Harnack estimates for nonlinear parabolic equations under the Ricci flow

EQUATIONS UNDER THE RICCI FLOW

YI LI AND XIAORUI ZHU

Abstract. In this paper, we consider first the Li-Yau Harnack estimates for a nonlinear parabolic equation∂tu= ∆tu−qu−au(lnu)^α under the Ricci flow, whereα >0 is a constant. To extend these estimates to a more general situation, in the second part, we consider the gradient estimates for a positive solution of the nonlinear parabolic equation∂tu= ∆tu+hu^pon a Riemannian manifold whose metrics evolve under the geometric flow∂tg(t) =−2S_g(t). To obtain these estimates, we introduce a quantitySalong the flow which mea- sures whether the tensorSij satisfies the second contracted Bianchi identity.

Under conditions on Ric_g(t), S_g(t), andS, we obtain the gradient estimates.

1. Introduction

The nonlinear parabolic equation is a classical subject that has been extensively studied, which leads to lots of important results especially in researches of differen- tial geometry. One of the important technique in studying the heat equation is the differential Harnack inequality developed by Li and Yau [8]. This is also applied to Ricci flow by Hamilton [6], and plays an important role in solving the Poincar´e conjecture [12].

1.1. Gradient estimates for (1.2) under the Ricci flow. Consider first pos- itive solutions of a nonlinear parabolic equation on an n-dimensional complete manifold M, which evolves under the Ricci flow. A series of gradient estimates are obtained for such solutions, including several Li-Yau-type inequalities. Let (M, g(t))_t∈[0,T] be a complete solution to the Ricci flow

(1.1) ∂tg(t) =−2 Ricg(t), t∈[0, T].

We assume that its Ricci curvature remains uniformly bounded for all t ∈ [0, T].

Consider a positive functionu=u(x, t) defined onM×[0, T] solving the equation (1.2) ∆_g(t)−q−∂t

u=au(lnu)^α, t∈[0, T],

which has been first studied in [19] whereg(t)≡g is a fixed metric. Qian [13] and Wu [18] got a series of similar conclusions. Here ∆g(t) stands for the Laplacian ofg(x, t) defined onM ×[0, T] and q(x, t) is a C² function defined on M×[0, T].

Notice that the Laplacian ∆g(t)depends on the parametert, and we should study

2010Mathematics Subject Classification. Primary 53C44.

Key words and phrases. Harnack estimate, Ricci flow, geometric flow.

Yi Li is partially supported by the Fonds National de la Recherche Luxembourg (FNR) unde the OPEN scheme (project GEOMREV O14/7628746). Xiaorui Zhu is partially supported by CPSF (grant) No. 2014M551721 and Zhejiang Province Natural Science Foundation of China (grant) No. Q14A010002.

1

the nonlinear parabolic equation (1.2) coupled with the Ricci flow (1.1).

We introduce notions used throughout this paper. Let Bρ,T ={(x, t) ∈ M × [0, T] : dist_g(t)(x, x₀)< ρ}, where distg(t)(x, x₀) denotes the distance between xto a fixed point x₀ with respect to g(t). ∇g(t) and | · |g(t) stand for the Levi-Civita connection and norm with respect tog(t) respectively. For the simplicity, we always omit the subscriptsg(t) ortin the concrete computations.

We divide the study of the equation (1.2) into two cases: (1) α = 1 and (2) α6= 1. For the first case, we have

Theorem 1.1. Suppose that (M, g(t))_t∈[0,T] is a complete solution to the Ricci flow (1.1) on ann-dimensional manifoldM withsup_Bρ,T|Ric_g(t)|_g(t)≤K for some K >0, and uis a smooth positive function onM ×[0, T] satisfying the nonlinear parabolic equation (1.2) where the functionq(x, t)is defined onM×[0, T] which is C² in thex-variable and C¹ in the t-variable. If u(x, t) ≤A for some A >0 on B_ρ,T,α= 1,|∇_g(t)q|_g(t)≤γ, and

b:=1

8 + min

M×[0,T]q−max{a,0} ∈

0,1 2

then there exists a constantC that depends only onn such that (1.3) |∇_g(t)u|_g(t)

u ≤ C

b 1

ρ+ 1

√t +√

K+ 1 +√ γ+p

|a| 1 + lnA u

onB_ρ/2,T with t6= 0.

Whenqis nonnegative anda≤0, the constantC/bcan be a universal constant which means a constant depending only on the dimensionn. The number 1/8 inb is not essential, because in the following proof we shall see that we can replace 1/8 by 1/2.

For the general value ofα, we have the following version of estimates.

Theorem 1.2. Suppose that(M, g(t))_t∈[0,T]is a complete solution to the Ricci flow (1.1) on ann-dimensional manifold M with −K1g(t)≤Ricg(t)≤K2g(t)for some K₁, K₂ > 0 on B_ρ,T. If u is a smooth positive function satisfying the nonlinear parabolic equation (1.2), then there exists a constant C depending only on nsuch that, on B_ρ/2,T,

(1) fora≥0, we have

|∇g(t)f|²_g(t)−βft−βq−βaf^α ≤ nβ

2c(1−)t+(A+γ)nβ

2c(1−) + n²β³C² 4c²(1−)(β−1)ρ² + nβ[βK1+a(β−1)α|f^α−1|∞]

c(1−)(β−1) + nβ²aα|α−1||f^α−2|_∞

2c(β−1)(1−) +

s

[βθ+ (β−1)γ+^nβ_2dK²]nβ 2c(1−)

(2) fora≤0, we have

|∇g(t)f|²_g(t)−βf_t−βq−βaf^α ≤ nβ

2c(1−)t+(A+γ)nβ

2(1−c) + n²β³C₁² 4c²(1−)(β−1)ρ² + nβ[βK1−^a₂(β−1)α|f^α−1|∞]

c(1−)(β−1) +

s

[βθ+ (β−1)γ+^nβ_2dK²]nβ

2c(1−) .

Here f := lnu, |f|_∞ := maxM|f|, K := max{K1, K2}, β > 1, 0 < < 1,

|∇_g(t)q|_g(t)≤γ,∆_g(t)q≤θ, A=C

1 ρ²+

√K1

ρ +1 t +K

andc, d >0 withc+d= 1/β.

