The Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity

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1 The Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity Mihaela Ignatova Igor Kukavica Lenya Ryzhik April 3, 03 Abstract We establish the Harnack inequality for advection-diffusion equations with divergencefree drifts of low regularity. While our previous work [IKR] considered the elliptic case, here we treat the more challenging parabolic problem by adapting the classical Moser technique to parabolic equations with drifts with regularity lower than the scale-invariant spaces. Introduction In this paper, we address the qualitative properties of solutions to the parabolic equation u t u + b u = 0 in Ω,. where b is a given divergence free vector field of low regularity, and Ω is a space-time domain. The study of such equations with non-smooth drifts bx, t is motivated by the need to understand the qualitative and quantitative properties of nonlinear partial differential equations, where the drift depends on the solution u and its first derivatives and for which we often do not have a priori bounds available except in some very low regularity spaces. Advection-diffusion equations of the form. often arise in applications with the additional divergence-free condition div b = 0, in particular, in problems involving incompressible fluids. Several important recent papers have addressed regularity of the solutions of the linear advection-diffusion equations with very little smoothness assumptions on the divergence free drift [CV, CV, FV, KNSS, NU, SSSZ, Z]. Here, we study this problem for the parabolic equation. with a divergence-free supercritical drift b. Criticality here referes to the following property: the usual parabolic rescaling x λx, t λ t leaves the norm of the drift term in the equation invariant in a space b L q t Lp x if /q + n/p =. Accordingly, we say that the drift is critical if this relation holds, is subcritical if /q + n/p < and is supercritical if /q + n/p >. Department of Mathematics, Stanford University, Stanford CA 94305, Department of Mathematics, University of Southern California, Los Angeles, CA 90089, Department of Mathematics, Stanford University, Stanford CA 94305,

2 Our main result is the Harnack-type inequality for parabolic advection-diffusion equations with supercritical drifts. We use the notation Q Rx 0, t 0 = {x, t R n+ : x x 0 < R, t 0 < t < t 0 + R }. for the parabolic cylinder centered at the bottom and Q R x 0, t 0 = {x, t R n+ : x x 0 < R, t 0 R < t < t 0 }.3 for the parabolic cylinder centered at the top, while we denote Q R = Q R 0, 0. Theorem.. Let u be a nonnegative Lipschitz solution to the parabolic equation that is, Ω u t u + b u = 0 in Ω,.4 t uϕ + j u j ϕ + b j j uϕ = 0.5 Ω Ω for any Lipschitz function ϕ 0 in Ω and ϕ = 0 in Ω c. Assume that b L q Ω L L Ω with n/ + < q n + and div b = 0 in the sense of distributions. Then for any Q R Ω, sup u C + CR n+/ q b L q / n+/ q Cn/p 0 inf u,.6 Q R/ 0, 3R Q R where p 0 = /CM C R and M R = + R n/ b L L + R n+/ q b L q. Here, and elsewhere in this paper, the symbol C denotes a large constant which depends on the parameters q and n, and on the domain Ω R n+. Also, we denote the anisotropic Lebesque spaces by L p L q Ω = L p t Lq xω, and in the case when p = q, by L q Ω = L q x,t Ω. The qualitative properties of solutions to the equation. have been extensively studied in the past. In particular, Harnack s inequality for the second order parabolic equation u t i a ij x, t j u = 0 in the self-adjoint form, with measurable strongly elliptic coefficients a ij was obtained in the seminal work of Moser [M] for subctricical drifts and no lower order terms. In [L], Liebermann established the Harnack inequality in the case of non-zero lower-order coefficients, when the drift belongs to a subcritical space. Recently, Nazarov and Ural tseva proved in [NU] that the assumptions on the divergence free drift b can be significantly relaxed to allow it to lie in the scale invariant critical Morrey spaces for all q and l satisfying n/q + /l <. Seregin et al. c.f. [SSSZ] established the Harnack inequality when b belongs to L BMO, which is also a critical scale-invariant space. In our previous paper [IKR], we obtained a Harnack inequality for elliptic equations with supercritical divergence-free drifts. The purpose of the present paper is to relax the assumptions M n/q+/l l,q

