Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x,t) -Laplacian type

Nonlinear Analysis 180 (2019) 225–235

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Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type Junjie Zhang a,b , Maomao Cai c , Shenzhou Zheng a ,∗ a

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei 050016, China c Department of Mathematics, Weber State University, Ogden UT 84408-2517, United States b

article

info

Article history: Received 30 May 2018 Accepted 26 October 2018 Communicated by S. Carl MSC: 35D30 35K55

abstract We prove a global weighted Lorentz estimate of the spatial gradients of weak solution to the Cauchy–Dirichlet problem for asymptotically regular parabolic equations of p(x, t)-Laplacian type in a Reifenberg flat domain. It is mainly assumed that the associated regular nonlinearity is measurable with respect to the time variable and satisfies the (δ, R)-vanishing condition with respect to the spatial variables. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Parabolic equations of p(x, t)-Laplacian type Asymptotically regular Weighted Lorentz spaces (δ, R)-vanishing condition Reifenberg flat domain

1. Introduction Let ΩT = Ω × [0, T ] ⊂ Rn+1 be a parabolic cylindrical domain with bounded open subset Ω ⊂ Rn≥1 and T ∈ (0, ∞). The aim of this present article is to attain a global estimate in the framework of weighted Lorentz spaces to the following zero Cauchy–Dirichlet problem of nonlinear parabolic equation { p(x,t)−2 ut − div a(x, t, Du) = div (|F | F ), in ΩT , (1.1) u = 0, on ∂ΩT , where ∂ΩT = (∂Ω × (0, T )) ∪ (Ω × {t = 0}) as a usual parabolic boundary, the vectorial-valued function a(x, t, ξ) : Rn × [0, T ] × Rn → Rn is asymptotically regular (for details see Definition 3.1), and F ∈ Lp(·) (ΩT ; Rn ) is a given vectorial-valued function with p(·) : Rn+1 → (1, ∞) being a continuous function satisfying the following assumptions ∗

Corresponding author. E-mail addresses: [email protected] (J. Zhang), [email protected] (M. Cai), [email protected] (S. Zheng).

https://doi.org/10.1016/j.na.2018.10.013 0362-546X/© 2018 Elsevier Ltd. All rights reserved.

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• H1. There exist two constants ϑ1 and ϑ2 such that 2n < ϑ1 ≤ p(x, t) ≤ ϑ2 < ∞, n+2

∀ (x, t) ∈ Rn+1 .

(1.2)

• H2. (strong log-H¨ older continuity) There exists a nondecreasing function φ : [0, ∞) → [0, ∞) such that (1) =0 (1.3) lim φ(ρ) log ρ ρ→0+ with |p(z1 ) − p(z2 )| ≤ φ(d(z1 , z2 )), (1.4) √ where the parabolic distance d(z1 , z2 ) = max{|x1 − x2 |, t1 − t2 } for all zi = (xi , ti ) ∈ Rn+1 , i = 1, 2. It is a well-known fact that partial differential equations with variable growth have received a great deal of attention from many mathematicians. Areas of its application actually include the study of electro-rheological fluids, temperature-dependent viscosity fluids, porous medium, homogenization of strongly anisotropic material, image restoration and so on. First of all, let us recall some recent studies on the topic of nonlinear parabolic equations of p(x, t)p(·) Laplacian type. Considering the case that F = 0, Antontsev–Zhikov [4] proved that |Du| ∈ L1+ε loc (ΩT ) with some ε ∈ (0, ∞) for the weak solutions of parabolic equation ut − div(|Du|

p(x,t)−2

Du) = 0

in ΩT .

p(·)

As for the case that F ̸= 0, if |F | ∈ L1 (ΩT ), then problem (1.1) has a unique weak solution u satisfying p(·) |Du| ∈ L1 (ΩT ). The existence of such weak solution is ensured by a result of Antontsev and Shmarev [2,3]. Recently, Baroni and B¨ ogelein [7] further studied a general nonlinear parabolic equation ut − div(a(x, t)|Du| p(·)

p(x,t)−2

p(x,t)−2

Du) = div(|F |

and proved that |Du| ∈ Lγloc (ΩT ), γ ∈ (1, ∞) when |F | when the exponent p(·) in (1.5) is a constant, the result p

p(·)

F)

in ΩT ,

(1.5)

