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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 163269, 16 pages
doi:10.1155/2010/163269
Research Article
Local Boundedness of Weak Solutions for
Nonlinear Parabolic Problem with px-Growth
Yongqiang Fu and Ning Pan
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Ning Pan, [email protected]
Received 23 October 2009; Revised 6 January 2010; Accepted 29 January 2010
Academic Editor: Shusen Ding
Copyright q 2010 Y. Fu and N. Pan. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the nonlinear parabolic problem with px-growth conditions in the space W 1,x Lpx Q
and give a local boundedness theorem of weak solutions for the following equation ∂u/∂t Au 0, where Au −divax, t, u, ∇u a0 x, t, u, ∇u, ax, t, u, ∇u and a0 x, t, u, ∇u satisfy
px-growth conditions with respect to u and ∇u.
1. Introduction
The study of variational problems with nonstandard growth conditions is an interesting
topic in recent years. px-growth problems can be regarded as a kind of nonstandard
growth problems and they appear in nonlinear elastic, electrorheological fluids and other
physics phenomena. Many results have been obtained on this kind of problems, for example
1–9.
Let Q be Ω × 0, T , where T > 0 is given. In 8, the authors studied the following
equation:
ut − div |Du|px,t−2 Du 0,
1.1
where p1 infx,t∈Q px, t > max{1; 2N/N 2}, px, t is dependent on the space variable x
1,px,t
Q ∩ C0, T ; L2loc Ω,
and the time variable t, u is the local weak solution in the space Wloc
2
Journal of Inequalities and Applications
and the authors proved the local boundedness of the local weak solution in Q. In this paper,
we will study the following more general problem:
∂u
Au 0,
∂t
ux, t 0,
in Q,
on ∂Ω × 0, T ,
ux, 0 ψx,
in Ω,
1.2
1.3
1.4
where ψx is a given function in L2 Ω and A : W01,x Lpx Q → W −1,x Lqx Q is an
elliptic operator of the form Au − div ax, t, u, ∇u a0 x, t, u, ∇u with the coefficients
a and a0 satisfying the classical Leray-Lions conditions. In 10, we have proved the
existence of the solutions of 1.2–1.4 and have gotten u ∈ W 1,x Lpx Q ∩ L∞ 0, T ; L2 Ω;
in this paper we will give the local boundedness theorem of the weak solutions in the
framework space W 1,x Lpx Q, which can be considered as a special case of the space
W 1,px,t Q.
Many authors have already studied the boundedness of weak solutions of parabolic
equation with p-growth conditions, where p is a constant, for example 8, 11–15. The
boundedness of the weak solutions plays a central role in many aspects. Based on the
boundedness, we can further study the regularity of the solutions. For example, first in 15
the author studied the equation
ut − div ax, t, u, ∇u bx, t, u, ∇u
1.5
and got L∞
-estimates of the degenerate parabolic equation with p-growth conditions for
loc
p > 1, where p is a constant, then in 16 the authors established the Hölder continuity
of the equation for the singular case 1 < p < 2, and in 17 the authors discussed
Harnack estimates for the bounded solutions of the above parabolic equation for p ≥
2.
The space W 1,x Lpx Q provides a suitable framework to discuss some physical
problems. In 18, the authors studied a functional with variable exponent, 1 ≤ px ≤ 2,
which provided a model for image denoising, enhancement, and restoration. Because in
18 the direction and speed of diffusion at each location depended on the local behavior,
px only depended on the location x in the image. Consider that the space W 1,x Lpx Q
was introduced and discussed in 10 and 19, we think that the space W 1,x Lpx Q is
a reasonable framework to discuss the px-growth problem 1.2–1.4, where px only
depends on the space variable x similar to 18.
