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Local boundedness of local weak solutions for doubly degenerate parabolic systems with singular weights
Applied Mathematics Letters 25 (2012) 1711–1715
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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Local boundedness of local weak solutions for doubly degenerate parabolic systems with singular weights✩ Yongzhong Wang ∗ , Pengcheng Niu, Xuewei Cui Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, China
article
abstract
info
Article history: Received 13 December 2010 Received in revised form 10 June 2011 Accepted 30 January 2012
In this paper, we prove that local weak solutions for doubly degenerate parabolic systems with singular weights are locally bounded. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Local boundedness Doubly degenerate Parabolic system Singular weight
1. Introduction We are concerned with the following parabolic system with singular weights:
∂t ui − div(|x|pγ |u|α |Du|p−2 Dui ) = C |x|pγ |u|α−1 |Du|p−1 ui (i = 1, 2 . . . , l), (x, t ) ∈ ΩT ,
(1.1)
where ΩT ≡ Ω × (0, T ), Ω is a bounded domain containing the origin O in R and having a smooth boundary, 0 < N
T < +∞, 2 < p < N , 1 −
N p
< γ ≤ 0, u = (u1 , u2 , . . . , ul ), |u| = (
1 ( li=1 Nj=1 | ∂∂ uxji |2 ) 2 , and C is a positive constant.
l
i =1
1
∂u
∂u
∂u
u2i ) 2 , Dui = ( ∂ x i , ∂ x i , . . . , ∂ x i ), |Du| = N 1 2
System (1.1) can be seen as a generalization of the model of motion of non-Newtonian fluids in [1]. The original work considering doubly degenerate parabolic equations is from [2–4]. Kinnunen and Kuusi [5] extend the result in [2] to some doubly degenerate parabolic equations with weights. Using the method given by [6], regularity of weak solutions to the equations in [5] was studied in [7]. We deal with the local boundedness of weak solutions to (1.1). The boundedness of weak solutions to some parabolic equations and systems contributes to the regularity of the solutions; see [3,4,8–12]. Weak solutions to (1.1) in special cases have been studied by some authors. According to the energy estimates and some interpolation inequalities without weights, DiBenedetto [11] proved that weak solutions to (1.1) with γ = 0 and α = 0 are locally bounded. Following the method in [11], Abdellaoui and Peral Alonso [8] showed that weak solutions to a single equation with γ ̸= 0 and α = 0 are locally bounded. Choosing a test function different from that in [11], Sango [13] investigated the local boundedness of weak solutions to (1.1) with γ = 0 but α ̸= 0. The boundedness of weak solutions to a doubly degenerate parabolic single equation with a degenerate weight was studied in [14].
✩ This work was supported by the National Natural Science Foundation of China (Grant No. 10871157, 11001221) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 200806990032). ∗ Corresponding author. E-mail address: [email protected] (Y. Wang).
0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.01.043
1712
Y. Wang et al. / Applied Mathematics Letters 25 (2012) 1711–1715
some which is similar to that in [8]. For any function ϕ ∈ C0∞ (Ω ), we define ∥ϕ∥γ = Letpγus give notation p pγ p |x| |Dϕ| dx + Ω |x| |ϕ| dx. We denote by Wγ1,p (Ω ) the completion of {ϕ ∈ C0∞ (Ω ) : ∥ϕ∥γ < ∞} with respect Ω to the norm ∥ϕ∥γ . If A ⊂ Ω × (0, T ), we set |A|dν×dt ≡ A dν dt, where dν = |x|pγ dx. In the whole paper, repeated indices i indicate summation over i from 0 to l. α
1,p
p
p
Definition 1.1. If a measurable function u(x, t ) satisfies |u| p−1 ui ∈ Lloc (0, T ; Wγ ,loc (Ω )), ui ∈ Cloc (0, T ; L2loc (Ω ))∩ Lloc (0, T ;
1,p Wγ ,loc
(Ω )) and
t
Ω′
ui φi |t21 dx +
t2
t1
Ω′
(−ui ∂t φi + |x|pγ |u|α |Du|p−2 Dui Dφi )dxdt =
t2
t1 1 ,p
p
Ω′
C |x|pγ |u|α−1 |Du|p−1 ui φi dxdt
(1.2)
1 ,2
for any [t1 , t2 ] ⊂ (0, T ), Ω ′ ⊂ Ω and φi ∈ Lloc (0, T ; Wγ ,loc (Ω ′ )) ∩ Wloc (0, T ; L2 (Ω ′ )), then u(x, t ) is called a local weak solution to (1.1). The main result in this paper is the following. Theorem 1.1. Denote a cylinder in ΩT by Q (ρ, θ ) = Bρ (x0 ) × (t0 − θ , t0 ), where (x0 , t0 ) is any point in ΩT , 0 constant that is sufficiently small, θ > 0, and Bρ (x0 ) = {x ∈ < ρ < 1 is a positive 2(p−1) N
Ω | |x − x0 | < ρ}. If 0 ≤ α < min
, (N +2−pp)(p−1) , then the local weak solution u to (1.1) is locally bounded in ΩT , i.e.,
ess sup |u| ≤ k,
Q (σ ρ,σ θ)
(p+α)(2+N )2 (p−1)2 N α p(p+α−1)2
0θ N α (1 − σ ) ( ρ p(1Y−γ )C′ ) ′ C is a constant depending only on the parameters C , N , p, γ , α and Ω .