∈ Lγloc (ΩT ). At this stage we mention that

p

|F | ∈ Lγloc (ΩT ) ⇒ |Du| ∈ Lγloc (ΩT ) p(·)

has been obtained in [1]. Moreover, the global regularity estimates of (1.1) for |Du| ∈ Lq(·) (ΩT ) and p(·) p(·) γ,q |Du| ∈ Lω (ΩT ) have been established recently by Byun–Ok in [12] when |F | ∈ Lq(·) (ΩT ) and by p(·) Bui–Duong in [8] when |F | ∈ Lγ,q ω (ΩT ), respectively. For more information on this topic see [6,5,13,16]. In this article, we are interested in looking for the weaker regular conditions on the nonlinearity a(x, t, ξ) and the underlying domain Ω so that the spatial gradient of its weak solution is as integrability in a certain function space as the inhomogeneous term. The problems with asymptotically regular nonlinearity are a class of very important issues for nonlinear partial differential equations. Note that the concept of asymptotically regular was originated from Chiop– Evans’s work [14]. Since then there is a large of literature on the topic of asymptotically regular problem, see [9,11,10,15–18]. Motivated by above-mentioned investigations, in this article we are devoted to the regularity for asymptotically regular problems with non-standard growth. Here, we would like to point out that Byun–Oh–Wang [11] studied the global Calder´on–Zygmund estimates for inhomogeneous asymptotically regular elliptic and parabolic problems of divergence form in a bounded Reifenberg flat domain by converting the given asymptotically regular problems to suitable regular problems via an appropriate transformation. Also, inspired by [11,8], we will utilize the main result in [8] to obtain the required weighted Lorentz estimate for the asymptotically regular parabolic problem of p(x, t)-Laplacian type in a Reifenberg flat domain. Here, we employ a method of transformation introduced in [11], and

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partially modify this approach to fit for the setting of generalized Lebesgue and Sobolev space with variable exponent. A main point of this paper is that we impose the so-called strong log-H¨ older continuity condition on the variable exponent p(·), which enables us to employ many crucial tools in classical analysis, such as the Hardy–Littlewood maximal boundedness, the Sobolev embedding and the Poincar´e inequality. Another main point is that the nonlinearity a(x, t, ξ) is asymptotically regular, which leads to that there is a more general kind of parabolic behavior near infinity. The rest of the paper is organized as follows. In Section 2, we review the generalized Lebesgue and Sobolev space with the variable exponent and the weighted Lorentz space. In Section 3, we introduce the definition of asymptotically regular operators and state the main result. Section 4 is devoted to proving our main conclusion via an appropriate transformation of asymptotically regular problem to a regular one. 2. Preliminaries We start this section with some notations and related facts. For a typical parabolic point (x, t) ∈ Rn+1 and r > 0, the Euclidean ball in Rn with center x and radius r will be denoted by Br (x) = {y ∈ Rn : |x − y| < r}. Also, the time derivative and the spatial gradient of u will be denoted by ut = ∂u/∂t and Du = (Dx1 u, . . . , Dxn u). Throughout this paper, C(n, ϑ1 , ϑ2 , . . .) stands for a universal constant depending only on prescribed quantities and possibly varying from line to line. First of all, let us recall the generalized Lebesgue–Sobolev space involving the variable exponent p(·). Given a bounded domain U ⊂ Rn+1 and a bounded measurable function p(·) : U → [1, ∞), the generalized Lebesgue space Lp(·) (U ; Rm ) for m ≥ 1 is defined to be the set of all measurable functions g : U → Rm obeying ˆ ϱLp(·) (U ;Rm ) (g) =

|g(z)|

p(z)

dz < +∞.