In this paper, let a : Q × R × RN → RN and a0 : Q × R × RN → R be the operators such
that for any s ∈ R and ξ ∈ RN , ax, t, s, ξ and a0 x, t, s, ξ are both continuous in t, s, ξ for
Journal of Inequalities and Applications
3
a.e. x ∈ Ω and measurable in x for all t, s, ξ ∈ 0, T × R × Rn . They also satisfy that for a.e.
x, t ∈ Q, any s ∈ R and ξ /
ξ∗ ∈ RN :
|ax, t, s, ξ| ≤ α |s|px−1 |ξ|px−1 ,
1.6
|a0 x, t, s, ξ| ≤ α |s|px−1 |ξ|px−1 ,
1.7
ax, t, s, ξ − ax, t, s, ξ∗ ξ − ξ∗ > 0,
ax, t, s, ξξ a0 x, t, s, ξs ≥ β |ξ|px |s|px ,
1.8
1.9
where α, β > 0 are constants.
Throughout this paper, unless special statement, we always suppose that px is ∗continuous on Ω, that is, limy → x,y∈Ω py px for every x ∈ Ω, and satisfy
1 < p− inf px ≤ px ≤ sup px p < ∞;
Ω
Ω
1.10
qx is the conjugate function of px.
Definition 1.1. A function u ∈ W 1,x Lpx Q ∩ L∞ 0, T ; L2 Ω is called a weak solution of
1.2–1.4 if
∂ϕ
− u dx dt Q ∂t
Ω
T
uϕ dx0 ax, t, u, ∇u∇ϕ a0 x, t, u, ∇uϕ dx dt 0
1.11
Q
for all ϕ ∈ C1 0, T ; C0∞ Ω.
We will prove the following local boundedness theorem.
Theorem 1.2. Let p− > max{1, 2N/N 2}. If u is a nonnegative local weak solution of 1.2–
1.4, then u is locally bounded in Q. Moreover, there exists a constant C CN, pρ , pρ− , ρ such that
for any Qρpρ , ρ ∈ Q and any σ ∈ 0, 1,
⎞pρ− /Nq−δ ⎫
⎪
⎪
⎬
1
⎟
−pρ Npρ− /Nq−δ ⎜
δ
sup u ≤ max 1, C1 − σ
u
dx
dt
,
⎝ ⎠
p
⎪
⎪
p
⎪
⎪
Q ρpρ , ρ Qρ ρ ,ρ
⎭
⎩
Qσρ ρ ,σρ
⎧
⎪
⎪
⎨
⎛
1.12
where for all x0 , t0 ∈ Q, Kρ {x ∈ Ω | max1≤i≤N |xi − x0,i | < ρ}, pρ supKρ px, pρ− infKρ px,
Qρpρ , ρ Kρ × t0 − ρpρ , t0 , and max{pρ , 2} ≤ δ < q N 2/Npρ− .
4
Journal of Inequalities and Applications
2. Preliminaries
We first recall some facts on spaces Lpx Ω, W m,px Ω, and W m,x Lpx Q. For the details,
see 19–21.
Although we assume 1.10 holds in this paper, in this section we introduce the general
spaces Lpx Ω, W m,px Ω, and W m,x Lpx Q.
Denote
E {ω : ω is a measurable function on Ω},
2.1
where Ω ⊂ RN is an open subset.
Let px : Ω → 1, ∞ be an element in E. Denote Ω∞ {x ∈ Ω : px ∞}. For u ∈ E,
we define
ρu Ω\Ω∞
|ux|px dx ess sup |ux|.
x∈Ω∞
2.2
The space Lpx Ω is
Lpx Ω u ∈ E : ∃λ > 0, ρλu < ∞
2.3
u
≤1 .
uLpx Ω inf λ > 0 : ρ
λ
2.4
endowed with the norm
We define the conjugate function qx of px by
⎧
⎪
∞,
⎪
⎪
⎪
⎪
⎨
qx 1,
⎪
⎪ px
⎪
⎪
⎪
,
⎩
px − 1
if px 1;
if px ∞;
2.5
if 1 < px < ∞.
Lemma 2.1 see 21. 1 The dual space of Lpx Ω is Lqx Ω if 1 ≤ px < ∞.
