Open Access

Harnack inequality for parabolic Lichnerowicz equations on complete noncompact Riemannian manifolds

Boundary Value Problems20132013:190

DOI: 10.1186/1687-2770-2013-190

Received: 31 May 2013

Accepted: 9 August 2013

Published: 27 August 2013

Abstract

In this paper, we study the gradient estimates for positive solutions to the following parabolic Lichnerowicz equations

u t = u + h u ( x , t ) + A u p ( x , t ) + B u q ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equa_HTML.gif

on complete noncompact Riemannian manifolds, where h, p, q, A, B are real constants and p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq1_HTML.gif, q > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq2_HTML.gif.

MSC:58J05, 58J35.

Keywords

Lichnerowicz equation positive solutions Harnack inequality

1 Introduction

Let M be an n-dimensional complete noncompact Riemannian manifold. In this paper, we study the following nonlinear parabolic equation
u t ( x , t ) = u ( x , t ) + h u ( x , t ) + A u p ( x , t ) + B u q ( x , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ1_HTML.gif
(1.1)

where h, p, q, A, B are real constants and p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq1_HTML.gif, q > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq2_HTML.gif.

Gradient estimates play an important role in the study of PDE, especially the Laplace equation and heat equation. Li [1] derived the gradient estimates and Harnack inequalities for positive solutions of nonlinear equations ( t ) u ( x , t ) + h ( x , t ) u α ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq3_HTML.gif and A u + b u + h u α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq4_HTML.gif on Riemannian manifolds. The author in [1] also obtained a theorem of Liouville-type for positive solutions of the nonlinear elliptic equation. Later, Yang [2] gave the gradient estimates for the solution to the elliptic equation with singular nonlinearity
u + c u α = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ2_HTML.gif
(1.2)

where α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq5_HTML.gif, c are two real constants. More precisely, the author [2] obtained the following result.

Theorem 1.1 (Yang [2])

Let M be a noncompact complete Riemannian manifold of dimension n without boundary. Let B p ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq6_HTML.gif be a geodesic ball of radius 2R around p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq7_HTML.gif. We denote K ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq8_HTML.gif, with K ( 2 R ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq9_HTML.gif, to be a lower bound of the Ricci curvature on B p ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq6_HTML.gif, i.e., Ric ( ξ , ξ ) K ( 2 R ) | ξ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq10_HTML.gif for all tangent field ξ on B p ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq6_HTML.gif. Suppose that u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq11_HTML.gif is a positive smooth solution of the equation (1.2) with α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq12_HTML.gif, c being two real constants. Then we have:
  1. (i)
    If c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq13_HTML.gif, then u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq11_HTML.gif satisfies the estimate
    | u | 2 u 2 + c u ( α + 1 ) n ( 2 n + 1 ) ϵ 2 R 2 + n ( n 1 ) ϵ 2 R K ( 2 R ) + n ν R 2 + 2 n K ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equb_HTML.gif
     
on B p ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq14_HTML.gif, where ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq15_HTML.gif and ν > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq16_HTML.gif are some universal constants independent of geometry of M.
  1. (ii)
    If c < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq17_HTML.gif, then u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq11_HTML.gif satisfies the estimate
    | u | 2 u 2 + c u ( α + 1 ) ( n ( α + 1 ) ( α + 2 ) + n ( α + 1 ) ) | c | ( inf B p ( 2 R ) u ) α 1 + n ν R 2 + ( 2 n + n α + 1 ) K ( 2 R ) + n ϵ 2 R 2 ( 2 n + 1 ) + n 2 ( α + 1 ) + ( n 1 ) K ( 2 R ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equc_HTML.gif
     

on B p ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq14_HTML.gif, where ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq15_HTML.gif and ν > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq16_HTML.gif are some universal constants independent of geometry of M.

For some interesting gradient estimates in this direction, we can refer to [37].

Recently, Song and Zhao [8] studied a generalized elliptic Lichnerowicz equation
u ( x ) + h ( x ) u ( x ) = A ( x ) u p ( x ) + B ( x ) u q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ3_HTML.gif
(1.3)
on compact manifold ( M , g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq18_HTML.gif. The authors in [8] got the local gradient estimate for the positive solutions of (1.3). Moreover, they considered the following parabolic Lichnerowicz equation
u t ( x , t ) + u ( x , t ) + h ( x ) u ( x , t ) = A ( x ) u p ( x , t ) + B ( x ) u q ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ4_HTML.gif
(1.4)

on manifold ( M , g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq18_HTML.gif and obtained the Harnack differential inequality.

