# Hamilton-Souplet-Zhang’s Gradient Estimates for Two Types of Nonlinear Parabolic Equations under the Ricci Flow

Authors: Guangyue Huang; Bingqing Ma
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We consider gradient estimates for two types of nonlinear parabolic equations under the Ricci flow: one is the equationut=Δu+aulog⁡u+bu with a,b being two real constants; the other is ut=Δu+λuα with λ,α being two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-Souplet-Zhang-type gradient estimates.


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## Body

## 1. Introduction

Since the nonlinear parabolic equation(1)ut=Δu+aulog⁡u+buon a given Riemannian manifold is related to gradient Ricci solitons which are self-similar solutions to the Ricci flow, many attentions are paid to the study on gradient estimates for (1); for example, see [1–7]. Here a,b in (1) are two real constants. Clearly, a heat equation(2)ut=Δuis a special case of (1) when a=b=0. Hence many known results on heat equations are generalized to the nonlinear parabolic equation (1). For gradient estimates of solutions to (1) of Li-Yau type, Davies type, Hamilton type, and Li-Xu type on a given Riemannian manifold, we refer to [1, 3, 7, 8] and the references therein. In a recent paper [9], Dung and Khanh obtained Hamilton-Souplet-Zhang-type gradient estimates on a given Riemannian manifold for (1). On a family of Riemannian metrics g(t) evolving from the Ricci flow(3)∂∂tgij=-2Rij,Hsu in [10] obtained Li-Yau-type gradient estimates of (1).In [11], generalizing Hamilton’s estimate in [12], Souplet and Zhang proved the following result.Theorem A (see [11]).
Let(Mn,g) be an n-dimensional Riemannian manifold with Ric(Mn)≥-k, where k is a nonnegative constant. Suppose that u is a positive solution to (2) in QR,T=x,t∣x∈M,dx,x0,t<R,t∈0,T with u≤A. Then in QR,T,(4)∇uu≤Ck+1R+1T1+log⁡Au,where constant C depends only on the dimension n.The key to prove Theorem A of Souplet and Zhang is the scalingu→u~=u/A. After this scaling, (2) becomes the following heat equation with respect to u~:(5)u~t=Δu~since the heat equation is linear. Under this case, we obtain that 0<u~≤1. Inspired by this method, in this paper, we also adopt the similar scaling method by u→u~=u/A to study the nonlinear parabolic equation (1). By the scaling, we can derive from (1) the following analogous equation:(6)u~t=Δu~+au~log⁡u~+b~u~,where constant b~ satisfies b~=b+alog⁡A. That is, we only need to study the nonlinear equation (6) with 0<u~≤1.Our first result is the following Hamilton-Souplet-Zhang-type gradient estimates of the nonlinear equation (1) under the Ricci flow.Theorem 1.
LetM be a complete Riemannian manifold with a family of Riemannian metrics g(t) evolving from the Ricci flow (3). Suppose that u is a positive solution to (1) in(7)BR,T=x,t∣x∈M,dx,x0,t<R,t∈0,Twith Ric≤k for some positive constant k and u≤A. Then there exists a constant C depending only on the dimension of M such that(8)∇uu≤C1R+1T+k+Ma,b1+log⁡Aufor all (x,t)∈BR/2,T with t≠0, where Ma,b=max⁡0,a1+log⁡A+b.The study of Li-Yau-type estimates of the following nonlinear parabolic equation:(9)ut=Δu+λuα,where λ,α are two real constants, can be traced back to Li [13]. Later, for 0<α<1, Zhu in [14] obtained Hamilton-Souplet-Zhang-type gradient estimates of (9) on a given Riemannian manifold. On gradient estimates of the elliptic case of (9), see [15, 16]. A natural subject is to study Hamilton-Souplet-Zhang-type gradient estimates of the nonlinear equation (9) under the Ricci flow. Our second result is the following.Theorem 2.
LetM be a complete Riemannian manifold with a family of Riemannian metrics g(t) evolving from the Ricci flow (3). Suppose that u is a positive solution to (9) in(10)BR,T=x,t∣x∈M,dx,x0,t<R,t∈0,Twith Ric≤k for some positive constant k and u≤B. Then there exists a constant C depending only on the dimension of M such that(1) if α≥1, then(11)∇uu≤C1R+1T+k+Mλα1+log⁡Bu,for all (x,t)∈BR/2,T/2 with t≠0, where Mλ=max⁡{0,λBα-1};(2) if α≤0, then(12)∇uu≤C1R+1T+k+Mλ-α+1uminα-11+log⁡Bu,for all (x,t)∈BR/2,T with t≠0, where Mλ=max⁡{0,-λ};(3) if α∈(0,1), then(13)∇uu≤C1R+1T+k+λuminα-11+log⁡Bufor all (x,t)∈BR/2,T with t≠0.Remark 3.
Takinga=b=0 in (8), we obtain the estimate (16) of Theorem  2.2 in [17] with respect to the heat equation under the Ricci flow. Hence, our estimates in Theorem 1 extend Bailesteanu, Cao, and Pulemotov’s Theorem 2.2.Remark 4.
There are many studies on gradient estimates of the heat equation (2) under geometric flows; we refer to [18, 19] among others.