2 The space Lpx Ω is reflexive if and only if 1.10 is satisfied.
Lemma 2.2 see 21. If 1 ≤ px < ∞, C0∞ Ω is dense in the space Lpx Ω and Lpx Ω is
separable.
Lemma 2.3 see 21. Let 1 ≤ px ≤ ∞, for every ux ∈ Lpx Ω and vx ∈ Lqx Ω, we have
Ω
|uxvx|dx ≤ CuxLpx Ω vxLqx Ω ,
where C is only dependent on px and Ω, not dependent on ux, vx.
2.6
Journal of Inequalities and Applications
5
Next let m > 0 be an integer. For each α α1 , α2 , . . . , αn , αi are nonnegative integers
and |α| Σni1 αi , and denote by Dα the distributional derivative of order α with respect to the
variable x.
We now introduce the generalized Lebesgue-Sobolev space W m,px Ω which is
defined as
W m,px Ω u ∈ Lpx Ω : Dα u ∈ Lpx Ω, |α| ≤ m .
2.7
W m,px Ω is a Banach space endowed with the norm
u Dα uLpx Ω .
2.8
|α|≤m
m,px
The space W0
m,px
W0
Ω∗
Ω is defined as the closure of C0∞ Ω in W m,px Ω. The dual space
is denoted by W −m,qx Ω equipped with the norm
f −m,qx
inf Σ|α|≤m fα Lqx Ω ,
W
Ω
2.9
where infimum is taken on all possible decompositions
f Σ|α|≤m −1|α| Dα fα ,
m,px
Lemma 2.4 see 21. 1 W m,px Ω and W0
m,px
2 W m,px Ω and W0
2.10
fα ∈ Lqx Ω.
Ω are separable if 1 ≤ px < ∞.
Ω are reflexive if 1.10 holds.
We define the space W m,x Lpx Q as the following:
W m,x Lpx Q u ∈ Lpx Q : Dα u ∈ Lpx Q, |α| ≤ m .
2.11
α
W m,x Lpx Q is a Banach space with the norm u |α|≤m D uLpx Q , where px is
independent of t.
The space W0m,x Lpx Q is defined as the closure of C0∞ Q in W m,x Lpx Q, and
W0m,x Lpx Q → Lpx Q is continuous embedding. Let M be the number of multiindexes α
which satisfies 0 ≤ |α| ≤ m, then the space W0m,x Lpx Q can be considered as a close subspace
M px
M px
of the product space Πi1
L Q. So if 1 < px < ∞, Πi1
L Q is reflexive and further we
m,x px
can get that the space W0 L Q is reflexive. The dual space W0m,x Lpx Q∗ is denoted
by W −m,x Lqx Q equipped with the norm
f −m,x qx
W
L Q
sup
f, u inf Σ|α|≤m fα Lqx Q
uW m,x Lpx Q ≤1
,
2.12
0
where infimum is taken on all possible decompositions
f Σ|α|≤m −1|α| Dxα fα ,
fα ∈ Lqx Q.
2.13
6
Journal of Inequalities and Applications
Next, we will introduce some results in 22.
Lemma 2.5. Let {Yn }, n 0, 1, 2, . . . , be a sequence of positive numbers, satisfying the inequalities
2
Yn1 ≤ Cbn Yn1α , where C, b > 1 and α > 0 are given numbers. If Y0 ≤ C−1/α b−1/α , then {Yn }
converges to 0 as n → ∞.
Lemma 2.6. There exists a constant C depending only on N, r, m, such that for every v ∈
L∞ 0, T ; Lm Ω ∩ Lr 0, T ; W01,r Ω,
q
|vx, t| dx dt ≤ C
r
|Dvx, t| dx dt
q
Q
sup
Q
r/N
0<t<T
m
Ω
|vx, t| dx
2.14
,
where q rN m/N.
