# Some gradient estimates for nonlinear heat-type equations on smooth metric
measure spaces with compact boundary

Abimbola Abolarinwa  Mathematical Analysis and Applications Research Group
(MAARG), Department of Mathematics, University of Lagos, Nigeria.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: July 25, 2023)

###### Abstract.

In this paper we prove some Hamilton type and Li-Yau type gradient estimates
on positive solutions to generalized nonlinear parabolic equations on smooth
metric measure space with compact boundary. The geometry of the space in terms
of lower bounds on the weighted Bakry-Émery Ricci curvature tensor and
weighted mean curvature of the boundary are key in proving generalized local
and global gradient estimates. Various applications of these gradient
estimates in terms of parabolic Harnack inequalities and Liouville type
results are discussed. Further consequences to some specific models informed
by the nature of the nonlinearities are highlighted.

###### Key words and phrases:

Gradient estimates; Harnack type inequalities; Liouville-type theorems;
weighted manifolds; Drifting Laplacian; Bakry-Émery Ricci tensor; mean
curvature

###### 2010 Mathematics Subject Classification:

35K55, 35K58, 58J35, 58J60, 60J60

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and statement of main results
    1. 2.1 Preliminaries
      1. 2.1.1 Weighted Laplacian and weighted Ricci tensor
      2. 2.1.2 Weighted Bochner formula
      3. 2.1.3 Mean curvature of the boundary
      4. 2.1.4 Laplacian comparison on weighted manifold with boundary
      5. 2.1.5 Cut-off functions
      6. 2.1.6 Some elementary inequalities
    2. 2.2 Statement of main results
      1. 2.2.1 Hamilton-Souplet-Zhang type gradient estimates I
      2. 2.2.2 Hamilton-Souplet-Zhang type gradient estimates II
      3. 2.2.3 Li-Yau type gradient estimates
  3. 3 Hamilton-Souplet-Zhang type gradient estimates I
    1. 3.1 proof of Theorem 3.1
  4. 4 Hamilton-Souplet-Zhang type gradient estimates II
  5. 5 Li-Yau type gradient estimates
    1. 5.1 Evolution equations
    2. 5.2 Fundamental lemma
    3. 5.3 A suitable cut-off function and its consequence
    4. 5.4 Proof of Theorem 5.1
  6. 6 Applications of Li-Yau type gradient estimates (Theorem 5.1)
    1. 6.1 Global Li-Yau gradient estimates
    2. 6.2 Li-Yau Harnack inequalities
    3. 6.3 Li-Yau type gradient estimates for elliptic equations and Liouville-type theorems
    4. 6.4 Some special examples of $H(u)$
  7. 7 Applications of elliptic gradient estimates
    1. 7.1 Global elliptic gradient estimates
    2. 7.2 Liouville-type results

##  1\. Introduction

Let the triple $(M,g,e^{-\phi}dv)$ be a smooth metric measure space whereby
the pair $(M,g)$ is a complete Riemannian manifold of dimension greater than
or equal to $2$ equipped with volume measure $dv$ and
$\phi\in\mathscr{C}^{\infty}(M)$ is a smooth potential function. The smooth
metric measure space $M:=(M,g,e^{-\phi}dv)$ is linked with a second order
self-adjoint diffusion operator called $\phi$-Laplacian, $\mathscr{L}_{\phi}$,
which is usually associated with weighted Ricci tensor,
$\mathscr{R}ic_{\phi}$. In this paper we are interested in gradient estimates
of parabolic and elliptic types and their consequences for positive solutions
to a class of nonlinear heat-type equations on $M$ with compact boundary
$\partial M$. Specifically, our attention is on the positive smooth solutions
$u=u(x,t)$ with $x\in M$ and $t\in(-\infty,+\infty)$ satisfying the nonlinear
parabolic partial differential equation

$\displaystyle\begin{cases}\displaystyle\frac{\partial u}{\partial
t}&=\mathscr{L}_{\phi}u+H(u)\\\ \displaystyle u|_{\partial
M}&=\text{const}.\end{cases}$ (1.1)

Here $H=H(s)$ is a sufficiently smooth nonlinear function for $s>0$. Our
interest is motivated by the classical gradient estimates pioneered by [39,
45] which have been of great importance in the analysis of partial
differential equations and geometric analysis. The antecedence of gradient
estimates and their accompanied consequences in the field of geometric
analysis and partial differential equations are well studied, and they can not
be underestimated. Gradient estimates have served as a basic tool in the study
of linear and nonlinear PDEs yielding several applications ranging from
Harnack estimates to Liouville theorems, Heat kernel estimates to eigenvalue
bounds, existence, compactness and regularity of solutions, and so on. See [1,
2, 3, 6, 8, 11, 27, 43, 53, 61, 66, 68] and the references cited therein for
some related theories and applications.

Specifically, Li and Yau [45] established local and global version of gradient
type Harnack inequality for positive solutions to the heat equation

$\displaystyle\frac{\partial u}{\partial t}=\Delta u$ (1.2)

on complete Riemannian manifolds with or without boundary. Their inequality
was used to derive bounds for the fundamental solution of (1.2) under the
nonnegativity assumption on the Ricci curvature tensor. Hamilton in [39]
proved another version of Li-Yau estimate which is time independent for the
heat equation (1.2) on compact Riemannian manifold. Hamilton’s result allows
to compare temperatures of two different points at the same time. Inspired by
Hamilton’s result, Souplet and Zhang [61] got different version of gradient
estimate which is also time independent yielding a new Liouville-type theorems
of Cheng-Yau type [28] for the heat equation on non-compact Riemannian
manifolds. Souplet-Zhang’s result allows the comparison of temperature
distribution simultaneously, without any lag in time, even for noncompact
manifolds. Since the appearance of these works by Li and Yau [45], Hamilton
[39] and Souplet and Zhang [61] several results on generalizing, extending and
improving classical gradient estimates with several applications have appeared
in literature., including but not limited to those ones cited above. See also
[4, 7, 18, 19, 22, 23, 28, 29, 30, 33, 37, 40, 44, 51, 55, 56, 61, 67, 69,
70]. The gradient estimates discussed in this paper are broadly categorized
into two, namely, elliptic type and parabolic type. The elliptic type is also
known as Hamilton-Souplet-Zhang type, they only involve the solution and its
gradient. The parabolic type which involve the time derivative of the solution
in addition is known as Li-Yau type.

The study of evolution equations and their steady states involving the
weighted Laplacian on manifolds on its own has been of tremendous usefulness
in recent research development due to their profound intimacy and connection
with differential geometry, probability theory, quantum theory and statistical
mechanics to list a few. An important aspect of this research line is the
linkage between the diffusion operator $\mathscr{L}_{\phi}$ and the geometry
of the underlying manifold in the form of generalized weighted Ricci curvature
tensor, popularly known as Bakry-Émery-Ricci tensor $\mathscr{R}ic_{\phi}^{m}$
in the literature [13, 49, 62]. This enables convenient and nice
generalizations of manifold results such as Li-Yau Harnack inequalities,
Laplacian comparison, Myers’ compactness, Bishop-Gromov’s volume comparison
and Cheeger-Gromoll’s splitting theorems, e.t.c. to smooth metric measure
spaces [12, 13, 14, 44, 46, 47, 48, 49, 59, 62, 64, 65]. The generalizations
are not usually trivial, as they may involve a careful consideration of
properties of the potential function and the choice of input data such that
special conditions would not be required on the potential function.

The nonlinear evolution equation (1.1) has attracted attention of analysts and
applied mathematicians owing to its numerous applications in geometry,
mathematical physics and several other areas of science. The simplest case
being when $H(u)\equiv 0$ in which case (1.1) is just the weighted heat
equation, $(\partial_{t}-\mathscr{L}_{\phi})u=0$, which is archetypical
diffusion processes. For instance it is linked with Markov processes, and the
fundamental solution of the weighted heat equation is actually the transition
probability density function of the diffusion processes associated with
$\mathscr{L}_{\phi}$ which can be constructed using Itô calculus via certain
stochastic differential equation [13, 44, 48]. When we have $H(u)=au(1-u)$ or
$H(u)=au(1-u^{2})$ for $a>0$ and $0<u\leq 1$, (1.1) is known in literature as
Fisher-KPP or Allen-Cahn equation, both of which arise in biological and
chemical dynamics [10, 9, 16, 25, 35, 41]. When $H(u)=au\log u$ (1.1) is
closely related to the gradient Ricci solitons and weighted Log-Sobolev
inequalities on smooth metric measure spaces. The stationary form of (1.1) is
related to weighted Yamabe-type equation for $H(u)=\lambda u+\mu u^{p}$ which
is applicable in conformal geometry, and also Einstein Scalar field equation
for $H(u)=au+bu^{\alpha}+cu^{\beta}$ which is widely applicable in Einstein
field theory of general relativity [20, 21, 33]. In general, the nonlinear
parabolic equation (1.1) is the so-called weighted reaction-diffusion equation
which can be found in many mathematical models in physics, biology, chemistry,
engineering in which case the term $\mathscr{L}_{\phi}$ accounts for diffusion
while the term $H(u)$ is the reaction term.

Motivated by the above works, most especially, Li-Yau [45], Hamilton [39],
Souplet-Zhang [61], Li [43], Fu and Wu [36] and Abolarinwa, Ali and Mofarreh
[4], we prove local and global elliptic-type and parabolic-type gradient
estimates for positive solutions to generalized nonlinear parabolic and
elliptic partial differential equations on smooth metric measure space with
boundary satisfying some lower bound assumptions on $m$-Bakry-Émery Ricci
tensor and weighted mean curvature of the boundary. Applications of these
estimates are made towards deriving the Harnack inequalities and Liouville-
type results.

This introductory section is set to provide an overview of the theme of and
motivation for this study. Previous studies and background information on the
topic are highlighted. Meanwhile, the rest of this paper is structured as
follows:

  * •

Section 2: Preliminaries and statement of main results: The purpose of this
section is in two folds. First, the preliminaries subsection (Subsection 2.1)
presents basic concepts and fundamental results of smooth metric measure
spaces with boundary which are relevant to this study, as well as fixing the
required notation. More concretely, descriptions of weighted Laplacian, Bakry-
Émery Ricci tensor, weighted Bochner formula, weighted Reilly-type formula,
mean curvature of the boundary, Laplacian comparison result on smooth metric
measure space with boundary are made. Secondly, the statement of main results
subsection (Subsection 2.2) states the three main results on gradient
estimates. The first two results (Theorem 2.3 and Theorem 2.4) are elliptic in
nature and are called Hamilton-Souplet-Zhang type gradient estimates, while
the third result (Theorem 2.5) is on Li-Yau gradient estimates.

  * •

Section 3: Hamilton-Souplet-Zhang type gradient estimates I: Theorem 2.3 is
proved in this section. Some fundamental lemmas needed in the proof are stated
and their detailed proofs are given.

  * •

Section 4: Hamilton-Souplet-Zhang type gradient estimates II: Here in this
section we prove Theorem 2.4. We also state and prove some fundamental lemmas
needed in the proof of Theorem 2.4. The approach adopted in this section is
similar to that of Section 3, but with a different auxilliary function.

  * •

Section 5: Li-Yau type gradient estimates: Theorem 2.5 is proved in this
section. The approach introduced by Li [43] is adopted by choosing a cut-off
function different from the one applied in Sections 3 and 4.

  * •

Section 6: Applications of Li-Yau type gradient estimates: This section
presents some applications of Theorem 2.5 to deriving global gradient
estimates and Li-Yau (parabolic) Harnack inequalities for positive solutions
of (1.1) as presented in Theorem 6.2 and Theorem 6.3. Li-Yau type gradient
estimates for weighted nonlinear elliptic equations on weighted manifold with
boundary and Liouville-type theorems are derived as applications of the
parabolic gradient etimstes. These results are consequently applied to special
cases in the form of logarithmic and power-like nonlinearities (Subsection
6.4).

  * •

Section 7: Applications of elliptic gradient estimates: Similar to Section 6,
this final section presents some applications of Hamilton-Souplet-Zhang
gradient estimates. The applications yield global gradient estimates and
Liouville-type results, among others. These results are also applied to
special cases in the form of logarithmic and power-like nonlinearities.

##  2\. Preliminaries and statement of main results

###  2.1. Preliminaries

As stated earlier we shall consider several gradient estimates for positive
solutions of (1.1) on smooth metric measure spaces with compact boundary. Here
in this section we want to give some preliminaries and notation as a way of
formal introduction of a weighted manifold with compact boundary. Let $(M,g)$
be an $n$-dimensional complete Riemannian manifold equipped with metric $g$
and its induced volume form $dv$. Let $\phi$ be a real valued
$\mathscr{C}^{\infty}$-function on $M$ then the triple $(M,g,e^{-\phi}dv)$
endowed with weighted volume form $e^{-\phi}dv$ is called a smooth metric
measure space (also called weighted manifold or manifold with density). The
concept of smooth metric measure spaces was introduced by Bakry and Émery [12]
in the mid 80’s as a natural extension of smooth manifolds and has since been
applied in several contexts such as functional inequalities, optimal
transport, Ricci solitons to mention but a few, see [12, 13, 50, 54, 62].

####  2.1.1. Weighted Laplacian and weighted Ricci tensor

The generalized Laplacian and generalized Ricci curvature tensor are defined
on a weighted manifold respectively by $\phi$-Laplacian (also called Weighted
Laplacian or Witten Laplacian) and Bakry-Émery Ricci curvature tensor
$\mathscr{R}ic_{\phi}$. For a $\mathscr{C}^{2}$ function $u$, the weighted
Laplacian $\mathscr{L}_{\phi}$, which is a symmetric diffusion operator with
respect to the invariant volume form $e^{-\phi}dv$, is given by

$\mathscr{L}_{\phi}u:=e^{\phi}\text{div}(e^{-\phi}\nabla u)=\Delta
u-\langle\nabla\phi,\nabla u\rangle,$

while the weighted $m$-Bakry-Émery Ricci curvature tensor
$\mathscr{R}ic_{\phi}^{m}$ is given by

$\displaystyle\mathscr{R}ic_{\phi}^{m}$
$\displaystyle=\mathscr{R}ic+\text{Hess}\phi-\frac{\nabla\phi\otimes\nabla\phi}{m-n},\
\ \ m\in[n,\infty)$ (2.1)

Here,

$\text{div}(X)=\frac{1}{\sqrt{|g|}}\sum_{i=1}^{n}\frac{\partial}{\partial
x^{i}}(\sqrt{|g|}X^{i}),\ \ \ \ (\nabla
u)^{i}=\sum_{i=1}^{n}g^{ij}\frac{\partial u}{\partial x^{j}}$

and

$\Delta u=\text{div}(\nabla
u)=\frac{1}{\sqrt{|g|}}\sum_{i,j=1}^{n}\frac{\partial}{\partial
x^{i}}\left(\sqrt{|g|}g^{ij}\frac{\partial u}{\partial x^{j}}\right),$

are respectively the usual divergence of a smooth vector field $X$, gradient
and Laplace-Beltrami operators on $(M,g)$ (in local coordinates), where
$|g|=\det(g_{ij})$ and $[g^{ij}]=[g_{ij}]^{-1}$ denote the determinant and the
inverse of the component matrix of the metric tensor $g$. While
$\langle\cdot,\cdot\rangle$, $\text{Hess}\phi=\nabla^{2}\phi$ and
$\mathscr{R}ic$ are respectively the inner product with respect to $g$,
Hessian of function $\phi$ and the Ricci curvature tensor of $M$. The case
$m=n$ is admissible in formula (2.1) only when $\phi$ is a constant function.
The Bakry-Émery Ricci tensor is interpreted as the limit of
$\mathscr{R}ic_{\phi}^{m}$ as $m\to\infty$ and one simply writes

$\lim_{m\to\infty}\mathscr{R}ic_{\phi}^{m}=\mathscr{R}ic+\text{Hess}\phi=:\mathscr{R}ic_{\phi}.$

####  2.1.2. Weighted Bochner formula

It is important to note that $\phi$-Laplacian and the Bakry-Émery Ricci tensor
are connected by the weighted Bochner formula

$\displaystyle\mathscr{L}_{\phi}(|\nabla v|^{2})=2\|\text{Hess}\
v\|_{HS}^{2}+2\langle\nabla
v,\nabla\mathscr{L}_{\phi}v\rangle+2\mathscr{R}ic_{\phi}(\nabla v,\nabla v),$
(2.2)

where $v$ is $\mathscr{C}^{\infty}$ and $\|\cdot\|_{HS}$ is Hilbert Schmidt
norm with respect to metric $g$. Applying the elementary inequality of the
form $(a+b)^{2}\geq\frac{a^{2}}{1+s}-\frac{b^{2}}{s},\ s>0$ and a Newton
inequality in the form $\|\text{Hess}\ v\|^{2}\geq\frac{1}{n}(\Delta v)^{2}$,
we clearly have