where σ ∈ (0, 1), k = max{1, 256
p
+2)(p−1) − 2p(p(N +α−1)N α
1
, ( ρθ ) p−2 }, Y0 = p
pα +p pγ p−1 dxdt Q (ρ,θ) |x| |u|
|Q (ρ,θ )|dν×dt
, and
Because of the presence of the weight |x|pγ , (1.1) is not translation invariant with respect to the variables x. Therefore, we distinguish two cases x0 = O and x0 ̸= O to prove Theorem 1.1. Taking the test function as in [11], we get the energy estimates of the weak solution u to (1.1). Note that the function |u|α appears in the energy estimates. This means that the lemma in [8] cannot be used directly here. The lemma in [8] is stated as follows. Lemma 1.2 ([8]). There exists a positive constant γ1 (N , p, γ ) such that
|U |
(N +q) Np
pγ
|x| dxdt ≤ γ1 ess sup t ∈(t1 ,t2 )
H
q
Np
|DU |p |x|pγ dxdt
|U | dx Ω′
H
for all U ∈ Lp (t1 , t2 ; Wγ1,p (Ω ′ )) ∩ L∞ (t1 , t2 ; Lq (Ω ′ )) and γ < 1. Here, H = (t1 , t2 ) × Ω ′ . To overcome the difficulty, we establish the lower boundedness of weak solutions to (1.1). Following the way given in [11], we arrive at our aim by using the following lemma. 1+β
Lemma 1.3 ([15]). Let {Yn } be a sequence of positive numbers such that Yn+1 ≤ Γ Bn Yn Y0 ≤ Γ
−1 −1 β B β2 ,
, where Γ , B > 1 and 0 < β < 1. If
then Yn → 0 as n → ∞.
2. Proof of Theorem 1.1 ρ +ρ θ +θ )ρ )θ , ρn = σ ρ + (1−σ , θn = σ θ + (1−σ , ρ˜ n = n 2 n+1 , θ˜n = n 2n+1 , Qn = Bρn (x0 ) × [t0 − θn , t0 ], 2n 2n and Q˜ n = Bρ˜ n (x0 ) × [t0 − θ˜n , t0 ], where n = 0, 1, 2, . . .. Obviously, Qn+1 ⊂ Q˜ n ⊂ Qn . Choose the cut-off function ξn (x, t ) 2n 2n satisfying 0 ≤ ξn ≤ 1, |Dξn | ≤ (1−σ and |∂t ξn | ≤ (1−σ in Qn , ξn = 0 on the lateral boundary of Qn , and ξn ≡ 1 in Q˜ n . )ρ )θ As on page 222 of [11], set fε (s) = 1 if s − kn+1 ≥ ε , fε (s) = ε −1 (s − kn+1 ) if 0 < s − kn+1 < ε , and fε (s) = 0 when s − kn+1 ≤ 0. We get from (1.2) that ∂t ui,h φi dxdt + (|x|pγ |u|α |Du|p−2 Dui )h Dφi dxdt − (C |x|pγ |u|α−1 |Du|p−1 ui )h φi dxdt = 0, (2.1)
Let kn = k −
Qn (t )
k 2n
Qn (t )
Qn (t )
p
where Qn (t ) = Bρn (x0 ) × [t0 − θn , t ], t ∈ [t0 − θn , t0 ], φi = ui,h fε (|uh |)ξn , vh is the Steklov average of v if v ∈ Lloc (ΩT ) (see [11]).
Y. Wang et al. / Applied Mathematics Letters 25 (2012) 1711–1715
1713
Integrating the first integral on the left-hand side of (2.1) by parts, and letting h → 0, we have
∂t ui,h φi dxdt →
Qn (t )
|u|
where F (|u|) =
0
F (|u|)ξnp dx − p
Bρn (x0 )
Qn (t )
F (|u|)ξnp−1 ∂t ξn dxdt ,
(2.2)
sfε (s)ds. Letting h → 0 in the remaining parts in (2.1), we have
(|x|pγ |u|α |Du|p−2 Dui )h Dφi dxdt − (C |x|pγ |u|α−1 |Du|p−1 ui )h φi dxdt Qn (t ) Qn (t ) |u|α |x|pγ |Du|p fε (|u|)ξnp dxdt + |u|α+1 |x|pγ |D|u| |2 |Du|p−2 fε′ (|u|)ξnp dxdt → Qn (t )
Qn (t )
α
+p
pγ
p−2
|u| |x| |Du| Qn (t )
ui Dui fε (|u|)Dξ ξ
p−1 dxdt n n
−C Qn (t )
|u|α+1 |x|pγ |Du|p−1 fε (|u|)ξnp dxdt .