3 on the drift in the parabolic case to lie in a supercritical space. Note that the approach from [IKR] does not apply here and the proof in the parabolic case is different. The paper is organized as follows. In Section, we establish the local boundedness of nonnegative Lipschitz subsolutions to. by using Moser s iteration. This result of independent interest was previously obtained in [NU]. However, the bound. with an explicit dependence on the parameters is needed for establishing the validity of Theorem., thus we provide our proof here for completeness. The rest of the paper, Section 3, is devoted to the lower bound of the infimum of Lipschitz supersolutions to., stated in Lemma 3.. We proceed by deriving consecutive estimates on the nonnegative supersolution w = log + u/k, where the constant K is determined in the initial step c.f. Lemma 3. and depends on the values of the supersolution u to.. Here, we follow the approach of Liebermann [L]. We emphasize that this initial step requires the additional assumption on the drift b L L Ω which was not needed in the elliptic case see [IKR]. In Lemma 3.3, we establish an estimate which allows us to bootstrap the initial bounds on w from Lemma 3. to higher L σ -norms for any σ [, n + /n. Using Lemma 3.3, we also obtain a bound on w L in Subsection 3.3, which is essential for estimating higher norms. Then, the aforementioned estimates on all the higher norms are deduced by using Moser s iteration technique see Subsection 3.4. The lower bound on the infimum then follows from the auxiliary assertion in Lemma 3.4. Our main result is a consequence of Lemmas. and 3.. Acknowledgments. I.K. was supported in part by the NSF grant DMS , L.R. was supported in part by the NSF grant DMS , and both M.I and L.R. were supported in part by NSF FRG grant DMS Local boundedness In this section, we show that any nonnegative Lipschitz subsolution of. is locally bounded when the divergence free drift belongs to the anisotropic Lebesgue spaces L l L q Ω for all l and q satisfying /l + n/ q <. Lemma.. Assume that u is a nonnegative Lipschitz subsolution to the equation u t u + b u = 0. with b L ll q Ω for /l + n/ q < and div b 0 in the sense of distributions. Then for any Q R Ω, p > 0, and 0 < θ < τ < / /l n/ q n+/p sup u C + R /l n/ q b L Ω l L q R n+/p u L p R,. Q θr where C = Cn, p, l, q, θ, τ is a positive constant. Proof of Lemma.. Let u be a nonnegative Lipschitz subsolution of. in Ω, that is, t uϕ + j u j ϕ + b j j uϕ 0.3 Ω Ω Ω 3

4 for any Lipschitz function ϕ 0 in Ω and ϕ = 0 in Ω c. We assume without loss of generality that R =. We will use in.3 test functions of the form β ϕ = + u β+ η γ χ {t T }, with a Lipschitz cut-off function η in, such that 0 η, and the constants β > 0 and γ > 0 to be set later we will let β + while γ will remain fixed. This gives, for T τ, 0 β β + t uu β+ η γ χ {t T } + + j u[ j u β+ ]η γ χ {t T }.4 Q τ β β + + u β+ [ j u][ j η γ ]χ {t T } + + b j u β+ [ j u]η γ χ {t T } 0. Set w = u β/+, so that j w = β + u β/ j u. Using.4, we get, integrating the first term by parts in time: w η γ + β + w η γ χ B τ t=t β/ + {t T } γ j wwη γ j ηχ {t T }.5 Q τ b j j wwη γ χ {t T } + γ w η γ t ηχ {t T }. Here, we have utilized the fact that ηx, τ = 0. For the first term in the right side of.5 we have γ j wwη γ j ηχ {t T } = γ w η γ η + γ η γ η χ {t T },.6 while for the second term: b j j wwη γ χ {t T } = j b j w η γ χ {t T } + γ b j w η γ j ηχ {t T }.7 Q τ γ b j w η γ j ηχ {t T }, since div b 0. Next, let γ 0 = /l + n/ q. Then, by assumption, we have γ 0 [,. We also choose γ = / γ 0, so that γγ 0 = γ. By Hölder s inequality we have the following estimate for the right side in.7: b j w η γ j ηχ {t T } b j wη γ γ 0 w γ 0 j η χ {t T }.8 b L l L q wηγ χ t x t T γ 0 L s t Lr w η / γ 0 χ x t T γ 0. L t,x 4