U

This space Lp(·) (U ; Rm ) with p(·) > 1 is a reflective Banach space endowed with the Luxemburg type norm { (g) } ∥g∥Lp(·) (U ;Rm ) = inf α > 0 : ϱLp(·) (U ;Rm ) ≤1 . α The generalized Sobolev space W 1,p(·) (U ; Rm ) is defined as the set of all measurable functions g ∈ Lp(·) (U ; Rm ) whose derivative Dg ∈ Lp(·) (U ; Rmn ). Its norm is naturally defined by ∥g∥W 1,p(·) (U ;Rm ) = ∥g∥Lp(·) (U ;Rm ) + ∥Dg∥Lp(·) (U ;Rmn ) . 1,p(·)

(U ; Rm ) = W 1,p(·) (U ; Rm ) ∩ L1 (0, T ; W01,1 (U ; Rm )). In particular, for m = 1, we simply We denote W0 1,p(·) write Lp(·) (U ), W 1,p(·) (U ) and W0 (U ). 1,p(·) We recall that a function u ∈ C(0, T ; L2 (Ω )) ∩ W0 (ΩT ) is said to be a weak solution to the zero Cauchy–Dirichlet problem (1.1) if it holds ˆ ˆ ˆ p(x,t)−2 uψt dxdt − ⟨a(x, t, Du), Dψ⟩dxdt = ⟨|F | F, Dψ⟩dxdt (2.1) ΩT

ΩT

ΩT

for every test function ψ ∈ C0∞ (ΩT ). We say that ω is a weight function in Muckenhoupt class, or an Ap weight for p ∈ (1, ∞), if ω is a positive locally integrable function on Rn+1 such that ( )( )p−1 −1 [ω]p := sup ω(x, t)dxdt ω(x, t) p−1 dxdt < +∞, Q⊂Rn+1

Q

Q

where the supremum runs over all cylinders Q in Rn+1 . If ω is an Ap weight, we write ω ∈ Ap , and [ω]p is called the Ap constant of ω. The Ap class is invariant with respect to translation, dilation and multiplication

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228

by a positive scalar. For each measurable set E ⊂ Rn+1 and a weight ω, we set ˆ ω(E) = ω(x, t)dxdt. E

Lγ,q ω (U )

The weighted Lorentz space functions g : U → R such that ∥g∥Lγ,q ω (U )

with two parameters (γ, q) ∈ [1, ∞) × (0, ∞] is the set of all measurable

( ˆ := γ 0

∞

dµ (µ ω({(x, t) ∈ U : |g(x, t)| > µ})) µ q γ

γ

) 1q < +∞,

while q ̸= ∞; for q = ∞ the space Lγ,∞ ω (U ) is set to be the usual Marcinkiewicz space with quasinorm 1

γ ∥g∥Lγ,∞ (U ) := sup µ ω({(x, t) ∈ U : |g(x, t)| > µ}) . ω

µ>0

γ,q If ω ≡ 1 we will write Lγ,q (U ). In particular, if q = γ then the weighted Lorentz space Lγ,γ ω (U ) as L ω (U ) is nothing but a weighted Lebesgue space Lγω (U ), which is equivalently defined as

g∈

Lγω (U )

(ˆ ⇔ ∥g∥Lγω (U ) :=

γ

) γ1

|g(x, t)| ω(x, t)dxdt

< +∞.

U

3. Main result In this paper, we are mainly interested in the case that a(x, t, ξ) is asymptotically regular nonlinearity for the zero Cauchy–Dirichlet problem (1.1). This is the case that it is getting closer to some regular vectorial-valued nonlinearity b(x, t, ξ) as |ξ| goes to infinity, where b(x, t, ξ) satisfies the following weak regular assumptions: • H3. b(x, t, ξ) : Rn × [0, T ] × Rn → Rn is measurable in (x, t) for every ξ and differentiate in ξ for a.e. (x, t), and satisfies the following conditions: there exist positive constants λ, Λ ∈ (0, ∞) and s ∈ [0, 1] such that { 2 ⟨Dξ b(x, t, ξ)η, η⟩ ≥ λ(s + |ξ|)p(x,t)−2 |η| , (3.1) |b(x, t, ξ)| + (s + |ξ|)|Dξ b(x, t, ξ)| ≤ Λ(s + |ξ|)p(x,t)−1 for a.e. (x, t) ∈ Rn+1 and any ξ, η ∈ Rn . Here, s ∈ [0, 1] is a parameter used to discern the degenerate case (s = 0) from the non-degenerate one (s > 0). • H4. ((δ, R)-vanishing) We say that b(x, t, ξ) is (δ, R)-vanishing, if t2

[b]2,R :=

0
where

2

|Θ(b, Br (y))(x, t)| dxdt ≤ δ 2 ,

sup t1

Br (y)