$\displaystyle\|\text{Hess}\ v\|_{HS}^{2}+\mathscr{R}ic_{\phi}(\nabla v,\nabla
v)$ $\displaystyle\geq\frac{1}{n}(\mathscr{L}_{\phi}v+\langle\nabla\phi,\nabla
v\rangle)^{2}+\mathscr{R}ic_{\phi}(\nabla v,\nabla v)$
$\displaystyle\geq\frac{1}{n}\left(\frac{n(\mathscr{L}_{\phi}v)^{2}}{m}-\frac{n\langle\nabla\phi,\nabla
v\rangle^{2}}{m-n}\right)+\mathscr{R}ic_{\phi}(\nabla v,\nabla v)$
$\displaystyle=\frac{1}{m}(\mathscr{L}_{\phi}v)^{2}+\mathscr{R}ic^{m}_{\phi}(\nabla
v,\nabla v).$ (2.3)

Then (2.2) yields a very important inequality that would be applied under the
name weighted Bochner inequality

$\displaystyle\mathscr{L}_{\phi}(|\nabla
v|^{2})\geq\frac{2}{m}(\mathscr{L}_{\phi}v)^{2}+2\langle\nabla
v,\nabla\mathscr{L}_{\phi}v\rangle+2\mathscr{R}ic^{m}_{\phi}(\nabla v,\nabla
v).$ (2.4)

The identity (2.4) is equivalent to the curvature dimension condition
$CD(\mathsf{k},m)$ with respect to the operator $\mathscr{L}_{\phi}$ under the
lower bound assumption $\mathscr{R}ic^{m}_{\phi}\geq\mathsf{k}g$,
$\mathsf{k}\in\mathbb{R}$, $m\in[n,\infty]$ embedded in the next given
inequality

$\displaystyle\frac{1}{2}\mathscr{L}_{\phi}(|\nabla v|^{2})-\langle\nabla
v,\nabla\mathscr{L}_{\phi}v\rangle\geq\frac{1}{m}(\mathscr{L}_{\phi}v)^{2}+\mathsf{k}|\nabla
v|^{2}.$ (2.5)

Obviously, the condition $\mathscr{R}ic_{\phi}\geq\mathsf{k}g$ implies
$CD(\mathsf{k},\infty)$ and one then sees that (2.5) implies

$\displaystyle\frac{1}{2}\mathscr{L}_{\phi}(|\nabla v|^{2})-\langle\nabla
v,\nabla\mathscr{L}_{\phi}v\rangle\geq\mathsf{k}|\nabla v|^{2}$ (2.6)

but (2.6) does not imply (2.5). Thus, the lower bound condition on
$\mathscr{R}ic_{\phi}^{m}$ implies corresponding condition on
$\mathscr{R}ic_{\phi}$ but not the other way round. For more details on
Bochner formula and curvature dimension condition see [12, 13, 44, 65]. We
note that the lower bound assumption is imposed on the generalized weighted
Bakry-Émery Ricci tensor $\mathscr{R}ic_{\phi}^{m}$, $m\in[n,\infty)$ in all
the reults of this paper.

The relation $\mathscr{R}ic_{\phi}=\kappa g$ for some constant $\kappa$ is
just the gradient Ricci solitons equation which is fundamental in the
singularity analysis of the Hamilton-Perelman Ricci flow [38, 54] (interested
readers can find [17] for detail exposition on Ricci solitons), whilst the
relation $\mathscr{R}ic_{\phi}^{m}=\kappa g$ corresponds to the quasi-Einstein
equation which consists gradient Ricci solitons for $m=\infty$, and Einstein
metrics when $\phi$ is constant. It also relates to warped product Einstein
metrics for $m$ being a positive integer. Quasi-Einstein metrics have been
studied by several authors see [20, 21] for examples.

####  2.1.3. Mean curvature of the boundary

Let $\nu$ be the outward pointing unit vector normal to $\partial M$. The
weighted mean curvature (also called $\phi$-mean curvature) of the boundary of
$M$ denoted by $\mathscr{H}_{\phi}$ is defined by

$\mathscr{H}_{\phi}:=\mathscr{H}-\langle\nabla\phi,\nu\rangle,$

where $\mathscr{H}$ is the mean curvature of the boundary of $(M,g)$ with
respect to $\nu$. If $\nu$ is a unit normal pointing inward, the weighted mean
curvature will be
$\mathscr{H}_{\phi}:=\mathscr{H}+\langle\nabla\phi,\nu\rangle$. Let
$\vartheta_{\nu}$ be the normal derivative of a smooth function $\vartheta$ on
$\partial M$ and $\rm{II}_{\phi}$ be the second fundamental form of $\partial
M$ with respect to $\nu$. The following identity which will be henceforth
referred to as Reilly-type formula holds for all smooth function $\vartheta$
on ${M}$ (see [36, Proposition 2.3] or [31, 34])

$\displaystyle(|\nabla\vartheta|^{2})_{\nu}$
$\displaystyle=2\vartheta_{\nu}\left[\mathscr{L}_{\phi}\vartheta-\mathscr{L}_{\partial
M,f}(\vartheta|_{\partial
M})-\mathscr{H}_{\phi}\vartheta_{\nu}\right]+2\langle\nabla_{\partial
M}(\vartheta|_{\partial M}),\nabla_{\partial
M}(\vartheta_{\nu})\rangle|_{g_{\partial M}}$ $\displaystyle\hskip
14.22636pt-2\rm{II}_{\phi}\langle\nabla_{\partial M}(\vartheta|_{\partial
M}),\nabla_{\partial M}(\vartheta|_{\partial M})\rangle|_{g_{\partial M}},$
(2.7)

where $\mathscr{L}_{\partial M,\phi}:=\Delta_{\partial
M}-\langle\nabla_{\partial M}\phi\nabla_{\partial M}\cdot\rangle$ and
$\nabla_{\partial M}$ are respectively the $\phi$-Laplacian and covariant
derivative with respect to the induced metric $g_{\partial M}$ on $\partial
M$. The formula (2.1.3) can be derived from the proof of the weighted Reilly
formula in [52] (see also [26, Appendix 4] while its classical version has
been proved in [58]. One can also observe that (2.1.3) does not involve Ricci
curvature tensor which is due to the fact that weighted Ricci tensor bound
concerns only the interior. Note that only the mean curvature bound is imposed
on the boundary.

####  2.1.4. Laplacian comparison on weighted manifold with boundary

Let

$\varrho(x):=\varrho_{\partial M}(x)=d(x,\partial M)$

be the distance function from the boundary $\partial M$ to any point $x$ in
$M$, we denote by

$B_{R}(\partial M):=\\{x:\ \varrho_{\partial M}(x)<R\\}$

for $R>0$ the $R$-neigbourhood of $\partial M$. Based on the result in [59]
one may suppose that $\varrho$ is smooth away from the cut locus for the
boundary, $\text{Cut}(\partial M)$. By the results in [64] and [59] (see also
[31, Theorem 2.1] and [36, Theorem 2.1]) we have weighted Laplacian comparison
theorem for the distance function $\varrho$ on $M$ under some assumptions.

######  Theorem 2.1.

([59] Let $M=(M,g,e^{-\phi}dv)$ be an $n$-dimensional complete smooth metric
measure space with compact boundary $\partial M$. Suppose the weighted Ricci
tensor of $M$ and the weighted mean curvature of $\partial M$ satisfy
$\mathscr{R}ic_{\phi}\geq-(n-1)\kappa$ and $\mathscr{H}_{\phi}\geq-\ell$ for
some constants $\kappa\geq 0$ and $\ell\in\mathbb{R}$. Then

$\displaystyle\mathscr{L}_{\phi}\varrho_{\partial M}(x)\leq(n-1)\kappa R+\ell$
(2.8)

for all $x\in B_{R}(\partial M)$ outside the $\rm{Cut}(\partial M)$.

The above comparison theorem for the generalized Laplacian in smooth metric
measure spaces with boundary was proved in [59]. The authors in [64] had
earlier proved a similar result on weighted manifold with boundary under some
assumption. This comparison theorem holds for all $R>0$ in contrast to Wei-
Wylie comparison theorem [65] on weighted manifold with boundary.

####  2.1.5. Cut-off functions

Some space-time localization argument will be required so as to prove the
elliptic type gradient estimates. As a routine we do this through the
construction of suitable cut-off function as in the next lemma based on the
idea introduced by Souplet and Zhang [61] for manifold without boundary. This
is the technique used by Li and Yau [45], see also [2, 3]. In the setting of
manifolds with boundary we refer to [31],[36] and [42] for similar
applications.

Define a set called parabolic (space-time) cylinder

$\mathcal{Q}_{R,T}:=B_{R}(\partial
M)\times[T_{0}-T,T_{0}]\subset\overline{M}\times(-\infty,\infty)$

for $R\geq 2$ and $T>0$. For a fixed point in $M$ at time $t$, we set

$\psi(x,t)=\bar{\psi}(\varrho_{\partial M}(x),t)$

to be a cut-off function depending on space and time. The existence and other
useful properties of $\bar{\psi}(\varrho,t)$ in this context are stated in the
following lemma.

######  Lemma 2.2.

Let $M=(M,g,e^{-\phi}dv)$ be an $n$-dimensional complete smooth metric measure
space with compact boundary $\partial M$. There exists a smooth non-increasing
cut-off-function $\psi=\psi(\rho_{\partial M},t)$ supported in the closure of
$\mathcal{Q}_{R,T}(\partial M)$, $T,R>0$, such that for

  1. (1)

$\psi(x,t)=\bar{\psi}(\varrho_{\partial M}(x),t)$ with
$0\leq\bar{\psi}(\varrho_{\partial M}(x),t)\leq 1$ in
$[0,R]\times[T_{0}-T,T_{0}]$ and $\bar{\psi}$ is supported in a subset of
$[0,R]\times[T_{0}-T,T_{0}]$.

  2. (2)

$\bar{\psi}(\varrho_{\partial M}(x),t)=1$ and
$\displaystyle\frac{\partial\bar{\psi}(\varrho,t)}{\partial\varrho}=0$ in
$[0,R/2]\times[T_{0}-T,T_{0}]$.

  3. (3)

$\displaystyle\left|\frac{\partial\bar{\psi}}{\partial\varrho}\right|\leq\frac{C_{a}\bar{\psi}^{a}}{R}$
and
$\displaystyle\left|\frac{\partial^{2}\bar{\psi}}{\partial\varrho^{2}}\right|\leq\frac{C_{a}\bar{\psi}^{a}}{R^{2}}$,
where $0<a<1$ and $C_{a}>0$ is a constant depending on $a$, hold on
$[0,\infty)\times[T_{0}-T,T_{0}]$.

  4. (4)

The estimate $\displaystyle\left|\frac{\partial\bar{\psi}}{\partial
t}\right|\leq\frac{C\bar{\psi}^{1/2}}{T}$ holds on
$[0,\infty)\times[T_{0}-T,T_{0}]$ for some constant $C>0$ independent of $R$
and $T$.

####  2.1.6. Some elementary inequalities

Some important elementary inequalities will be applied in estimating certain
bounds during the proof of our results, namely Cauchy-Schwarz, Young’s and
Newton inequalities. For clarity we state them in a simple form, though they
will be applied with little modification. Cauchy-Schwarz inequality says that
the product of any two real numbers is not bigger than the product of their
absolute value, that is, if $a,b\in\mathbb{R}$, it holds that $ab\leq|a||b|$,
where $|\cdot|$ is the absolute value. Young inequality states that for any
nonnegative real numbers $A,B\geq 0$ and real numbers $r,s>1$ such that
$1/s+1/r=1$, it holds that

$AB\leq(1/s)A^{s}+(1/r)B^{r},$

with equality if $A^{s}=B^{r}$. Combining Cauchy-Schwarz and Young’s
inequality therefore we have for product of any two real numbers $a$ and $b$

$ab\leq|a||b|\leq(1/s)|a|^{s}+(1/r)|b|^{r}.$

A Newton inequality (as applied in this paper) states that

$\displaystyle\|\text{Hess}(f)\|^{2}\geq(1/n)(\Delta f)^{2}.$ (2.9)

This is derived as a consequence of the generalized AM-GM inequality
$\|\mathbb{A}\|^{2}\geq(1/n)(\text{tr}\mathbb{A})^{2}$, where $\mathbb{A}$ is
an $n\times n$ real symmetric matrix. For instance, take the $n$-dimensional
vector $\mathbf{A}=(A_{ii}),i=1,\cdots n,$ and apply Cauchy-Schwarz inequality
we can derive

$\left(\sum_{i=1}^{n}A_{ii}\right)^{2}\leq n\sum_{i=1}^{n}(A_{ii})^{2}\leq
n\sum_{i,j=1}^{n}(A_{ij})^{2}.$

###  2.2. Statement of main results

In the next we state the main results of this paper while their detailed
proofs and applications are postponed till later sections. First consider the
following notation that will appear in the results and their proofs. For time
dependent smooth function $u=u(x,t)$ we denote time (partial) derivative by
$u_{t}=\partial_{t}u=\frac{\partial u}{\partial t}$. For a real valued
quantity $[h]$ we denote its positive and negative parts by
$[h]_{+}:=\max\\{0,h\\}$ and $[h]_{-}:=\min\\{0,h\\}$, respectively. Ordinary
derivative is denoted by ′, that is $H^{\prime}(u)=\frac{d}{du}H(u)$ and
$H^{\prime\prime}(u)=\frac{d^{2}}{du^{2}}H(u)$. Furthermore, we denote
$\widehat{H}(u)=H(u)/u$ and $H_{u}=H^{\prime}(u)-H(u)/u$. The notation $\log$
represents natural logarithm. The letters $C,C_{1},C_{2},\cdots,$ represent
universal positive constants which may depend only on $m$ and $n$, with
varying value from line to line.

####  2.2.1. Hamilton-Souplet-Zhang type gradient estimates I

######  Theorem 2.3.

Let $M=(M,g,e^{-\phi}dv)$ be an $n$-dimensional complete smooth metric measure
space with compact boundary satisfying the bounds

$\displaystyle\mathscr{R}ic_{\phi}^{m}\geq-(m-1)\kappa\ \ \text{and}\ \
\mathscr{H}_{\phi}\geq-\ell$

for some constants $\kappa,\ell\geq 0$. Let $u=u(x,t)$ be a smooth positive
solution to (1.1) (i.e. $(\partial_{t}-\mathscr{L}_{\phi})u=H(u)$ in
$\mathcal{Q}_{R,T}(\partial M$). Suppose further that $u$ satisfies the
Dirichlet boundary condition (i.e., $u(\cdot,t)|_{\partial M}$ is constant for
each time $t\in(-\infty,+\infty)$),

$\displaystyle u_{\nu}\geq 0\ \ \text{and}\ \ u_{t}\leq H(u)$

over $\partial M\times[T_{0}-T,T_{0}]$. Then there exists a constant $C>0$
depending only on $n,m$, such that for all
$(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial M)$

$\displaystyle\frac{|\nabla u|}{1+\sqrt{u}}$ $\displaystyle\leq
C\Bigg{\\{}\left(\frac{1}{R}+\frac{1}{\sqrt{T}}+\sqrt{L}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}(\partial
M)}\\{u\\}\Big{)}$ $\displaystyle\hskip
56.9055pt+\Big{(}\sup_{\mathcal{Q}_{R,T}(\partial
M)}\\{\sqrt{\mathbb{M}_{+}[u]}\\}\Big{)}\Big{(}\sup_{\mathcal{Q}_{R,T}(\partial
M)}\\{\sqrt{u}\\}\Big{)}\Bigg{\\}},$ (2.10)

where $L=(\kappa^{2}+\ell^{4})^{1/2}$,

$\mathbb{M}_{+}[u]=\left[\frac{u[(1/2+\sqrt{u})\widehat{H}(u)-(1+\sqrt{u})H^{\prime}(u)]}{(1+\sqrt{u})^{2}}\right]_{+}\
\ \text{and}\ \ \widehat{H}(u):=\frac{H(u)}{u}.$

####  2.2.2. Hamilton-Souplet-Zhang type gradient estimates II

######  Theorem 2.4.

Let $M=(M,g,e^{-\phi}dv)$ be an $n$-dimensional complete smooth metric measure
space with compact boundary satisfying the bounds

$\displaystyle\mathscr{R}ic_{\phi}^{m}\geq-(m-1)\kappa\ \ \text{and}\ \
\mathscr{H}_{\phi}\geq-\ell$

for some constants $\kappa,\ell\geq 0$. Let $u=u(x,t)$ be a smooth positive
solution to (1.1) in $\mathcal{Q}_{R,T}(\partial M)$. Suppose further that $u$
satisfies the Dirichlet boundary condition (i.e., $u(\cdot,t)|_{\partial M}$
is constant for each time $t\in(-\infty,+\infty)$),