5 Here s and r are determined by q + γ 0 r + γ 0 =, and l + γ 0 s + γ 0 =. It is easy to verify that /s + n/r = n/ this is how γ 0 was chosen. Now, Young s and the interpolation inequality f L s t L r x C f α L t L x f α L t,x with /s + n/r = n/ and α = n/ n/r, applied to the right side of.8, imply b j w η γ j ηχ {t T } ɛ wη γ χ t T L s + t Q Lr C b / γ 0 w η / γ0 χ x L ll q t T t x L.0 t,x τ wη γ χ t T L + t L wηγ χ x t T L + C b / γ 0 w η / γ0 χ t,x L ll q t T t x Lt,x. By.5,.6, and.0, we obtain, for any τ < T < 0: u β+ T η γ T + u β/+ η γ χ {t T }. B τ Q τ C u β+ η γ η χ {t T } + C u β+ η γ η χ {t T } + C u β+ η γ t η χ {t T } + C b / γ 0 u β/+ η / γ0 χ L ll q t T t x L + t,x uβ/+ η γ χ t T L. t L x As this inequality holds for all τ < T < 0, we may take the supremum over T to eliminate the L t L x-norm in the right side. Namely, from., we have sup T [ τ,0] u β+ T η γ T C u β+ η γ η + C u β+ η γ η + C u β+ η γ t η B τ + C b / γ 0 u β/+ η / γ0 L ll q t x L + t,x uβ/+ η γ L t L x and u β/+ η γ C u β+ η γ η + C u β+ η γ η + C u β+ η γ t η.9. + C b / γ 0 u β/+ η / γ0 L ll q t x L + t,x uβ/+ η γ L..3 t L x Adding the last two estimates and absorbing the L t L x-norm, we obtain sup u β+ T η γ T + u β/+ η γ.4 τ T 0 B τ C u β+ η γ η + C u β+ η γ η + C u β+ η γ t η + C b / γ 0 u β/+ η / γ0 L ll q t x Lt,x, 5

6 with an increased constant C > 0. By the interpolation inequality.9, used on the left side of.4 with r = s = n + /n, and α = n/n +, used together with Young s inequality, we get the following estimate: / / u β/+ η γ L χ C u β+ η γ η + C u β+ η γ η.5 / + C u β+ η γ t η + C b / γ 0 u β/+ η / γ0 L Q ll q L, τ with χ = n + /n. We will now use.5 iteratively. We take a decreasing sequence r i > 0, and at each step choose the cut-off function η C0 Ω such that η in Q ri+, and η 0 in Q c r i, and Then.5 gives η C r i r i+, η C r i r i+, tη C r i r i+. u β/+ L χ Q ri+ C C b / γ 0 u β/+ L r i r L Q ri + ll q Q ri i+ r i r i+ / γ 0 uβ/+ L Q ri..6 Let us choose β i in.6 so that χ i = β i / +. In addition, we set r i = θ + so that r i r i+ = τ θ/ i+. Thus, we obtain u L χ i+ Q ri+ C i+ C /χi i+/γ χ i τ θ + τ θ i for i = 0,,,..., τ θ + Ci+/ γ0 τ θ / γ 0 b / γ 0 L l L q Q ri τ θ b L l L q Q ri /χ i u L χ i Q ri.7 / γ0 /χ i u L χ i Q ri, where γ = min{ γ 0, }. By iteration, starting from i = 0, we conclude that the estimate. holds for p. Now, let p 0,. The previous argument has shown that sup u C τ θ + τ θ / γ0 n+/ b L ll q u L B τ.8 Q θ C τ θ + τ θ / γ0 n+/ b L ll q u p/ L u p/ L p, 6