1.2. Gradient estimates for (1.4) under the geometric flow. In the sec- ond part, we consider the gradient estimates of (1.2) under a general geomet- ric flow; these results generalize our previous works [9, 21]. More generally, let (M, g(t))_t∈[0,T] be a complete solution to the geomtric flow

(1.4) ∂_tg(t) =−2S_g(t), t∈[0, T].

on a complete and noncompactn-dimensional manifoldM and consider a positive functionu=u(x, t) defined onM×[0, T] solving the equation

(1.5) ∂tu= ∆g(t)u+hu^p, t∈[0, T],

where ∆_g(t) stands for the Laplacian ofg(t),his a function defined onM ×[0, T] which isC²inxandC¹ int, andpis a positive constant. When metrics are fixed, the study on the gradient estimates of (1.5) arose from [7]. If h= 0, Sun [17] de- rived the gradient estimates and the Harnack inequalities for the positive solutions of the linear parabolic equation ∂tu = ∆_g(t)u under the geometric flow. In this paper, we consider the general case for the nonlinear parabolic equation. Notice that the ∆g(t)depends on the parametert, and we should study the equation (1.5) coupled with the geometric flow (1.4).

Introduce a 1-formS_g(t) onM×[0, T] by S_g(t):= div_g(t)S_g(t)−1

2∇_g(t) tr_g(t)S_g(t) .

Locally, one has

S_i=∇^jSij−1

2∇i tr_g(t)S_g(t) .

For example, ifSij =Rij, that is, (1.4) is the Ricci flow, we have S_i=∇^jRij−1

2∇iR_g(t)=1

2∇iR_g(t)−1

2∇iR_g(t)= 0

by the second contracted Bianchi identity. Thus, the quantity S_g(t) measures whetherS_ij satisfies the second contracted Bianchi identity.

Theorem 1.3. Suppose that(M, g(t))_t∈[0,T]is a solution to the geometric flow (1.4) on M with −K1g(t)≤ Sg(t) ≤K2g(t), −K3g(t) ≤Ricg(t)−Sg(t) ≤K3g(t), and

|S_g(t)|g(t)≤K4 for some K1, K2, K3, K4>0 on B2R,T, with K := max{K1, K2}.

Let h(x, t) be a function defined on M ×[0, T] which is C² in x and C¹ in t, satisfying∆g(t)h≥ −θ and|∇g(t)h|g(t)≤γ onB2R,T×[0, T]for some nonnegative constants θ andγ. Ifu(x, t) is a positive smooth solution of (1.5) on M ×[0, T], then

(i) for0< p <1, we have

|∇g(t)u|²_g(t) u² +h

pu^p−1−1 p

ut

u ≤ C1

p²t +n(1−p)

p² M1M2+n[3K1+ 2(K3+K4)p]

2p²(1−p) + C1

p² 1

R² +

√K1+K3

R +K+ n

p(1−p) (1.6)

+ n

p 3/2

pθM2+

pn/K₁

p γM2+ n p²

rK4

2n, whereC1 is a positive constant depending only on nand

M1:= max

B2R,T

h₋, M2:= max

B2R,T

u^p−1, h₋:= max(−h,0).

(ii) forp≥1,we have

|∇_g(t)u|²_g(t) u² +h

pu^p−1−1 p

u_t

t ≤ k²C₂

p²t +nk²(p−1)

p² M₄M₅+ k³n k−pM₃M₄ + k²C₂

p² 1

R² +

√K₁+K₃

R +K+ k²n p(k−p)

(1.7)

+ 2k³n (k−p)p²

h K1+p

k(K3+K4)i +k²√

nγ p M4

+ kn

p ^3/2

pθM4+k²n p² K+

rK4

2n

! ,

wherek > p,C₂ is a positive constant depending only onn and M₃:= max

B_2R,Th₋, M₄:= max

B_2R,Tu^p−1, M₅:= max

B_2R,Th.

As an immediate consequence of the above theorem we have

Theorem 1.4. Suppose that (M, g(t))_t∈[0,T] is a solution to the geometric flow (1.4) on M. Let h(x, t) be a function defined on M ×[0, T] which is C²in xand C¹ int.

(i) For 0 < p < 1, assume that h ≥ 0, |∇g(t)h|g(t) ≤ γ, ∆_g(t)h ≥ 0 along the geometric flow with −K1g(t)≤ S_g(t) ≤ K₂g(t), −K3g(t) ≤Ric_g(t)−S_g(t) ≤ K₃g(t), |S_g(t)|g(t) ≤K₄ for some positive constants γ, K₁, K₂, K₃, K₄ with K :=

max{K1, K₂}, along the geometric flow. Ifuis a smooth positive function satisfying

the nonlinear parabolic equation (1.5), then

|∇g(t)u|²_g(t) u² +h

pu^p−1−1 p

ut

u ≤ C1

p²t+ C1

p³(1−p)+C1

p²K+ 2nK1

p²(1−p) +

pn/K₁

p γM+ n p²

rK₄

2n +n(K₃+K₄) p(1−p) (1.8)

for some positive constantC1 depending only on n, whereM := max_M×[0,T]u^p−1. (ii) For p = 1, assume that −K1g(t) ≤ Sg(t) ≤K2g(t), −K3g(t) ≤ Ricg(t)− Sg(t) ≤ K3g(t), |S_g(t)|g(t) ≤ K4 for some positive constants K1, K2, K3, K4 with K := max{K1, K2}, h≥0, ∆g(t)h≥ −θ (θ is nonnegative), and |∇g(t)h|g(t) ≤γ (γ is nonnegative), along the geometric flow. If u is a smooth positive function satisfying the nonlinear parabolic equation (1.5), then

(1.9) |∇g(t)u|²_g(t)

u² +h−ut

u ≤ C2

t +C2

1 +K1+K2+K3+K4+K+γ+

√ θ for some positive constantC₂ depending only on n.