Remark 2.7. In 10, we have gotten that for the Galerkin solutions un ∈ C1 0, T ; C0∞ Ω,
un → u strongly in L1 Q, un u weakly in W 1,x Lpx Q, ax, t, un , ∇un ax, t, u, ∇u
weakly in Lqx Q and a0 x, t, un , ∇un a0 x, t, u, ∇u weakly in Lqx Q.
3. Proof of the Theorem
Suppose that u is a weak solution of 1.2–1.4, then there exists δ > max{p , 2} such that
|u|δ dx dt < ∞.
3.1
Q
Indeed, by Young’s inequality, we have
p−
Q∩{p− <px}
|∇u| dx dt p−
Q∩{p− px}
|∇u|px dx dt < ∞,
|∇u| dx dt ≤ |Q| where |Q| is the Lebesgue measure of Q. Since W01,x Lpx Q
1,p−
L 0, T ; W0 Ω and u ∈ W 1,x Lpx Q ∩ L∞ 0, T ; L2 Ω,
−
1,p−
Lp 0, T ; W0 Ω. Then by Lemma 2.6, we get
p−
δ
|u| dx dt ≤ C
p−
|Du| dx dt
δ
Q
3.2
Q
Q
sup
−
W01,x Lp Q
∞
we can get u ∈ L 0, T ; L Ω ∩
2
2/N
0<t<T
→
2
Ω
|u| dx
,
3.3
where δ N 2/Np− . Thus the desired result is obtained.
We define u max{u, 0}. Fix a point x0 , t0 in Q. Let 0 < ρ < 1, 0 < θ < 1, and
Qθ, ρ ≡ Kρ × t0 − θ, t0 ⊂ Q. Fix σ ∈ 0, 1 and consider the sequences
ρm σρ 1−σ
ρ,
2m
θm σθ 1−σ
θ,
2m
m 0, 1, 2, . . .,
3.4
Journal of Inequalities and Applications
7
and the corresponding cylinders Qm Qθm , ρm . It follows from the definitions that
!
Q0 Q θ, ρ ,
!
Q∞ Q σθ, σρ .
3.5
" m Qθ"m , ρ"m , where for m 0, 1, 2, . . . ,
We consider also the boxes Q
ρ"m ρm ρm1
,
2
θm θm1
θ"m .
2
3.6
m 0, 1, 2, . . . .
3.7
For these boxes, we have the inclusion
" m ⊂ Qm ,
Qm1 ⊂ Q
We introduce the sequence of increasing levels
km k −
k
,
2m
3.8
m 0, 1, 2, . . . , k > 0 to be chosen.
Let {un } be the Galerkin solutions in 10. Similarly, we can get un − u is bounded in
Lδ Q. Since un − u converges to 0 in L1 Q, by interpolation inequality, we have
un − uLp Q ≤ un − uλL1 Q un − u1−λ
Lδ Q ,
3.9
where 0 < λ < 1, 1/p λ δ/1 − λ. Furthermore, un → u strongly in Lp Q. Since
Lp Q → Lpx Q, un → u strongly in Lpx Q. In the same way, we obtain that un → u
strongly in L2 Q; furthermore, we get un t − utL2 Ω → 0 for a.e. t ∈ 0, T .
t
Kρm × t0 − θm , t and ζ be the smooth cutoff function satisfying
Let Qm
0 ≤ ζ ≤ 1,
ζ ≡ 0 on ∂Kρm × t0 − θm , t0 ∪ Kρm × {t},
|∇ζ| ≤
2m2
,
1 − σρ
0 ≤ ζt ≤
" m,
ζ ≡ 1 in Q
2m2
.
1 − σθ
3.10
Take ϕ un − km1 ζpρ as the testing function in the following equation:
t
Qm
ϕ
∂un
dx dt ∂t
t
Qm
ax, t, un , ∇un ∇ϕ dx dt t
Qm
a0 x, t, un , ∇un ϕ dx dt 0.
