (1.6). Therefore, (2.7) and (2.8) imply that ( )sa +3 ( )sa +3 1 1 2 a (|ξ|) ξ · ξ ≥ (1 + ia ) a (|ξ|) |ξ| ≥ (1 + ia ) G (|ξ|) , 2 2 since ∫ G (|ξ|) =

|ξ|

∫

0

in view of (1.6). Thus, this finishes our proof. □

|ξ|

2

ta(t)dt ≤ a (|ξ|) |ξ|

g(t)dt = 0

(2.9)

F. Yao / Nonlinear Analysis 176 (2018) 84–99

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Lemma 2.5 (see [35], Lemma 2.4). Assume that a(t) satisfies (1.2) and G(t) is defined in (1.5). Then there exists C = C(ia , sa ) > 0 we have G (|ξ − η|) ≤ C [ξa (|ξ|) − ηa (|η|)] · (ξ − η) for any ξ, η ∈ Rn and ϵ ∈

(0, A−1 1 ),

(2.10)

where A1 is defined in Remark 1.4.

Moreover, we first give the following result. Lemma 2.6. Assume that u is a local weak solution of (1.1) in ΩT with Q2 ⊂ ΩT and (1.2). Then we have {∫ } ∫ ∫ 2 G (|∇u|) dz ≤ C G(|u|) + |u| dz + G(|f |)dz . (2.11) Q1

Q2

Q2

Proof . We may as well select the test function φ = ζu, where ζ ∈ C0∞ (Rn+1 ) is a cut-off function satisfying 0 ≤ ζ ≤ 1,

ζ ≡ 1 in Q1 ,

ζ ≡ 0 in Rn+1 /Q2 .

Then by Definition 1.7, after a direct calculation we show the resulting expression as I1 + I2 = I3 + I4 , where I1 =

1 2 ∫

∫ B2

I2 = Q2

I3 =

1 2

∫

⏐t=4 ⏐ 2 |u(x, t)| ζ(x, t)dx⏐ = 0, t=−4

ζa (|∇u|) ∇u · ∇udz, ∫ 2 ua (|∇u|) ∇u · ∇ζdz, ζt u dz − Q2

∫Q2

I4 = −

ζa (|f |) f · ∇u + ua (|f |) f · ∇ζdz. Q2

Estimate of I2 . It follows from Lemma 2.4 and the definition of ζ that ∫ ∫ ζG (|∇u|) dz ≥ C1 I2 ≥ C1 G (|∇u|) dz. Q1

Q2

Estimate of I3 . From Lemmas 1.5 and 2.1 (1), we conclude that ∫ 2 |I3 | ≤ C a (|∇u|) |∇u| |u| + |u| dz Q2 ∫ ∫ τ 2 ˜ (a (|∇u|) |∇u|) dz + C(τ ) ≤ G G (|u|) + |u| dz C0 Q2 Q2 ∫ ∫ 2 ≤τ G (|∇u|) dz + C(τ ) G (|u|) + |u| dz for any Q2

τ > 0.

Q2

Estimate of I4 . Similarly to the estimate of I3 , we have ∫ ∫ |I4 | ≤ τ G (|∇u|) dz + C(τ ) G (|f |) + G (|u|) dz Q2

for any

τ > 0.

Q2

Combining all the estimates of Ii (1 ≤ i ≤ 4), we deduce that ∫ ∫ ∫ C1 G (|∇u|) dz ≤ 2τ G (|∇u|) dz + C(τ ) Q1

Q2 C1 4 ,

2

G (|f |) + G (|u|) + |u| dz.

Q2

Selecting suitable τ such that τ = and removing the right-hand side first integral in the inequality above by a covering and iteration argument (see Lemma 4.1, Chapter 2 in [12] or, Lemma 2.1, Chapter 3 in [21]), we deduce that (2.11) is true. This finishes the proof. □

F. Yao / Nonlinear Analysis 176 (2018) 84–99

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In the following it is sufficient to finish the proof of Theorem 1.8 as an a priori estimate, therefore assuming a priori that G (|∇u|) ∈ Lqloc . After an approximation argument by choosing |∇u|k := min{k, |∇u|} like the one in Page 2946 of [3] we can remove this assumption. Next, we shall give one important lemma (the iteration-covering procedure). To start with, let u be a local weak solution of (1.1). By a scaling and shifting argument we may as well assume that QR (z0 ) = Q1 in Theorem 1.8. We write ∫ ∫ 1 2 λ0 = − G (|∇u|) dz + − G (|f |) dz + 1, (2.12) δ Q2 Q2 where δ ∈ (0, 1) will be chosen later. Moreover, for any domain Q ⊂ Rn+1 we write ∫ ∫ 1 J [Q] = − G (|∇u|) dz + − G (|f |) dz δ Q Q

(2.13)

and the level set E(λ) = {z ∈ Q1 : G (|∇u|) > G (λ)} .