(2.3)
Using Young’s inequality, the third and fourth terms on the right-hand side of (2.3) can be estimated by
|u|α |x|pγ |Du|p−2 ui Dui fε (|u|)Dξn ξnp−1 dxdt
p Qn (t )
p
≤ τ p−1
Qn (t )
|u|α |x|pγ |Du|p fε (|u|)ξnp dxdt + τ −p
Qn (t )
|u|α+p |x|pγ fε (|u|)|Dξn |p dxdt
(2.4)
and
|u|α+1 |x|pγ |Du|p−1 fε (|u|)ξnp dxdt
−C Qn (t )
≤ι
p p−1
α
Qn (t )
where τ = (3p) we have
1 − p− p
Bρn (x0 )
|u| |x| |Du| fε (|u|)ξ p−1 p
+ι
−p
p n dxdt
sds ξnp χ (|u| − kn+1 )dx + sfε (s)ds ξ
≤ Bρn (x0 )
p n dx
+
0
Qn (t )
Qn (t )
|u|α+p |x|pγ fε (|u|)ξnp dxdt ,
(2.5)
Qn (t )
|u|α |x|pγ |D|u| |p χ (|u| − kn+1 )ξnp dxdt
Qn (t )
|u|α |x|pγ |D|u| |p fε (|u|)ξnp dxdt
|u|α+p |x|pγ fε (|u|)(ξnp + |Dξn |p )dxdt +
|u|
Qn (t )
sfε (s)ds |∂t ξn |dxdt
0
|u|α+p |x|pγ χ (|u| − kn+1 )(ξnp + |Dξn |p )dxdt
Qn (t )
kn+1 +ε
|u|
. Obviously, |Du| ≥ |D|u| | and fε′ (|u|) ≥ 0. Using (2.1)–(2.5) and the selection of fε (s),
≤ C′
p
, ι = (3C )−
|u|
≤ C′
pγ
sds |∂t ξn |χ (|u| − kn+1 )dxdt
+ Qn (t )
|u|
,
(2.6)
kn+1
where χ (z ) = 1 when z > 0 and χ (z ) = 0 when z ≤ 0. Therefore, letting ε → 0 in (2.6) and taking the supremum for t over (t0 − θn , t0 ), from the selection of ξn , we conclude that
ess sup
t ∈(t0 −θn ,t0 )
≤ C′
Bρn (x0 )
2np
(1 − σ )p
(|u| −
) ξ 2
kn+1 + np dx
+k
α
|x|pγ |D(|u| − kn+1 )+ |p ξnp dxdt Qn
1
ρp
|u|α+p |x|pγ χ (|u| − kn+1 )dxdt +
Qn
1
θ
|u|(|u| − kn+1 )+ dxdt .
(2.7)
Qn
Since Ω is bounded, there exists a positive constant L such that |x| ≤ L for any x ∈ Ω . Let x ∈ Bρn (x0 ). If x0 = O, then
|x|pγ ≥ (ρn )pγ > 1; if x0 ̸= O, then 1 ≤ C ′ |x|pγ .
2 −pγ 3
Lpγ ≤
2 −pγ 3
|x0 |pγ ≤ |x|pγ ≤ 2−pγ |x0 |pγ ≤ (ρn )pγ . Thus, for any x ∈ Bρn (x0 ),
(2.8)
1714
Y. Wang et al. / Applied Mathematics Letters 25 (2012) 1711–1715 2n+1 −2
2n+1 −2 2n+1 −1
Note that kn = kn+1 2n+1 −1 and 1 −
>
Qn
Therefore,
| u| dxdt (|u| − kn+1 )+ |u| − kn
|u|(|u| − kn+1 )+ dxdt ≤ 2n+1
1 . 2n+1
kn+1
Qn
(|u| − kn )2+ χ (|u| − kn+1 )dxdt ≤ C ′
≤ Qn
(|u| − kn )2+ |x|pγ χ (|u| − kn+1 )dxdt .
(2.9)
Qn
Similarly to the derivation of (7.2)–(7.4) on page 132 of [11], we obtain
λ
pγ
(n+1)λ
|u| χ (|u| − kn+1 )|x| dxdt ≤ 2
Qn
(|u| − kn )λ+ |x|pγ χ (|u| − kn+1 )dxdt ,
(2.10)
Qn
2(δ−m)n
pγ (|u| − kn )m + |x| χ (|u| − kn+1 )dxdt ≤
kδ−m
Qn
(|u| − kn )δ+ |x|pγ χ (|u| − kn+1 )dxdt ,
(2.11)
Qn
and 2(n+1)λ
|An+1 |dν×dt ≤
kλ
(|u| − kn )λ+ |x|pγ χ (|u| − kn+1 )dxdt ,
(2.12)
Qn
where λ > 0, δ ≥ m > 0, and An+1 = {(x, t ) ∈ Qn ||u| > kn+1 }. Using (2.7), and (2.9)–(2.11), it follows that
ess sup t ∈(t0 −θ˜n ,t0 ) Bρ˜ n (x0 )
≤ C′ ≤C
′
4n(p+α)
( 1 − σ )p
1
ρp
16n(p+α) α k p−1
(|u| − kn+1 )2+ dx + kα +
(1 − σ )
θ kp+α−2
|x|pγ |D(|u| − kn+1 )+ |p dxdt
p pγ (|u| − kn )α+ + |x| χ (|u| − kn+1 )dxdt
Qn pα
1
+
ρp
p
1
1
Q˜ n
θ kp+α−2
(|u| − kn )+p−1
+p
|x|pγ dxdt .