$\displaystyle u_{\nu}\geq 0\ \ \text{and}\ \ u_{t}\leq H(u)$

over $\partial M\times[T_{0}-T,T_{0}]$. Then there exists a constant $C>0$
depending only on $n,m$, such that for all
$(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial M)$

$\displaystyle\frac{|\nabla u|}{\sqrt{u}}$ $\displaystyle\leq
C\Bigg{\\{}\frac{1}{R}+\frac{1}{\sqrt{T}}+\sqrt{L}+\Gamma^{H}_{+}[u]\Bigg{\\}}\Big{(}\sup_{\mathcal{Q}_{R,T}(\partial
M)}\\{\sqrt{u}\\}\Big{)},$ (2.11)

where $L=(\kappa^{2}+\ell^{4})^{1/2}$,

$\Gamma^{H}_{+}[u]=\sup_{\mathcal{Q}_{R,T}(\partial
M)}\left\\{\left[2H^{\prime}(u)-\widehat{H}\right]_{+}\right\\}^{1/2}\ \
\text{and}\ \ \widehat{H}(u):=\frac{H(u)}{u}.$

####  2.2.3. Li-Yau type gradient estimates

######  Theorem 2.5.

Let $M=(M,g,e^{-\phi}dv)$ be an $n$-dimensional complete smooth metric measure
space with compact boundary satisfying the bounds

$\displaystyle\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k\ \ \text{and}\ \
\mathscr{H}_{\phi}\geq-l$

for some constants $k,l\geq 0$. Let $u=u(x,t)$ be a smooth positive solution
to (1.1) in $\mathcal{Q}_{2R,T}(\partial M)$. Suppose further that $u$
satisfies the Dirichlet boundary condition (i.e., $u(\cdot,t)|_{\partial M}$
is constant for each time $t\in(-\infty,+\infty)$),

$\displaystyle u_{\nu}\geq 0\ \ \text{and}\ \ u_{t}\leq H(u)$

over $\partial M\times[T_{0}-T,T_{0}]$. Then have for all
$(x,t)\in\mathcal{Q}_{R,T}(\partial M)$, $t>0$

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}$
$\displaystyle+\delta\frac{H(u)}{u}-\delta\frac{u_{t}}{u}\leq\frac{m\delta^{2}}{2(1-\epsilon)}\Big{(}\frac{1}{t}+\frac{k}{\delta-1}+\widetilde{\Phi}+\mu^{H}\Big{)}+\frac{\delta}{p}\sqrt{\frac{m}{2(1-\epsilon)}}\gamma^{H}$
(2.12)

on $\partial M\times[T_{0}-T,T_{0}]$. Here $\epsilon\in(0,1)$, $\delta>1$ and
$p>0$ such that

$\frac{2(1-\epsilon)}{m\delta
p}\geq\frac{1}{\epsilon}-1+\frac{(1-\delta)^{2}}{8}$

and

$\widetilde{\Phi}=\frac{1}{R^{2}}\Big{[}\Big{(}\frac{m\delta^{2}}{4(1-\epsilon)(\delta
p+\delta-1)}\Big{)}C_{1}^{2}+C_{2}+C_{1}[(m-1)kR+l]R\Big{]}\Phi\Big{]}$

for some constants positive $C_{1}$ and $C_{2}$. Furthermore,

$\mu^{H}:=\sup_{\mathcal{Q}_{2R,T}}\\{[H_{u}]_{+}\\}\ \ \text{and}\ \
\gamma^{H}:=\sup_{\mathcal{Q}_{2R,T}}\\{[\Omega^{H}]_{+}\\},$

where $[H_{u}]_{+}=[H^{\prime}(u)-H(u)/u]_{+}$ and
$[\Omega^{H}]_{+}=[H^{\prime}(u)-H(u)/u-(\alpha/p^{2})uH^{\prime\prime}(u)]_{+}$.

In the above results the lower bound on the weighted Bakry-Émery Ricci tensor,
$\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k$, concerns the infimum of
$\mathscr{R}ic_{\phi}^{m}$ only in the interior of $M$, whilst the lower bound
on the weighted mean curvature of the boundary implies some weakconvexity
condition on the boundary $\partial M$. In [45] Li and Yau have considered
gradient estimates on linear heat equation with Neumann boundary condition on
manifold whose boundary is convex in the sense that the second fundamental
form of the boundary is nonnegative. (See also Wang [63], Chen [24] and Ramos
Olivé [57]). In some recent works Dung, Dung and Wu [31] studied Souplet-Zhang
type gradient estimates on weighted heat equation with Dirichlet boundary
condition on a smooth metric space with compact boundary satisfying
$\mathscr{H}_{\phi}\geq-l$ based on the idea of Kunikawa and Sakurai [42]. Fu
and Wu [36] extended this result to nonlinear parabolic equation which is
somewhat a special case of ours with $H(u)=au\log u$, $a\in\mathbb{R}$. In
[4], we established Souplet-Zhang type gradient estimates on a generalized
nonlinear parabolic equation smooth metric measure spaces with compact
boundary. A generalized Laplacian comparison result (Theorem 2.1) plays a key
role in the proofs of all these results. The Sakurai generalized Laplacian
comparison result is quite different from Wei-Wylie comparison [65] without
boundary. Fu and Wu [36] remarked that Wei-Wylie comparison theorem requires
the radius of geodesic ball $R\geq R_{0}$ for some constant $R_{0}>0$.

##  3\. Hamilton-Souplet-Zhang type gradient estimates I

The proof of the following theorem is presented in this section.

######  Theorem 3.1.

Let $M=(M,g,e^{-\phi}dv)$ be an $n$-dimensional complete smooth metric measure
space with compact boundary satisfying the bounds

$\displaystyle\mathscr{R}ic_{\phi}^{m}\geq-(m-1)\kappa\ \ \text{and}\ \
\mathscr{H}_{\phi}\geq-\ell$

for some constants $\kappa,\ell\geq 0$. Let $u=u(x,t)$ be a smooth positive
solution to (1.1) (i.e. $(\partial_{t}-\mathscr{L}_{\phi})u=H(u)$ in
$\mathcal{Q}_{R,T}(\partial M$). Suppose further that $u$ satisfies the
Dirichlet boundary condition (i.e., $u(\cdot,t)|_{\partial M}$ is constant for
each time $t\in(-\infty,+\infty)$),

$\displaystyle u_{\nu}\geq 0\ \ \text{and}\ \ u_{t}\leq H(u)$

over $\partial M\times[T_{0}-T,T_{0}]$. Then there exists a constant $C>0$
depending only on $n,m$, such that for all
$(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial M)$

$\displaystyle\frac{|\nabla u|}{1+\sqrt{u}}$ $\displaystyle\leq
C\Bigg{\\{}\left(\frac{1}{R}+\frac{1}{\sqrt{T}}+\sqrt{L}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}(\partial
M)}\\{u\\}\Big{)}$ $\displaystyle\hskip
56.9055pt+\Big{(}\sup_{\mathcal{Q}_{R,T}(\partial
M)}\\{\sqrt{\mathbb{M}_{+}[u]}\\}\Big{)}\Big{(}\sup_{\mathcal{Q}_{R,T}(\partial
M)}\\{\sqrt{u}\\}\Big{)}\Bigg{\\}},$ (3.1)

where $L=(\kappa^{2}+\ell^{4})^{1/2}$,

$\mathbb{M}_{+}[u]=\left[\frac{u[(1/2+\sqrt{u})\widehat{H}(u)-(1+\sqrt{u})H^{\prime}(u)]}{(1+\sqrt{u})^{2}}\right]_{+}\
\ \text{and}\ \ \widehat{H}(u):=\frac{H(u)}{u}.$

To prove the above theorem we need some fundamental lemmas which we state and
prove below.

######  Lemma 3.2.

Let $f(x,t)=u^{\beta}(x,t)$, $\beta\in(0,1)$, be a smooth function where
$u(x,t)$ is a positive smooth solution to (1.1). Then $f$ satisfies the
nonlinear equation

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})f=\beta^{-1}(\beta-1)|\nabla
f|^{2}/f-\beta f\widehat{H}(f),$ (3.2)

where $\widehat{H}(f)=H(f^{1/\beta})/f^{1/\beta}=H(u)/u.$

###### Proof.

By direct computation for $f=u^{\beta}$ we have $f_{t}=\beta
u^{\beta-1}u_{t}$, $\nabla f=\beta u^{\beta-1}\nabla u$, $|\nabla
f|^{2}/f^{2}=\beta^{2}|\nabla u|^{2}/u^{2}$ and $\Delta f=\beta
u^{\beta-1}\Delta u+\beta(\beta-1)u^{\beta-2}|\nabla u|^{2}.$ Therefore

$\displaystyle\mathscr{L}_{\phi}f$ $\displaystyle=\Delta f-\nabla\phi\nabla f$
$\displaystyle=\beta
u^{\beta-1}\mathscr{L}_{\phi}u+\beta(\beta-1)u^{\beta-2}|\nabla u|^{2}$
$\displaystyle=\beta u^{\beta-1}(u_{t}-H(u))+(\beta-1)u^{\beta-2}|\nabla
u|^{2}$ $\displaystyle=f_{t}-\beta
fH(f^{1/\beta})/f^{1/\beta}+\beta^{-1}(\beta-1)|\nabla f|^{2}/f$

and consequently we arrive at (3.2).

∎

######  Lemma 3.3.

Let $f(x,t)=u^{\beta}(x,t)$, $\beta\in(0,1)$, be a smooth function where
$u(x,t)$ is a positive smooth solution to (1.1). Set $w:=|\nabla
f|^{2}/(1+f)^{2}$, then $w$ satisfies

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w=$
$\displaystyle\frac{|\nabla^{2}f|^{2}}{(1+f)^{2}}+\frac{2\mathscr{R}ic_{\phi}(\nabla
f,\nabla f)}{(1+f)^{2}}-\frac{4\nabla f\nabla(|\nabla
f|^{2})}{(1+f)^{3}}+\frac{6|\nabla f|^{4}}{(1+f)^{4}}$
$\displaystyle-2\beta^{-1}(\beta-1)\left[\frac{|\nabla
f|^{4}}{(1+f)^{3}f}-\frac{\nabla f\nabla(|\nabla
f|^{2})}{(1+f)^{2}f}+\frac{|\nabla f|^{4}}{(1+f)^{2}f^{2}}\right]$
$\displaystyle+2\beta\left[\frac{|\nabla
f|^{2}f\widehat{H}(f)}{(1+f)^{3}}-\frac{\nabla
f\nabla(f\widehat{H}(f))}{(1+f)^{2}}\right],$ (3.3)

where $\widehat{H}(f)=H(f^{1/\beta})/f^{1/\beta}.$

###### Proof.

From the lemma hypothesis $w$ is given by $w=|\nabla f|^{2}/(1+f)^{2}$ with
$f=u^{\beta}$, $0<\beta<1$. Based on this we can directly compute
$\mathscr{L}_{\phi}w$ and $\partial_{t}w$. For this course, we first note that

$\displaystyle\partial_{t}w=\partial_{t}\left(|\nabla
f|^{2}/(1+f)^{2}\right)=\frac{2\nabla f\nabla f_{t}}{(1+f)^{2}}-\frac{2|\nabla
f|^{2}f_{t}}{(1+f)^{3}}.$ (3.4)

Similarly, we compute the gradient and Laplacian of $w$ as follows

$\displaystyle\nabla w$ $\displaystyle=\frac{\nabla|\nabla
f|^{2}}{(1+f)^{2}}-\frac{2|\nabla f|^{2}\nabla f}{(1+f)^{3}},$
$\displaystyle\Delta w$ $\displaystyle=\frac{\Delta|\nabla
f|^{2}}{(1+f)^{2}}-\frac{4\nabla|\nabla f|^{2}\nabla
f}{(1+f)^{3}}-\frac{2|\nabla f|^{2}\Delta f}{(1+f)^{3}}+\frac{6|\nabla
f|^{4}}{(1+f)^{4}}$

which then yields

$\displaystyle\mathscr{L}_{\phi}w$ $\displaystyle=\Delta w-\nabla\phi\nabla w$
$\displaystyle=\frac{\mathscr{L}_{\phi}(|\nabla
f|^{2})}{(1+f)^{2}}-\frac{4\nabla|\nabla f|^{2}\nabla
f}{(1+f)^{3}}-\frac{2|\nabla
f|^{2}\mathscr{L}_{\phi}f}{(1+f)^{3}}+\frac{6|\nabla f|^{4}}{(1+f)^{4}}.$
(3.5)

Now putting together the individual descriptions of $\partial_{t}w$ and
$\mathscr{L}_{\phi}w$ from (3.4) and (3) respectively, and taking into account
the relevant cancellations, we get

$\displaystyle(\mathscr{L}_{\phi}-\partial)w$
$\displaystyle=\frac{\mathscr{L}_{\phi}(|\nabla
f|^{2})}{(1+f)^{2}}-\frac{4\nabla|\nabla f|^{2}\nabla
f}{(1+f)^{3}}-\frac{2|\nabla
f|^{2}(\mathscr{L}_{\phi}f-f_{t})}{(1+f)^{3}}+\frac{6|\nabla
f|^{4}}{(1+f)^{4}}-\frac{2\nabla f\nabla f_{t}}{(1+f)^{2}}.$

Combining the last equation with the weighted Bochner formula (2.2), we arrive
at

$\displaystyle(\mathscr{L}_{\phi}-\partial)w=$
$\displaystyle\frac{2|\nabla^{2}f|^{2}}{(1+f)^{2}}+\frac{2\mathscr{R}ic_{\phi}(\nabla
f,\nabla f)}{(1+f)^{2}}+\frac{2\nabla f(\mathscr{L}_{\phi}f-\nabla
f_{t})}{(1+f)^{2}}$ $\displaystyle-\frac{2|\nabla
f|^{2}(\mathscr{L}_{\phi}f-f_{t})}{(1+f)^{3}}-\frac{4\nabla|\nabla
f|^{2}\nabla f}{(1+f)^{3}}+\frac{6|\nabla f|^{4}}{(1+f)^{4}}.$ (3.6)

Referring to (3.2) we compute directly

$\displaystyle\nabla f\nabla(\mathscr{L}_{\phi}f-f_{t})$ $\displaystyle=\nabla
f\nabla(\beta^{-1}(\beta-1)|\nabla f|^{2}/f-\beta f\widehat{H}(f))$
$\displaystyle=\beta^{-1}(\beta-1)\left[\nabla f\nabla(|\nabla
f|^{2})/f-|\nabla f|^{2}/f^{2}\right]-\beta\nabla f\nabla(f\widehat{H}(f)).$
(3.7)

Finally, plugging in (3.2) and (3) into (3), and rearranging terms gives the
desired conclusion.