7 which implies sup u Q θ u L + C τ θ + τ θ / γ0 n+/p b L ll q u L p..9 Now, the iteration argument of [HL, Lemma 4.3] can be applied to complete the proof of Lemma. for 0 < p <. 3 The lower bound The goal of this section is to establish a lower bound of the infimum of a Lipschitz supersolution to., given in Lemma 3. below. Then, from Lemmas. and 3., we obtain the Harnack inequality sup u C + CR n+/ q b L q / n+/ q Cn/p 0 inf u 3. Q R/ 0, 3R Q R for any Lipschitz solutions u to., where p 0 = /CM C R and M R = + R n/ b L L + R n+/ q b L q. 3. Recall see. and.3 that we use the notation Q R x 0, t 0 for the cylinder centered at the bottom and Q R x 0, t 0 for the cylinder centered at the top, and Q R = Q R 0, 0. Lemma 3.. Assume that u is a nonnegative Lipschitz supersolution to., and b L q Ω L L Ω with n/ + < q n + and div b = 0 in the sense of distributions. Then there exists a small positive number p 0 = p 0 n, q, R, M R such that /p0 CR n u 0 p exp + R n+/ q b L q / n+/ q Cn inf u, Q R 0, 4R Q R 3.3 with M R given by 3.. We establish the proof of Lemma 3. in several steps, successively improving the estimate. We will primarily work with the function v = logu/k, with a constant K to be determined. If u is a supersolution to., then v is also a supersolution to.. More precisely, v satisfies the inequality We will obtain various bounds on w = v + below. v v t v + b v, in Ω

8 3. A bound on w α for α 0,. We begin with the following initial estimate on w. Note that the constant K we choose in 3.5 below does depend on the solution ux, t. Lemma 3.. Let ηx = C x /9R + be normalized so that η x dx =, and set R n K = exp η x log ux, 4R dx B 3R. 3.5 Then for α 0, we have with M 0 = + R n/ b L L. R w α dx dt CM 0 R n+ 3.6 Proof of Lemma 3.. Again, without loss of generality, we assume that R =. We multiply 3.4 by the cut-off η x and integrate over B 3 t, t with 0 t < t 4 in order to obtain t vx, t η x dx dt vx, t η x dx vx, t η x dx 3.7 t B 3 B 3 B 3 t t + j vx, tηx j ηx dx dt + b j x, t j vx, tη x dx dt. B 3 B 3 t After rearranging the terms and using the Cauchy-Schwarz inequality, we have t vx, t η x dx vx, t η x dx + vx, t η x dx dt 3.8 B 3 B 3 t B 3 t t j vx, tηx j ηx dx dt + b j x, t j vx, tη x dx dt t B 3 t B 3 t vx, t η x dx dt + C ηx L B 3 t,t t B 3 + C b L t L xb 3 t,t η L t L x B 3 t,t. Absorbing the first term on the far right, 3.8 leads to vx, t η x dx vx, t η x dx + B 3 B 3 C + b L L Ω t t, t t t B 3 vx, t η x dx dt 3.9 since 0 η. Now, we set M 0 = + b L L Ω. Using weighted Poincaré s inequality c.f. [Lieberman, Lemma 6.] in the left side of 3.9, we get vx, t η x dx vx, t η x dx 3.0 B 3 B 3 + t C B vx, t vx, tη x dx η x dx dt CM 0 t t. 3 t B 3 8