(2.14)

Lemma 2.7. Given λ ≥ λ∗ =: 20(n+2)/2 λ0 , (2.15) { } there exists a family of disjoint cylinders Q0i i∈N with zi = (xi , ti ) ∈ E(λ), 0 < ρi = ρ(zi ) ≤ 1/10 and ( 2 ) λ 0 2 Qi = Qzi ρ , ρi , G(λ) i where Q(θ, ρ) = Bρ × (−θ, θ], such that [ ] [ ] J Q0i = G(λ), J Qzi (θ, ρ) < G(λ) for any Qzi (θ, ρ) ⊃ Q0i ,

(2.16)

and ⋃

Q1i ∪ negligible set,

(2.17)

) ( 2 2 2 λ ρ , 5jρi . = Qzi (5j) G(λ) i

(2.18)

E(λ) ⊂

i∈N

where for j = 1,2 , Qji

=:

5jQ0i

Moreover, we have ⏐ 0⏐ ⏐Qi ⏐ ≤

2 G(λ)

(∫

) ∫ 1 G (|∇u|) dz + G (|f |) dz . δ {z∈Q0 :G(|f |)>δG(λ)/4} {z∈Q0i :G(|∇u|)>G(λ)/4} i

Proof . Let λ ≥ λ∗ = 20(n+2)/2 λ0 . For any w ∈ Q1 and 1/10 ≤ ρ ≤ 1 we have ∫ ∫ |Q2 | ⏐ ( )⏐ − ( 2 ) G (|∇u|) dz ≤ ⏐ ⏐ − G (|∇u|) dz λ λ2 Qw G(λ) ρ2 ,ρ ρ2 , ρ ⏐ Q2 ⏐Qw G(λ) ∫ G(λ) ≤ 20n+2 2 − G (|∇u|) dz λ Q2 and then ∫ − (

2

n+2 ) G (|f |) dz ≤ 20

λ Qw G(λ) ρ2 ,ρ

∫ G(λ) − G (|f |) dz. λ2 Q2

(2.19)

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Furthermore, from the definition of λ0 we find that [ ( 2 )] G(λ) λ G(λ) J Qw ρ2 , ρ ≤ 20n+2 λ20 2 = λ2∗ 2 ≤ G(λ) G(λ) λ λ for any w ∈ Q1 and 1/10 ≤ ρ ≤ 1, which implies that )] [ ( 2 λ 2 ρ ,ρ sup ≤ G(λ). sup J Qw G(λ) w∈Q1 1/10≤ρ≤1

(2.20)

For a.e. w ∈ E(λ), from Lebesgue’s differentiation theorem we know that [ ( 2 )] λ lim J Qw ρ2 , ρ > G(λ), ρ→0 G(λ) which implies that there exists some ρ ∈ (0, 1] satisfying )] [ ( 2 λ 2 ρ ,ρ > G(λ). J Qw G(λ) Therefore, from (2.20) we can select a radius ρw ∈ (0, 1/10] such that { [ ( 2 )] } λ 2 ρw =: max ρ | J Qw ρ ,ρ = G(λ), 0 < ρ ≤ 1/10 , G(λ) which implies that )] [ )] [ ( 2 ( 2 λ λ 2 2 ρ , ρw ρ ,ρ < G(λ) J Qw = G(λ) and J Qw G(λ) w G(λ)

for ρw < ρ ≤ 1.