(iii) For p > 1, assume that −K1g(t)≤ S_g(t) ≤K2g(t), −K3g(t) ≤ Ric_g(t)− S_g(t)≤K3g(t),|S_g(t)|_g(t)≤K4 for some positive constantsγ, K1, K2, K3, K4 with K := max{K1, K2}. ∆_g(t)h ≥ −θ, |∇_g(t)h|_g(t) ≤ γ, and −k1 ≤ h ≤ k2, where θ, γ, k1, k2>0, along the geometric flow. Ifuis a bounded smooth positive function satisfying the nonlinear parabolic equation (1.5), then

|∇_g(t)u|²_g(t) u² +h

pu^p−1−1 p

ut

u ≤ k

p ²C3

t + k

p ³ k

k−pC3+ k

p ²

C3

K+

+ k

k−p(K₁+K₃+K₄) + rK4

2n

+ k

p ²

n(p−1)k2M + k³n k−pk1M (1.10)

+k²√ n p γM+

kn p

^3/2√ θM ,

for some positive constantC₃ depending only onn, where M := max_M×[0,T]u^p−1 andk > p. In particular, takingk= 2p, we get

|∇_g(t)u|²_g(t) u² +h

pu^p−1−1 p

u_t

u ≤ C₄

t +C₅ 1 +K₁+K₂+K₃+K₄+K +C4p²h

(k1+k2)M +γM+

√ θMi

, (1.11)

for some positive constantC4 depending only on n.

Another type of Harnack inequality is the following

Theorem 1.5. Suppose that(M, g(t))_t∈[0,T]is a solution to the geometric flow (1.4) on M, satisfying−K1g(t)≤S_g(t) ≤K2g(t), −K3g(t)≤Ric_g(t)−S_g(t) ≤K3g(t),

|∇_g(t)S_g(t)|_g(t)≤K4, for some K1, K2, K3, K4>0, withK := max{K1, K2}. Let h(x, t)be a nonnegative function defined onM×[0, T]which isC² inxandC¹ in t,∆_g(t)h+h_t−2C_n,pp^|∇g(t)h|

2 g(t)

h ≥0onM×[0, T](whereC_n,p=_p−1^p ifp >1and

Cn,p=nif p≤1), and0< p≤ _2n−1²ⁿ (n≥3). If uis a positive solution of (1.5), then

|∇g(t)u|²_g(t) u² +h

pu^p−1−2 p

ut

u ≤ C

p²t+8n

p²K+8n p²

s 2n p(2−p)K1

+ 4n

p(2−p)(K1+K3+K4) + 1 p²

p8nK4, (1.12)

for some positive constantC depending only onn.

This theorem has three important consequences.

Corollary 1.6. Suppose that (M, g(t))_t∈[0,T] is a solution to the geometric flow (1.4) on M, satisfying 0 ≤ S_g(t) ≤ K2g(t), −K3 ≤ Ric_g(t)−S_g(t) ≤ K3g(t),

|∇_g(t)S_g(t)|_g(t) ≤ K4, for some positive constants K2, K3, K4. Let h(x, t) be a nonnegative function defined onM×[0, T]which is C²in xandC¹ int,∆_g(t)h+ ht−2Cn,pp^|∇g(t)h|

2 g(t)

h ≥0 onM ×[0, T](where Cn,p =_p−1^p if p >1 andCn,p=n if p≤1), and 0< p≤_2n−1²ⁿ (n≥3). If uis a positive solution of (1.5), then (1.13) |∇_g(t)u|²_g(t)

u² +h

pu^p−1−2 p

u_t t ≤ C

p²t+8n

p²K₂+ 4n

p(2−p)(K₃+K₄)+ 1 p²

p8nK₄ for some positive constantC depending only onn.

Corollary 1.7. Suppose that (M, g(t))_t∈[0,T] is a solution to the geometric flow (1.4) on M, satisfying −K1g(t) ≤ Sg(t) ≤ K2g(t), −K3g(t) ≤ Ricg(t)−Sg(t) ≤ K3g(t),|∇g(t)S_g(t)|g(t)≤K4, for someK1, K2, K3, K4>0, withK:= max{K1, K2}.

Leth(x, t)be a nonnegative function defined onM×[0, T]which isC²inxandC¹ int,∆_g(t)h+ht−2Cn,pp^|∇g(t)h|

2 g(t)

h ≥0 onM ×[0, T](where Cn,p= _p−1^p ifp >1 and Cn,p =n if p≤1), and 0 < p≤ _2n−1²ⁿ (n≥3). If uis a positive solution of (1.5), then

u(x₂, t₂) u(x1, t1) ≥

t₂ t1

−C/p

exp

"

− 1

2p min

γ∈Θ(x₁,t₁,x₂,t₂)

Z t₂ t₁

|γ(t)|˙ ²_g(t)dt−2n(t2−t1) 1

pK+2 p

s 2n

p(2−p)K1+ 1

2−p(K1+K3+K4) +1 p

p2nK4

! # (1.14)

for some positive constant C depending only on n, where (x1, t1),(x2, t2)∈ M × [0, T] witht1< t2.

WhenK1= 0, we have the following

Corollary 1.8. Suppose that (M, g(t))_t∈[0,T] is a solution to the geometric flow (1.4) on M, satisfying 0 ≤S_g(t) ≤ K₂g(t), −K3g(t) ≤ Ric_g(t)−S_g(t) ≤K₃g(t),

|∇g(t)S_g(t)|g(t) ≤ K₄, for some K₂, K₃, K₄ > 0. Let h(x, t) be a nonnegative function defined on M ×[0, T] which is C² in x and C¹ in t, ∆_g(t)h+h_t− 2C_n,pp^|∇g(t)h|

2 g(t)

h ≥ 0 on M ×[0, T] (where C_n,p = _p−1^p if p > 1 and C_n,p = n

if p≤1), and 0< p≤_2n−1²ⁿ (n≥3). If uis a positive solution of (1.5), then u(x2, t2)

u(x1, t1) ≥ t2

t1

−C/p

exp

"

− 1

2p min

γ∈Θ(x₁,t₁,x₂,t₂)

Z t₂ t₁

|γ(t)|˙ ²_g(t)dt

−2n(t₂−t₁) K₂

p +K₃+K₄ 2−p +

√2nK₄ p

#

for some positive constant C depending only on n, where (x1, t1),(x2, t2)∈ M × [0, T] witht1< t2.

2. Gradient estimates for (1.2) under the Ricci flow

Firstly, we introduce a cut-off function (see [3, 8, 10, 11, 19]) onBρ,T :={(χ, t)∈ M×[0, T] : distg(t)(χ, x0)< ρ}, where distg(t)(χ, x0) stands for the distance between χ and x0 with respect to the metric g(t), which satisfies a basic analytical result stated in the following lemma.

Lemma 2.1. Givenτ ∈(0, T],there exists a smooth functionΨ : [0,∞)×[0, T]→ R satisfying the following requirements:

(1) The support of Ψ(r, t) is a subset of [0, ρ]×[0, T], 0 ≤ Ψ(r, t) ≤ 1 in [0, ρ]×[0, T], andΨ(r, t) = 1 holds in[0,^ρ₂]×[τ, T].