(2.13)
Qn n
2 Choose a new cut-off function ξ˜n (x) satisfying 0 ≤ ξ˜n ≤ 1, |Dξ˜n | ≤ (1−σ in Q˜ n , and ξ˜n ≡ 1 in Qn+1 . Lemma 1.2, (2.11) and )ρ (2.13) lead to
Q˜ n
2p
p+ N n
pγ
|x| ξ˜
(|u| − kn+1 )+ pγ
× Q˜ n
|x| |D(|u| −
−α
≤k
C
α
1
ρp
k p−1 (1 − σ )p
pα +p p−1 pγ dxdt Qn |x| (|u|−kn )+
+
Bρ˜ n (x0 )
pγ
+ Q˜ n
θ kp+α−2
p
|x| (|u| − kn+1 )+ |Dξ˜n |p dxdt pα +p p−1
1
(|u| − kn+1 )+ dx 2
ess sup t ∈(t0 −θ˜n ,t0 )
) | ξ˜
Set Yn =
dxdt ≤ C
Np
′
kn+1 + p np dxdt
16n(p+α)
′
2p
p+ N
(|u| − kn )+
1+ Np pγ
.
|x| dxdt
(2.14)
Qn
N (p+α−1)
p+
2p
N = and q1 = (p−1)(N +2) . The assumption on α implies that pα p+ p−1
|Qn |dν×dt
(N +2)(p−1) N (α+p−1)
> 1. From
Hölder’s inequality, Qn+1 ⊂ Q˜ n ⊂ Qn , (2.12) and (2.14), we get q1 2p 2p p+ p+ |x|pγ ξ˜n N (|u| − kn+1 )+ N dxdt |An+1 |dν×dt 1−q1 |Qn |dν×dt Q˜ n Yn+1 ≤ |Qn |dν×dt |Qn |dν×dt |Qn+1 |dν×dt
≤ C′
2p
p+ N
|x|pγ ξ˜n Q˜ n
2p
p+ N
(|u| − kn+1 )+
dxdt
N +pγ
pα n p+ p−1
2
|Qn |dν×dt ρ
1) (Np−(p1+α− )(N +2)
k
pα p+ p−1
Nα (2p(−p−1)(1)− N +2)
.
Yn
(2.15)
Here, we use that |Q n d|ν×dt ≤ C ′ θ θn nN +pγ ≤ C ′ if x0 = O and |Q n d|ν×dt ≤ C ′ |Q n dx| ×dt ≤ C ′ if x0 ̸= O. Thus from (2.14) n+1 dν×dt n+1 ρ n+1 dν×dt n+1 dx×dt n+1 pα and (2.15), p−1 ≤ 2(p + α), and kp1−2 ≤ ρθp , it follows that |Q |
Yn+1 ≤
−α
k
C
′
16n(p+α)
(1 − σ )p
1
ρp
+
|Q |
1
θ kp−2
1+ Np
Yn
p N
|Qn |dν×dt
q1
|Q |
pα n p+ p−1
2
1−q1
Yn
Y. Wang et al. / Applied Mathematics Letters 25 (2012) 1711–1715
≤C
′
(1 − σ )
≤ C′
≤C
256n(p+α)
′
2p
n(p+α)
256
(1 − σ )2p 256n(p+α)
(1 − σ )
2p
p(p+α−1) 1+ (N +2)(p−1)
Yn
1
ρ
p(p+α−1) 1+ (N +2)(p−1)
Yn
p(p+α−1) 1+ (N +2)(p−1)
Yn
+
θ
(pp−(p1−)(1N+α) +2)
1+ Np |Qn |dν×dt
kp−2
p(p−1+α)
p−1) − (NNα(α+ +2)(p−1)
k
p−1) − (NNα(α+ +2)(p−1)
[ρ −p−N θ ρ N +pγ ] (p−1)(N +2) k θ
ρ
Nα
p
1
1715
C ′ ρ p(1−γ )
p(1−γ )
p(p−1+α) (p−1)(N +2)
p−1) − (NNα(α+ +2)(p−1)
k
2 2 − (p+α)(2+N ) (p−1)
.
2(N +2)(p−1)
p2 (p+α−1)2 The condition on k gives Y0 ≤ k p 256 (1 − σ ) p+α−1 θ that Yn → 0(n → ∞), i.e., ess supQ (σ ρ,σ θ) |u| ≤ k. This completes the proof.