9 For the rest of the proof we may proceed as in the proof of Lemma 6. from Lieberman. Consider the function px, t = vx, t CM 0 t, defined as a translation of v in time by the factor coming from the right side of 3.0. Note that the constant K in 3.5 was chosen so that B 3 vx, 4η x dx = 0, 3. and 3.0 and 3. imply that B 3 vx, tη x dx CM 0 4 t, for all 0 t As ηx is uniformly positive for x, we deduce the upper bound {x, t : px, t > µ} Q Cµ, 3.3 on the size of the level sets of p that holds for any µ. This leads to the bound p α dx dt = α µ α {x, t Q {x,t : px,t>} : px, t > µ} dµ 3.4 α Q µ α dµ = C Q C since α 0,. We conclude the proof of 3.6 by noticing that the function w satisfies w α Cp α + CM α 0 if p and w α C + CM α 0 if p <. 3. Bound on w σ for any σ [, n + /n. From now on, without loss of generality, we assume that R =. As before, we will work with w = log + u/k with a constant K defined as in 3.5. The function w = v + is a supersolution to the equation for v, that is, w w t w + b w, 3.5 since it is a maximum of two supersolutions, v = vx, t and v 0. We need the following bound that will bootstrap bounds for the L α -norms with α 0, we have obtained in Lemma 3., to higher norms. 9

10 Lemma 3.3. For any σ [, n + /n and any α 0,, we have where C = Cα, σ, n, q. w L σ Q C + b L q C w L α, 3.6 Proof of Lemma 3.3. Let η be a Lipschitz cut-off in with 0 η note that unlike in the proof of Lemma 3. we use a cut-off that also depends on time. We multiply 3.5 by the function w + β η γ χ {t T } with β /, 0, γ > to be determined, and T 0, 4, and integrate over to obtain w + β+ η γ + w w + β η γ χ β + {t T } 3.7 B t=t β w w + β η γ χ {t T } + γ j ww + β η γ j ηχ {t T } γ b j w + β+ η γ j ηχ β + {t T } γ w + β+ η γ t ηχ β + {t T }. Here we have used the condition div b = 0. The first term on the right side is negative since β /, 0, while integration by parts in the second term on the right gives γ j ww + β η γ j ηχ {t T } = γ j w + β+ η γ j ηχ β + {t T } 3.8 γγ = β + w + β+ η γ η χ {t T } γ β + w + β+ η γ ηχ {t T }. This, together with 3.7 leads to w + β+ η γ + w + β w η γ χ β + {t T } 3.9 B t=t γ b j w + β+ η γ j ηχ β + {t T } γ w + β+ η γ t η + γ η γ η + η γ η χ β + {t T }. We may use the inequality w + β+/ η γ β + / w + β w η γ + γ w + β+ η γ η 3.0 in the left side of 3.9. In addition, as w > 0, we have w+ β w+ β, which altogether gives w + β+ η γ t=t + w + β+/ η γ χ {t T } 3. B Cγ b j w + β+ η γ j η χ {t T } + Cγ w + β+ η γ t η + η γ η + η γ η χ {t T }. 0

11 An application of the interpolation inequality.9 with r = s = n + /n leads to w + β+/ η γ C w + β+/ η γ L n+/n L L + C w + β+/ η γ L 3. Cγ b j w + β+ η γ j η χ {t T } + Cγ w + β+ η γ t η + η γ η + η γ η χ {t T }. Next, we may estimate the drift term in 3. with the help of Hölder s inequality as follows Cγ b w + β+ η γ η = Cγ b w + β+ λ w + β+λ η γ η 3.3 where Cγ b L q w + β+ λ L / λ w + β+λ η γ L n+/nλ η L, q + λ + nλ n + =. Therefore, λ is given by λ = n + / q and λ [/,, as n + / q < by assumption. Using Young s inequality, this leads to Cγ b w + β+ η γ η Cγ b L q w + β+ λ w + β+ η γ /λ λ η L L n+ L w + β+/ η γ /λ L n+/n + Cγ b L q η L / λ w + β+/ L. 3.4 Setting γ = / λ so that γ /λ = γ and using 3. and 3.4, we obtain n w + β+ η γ L n+ n w + β+ η γ + C b L q η L n+ L γ w + β+ L n +C w + β+ η γ t η + η γ η + η γ η. 3.5 The first term on the right can be absorbed into the left side: w + β+ η γ C b L q η L n+ L γ w + β+ L n + C w + β+ η γ t η + η γ η + η γ η. 3.6 We will now once again use an iteration procedure, applied to a decreasing sequence of parabolic cylinders Q ri with r i+ < r i. Choosing the cut-off η such that η in Q r i+ and η 0 in Q r i Q ri c, we have, from 3.6: w+ β+ L n+ n Q r i+ Cγ b L q r i r i+ γ w+ β+ L Q r i + C r i r i+ w+β+ L Q r i