(2) Ψ is decreasing as a radial function in the spatial variables.

(3) The estimate|∂_tΨ| ≤ ^C_τΨ^1/2is satisfied on [0,∞)×[0, T] for someC >0.

(4) The inequalities −^C_ρ^αΨ^α≤∂_rΨ≤0 and |∂_r²Ψ| ≤ ^C_ρ2^αΨ^α hold on [0,∞)× [0, T] for everya∈(0,1) with some constantCαdependent on a.

Proof. See [1].

These properties are derived from Calabi’s argument(see, e.g., [2, 4, 15]). Using this auxiliary function and applying the maximum principle, we are able to estab- lish Li-Yau-type inequality for the system (1.1) – (1.2).

In the following, we always omit the subscriptsg(t) ortin concrete computations.

For example, we write ∆ instead of ∆_g(t).

2.1. Gradient estimates I: α= 1. To prove Theorem 1.1, we need the following crucial lemma.

Lemma 2.2. Let (M, g(t))_t∈[0,T] be a complete solution to the Ricci flow (1.1) on an n-dimensional manifold M with sup_B

ρ,T|Ricg(t)|g(t)≤K for some K >0, anduis a smooth positive function onM×[0, T] satisfying the nonlinear parabolic equation (1.2) with α= 1, a <0. We assume thatu≤1 on Bρ,T. If f := lnuand w:=|∇g(t)ln(1−f)|²_g(t)=|∇g(t)f|²_g(t)/(1−f)², then the inequality

(∆−∂t)w ≥ 2 f

1−fh∇f,∇wi+ 2(1−f)w² + 2h∇f,∇qi

(1−f)² + 2a |∇f|²

(1−f)² + 2|∇f|²(q+af) (1−f)³ (2.1)

holds on B_ρ,T.

Proof. Sinceuis a positive solution to the nonlinear parabolic equation (1.2) with α= 1 anda <0, direct calculation shows that

∆f+|∇f|²−ft−q−af = 0, ft:=∂tf.

The partial derivative ofw with respect totis given by w_t = 2h∇f,∇fti

(1−f)² +2|∇f|²ft

(1−f)³ +2Ric(∇f,∇f) (1−f)²

= 2h∇f,∇(∆f+|∇f|²−q−af)i

(1−f)² +2Ric(∇f,∇f) (1−f)² + 2|∇f|²(∆f +|∇f|²−q−af)

(1−f)³ .

Using Bochner’s identityh∇f,∇∆fi=h∇f,∆∇fi −Ric(∇f,∇f) we obtain w_t = 2h∇f,∆∇fi+ 2h∇f,∇(|∇f|²−q−af)i

(1−f)² + 2|∇f|²(∆f+|∇f|²−q−af)

(1−f)³ . The partial derivative ofw with respect toxis given by

∆w= 2 |∇²f|²

(1−f)² + 2h∇f,∆∇fi

(1−f)² + 4h∇f,∇|∇f|²i

(1−f)³ + 6 |∇f|⁴

(1−f)⁴ + 2|∇f|²∆f (1−f)³. Combining those partial derivatives imply

(∆−∂t)w = 2 |∇²f|²

(1−f)² + 4h∇f,∇|∇f|²i

(1−f)³ + 6 |∇f|⁴

(1−f)⁴ −2h∇f,∇|∇f|²i (1−f)²

−2 |∇f|⁴

(1−f)³ + 2a |∇f|²

(1−f)² + 2h∇f,∇qi

(1−f)² + 2 q|∇f|²

(1−f)³ + 2a f|∇f|² (1−f)³. On the other hand, the gradient termh∇f,∇wiis determined by

h∇f,∇wi=h∇f,∇|∇f|²i

(1−f)² + 2 |∇f|⁴ (1−f)³. Plugging it into the evolution of (∆−∂_t)wwe conclude that

(∆−∂t)w = 2 |∇²f|²

(1−f)² + 2h∇f,∇|∇f|²i (1−f)³ + 2

1−fh∇f,∇wi + 2 |∇f|⁴

(1−f)⁴ −2h∇f,∇wi+ 2 |∇f|⁴

(1−f)³ + 2a |∇f|² (1−f)² + 2h∇f,∇qi

(1−f)² + 2 q|∇f|²

(1−f)³+ 2af|∇f|² (1−f)³. Because of the identity

|∇²f|²

(1−f)² +h∇f,∇|∇f|²i

(1−f)³ + |∇f|⁴ (1−f)⁴ =

∇²f

1−f +∇f ⊗ ∇f (1−f)²

2

we therefore arrive at (∆−∂_t)w = 2

∇²f

1−f +∇f⊗ ∇f (1−f)²

2

+ 2

1−fh∇f,∇wi+ 2 |∇f|⁴ (1−f)³

−2h∇f,∇wi+2h∇f,∇qi

(1−f)² + 2a |∇f|²

(1−f)² +2|∇f|²(q+af) (1−f)³ which immediately implies

(∆−∂t)w ≥ 2f

1−fh∇f,∇wi+ 2|∇f|⁴

(1−f)³+2h∇f,∇qi (1−f)² + 2a |∇f|²

(1−f)² +2|∇f|²(q+af) (1−f)³ .

This complete the proof.

Theorem 2.3. Suppose that (M, g(t))_t∈[0,T] is a complete solution to the Ricci flow (1.1) on ann-dimensional manifoldM withsup_Bρ,T|Ric_g(t)|_g(t)≤K for some K >0, and uis a smooth positive function onM ×[0, T] satisfying the nonlinear parabolic equation (1.2) where the functionq(x, t)is defined onM×[0, T] which is C² in thex-variable and C¹ in the t-variable. If u(x, t) ≤A for some A >0 on Bρ,T,α= 1,|∇_g(t)q|_g(t)≤γ, and

b:=1

8 + min

M×[0,T]q−max{a,0} ∈

0,1 2

then there exists a constantC that depends only onn such that

(2.2) |∇g(t)u|g(t)

u ≤ C

b 1

ρ+ 1

√t +

√

K+ 1 +√ γ+p

|a| 1 + lnA u

onB_ρ/2,T with t6= 0.