∎

######  Lemma 3.4.

Under the assumptions of Lemma 3.3, suppose
$\mathscr{R}ic_{\phi}^{m}\geq-(m-1)\kappa$, $\kappa\geq 0$, then

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w$
$\displaystyle\geq-\frac{2[f-\beta^{-1}(\beta-1)(1+f)]}{1+f}\langle\nabla
f,\nabla w\rangle$ $\displaystyle\hskip
28.45274pt-2\beta^{-1}(\beta-1)\frac{(1+f)}{f}w^{2}-2(m-1)\kappa
w+2\mathbb{M}_{\beta}(f)w,$ (3.8)

where

$\mathbb{M}_{\beta}(f):=\left[\frac{(1-\beta+f)\widehat{H}(f)-(1+f)H^{\prime}(f)}{1+f}\right]\
\ \text{and}\ \ \widehat{H}(f)=H(f^{1/\beta})/f^{1/\beta}.$

###### Proof.

The proof is based on further simplification of (3.3) and application of
weighted $m$-Ricci tensor lower bound. First recall that
$\widehat{H}(f)=H(f^{1/\beta})/f^{1/\beta}$, so that

$\displaystyle\nabla(f\widehat{H}(f))$
$\displaystyle=\nabla(H(f^{1/\beta})f^{-1/\beta+1})$
$\displaystyle=(1/\beta)\nabla fH^{\prime}(f^{1/\beta})-(1/\beta-1)\nabla
fH(f^{1/\beta})/f^{1/\beta}$

and then

$\displaystyle 2\beta\frac{\nabla f\nabla(f\widehat{H}(f))}{(1+f)^{2}}$
$\displaystyle=2\frac{|\nabla
f|^{2}}{(1+f)^{2}}H^{\prime}(f^{1/\beta})+2(\beta-1)\frac{|\nabla
f|^{2}}{(1+f)^{2}}\widehat{H}(f^{1/\beta})$
$\displaystyle=2\left[H^{\prime}(f^{1/\beta})+(\beta-1)\widehat{H}(f^{1/\beta})\right]w.$

By this, it can be seen clearly that

$\displaystyle 2\beta\left[\frac{|\nabla
f|^{2}f\widehat{H}(f)}{(1+f)^{3}}-\frac{\nabla
f\nabla(f\widehat{H}(f))}{(1+f)^{2}}\right]$
$\displaystyle=2\left[\Big{(}\frac{\beta
f}{(1+f)}-(\beta-1)\Big{)}\widehat{H}(f^{1/\beta})-H^{\prime}(f^{1/\beta})\right]w$
$\displaystyle=2\mathbb{M}_{\beta}(f)w,$ (3.9)

where
$\displaystyle\mathbb{M}_{\beta}(f):=\left[\frac{(1-\beta+f)\widehat{H}(f^{1/\beta})-(1+f)H^{\prime}(f^{1/\beta})}{1+f}\right].$

Next considering the following identity (which can be verified by direct
derivation)

$\displaystyle\frac{4\nabla^{2}f|\nabla f|^{2}}{(1+f)^{3}}=\frac{2\nabla
f\nabla(|\nabla f|^{2})}{(1+f)^{3}}$

one can then simplify some other terms of (3.3) as follows

$\displaystyle\frac{2|\nabla^{2}f|^{2}}{(1+f)^{2}}-$
$\displaystyle\frac{4\nabla|\nabla f|^{2}\nabla f}{(1+f)^{3}}+\frac{6|\nabla
f|^{4}}{(1+f)^{4}}$
$\displaystyle=\frac{2}{(1+f)^{2}}\left|\nabla^{2}f-\frac{\nabla
f\otimes\nabla f}{1+f}\right|^{2}-\frac{2\nabla|\nabla f|^{2}\nabla
f}{(1+f)^{3}}+\frac{4|\nabla f|^{4}}{(1+f)^{4}}.$ (3.10)

Refering to the definition of weighted $m$-Ricci tensor given in the
introduction we write

$\displaystyle\frac{2\mathscr{R}ic_{\phi}(\nabla f,\nabla f)}{(1+f)^{2}}$
$\displaystyle=\frac{2[(m-n)\mathscr{R}ic_{\phi}^{m}(\nabla f,\nabla
f)+\nabla\phi\otimes\nabla\phi(\nabla f,\nabla f)}{(m-n)(1+f)^{2}}$
$\displaystyle\geq-2(m-1)\kappa w$ (3.11)

since $\mathscr{R}ic_{\phi}^{m}(\nabla f,\nabla f)\geq-(m-1)\kappa|\nabla
f|^{2}$ from the hypothesis and the observation that

$\frac{\nabla\phi\otimes\nabla\phi(\nabla f,\nabla
f)}{(m-n)(1+f)^{2}}=\frac{|\langle\nabla\phi,\nabla
f\rangle|^{2}}{(m-n)(1+f)^{2}}\geq 0$

for $m\geq n$ ($m=n$ only for $f$ constant). Putting (3)-(3) into (3.3) we
have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w\geq$
$\displaystyle-\frac{2\nabla f\nabla(|\nabla
f|^{2})}{(1+f)^{3}}+\frac{4|\nabla
f|^{4}}{(1+f)^{4}}-2\beta^{-1}(\beta-1)\Bigg{[}\frac{|\nabla
f|^{4}}{(1+f)^{3}f}$ $\displaystyle-\frac{\nabla f\nabla(|\nabla
f|^{2})}{(1+f)^{2}f}+\frac{|\nabla
f|^{4}}{(1+f)^{2}f^{2}}\Bigg{]}-2(m-1)\kappa w+2\mathbb{M}_{\beta}(f)w.$
(3.12)

To simplify further some terms of (3) we take into consideration the following
identity

$2\nabla f\nabla w=\frac{2\nabla f\nabla(|\nabla
f|^{2})}{(1+f)^{2}}-\frac{4|\nabla f|^{4}}{(1+f)^{3}}$

so that we compute the following identities

$\displaystyle-\frac{2\nabla f\nabla(|\nabla
f|^{2})}{(1+f)^{3}}+\frac{4|\nabla f|^{4}}{(1+f)^{4}}=-\frac{2}{1+f}\nabla
f\nabla w$ (3.13)

and

$\displaystyle 2\beta^{-1}(\beta-1)\left[\frac{\nabla f\nabla(|\nabla
f|^{2})}{(1+f)^{2}f}-\frac{|\nabla
f|^{4}}{(1+f)^{3}f}\right]=2\beta^{-1}(\beta-1)\frac{1}{f}\nabla f\nabla w.$
(3.14)

To this end, (3.13) and (3.14) show that the first four terms on the right
hand side of (3) are simplified to

$\displaystyle-\frac{2}{1+f}\nabla f\nabla
w+2\beta^{-1}(\beta-1)\frac{1}{f}\nabla f\nabla
w=-\frac{2[f-\beta^{-1}(\beta-1)(1+f)]}{1+f}\nabla f\nabla w.$ (3.15)

Replacing the first four terms on the right hand side of (3) by (3.15) gives
the required inequality which is (3.4).

∎

######  Remark 3.5.

Informed by the sign of the constant $\beta^{-1}(\beta-1)$ appearing in the
inequality (3.4) which is always negative in the range $0<\beta<1$, we can
normalize the constant as $\beta^{-1}(\beta-1)=1$. This gives a good
impression that $\beta$ can be chosen to be $\beta=1/2$ in the later analysis.

###  3.1. proof of Theorem 3.1

###### Proof.

Suppose $f=u^{\beta}$, $\beta\in(0,1)$ and $w=|\nabla\log(1+f)|^{2}$
satisfying the assumptions of Lemma 3.2. In the case $w$ is constant on
$\mathcal{Q}_{R/2,T}(\partial M)$ there remain nothing to prove. To this end
we assume that $w$ is non-constant in $\mathcal{Q}_{R/2,T}(\partial M)$. Let
there exists a smooth cut-off function $\psi$ verifying those properties
listed in Lemma 2.2. Clearly $\psi w$ is positive and attains its maximum at a
point in $\mathcal{Q}_{R/2,T}(\partial M)$, say $(x_{1},t_{1})$. Our argument
will henceforth split into two cases, according to whether
$x_{1}\not\in\partial M$ or $x_{1}\in\partial M$.

Case 1: If $x_{1}\not\in\partial M$, we may assume without loss of generality
that $x_{1}\not\in\partial M\cup\ \text{cut}(\partial M)$ (where
$\text{cut}(\partial M)$ is the cut locus of $\partial M$) by invoking the
well known’s Calabi’s trick (see [15, 45, 60]). Then at $(x_{1},t_{1})$ we
have

$\nabla(\psi w)=0,\ \ \mathscr{L}_{\phi}(\psi w)\leq 0\ \ \text{and}\ \
\partial_{t}(\psi w)\geq 0.$

To start with we first observe that an elementary calculation yields

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})(\psi w)$
$\displaystyle=\psi(\mathscr{L}_{\phi}-\partial_{t})w+2\nabla\psi\nabla
w+w(\mathscr{L}_{\phi}-\partial_{t})\psi.$ (3.16)

Combining Lemma 3.2 with (3.16) while denoting
$A:=2[(f-\beta^{-1}(\beta-1)(1+f))/(f(1+f))]\nabla f$ and
$B:=2\beta^{-1}(\beta-1)(1+f)/f^{2}$ for the sake of convenience and brevity,
we obtain

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})(\psi w)$
$\displaystyle\geq-A\psi\nabla w-B\psi w^{2}+2\mathbb{M}_{\beta}[f]\psi
w-2(m-1)k\psi w$ $\displaystyle\hskip 28.45274pt+2\nabla\psi\nabla
w+w(\mathscr{L}_{\phi}-\partial_{t})\psi.$ (3.17)

Proceeding forward, we note the following basic vector identities using
Leibniz rule:

$A\psi\nabla w=A\nabla(\psi w)-(A\nabla\psi)w\ \ \text{and}\ \
\nabla\psi\nabla w=(\nabla\psi/\psi)\nabla(\psi w)-(|\nabla\psi|^{2}/\psi)w.$

Thus upon substitution of these identities into (3.1) we have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})(\psi w)\geq$
$\displaystyle-A\nabla(\psi w)+(A\nabla\psi)w-B\psi
w^{2}+2\frac{\nabla\psi}{\psi}\nabla(\psi w)-2\frac{|\nabla\psi|^{2}}{\psi}w$
$\displaystyle+2\mathbb{M}_{\beta}[f]\psi w+2\nabla\psi\nabla
w+w[\mathscr{L}_{\phi}-\partial_{t}-2(m-1)k]\psi.$ (3.18)

Rearranging (3.1) and making use of the fact that localized function $\psi w$
attains its maximum at $(x_{1},t_{1})$, where we note that $\nabla(\psi w)=0$
and $(\mathscr{L}_{\phi}-\partial_{t})(\psi w)\leq 0$, and obtain

$\displaystyle-B\psi
w^{2}\leq(A\nabla\psi)w+2\frac{|\nabla\psi|^{2}}{\psi}w-2\mathbb{M}_{\beta}[f]\psi
w-w[\mathscr{L}_{\phi}-\partial_{t}-2(m-1)k]\psi.$ (3.19)

Since $B:=2\beta^{-1}(\beta-1)(1+f)/f^{2}$ and $0<\beta<1$, we note that
$\beta^{-1}(\beta-1)<0$, meaning that $B$ is negative and $-B>0$. So we
multiply through (3.19) by $(-B)^{-1}$, and we then have

$\displaystyle\psi w^{2}$
$\displaystyle\leq\left(\frac{A}{B}\nabla\psi\right)w-\beta(\beta-1)^{-1}\frac{|\nabla\psi|^{2}}{\psi}\frac{f^{2}}{1+f}w+\beta(\beta-1)^{-1}\mathbb{M}_{\beta}[f]\frac{f^{2}}{1+f}\psi
w$ $\displaystyle\hskip
28.45274pt+\beta(\beta-1)^{-1}\frac{|\nabla\psi|^{2}}{\psi}\frac{f^{2}}{2(1+f)}w(\mathscr{L}_{\phi}-\partial_{t}-2(m-1)k)\psi.$
(3.20)

The next step is to apply inequality (3.1) to derive estimates at $(x,t)$.
Here repeated applications of Cauchy-Schwarz inequality, Young’sinequality and
the properties of the cut-off function $\psi$ presented in Lemma 2.2 will give
appropriate bounds for each term on the right hand side of (3.1). Before we
proceed, note again that $u^{\beta}=f>0$, $0<\beta<1$ implies $0<f/(1+f)\leq
1$, $\beta^{-1}(\beta-1)<0,$ $\beta(\beta-1)^{-1}<0$ and
$f/(1+f)-\beta(\beta-1)^{-1}>0$.

By definition

$\frac{A}{B}=\beta(\beta-1)^{-1}\left[\frac{f-\beta^{-1}(\beta-1)(1+f)}{1+f}\right]\frac{f\nabla
f}{(1+f)}.$

Since the quantity in front of $(f\nabla f)/(1+f)$ in the last equality is
negative we estimate the first term on the right hand side of (3.1) as
follows:

$\displaystyle\left(\frac{A}{B}\nabla\psi\right)w$
$\displaystyle=\beta(\beta-1)^{-1}\left[\frac{f-\beta^{-1}(\beta-1)(1+f)}{1+f}\right]\frac{f\nabla
f}{(1+f)}\nabla\psi w$
$\displaystyle\leq\beta(1-\beta)^{-1}\left[\frac{f-\beta^{-1}(\beta-1)(1+f)}{1+f}\right]f\frac{|\nabla
f|}{(1+f)}|\nabla\psi|w$
$\displaystyle\leq(1-\beta)^{-1}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}|\nabla\psi|w^{3/2}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+C_{\beta}\left(\frac{|\nabla\psi|\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}}{\psi^{3/4}}\right)^{4}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+\frac{C_{\beta}}{R^{4}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4},$
(3.21)

where $\mathcal{Q}_{R,T}:=\mathcal{Q}_{R,T}(\partial M)$ and $C_{\beta}$ is a
constant which depends only on $\beta$, $n$ and $m$, the value of which may
vary from line to line as remarked before. (Similarly any $C_{\beta}$
appearing in the rest analysis depends only on $\beta$, $n$ and $m$ with
varying values).

Estimating the second term on the right hand side of (3.1) we proceed in a
likewise manner as follows

$\displaystyle-\beta(\beta-1)^{-1}\frac{|\nabla\psi|^{2}}{\psi}\frac{f^{2}}{1+f}w$
$\displaystyle=\beta(1-\beta)^{-1}\frac{|\nabla\psi|^{2}}{\psi}\frac{f^{2}}{1+f}w$
$\displaystyle\leq\beta(1-\beta)^{-1}\psi^{1/2}\frac{|\nabla\psi|^{2}}{\psi^{3/2}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{2}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+C_{\beta}\Big{(}\frac{|\nabla\psi|^{2}}{\psi^{3/2}}\Big{)}^{2}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+\frac{C_{\beta}}{R^{4}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}.$
(3.22)

For the third term on the right hand side of (3.1) we first write
$\bar{\mathbb{M}}_{\beta}[f]=f^{2}\mathbb{M}_{\beta}[f]/(1+f)$, ( where
$\mathbb{M}_{\beta}[f]$ is as defined in Lemma 3.4) and then estimate as
follows:

$\displaystyle\beta(\beta-1)^{-1}\mathbb{M}_{\beta}[f]\frac{f^{2}}{1+f}\psi w$
$\displaystyle=\beta(\beta-1)^{-1}\bar{\mathbb{M}}_{\beta}[f]\psi w$
$\displaystyle\leq\beta(1-\beta)^{-1}\bar{\mathbb{M}}_{\beta+}[f]\psi w$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+C_{\beta}\Big{(}\sup_{\mathcal{Q}_{R,T}}\bar{\mathbb{M}}_{\beta+}[f]\Big{)}^{2},$
(3.23)

where

$\bar{\mathbb{M}}_{\beta+}[f]:=\left[\frac{f^{2}[(1-\beta+f)\widehat{H}(f^{\frac{1}{\beta}})-(1+f)H^{\prime}(f^{\frac{1}{\beta}})]}{(1+f)^{2}}\right]_{+}.$

We break the fourth term to individual item and estimate each of them and
later sum up their estimates. For the term with time derivative of $\psi$:

$\displaystyle-\beta(\beta-1)^{-1}\frac{f^{2}}{2(1+f)}w\psi_{t}$
$\displaystyle=\beta(1-\beta)^{-1}\frac{f^{2}}{2(1+f)}w|\psi_{t}|$
$\displaystyle\leq\frac{1}{2}\beta(1-\beta)^{-1}\psi^{\frac{1}{2}}w\frac{|\psi_{t}|}{\psi^{\frac{1}{2}}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{2}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+C_{\beta}\frac{|\psi_{t}|^{2}}{\psi}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+\frac{C_{\beta}}{T^{2}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}.$
(3.24)

Secondly, the term with $(m-1)\kappa$ is estimated in a similar way:

$\displaystyle-\beta(\beta-1)^{-1}(m-1)\kappa\psi w\frac{f^{2}}{1+f}$
$\displaystyle=\beta(1-\beta)^{-1}(m-1)\kappa\psi w\frac{f^{2}}{1+f}$
$\displaystyle\leq\beta(1-\beta)^{-1}(m-1)\kappa(\psi^{\frac{1}{2}}w)\psi^{\frac{1}{2}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+C_{\beta}\kappa^{2}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}.$ (3.25)

Lastly, we compute the term involving the weighted Laplacian of $\psi$ as
follows. Here we apply the weighted Laplacian comparison
$\mathscr{L}_{\phi}\varrho_{\partial M}(x)\leq(n-1)\kappa R+\ell$,
$\kappa,\ell\geq 0$, for all $x\in B_{R}(\partial M)$ since
$\mathscr{R}ic_{\phi}^{m}\geq-(m-1)\kappa$ and $\mathscr{H}_{\phi}\geq-\ell$
(that is, the weighted Ricci tensor and the boundary mean curvature bounds
(see Theorem 2.1)):

$\displaystyle\frac{\beta f^{2}}{2(\beta-1)(1+f)}w\mathscr{L}_{\phi}\psi$
$\displaystyle=-\frac{\beta
f^{2}}{2(1-\beta)(1+f)}\left[\psi_{\varrho}\mathscr{L}_{\phi}\varrho+\psi_{\varrho\varrho}|\nabla\varrho|^{2}\right]w$
$\displaystyle\leq\frac{\beta
f^{2}}{2(1-\beta)(1+f)}\left[|\psi_{\varrho}|((n-1)\kappa
R+\ell)+\psi_{\varrho\varrho}|\nabla\varrho|^{2}\right]w$
$\displaystyle\leq\frac{\beta}{2(1-\beta)}\psi^{1/2}w\left[\frac{|\psi_{\varrho\varrho}|}{\psi^{1/2}}+\frac{|\psi_{\varrho}|}{\psi^{1/2}}\Big{(}(n-1)\kappa
R+\ell\Big{)}\right]\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{2}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+C_{\beta}\left[\frac{|\psi_{\varrho\varrho}|^{2}}{\psi}+\frac{|\psi_{\varrho}|^{2}}{\psi}(n^{2}\kappa^{2}R^{2}+l^{2})\right]\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}$
$\displaystyle\leq\frac{1}{8}\psi
w^{2}+C_{\beta}\left[\frac{1}{R^{4}}+\kappa^{2}+\ell^{4}\right]\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}.$
(3.26)