3.11
8
Journal of Inequalities and Applications
First, by un t − utL2 Ω → 0 for a.e. t ∈ 0, T and un → u strongly in L2 Q, we
get
lim
∂un
dx dt
∂t
1
∂
lim
un − km1 2 ζpρ dx dt
n → ∞ 2 Qt ∂t
m
1
1
2 pρ
lim
un − km1 ζ x, tdx −
un − km1 2 ζpρ x, t0 − θm dx
n→∞ 2 K
2 Kρm
ρm
pρ
2 pρ −1
−
un − km1 ζ
|ζt |dx dt
2 Qmt
1
1
2 pρ
u − km1 ζ x, tdx −
u − km1 2 ζpρ x, t0 − θm dx
2 Kρm
2 Kρm
n → ∞ Qt
m
ϕ
pρ
−
2
3.12
t
Qm
u − km1 2 ζpρ −1 |ζt |dx dt.
By Fatou’s lemma, we get
lim
n→∞
t
Qm
ax, t, un , ∇un ∇un − km1 ζ dx dt ≥
t
Qm
ax, t, u, ∇u∇u − km1 ζ dx dt pρ
t
Qm
∩{un >km1 }
pρ
pρ
a0 x, t, un , ∇un un ζ dx dt
t
Qm
∩{u>km1 }
a0 x, t, u, ∇uuζpρ dx dt.
3.13
Because un → u strongly in Lpx Q and ax, t, un , ∇un ax, t, u, ∇u weakly in
Lqx Q, we get
lim
n → ∞ Qt
m
ax, t, un , ∇un un − km1 ζpρ −1 ∇ζ dx dt t
Qm
ax, t, u, ∇uu − km1 ζpρ −1 ∇ζ dx dt.
3.14
Since un → u strongly in Lpx Q and a0 x, t, un , ∇un a0 x, t, u, ∇u weakly in
Lqx Q, we have
lim
n→∞
t
Qm
a0 x, t, un , ∇un un − km1 ζ dx dt −
pρ
t
Qm
a0 x, t, u, ∇uu − km1 ζ dx dt −
pρ
pρ
t
Qm
∩{un >km1 }
a0 x, t, un , ∇un un ζ dx dt
t
Qm
∩{u>km1 }
a0 x, t, u, ∇uuζpρ dx dt.
3.15
Journal of Inequalities and Applications
9
Then for the remaining parts of 3.11, we get
I lim
n → ∞ Qt
m
ax, t, un , ∇un ∇ϕ a0 x, t, un , ∇un ϕ dx dt
lim
n→∞
ax, t, un , ∇un ∇un − km1 ζpρ dx dt
t
Qm
pρ
t
Qm
ax, t, un , ∇un un − km1 ζpρ −1 ∇ζ dx dt
a0 x, t, un , ∇un un − km1 ζpρ dx dt
t
Qm
t
Qm
∩{un >km1 }
a0 x, t, un , ∇un un ζpρ dx dt
−
t
Qm
∩{un >km1 }
≥
pρ
t
Qm
ax, t, u, ∇u∇u − km1 ζpρ dx dt
t
Qm
∩{u>km1 }
pρ
−
3.16
a0 x, t, un , ∇un un ζ dx dt
a0 x, t, u, ∇uuζpρ dx dt
t
Qm
ax, t, u, ∇uu − km1 ζpρ −1 ∇ζ dx dt
t
Qm
a0 x, t, u, ∇uu − km1 ζpρ dx dt
t
Qm
∩{u>km1 }
a0 x, t, u, ∇uuζpρ dx dt.
By 1.6, 1.7, and 1.9,
I≥β
px pρ
t
Qm
− pρ α
−
pρ α
−α
|∇u − km1 |
ζ dx dt t
Qm
∩{u>km1 }
|u|
px pρ
ζ dx dt
t
Qm
∩{u>km1 }
|u|px ζpρ −1 |∇ζ|dx dt
t
Qm
|∇u − km1 |px−1 u − km1 ζpρ −1 |∇ζ|dx dt
t
Qm
∩{u>km1 }
|u|px ζpρ dx dt − α
t
Qm
|∇u − km1 |px−1 |u|ζpρ dx dt.