Whenqis nonnegative anda≤0, the constantC/bcan be a universal constant which means a constant depending only on the dimensionn. The number 1/8 inb is not essential, because in the following proof we shall see that we can replace 1/8 by 1/2.

Proof. Without loss of generality, we may assume thatA = 1; otherwise, we can replace uby u/A. Pick a number τ ∈ (0, T] and fix a function Ψ(x, t) satisfying the conditions of Lemma 2.1. We will establish (2.2) at (x, τ) for all xsuch that dist_g(τ)(x, x0)< ρ/2. Define Ψ :M×[0, T]→Rby setting

Ψ(x, t) := Ψ dist_g(t)(x, x₀), t .

Then, using the identity (2.1) and a straightforward calculation, one has (∆−∂t) (Ψw) ≥ Ψh−Λ,∇wi+

2(1−f)w²+2h∇f,∇qi

(1−f)² + 2a |∇f|² (1−f)² + 2|∇f|²(q+af)

(1−f)³

Ψ + 2h∇w,∇Ψi+w∆Ψ−w∂tΨ

≥ h−Λ,∇(Ψw)i+ 2Ψ(1−f)w²+whΛ,∇Ψi +w∆Ψ−wΨ_t+ 2

Ψh∇Ψ,∇(Ψw)i −2|∇Ψ|²

Ψ w

+

2h∇f,∇qi

(1−f)² + 2a |∇f|²

(1−f)² +2|∇f|²(q+af) (1−f)³

Ψ where Λ =−_1−f^2f ∇f. By our assumption that |Ric| ≤K onBρ,T and Lemma 2.1 that−^C_ρ¹Ψ^1/2≤Ψr≤0, and the identity

−wΨt=−

Ψt+ Ψr∂tdist_g(t)(·, x0) w,

we have (because−∂tdist_t(·, x0)≤4p

(m−1)K, c.f. Lemma 8.33 in [5])

−wΨ_t≥ −Ψ_tw−4C1

p(m−1)K ρ wΨ^1/2.

Suppose that Ψwachieves its maximum at (x0, t0).By [8], without loss of generality, we may assume thatx0is not in the cut-locus of M.At the point (x0, t0), one has

∆(Ψw)≤0,∇(Ψw) = 0,(Ψw)t≥0.Therefore 2Ψ(1−f)w² ≤ −whΛ,∇Ψi+ 2|∇Ψ|²

Ψ w−w∆Ψ +wΨt

−

2h∇f,∇qi

(1−f)² + 2a |∇f|²

(1−f)² +2|∇f|²(q+af) (1−f)³

Ψ (2.3)

at (x₀, t₀). We need to bound each term on the right-hand side of (2.3):

|whΛ,∇Ψi| ≤2 w|f|

1−f|∇f||∇Ψ|= 2w^3/2|f||∇Ψ| ≤Ψ(1−f)w²+27 16

|f|⁴|∇Ψ|⁴ [Ψ(1−f)]³ where we used the Young’s inequality thatab≤a^p+b^q/(q(p)^q/p) for anya, b, >0 andp, q >1 withp⁻¹+q⁻¹= 1. This together with Lemma 2.1 implies

(2.4) |whΛ,∇Ψi| ≤Ψ(1−f)w²+C2

f⁴ ρ⁴(1−f)³. Using again Lemma 2.1 we have

(2.5) |∇Ψ|²

Ψ w= Ψ^1/2w|∇Ψ|² Ψ^3/2 ≤1

8Ψw²+ 2|∇Ψ|⁴ Ψ³ ≤1

8Ψw²+C₃ ρ⁴.

Furthermore, by the properties of Ψ and the assumption of the Ricci curvature, one has (c.f., [16, 20])

(2.6) −w∆Ψ≤1

8Ψw²+C4

ρ⁴ +C4K². The estimation forwΨt is given by (c.f. [11])

(2.7) |wΨ_t| ≤ 1

8Ψw²+C5

τ² +C₅K².

Sincef ≤0 it follows that

2h∇f,∇qi (1−f)² Ψ

≤ |∇f|²+|∇q|²

(1−f)² Ψ ≤ wΨ +γ²Ψ

≤ 1

8Ψw²+ (2 +γ²)Ψ, (2.8)

2a |∇f|² (1−f)²Ψ

= 2awΨ ≤ 1

8Ψw²+ 2a²Ψ (2.9)

and

−2|∇f|²(q+af)

(1−f)³ Ψ = 2Ψw² −q

1−f +a −f 1−f

≤ 2Ψw²

− min

M×[0,T]q+a −f 1−f

(2.10)

≤ 2

max{a,0} − min

M×[0,T]q

Ψw². Substituting (2.5) – (2.10) to the right-hand side of (2.3), we deduce that

Ψ(1−f)w² ≤ C6

f⁴

ρ⁴(1−f)³ +3

4Ψw²+ 2

max{a,0} − min

M×[0,T]q

Ψw² + C₆

τ² +C₆

ρ⁴ +C₆K²+ 2 +γ²+ 2a² at (x0, t0). Sincef <0, it follows thatf⁴/(1−f)⁴≤1 and then

Ψw² ≤ C₆ ρ⁴ + 1

1−f 3

4Ψw²+ 2

max{a,0} − min

M×[0,T]q

Ψw² + C₆

τ² +C₆

ρ⁴ +C6K²+ (2 +γ²+ 2a²)

≤ C₆ ρ⁴ +3

4Ψw²+ 2

max{a,0} − min

M×[0,T]q

Ψw² + C6

τ² +C6

ρ⁴ +C₆K²+ 2 +γ²+ 2a² at (x0, t0), when

(2.11) min

M×[0,T]q−max{a,0} ≤ 3 8. Therefore, we can conclude that

1 4 + 2

min

M×[0,T]q−max{a,0}

Ψw²≤2C6

ρ⁴ +C6

τ² +C6K²+ (2 +γ²+ 2a²) at (x0, t0). If we assume in addition that

(2.12) 1

8+ min

M×[0,T]q−max{a,0}>0, we arrive at

Ψw²≤ C7

1

8+ min_M×[0,T]q−max{a,0}

1 ρ⁴ + 1

τ²+K²+ 1 +γ²+a²

at (x0, t0). Because Ψ(x, τ) = 1 when distτ(x, x0)< ρ/2, we finally arrive at w²(x, τ)≤ C₇

1

8+ min_M××[0,T]q−max{a,0}

1 ρ⁴ + 1

τ² +K²+ 1 +γ²+a²

onBρ/2,T, which, sinceτ∈(0, T] was arbitrary, implies

|∇f|

1−f ≤ C8

1

8+ min_M×[0,T]q−max{a,0}

1 ρ+ 1

√t +√

K+ 1 +√ γ+p

|a|

,

provided (2.11) and (2.12), or provided

(2.13) −1

8 < min

M×[0,T]q−max{a,0} ≤ 3 8.