⏐ ⏐ ) ( ⏐ ⏐ b(·, t, ξ) b(x, t, ξ) ⏐ ⏐ − Θ(b, Br (y))(x, t) = sup ⏐ ⏐, p(x,t)−1 p(·,t)−1 ⏐ ⏐ n (s + |ξ|) (s + |ξ|) ξ∈R \{0} Br (y) ( ) ˆ b(·, t, ξ) 1 b(x, t, ξ) = dx. p(·,t)−1 |Br (y)| Br (y) (s + |ξ|)p(x,t)−1 (s + |ξ|) Br (y)

We are now in a position to introduce the concept of the so-called asymptotically δ-regular condition on the nonlinearity a(x, t, ξ). Definition 3.1 (Asymptotically δ-regular). Let b(x, t, ξ) : Rn × [0, T ] × Rn satisfy the assumptions H3–H4. Then we say that a(x, t, ξ) is asymptotically δ-regular with b(x, t, ξ) if there exists a uniformly bounded

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nonnegative function θ : [0, ∞) → [0, ∞) such that lim θ(ρ) ≤ δ,

ρ→0

and |a(x, t, ξ) − b(x, t, ξ)| ≤ θ(|ξ|)(1 + |ξ|

p(x,t)−1

)

for almost every (x, t) ∈ Rn+1 and all ξ ∈ Rn . Definition 3.2. We say that Ω is a (δ, R)-Reifenberg flat domain if for every x ∈ ∂Ω and every r ∈ (0, R], there exists a coordinate system {y1 , . . . , yn } with the center x, which can depend on x and r, such that Br (0) ∩ {yn > δr} ⊂ Br (0) ∩ Ω ⊂ Br (0) ∩ {yn > −δr}.

(3.2)

Finally, let us summarize our main result of this paper as follows. p(·)

Theorem 3.3. For any given (γ, q) ∈ (1, ∞) × (0, ∞], we assume that ω ∈ Aγ and |F | ∈ Lγ,q ω (ΩT ) with p(x, t) satisfying the assumptions H1–H2. Then there exists a constant δ = δ(n, λ, Λ, ϑ1 , ϑ2 ) > 0 such that if Ω is (δ, R)-Reifenberg flat and a(x, t, ξ) is asymptotically δ-regular with b(x, t, ξ) satisfying the assumptions H3–H4, then the zero Cauchy–Dirichlet problem (1.1) has a unique weak solution u satisfying p(·) |Du| ∈ Lγ,q ω (ΩT ) with the estimate    p(·)  |Du| 

γ,q

Lω (ΩT )

)d

(   p(·)  ≤ C |F | 

,

+1

γ,q

(3.3)

Lω (ΩT )

where d=

sup d(x, t), (x,t)∈ΩT

⎧ p(x, t) ⎪ ⎨ , 2 d(x, t) = 2p(x, t) ⎪ ⎩ , p(x, t)(n + 2) − 2n

if p(x, t) ≥ 2, 2n if < p(x, t) < 2, n+2

(3.4)

and C = C(n, λ, Λ, ϑ1 , ϑ2 , γ, q, φ, ω, Ω , T, θ). Remark 3.4. Three comments on Theorem 3.3 are in order. (1) The assumption H3 implies the following monotonicity condition: for any ξ, η ∈ Rn and for almost every (x, t) ∈ Rn+1 , 2

(b(x, t, ξ) − b(x, t, η)) · (ξ − η) ≥ ν|ξ − η| (s + |ξ| + |η|)p(x,t)−2 , where ν is a positive constant depending only on n, ϑ1 , ϑ2 and λ. In particular, for the case that p(x, t) ≥ 2, it can be reduced to p(x,t)

(b(x, t, ξ) − b(x, t, η)) · (ξ − η) ≥ ν|ξ − η|

.

(2) The assumption that b(x, t, ξ) is (δ, R)-vanishing refines the assumption that b(x, t, ξ) is V M Ox in other papers (cf. [6]), that is to say, the nonlinearity b(x, t, ξ) has a small BMO semi-norm for the spatial variable x and it is allowed to be merely measurable in the time variable, uniformly in ξ variable. (3) A Reifenberg flat domain can go beyond a Lipschitz domain, not necessarily given by graphs. The boundary of a Reifenberg flat domain is so rough that even the unit normal vector cannot be well defined there in the usual sense. However, its boundary can be well approximated by hyperplanes at every point and at every scale.