With bounds (3.1)–(3.1) at our disposal, we continue the analysis for
$x\not\in\partial M\cup\ \text{cut}(\partial M)$ by substituting the bounds
back into (3.1), simplifying further and rearranging terms to get the
following inequality

$\displaystyle\psi w^{2}$ $\displaystyle\leq\frac{3}{4}\psi
w+C_{\beta}\left\\{\Big{(}\frac{1}{R^{4}}+\frac{1}{T^{2}}+L^{2}\Big{)}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}+\Big{(}\sup_{\mathcal{Q}_{R,T}}\mathbb{M}_{\beta+}[f]\Big{)}^{2}\right\\},$

where $L=(\kappa^{2}+\ell^{4})^{1/2}$ at $(x_{1},t_{1})$. It therefore follows
by reverting to the maximal characterization of the point $(x_{1},t_{1})$ that
we have for all $(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial M)$

$\displaystyle\psi^{2}w^{2}(x,t)$
$\displaystyle\leq\psi^{2}w^{2}(x_{1},t_{1})\leq\psi w^{2}(x_{1},t_{1})$
$\displaystyle\leq
C_{\beta}\left\\{\frac{1}{R^{4}}\Big{(}1+\frac{R^{4}}{T^{2}}+L^{2}R^{4}\Big{)}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{4}+\Big{(}\sup_{\mathcal{Q}_{R,T}}\mathbb{M}_{\beta+}[f]\Big{)}^{2}\right\\}.$

Since $\psi(x,t)\equiv 1$ on $\mathcal{Q}_{R/2,T/2}(\partial M)$ and by
definition $w=|\nabla f|^{2}/(1+f)^{2}$ we obtain

$\displaystyle\frac{|\nabla f|}{1+f}\leq
C_{\beta}\left\\{\frac{1}{R}\Big{(}1+\frac{R}{\sqrt{T}}+\sqrt{L}R\Big{)}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}+\Big{(}\sup_{\mathcal{Q}_{R,T}}\mathbb{M}_{\beta+}[f]\Big{)}^{1/2}\right\\}.$

Considering the range of $\beta\in(0,1)$ and referring to the comments made in
Remark 3.5 above, we can choose $\beta=1/2$. Noting that $f=u^{\beta}$ with
$\beta=1/2$ yields $f=\sqrt{u}$ and $|\nabla f|=|\nabla u|/2\sqrt{u}$, we
finally arrive at

$\displaystyle\frac{|\nabla\sqrt{u}|}{1+\sqrt{u}}\leq
C\left\\{\frac{1}{R}\Big{(}1+\frac{R}{\sqrt{T}}+\sqrt{L}R\Big{)}\Big{(}\sup_{\mathcal{Q}_{R,T}}u\Big{)}+\Big{(}\sup_{\mathcal{Q}_{R,T}}\sqrt{\mathbb{M}_{+}[u]}\Big{)}\Big{(}\sup_{\mathcal{Q}_{R,T}}\sqrt{u}\Big{)}\right\\},$

which says the desired gradient estimate (2.2) holds in the interior for all
$(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial M)$. It remains to prove that the
gradient estimate is still valid on the boundary. We therefore consider the
case $x_{1}\in\partial M$.

Case 2: Assume $x_{1}\in\partial M$. Indeed, at the maximum point
$(x_{1},t_{1})$ of $(\psi w)$, we have $(\psi w)_{\nu}\geq 0$ and
consequently, $\psi_{\nu}w+\psi w_{\nu}\geq 0$, which in particular gives

$w_{\nu}\geq 0.$

Since $w=|\nabla\log(1+f)|^{2}$, and $\log(1+f)$ satisfies the Dirichlet
boundary condition, we use the Reilly-type formula (2.1.3) and get

$\displaystyle 0\leq w_{\nu}$ $\displaystyle=(|\nabla\log(1+f)|^{2})_{\nu}$
$\displaystyle=2(\log(1+f))_{\nu}\left[\mathscr{L}_{\phi}(\log(1+f))-(\log(1+f))_{\nu}\mathscr{H}_{\phi}\right].$
(3.27)

Notice also that $f$ satisfies the Dirichlet boundary condition since $u$
satisfies the Dirichlet boundary condition, and then $|\nabla f|=f_{v}=\beta
u^{\beta}u_{\nu}/u\geq 0$ since $u>0$ and $u_{v}\geq 0$ on $\partial M$. By
smoothness of the function $f$ we have

$\displaystyle\mathscr{L}_{\phi}(\log(1+f))$
$\displaystyle=\Delta(\log(1+f))-\nabla\phi\nabla(\log(1+f))$
$\displaystyle=-\frac{|\nabla f|^{2}}{(1+f)^{2}}+\frac{\Delta
f}{1+f}-\frac{1}{1+f}\nabla\phi\nabla f$ $\displaystyle=-\frac{|\nabla
f|^{2}}{(1+f)^{2}}+\frac{\mathscr{L}_{\phi}f}{1+f}.$ (3.28)

Setting $\beta=1/2$, we have from Lemma 3.2 that

$\displaystyle f_{t}=\mathscr{L}_{\phi}f+|\nabla f|^{2}/f+f\widehat{H}(f)/2.$
(3.29)

Thus

$\displaystyle\mathscr{L}_{\phi}(\log(1+f))=-\frac{(1+2f)}{f}(\log(1+f))_{\nu}^{2}+\frac{f_{t}-\frac{1}{2}f\widehat{H}(f)}{1+f}$
(3.30)

by substituting (3.29) into (3.1) and then simplifying the resulting
expression. Noting also that on $\partial M$,
$(\log(1+f))_{\nu}=f_{v}/(1+f)=|\nabla f|/(1+f)=\sqrt{w}$. It then follows
from (3.1) that

$\displaystyle 0\leq
2\sqrt{w}\left(-\frac{(1+2f)}{f}w+\frac{f_{t}-\frac{1}{2}f\widehat{H}(f)}{1+f}-\sqrt{w}\mathscr{H}_{\phi}\right)$

which upon rearranging and considering the assumption $u_{t}\leq H(u)$ over
$\partial M\times[T_{0}-T,T_{0}]$ yields $f_{t}-\frac{1}{2}f\widehat{H}(f)\leq
0$ and the inequality

$\displaystyle\frac{(1+2f)}{f}w^{3/2}+\mathscr{H}_{\phi}w\leq 0$ (3.31)

at $(x_{1},t_{1})$. Taking into account that $(1+2f)/f>1/f>0$, (3.31) is then
solved at $(x_{1},t_{1})$ to give

$w(x_{1},t_{1})=0\ \ \ \text{or}\ \ \
w^{1/2}(x_{1},t_{1})\leq\ell\cdot\sup_{\mathcal{Q}_{R,T}}f$

since $\mathscr{H}_{\phi}\geq-\ell$ by the theorem hypothesis. This thus means

$\psi w(x_{1},t_{1})=0\ \ \ \text{or}\ \ \ \psi
w(x_{1},t_{1})\leq\ell^{2}\cdot(\sup_{\mathcal{Q}_{R,T}}f)^{2}$

on $\mathcal{Q}_{R,T}(\partial M)$. Note that $\psi w=0$ indicates that $f$
(respectively $u$) is a constant from where the conclusion follows at once,
whereas the case $(\psi w)\leq\ell^{2}(\sup f)^{2}$ for all
$(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial M)$ together with the property
$\psi(x,t)\equiv 1$ implies

$\displaystyle\frac{|\nabla f|^{2}}{(1+f)^{2}}(x,t)=w(x,t)=\psi
w(x,t)\leq(\psi
W)(x_{1},t_{1})\leq\ell^{2}\cdot(\sup_{\mathcal{Q}_{R,T}}f)^{2}.$

Taking into consideration that $f=\sqrt{u}$ and $|\nabla
f|/(1+f)=|\nabla\sqrt{u}|/(1+\sqrt{u})$, the last inequality implies the
conclusion.

∎

##  4\. Hamilton-Souplet-Zhang type gradient estimates II

In this section we discuss the proof of Theorem 2.4 by stating and proving the
following theorem.

######  Theorem 4.1.

Let $M=(M,g,e^{-\phi}dv)$ be an $n$-dimensional complete smooth metric measure
space with compact boundary satisfying the bounds

$\displaystyle\mathscr{R}ic_{\phi}^{m}\geq-(m-1)\kappa\ \ \text{and}\ \
\mathscr{H}_{\phi}\geq-\ell$

for some constants $\kappa,\ell\geq 0$. Let $u=u(x,t)$ be a smooth positive
solution to (1.1) in $\mathcal{Q}_{R,T}(\partial M)$. Suppose further that $u$
satisfies the Dirichlet boundary condition (i.e., $u(\cdot,t)|_{\partial M}$
is constant for each time $t\in(-\infty,+\infty)$),

$\displaystyle u_{\nu}\geq 0\ \ \text{and}\ \ u_{t}\leq H(u)$

over $\partial M\times[T_{0}-T,T_{0}]$. Then for $\varepsilon\in(0,1/2)$,
there exists a constant $C_{\varepsilon}>0$ depending only on
$n,m,\varepsilon$, such that for all $(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial
M)$

$\displaystyle\frac{|\nabla u|}{u^{1-3\varepsilon/2}}$ $\displaystyle\leq
C_{\varepsilon}\Bigg{\\{}\frac{1}{R}+\frac{1}{\sqrt{T}}+\sqrt{L}+\Gamma^{\varepsilon}_{+}[u]\Bigg{\\}}\Big{(}\sup_{\mathcal{Q}_{R,T}(\partial
M)}\\{\sqrt{u^{3\varepsilon}}\\}\Big{)},$ (4.1)

where $L=(\kappa^{2}+\ell^{4})^{1/2}$,

$\Gamma^{\varepsilon}_{+}[u]=\sup_{\mathcal{Q}_{R,T}(\partial
M)}\left\\{\left[2H^{\prime}(u)+(3\varepsilon-2)\widehat{H}(u)\right]_{+}\right\\}^{1/2}\
\ \text{and}\ \ \widehat{H}(u):=\frac{H(u)}{u}.$

######  Remark 4.2.

The proof of Theorem 2.4 follows from the above theorem by choosing
$\varepsilon=1/3$ in the range $0<\varepsilon<1/2$.

On the other hand, passing to the limit $\varepsilon\searrow 0$ in (4.1) we
have the following estimates for all $(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial
M)$ under the assumption of Theorem 4.1:

$\displaystyle\frac{|\nabla u|}{u}$ $\displaystyle\leq
C_{0}\Bigg{\\{}\frac{1}{R}+\frac{1}{\sqrt{T}}+\sqrt{L}+\Gamma^{0}_{+}[u]\Bigg{\\}},$
(4.2)

where the constant $C_{0}>0$ depends only on $n$ and $m$, and

$\Gamma^{0}_{+}[u]=\sup_{\mathcal{Q}_{R,T}(\partial
M)}\left\\{\left[2(H^{\prime}(u)-\widehat{H}(u))\right]_{+}\right\\}^{1/2}.$

Next we prove some fundamental lemmas which will be applied in the proof of
Theorem 4.1.

## Fundamental lemmas

In this section,we define a qunatity $w$ as $w(x,t)=f(x,t)|\nabla f(x,t)|^{2}$
with $f=u^{\varepsilon}$ and $\varepsilon\in(0,1)$, where $u>0$ solves the
nonlinear heat-type equation (1.1). Going by Lemma 3.2 of the previous
section, it is clear that $f_{t}=\varepsilon u^{\varepsilon-1}u_{t}$, $\nabla
f=\varepsilon u^{\varepsilon-1}\nabla u$, $|\nabla
f|^{2}/f^{2}=\varepsilon^{2}|\nabla u|^{2}/u^{2}$ and $f$ satisfies

$\displaystyle
f_{t}=\mathscr{L}_{\phi}f-\varepsilon^{-1}(\varepsilon-1)|\nabla
f|^{2}/f+\varepsilon fH(f^{1/\varepsilon})/f^{1/\varepsilon}$ (4.3)

Furthermore, we will denote
$\widehat{H}(f)=H(f^{1/\varepsilon})/f^{1/\varepsilon}$ and note that the
quantity $\widehat{H}(f)$ is the same as $\widehat{H}(u)=H(u)/u$.

######  Lemma 4.3.

Let $f=u^{\varepsilon}$, $\varepsilon\in(0,1)$, be a smooth function where
$u(x,t)$ is a positive smooth solution to (1.1). Then $w=f|\nabla f|^{2}$
satisfies the nonlinear equation

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w=$ $\displaystyle
2f|\nabla^{2}f|^{2}+2\varepsilon^{-1}(\varepsilon-1)\nabla f\nabla(|\nabla
f|^{2})-\varepsilon^{-1}(\varepsilon-1)|\nabla f|^{4}/f$
$\displaystyle+2f\mathscr{R}ic_{\phi}(\nabla f,\nabla f)+2\nabla
f\nabla(|\nabla f|^{2})-\varepsilon|\nabla f|^{2}f\widehat{H}(f)$ (4.4)
$\displaystyle-2\epsilon f\nabla f\nabla(f\widehat{H}(f)).$

###### Proof.

By direct computation

$\displaystyle w_{t}=f_{t}|\nabla f|^{2}+2f\langle\nabla f,\nabla
f_{t}\rangle.$ (4.5)

Using the weighted Bochner formula

$\displaystyle\mathscr{L}_{\phi}w=f\mathscr{L}_{\phi}(|\nabla f|^{2})+|\nabla
f|^{2}\mathscr{L}_{\phi}f+2\nabla f\nabla(|\nabla f|^{2}).$ (4.6)

Putting (4.5) and (4.6) together gives

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w=$ $\displaystyle
2f|\nabla^{2}f|^{2}+2f\langle\nabla
f,\nabla(\mathscr{L}_{\phi}-\partial_{t})f\rangle+2f\mathscr{R}ic_{\phi}(\nabla
f,\nabla f)$ $\displaystyle+|\nabla
f|^{2}(\mathscr{L}_{\phi}-\partial_{t})f+2\nabla f\nabla(|\nabla f|^{2}).$
(4.7)

Referring to (4.3) which simply reads
$(\mathscr{L}_{\phi}-\partial_{t})f=\varepsilon^{-1}(\varepsilon-1)|\nabla
f|^{2}/f-\varepsilon fH(f^{1/\varepsilon})/f^{1/\varepsilon}$ with
$\widehat{H}(f)=H(f^{1/\varepsilon})/f^{1/\varepsilon}$, we can compute

$\displaystyle\nabla(\mathscr{L}_{\phi}-\partial_{t})f=\varepsilon^{-1}(\varepsilon-1)\nabla(|\nabla
f|^{2})/f-\varepsilon^{-1}(\varepsilon-1)\nabla f|\nabla
f|^{2}/f^{2}-\varepsilon\nabla(f\widehat{H}(f))$

so that

$\displaystyle 2f$ $\displaystyle\langle\nabla
f,\nabla(\mathscr{L}_{\phi}-\partial_{t})f\rangle$
$\displaystyle=2\varepsilon^{-1}(\varepsilon-1)\nabla f\nabla(|\nabla
f|^{2})-2\varepsilon^{-1}(\varepsilon-1)|\nabla f|^{4}/f-2\varepsilon f\nabla
f\nabla(f\widehat{H}(f))$ (4.8)

and

$\displaystyle|\nabla
f|^{2}(\mathscr{L}_{\phi}-\partial_{t})f=\varepsilon^{-1}(\varepsilon-1)|\nabla
f|^{4}/f-\varepsilon|\nabla f|^{2}f\widehat{H}(f).$ (4.9)

Substituting (Proof.) and (4.9) into (Proof.) gives the desired result.

∎

######  Lemma 4.4.

With the assumption of Lemma 4.3 we have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w\geq$ $\displaystyle
2\varepsilon^{-1}(\varepsilon-1)\frac{1}{f}\langle\nabla f,\nabla
w\rangle-\varepsilon^{-1}(5\varepsilon-3)\frac{1}{f^{3}}w^{2}$
$\displaystyle-2(m-1)\kappa w-\Gamma^{\varepsilon}[u]w,$ (4.10)

where
$\Gamma^{\varepsilon}[u]=[2H^{\prime}(u)+(3\varepsilon-2)\widehat{H}(u)]$.