(2.16)
. Then, from (2.16) and Lemma 1.3, we know
Acknowledgments We thank all the anonymous reviewers for their useful suggestions. References [1] O.A. Ladyzenskaja, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Steklov Inst. Math. 102 (1967) 95–118. [2] N.S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968) 205–226. [3] V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math. 75 (1) (1992) 65–80. [4] M.M. Porzio, V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations 103 (1) (1993) 146–178. [5] J. Kinnunen, T. Kuusi, Local behaviour of solutions to doubly nonlinear parabolic equations, Math. Ann. 337 (3) (2007) 705–728. [6] E. DiBenedetto, A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 349 (1984) 83–128. [7] J. Siljander, Boundedness of the gradient for a doubly nonlinear parabolic equation, J. Math. Anal. Appl. 371 (1) (2010) 158–167. [8] B. Abdellaoui, I. Peral Alonso, Hölder regularity and Harnack inequality for degenerate parabolic equations related to Caffarelli–Kohn–Nirenberg inequalities, Nonlinear Anal. 57 (2004) 971–1006. [9] A.V. Ivanov, Hölder estimates for quasilinear doubly degenerate parabolic equations, J. Math. Sci. 56 (1991) 2320–2347. [10] H.J. Yuan, Hölder continuity of weak solutions for a nonlinear degenerate parabolic equation, J. Jilin Univ. Sci. 2 (1991) 36–51. [11] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. [12] T. Kuusi, J. Siljander, J.M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J. Preprints. [13] M. Sango, Local boundedness for doubly degenerate quasi-linear parabolic systems, Appl. Math. Lett. 16 (2003) 465–468. [14] Y.Z. Wang, Local Hölder continuity of nonnegative weak solutions of degenerate parabolic equations, Nonlinear Anal. 72 (2010) 3289–3302. [15] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (3) (1984) 259–275.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Local boundedness of local weak solutions for doubly degenerate parabolic systems with singular weights✩ Yongzhong Wang ∗ , Pengcheng Niu, Xuewei Cui Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710129, China
article
abstract
info
Article history: Received 13 December 2010 Received in revised form 10 June 2011 Accepted 30 January 2012
In this paper, we prove that local weak solutions for doubly degenerate parabolic systems with singular weights are locally bounded. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Local boundedness Doubly degenerate Parabolic system Singular weight
1. Introduction We are concerned with the following parabolic system with singular weights:
∂t ui − div(|x|pγ |u|α |Du|p−2 Dui ) = C |x|pγ |u|α−1 |Du|p−1 ui (i = 1, 2 . . . , l), (x, t ) ∈ ΩT ,
(1.1)
where ΩT ≡ Ω × (0, T ), Ω is a bounded domain containing the origin O in R and having a smooth boundary, 0 < N
T < +∞, 2 < p < N , 1 −
N p
< γ ≤ 0, u = (u1 , u2 , . . . , ul ), |u| = (
1 ( li=1 Nj=1 | ∂∂ uxji |2 ) 2 , and C is a positive constant.
l
i =1
1
∂u
∂u
∂u
u2i ) 2 , Dui = ( ∂ x i , ∂ x i , . . . , ∂ x i ), |Du| = N 1 2
System (1.1) can be seen as a generalization of the model of motion of non-Newtonian fluids in [1]. The original work considering doubly degenerate parabolic equations is from [2–4]. Kinnunen and Kuusi [5] extend the result in [2] to some doubly degenerate parabolic equations with weights. Using the method given by [6], regularity of weak solutions to the equations in [5] was studied in [7]. We deal with the local boundedness of weak solutions to (1.1). The boundedness of weak solutions to some parabolic equations and systems contributes to the regularity of the solutions; see [3,4,8–12]. Weak solutions to (1.1) in special cases have been studied by some authors. According to the energy estimates and some interpolation inequalities without weights, DiBenedetto [11] proved that weak solutions to (1.1) with γ = 0 and α = 0 are locally bounded. Following the method in [11], Abdellaoui and Peral Alonso [8] showed that weak solutions to a single equation with γ ̸= 0 and α = 0 are locally bounded. Choosing a test function different from that in [11], Sango [13] investigated the local boundedness of weak solutions to (1.1) with γ = 0 but α ̸= 0. The boundedness of weak solutions to a doubly degenerate parabolic single equation with a degenerate weight was studied in [14].
✩ This work was supported by the National Natural Science Foundation of China (Grant No. 10871157, 11001221) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 200806990032). ∗ Corresponding author. E-mail address: [email protected] (Y. Wang).
0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.01.043
1712
Y. Wang et al. / Applied Mathematics Letters 25 (2012) 1711–1715
some which is similar to that in [8]. For any function ϕ ∈ C0∞ (Ω ), we define ∥ϕ∥γ = Letpγus give notation p pγ p |x| |Dϕ| dx + Ω |x| |ϕ| dx. We denote by Wγ1,p (Ω ) the completion of {ϕ ∈ C0∞ (Ω ) : ∥ϕ∥γ < ∞} with respect Ω to the norm ∥ϕ∥γ . If A ⊂ Ω × (0, T ), we set |A|dν×dt ≡ A dν dt, where dν = |x|pγ dx. In the whole paper, repeated indices i indicate summation over i from 0 to l. α
1,p
p
p
Definition 1.1. If a measurable function u(x, t ) satisfies |u| p−1 ui ∈ Lloc (0, T ; Wγ ,loc (Ω )), ui ∈ Cloc (0, T ; L2loc (Ω ))∩ Lloc (0, T ;
1,p Wγ ,loc
(Ω )) and
t
Ω′
ui φi |t21 dx +
t2
t1
Ω′
(−ui ∂t φi + |x|pγ |u|α |Du|p−2 Dui Dφi )dxdt =
t2
t1 1 ,p
p
Ω′
C |x|pγ |u|α−1 |Du|p−1 ui φi dxdt
(1.2)
1 ,2
for any [t1 , t2 ] ⊂ (0, T ), Ω ′ ⊂ Ω and φi ∈ Lloc (0, T ; Wγ ,loc (Ω ′ )) ∩ Wloc (0, T ; L2 (Ω ′ )), then u(x, t ) is called a local weak solution to (1.1). The main result in this paper is the following. Theorem 1.1. Denote a cylinder in ΩT by Q (ρ, θ ) = Bρ (x0 ) × (t0 − θ , t0 ), where (x0 , t0 ) is any point in ΩT , 0 constant that is sufficiently small, θ > 0, and Bρ (x0 ) = {x ∈ < ρ < 1 is a positive 2(p−1) N
Ω | |x − x0 | < ρ}. If 0 ≤ α < min
, (N +2−pp)(p−1) , then the local weak solution u to (1.1) is locally bounded in ΩT , i.e.,
ess sup |u| ≤ k,
Q (σ ρ,σ θ)
(p+α)(2+N )2 (p−1)2 N α p(p+α−1)2
0θ N α (1 − σ ) ( ρ p(1Y−γ )C′ ) ′ C is a constant depending only on the parameters C , N , p, γ , α and Ω .