230

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4. Proof of the main theorem In this section, we are devoted to the proof of our main result. To this end, we introduce a global weighted Lorentz estimate for the spatial gradients of weak solution to the Cauchy–Dirichlet problem of nonlinear parabolic equations of p(x, t)-Laplacian type with the regular nonlinearity b(x, t, ξ) satisfying the assumptions H3–H4, which is from the reference [8]. p(·)

Lemma 4.1. For any given (γ, q) ∈ (1, ∞) × (0, ∞], we assume that ω ∈ Aγ and |F | ∈ Lγ,q ω (ΩT ) with p(x, t) satisfying the assumptions H1–H2. Then there exists a constant δ = δ(n, λ, Λ, ϑ1 , ϑ2 ) > 0 such that if Ω is (δ, R)-Reifenberg flat and b(x, t, ξ) satisfies the assumptions H3–H4, then the problem { p(x,t)−2 ut − div b(x, t, Du) = div (|F | F ), in ΩT , (4.1) u = 0, on ∂ΩT p(·)

has a unique weak solution u satisfying |Du|    p(·)  |Du| 

γ,q

Lω (ΩT )

∈ Lγ,q ω (ΩT ) with the estimate (   p(·)  ≤ C |F | 

)d γ,q

+1

,

(4.2)

Lω (ΩT )

where the constant C depends only on n, λ, Λ, ϑ1 , ϑ2 , γ, q, ω, Ω , T, φ; and d is the same as (3.4). Let a(x, t, ξ) be asymptotically δ-regular to the nonlinearity b(x, t, ξ) which is a regular function satisfying the assumptions H3–H4. Then by Definition 3.1, we can easily conclude that lim

|ξ|→∞

|a(x, t, ξ) − b(x, t, ξ)| ≤ 2δ (s + |ξ|)p(x,t)−1

(4.3)

uniformly with respect to (x, t) ∈ Rn+1 . Now, we define a vectorial-valued function c(x, t, ξ) : Rn × [0, T ] × Rn → Rn by (s + |ξ|)p(x,t)−1 c(x, t, ξ) = a(x, t, ξ) − b(x, t, ξ). (4.4) Then from (4.3), it yields that for any sufficiently small δ > 0, there exists a positive constant M = M (δ) such that |ξ| ≥ M ⇒ |c(x, t, ξ)| ≤ 2δ (4.5) uniformly in (x, t) ∈ Rn+1 . For any fixed point (x, t) ∈ Rn+1 , we consider the Poisson integral ˆ P [c(x, t, ·)](ξ) := c(x, t, η)K(ξ, η)dσ(η), ∂B

M

where

2

K(ξ, η) =

M 2 − |ξ| n M ωn−1 |ξ − η|

for ξ ∈ BM and η ∈ ∂BM

is the Poisson kernel for ball BM ⊂ Rn with the radius M , and ωn−1 is the surface area of the unit sphere B1 ⊂ Rn . We introduce a new vectorial-valued function c˜(x, t, ξ) defined by { c(x, t, ξ), if |ξ| ≥ M , c˜(x, t, ξ) = (4.6) P [c(x, t, ·)](ξ), if |ξ| < M . Then it follows from the maximum principle and (4.5) that for any ξ ∈ Rn and any (x, t) ∈ Rn+1 , there holds |˜ c(x, t, ξ)| ≤ 2δ. (4.7)

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Now, we combine (4.4) and (4.6) to get a(x, t, ξ) = b(x, t, ξ) + (s + |ξ|)p(x,t)−1 c(x, t, ξ) = b(x, t, ξ) + (s + |ξ|)p(x,t)−1 c˜(x, t, ξ) + (s + |ξ|)p(x,t)−1 χ{s+|ξ|
Lemma 4.2. Let u ∈ C(0, T ; L2 (Ω )) ∩ W0 (ΩT ) be a weak solution of the Cauchy–Dirichlet problem (1.1). Assume that a(x, t, ξ) is asymptotically δ-regular to the nonlinearity b(x, t, ξ) satisfying the regular assumptions H3–H4. Define a function a ˜(x, t, ξ) : Rn × [0, T ] × Rn → Rn by a ˜(x, t, ξ) := b(x, t, ξ) + (s + |ξ|)p(x,t)−1 c˜(x, t, Du(x, t)),