12 or equivalently w + β+ L n+/n Q r i+ Cr i r i+ γ b γ L q + w + β+ L Q r i, 3.7 since γ. Set χ = n + /n, pick α 0,, and consider σ [, n + /n. Possibly increasing σ and decreasing α we may assume that σ = χ j α with j N. We will use 3.7 with β i = χi α, for i = 0,..., j so that β 0 + = α and β j + = σ, and r i = + i. Then 3.7 implies the recursive relation w + L χ i+α Q r i+ Cγi+/χi b γ L q + /χi w + L χ iα Q r i 3.8 and a finite number of iteration gives An estimate for w The next step is to obtain bounds on w L. Recall that w satisfies Multiplying 3.9 by η χ {t T } and integrating over gives wη + B t=t w w t w + b w, 3.9 w η j wη j η b j wη j η wη t η, 3.30 where we used div b = 0. After estimating the right side, we get wη + w η C η L + C b L q wη L q η L + C η t L wη L 3.3 B t=t C + b L L + b L q C = CM C, where M = + b L L + b L q. In the last inequality, we used Lemmas 3. and 3.3 with σ = q and σ =, respectively, where q < n + /n, as / q + / q = and n + / < q n +. Note that with the bound 3.3 in hand we may extend the argument in the proof of Lemma 3.3 to include β [0, /]. Namely, in that proof we have considered β /, 0 and dropped the first term in the right side of 3.7 simply because β was negative. Now, we can rely on 3.3 to bound this term in 3.7. The rest of the argument in the proof of Lemma 3.3 did not rely on the negativity of β. As the aforementioned term in 3.7 involves the product w w + β while 3.3 estimates w, we would still need the restriction β /.

13 3.4 Bound on w β+ for β / We now extend the bound for w β+ to β /. As in the proof of Lemma 3.3, we let η be a Lipschitz cut-off in with 0 η. This time, we multiply 3.5 by the function w β η γ χ {t T }, with β / and T 0, 4, and integrate over, using the divergence-free condition on b: w β+ η γ + w β w η γ β w w β η γ + γ j ww β η γ j η β + t=t B γ β + b j w β+ η γ j η γ β + For the first term in the right side of 3.3, we use the inequality w β+ η γ t η. 3.3 β w β w β + 4β β, 3.33 and for the second: γ j ww β η γ j η = γ j w β+ η γ j η 3.34 β + γγ = w β+ η γ η γ w β+ η γ η. β + β + Together with 3.3 this gives w β+ η γ + β + t=t B γ β + b j w β+ η γ j η γ β + w β w η γ 4β β w η γ 3.35 w β+ η γ t η + γ η γ η + η γ η. Applying the estimate 3.33 for the second term on the left side of 3.35, we obtain w β+ η γ t=t + β w β w η γ 3.36 β + B 4β β w η γ + γ b j w β+ η γ j η β + + γ w β+ η γ t η + γ η γ η + η γ η. β + 3