###### Proof.

Let us put into consideration the following identity

$0\leq 2\left|\sqrt{f}\ \nabla^{2}f+\frac{\nabla f\otimes\nabla
f}{\sqrt{f}}\right|^{2}=2f|\nabla^{2}f|^{2}+2\nabla f\nabla|\nabla
f|^{2}+\frac{2|\nabla f|^{4}}{f}$

and the weighted $m$-Ricci tensor bound
$\mathscr{R}ic^{m}_{\phi}\geq-(m-1)\kappa$ we have from Lemma 4.3 as follows

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w$
$\displaystyle=2\left|\sqrt{f}\ \nabla^{2}f+\frac{\nabla f\otimes\nabla
f}{\sqrt{f}}\right|^{2}+2\varepsilon^{-1}(\varepsilon-1)\nabla f\nabla(|\nabla
f|^{2})$ $\displaystyle\ \ \ \
-(\varepsilon^{-1}(\varepsilon-1)+2)\frac{|\nabla
f|^{4}}{f}+2f\mathscr{R}ic^{m}_{\phi}(\nabla f,\nabla
f)+\frac{2f\langle\nabla\phi,\nabla f\rangle^{2}}{m-n}$ $\displaystyle\ \ \ \
-\varepsilon|\nabla f|^{2}f\widehat{H}(f)-2\epsilon f\nabla
f\nabla(f\widehat{H}(f))$ $\displaystyle\geq
2\varepsilon^{-1}(\varepsilon-1)\nabla f\nabla(|\nabla
f|^{2})-\varepsilon^{-1}(3\varepsilon-1)\frac{|\nabla f|^{4}}{f}-2(m-1)\kappa
f|\nabla f|^{2}$ $\displaystyle\ \ \ \ -\varepsilon|\nabla
f|^{2}f\widehat{H}(f)-2\epsilon f\nabla f\nabla(f\widehat{H}(f)).$ (4.11)

Taking into account the following identity

$\nabla f\nabla(f|\nabla f|^{2})=|\nabla f|^{4}+f\nabla f\nabla(|\nabla
f|^{2})$

which implies

$\displaystyle\nabla f\nabla(|\nabla f|^{2})=(\nabla f/f)\nabla(f|\nabla
f|^{2})-|\nabla f|^{4}/f$ (4.12)

and upon substituting (Proof.) into (4.12) we have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w\geq$ $\displaystyle
2\varepsilon^{-1}(\varepsilon-1)\frac{1}{f}\langle\nabla f,\nabla
w\rangle-\varepsilon^{-1}(5\varepsilon-3)\frac{1}{f^{3}}w^{2}-2(m-1)\kappa w$
$\displaystyle-\varepsilon|\nabla f|^{2}f\widehat{H}(f)-2\epsilon f\nabla
f\nabla(f\widehat{H}(f)).$ (4.13)

The explicit value of the last two terms of (Proof.) can be computed. Recall
that $\widehat{H}(f)=H(f^{1/\varepsilon})/f^{1/\varepsilon}$. We have

$\displaystyle\nabla(f\widehat{H}(f))$
$\displaystyle=\nabla(H(f^{1/\varepsilon})/f^{1/\varepsilon-1})$
$\displaystyle=\varepsilon^{-1}\nabla
fH^{\prime}(u)+\varepsilon^{-1}(\varepsilon-1)\nabla f\widehat{H}(u)$

so that

$\displaystyle 2\epsilon f\nabla f\nabla(f\widehat{H}(f))$
$\displaystyle=2f|\nabla f|^{2}H^{\prime}(u)+2(\varepsilon-1)|f\nabla
f|^{2}\widehat{H}(u)$
$\displaystyle=2[H^{\prime}(u)+(\varepsilon-1)\widehat{H}(u)]w.$

Therefore we have

$\displaystyle-\varepsilon|\nabla f|^{2}f\widehat{H}(f)$
$\displaystyle-2\epsilon f\nabla
f\nabla(f\widehat{H}(f))=-2\left[H^{\prime}(u)+\frac{(3\varepsilon-2)}{2}\widehat{H}(u)]\right]w,$

$\widehat{H}(u)=H(u)/u=H(f^{1/\varepsilon})/f^{1/\varepsilon}=\widehat{H}(f)$.
Replacing this into (Proof.) we finally arrive at the required inequality
(4.4).

∎

### Proof of Theorem 4.1

With recourse to the procedure adopted in the proof of Theorem 3.1 with the
same cut-off function $\psi$ described in Lemma 2.2, we can write using Lemma
4.4 that

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})(\psi w)\geq$ $\displaystyle
2\varepsilon^{-1}(\varepsilon-1)\frac{1}{f}\left[\langle f,\nabla(\psi
w)\rangle-w\langle\nabla f,\nabla\psi\rangle\right]$
$\displaystyle+2\left[\frac{\nabla\psi}{\psi}\nabla(\psi
w)-\frac{|\nabla\psi|^{2}}{\psi}w\right]-\varepsilon^{-1}(5\varepsilon-3)\frac{1}{f^{3}}\psi
w^{2}$
$\displaystyle+w[\mathscr{L}_{\phi}-\partial_{t}-2(m-1)\kappa]\psi-\Gamma^{\varepsilon}[u]\psi
w.$

Clearly $\psi w$ is positive and attain its maximum at a point in
$\mathcal{Q}_{R/2,T/2}(\partial M)$ say $(x_{0},t_{0})$. Our argument will
henceforth split into two cases as in the proof of Theorem 3.1.

Case 1: If $x_{0}\not\in\partial M$, we may assume withot loss of generality
that $x_{0}\not\in\text{cut}(\partial M)$ by Calabi’s argument. Then at the
maximum point $(x_{0},t_{0})$ we have $\nabla(\psi w)=0$, $\partial_{t}(\psi
w)\geq 0$, $\mathscr{L}_{\phi}(\psi w)\leq 0$ and then

$\displaystyle 0\geq$
$\displaystyle-2\varepsilon^{-1}(\varepsilon-1)\frac{1}{f}\langle\nabla
f,\nabla\psi\rangle
w-2\frac{|\nabla\psi|^{2}}{\psi}w-\varepsilon^{-1}(5\varepsilon-3)\frac{1}{f^{3}}\psi
w^{2}$
$\displaystyle+w[\mathscr{L}_{\phi}-\partial_{t}-2(m-1)\kappa]\psi-\Gamma^{\varepsilon}[u]\psi
w$

which can be rearranged to give

$\displaystyle-\varepsilon^{-1}(5\varepsilon-3)\psi w^{2}\leq$ $\displaystyle
2\varepsilon^{-1}(\varepsilon-1)f^{2}\langle\nabla f,\nabla\psi\rangle
w+2\frac{|\nabla\psi|^{2}}{\psi}f^{3}w$
$\displaystyle-f^{3}w[\mathscr{L}_{\phi}-\partial_{t}-2(m-1)\kappa]\psi+f^{3}\Gamma^{\varepsilon}[u]\psi
w.$ (4.14)

We can now proceed to estimate each term on the right hand side of (Proof of
Theorem 4.1) as before. This procedure follows directly from the applications
of Cauchy-Schwarz and Young’s inequalities and properties of $\psi$ as stated
in Lemma 2.2. To this end we estimate the first and second term on the right
hand side of (Proof of Theorem 4.1) respectively as follows:

$\displaystyle 2\frac{(\varepsilon-1)}{\varepsilon}f^{2}\langle\nabla
f,\nabla\psi\rangle w$ $\displaystyle\leq
2\frac{(1-\varepsilon)}{\varepsilon}\psi^{3/4}w^{3/2}f^{3/2}\frac{|\nabla\psi|}{\psi^{3/4}}$
$\displaystyle\leq\frac{(1-\varepsilon)}{\varepsilon}\psi
w^{2}+C_{\varepsilon}\frac{|\nabla\psi|^{4}}{\psi^{3}}(\sup_{\mathcal{Q}_{R,T}}f)^{6}$
$\displaystyle\leq\frac{(1-\varepsilon)}{6\varepsilon}\psi
w^{2}+\frac{C_{\varepsilon}}{R^{4}}(\sup_{\mathcal{Q}_{R,T}}f)^{6}$ (4.15)

and

$\displaystyle 2\frac{|\nabla\psi|^{2}}{\psi}f^{3}w$ $\displaystyle\leq
2\psi^{1/2}w\frac{|\nabla\psi|^{2}}{\psi^{3/2}}(\sup_{\mathcal{Q}_{R,T}}f)^{3}$
$\displaystyle\leq\frac{(1-\varepsilon)}{6\varepsilon}\psi
w^{2}+C_{\varepsilon}\left(\frac{|\nabla\psi|^{2}}{\psi^{3/2}}\right)^{2}(\sup_{\mathcal{Q}_{R,T}}f)^{6}$
$\displaystyle\leq\frac{(1-\varepsilon)}{6\varepsilon}\psi
w^{2}+\frac{C_{\varepsilon}}{R^{4}}(\sup_{\mathcal{Q}_{R,T}}f)^{6}$ (4.16)

at $(x_{0},t_{0})$.

Similarly to (3.1) - (3.1) with little modification we have for the term with
time derivative of $\psi$:

$\displaystyle f^{3}w\psi_{t}$
$\displaystyle\leq\psi^{\frac{1}{2}}w\frac{|\psi_{t}|}{\psi^{\frac{1}{2}}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{3}$
$\displaystyle\leq\frac{(1-\varepsilon)}{6\varepsilon}\psi
w^{2}+C_{\varepsilon}\frac{|\psi_{t}|^{2}}{\psi}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}$
$\displaystyle\leq\frac{(1-\varepsilon)}{6\varepsilon}\psi
w^{2}+\frac{C_{\varepsilon}}{T^{2}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}.$
(4.17)

For the term with $(m-1)\kappa$ we have

$\displaystyle 2f^{3}(m-1)\kappa\psi w$ $\displaystyle\leq
2(m-1)\kappa(\psi^{\frac{1}{2}}w)\psi^{\frac{1}{2}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{3}$
$\displaystyle\leq\frac{(1-\varepsilon)}{6\varepsilon}\psi
w^{2}+C_{\varepsilon}m^{2}\kappa^{2}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}.$
(4.18)

For the term involving the weighted Laplacian of $\psi$ we have

$\displaystyle-f^{3}w\mathscr{L}_{\phi}\psi$
$\displaystyle=-f^{3}\left[\psi_{\varrho}\mathscr{L}_{\phi}\varrho+\psi_{\varrho\varrho}|\nabla\varrho|^{2}\right]w$
$\displaystyle\leq\left[|\psi_{\varrho}|((n-1)\kappa
R+\ell)+\psi_{\varrho\varrho}|\nabla\varrho|^{2}\right]w\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{3}$

which through similar step as in (3.1) yields the bound

$\displaystyle-f^{3}w\mathscr{L}_{\phi}\psi$
$\displaystyle\leq\frac{(1-\varepsilon)}{6\varepsilon}\psi
w^{2}+\left(\frac{C_{\varepsilon}}{R^{4}}+C_{\varepsilon}\kappa^{2}+C_{\varepsilon}\ell^{4}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}.$
(4.19)

since $\mathscr{R}ic_{\phi}^{m}\geq-(m-1)\kappa$ and
$\mathscr{H}_{\phi}\geq-\ell$. Combining (Proof of Theorem 4.1) - (4.19) we
have the bound on the third term on the right hand side of (Proof of Theorem
4.1) as follows

$\displaystyle-f^{3}w[\mathscr{L}_{\phi}$
$\displaystyle-\partial_{t}-2(m-1)\kappa]\psi$
$\displaystyle\leq\frac{(1-\varepsilon)}{2\varepsilon}\psi
w^{2}+C_{\varepsilon}\left(\frac{1}{R^{4}}+\kappa^{2}+\ell^{4}+\frac{1}{T^{2}}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}.$
(4.20)

at $(x_{0},t_{0})$.

For the last term we compute at $(x_{0},t_{0})$

$\displaystyle f^{3}\Gamma^{\varepsilon}[u]\psi w$
$\displaystyle\leq\psi^{\frac{1}{2}}w[\Gamma^{\varepsilon}[u]]_{+}\psi^{\frac{1}{2}}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{3}$
$\displaystyle\leq\frac{(1-\varepsilon)}{6\varepsilon}\psi
w^{2}+C_{\varepsilon}[\Gamma^{\varepsilon}[u]]_{+}^{2}\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6},$
(4.21)

where
$[\Gamma^{\varepsilon}[u]]_{+}=[2H^{\prime}(u)+(3\varepsilon-2)\widehat{H}(u)]_{+}$.

Putting these estimates back into (Proof of Theorem 4.1) and rearranging we
obtain

$\displaystyle\varepsilon^{-1}(3-5\varepsilon)\psi w^{2}\leq$ $\displaystyle
2\varepsilon^{-1}(1-\varepsilon)\psi
w^{2}+C_{\varepsilon}\left(\frac{1}{R^{4}}+\frac{1}{T^{2}}+\kappa^{2}+\ell^{4}+[\Gamma^{\varepsilon}[u]]_{+}^{2}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}$

at $(x_{0},t_{0})$. It therefore follows that

$\displaystyle 2\varepsilon^{-1}(1-2\varepsilon)\psi w^{2}\leq$ $\displaystyle
C_{\varepsilon}\left(\frac{1}{R^{4}}+\frac{1}{T^{2}}+\kappa^{2}+\ell^{4}+[\Gamma^{\varepsilon}[u]]_{+}^{2}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}$

by restricting $\varepsilon$ to the range $0<\varepsilon<1/2$ at
$(x_{0},t_{0})$. Hence

$\displaystyle\psi w^{2}\leq
C_{\varepsilon}\left(\frac{1}{R^{4}}+\frac{1}{T^{2}}+\kappa^{2}+\ell^{4}+[\Gamma^{\varepsilon}[u]]_{+}^{2}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}$

at $(x_{0},t_{0})$ Reverting to the maximal characterization of the point
$(x_{0},t_{0})$ we have for all $(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial M)$

$\displaystyle\psi^{2}w^{2}(x,t)$
$\displaystyle\leq\psi^{2}w^{2}(x_{0},t_{0})\leq\psi w^{2}(x_{0},t_{0}).$

Since $\psi(x,t)\equiv 1$ on $\mathcal{Q}_{R/2,T}(\partial M)$ and by
definition of $w$, i.e., $w=f|\nabla f|^{2}$ with $f=u^{\varepsilon}$,
$\varepsilon\in(0,1/2)$ we obtain

$\displaystyle(f|\nabla f|^{2})^{2}\leq
C_{\varepsilon}\left(\frac{1}{R^{4}}+\frac{1}{T^{2}}+\kappa^{2}+\ell^{4}+[\Gamma^{\varepsilon}[u]]_{+}^{2}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}}f\Big{)}^{6}.$

Noting that $f|\nabla f|^{2}=\varepsilon^{2}f^{3}|\nabla u|^{2}/u^{2}$ and
$(\sup\\{f\\})=(\sup\\{u^{\varepsilon}\\})$. Then we arrive at

$\displaystyle\frac{|\nabla u|^{2}}{u^{2-3\varepsilon}}\leq
C_{\varepsilon}\left(\frac{1}{R^{2}}+\frac{1}{T}+L+\sup_{\mathcal{Q}_{R,T}}[\Gamma^{\varepsilon}[u]]_{+}\right)\Big{(}\sup_{\mathcal{Q}_{R,T}}u^{\varepsilon}\Big{)}^{3}$

for all $(x,t)\in\mathcal{Q}_{R/2,T/2}(\partial M)$, where
$L=(\kappa^{2}+\ell^{4})^{1/2}$. In conclusion, rearranging gives the desired
estimates.