Theorem 1.2 (Song and Zhao [8])

Let M be a compact Riemannian manifold without boundary, Ric ( M ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq19_HTML.gif. Let c ( t ) C 1 ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq20_HTML.gif. Assume that u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq21_HTML.gif is any positive solution of (1.4) on M with A ( x ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq22_HTML.gif, B ( x ) B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq23_HTML.gif, and h ( x ) h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq24_HTML.gif. Denote φ = ln u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq25_HTML.gif, suppose that A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq26_HTML.gif, B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq27_HTML.gif, c ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq28_HTML.gif, c ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq29_HTML.gif. If | φ | 2 1 p φ t + 1 p H ˜ c ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq30_HTML.gif with H ˜ = A e ( p 1 ) φ + B e ( q + 1 ) φ h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq31_HTML.gif at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq32_HTML.gif, then we have
| φ | 2 1 p φ t + 1 p H ˜ c ( t ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equd_HTML.gif

While the author considered the gradient estimates on compact Riemannian manifolds in Theorem 1.2, it is natural to study this problem on complete noncompact manifolds. Motivated by the work above, we present our main results as follows.

Theorem 1.3 Let ( M , g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq18_HTML.gif be a complete noncompact n-dimensional Riemannian manifold with Ricci tensor bounded from below by the constant K = : K ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq33_HTML.gif, where R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq34_HTML.gif and K ( 2 R ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq35_HTML.gif in the metric ball B 2 R ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq36_HTML.gif around p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq7_HTML.gif. Assume that u is a positive solution of (1.1) with u M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq37_HTML.gif for all ( x , t ) B R ( p ) × ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq38_HTML.gif. Then
  1. (1)
    if A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq26_HTML.gif, B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq27_HTML.gif, we have
    β | u | 2 u 2 + h + A u p 1 + B u ( q + 1 ) u t u n 2 ( 1 δ ) β ( n c 1 2 4 δ β ( 1 β ) R 2 M 1 p 1 A ( p 1 ) ( p β ) 4 ( 1 β ) 2 + H + 1 t ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Eque_HTML.gif
     
  2. (2)
    if A > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq39_HTML.gif, B > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq40_HTML.gif, we get
    β | u | 2 u 2 + h + A u p 1 + B u ( q + 1 ) u t u n 2 ( 1 δ ) β ( n c 1 2 4 δ β ( 1 β ) R 2 + M 1 p 1 A ( p 1 ) + H + 1 t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equf_HTML.gif
     

where H = ( n 1 ) ( 1 + K R ) c + c 2 + 2 c 1 2 R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq41_HTML.gif, c, c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq42_HTML.gif, c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq43_HTML.gif, δ are positive constants with 0 < δ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq44_HTML.gif and β = e 2 K t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq45_HTML.gif.

Let R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq46_HTML.gif, we can get the following global gradient estimates for the nonlinear parabolic equation (1.1).

Corollary 1.4 Let ( M , g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq18_HTML.gif be a complete noncompact n-dimensional Riemannian manifold with Ricci tensor bounded from below by the constant K = : K ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq47_HTML.gif, where K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq48_HTML.gif. Assume that u is a positive solution of (1.1) with u M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq37_HTML.gif for all ( x , t ) M × ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq49_HTML.gif. Then
  1. (1)
    if A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq26_HTML.gif, B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq27_HTML.gif, we have
    β | u | 2 u 2 + h + A u p 1 + B u ( q + 1 ) u t u n 2 ( 1 δ ) β ( M 1 p 1 A ( p 1 ) ( p β ) 4 ( 1 β ) 2 + 1 t ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equg_HTML.gif
     
  2. (2)
    if A > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq39_HTML.gif, B > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq40_HTML.gif, we get
    β | u | 2 u 2 + h + A u p 1 + B u ( q + 1 ) u t u n 2 ( 1 δ ) β ( M 1 p 1 A ( p 1 ) + 1 t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equh_HTML.gif
     

δ are positive constants with 0 < δ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq44_HTML.gif and β = e 2 K t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq45_HTML.gif.

Let δ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq50_HTML.gif, A = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq51_HTML.gif in Corollary 1.4, we get a Li-Yau-type gradient estimate.