We have completed the proof of Theorem 1.1 sincef = ln(u/A) with A scaled to

1.

The number 1/8 in (2.12) is not essential, because in the above argument we can replace 1/8 in (2.5) – (2.9) by a given positive number , and hence we need only to require that

1−6+ 2 min

M×[0,T]q−2 max{a,0}>0, 6+ 2

max{a,0} − min

M×[0,T]q

≥0 or

3−1

2 < min

M×[0,T]q−max{a,0} ≤3 instead of (2.13). When

1

2 + min

M×[0,T]q−max{a,0} ∈

0,1 2

we can choose to be any positive number in the interval [^A₃,^A₃ +¹₆), where A:=

min_M×[0,T]q−max{a,0} ∈(−1/2,0].

2.2. Gradient estimates II: general case. In this section we extend Theorem 1.1 withα= 1 to the general case.

Lemma 2.4. Suppose(M, g(t))_t∈[0,T] is a complete solution to the Ricci flow (1.1) on an n-dimensional manifold M, with −K1g(t) ≤ Ricg(t) ≤ K2g(t) for some K₁, K₂ > 0 on B_ρ,T. If u is a smooth positive function satisfying the nonlinear parabolic equation (1.2), then, for given β ≥1 and any c, d >0 withc+d= 1/β, we have

F ≥ −2h∇f,∇Fi −F

t +2cβt

n |∇f|²−q−f_t−af^α2

−2(β−1)th∇f,∇qi −2(β−1)taαf^α−1|∇f|²

−βtaα(α−1)f^α−2|∇f|²−2βtK1|∇f|²−nβt 2dK² (2.14)

−βaαtf^α−1 −|∇f|²+ft+q+af^α

−βt∆q,

whereK:= max{K1, K2},f := lnu, andF:=t(|∇f|²−βft−βq−βaf^α).

Proof. The proof of this lemma was original from [11]. Now we will find a convenient bound on ∆F like the way in [21].Notice that

∇iF =t 2∇^jf∇i∇jf −β∇ift−β∇iq−βaαf^α−1∇if .

Then the Laplace ofF equals

∆F = ∇ⁱ∇iF

= t

2|∇²f|²+ 2h∇f,∆∇fi −β∆ft−β∆q

−βaα (α−1)f^α−2|∇f|²+f^α−1∆f

.

Using the Bochner’s formula ∆∇f =∇∆f+ Ric(∇f,·), we get

∆F = t

2|∇²f|²+ 2h∇f,∇∆fi+ 2Ric(∇f,∇f)−β∆ft

−β∆q−βaα

(α−1)f^α−2|∇f|²+f^α−1∆f

≥ t

2(∆f)²

n + 2h∇f,∇∆fi −2K1|∇f|²−β(∆f)t−β∆q

−βaα

(α−1)f^α−2|∇f|²+f^α−1∆f

+ 2βRij∇ⁱ∇^jf

since|∇²f|²≥_n¹(∆f)²and ∆ft= (∆f)t−2Rij∇ⁱ∇^jf. Recalling from the result

∆f =−|∇f|²+q+f_t+af^α=−F

t −(β−1) (q+f_t+af^α), we arrive at

∆F ≥ 2cβt

n |∇f|²−q−ft−af^α2

+ 2dβt

n (∆f)²+ 2tβRij∇ⁱ∇^jf

−2t

∇f,∇ F

t + (β−1)(q+ft+af^α)

−2K₁t|∇f|²−tβ

−F

t −(β−1)(q+f_t+af^α)

t

−βt∆q (2.15)

−βaαt

(α−1)f^α−2|∇f|²+f^α−1∆f in the setBρ,T. Because

2dβt

n (∆f)²+ 2tβRij∇ⁱ∇^jf = 2dβt n

h

(∆f)²+n

dRij∇ⁱ∇^jfi

= 2dβt n

∇²f+ n 2dRic

2

−nβt 2d |Ric|²

≥ −nβt 2d |Ric|²,

the inequality (2.15) can be written as

∆F ≥ 2cβt

n |∇f|²−q−ft−af^α2

−nβt 2d |Ric|²

−2t

∇f,∇ F

t + (β−1)(q+ft+af^α)

−2K1t|∇f|²−tβ

−F

t −(β−1)(q+ft+af^α)

t

−βt∆q (2.16)

−βaαt

(α−1)f^α−2|∇f|²+f^α−1∆f To get the time derivative ofF, we shall use the identity

F_t= F t +t

|∇f|²−βf_t−βq−aβf^α

t

.

Subtracting this from (2.16), we get (∆−∂t)F ≥ −2h∇f,∇Fi −F

t +2cβt

n |∇f|²−q−ft−af^α2

−βt∆q−2(β−1)th∇f,∇qi −nβt 2d |Ric|²

−2(β−1)taαf^α−1|∇f|²−βtaα(α−1)f^α−2|∇f|²

−βaαtf^α−1

− |∇f|²+f_t+q+af^α

−2βK₁t|∇f|².

Now the inequality (2.14) follows immediately by noting that|Ric| ≤K.

Now we can consider the local space-time gradient estimate with Lemma 2.3. In the following part,nis the dimension of M.

Theorem 2.5. Suppose that(M, g(t))_t∈[0,T]is a complete solution to the Ricci flow (1.1) on ann-dimensional manifold M with −K1g(t)≤Ricg(t)≤K2g(t)for some K1, K2 > 0 on Bρ,T. If u is a smooth positive function satisfying the nonlinear parabolic equation (1.2), then there exists a constant C depending only on nsuch that, on B_ρ/2,T,

(1) fora≥0, we have

|∇g(t)f|²_g(t)−βft−βq−βaf^α ≤ nβ

2c(1−)t+(A+γ)nβ

2c(1−) + n²β³C² 4c²(1−)(β−1)ρ² + nβ[βK1+a(β−1)α|f^α−1|∞]

c(1−)(β−1) + nβ²aα|α−1||f^α−2|_∞

2c(β−1)(1−) +

s

[βθ+ (β−1)γ+^nβ_2dK²]nβ 2c(1−)

(2) fora≤0, we have

|∇_g(t)f|²_g(t)−βft−βq−βaf^α ≤ nβ

2c(1−)t+(A+γ)nβ

2(1−c) + n²β³C₁² 4c²(1−)(β−1)ρ² + nβ[βK1−^a₂(β−1)α|f^α−1|_∞]

c(1−)(β−1) +

s

[βθ+ (β−1)γ+^nβ_2dK²]nβ

2c(1−) .