(4.9)

where c˜(x, t, ξ) is given { as (4.6).}Then we have ˜(x, t, ξ) satisfies (i) If 0 < δ ≤ min 4(ϑ2λ−1) , 1 , then a ⎧ λ ⎪ 2 ⎨⟨Dξ a ˜(x, t, ξ)η, η⟩ ≥ (s + |ξ|)p(x,t)−2 |η| , 2 ⎪ ⎩ ˜ + |ξ|)p(x,t)−1 , |˜ a(x, t, ξ)| + (s + |ξ|)|Dξ a ˜(x, t, ξ)| ≤ Λ(s

(4.10)

˜ = Λ + 2ϑ2 . for almost every (x, t) ∈ Rn+1 and all ξ, η ∈ Rn , where Λ √ (ii) a ˜(x, t, ξ) is ( 34δ, R)-vanishing. Proof . (i) For any given 0 < δ ≤ min

{

} , by employing (4.9) and (4.7) it follows that

λ 4(ϑ2 −1) , 1

|˜ a(x, t, ξ)| ≤ |b(x, t, ξ)| + 2(s + |ξ|)p(x,t)−1 .

(4.11)

Direct calculation gives Dξ a ˜(x, t, ξ) = Dξ b(x, t, ξ) + c˜(x, t, Du(x, t))[Dξ (s + |ξ|)p(x,t)−1 ]T = Dξ b(x, t, ξ) + c˜(x, t, Du(x, t))(p(x, t) − 1)(s + |ξ|)p(x,t)−2 |ξ|

−1 T

ξ .

(4.12)

Using (4.7) and δ ≤ 1 we get |Dξ a ˜(x, t, ξ)| ≤ |Dξ b(x, t, ξ)| + 2(ϑ2 − 1)(s + |ξ|)p(x,t)−2 . Then, combining (4.11) (4.13) and (3.1) leads to that |˜ a(x, t, ξ)| + (s + |ξ|)|Dξ a ˜(x, t, ξ)| ≤ |b(x, t, ξ)| + 2(s + |ξ|)p(x,t)−1 + (s + |ξ|)|Dξ b(x, t, ξ)| + 2(ϑ2 − 1)(s + |ξ|)p(x,t)−1 ≤ |b(x, t, ξ)| + (s + |ξ|)|Dξ b(x, t, ξ)| + 2ϑ2 (s + |ξ|)p(x,t)−1 ≤ Λ(s + |ξ|)p(x,t)−1 + 2ϑ2 (s + |ξ|)p(x,t)−1 ˜ + |ξ|)p(x,t)−1 , = Λ(s

(4.13)

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˜ = Λ + 2ϑ2 . On the other hand, by (4.12), (3.1) and (4.7) we conclude that where Λ −1

⟨Dξ a ˜(x, t, ξ)η, η⟩ = ⟨Dξ b(x, t, ξ)η, η⟩ + (p(x, t) − 1)(s + |ξ|)p(x,t)−2 |ξ| 2

c˜(x, t, Du(x, t))ξ T η · η

≥ λ(s + |ξ|)p(x,t)−2 |η| − 2δ(ϑ2 − 1)(s + |ξ|)p(x,t)−2 |η| = [λ − 2δ(ϑ2 − 1)](s + |ξ|)p(x,t)−2 |η| λ 2 ≥ (s + |ξ|)p(x,t)−2 |η| , 2

2

2

where we used 0 < δ ≤ 4(ϑ2λ−1) in the last inequality. So (i) is proved. (ii) Let r ∈ (0, R) and (x, t) ∈ Rn+1 . Then for any ξ ∈ Rn \{0} and any δ > 0, it follows from (4.9) that a ˜(x, t, ξ) b(x, t, ξ) = + c˜(x, t, Du(x, t)). (s + |ξ|)p(x,t)−1 (s + |ξ|)p(x,t)−1

(4.14)