## 2. Proof of Theorem1

In order to prove our Theorem1, we first give a lemma which will play an important role in the proof.Lemma 5.
LetM be a complete Riemannian manifold with a family of Riemannian metrics g(t) evolving from the Ricci flow (3). Let u be a positive solution to (1) with u≤A. Denoting by u~=u/A, f=log⁡u~≤0 and w=∇log⁡1-f2. Then, it holds that(14)Δ-∂tw≥2f1-f∇f∇w+21-fw2-2a+B1-fw,where B=b+alog⁡A.Proof.
Under the scalingu→u~=u/A, we have 0<u~≤1. From (1), we obtain that u~ satisfies the following equation:(15)u~t=Δu~+au~log⁡u~+Bu~,where constant B satisfies B=b+alog⁡A. Let f=log⁡u~≤0 and w=|∇log⁡(1-f)|2=|∇f|2/(1-f)2. Then we have(16)ft=Δf+∇f2+af+B.Using (3), we can obtain(17)∇f2t=2Rijfifj+2fifti.By the definition of w, we have(18)wt=21-f2Rijfifj+fifti+21-f3fj2ft=21-f2Rijfifj+fifjji+2fijfifj+afi2+21-f3fj2fii+fi2+af+B.On the other hand,(19)Δw=61-f4fi2fj2+21-f3fiifj2+81-f3fjififj+21-f2fji2+21-f2fjfjii=61-f4fi2fj2+21-f3fiifj2+81-f3fjififj+21-f2fji2+21-f2fjfiij+21-f2Rijfifj,where, in the second equality, we used the Ricci formula(20)fjii=fiji=fiij+Rijfi.By the formulas (18) and (19), we arrive at(21)Δ-∂tw=21-f2fji2+81-f3fjififj-41-f2fijfifj+61-f4fi2fj2-21-f3fi2fj2-2a+B1-ffi21-f2.Note that(22)∇f∇w=21-f3fi2fj2+21-f2fjififj.Therefore, (21) can be written as (23)Δ-∂tw=21-f2fji2+41-f3fjififj-41-f2fijfifj+21-f∇f∇w+2∇f41-f4-2∇f41-f3-2a+B1-f∇f21-f2=21-f2fji2+41-f3fjififj-2∇f∇w+21-f∇f∇w+2∇f41-f4+2∇f41-f3-2a+B1-f∇f21-f2=21-f2fji+11-ffifj2+2f1-f∇f∇w+2∇f41-f3-2a+B1-f∇f21-f2≥2f1-f∇f∇w+21-fw2-2a+B1-fw. Then, the desired estimate (14) follows.Proof of Theorem1.
Letψ=ψ(x,t) be a smooth cutoff function supported in BR,T which satisfies the following properties (see page  13 in [20] or Lemma  2.1 in [17]):(1)
ψ=ψ(d(x,x0,t),t)≡ψ(r,t); ψ(x,t)=1 in BR/2,T/2 and 0≤ψ≤1.(2)
ψ is decreasing as a radial function in the spatial variables.(3)
∂ψ/∂r≤Cγ/Rψγ and ∂2ψ/∂r2≤Cγ/R2ψγ for every γ∈(0,1).(4)
∂ψ/∂t≤C/Tψ1/2.
We use (14) to conclude that(24)Δ-∂tψw=ψΔ-∂tw+wΔ-∂tψ+2∇ψ∇w=ψΔ-∂tw+wΔ-∂tψ+2∇ψψ∇ψw-2w∇ψ2ψ≥2f1-f∇f∇ψw-w∇f∇ψ+21-fψw2-2a+B1-fψw+wΔ-∂tψ+2∇ψψ∇ψw-2w∇ψ2ψ.Now we let (x1,t1) be a maximum point of ψw in the closure of BR,T, and where ψw>0 (otherwise the proof is trivial). Then at the point (x1,t1), we have Δ(ψw)≤0, (ψw)t≥0, and ∇(ψw)=0. Thus, from (24), we deduce(25)21-fψw2≤2f1-fw∇f∇ψ+2a+B1-fψw-wΔ-∂tψ+2w∇ψ2ψat (x1,t1). Next, we will find an upper bound for each term of the right-hand side of (25). By the Cauchy inequality, we have(26)2f1-fw∇f∇ψ≤2fw3/2∇ψ=2ψ1-fw23/4f∇ψψ1-f3/4≤1-fψw2+C1R4f41-f3.It has been shown in [20] (see formulas (3.30), (3.32), and (3.34) in [20]) that(27)w∇ψ2ψ≤18ψw2+C1R4,-Δψw≤18ψw2+C1R4+CkR2,∂ψ∂tw≤18ψw2+C1T2+Ck2.Putting (26)-(27) into (25), we obtain(28)1-fψw2≤C1R4f41-f3+2a+B1-fψw+12ψw2+C1R4+C1T2+Ck2.Since f≤0, the inequality (28) implies(29)ψw2≤C1R4f41-f4+1R4+1T2+k2+max⁡0,a+B2=C1R4f41-f4+1R4+1T2+k2+Ma,b2.Therefore, we deduce that, for any (x,t)∈BR,T,(30)ψ2w2x,t≤ψw2x1,t1≤C1R4+1T2+k2+Ma,b2from f4/(1-f)4<1. Noticing ψ(x,t)=1 in BR/2,T/2 and w=|∇f|2/(1-f)2, we have the fact that(31)∇f1-f≤C1R+1T+k+Ma,bholds at any (x,t)∈BR/2,T, which shows that(32)∇uu≤C1R+1T+k+Ma,b1+log⁡Au.We complete the proof of the estimate (8) in Theorem 1.