3.17
10
Journal of Inequalities and Applications
As pρ − 1px/px − 1 > pρ , by Young’s inequality and Hölder’s inequality, we
have
t
Qm
|∇u − km1 |px−1 u − km1 ζpρ −1 |∇ζ|dx dt
≤ε
≤ε
t
Qm
t
Qm
ζ dx dt Cε
|∇u − km1 |
px pρ
|∇u − km1 |
px pρ
Cε
ζ dx dt Cε
t
Qm
px
t
Qm
t
Qm
u − km1 |∇ζ|px dx dt
u −
pρ
km1 |∇ζ|pρ dx dt
3.18
χu − km1 > 0dx dt.
In the same way, by pρ px/px − 1 > pρ and Young’s inequality, we have
t
Qm
|∇u − km1 |
px−1
pρ
|u|ζ dx dt ≤ ε
t
Qm
|∇u − km1 |px ζpρ dx dt
Cε
3.19
t
Qm
∩{u>km1 }
|u|
px
dx dt.
For a set A, meas A is the Lebesgue measure of A. Let |Am1 | ≡ meas{x, t ∈ Qm |
ux, t > km1 } and εα β/4. By 3.11–3.19, we get
u −
sup
t0 −θm <t<t0
Kρm
km1 2 ζpρ dx
Qm
|∇u − km1 |px ζpρ dx dt
≤
Qm
u − km1 2 ζpρ −1 |ζt |dx dt C
C
px pρ −1
Qm ∩{u>km1 }
|u|
ζ
pρ
Qm
u − km1 |∇ζ|pρ dx dt C|Am1 |
3.20
|∇ζ|dx dt C
Qm ∩{u>km1 }
|u|px dx dt.
Moreover, we observe that for s > 0 to be determined later,
Qm
u − km s dx dt ≥
Qm
u − km s χu > km1 dx dt
≥ km1 − km s |Am1 |
ks
2m1s
3.21
|Am1 |,
thus we get
|Am1 | ≤
2m1s
ks
Qm
u − km s dx dt.
3.22
Journal of Inequalities and Applications
11
Then for s 2 and s pρ in 3.22, by Hölder inequality, we obtain respectively
u −
Qm
km1 2 dx dt
≤
u −
Qm
≤C
u −
Qm
pρ
km1 dx dt
2/δ
km1 δ dx dt
δ−2m
2
3.23
kδ−2
u − km δ dx dt,
Qm
pρ /δ
≤
u −
Qm
≤C
|Am1 |1−2/δ
km1 δ dx dt
3.24
δ−pρ m 2
kδ−pρ
|Am1 |1−pρ /δ
Qm
u − km δ dx dt.
For the integral involving |u|px , first we write km km1 2m1 − 2/2m1 − 1, then
we obtain
u −
Qm
km δ dx dt
≥
Qm
u − km δ χu > km1 dx dt
δ
≥
|u|
Qm
C
≥ mδ
2
2m1 − 2
1 − m1
2
−1
δ
χu > km1 dx dt
3.25
|u|δ χu > km1 dx dt.
Qm
By Young’s inequality and 3.25, we get
Qm ∩{u>km1 }
|u|px ζpρ −1 |∇ζ| |u|px dx dt
2m
≤C
1 − σρ
2m
≤C
1 − σρ
2m
≤C
1 − σρ
Qm ∩{u>km1 }
|u|px dx dt
Qm ∩{u>km1 }
|u|δ dx dt |Am1 |
u −
mδ
2
Qm
km δ dx dt
|Am1 | .