Then we deduce from (4.7) that ⏐ ⏐ ( ) ⏐ ⏐ a ˜(x, t, ξ) a ˜(·, t, ξ) ⏐ ⏐ Θ(˜ a, Br (y))(x, t) = sup ⏐ − ⏐ p(x,t)−1 (s + |ξ|)p(·,t)−1 Br (y) ⏐ ξ∈Rn \{0} ⏐ (s + |ξ|) ⏐ ⏐ ( ) ⏐ ⏐ b(x, t, ξ) b(·, t, ξ) ⏐ ⏐ ≤ sup ⏐ − ⏐ p(x,t)−1 p(·,t)−1 ⏐ ⏐ n (s + |ξ|) (s + |ξ|) ξ∈R \{0} B (y) ⏐( ⏐ r ) ⏐ ⏐ + |˜ c(x, t, Du(x, t))| + ⏐ c˜(x, t, Du(x, t)) ⏐ Br (y) ⏐ ⏐ ( ) ⏐ ⏐ b(x, t, ξ) b(·, t, ξ) ⏐ ⏐ ≤ sup ⏐ − ⏐ + 4δ p(x,t)−1 p(·,t)−1 ⏐ ⏐ n (s + |ξ|) (s + |ξ|) ξ∈R \{0} Br (y) = Θ(b, Br (y))(x, t) + 4δ. Consequently, we have t2

[˜ a]2,R :=

2

|Θ(˜ a, Br (y))(x, t)| dxdt

sup 0
t1

Br (y)

t2

≤

2

2|Θ(b, Br (y))(x, t)| dxdt + 32δ 2

sup 0
t1

Br (y)

≤ 34δ ,

√ which proves our assertion that a ˜(x, t, ξ) is ( 34δ, R)-vanishing. So (ii) is proved. □ We are now ready to prove our main result. The proof of Theorem 3.3. Assume that a(x, t, ξ) is asymptotically δ-regular to the nonlinearity b(x, t, ξ) 1,p(·) (ΩT ) be a weak solution of the satisfying the assumptions H3–H4, and let u ∈ C(0, T ; L2 (Ω )) ∩ W0 Cauchy–Dirichlet problem (1.1). From (4.8) and (4.9), we have that for any given δ ∈ (0, 1), there exist a positive constant M > 1 and a vectorial-valued function c˜(x, t, Du) such that |˜ c(x, t, Du)| ≤ 2δ and a(x, t, Du) = a ˜(x, t, Du) + (s + |Du|)p(x,t)−1 χ{s+|Du|
ut − div a ˜(x, t, Du) = div (|G|

G),

(4.15)

J. Zhang, M. Cai and S. Zheng / Nonlinear Analysis 180 (2019) 225–235

233

where G is defined by p(x,t)−2

|F | F + (s + |Du|)p(x,t)−1 χ{s+|Du|
⏐ ⏐ ⏐ p(x,t)−2 ⏐ F + (s + |Du|)p(x,t)−1 χ{s+|Du|
while G = 0 if ⏐ ⏐ ⏐ p(x,t)−2 ⏐ F + (s + |Du|)p(x,t)−1 χ{s+|Du|
p(·)

∈ Lγ,q ω (ΩT ) with the estimate (     p(·)   p(·)  ≤ C |F |  |G|  γ,q Lω (ΩT )

) +1 .

γ,q

(4.16)

Lω (ΩT )

Notice that |˜ c(x, t, Du)| ≤ 2δ < 2

(4.17)

for almost all (x, t) ∈ Rn+1 . From (4.4) and Definition 3.1, it yields that ⏐ ⏐ ⏐ ⏐ ⏐(s + |Du|)p(x,t)−1 χ{s+|Du|
≤ ∥θ∥∞ (1 + |Du| ≤ ∥θ∥∞ (1 + M ≤ 2∥θ∥∞ M

p(x,t)−1

p(x,t)−1

) χ{s+|Du|
)

,

where we use M ≥ 1 in the last inequality. Let H(x, t) = |F |

p(x,t)−2

F + (s + |Du|)p(x,t)−1 χ{s+|Du|
then we have

1

|G(x, t)| = |H(x, t)| p(x,t)−1 ⇒ |G(x, t)|

p(x,t)−1

= |H(x, t)|.