Corollary 1.5 Let ( M , g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq18_HTML.gif be a complete noncompact n-dimensional Riemannian manifold with Ric ( M ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq19_HTML.gif. Assume that u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq21_HTML.gif is a positive solution to the equation
u t = u + h u ( x , t ) + B u q ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equi_HTML.gif
on complete noncompact manifolds, where h, q, B are real constants and q > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq2_HTML.gif. Then we have
| u | 2 u 2 + h + B u ( q + 1 ) u t u n 2 t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ5_HTML.gif
(1.5)

As an application, we have the following Harnack inequality.

Theorem 1.6 Let ( M , g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq18_HTML.gif be a complete noncompact n-dimensional Riemannian manifold with Ric ( M ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq19_HTML.gif. Assume that u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq21_HTML.gif is a positive solution to the equation
u t = u + h u ( x , t ) + B u q ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equj_HTML.gif
on complete noncompact manifolds, where h, q, B are real constants and q > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq2_HTML.gif, B > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq52_HTML.gif. Then for any points ( x 1 , t 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq53_HTML.gif and ( x 2 , t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq54_HTML.gif on M × [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq55_HTML.gif with 0 < t 1 < t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq56_HTML.gif, we have the following Harnack inequality:
u ( x 1 , t 1 ) u ( x 2 , t 2 ) ( t 2 t 1 ) n 2 e ϕ ( x 1 , x 2 , t 1 , t 2 ) + D , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equk_HTML.gif

where ϕ ( x 1 , x 2 , t 1 , t 2 ) = d 2 ( x 1 , x 2 ) t 2 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq57_HTML.gif, D = h ( t 1 t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq58_HTML.gif.

2 Proof of Theorem 1.3

Assume that u is a positive solution to (1.1). Set w = ln u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq59_HTML.gif, then w satisfies the equation
w t = w + | w | 2 + h + A e ( p 1 ) w + B e w ( q + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ6_HTML.gif
(2.1)
Lemma 2.1 Let ( M , g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq18_HTML.gif be a complete noncompact n-dimensional Riemannian manifold with Ricci curvature bounded from below by the constant K = : K ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq33_HTML.gif, where R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq34_HTML.gif and K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq48_HTML.gif in the metric ball B 2 R ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq36_HTML.gif around p M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq7_HTML.gif. Let w be a positive solution of (2.1), then
( t ) F 2 w F + t { 2 β n ( ( β 1 ) | w | 2 F t ) 2 + A ( p β ) ( p 1 ) e ( p 1 ) w | w | 2 + B ( q + 1 ) ( q + β ) e w ( q + 1 ) | w | 2 } + [ B ( q + 1 ) e ( q + 1 ) w A ( p 1 ) e ( p 1 ) w ] F F t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equl_HTML.gif
where
F = t ( β | w | 2 + h + A e ( p 1 ) w + B e ( q + 1 ) w w t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equm_HTML.gif

and β = e 2 K t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq45_HTML.gif.