3.26
12
Journal of Inequalities and Applications
Let 1 < k ≤ 1/ρpρ −1 1/δ−pρ , then 1/ρ ≤ 1/ρpρ kδ−pρ . By 3.20–3.24 and 3.26, we
obtain
t0 −θm <t<t0
Kρm
≤C
u − km1 2 ζpρ dx Qm
|∇u − km1 |px ζpρ dx dt
2δ−pρ m 2m2
2m1δ
2δ−2m 2m2
2m
2m 2m1δ
mδ
2
1 − σρ
1 − σρ kδ
kδ−2 1 − σθ
kδ
kδ−pρ 1 − σρ
×
Qm
≤C
sup
u − km δ dx dt
2m1δ
1 − σ
pρ
1
1
θkδ−2 ρpρ kδ−pρ
Qm
u − km δ dx dt.
3.27
By Young’s inequality,
−
"m
Q
|∇u − km1 |pρ dx dt ≤
≤
"m
Q
" m |∇u − km1 |px dx dt Am1 ∩ Q
3.28
px
"m
Q
|∇u − km1 |
dx dt |Am1 |.
Moreover, by 3.27, we can get
u −
sup
t0 −θm <t<t0
≤C
km1 2 dx
Kρ"m
2m1δ
1 − σpρ
−
"m
Q
|∇u − km1 |pρ dx dt
1
1
p δ−p
δ−2
ρ
θk
ρ k ρ
3.29
u −
Qm
km δ dx dt.
"m
Next we define the smooth cutoff function ζ"m in Q
0 ≤ ζ"m ≤ 1,
ζ"m ≡ 1
ζ"m ≡ 0 on ∂Kρ"m × t0 − θ"m , t0 ,
in Qm1 ,
" ∇ζm ≤
2m2
.
1 − σρ
3.30
Journal of Inequalities and Applications
13
For the function u − km1 ζ"m , by Lemma 2.6 and 3.29, we get
"m
Q
q q
u − km1 ζ"m dx dt
≤C
"m
Q
|∇u − km1 | dx dt ×
≤C
"m
Q
3.31
km1 2 dx
u −
sup
Kρ"m
2m1δ
pρ−
"
|u − km1 | ∇ζm dx dt
pρ− pρ− /N
t0 −θm <t<t0
pρ−
1 − σpρ
1
1
p δ−p
δ−2
ρ
θk
ρ k ρ
Finally, we define Ym 1/|Qm |
Hölder inequality, we obtain
1
1pρ− /N #
1pρ− /N
u −
Qm
km δ dx dt
.
Qm
u − km δ dx dt, m 0, 1, 2, . . . . Let θ ρpρ ; by
u − km1 δ dx dt
|Qm1 | Qm1
⎛
⎞
1
δ
≤ C⎝ dx dt⎠
u − km1 δ ζ"m
" Q"
Qm m
Ym1 ⎛
1
≤ C⎝ " Qm ⎛
1
≤ C⎝ " Qm ≤
ρ1 − σ
−
⎞δ/q
"m
Q
q q
u − km1 ζ"m dx dt⎠
$
|Am1 |
|Qm |
⎞δ/q "m
Q
u −
q q
km1 ζ"m dx dt⎠
Cbm
!pρ Npρ− /Nδ/q
2mδ
Ym
kδ
1δpρ− /Nq
kδ/qq−δ
Ym
%1−δ/q
3.32
1−δ/q
,
−
where b 2δ1δpρ /qN1/q1pρ /N . Then by Lemma 2.5, we have Ym → 0 as m → ∞,
provided k max{k, 1} is chosen to satisfy
1
Y 0 Q ρpρ , ρ p
Qρ ρ ,ρ
uδ dx dt Ck
q−δN/pρ−
−
−
1 − σNpρ /pρ pρ .
#
3.33
u − km δ χQm dx dt → 0 as m → ∞. Since u − km δ χQm ≤ |u| #
→ u − kδ χQ σθ,σρ a.e. in Q0 , by Lebesuge’s theorem we get Q0 u −
By Ym → 0, we can get
Q0
kδ and u − km δ χQm
#
km δ χQm dx dt → Q0 u − kδ χQ σθ,σρ dx dt 0. So we obtain u ≤ k a.e. in Qσθ, σρ.