Case 2: If $x_{0}\in\partial M$, we will show that the gradient estimates
(4.1) holds on $\partial M\times[0,T]$. Indeed, at the maximum point
$(x_{1},t_{1})\in\partial M\times[0,T]$ of $\psi w$ we have $(\psi
w)_{\nu}\geq 0$ which implies $\psi_{\nu}w+\psi w_{\nu}=\psi w_{\nu}\geq 0$,
and in particular

$w_{\nu}\geq 0.$

Since $w=f|\nabla f|^{2}$, and $f=u^{\varepsilon}$ with $\varepsilon\in(0,1)$
satisfies the Dirichlet boundary condition we have

$\displaystyle 0\leq w_{\nu}=(f|\nabla f|^{2})_{\nu}=f_{\nu}|\nabla
f|^{2}+f(|\nabla f|^{2})_{\nu}.$ (4.22)

Noting that $|\nabla f|=f_{\nu}=\varepsilon u^{\varepsilon}u_{\nu}/u\geq 0$
since $u>0$ and $u_{\nu}\geq 0$. Now referring to the Reilly-type formula
formula (2.1.3) then (4.22) gives

$\displaystyle 0\leq
f_{\nu}f_{\nu}^{2}+2ff_{\nu}\left[\mathscr{L}_{\phi}f-f_{\nu}\mathscr{H}_{\phi}\right].$
(4.23)

To keep the analysis simple, we write (c.f. (4.3) and note that
$\widehat{H}(f^{1/\varepsilon})=H(f^{1/\varepsilon})/f^{1/\varepsilon}$)

$\displaystyle\mathscr{L}_{\phi}f$
$\displaystyle=\varepsilon^{-1}(\varepsilon-1)|\nabla
f|^{2}/f+f_{t}-\varepsilon f\widehat{H}(f^{1/\varepsilon})$
$\displaystyle=\varepsilon^{-1}(\varepsilon-1)f_{\nu}^{2}/f+f_{t}-\varepsilon
f\widehat{H}(f^{1/\varepsilon}).$

Noting that $f^{2}_{\nu}=w/f$ and $f_{\nu}f_{\nu}^{2}=(w/f)^{3/2}.$ It the
follows by plugging in these data into (4.23) that

$\displaystyle
0\leq\frac{w^{3/2}}{f^{3/2}}+\frac{2w^{1/2}}{f^{-1/2}}\left[\frac{\varepsilon^{-1}(\varepsilon-1)w}{f^{2}}+f_{t}-\varepsilon
f\widehat{H}(f^{1/\varepsilon})-\frac{w^{1/2}}{f^{1/2}}\mathscr{H}_{\phi}\right]$
(4.24)

at $(x_{0},t_{0})$. Within the consideration of the condition that $u_{t}\leq
H(u)$ which implies $f_{t}-\varepsilon f\widehat{H}(f^{1/\varepsilon})\leq 0$
over $\partial M\times[0,T]$ we see that (4.24) is reduced to

$\displaystyle-[1+2\varepsilon^{-1}(\varepsilon-1)]\frac{w^{3/2}}{f^{3/2}}+2w\mathscr{H}_{\phi}\leq
0,$ (4.25)

Since $\varepsilon\in(0,1/2)$ then
$-[1+2\varepsilon^{-1}(\varepsilon-1)]=2\varepsilon^{-1}(1-\varepsilon)-1>0$.
Clearly (4.25) can be solved to give

$w(x_{0},t_{0})=0\ \ \text{or}\ \ \
w^{1/2}(x_{0},t_{0})\leq\frac{2}{2\varepsilon^{-1}(1-\varepsilon)-1}\ell(\sup
f)^{3/2}$

since $\mathscr{H}_{\phi}\geq-\ell$ by the theorem hypothesis. It therefore
means that

$\psi w=0\ \ \text{or}\ \ \psi
w(x_{0},t_{0})\leq\frac{4}{(2\varepsilon^{-1}(1-\varepsilon)-1)^{2}}\ell^{2}(\sup
f)^{3}$

on $\mathcal{Q}_{R,T}(\partial M)$. The rest of the proof is exactly the same
as that of Theorem 3.1. Thus

$\displaystyle f|\nabla f|^{2}(x,t)=w(x,t)=\psi w(x,t)\leq\psi
w(x_{0},t_{0})\leq\frac{4}{(2\varepsilon^{-1}(1-\varepsilon)-1)^{2}}\ell^{2}(\sup
f)^{3},$

which also implies the conclusion by using the relation $f=u^{\varepsilon}$
with $\varepsilon$ restricted to the range $(0,1/2)$.

##  5\. Li-Yau type gradient estimates

######  Theorem 5.1.

Let $M=(M,g,e^{-\phi}dv)$ be an $n$-dimensional complete smooth metric measure
space with compact boundary satisfying the bounds

$\displaystyle\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k\ \ \text{and}\ \
\mathscr{H}_{\phi}\geq-l$

for some constants $k,l\geq 0$. Let $u=u(x,t)$ be a smooth positive solution
to (1.1) in $\mathcal{Q}_{2R,T}(\partial M)$. Suppose further that $u$
satisfies the Dirichlet boundary condition (i.e., $u(\cdot,t)|_{\partial M}$
is constant for each time $t\in(-\infty,+\infty)$),

$\displaystyle u_{\nu}\geq 0\ \ \text{and}\ \ u_{t}\leq H(u)$

over $\partial M\times[T_{0}-T,T_{0}]$. Then have for all
$(x,t)\in\mathcal{Q}_{R,T}(\partial M)$, $t>0$

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}$
$\displaystyle+\delta\frac{H(u)}{u}-\delta\frac{u_{t}}{u}\leq\frac{m\delta^{2}}{2(1-\epsilon)}\Big{(}\frac{1}{t}+\frac{k}{\delta-1}+\widetilde{\Phi}+\mu^{H}\Big{)}+\frac{\delta}{p(\delta-1)}\sqrt{\frac{m}{2(1-\epsilon)}}\gamma^{H}$
(5.1)

on $\partial M\times[T_{0}-T,T_{0}]$. Here $\epsilon\in(0,1)$, $\delta>1$ and
$p>0$ such that

$\displaystyle\frac{2(1-\epsilon)(\delta-1)}{m\delta
p}\geq\frac{1}{\epsilon}-1+\frac{(1-\epsilon)(1-\delta)^{2}}{4},$ (5.2)

and

$\widetilde{\Phi}=\frac{1}{R^{2}}\Big{[}\Big{(}\frac{m\delta^{2}}{4(1-\epsilon)(\delta
p+\delta-1)}\Big{)}C_{1}^{2}+C_{2}+C_{1}[(m-1)kR+l]R\Big{]}\Big{]}$

for some constants positive $C_{1}$ and $C_{2}$. Furthermore,

$\mu^{H}:=\sup_{\mathcal{Q}_{2R,T}}\\{[H_{u}]_{+}\\}\ \ \text{and}\ \
\gamma^{H}:=\sup_{\mathcal{Q}_{2R,T}}\\{[\Omega^{H}]_{+}\\},$

where $[H_{u}]_{+}=[H^{\prime}(u)-H(u)/u]_{+}$ and
$[\Omega^{H}]_{+}=[-[H^{\prime}(u)-H(u)/u-(\alpha/p^{2})uH^{\prime\prime}(u)]]_{+}$.

###  5.1. Evolution equations

Here in this section we consider the nonlinear heat-type equation

$\displaystyle u_{t}=\mathscr{L}_{\phi}u+H(u).$ (5.3)

Let

$w=u^{-p},$

where $p>0$ is to be determined later. By direct computation as before we
obtain

$\displaystyle w_{t}=-pu^{-p-1}u_{t},\ \ \nabla w=-pu^{-p-1}\nabla u,\ \
|\nabla w|^{2}/w^{2}=p^{2}|\nabla u|^{2}/u^{2}$

and

$\displaystyle\Delta w=-pu^{-p-1}\Delta u+p(p-1)u^{-p-2}|\nabla u|^{2}.$

Clearly by the definition of $\mathscr{L}_{\phi}w=\Delta w-\nabla\phi\nabla w$
we have

$\displaystyle\mathscr{L}_{\phi}w$
$\displaystyle=-pu^{-p-1}\mathscr{L}_{\phi}u+p(p-1)u^{-p-2}|\nabla u|^{2}$
$\displaystyle=-pu^{-p-1}(u_{t}-H(u))+((p+1)/p)|\nabla w|^{2}/w.$

Then we have proved the following lemma.

######  Lemma 5.2.

Suppose $u>0$ is a smooth solution of (5.3). The the function $w=u^{-p}$,
$p>0$, satisfies the following equation

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})w$
$\displaystyle=\frac{P+1}{p}\frac{|\nabla w|^{2}}{w}+pw\widehat{H}(w),$ (5.4)

where $\widehat{H}(w)=H(w^{-1/p})/w^{-1/p}=H(u)/u$.

For the purpose of Li-Yau type gradient estimate to be established in this
section, we introduce the following Harnack quantity

$\displaystyle F(x,t)=F_{0}(x,t)+\beta F_{1}(x,t).$ (5.5)

For computational purpose, we also introduce $F_{0}$ and $F_{1}$ in the
following quantities

$\displaystyle F_{0}(x,t)=\frac{|\nabla w|^{2}}{w^{2}}+\alpha\widehat{H}(w)\ \
\text{and}\ \ \ F_{1}(x,t)=\frac{w_{t}}{w},$

where $\alpha,\beta>0$ are constants to be chosen later. Before proceeding to
prove the desired Li-Yau type gradient estimate we first prove some
computational lemmas.

######  Lemma 5.3.

Suppose $u>0$ is a smooth solution of (5.3) and $w$ satisfies (5.4). Then

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F_{1}(x,t)$
$\displaystyle=\frac{2}{p}\nabla F_{1}(x,t)\nabla\log
w(x,t)+p\widehat{H}_{t}.$ (5.6)
$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F_{0}(x,t)$
$\displaystyle=\frac{\mathscr{L}_{\phi}|\nabla
w|^{2}}{w^{2}}+2\Big{(}3-\frac{p+1}{p}\Big{)}\frac{|\nabla
w|^{4}}{w^{4}}-\frac{8\langle\nabla^{2}w,\nabla w\otimes\nabla
w\rangle}{w^{3}}$ $\displaystyle\hskip 28.45274pt-\frac{2\nabla w\nabla
w_{t}}{w^{2}}+\alpha\Big{(}\mathscr{L}_{\phi}-\partial_{t}-\frac{2p}{\alpha}\frac{|\nabla
w|^{2}}{w^{2}}\Big{)}\widehat{H}.$ (5.7)

###### Proof.

A simple computation gives

$\displaystyle\partial_{t}F_{1}(x,t)$
$\displaystyle=\frac{1}{w^{2}}(ww_{tt}-w^{2}_{t})$ $\displaystyle\nabla
F_{1}(x,t)$ $\displaystyle=\frac{\nabla w_{t}}{w}-\frac{w_{t}\nabla w}{w^{2}}$
$\displaystyle\mathscr{L}_{\phi}F_{1}(x,t)$
$\displaystyle=e^{\phi}\text{div}(e^{-\phi}F_{1})=\text{div}\nabla
F_{1}-\nabla\phi\nabla F_{1}$
$\displaystyle=\frac{1}{w^{2}}(w\mathscr{L}_{\phi}w_{t}-2\nabla w\nabla
w_{t}-w_{t}\mathscr{L}_{\phi}w+2w_{t}|\nabla w|^{2}/w).$

Combining the expressions for $\partial_{t}F_{1}$ and
$\mathscr{L}_{\phi}F_{1}$ and rearranging terms of the resulting expression,
we have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F_{1}(x,t)=\frac{1}{w}(\mathscr{L}_{\phi}w-w_{t})_{t}-\frac{w_{t}}{w^{2}}(\mathscr{L}_{\phi}w-w_{t})-\frac{2\nabla
w\nabla w_{t}}{w^{2}}+2w_{t}\frac{|\nabla w|^{2}}{w^{3}}.$ (5.8)

Substituting (5.4) into (5.8) and simplifying gives

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F_{1}(x,t)$
$\displaystyle=\frac{1}{w}\Big{(}\frac{P+1}{p}\frac{|\nabla
w|^{2}}{w}+pw\widehat{H}\Big{)}_{t}-\frac{w_{t}}{w^{2}}\Big{(}\frac{P+1}{p}\frac{|\nabla
w|^{2}}{w}+pw\widehat{H}\Big{)}$ $\displaystyle\hskip 28.45274pt-\frac{2\nabla
w\nabla w_{t}}{w^{2}}+2w_{t}\frac{|\nabla w|^{2}}{w^{3}}$
$\displaystyle=\frac{p+1}{p}\Big{(}\frac{2\nabla w\nabla
w_{t}}{w^{2}}-\frac{2|\nabla
w|^{2}w_{t}}{w^{3}}\Big{)}+p\widehat{H}_{t}-\frac{p+1}{p}\frac{|\nabla
w|^{2}w_{t}}{w^{3}}$ $\displaystyle\hskip 28.45274pt-\frac{2\nabla w\nabla
w_{t}}{w^{2}}+2w_{t}\frac{|\nabla w|^{2}}{w^{3}}$
$\displaystyle=\frac{2}{p}\nabla F_{1}\nabla\log w+p\widehat{H}_{t}.$

Similarly

$\displaystyle\partial_{t}F_{0}(x,t)=\frac{2\nabla w\nabla
w_{t}}{w^{2}}-\frac{2|\nabla w|^{2}w_{t}}{w^{3}}+\alpha\widehat{H}_{t}.$

$\displaystyle\mathscr{L}_{\phi}F_{0}(x,t)$
$\displaystyle=\frac{\mathscr{L}_{\phi}|\nabla w|^{2}}{w^{2}}+|\nabla
w|^{2}\mathscr{L}_{\phi}\left(\frac{1}{w^{2}}\right)+2\Big{\langle}\nabla|\nabla
w|^{2},\nabla\left(\frac{1}{w^{2}}\right)\Big{\rangle}+\alpha\widehat{H}_{t}$
$\displaystyle=\frac{\mathscr{L}_{\phi}|\nabla w|^{2}}{w^{2}}+\frac{6|\nabla
w|^{4}}{w^{4}}-\frac{2|\nabla
w|^{2}\mathscr{L}_{\phi}w}{w^{3}}-\frac{8\langle\nabla^{2}w,\nabla
w\otimes\nabla w\rangle}{w^{3}}+\alpha\widehat{H}_{t}$

and then

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F_{0}(x,t)$
$\displaystyle=\frac{\mathscr{L}_{\phi}|\nabla w|^{2}}{w^{2}}+\frac{6|\nabla
w|^{4}}{w^{4}}-\frac{8\langle\nabla^{2}w,\nabla w\otimes\nabla
w\rangle}{w^{3}}-\frac{2\nabla w\nabla w_{t}}{w^{2}}$ $\displaystyle\hskip
28.45274pt-\frac{2|\nabla
w|^{2}}{w^{3}}(\mathscr{L}_{\phi}w-w_{t})+\alpha(\mathscr{L}_{\phi}-\partial_{t})\widehat{H}.$

Substituting (5.4) into the last expression gives (5.3). This complete the
proof.

∎

######  Lemma 5.4.

With the assumption of Lemma 5.3 we have for $\epsilon\in(0,1)$

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F_{0}(x,t)$
$\displaystyle\geq\frac{2(1-\epsilon)}{m}\Big{(}\frac{\mathscr{L}_{\phi}w}{w}\Big{)}^{2}-2\Big{(}\frac{1}{\epsilon}-1\Big{)}\frac{|\nabla
w|^{2}}{w^{4}}+\frac{2}{w^{2}}\mathscr{R}ic_{\phi}^{m}(\nabla w,\nabla w)$
$\displaystyle\hskip 28.45274pt+\frac{2}{p}\nabla F_{0}\nabla\log
w+2\Big{(}p-\frac{\alpha}{p}\Big{)}\frac{\nabla
w}{w}\nabla\widehat{H}+\alpha(\mathscr{L}_{\phi}-\partial_{t})\widehat{H}.$
(5.9)

###### Proof.

First, by applying the weighted Bochner formula (2.2) we write the evolution
equation (5.3) for $F_{0}(x,t)$ to have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F_{0}(x,t)$
$\displaystyle\geq\Big{(}\frac{2|\nabla^{2}w|^{2}}{w^{2}}+\frac{6|\nabla
w|^{4}}{w^{4}}-\frac{8\langle\nabla^{2}w,\nabla w\otimes\nabla
w\rangle}{w^{3}}\Big{)}+\frac{2\mathscr{R}ic_{\phi}(\nabla w,\nabla
w)}{w^{2}}$ $\displaystyle\hskip 28.45274pt+\frac{2\langle\nabla
w,\nabla(\mathscr{L}_{\phi}w-w_{t})\rangle}{w^{2}}-\frac{2(p+1)}{p}\frac{|\nabla
w|^{4}}{w^{4}}$ $\displaystyle\hskip
28.45274pt+\alpha\Big{(}\mathscr{L}_{\phi}-\partial_{t}-\frac{2p}{\alpha}\frac{|\nabla
w|^{2}}{w^{2}}\Big{)}\widehat{H}.$ (5.10)

We will combine the evolution equations for $F_{0}(x,t)$ and $F_{1}(x,t)$ as
presented in Lemma 5.3, but before we proceed lets do further analysis on
(5.1). To do this we find some lower bounds as follows. By Young’s inequality
for $\epsilon\in(0,1)$ we have