{ { } Thus, using Vitali’s covering lemma, we can find a family of disjoint cylinders Q0i i∈N = Qzi ( 2 )} λ 2 , zi = (xi , ti ) ∈ E(λ) and ρi = ρ(zi ) ≤ 1/10 so that (2.16) and (2.17) are true. Furthermore, G(λ) ρi , ρi i∈N

from the first equality of (2.16) we find that ∫ ∫ [ 0] 1 J Qi = − G (|∇u|) dz + − G (|f |) dz = G(λ). δ Q0 Q0 i

i

Therefore, by splitting the two integrals above as follows we have ∫ ∫ 1 G(λ)|Q0i | ≤ G (|∇u|) dz + G (|f |) dz + G(λ)|Q0i |/2, 0 :G(|∇u|)>G(λ)/4 0 :G(|f |)>δG(λ)/4 δ z∈Q z∈Q { } { } i i □

which implies that the desired estimate (2.19) holds.

Remark 2.8. Given λ ≥ λ∗ , from (2.16) and (2.18) in Lemma 2.7 we obtain ∫ ∫ − G (|∇u|) dz ≤ G(λ) and − G (|f |) dz ≤ δG(λ) ∀ j = 1, 2. j

j

Qi

(2.21)

Qi

Moreover, from Theorem 2.2 in [20] we can obtain the following interior Lipschitz regularity. Lemma 2.9. If v is a local weak solution of vt − div (a (|∇v|) ∇v) = 0

in Qr ,

then there exists a constant C = C(G(1), ia , sa ) > 0 such that a [∫ ] 2+s 2 sup G (|∇v|) ≤ C − G(|∇v|) + 1dz . Qr/2

Qr

(2.22)

(2.23)

F. Yao / Nonlinear Analysis 176 (2018) 84–99

Proof . From Theorem 2.2 in [20] we have { sup ρ(|∇v|), sup |∇v|

min

Qr/2 n 2−n 2

where ρ(t) = (G(t)) t

} 2

93

∫ 2 ≤ C− |∇v| + G(|∇v|)dz,

Qr/2

Qr

and C = (G(1), ia , sa ), which implies that { } ∫ sup ρ(|∇v|), sup |∇v|

min

Qr/2

2

≤ C− G(|∇v|) + 1dz,

Qr/2

(2.24)

Qr

since ∫

∫

2

2

|∇v| dz ≤

|∇v| dz + |Qr | {z∈Qr :|∇v(z)|≥1}

Qr

∫

2+ia

|∇v|

≤

dz + |Qr |

{z∈Qr :|∇v(z)|≥1}

∫ ≤C

G(|∇v|) + 1dz

(2.25)

Qr

in view of (2.4) and the fact that ia ≥ 0. Since n

n

ρ(t) = (G(t)) 2 t2−n ≥ C(t2+ia ) 2 t2−n ≥ Ctn t2−n = Ct2

for any

t≥1

in view of (2.4) and the fact that ia ≥ 0, from (2.24) we find that [∫ ] 21 sup |∇v| ≤ C − G(|∇v|) + 1dz . Qr/2

Qr

Furthermore, from (2.4) we deduce that a ] 2+s [∫ 2 sup G (|∇v|) ≤ C − G(|∇v|) + 1dz .

Qr/2

Qr

Thus, we finish the proof. □ Lemma 2.10. Assume that u is a local weak solution of (1.1) in ΩT with Q2 ⊂ ΩT and (1.2). If v is the weak solution of (2.22) in Q2 with v = u on ∂p Q2 , then we have {∫ } ∫ ∫ G (|∇v|) dz ≤ C G(|∇u|)dz + G(|f |)dz . (2.26) Q2

Q2

Q2

Proof . Noting that u and v are the weak solutions of (1.1) and (2.22) respectively, we may as well select the test function φ = v − u. Then a direct calculation shows the resulting expression as I1 + I2 = I3 + I4 + I5 , where I1 I2 I3 I4 I5

∫ 1 2 = |v(x, 4) − u(x, 4)| dx ≥ 0, 2 B2 ∫ = a (|∇v|) ∇v · ∇v dz, ∫Q2 = a (|∇v|) ∇v · ∇u dz, Q2 ∫ = a (|∇u|) ∇u · ∇v − a (|∇u|) ∇u · ∇u dz, Q2 ∫ = a(|f |) f · ∇v − a(|f |) f · ∇u dz. Q2

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Estimate of I2 . From Lemma 2.4 we have ∫ I2 ≥ C1