where σ ∈ (0, 1), k = max{1, 256
p
+2)(p−1) − 2p(p(N +α−1)N α
1
, ( ρθ ) p−2 }, Y0 = p
pα +p pγ p−1 dxdt Q (ρ,θ) |x| |u|
|Q (ρ,θ )|dν×dt
, and
Because of the presence of the weight |x|pγ , (1.1) is not translation invariant with respect to the variables x. Therefore, we distinguish two cases x0 = O and x0 ̸= O to prove Theorem 1.1. Taking the test function as in [11], we get the energy estimates of the weak solution u to (1.1). Note that the function |u|α appears in the energy estimates. This means that the lemma in [8] cannot be used directly here. The lemma in [8] is stated as follows. Lemma 1.2 ([8]). There exists a positive constant γ1 (N , p, γ ) such that
|U |
(N +q) Np
pγ
|x| dxdt ≤ γ1 ess sup t ∈(t1 ,t2 )
H
q
Np
|DU |p |x|pγ dxdt
|U | dx Ω′
H
for all U ∈ Lp (t1 , t2 ; Wγ1,p (Ω ′ )) ∩ L∞ (t1 , t2 ; Lq (Ω ′ )) and γ < 1. Here, H = (t1 , t2 ) × Ω ′ . To overcome the difficulty, we establish the lower boundedness of weak solutions to (1.1). Following the way given in [11], we arrive at our aim by using the following lemma. 1+β
Lemma 1.3 ([15]). Let {Yn } be a sequence of positive numbers such that Yn+1 ≤ Γ Bn Yn Y0 ≤ Γ
−1 −1 β B β2 ,
, where Γ , B > 1 and 0 < β < 1. If
then Yn → 0 as n → ∞.
2. Proof of Theorem 1.1 ρ +ρ θ +θ )ρ )θ , ρn = σ ρ + (1−σ , θn = σ θ + (1−σ , ρ˜ n = n 2 n+1 , θ˜n = n 2n+1 , Qn = Bρn (x0 ) × [t0 − θn , t0 ], 2n 2n and Q˜ n = Bρ˜ n (x0 ) × [t0 − θ˜n , t0 ], where n = 0, 1, 2, . . .. Obviously, Qn+1 ⊂ Q˜ n ⊂ Qn . Choose the cut-off function ξn (x, t ) 2n 2n satisfying 0 ≤ ξn ≤ 1, |Dξn | ≤ (1−σ and |∂t ξn | ≤ (1−σ in Qn , ξn = 0 on the lateral boundary of Qn , and ξn ≡ 1 in Q˜ n . )ρ )θ As on page 222 of [11], set fε (s) = 1 if s − kn+1 ≥ ε , fε (s) = ε −1 (s − kn+1 ) if 0 < s − kn+1 < ε , and fε (s) = 0 when s − kn+1 ≤ 0. We get from (1.2) that ∂t ui,h φi dxdt + (|x|pγ |u|α |Du|p−2 Dui )h Dφi dxdt − (C |x|pγ |u|α−1 |Du|p−1 ui )h φi dxdt = 0, (2.1)
Let kn = k −
Qn (t )
k 2n
Qn (t )
Qn (t )
p
where Qn (t ) = Bρn (x0 ) × [t0 − θn , t ], t ∈ [t0 − θn , t0 ], φi = ui,h fε (|uh |)ξn , vh is the Steklov average of v if v ∈ Lloc (ΩT ) (see [11]).
Y. Wang et al. / Applied Mathematics Letters 25 (2012) 1711–1715
1713
Integrating the first integral on the left-hand side of (2.1) by parts, and letting h → 0, we have
∂t ui,h φi dxdt →
Qn (t )
|u|
where F (|u|) =
0
F (|u|)ξnp dx − p
Bρn (x0 )
Qn (t )
F (|u|)ξnp−1 ∂t ξn dxdt ,
(2.2)
sfε (s)ds. Letting h → 0 in the remaining parts in (2.1), we have
(|x|pγ |u|α |Du|p−2 Dui )h Dφi dxdt − (C |x|pγ |u|α−1 |Du|p−1 ui )h φi dxdt Qn (t ) Qn (t ) |u|α |x|pγ |Du|p fε (|u|)ξnp dxdt + |u|α+1 |x|pγ |D|u| |2 |Du|p−2 fε′ (|u|)ξnp dxdt → Qn (t )
Qn (t )
α
+p
pγ
p−2
|u| |x| |Du| Qn (t )
ui Dui fε (|u|)Dξ ξ
p−1 dxdt n n
−C Qn (t )
|u|α+1 |x|pγ |Du|p−1 fε (|u|)ξnp dxdt .