## 3. Proof of Theorem2

As in the proof of Theorem1, we first give a key lemma.Lemma 6.
LetM be a complete Riemannian manifold with a family of Riemannian metrics g(t) evolving from the Ricci flow (3). Let u be a positive solution to (9) with u≤B. Denoting by u~=u/B, f=log⁡u~≤0 and w=|∇log⁡(1-f)|2. Then, it holds that(33)Δ-∂tw≥2f1-f∇f,∇w+21-fw2-2λ~α--f1-feα-1fw,where λ~=λBα-1.Proof.
By the scalingu→u~=u/B, we have 0<u~≤1. Therefore, we obtain from (9) that u~ satisfies(34)u~t=Δu~+λ~u~α,where λ~=λBα-1. Let f=log⁡u~≤0 and(35)w=∇log⁡1-f2.Then, the function f satisfies(36)ft=Δf+∇f2+λ~eα-1f.Using (3) again, one has(37)∇f2t=2Rijfifj+2fifti.It follows from (35) that(38)wt=21-f2Rijfifj+fifti+21-f3fj2ft=21-f2Rijfifj+fifjji+2fijfifj+λ~α-1eα-1ffi2+21-f3fj2fii+fi2+λ~eα-1f.Similarly, by the Ricci formula, we obtain(39)Δw=61-f4fi2fj2+21-f3fiifj2+81-f3fjififj+21-f2fji2+21-f2fjfiij+21-f2Rijfifj.Thus, we derive from (38) and (39) the following:(40)Δ-∂tw=21-f2fji2+61-f4-21-f3fi2fj2+81-f3-41-f2fjififj-2λ~α-111-f2+11-f3eα-1ffi2.Using the relationship(41)∇f,∇w=21-f3fi2fj2+21-f2fjififj, (40) can be written as(42)Δ-∂tw-ɛ∇f,∇w=21-f2fji2+231-f4-1+ɛ11-f3fi2fj2+241-f3-2+ɛ11-f2fjififj-2λ~α-111-f2+11-f3eα-1ffi2,where ɛ=ɛ(f) is a function depending on f which will be determined. Applying the inequality (43)21-f2fji2+241-f3-2+ɛ11-f2fjififj=21-f2fji2+41-f-2+ɛfjififj≥-121-f241-f-2+ɛ2fi2fj2 into (42) gives(44)Δ-∂tw-ɛ∇f,∇w≥-21-f4+6+2ɛ11-f3-2+ɛ2121-f2fi2fj2-2λ~α-111-f2+11-f3eα-1ffi2=11-f4-121-f2ɛ2-21-f2-1-fɛ-21-f2-61-f+2fi2fj2-2λ~α-111-f2+11-f3eα-1ffi2.Taking (45)ɛ=-2+21-fin (44), we derive(46)Δ-∂tw-2f1-f∇f,∇w≥21-f3∇f4-2λ~α-111-f2+11-f3eα-1ffi2=21-fw2-2λ~α--f1-feα-1fw.Thus, the desired estimate (33) is attained.Proof of Theorem2.