14
Journal of Inequalities and Applications
Thus we get
⎞pρ− /Nq−δ ⎫
⎪
⎪
⎬
1
⎟
−pρ Npρ− /Nq−δ ⎜
δ
u
dx
dt
sup u ≤ max 1, C1 − σ
.
⎝ ⎠
p
⎪
⎪
p
⎪
⎪
Q ρpρ , ρ Qρ ρ ,ρ
⎭
⎩
Qσρ ρ ,σρ
⎧
⎪
⎪
⎨
⎛
3.34
Remark 3.1. In this paper, we study the boundedness of weak solution in the case p− >
max{1, 2N/N 2}. For the singular case 1 < p− ≤ max{1, 2N/N 2}, the conditions
in the paper are not enough. In 22, there is a counterexample in §13 of Chapter XII. The
author studied the solutions of the homogeneous equation
ut − div |Du|p−2 Du 0, in Q,
p
1,p
u ∈ Cloc 0, T ; L2loc Ω ∩ Lloc 0, T ; Wloc Ω ,
3.35
p > 1,
where
u ∈ L1loc Q,
u∈L1ε
loc Q
∀ε ∈ 0, 1,
p
2N
,
N1
3.36
and proved that the solution u is unbounded in Q.
Remark 3.2. In general, we consider the equation
∂u
Au fx, t ≥ 0,
∂t
in Q,
3.37
where
fx, tδ/δ−1 ∈ LNp
−
/1−h0 p−
Q,
3.38
h0 ∈ 0, 1 and A : W01,x Lpx Q → W −1,x Lqx Q is an elliptic operator of the form Au − div ax, t, u, ∇u a0 x, t, u, ∇u. ax, t, s, ξ and a0 x, t, s, ξ satisfy that for a.e. x, t ∈ Q,
any s ∈ R and ξ /
ξ∗ ∈ RN :
|ax, t, s, ξ| ≤ α Cx, t |s|px−1 |ξ|px−1 ,
|a0 x, t, s, ξ| ≤ α Cx, t |s|px−1 |ξ|px−1 ,
∗
∗
ax, t, s, ξ − ax, t, s, ξ ξ − ξ > 0,
ax, t, s, ξξ a0 x, t, s, ξs ≥ β |ξ|px |s|px ,
where Cx, t ≥ 0, Cx, tpx/px−1 ∈ LNp
−
/1−h0 p−
Q, and α, β > 0 are constants.
3.39
Journal of Inequalities and Applications
15
Similarly, we can get the following theorem.
Theorem 3.3. Let p− > max{1, 2N/N 2}. If u is a nonnegative local weak solution of 3.37,
1.3, and 1.4, then u is locally bounded in Q. Moreover, there exists a constant C CN, pρ , pρ− , ρ
such that for any Qρpρ , ρ ∈ Q and any σ ∈ 0, 1,
⎫
⎞h/q−δ
"
⎪
⎪
⎬
1
⎟
−pρ Npρ− /Nq−δ ⎜
δ
,
u
dx
dt
sup u ≤ max 1, C1 − σ
⎝ ⎠
p
⎪
⎪
p
⎪
⎪
Q ρpρ , ρ Qρ ρ ,ρ
⎭
⎩
Qσρ ρ ,σρ
⎧
⎪
⎪
⎨
⎛
3.40
where for all x0 , t0 ∈ Q, Kρ {x ∈ Ω | max1≤i≤N |xi − x0,i | < ρ}, pρ supKρ px, pρ− infKρ px,
" h0 p− /N ∈
Qρpρ , ρ Kρ × t0 − ρpρ , t0 , and max{p , 2} ≤ δ < q N 2/Np− , h
0, pρ− /N.
ρ
ρ
ρ
Acknowledgments
This work is supported by Science Research Foundation in Harbin Institute of Technology
HITC200702, and The Natural Science Foundation of Heilongjiang Province A2007-04 and
the Program of Excellent Team in Harbin Institute of Technology.
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