(2.3)
Using Young’s inequality, the third and fourth terms on the right-hand side of (2.3) can be estimated by
|u|α |x|pγ |Du|p−2 ui Dui fε (|u|)Dξn ξnp−1 dxdt
p Qn (t )
p
≤ τ p−1
Qn (t )
|u|α |x|pγ |Du|p fε (|u|)ξnp dxdt + τ −p
Qn (t )
|u|α+p |x|pγ fε (|u|)|Dξn |p dxdt
(2.4)
and
|u|α+1 |x|pγ |Du|p−1 fε (|u|)ξnp dxdt
−C Qn (t )
≤ι
p p−1
α
Qn (t )
where τ = (3p) we have
1 − p− p
Bρn (x0 )
|u| |x| |Du| fε (|u|)ξ p−1 p
+ι
−p
p n dxdt
sds ξnp χ (|u| − kn+1 )dx + sfε (s)ds ξ
≤ Bρn (x0 )
p n dx
+
0
Qn (t )
Qn (t )
|u|α+p |x|pγ fε (|u|)ξnp dxdt ,
(2.5)
Qn (t )
|u|α |x|pγ |D|u| |p χ (|u| − kn+1 )ξnp dxdt
Qn (t )
|u|α |x|pγ |D|u| |p fε (|u|)ξnp dxdt
|u|α+p |x|pγ fε (|u|)(ξnp + |Dξn |p )dxdt +
|u|
Qn (t )
sfε (s)ds |∂t ξn |dxdt
0
|u|α+p |x|pγ χ (|u| − kn+1 )(ξnp + |Dξn |p )dxdt
Qn (t )
kn+1 +ε
|u|
. Obviously, |Du| ≥ |D|u| | and fε′ (|u|) ≥ 0. Using (2.1)–(2.5) and the selection of fε (s),
≤ C′
p
, ι = (3C )−
|u|
≤ C′
pγ
sds |∂t ξn |χ (|u| − kn+1 )dxdt
+ Qn (t )
|u|
,
(2.6)
kn+1
where χ (z ) = 1 when z > 0 and χ (z ) = 0 when z ≤ 0. Therefore, letting ε → 0 in (2.6) and taking the supremum for t over (t0 − θn , t0 ), from the selection of ξn , we conclude that
ess sup
t ∈(t0 −θn ,t0 )
≤ C′
Bρn (x0 )
2np
(1 − σ )p
(|u| −
) ξ 2
kn+1 + np dx
+k
α
|x|pγ |D(|u| − kn+1 )+ |p ξnp dxdt Qn
1
ρp
|u|α+p |x|pγ χ (|u| − kn+1 )dxdt +
Qn
1
θ
|u|(|u| − kn+1 )+ dxdt .
(2.7)
Qn
Since Ω is bounded, there exists a positive constant L such that |x| ≤ L for any x ∈ Ω . Let x ∈ Bρn (x0 ). If x0 = O, then
|x|pγ ≥ (ρn )pγ > 1; if x0 ̸= O, then 1 ≤ C ′ |x|pγ .
2 −pγ 3
Lpγ ≤
2 −pγ 3
|x0 |pγ ≤ |x|pγ ≤ 2−pγ |x0 |pγ ≤ (ρn )pγ . Thus, for any x ∈ Bρn (x0 ),
(2.8)
1714
Y. Wang et al. / Applied Mathematics Letters 25 (2012) 1711–1715 2n+1 −2
2n+1 −2 2n+1 −1
Note that kn = kn+1 2n+1 −1 and 1 −
>
Qn
Therefore,
| u| dxdt (|u| − kn+1 )+ |u| − kn
|u|(|u| − kn+1 )+ dxdt ≤ 2n+1
1 . 2n+1
kn+1
Qn
(|u| − kn )2+ χ (|u| − kn+1 )dxdt ≤ C ′
≤ Qn
(|u| − kn )2+ |x|pγ χ (|u| − kn+1 )dxdt .
(2.9)
Qn
Similarly to the derivation of (7.2)–(7.4) on page 132 of [11], we obtain
λ
pγ
(n+1)λ
|u| χ (|u| − kn+1 )|x| dxdt ≤ 2
Qn
(|u| − kn )λ+ |x|pγ χ (|u| − kn+1 )dxdt ,
(2.10)
Qn
2(δ−m)n
pγ (|u| − kn )m + |x| χ (|u| − kn+1 )dxdt ≤
kδ−m
Qn
(|u| − kn )δ+ |x|pγ χ (|u| − kn+1 )dxdt ,
(2.11)
Qn
and 2(n+1)λ
|An+1 |dν×dt ≤
kλ
(|u| − kn )λ+ |x|pγ χ (|u| − kn+1 )dxdt ,
(2.12)
Qn
where λ > 0, δ ≥ m > 0, and An+1 = {(x, t ) ∈ Qn ||u| > kn+1 }. Using (2.7), and (2.9)–(2.11), it follows that
ess sup t ∈(t0 −θ˜n ,t0 ) Bρ˜ n (x0 )
≤ C′ ≤C
′
4n(p+α)
( 1 − σ )p
1
ρp
16n(p+α) α k p−1
(|u| − kn+1 )2+ dx + kα +
(1 − σ )
θ kp+α−2
|x|pγ |D(|u| − kn+1 )+ |p dxdt
p pγ (|u| − kn )α+ + |x| χ (|u| − kn+1 )dxdt
Qn pα
1
+
ρp
p
1
1
Q˜ n
θ kp+α−2
(|u| − kn )+p−1
+p
|x|pγ dxdt .