Here f := lnu, |f|∞ := maxM|f|, K := max{K1, K2}, β > 1, 0 < < 1,

|∇g(t)q|g(t)≤γ,∆g(t)q≤θ, A=C

1 ρ²+

√K1

ρ +1 t +K

andc, d >0 withc+d= 1/β.

Proof. We will use the same notationf = lnuandF =t(|∇f|²−βf_t−βq−βaf^α) as in lemma 2.4. For the fixedτ ∈(0, T], chose the cut-off function Ψ constructed in Lemma 2.1. Define Ψ :M×[0, T]→Rby setting

Ψ(x, t) = Ψ dist_g(t)(x, x0), t .

Lemma 2.4 implies that

(∆−∂t) (ΨF) ≥ −2h∇f,∇(ΨF)i+ 2Fh∇f,∇Ψi +

2cβt

n |∇f|²−q−f_t−af^α2

−F

t − βt∆q

−2(β−1)th∇f,∇qi −2(β−1)taαf^α−1|∇f|²

−βtaα(α−1)f^α−2|∇f|²−2βtK1|∇f|²−nβt 2d K²

−βaαtf^α−1 −|∇f|²+ft+q+af^α

Ψ +F∆Ψ + 2

Ψh∇Ψ,∇(ΨF)i −2|∇Ψ|²

Ψ F−F∂Ψ

∂t.

Let (x0, t0) be a maximum point for the function ΨF in the set {(x, t)|0≤t≤τ, dt(x, x0)≤ρ}.Then at the point (x0, t0) we have

0 ≥ 2Fh∇f,∇Ψi+F(∆−∂t) Ψ−2|∇Ψ|²

Ψ F

+

−F

t +2cβt

n |∇f|²−q−ft−af^α2

−βt∆q

−2(β−1)th∇f,∇qi −2(β−1)taαf^α−1|∇f|²

−βtaα(α−1)f^α−2|∇f|²−βaαtf^α−1 −|∇f|²+f_t+q+af^α

−2βtK₁|∇f|²−nβt 2d K²

Ψ.

By Lemma 2.1 and the Laplacian comparison theorem, we have

|∇Ψ|²

Ψ ≤ C_1/2² ρ² ,

∆Ψ ≥ −C_1/2Ψ¹²

ρ² −C_1/2Ψ¹²

ρ (n−1)p

K1coth(p k1ρ)

≥ −d1

ρ² −d1Ψ¹² ρ

pK1,

−∂tΨ ≥ −CΨ¹²

τ −C1/2KΨ¹²

where C_1/2, C andd1 are positive constants depending only onn. Plugging those estimates into above inequality yields that

0 ≥ d₂

−1

ρ²−Ψ^1/2 ρ

pK₁−Ψ^1/2

τ −KΨ^1/2

F−2F|∇f||∇Ψ|

+ 2cβt

n |∇f|²−q−ft−af^α2

−F

t − βt∆q−nβt 2d K²

−2(β−1)th∇f,∇qi −2(β−1)taαf^α−1|∇f|²−2βtk₁|∇f|² (2.17)

−βtaα(α−1)f^α−2|∇f|²−βaαtf^α−1 −|∇f|²+ft+q+af^α

Ψ at (x0, t0), whered2 is equal to max{d1, C, C_1/2,2C_1/2² }. Introduce a function

A:=d2

1

ρ²+Ψ^1/2 ρ

pK1+Ψ^1/2

τ +KΨ^1/2

.

If one multiplies bytΨ and makes a few elementary manipulations, one will obtain 0 ≥ −2F|∇f||∇Ψ|Ψt−AFΨt+

2cβt

n |∇f|²−q−ft−af^α2

−βt∆q−2(β−1)th∇f,∇qi −2(β−1)taαf^α−1|∇f|²

−βtaα(α−1)f^α−2|∇f|²−βaαtf^α−1 −|∇f|²+ft+q+af^α (2.18)

−2βtK1|∇f|²−nβt 2d K²

Ψ²t−FΨ² at (x0, t0). As in [3, 19], we set

µ:= |∇f|²(x0, t0) F(x₀, t₀) ≥0.

Because|∇f|=µ^1/2F^1/2 and

|∇f|²−ft−q−af^α = F

µ−µt−1 βt

,

h∇f,∇Ψi ≤ |∇f||∇Ψ| ≤ C1

ρ Ψ^1/2|∇f|

we can simplify (2.18) into the following inequality

AF tΨ ≥ −2C1t

ρ Ψ^3/2µ^1/2F^3/2−Ψ²F+2cΨ²

nβ [1 + (β−1)µt]²F²−nβ

2dK²(Ψt)²

−2(tΨ)²[βK1+a(β−1)αf^α−1]µF+atΨ²αf^α−1[1 + (β−1)tµ]F (2.19)

−β(tΨ)²θ−2(β−1)(tΨ)²γ(µF)^1/2−β(tΨ)²aα(α−1)f^α−2µF

at (x0, t0). If we set G := ΨF, then at the point (x0, t0) the inequality (2.19) becomes

AtG ≥ −2C1t

ρ µ^1/2G^3/2−ΨG+ 2c

nβ[1 + (β−1)µt]²G²−nβ

2dK²(Ψt)²

−2Ψt²[βK₁+a(β−1)αf^α−1]µG+aΨtαf^α−1[1 + (β−1)µt]G (2.20)