G (|∇v|) dz. Q2

Estimate of I3 . From Lemmas 1.5 and 2.1 (1) we have ∫ ∫ τ ˜ (a (|∇v|) |∇v|) dz + C(τ ) I3 ≤ G G (|∇u|) dz C0 Q2 Q2 ∫ ∫ ≤τ G (|∇v|) dz + C(τ ) G (|∇u|) dz for any Q2

Estimate of Ii ( 4 ≤ i ≤ 5). Similarly to the estimate of I3 , we have ∫ ∫ I4 ≤ τ G (|∇v|) dz + C(τ ) G (|∇u|) dz, Q2 Q2 ∫ ∫ G (|∇u|) + G (|f |) dz G (|∇v|) dz + C(τ ) I5 ≤ τ Combining the estimates of Ii (1 ≤ i ≤ 5), we deduce that ∫ ∫ ∫ C1 G (|∇v|) dz ≤ 3τ G (|∇v|) dz + C Q2 C1 6 ,

τ > 0.

for any

Q2

Q2

Selecting suitable τ =

τ > 0.

Q2

Q2

G (|∇u|) + G (|f |) dz.

Q2

we deduce that (2.26) is true. This finishes our proof.

□

Moreover, we shall give the following important comparison result. Lemma 2.11. For any ϵ > 0, there exists a small δ = δ(ϵ) > 0 such that if u is a local weak solution of (1.1) in ΩT with (1.2), Q2 ⊂ ΩT , ∫ ∫ − G (|∇u|) dz ≤ 1 and − G (|f |) dz ≤ δ, (2.27) Q2

Q2

then there exists a weak solution v of (2.22) with v = u on ∂p Q2 such that ∫ − G (|∇u − ∇v|) dz ≤ ϵ.

(2.28)

Q2

Moreover, there exists a constant N0 > 1 such that sup G (|∇v|) ≤ N0 .

(2.29)

Q3 2

Proof . The conclusion (2.29) can easily follow from (2.23), (2.26) and (2.27). Next, we set out to prove (2.28). Since u and v are weak solutions of (1.1) and (2.22) respectively, then by selecting the test function φ = u − v, after a direct calculation we show the resulting expression as I1 + I2 = I3 , where I1 =

1 2 ∫

∫

(u − v)2 (x, 4)dx ≥ 0,

B2

[ ] a (|∇u|) ∇u − a (|∇v|) ∇v · (∇u − ∇v) dz,

I2 = Q2

∫ I3 = −

a (|f |) f · (∇u − ∇v) dz. Q2

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Estimate of I2 . From Lemma 2.5 we observe that ∫ I2 ≥ C G (|∇u − ∇v|) dz. Q2

Estimate of I3 . From Lemmas 1.5 and 2.1 (1), we conclude that ∫ |I3 | ≤ a (|f |) |f | |∇u − ∇v| dz Q2 ∫ ∫ ˜ (a (|f |) |f |) dz ≤τ G (|∇u − ∇v|) dz + C(τ ) G Q2 Q2 ∫ ∫ ≤τ G (|∇u − ∇v|) dz + C(τ ) G (|f |) dz Q2

Q2

for any τ > 0. Combining the estimates of I2 and I3 , we obtain ∫ ∫ ∫ C G (|∇u − ∇v|) dz ≤ τ G (|∇u − ∇v|) dz + C(τ ) Q2

Q2

G (|f |) dz.

Q2

Selecting small constants τ, δ > 0 such that 0 < δ ≪ τ < 1 and using (2.27), we deduce that ∫ ∫ − G (|∇u − ∇v|) dz ≤ C(τ )− G (|f |) dz ≤ C(τ )δ = ϵ Q2

Q2

by selecting δ satisfying the last inequality above. This completes our proof. □ For each λ ≥ 1, we define uλ (x, t) =

fλ (x, t) =

) ( λ2 (5ρi )2 t u 5ρi x, G(λ) λ5ρi

,

(2.30)

( ) λ2 f 5ρi x, G(λ) (5ρi )2 t

(2.31)

λ

and aλ (ξ) =

a(λξ)

.