Together with (4.5) and (4.7), it yields p(x,t)

|G(x, t)|

p(x,t)

= |H(x, t)| p(x,t)−1 ( ) p(x,t) p(x,t)−1 p(x,t)−1 ≤ |F (x, t)| + (2∥θ∥∞ + 2)M p(x,t)−1 ( ) ϑ2 p(x,t) p(x,t) ϑ −1 ≤ C |F (x, t)| + 2(∥θ∥∞ + 1) 1 M

for a positive constant C = C(ϑ1 , ϑ2 ). Therefore, ˆ ∞(   ) γq dµ  p(·)  p(x,t) γ = q µ ω({(x, t) ∈ Ω : |G(x, t)| > µ}) |G|  γ,q T µ Lω (ΩT ) 0 ˆ ∞( ) γq dµ p(x,t) ≤ Cq µγ ω({(x, t) ∈ ΩT : |F (x, t)| > 2−1 µ}) µ 0 ) γq ˆ ∞( ϑ2 dµ γ p(x,t) −1 + Cq µ ω({(x, t) ∈ ΩT : 2(∥θ∥∞ + 1) ϑ1 −1 M > 2 µ}) µ 0

(4.18)

J. Zhang, M. Cai and S. Zheng / Nonlinear Analysis 180 (2019) 225–235

234

   p(·)  = C2q |F | 

γ,q

Lω (ΩT )

ˆ ∞(

γ

µ ω({(x, t) ∈ ΩT : 4(∥θ∥∞ + 1)

+ Cq

ϑ2 ϑ1 −1

M

ϑ2

0

) γq dµ > µ}) . µ

Since ˆ

∞

(

γ

µ ω({(x, t) ∈ ΩT : 4(∥θ∥∞ + 1)

q

ϑ2 ϑ1 −1

M

ϑ2

) γq

dµ µ

> µ})

0

ˆ

ϑ2

4(∥θ∥∞ +1) ϑ1 −1 M ϑ2

(

γ

µ ω({(x, t) ∈ ΩT : 4(∥θ∥∞ + 1)

=q

ϑ2 ϑ1 −1

M

ϑ2

0 q γ

ˆ

) γq dµ > µ}) µ

ϑ2

4(∥θ∥∞ +1) ϑ1 −1 M ϑ2

µq−1 dµ ( )q ϑ2 q ϑ ϑ −1 2 γ = ω(ΩT ) 4(∥θ∥∞ + 1) 1 M , ≤ qω(ΩT )

0

we have    p(·)  |G| 

γ,q

Lω (ΩT )

)q (   ϑ2 q  p(·)  ϑ2 ϑ −1 γ 1 M ≤ C |F |  γ,q + Cω(ΩT ) 4(∥θ∥∞ + 1) Lω (ΩT ) ) (   p(·)  +1 ≤ C |F |  γ,q

(4.19)

Lω (ΩT )

with the positive constant C = C(n, ϑ1 , ϑ2 , γ, q, ω, θ, Ω , T ). We finally consider the following zero initial–boundary problem { p(x,t)−2 vt − div a ˜(x, t, Dv) = div (|G| G), in ΩT , v = 0, on ∂ΩT ,

(4.20)

where a ˜(x, t, ξ) is given as (4.9). Note from Lemma 4.2 that a ˜(x, t, ξ) satisfies the assumptions H3–H4. We employ Lemma 4.1 with b(x, t, ξ) replaced by a ˜(x, t, ξ) and F replaced by G, respectively, to discover that p(·) |Dv| ∈ Lγ,q (Ω ) with the estimate T ω    p(·)  |Dv| 

γ,q

Lω (ΩT )

(   p(·)  ≤ C |G| 

)d γ,q

+1

(4.21)

Lω (ΩT )

for a positive constant C = C(n, λ, Λ, ϑ1 , ϑ2 , γ, q, ω, φ, Ω , T ). On the other hand, from (4.15) and the uniqueness of the solution to the problem (4.20) we conclude that u ≡ v in ΩT . Therefore, it follows from (4.19) and (4.21) that    p(·)  |Du| 

γ,q

Lω (ΩT )

   p(·)  = |Dv| 

γ,q

Lω (ΩT )

(   p(·)  ≤ C |G| 

)d γ,q

+1

Lω (ΩT )

(   p(·)  ≤ C |F | 

)d γ,q

+1

,

Lω (ΩT )

(4.22) where C = C(n, λ, Λ, ϑ1 , ϑ2 , γ, q, ω, φ, θ, Ω , T ). This completes the proof.

□

Acknowledgments The authors thank the anonymous referees for the very valuable suggestions and comments that led to improvement of this paper. This paper was supported by the National Natural Science Foundation of China Grant No. 11371050.

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235

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