−β(Ψt)²θ−2(β−1)t²Ψ^3/2γµ^1/2G^1/2−βt²Ψaα(α−1)f^α−2µG

at (x₀, t₀). According to the Cauchy inequality, where 0< <1, 2C₁t

R µ^1/2G^3/2 ≤ 2c

nβ[1 + (β−1)µt]²G²+ nβC₁²t²µG 2cρ²[1 + (β−1)µt]², 2µ^1/2G^1/2 ≤ 1 +µG,

we can simplify (2.20)as

AtG ≥ 2c(1−)

nβ [1 + (β−1)µt]²G²−ΨG− nβ²C₁²t²µ 2cρ²[1 + (β−1)µt]²G

−2Ψt²[βK₁+a(β−1)αf^α−1]µG+aΨtαf^α−1[1 + (β−1)µt]G

−βΨ²t²θ−(β−1)t²Ψ^3/2γ−(β−1)t²Ψ^3/2γµG

−βt²Ψaα(α−1)f^α−2µG−nβ

2dK²(Ψt)², at (x0, t0), or equivalently,

2c(1−)[1 + (β−1)µt]²G²

nβ ≤

nβ²C₁²t²µ

2cρ²[1 + (β−1)µt]² +βt²Ψaα(α−1)f^α−2µ

−aΨtαf^α−1[1 + (β−1)µt] + (β−1)t²Ψ^3/2γµ + 2Ψt²[βK₁+a(β−1)αf^α−1]µ+At+ Ψ

G

+

βΨ²θ+ (β−1)Ψ^3/2γ+nβ 2dK²Ψ²

t²

at (x0, t0). Note that 0≤Ψ≤1 and 1 + (β−1)µt0≥1. Therefore 2c(1−)G²

nβ ≤

At+ 1 + nβ²C₁²t²µ

2cρ²[1 + (β−1)µt] +2Ψt²[βK1+a(β−1)αf^α−1]µ [1 + (β−1)µt]²

− aΨtαf^α−1

1 + (β−1)µt + (β−1)γt²µ

1 + (β−1)µt+βt²Ψaα|α−1|f^α−2µ 1 + (β−1)µt

G

+

βθ+ (β−1)γ+nβ 2dK²

t²

≤

At+ 1 + nβ²C₁²t

2cρ²(β−1)+2Ψt²[βK1+a(β−1)α|f^α−1|_∞]µ [1 + (β−1)µt]²

(2.21)

+γt− aΨtαf^α−1

1 + (β−1)µt+βtΨaα|α−1||f^α−2|_∞ β−1

G

+

βθ+ (β−1)γ+nβ 2dK²

t².

Before completing the proof, we recall a fact: if x² ≤ax+b for somea, b, x ≥0, then

(2.22) x≤ a

2 + r

b+a 2

²

≤ a 2 +√

b+a

2 =a+√ b.

Ifa≥0 in (2.21), then from (2.21) we deduce that G² ≤

(A+γ)nβt

2c(1−) + nβ

2c(1−)+ n²β³C₁²t 4c²(1−)ρ²(β−1) + nβ²aα|α−1||f^α−2|∞t

2c(β−1)(1−) +nβ[βK1+a(β−1)α|f^α−1|∞]t c(1−)(β−1)

G (2.23)

+ [βθ+ (β−1)γ+^nβ_2dK²]nβt²

2c(1−) .

Applying (2.22) to the inequality (2.23), we get an upper bound forG:

G ≤

(A+γ)nβ

2c(1−) + n²β³C₁²

4c²(1−)(β−1)ρ² +nβ[βK₁+a(β−1)α|f^α−1|_∞] c(1−)(β−1) + nβ²aα|α−1||f^α−2|∞

2c(β−1)(1−)

τ+ s

[βθ+ (β−1)γ+^nβ_2dK²]nβ

2c(1−) τ+ nβ

2c(1−), since t1 ≤ τ. By the construction of Ψ, we have sup_B

ρ/2,TF ≤ sup_Bρ,T(ΨF) ≤ G(x0, t0) for allt∈[0, τ]. Becauseτ ≤T is arbitrary, it follows that

|∇f|²−βft−βq−βaf^α ≤ nβ

2c(1−)t +(A+γ)nβ

2c(1−) + n²β³C₁² 4c²(1−)(β−1)ρ² + nβ[βK1+a(β−1)α|f^α−1|∞]

c(1−)(β−1) + nβ²aα|α−1||f^α−2|_∞

2c(β−1)(1−) +

s

[βθ+ (β−1)γ+^nβ_2dK²]nβ 2c(1−)

where|f|∞:= maxM|f|. Similarly, whena≤0, we have G² ≤

(A+γ)nβt

2c(1−) + nβ

2c(1−)+ n²β³C₁²t

4c²(1−)ρ²(β−1) + nβ²K₁t c(1−)(β−1)

− nβatα|f^α−1|_∞ 2c(1−)

G+[βθ+ (β−1)γ+^nβ_2dK²]nβt²

2c(1−) .

(2.24)

From (2.22), (2.24), and above argument, an upper bound for desired quantity in this case is

|∇f|²−βft−βq−βaf^α ≤ nβ

2c(1−)t +(A+γ)nβ

2c(1−) + n²β³C₁² 4c²(1−)(β−1)ρ² + nβ[βK₁−^a₂(β−1)α|f^α−1|_∞]

c(1−)(β−1) +

s

[βθ+ (β−1)γ+^nβ_2dK²]nβ

2c(1−) .

Hence, we complete the proof.

3. Gradient estimates for (1.4) under the geometric flow Suppose now u is a positive solution of (1.5), and as in [7], we introduce a function

(3.1) W :=u^−q,

where qis a positive constant to be determined later. For convenience, we always omit time variable tand writeQt for the partial derivative ofQrelative tot. For example, throughout this paper, ∆,∇,| · |mean the correspondence quantities with respect tog(t). Write

:= ∆−∂t. A simple computation shows that

∇W = −qu^−q−1∇u, |∇W|² = q²u^−2q−2|∇u|²,

Wt = −qu^−q−1ut, ∆W = q(q+ 1)u^−q−2|∇u|²−qu^−q−1∆u.

The relation (3.1) yields (see [7, 9])

(3.2) |∇u|²= |∇W|²

q²W^2+2/q, ut=− Wt

qW^1+1/q, and hence

(3.3) W =q+ 1

q

|∇W|²

W +qhW^1+1−pq .

Since |∇W|²/W² = q²|∇u|²/u² and hW^(1−p)/q = hu^p−1, we consider again the same quantities as in [7, 9],

F₀ := |∇W|²

W² +αhW^(1−p)/q = |∇lnW|²+αhW^(1−p)/q, (3.4)

F1 := Wt

W = ∂tlnW, (3.5)

F := F₀+βF₁. (3.6)