Proof Define
F = t ( β | w | 2 + h + A e ( p 1 ) w + B e ( q + 1 ) w w t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equn_HTML.gif
where β = e 2 K t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq45_HTML.gif. By the Bochner formula, we have
| w | 2 2 n | w | 2 + 2 w ( w ) 2 K | w | 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ7_HTML.gif
(2.2)
By a direct computation, we have
w t = ( w ) t = 2 w w t A ( p 1 ) e ( p 1 ) w w t + B ( q + 1 ) e ( q + 1 ) w w t + w t t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ8_HTML.gif
(2.3)
and
w = | w | 2 h A e ( p 1 ) w B e ( q + 1 ) w + w t = ( 1 1 β ) ( h A e ( p 1 ) w B e ( q + 1 ) w + w t ) F β t = ( β 1 ) | w | 2 F t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equo_HTML.gif
and we know
F = t { β | w | 2 + A [ ( p 1 ) 2 e ( p 1 ) w | w | 2 + ( p 1 ) e ( p 1 ) w w ] + B [ ( q + 1 ) 2 e ( q + 1 ) w | w | 2 ( q + 1 ) e ( q + 1 ) w w ] w t } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equp_HTML.gif
Therefore, by equalities (2.2) and (2.3), we obtain
β | w | 2 2 β n ( ( β 1 ) | w | 2 F t ) 2 + 2 β w ( w ) 2 β K | w | 2 = 2 β n ( ( β 1 ) | w | 2 F t ) 2 + 2 β w [ ( 1 1 β ) ( h A e ( p 1 ) w B e ( q + 1 ) w + w t ) F β t ] 2 β K | w | 2 = 2 β n ( ( β 1 ) | w | 2 F t ) 2 2 A ( β 1 ) ( p 1 ) e ( p 1 ) w | w | 2 + 2 B ( β 1 ) ( q + 1 ) e ( q + 1 ) w | w | 2 + 2 ( β 1 ) w w t 2 t w F 2 β K | w | 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equq_HTML.gif
This implies that,
F t { 2 β n ( ( β 1 ) | w | 2 F t ) 2 2 A ( β 1 ) ( p 1 ) e ( p 1 ) w | w | 2 + 2 B ( β 1 ) ( q + 1 ) e ( q + 1 ) w | w | 2 + 2 ( β 1 ) w w t 2 t w F 2 β K | w | 2 + A ( p 1 ) 2 e ( p 1 ) w | w | 2 + A ( p 1 ) e ( p 1 ) w [ ( β 1 ) | w | 2 F t ] + B ( q + 1 ) 2 e ( q + 1 ) w | w | 2 B ( q + 1 ) e ( q + 1 ) w [ ( β 1 ) | w | 2 F t ] + 2 w w t + A ( p 1 ) e ( p 1 ) w w t B ( q + 1 ) e ( q + 1 ) w w t w t t } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equr_HTML.gif
and
F t = F t + t ( 2 β w w t + A ( p 1 ) e ( p 1 ) w w t B ( q + 1 ) e ( q + 1 ) w w t w t t 2 K β | w | 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equs_HTML.gif
Therefore, it follows that
( t ) F 2 w F + t { 2 β n ( ( β 1 ) | w | 2 F t ) 2 + A ( p β ) ( p 1 ) e ( p 1 ) w | w | 2 + B ( q + 1 ) ( q + β ) e w ( q + 1 ) | w | 2 } + [ B ( q + 1 ) e ( q + 1 ) w A ( p 1 ) e ( p 1 ) w ] F F t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equt_HTML.gif

which completes the proof of Lemma 2.1. □

We take a C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq60_HTML.gif cut-off function φ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq61_HTML.gif defined on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq62_HTML.gif such that φ ˜ ( r ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq63_HTML.gif for r [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq64_HTML.gif, φ ˜ ( r ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq65_HTML.gif for r [ 2 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq66_HTML.gif, and 0 φ ˜ ( r ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq67_HTML.gif. Furthermore, φ ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq61_HTML.gif satisfies
φ ˜ ( r ) φ ˜ 1 2 ( r ) c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equu_HTML.gif
and
φ ˜ ( r ) c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equv_HTML.gif
for some absolute constants c 1 , c 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq68_HTML.gif. Denote by r ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq69_HTML.gif the distance between x and p in M. Set
φ ( x ) = φ ˜ ( r ( x ) R ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equw_HTML.gif
Using an argument of Cheng and Yau [9], we can assume that φ ( x ) C 2 ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq70_HTML.gif with support in B p ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq6_HTML.gif. Direct calculation shows that on B p ( 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq6_HTML.gif
| φ | 2 φ c 1 2 R 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ9_HTML.gif
(2.4)
By the Laplacian comparison theorem in [10],
φ ( n 1 ) ( 1 + K R ) c 1 2 + c 2 R 2 ( c 1 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equ10_HTML.gif
(2.5)

In inequality (2.5), if c 1 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq71_HTML.gif, then φ can be controlled by ( n 1 ) ( 1 + K R ) c 1 + c 2 R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq72_HTML.gif, so in any case, φ ( n 1 ) ( 1 + K R ) c + c 2 R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq73_HTML.gif, where c is some positive constant.