(2.13)
Qn n
2 Choose a new cut-off function ξ˜n (x) satisfying 0 ≤ ξ˜n ≤ 1, |Dξ˜n | ≤ (1−σ in Q˜ n , and ξ˜n ≡ 1 in Qn+1 . Lemma 1.2, (2.11) and )ρ (2.13) lead to
Q˜ n
2p
p+ N n
pγ
|x| ξ˜
(|u| − kn+1 )+ pγ
× Q˜ n
|x| |D(|u| −
−α
≤k
C
α
1
ρp
k p−1 (1 − σ )p
pα +p p−1 pγ dxdt Qn |x| (|u|−kn )+
+
Bρ˜ n (x0 )
pγ
+ Q˜ n
θ kp+α−2
p
|x| (|u| − kn+1 )+ |Dξ˜n |p dxdt pα +p p−1
1
(|u| − kn+1 )+ dx 2
ess sup t ∈(t0 −θ˜n ,t0 )
) | ξ˜
Set Yn =
dxdt ≤ C
Np
′
kn+1 + p np dxdt
16n(p+α)
′
2p
p+ N
(|u| − kn )+
1+ Np pγ
.
|x| dxdt
(2.14)
Qn
N (p+α−1)
p+
2p
N = and q1 = (p−1)(N +2) . The assumption on α implies that pα p+ p−1
|Qn |dν×dt
(N +2)(p−1) N (α+p−1)
> 1. From
Hölder’s inequality, Qn+1 ⊂ Q˜ n ⊂ Qn , (2.12) and (2.14), we get q1 2p 2p p+ p+ |x|pγ ξ˜n N (|u| − kn+1 )+ N dxdt |An+1 |dν×dt 1−q1 |Qn |dν×dt Q˜ n Yn+1 ≤ |Qn |dν×dt |Qn |dν×dt |Qn+1 |dν×dt
≤ C′
2p
p+ N
|x|pγ ξ˜n Q˜ n
2p
p+ N
(|u| − kn+1 )+
dxdt
N +pγ
pα n p+ p−1
2
|Qn |dν×dt ρ
1) (Np−(p1+α− )(N +2)
k
pα p+ p−1
Nα (2p(−p−1)(1)− N +2)
.
Yn
(2.15)
Here, we use that |Q n d|ν×dt ≤ C ′ θ θn nN +pγ ≤ C ′ if x0 = O and |Q n d|ν×dt ≤ C ′ |Q n dx| ×dt ≤ C ′ if x0 ̸= O. Thus from (2.14) n+1 dν×dt n+1 ρ n+1 dν×dt n+1 dx×dt n+1 pα and (2.15), p−1 ≤ 2(p + α), and kp1−2 ≤ ρθp , it follows that |Q |
Yn+1 ≤
−α
k
C
′
16n(p+α)
(1 − σ )p
1
ρp
+
|Q |
1
θ kp−2
1+ Np
Yn
p N
|Qn |dν×dt
q1
|Q |
pα n p+ p−1
2
1−q1
Yn
Y. Wang et al. / Applied Mathematics Letters 25 (2012) 1711–1715
≤C
′
(1 − σ )
≤ C′
≤C
256n(p+α)
′
2p
n(p+α)
256
(1 − σ )2p 256n(p+α)
(1 − σ )
2p
p(p+α−1) 1+ (N +2)(p−1)
Yn
1
ρ
p(p+α−1) 1+ (N +2)(p−1)
Yn
p(p+α−1) 1+ (N +2)(p−1)
Yn
+
θ
(pp−(p1−)(1N+α) +2)
1+ Np |Qn |dν×dt
kp−2
p(p−1+α)
p−1) − (NNα(α+ +2)(p−1)
k
p−1) − (NNα(α+ +2)(p−1)
[ρ −p−N θ ρ N +pγ ] (p−1)(N +2) k θ
ρ
Nα
p
1
1715
C ′ ρ p(1−γ )
p(1−γ )
p(p−1+α) (p−1)(N +2)
p−1) − (NNα(α+ +2)(p−1)
k
2 2 − (p+α)(2+N ) (p−1)
.
2(N +2)(p−1)
p2 (p+α−1)2 The condition on k gives Y0 ≤ k p 256 (1 − σ ) p+α−1 θ that Yn → 0(n → ∞), i.e., ess supQ (σ ρ,σ θ) |u| ≤ k. This completes the proof.
(2.16)
. Then, from (2.16) and Lemma 1.3, we know
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