Letψ=ψ(x,t) be a smooth cutoff function supported in BR,T which is defined in Section 2. We use (33) to conclude that(47)Δ-∂tψw=ψΔ-∂tw+wΔ-∂tψ+2∇ψψ∇ψw-2w∇ψ2ψ≥-2-f1-f∇f∇ψw-w∇f∇ψ+21-fψw2-2λ~α--f1-feα-1fψw+wΔ-∂tψ+2∇ψψ∇ψw-2w∇ψ2ψ.Now we let (x2,t2) be a maximum point of ψw in the closure of BR,T, where ψw>0 (otherwise the proof is trivial). Then at the point (x2,t2), we have Δ(ψw)≤0, (ψw)t≥0, and ∇(ψw)=0. Thus, from (47), we deduce that(48)21-fψw2≤2f1-fw∇f∇ψ+2λ~α--f1-feα-1fψw-wΔ-∂tψ+2w∇ψ2ψat (x2,t2).
Case One(α≥1). Note that(49)-f1-f=1-11-f∈0,1.In this case, we have α--f/(1-f)>0 and e(α-1)f∈(0,1). Hence(50)2λ~α--f1-feα-1fψw≤CMλ21-fα--f1-f2+1161-fψw2,where Mλ=max⁡{0,λ~}. Putting (50), (26)-(27) into (48), we obtain(51)1-fψw2≤C1R4f41-f3+Mλ21-fα--f1-f2+1R4+1T2+k2.Therefore, we deduce, for any (x,t)∈BR,T, that(52)ψ2w2x,t≤ψw2x1,t1≤C1R4+1T2+k2+Mλ2α2from f2/(1-f)2<1. Noticing ψ(x,t)=1 in BR/2,T/2 and w=|∇f|2/(1-f)2, we have the fact that(53)∇f1-f≤C1R+1T+k+Mλαholds at any (x,t)∈BR/2,T, which shows that(54)∇uu≤C1R+1T+k+Mλα1+log⁡Bu.
Case Two (α≤0). In this case, we have α--f/(1-f)<0 and e(α-1)f>1. Hence(55)2λ~α--f1-feα-1fψw≤CM~λ21-fα--f1-f2u~min2α-1+1161-fψw2,where M~λ=max⁡{0,-λ~}. Similarly, putting (55), (26)-(27) into (48), we obtain(56)1-fψw2≤C1R4f41-f3+M~λ21-fα--f1-f2u~min2α-1+1R4+1T2+k2.Thus, we can obtain that(57)∇f1-f≤C1R+1T+k+M~λ-α+1u~minα-1holds at any (x,t)∈BR/2,T, which shows that(58)∇uu≤C1R+1T+k+Mλ-α+1uminα-11+log⁡Bu.
Case Three
(
α
∈
0,1
). In this case, we also have e(α-1)f>1. Hence(59)2λ~α--f1-feα-1fψw≤Cλ~21-fα--f1-f2u~min2α-1+1161-fψw2.Putting (59), (26)-(27) into (48), we obtain(60)1-fψw2≤C1R4f41-f3+λ~21-fα--f1-f2u~min2α-1+1R4+1T2+k2.Because of (α--f/(1-f))2≤1, we can obtain that(61)∇f1-f≤C1R+1T+k+λ~u~minα-1holds at any (x,t)∈BR/2,T, which shows that(62)∇uu≤C1R+1T+k+λuminα-11+log⁡Bu.We complete the proof of Theorem 2.



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