For T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq74_HTML.gif, let ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq75_HTML.gif be a point in B 2 R ( p ) × [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq76_HTML.gif, at which φF attains its maximum value P, and we assume that P is positive (otherwise the proof is trivial). At the point ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq75_HTML.gif, we have
( φ F ) = 0 , ( φ F ) 0 , F t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equx_HTML.gif
It follows that
φ F + F φ 2 F φ 1 | φ | 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equy_HTML.gif
This inequality, together with inequalities (2.4) and (2.5), yields
φ F H F , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equz_HTML.gif
where
H = ( ( n 1 ) ( 1 + K R ) ) c + c 2 + 2 c 1 2 R 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equaa_HTML.gif
At ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq75_HTML.gif, by Lemma 2.1, we have
φ F 2 φ w F + s φ { 2 β n ( ( β 1 ) | w | 2 F s ) 2 + A ( p β ) ( p 1 ) e ( p 1 ) w | w | 2 + B ( q + 1 ) ( q + β ) e w ( q + 1 ) | w | 2 } + φ [ B ( q + 1 ) e ( q + 1 ) w A ( p 1 ) e ( p 1 ) w ] F φ F s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equab_HTML.gif
it follows that
2 s φ β n ( ( β 1 ) | w | 2 F s ) 2 2 c 1 R φ 1 2 F | w | + H F A ( p β ) ( p 1 ) e ( p 1 ) w | w | 2 s φ B ( q + 1 ) ( q + β ) e w ( q + 1 ) | w | 2 s φ [ B ( q + 1 ) e ( q + 1 ) w A ( p 1 ) e ( p 1 ) w ] φ F + φ F s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equac_HTML.gif
here we used
2 φ w F = 2 F w φ 2 F | w | | φ | 2 c 1 R φ 1 2 F | w | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equad_HTML.gif
Following Davies [11] (see also Negrin [12]), we set
μ = | w | 2 F . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equae_HTML.gif
Then we have
2 φ β ( ( β 1 ) s μ 1 ) 2 F 2 n s 2 c 1 R φ 1 2 μ 1 2 F 3 2 + H F A ( p β ) ( p 1 ) e ( p 1 ) w s φ μ F B ( q + 1 ) ( q + β ) e w ( q + 1 ) s μ φ F [ B ( q + 1 ) e ( q + 1 ) w A ( p 1 ) e ( p 1 ) w ] φ F + φ F s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equaf_HTML.gif
Next, we consider the following two cases:
  1. (1)
    if A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq26_HTML.gif, B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq27_HTML.gif, then we have
    2 φ β ( ( β 1 ) s μ 1 ) 2 F 2 n s 2 c 1 R φ 1 2 μ 1 2 F 3 2 + H F M 1 p 1 A ( p 1 ) ( p β ) μ s φ F + φ F s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equag_HTML.gif
     
multiplying both sides of the inequality above by , we have
2 β ( ( β 1 ) s μ 1 ) 2 n ( φ F ) 2 2 c 1 R s μ 1 2 ( φ F ) 3 2 + H s φ F M 1 p 1 A ( p 1 ) ( p β ) μ s 2 φ F + φ F 2 δ β ( ( β 1 ) s μ 1 ) 2 n ( φ F ) 2 + n c 1 2 s 2 μ 2 δ β ( ( β 1 ) s μ 1 ) 2 R 2 φ F + H s φ F M 1 p 1 A ( p 1 ) ( p β ) μ s 2 φ F + φ F . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equah_HTML.gif
So, it follows that
P n 2 ( 1 δ ) β ( ( β 1 ) s μ 1 ) 2 × ( n c 1 2 s 2 μ 2 δ β ( ( β 1 ) s μ 1 ) 2 R 2 M 1 p 1 A ( p 1 ) ( p β ) μ s 2 2 ( 1 β ) + H s + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equai_HTML.gif
Since
( ( β 1 ) s μ 1 ) 2 2 ( 1 β ) s μ + 1 2 ( 1 β ) s μ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equaj_HTML.gif
we get
P n 2 ( 1 δ ) β ( n c 1 2 s 4 δ β ( 1 β ) R 2 M 1 p 1 A ( p 1 ) ( p β ) s 4 ( 1 β ) 2 + H s + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equak_HTML.gif
Now, (1) of Theorem 1.3 can be easily deduced from the inequality above;
  1. (2)
    if A > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq39_HTML.gif, B > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq40_HTML.gif, then we have
    2 φ β ( ( β 1 ) s μ 1 ) 2 F 2 n s 2 c 1 R φ 1 2 μ 1 2 F 3 2 + M 1 p 1 A ( p 1 ) φ F + H F + φ F s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equal_HTML.gif
     
multiplying both sides of the inequality above by , we have
2 β ( ( β 1 ) s μ 1 ) 2 n ( φ F ) 2 2 c 1 R s μ 1 2 ( φ F ) 3 2 + M 1 p 1 A ( p 1 ) φ F s + H s φ F + φ F 2 δ β ( ( β 1 ) s μ 1 ) 2 n ( φ F ) 2 + n c 1 2 s 2 μ 2 δ β ( ( β 1 ) s μ 1 ) 2 R 2 φ F + M 1 p 1 A ( p 1 ) φ F s + H s φ F + φ F . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equam_HTML.gif
So, it follows that
P n 2 ( 1 δ ) β ( n c 1 2 s 4 δ β ( 1 β ) R 2 + M 1 p 1 A ( p 1 ) s + H s + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equan_HTML.gif

Similarly, we can obtain (2) of Theorem 1.3.

