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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. References 1 E. Acerbi and G. Mingione, “Regularity results for stationary electro-rheological fluids,” Archive for Rational Mechanics and Analysis, vol. 164, no. 3, pp. 213–259, 2002. 2 E. Acerbi and G. 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