14 Next, we multiply 3.36 by β + and use the inequality w β+/ η γ β + / w β w η γ + γ w β+ η γ η 3.37 for the left side to get w β+ η γ + t=t w β+/ η γ C4β β w η γ + Cγ b j w β+ η γ j η B + Cγ w β+ η γ t η + η γ η + η γ η We use the interpolation inequality.9 with r = s = n + /n to write w β+/ η γ C w β+/ η γ L n+/n L L + C w β+/ η γ L 3.39 C4β β w η γ + Cγ b j w β+ η γ j η + Cγ w β+ η γ t η + η γ η + η γ η. First, we note that unlike in the proof of Lemma 3.3 we now have the uniform estimate 3.3 for the gradient: w CM C Next, we estimate the drift term in 3.39 similarly to what we did in the proof of Lemma 3.3. Namely, we may write Cγ b w β+ η γ η = Cγ b w β+ λ w β+λ η γ η 3.4 Cγ b L q w β+ λ L / λ w β+λ η γ L n+/nλ η L, with λ = n + / q [/,. An application of Young s inequality gives Cγ b w β+ η γ η Cγ b L q w β+ λ w β+ η γ /λ λ η L L n+/n L 3.4 wβ+/ η γ /λ L n+/n + Cγ b L q η L / λ w β+/ L. As before, choosing γ = / λ, we may absorb the first term in the right side of 3.4 into the left side of Thus, we obtain wη β+/ CM C 4β β + Cγ b L L q η n+/n L γ w β+/ L Cγ w β+ η γ t η + η γ η + η γ η. 4

15 We are ready to do the iteration process. We set r i = + i for i = 0,,,... and choose the cut-off η such that η in Q r i+ and η 0 in Q r i Q ri c. Then 3.43 gives at each iteration step: Cγ γ w β+/ L n+/n Q r i+ CM C 4β β + b L q w β+/ L r i r Q i+ r i 3.44 Cγ Cγ γ + r i r i+ wβ+/ L Q r i CM C 4β β + b γ + w β+/ r i r L q L Q i+ r i, since γ. Thus, we have the following relation between consecutive scales: w β+/ L n+/n Q r i+ CM C Cγ r i r i+ γ 4β β + w β+/ L Q r i As in the proof of Lemma 3.3 we will use it with β i = χ i / where χ = n + /n but this time we may allow β and thus i to be arbitrarily large. We obtain χ w χi CM C γi+ χ i χi + w χi CM C γi+ χ i i + w L χi+ Q r i+ L χi Q r i L χ i, Q r i 3.46 for i = 0,,,... Iterating the inequality w L χ i+ Q r i+ CMC/χi γi+/χi χ i + w L χ i, 3.47 Q r i obtained from 3.45 by taking /χ i power on both sides, we get By Lemma 3. and Lemma 3.3, we have which together with 3.48 implies Thus, we may conclude w L χ i+ Q r i+ CM C χ i+ + w L Q w L Q CM C w L α CM C 3.49 Q w L χ i+ Q r i+ CM C χ i w β+ /β+ CM C β for all β > 0, and Q p 0 w β+ β +! Cp 0 M C e β+ β

16 provided p 0 = CM C e. The last inequality leads to the estimate u p0 CR n+, 3.53 Q K R where the constant K is defined in 3.5 and M R = + R n/ b L L + R n+/ q b L q. We apply Lemma 3. and 3.53 to the translated in time cylinder Q R 0, 4R and obtain u p0 CR n Q R 0, 4R K with K = exp B 3R η x log ux, 0 dx. If u is a supersolution to., then log + K/u is a subsolution to.. The last ingredient in the proof of Lemma 3. is the following result. Lemma 3.4. We have where sup Q R log + K u C + R n+/ q b L q / n+/ q Cn, 3.55 K = exp η x log ux, 0 dx B 3R Proof of Lemma 3.4. We apply Lemma. for the positive subsolution log + K/u to. with p 0, to obtain K sup log + Q R u C + R n+/ q b L q / n+/ q n+/p R n+/p log K + u. L p Q R 3.57 Now, let v = logu/k with K given by We have v = logk/u and log + K/u = log u/k. The choice of K implies that B 3R η xvx, 0 dx = 0. We may proceed as in the proof of Lemma 3. to conclude log K + u CR n+/p, 3.58 L p Q R which, combined with 3.57 proves

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