G(λ) λ2

(2.32)

Moreover, we denote ∫ gλ (t) = taλ (t) and Gλ (t) =

t

∫ τ aλ (τ )dτ =

0

t

gλ (τ )dτ = 0

G (λt) G (λ)

for t ≥ 0,

(2.33)

which implies that Gλ (1) =

G (λ) =1 G (λ)

(2.34)

and Gλ (t) satisfies (2.4). Furthermore, we shall give the following result. Lemma 2.12. Assume that λ ≥ λ∗ . For any ϵ > 0, there exists a small δ = δ(ϵ) > 0 such that if u is a local weak solution of (1.1) in ΩT with Q2i ⊂ ΩT , then there exists N0 > 1 such that ∫ sup G (|∇v|) ≤ N0 G (λ) and − G (|∇(u − v)|) dz ≤ ϵG (λ) , (2.35) Q1 i

Q2 i

where v is the weak solution of (2.22) in Q2i with v = u on ∂p Q2i .

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Proof . From the definitions of uλ , fλ and aλ , we find that uλ is a local weak solution of (uλ )t − div (aλ (|∇uλ |) ∇uλ ) = div (aλ (|fλ |) fλ )

in Q2 .

Moreover, from Remark 2.8 one can readily check that ∫ ∫ − Gλ (|∇uλ |)dz ≤ 1 and − Gλ (|fλ |)dz ≤ δ. Q2

Q2

Then according to Lemma 2.11, there exists a weak solution v of { (vλ )t − div (aλ (|∇vλ |) ∇vλ ) = 0 vλ = uλ

in Q2 , on ∂p Q2

such that sup Gλ (|∇vλ |) ≤ N0 Q3/2

∫ and − Gλ (|∇(uλ − vλ )|)dz ≤ ϵ. Q2

Then changing variables, we recover the conclusion of Lemma 2.12. Thus we complete the proof.

□

Now we are ready to prove the main result, Theorem 1.8. Proof . From Lemma 2.12 and the fact that G ∈ ∆2 ∩ ∇2 is convex, for any λ ≥ λ∗ we have ⏐ ⏐ ⏐{z ∈ Q1i : G(|∇u|) > 2C∗ N0 G(λ)}⏐ ⏐ ⏐ ⏐ ⏐ ≤ ⏐{z ∈ Q1i : G(|∇(u − v)|) > N0 G(λ)}⏐ + ⏐{z ∈ Q1i : G(|∇v|) > N0 G(λ)}⏐ ⏐ ⏐ = ⏐{z ∈ Q1i : G(|∇(u − v)|) > N0 G(λ)}⏐ ∫ 1 G(|∇(u − v)|)dz ≤ Cϵ|Q2i | ≤ Cϵ|Q0i |, ≤ N0 G(λ) Q2 i

since G(a + b) ≤

1 1 G(2a) + G(2b) ≤ C∗ G(a) + C∗ G(b) 2 2

for any a, b ≥ 0,

which follows from Lemma 2.7 that ⏐ ⏐ ⏐{z ∈ Q1i : G(|∇u|) > 2C∗ N0 G(λ)}⏐ (∫ ) ∫ Cϵ 1 ≤ G (|∇u|) dz + G (|f |) dz . G (λ) δ {z∈Q0 :G(|f |)>δG(λ)/4} {z∈Q0i :G(|∇u|)>G(λ)/4} i Recalling the fact that the cylinders {Q0i } are disjoint and ⋃ Q1i ∪ negligible set ⊃ E(λ) = {z ∈ Q1 : G (|∇u|) > G(λ)} i∈N

for any λ ≥ λ∗ , and then summing up on i ∈ N in the inequality above, we have ⏐{ }⏐ ⏐ z ∈ Q1 : G(|∇u|) > 2C∗ N0 G(λ) ⏐ ∑ ⏐{ }⏐ ⏐ z ∈ Q1i : G(|∇u|) > 2C∗ N0 G(λ) ⏐ ≤ i

Cϵ ≤ G(λ)

(∫

1 G (|∇u|) dz + δ {z∈Q2 :G(|∇u|)>G(λ)/4}

)

∫ G (|f |) dz {z∈Q2 :G(|f |)>δG(λ)/4}

(2.36)

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for any λ ≥ λ∗ . Moreover, recalling the standard measure theory, for any q > 1 we compute ∫ ∫ q q−1 q [G(|∇u|)] dz = q(2C∗ N0 ) [G (λ)] |{z ∈ Q1 : G(|∇u|) > 2C∗ N0 G(λ)}| dG(λ) Q1 q

λ>0 λ∗

∫

= q(2C∗ N0 )