Proof of Theorem 1.6 For any points ( x 1 , t 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq53_HTML.gif and ( x 2 , t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq54_HTML.gif on M × [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq77_HTML.gif with 0 < t 1 < t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq56_HTML.gif, we take a curve γ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq78_HTML.gif parameterized with γ ( t 1 ) = x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq79_HTML.gif and γ ( t 2 ) = x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq80_HTML.gif. One gets from Corollary 1.5 that
log u ( x 2 , t 2 ) log u ( x 1 , t 1 ) = t 1 t 2 ( ( log u ) t + log u , γ ˙ ) d t t 1 t 2 ( | log u | 2 n 2 t + h + B u ( q + 1 ) | log u | | γ ˙ | ) d t t 1 t 2 ( 1 4 | γ ˙ | 2 + n 2 t h ) d t = ( t 1 t 2 1 4 | γ ˙ | 2 d t + log ( t 2 t 1 ) n 2 + h ( t 1 t 2 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equao_HTML.gif
which means that
log u ( x 1 , t 1 ) u ( x 2 , t 2 ) t 1 t 2 1 4 | γ ˙ | 2 d t + log ( t 2 t 1 ) n 2 + h ( t 1 t 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equap_HTML.gif
Therefore,
u ( x 1 , t 1 ) u ( x 2 , t 2 ) ( t 2 t 1 ) n 2 e ϕ ( x 1 , x 2 , t 1 , t 2 ) + D , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_Equaq_HTML.gif

where ϕ ( x 1 , x 2 , t 1 , t 2 ) = d 2 ( x 1 , x 2 ) t 2 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq57_HTML.gif, D = h ( t 1 t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq58_HTML.gif. □

Declarations

Acknowledgements

The author would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions that improved the quality of the paper. Moreover, the author would like to thank his supervisor Professor Kefeng Liu for his constant encouragement and help. This work is supported by the Postdoctoral Science Foundation of China (2013M531342) and the Fundamental Research Funds for the Central Universities (NS2012065).

Authors’ Affiliations

(1)
Department of Mathematics, Nanjing University of Aeronautics and Astronautics

References

  1. Li JY: Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J. Funct. Anal. 1991, 100: 233-256. 10.1016/0022-1236(91)90110-QMathSciNetView ArticleMATH
  2. Yang YY:Gradient estimates for the equation u + c u α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq81_HTML.gif on Riemannian manifolds. Acta Math. Sin. 2010, 26: 1177-1182. 10.1007/s10114-010-7531-yView ArticleMathSciNetMATH
  3. Chen L, Chen WY: Gradient estimates for positive smooth f -harmonic functions. Acta Math. Sci., Ser. B 2010, 30: 1614-1618.View ArticleMathSciNetMATH
  4. Huang, GY, Li, HZ: Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian. arXiv:​1203.​5482v1 [math.DG] (2012). MATH
  5. Li XD: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. 2005, 84: 1295-1361.MathSciNetView ArticleMATH
  6. Zhang J, Ma BQ:Gradient estimates for a nonlinear equation f u + c u α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-190/MediaObjects/13661_2013_Article_446_IEq82_HTML.gif on complete noncompact manifolds. Commun. Math. 2011, 19: 73-84.MathSciNet
  7. Zhu XB: Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds. Proc. Am. Math. Soc. 2011, 139: 1637-1644. 10.1090/S0002-9939-2010-10824-9View ArticleMATH
  8. Song XF, Zhao L: Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds. Z. Angew. Math. Phys. 2010, 61: 655-662. 10.1007/s00033-009-0047-6MathSciNetView ArticleMATH
  9. Cheng SY, Yau ST: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 1975, 28: 333-354. 10.1002/cpa.3160280303MathSciNetView ArticleMATH
  10. Aubin T: Nonlinear Analysis on Manifolds. Springer, New York; 1982.MATH
  11. Davies EB: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge; 1989.View ArticleMATH
  12. Negrin ER: Gradient estimates and a Liouville type theorem for the Schrödinger operator. J. Funct. Anal. 1995, 127: 198-203. 10.1006/jfan.1995.1008MathSciNetView ArticleMATH

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