0

+ q(2C∗ N0 )

q−1 ⏐⏐

[G (λ)] ∞

∫

q

⏐ {z ∈ Q1 : G(|∇u|) > 2C∗ N0 G(λ)}⏐dG(λ)

q−1 ⏐⏐

[G (λ)]

⏐ {z ∈ Q1 : G(|∇u|) > 2C∗ N0 G(λ)}⏐dG(λ)

λ∗

=: J1 + J2 . Estimate of J1 . From (2.4), (2.12) and H¨ older’s inequality we find that { [(∫ ) 21 ]}q ∫ 1 q J1 ≤ C[G(λ∗ )] ≤ C G G (|∇u|) dz + G (|f |) dz + 1 δ Q2 Q2 a [(∫ )q ∫ ] 2+s 2 q ≤ C2 , G(|∇u|)dz + (G (|f |)) + 1dz Q2

Q2

where C2 = C2 (n, ia , sa , q, δ). Estimate of J2 . Recalling that ∫ ∫ β |g| dz = (β − α) Rn+1

∞

λ

β−α−1

∫

α

|g| dzdλ {

}

z∈Rn+1 :|g|>λ

0

for any β > α > 1, from (2.36) we have ∫ ∞ ∫ q−2 J2 ≤ Cϵ [G (λ)] G (|∇u|) dzdG(λ) 0 {z∈Q2 :G(|∇u|)>G(λ)/4} ∫ ∞ ∫ Cϵ q−2 + G (|f |) dzdG(λ) [G (λ)] δ 0 {z∈Q2 :G(|f |)>δG(λ)/4} ∫ ∫ q q (G (|f |)) dz, (G (|∇u|)) dz + C4 ≤ C3 ϵ Q2

Q2

where C3 = C3 (n, ia , sa , q) and C4 = C4 (n, ia , sa , q, δ, ϵ). Combining the estimates of J1 and J2 , we obtain ∫ ∫ ∫ q q q [G(|∇u|)] dz ≤ C3 ϵ (G (|∇u|)) dz + C4 (G (|f |)) dz Q1

Q2

Q2

)q

[(∫ + C2

G(|∇u|)dz

∫ Q2

∫ ≤ C3 ϵ

(G (|f |)) + 1dz

+

Q2

a ] 2+s 2

q

)q

[(∫

q

(G (|∇u|)) dz + C5

G(|∇u|)dz

Q2

∫

q

(G (|f |)) + 1dz

+

Q2

a ] 2+s 2

,

Q2

where C5 = C5 (n, ia , sa , q, δ, ϵ). Selecting suitable ϵ such that C3 ϵ = 1/2, and reabsorbing at the right-side first integral in the inequality above by a covering and iteration argument (see Lemma 4.1, Chapter 2 in [12] or, Lemma 2.1, Chapter 3 in [21]), we have a [(∫ )q ∫ ] 2+s ∫ 2 q q [G(|∇u|)] dz ≤ C G(|∇u|)dz + (G (|f |)) + 1dz . Q1

Q2

Q2

Furthermore, from Lemma 2.6 and H¨ older’s inequality we obtain [(∫ )q ∫ ∫ 2 q [G(|∇u|)] dz ≤ C G(|u|) + |u| dz + Q1

Q4

)q G(|u|)dz

Q4

(G (|f |)) + 1dz

Q4

[(∫ ≤C

a ] 2+s 2

q

∫

q

(G (|f |)) + 1dz

+ Q4

a ] 2+s 2

,

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since ∫

2

∫

2

|u| dz = Q4

∫

|u| dz + |Q4 | ≤ {z∈Q4 :|u(z)|≥1}

|u| {z∈Q4 :|u(z)|≥1}

2+ia

∫ dz + |Q4 | ≤ C

G(|u|) + 1dz Q4

in view of (2.4) and the fact that ia ≥ 0. Moreover, from the above inequality and Lemma 2.6 we obtain {∫ } ∫ ∫ G (|∇u|) dz ≤ C G(|u|)dz + G(|f |) + 1dz . Q1

Q4

Q4

Then by a scaling and shifting argument we can finish the proof of the main result. □ Acknowledgment The author wishes to thank the anonymous reviewer for many valuable comments and suggestions to improve the expressions. This work is supported in part by the NSFC (11471207). References

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