$\displaystyle\frac{4\langle\nabla^{2}w,\nabla w\otimes\nabla
w\rangle}{w^{3}}\leq\frac{4|\nabla^{2}w|}{w}\frac{|\nabla w\otimes\nabla
w|}{w^{2}}\leq\frac{2\epsilon|\nabla^{2}w|^{2}}{w^{2}}+\frac{2}{\epsilon}\frac{|\nabla
w|^{4}}{w^{2}}$

and then

$\displaystyle\frac{2|\nabla^{2}w|^{2}}{w^{2}}+\frac{6|\nabla
w|^{4}}{w^{4}}-\frac{8\langle\nabla^{2}w,\nabla w\otimes\nabla
w\rangle}{w^{3}}$ (5.11) $\displaystyle\geq
2(1-\epsilon)\frac{|\nabla^{2}w|^{2}}{w^{2}}-4\Big{(}\frac{\langle\nabla^{2}w,\nabla
w\otimes\nabla w\rangle}{w^{3}}-\frac{|\nabla
w|^{4}}{w^{4}}\Big{)}-2\Big{(}\frac{1}{\epsilon}-1\Big{)}\frac{|\nabla
w|^{4}}{w^{4}}.$

Noting also by Newton inequality (2.9) (also check for (2.1.2)) we can write

$\displaystyle 2(1-\epsilon)\frac{|\nabla^{2}w|^{2}}{w^{2}}$
$\displaystyle\geq\frac{2(1-\epsilon)}{w^{2}}\Big{(}\frac{1}{m}(\mathscr{L}_{\phi}w)^{2}-\frac{\langle\nabla\phi,\nabla
w\rangle^{2}}{(m-n)}\Big{)}$
$\displaystyle\geq\frac{2(1-\epsilon)}{m}\Big{(}\frac{\mathscr{L}_{\phi}w}{w}\Big{)}^{2}-\frac{2\langle\nabla\phi,\nabla
w\rangle^{2}}{w^{2}(m-n)}$

since $2\epsilon\langle\nabla\phi,\nabla w\rangle^{2}/(w^{2}(m-n))\geq 0$ for
$0<\epsilon<1$. Thus we have

$\displaystyle
2(1-\epsilon)\frac{|\nabla^{2}w|^{2}}{w^{2}}+\frac{2\mathscr{R}ic_{\phi}(\nabla
w,\nabla
w)}{w^{2}}\geq\frac{2(1-\epsilon)}{m}\Big{(}\frac{\mathscr{L}_{\phi}w}{w}\Big{)}^{2}+\frac{2\mathscr{R}ic_{\phi}^{m}(\nabla
w,\nabla w)}{w^{2}}.$ (5.12)

In consideration of Lemma 5.2 we compute the term

$\displaystyle\frac{2\langle\nabla
w,\nabla(\mathscr{L}_{\phi}w-w_{t})\rangle}{w^{2}}$
$\displaystyle=\frac{4(p+1)}{p}\frac{\langle\nabla^{2}w,\nabla w\otimes\nabla
w\rangle}{w^{3}}-\frac{2(p+1)}{p}\frac{|\nabla w|^{4}}{w^{4}}$
$\displaystyle\hskip 28.45274pt+2p\frac{|\nabla
w|^{2}}{w^{2}}\widehat{H}+2p\frac{\langle\nabla
w,\nabla\widehat{H}\rangle}{w}.$ (5.13)

Substituting these equations (5.11)-(5.1) into (5.1) we have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F_{0}(x,t)$
$\displaystyle\geq\frac{2(1-\epsilon)}{m}\Big{(}\frac{\mathscr{L}_{\phi}w}{w}\Big{)}^{2}-2\Big{(}\frac{1}{\epsilon}-1\Big{)}\frac{|\nabla
w|^{4}}{w^{4}}+\frac{2\mathscr{R}ic_{\phi}^{m}(\nabla w,\nabla w)}{w^{2}}$
$\displaystyle+\frac{4}{p}\Big{(}\frac{\langle\nabla^{2}w,\nabla
w\otimes\nabla w\rangle}{w^{3}}-\frac{|\nabla
w|^{4}}{w^{4}}\Big{)}+2p\frac{\nabla w\nabla\widehat{H}}{w}$
$\displaystyle+\alpha\Big{(}\mathscr{L}_{\phi}-\partial_{t}\Big{)}\widehat{H}.$
(5.14)

The middle line on the right hand side of (5.1) (that is, the fourth and fifth
terms on the right hand side of (5.1)) are simplified to get

$\displaystyle\frac{4}{p}\Big{(}\frac{\langle\nabla^{2}w,\nabla w\otimes\nabla
w\rangle}{w^{3}}$ $\displaystyle-\frac{|\nabla
w|^{4}}{w^{4}}\Big{)}+2p\frac{\nabla w,}{w}\nabla\widehat{H}$
$\displaystyle=\frac{2}{p}\nabla F_{0}\nabla\log
w+2\Big{(}p-\frac{\alpha}{p}\Big{)}\frac{\nabla w\nabla\widehat{H}}{w}.$
(5.15)

Putting (5.1) into (5.1) gives (5.4) which is the desired evolution inequality
for $F_{0}(x,t)$.

∎

Now that we have evolution formulas for $F_{0}$ and $F_{1}$ on our hands the
next is to obtain evolution formula for $F$. Recall that $F=F_{0}+\beta F_{1}$
for $\beta>0$. Then combining (5.6) of Lemma 5.3 and (5.4) of Lemma 5.4
choosing $\beta=\alpha/p$ we have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F$
$\displaystyle\geq\frac{2(1-\epsilon)}{m}\Big{(}\frac{\mathscr{L}_{\phi}w}{w}\Big{)}^{2}-2\Big{(}\frac{1}{\epsilon}-1\Big{)}\frac{|\nabla
w|^{4}}{w^{4}}+\frac{2\mathscr{R}ic_{\phi}^{m}(\nabla w,\nabla w)}{w^{2}}$
$\displaystyle\hskip 28.45274pt+\frac{2}{p}\nabla F\nabla\log
w+2\Big{(}p-\frac{\alpha}{p}\Big{)}\frac{\nabla
w}{w}\nabla\widehat{H}+\alpha\mathscr{L}_{\phi}\widehat{H}.$ (5.16)

Clearly by Lemma 5.2 with $\beta=\alpha/p$ we have

$\displaystyle(\mathscr{L}_{\phi}w)/w=[(p+1)/p-p/\alpha](|\nabla
w|^{2}/w^{2})+(p/\alpha)F.$

Therefore, letting $\alpha=\delta p^{2}$ for $\delta>1$ yields

$\displaystyle(\mathscr{L}_{\phi}w)/w=(\delta p+\delta-1)/(\delta p)(|\nabla
w|^{2}/w^{2})+(1/\delta p)F.$ (5.17)

Substituting (5.17) into (5.1) we arrive at

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F\geq$
$\displaystyle\frac{2(1-\epsilon)}{m\delta^{2}p}F^{2}+\Big{[}\frac{2(1-\epsilon)}{m}\frac{(\delta
p+\delta-1)^{2}}{\delta^{2}p^{2}}-2\Big{(}\frac{1}{\epsilon}-1\Big{)}\Big{]}\frac{|\nabla
w|^{4}}{w^{4}}$ $\displaystyle+\frac{4(1-\epsilon)}{m}\frac{(\delta
p+\delta-1)}{\delta^{2}p^{2}}F\frac{|\nabla w|^{2}}{w^{2}}+\frac{2}{p}\nabla
F\nabla\log w$ $\displaystyle+\frac{2\mathscr{R}ic_{\phi}^{m}(\nabla w,\nabla
w)}{w^{2}}+2p(1-\delta)\frac{\nabla w}{w}\nabla\widehat{H}+\delta
p^{2}\mathscr{L}_{\phi}\widehat{H}.$ (5.18)

We can also compute the explicit value of the sum of the last two terms on the
right hand side of (5.1) using $\widehat{H}=H(u)/u$, $w=u^{-p}$. First note
that

$\nabla\widehat{H}=\nabla(H(u)/u)=H_{u}(\nabla u/u)=-(1/p)H_{u}\nabla w/w,$

where $H_{u}:=H^{\prime}(u)-H(u)/u$. Using this idea we thus compute the
second to the last term of (5.1) as

$\displaystyle 2p(1-\delta)\frac{\nabla
w}{w}\nabla\widehat{H}=-2(1-\delta)H_{u}|\nabla w|^{2}/w^{2}.$ (5.19)

Likewise we compute the last term of (5.1) as follows

$\displaystyle\mathscr{L}_{\phi}\widehat{H}$
$\displaystyle=e^{\phi}\text{div}(e^{-\phi}\nabla\widehat{H})=\nabla\cdot(\nabla\widehat{H})-\nabla\phi\nabla\widehat{H}$
$\displaystyle=\nabla\cdot(H_{u}(\nabla u/u))-H_{u}\nabla\phi\nabla u/u$
$\displaystyle=(uH^{\prime\prime}(u)-H_{u})|\nabla
u|^{2}/u^{2}+\nabla\cdot(\nabla u/u)-H_{u}\nabla\phi\nabla u/u$
$\displaystyle=(uH^{\prime\prime}(u)-2H_{u})|\nabla
u|^{2}/u^{2}+H_{u}(\mathscr{L}_{\phi}u)/u.$

Using $|\nabla u|^{2}/u^{2}=(1/p^{2})|\nabla w|^{2}/w^{2}$ and
$\mathscr{L}_{\phi}u=u_{t}-H(u)$ we have

$\displaystyle\alpha\mathscr{L}_{\phi}\widehat{H}=(\alpha/p^{2})(uH^{\prime\prime}(u)-2H_{u})|\nabla
w|^{2}/w^{2}-(\alpha/p)H_{u}w_{t}/w-\alpha H_{u}H(u)/u.$ (5.20)

Adding the expressions (5.19) and (5.20) and using $\alpha=\delta p^{2}$ and
$\beta=\alpha/p$, rearranging terms and observing some cancellation we obtain

$\displaystyle\alpha\mathscr{L}_{\phi}\widehat{H}$
$\displaystyle+2p(1-\delta)\frac{\nabla
w}{w}\nabla\widehat{H}=\Omega_{H}|\nabla w|^{2}/w^{2}-H_{u}F,$

where $\Omega_{H}:=-[H_{u}-(\alpha/p^{2})uH^{\prime\prime}(u)]$ and
$H_{u}:=H^{\prime}(u)-H(u)/u$. Combining the above discussion with Bakry-Émery
Ricci tensor lower bound, $\mathscr{R}ic_{\phi}^{m}\geq-(m-k)g$, $k\geq 0$ we
arrive at the final lemma in this section.

###  5.2. Fundamental lemma

######  Lemma 5.5.

With the assumption of Lemma 5.4 and if in addition,
$\mathscr{R}ic_{\phi}^{m}\geq-(m-k)g$, $k\geq 0$ holds we have

$\displaystyle(\mathscr{L}_{\phi}-\partial_{t})F\geq$
$\displaystyle\frac{2(1-\epsilon)}{m\delta^{2}p}F^{2}+\Big{[}\frac{2(1-\epsilon)}{m}\frac{(\delta
p+\delta-1)^{2}}{\delta^{2}p^{2}}-2\Big{(}\frac{1}{\epsilon}-1\Big{)}\Big{]}\frac{|\nabla
w|^{4}}{w^{4}}$ $\displaystyle+\frac{4(1-\epsilon)}{m}\frac{(\delta
p+\delta-1)}{\delta^{2}p^{2}}F\frac{|\nabla w|^{2}}{w^{2}}+\frac{2}{p}\nabla
F\nabla\log w$ $\displaystyle-2(m-1)k\frac{|\nabla
w|^{2}}{w^{2}}+\Omega^{H}|\nabla w|^{2}/w^{2}-H_{u}F,$ (5.21)

where $\Omega^{H}:=-[H_{u}-(\alpha/p^{2})uH^{\prime\prime}(u)]$ and
$H_{u}:=H^{\prime}(u)-H(u)/u$.

The last lemma will be combined with suitable cut-off function to finally
prove Theorem 5.1 in the remaining part of this section. Before we proceed we
briefly summarize the properties of such cut-off function as propagated by Li-
Yau [45].

###  5.3. A suitable cut-off function and its consequence

As in the previous sections, the distance function from the boundary
$\varrho_{\partial M}:M\to\mathbb{R}^{+}$ is defined $\varrho_{\partial
M}=d(\cdot,\partial M)$ which is smooth outside the cut locus of the boundary
$\text{Cut}(\partial M)$. For a reference point in $M$ we fix $T,R>0$ , and
define a cut-off function $\chi(r)$ in the interval $[0,+\infty)$. We set

$\displaystyle\psi(\varrho_{\partial M}(x))=\chi\left(\frac{\varrho_{\partial
M}(x)}{R}\right)$ (5.22)

for $x\in\partial M$ and $t>0$. The following existence result due to Li-Yau
[45] states its useful properties to the next localization procedure.

######  Lemma 5.6.

There exists a smooth non-increasing cut-off-function $\chi(r)$ atleast
$C^{2}([0,+\infty))$ such that

  1. (1)

$0\leq\chi(r)\leq 1$ for $0\leq r\leq 1$, $\chi(r)\equiv 1$ for $r\leq 1$ and
$\chi(r)\equiv 0$ for $r\geq 2$.

  2. (2)

$-c_{1}\chi^{1/2}\leq\chi^{\prime}(r)\leq 0$ and $\chi^{\prime\prime}(r)\geq-
C_{2}$ for $r\in[0,+\infty)$ and some constants $C_{1},C_{2}>0$.

  3. (3)

$\chi(r)$ is non-increasing as a radial function of r on $[0,+\infty)$.

Since the weighted Ricci curvature tensor is bounded below by
$\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k$, $k\geq 0$ and the weighted mean
curvature is bounded from below by $\mathscr{H}_{\phi}\geq-l$, $l\geq 0$, then
by the weighted Laplacian comparison theorem 2.1 we have
$\mathscr{L}_{\phi}\varrho_{\partial M}(x)\leq(m-1)kR+l$ for all $x\in
B_{R}(\partial M)$. Using the properties of the function defined by (5.22) as
stated in Lemma 5.6 we derive
$\nabla\psi=(\chi^{\prime}/R)\nabla\varrho_{\partial M}$ which implies

$\displaystyle|\nabla\psi|^{2}/\psi\leq C^{2}_{1}/R^{2}.$ (5.23)

We also derive $\Delta\psi=(\chi^{\prime}/R)\Delta\varrho_{\partial
M}+(\chi^{\prime\prime}/R)|\nabla\varrho_{\partial M}|^{2}$ which as well
implies

$\displaystyle\mathscr{L}_{\phi}\psi$
$\displaystyle=(\chi^{\prime}/R)\mathscr{L}_{\phi}\varrho_{\partial
M}+(\chi^{\prime\prime}/R)|\nabla\varrho_{\partial M}|^{2}$
$\displaystyle\geq-\frac{C_{1}}{R}[(m-1)kR+l]\chi^{\frac{1}{2}}-\frac{C_{2}}{R^{2}}=-\frac{1}{R^{2}}[C_{1}((m-1)kR+l)R+C_{2}].$
(5.24)

###  5.4. Proof of Theorem 5.1

Since $\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k$ and $\mathscr{H}_{\phi}\geq-l$,
$k,l\geq 0$, we will make use of the weighted Laplacian comparison theorem as
in the previous section, and consequently, a cut-off function described in
Lemma 5.6 will be applied. First, we define a function $G(x,t)=tF(x,t)$. Again
in the consideration of Calabi’s trick we assume that the function
$\psi(x)G(x,t)$ is smoothly supported in $B_{2R}(\partial M)$, where $\psi$ is
the function in Lemma 5.6 and $\psi(x)G(x,t)$ is spatially localized. Suppose
also that $\psi G$ attains its maximum at the point $(x_{0},t_{0})\in
B_{2R}(\partial M)\times[0,T]$ for any fixed $t>0$. Without loss of
generality, we will assume that $\psi(x_{0})G(x_{0},t_{0})>0$, otherwise the
result is obviously trivial. As before, the proof is divided into two cases,
according to whether $x_{0}\not\in\partial M$ or $x_{0}\in\partial M$.

Case 1: If $x_{0}\not\in\partial M$,, we assume
$x_{0}\not\in\text{Cut}(\partial M)$ without loss of generality. By the
maximum principle we have at $(x_{0},t_{0})$

$\displaystyle\nabla(\psi G)=0,\ \ \mathscr{L}_{\phi}(\psi G)\leq 0\ \ \
\partial_{t}(\psi G)\geq 0.$ (5.25)

Noting that the first item in (5.25) yields $\psi\nabla G=-\nabla\psi G$ which
implies $\nabla G=-(\nabla\psi/\psi)G$, while the second item in (5.25) yields
$\psi\mathscr{L}_{\phi}G+2\nabla\psi\nabla G+\mathscr{L}_{\phi}\psi\cdot G\leq
0.$ After combining these two items we obtain

$\displaystyle\psi\mathscr{L}_{\phi}G-2(|\nabla\psi|^{2}/\psi)G+\mathscr{L}_{\phi}\psi\cdot
G\leq 0.$ (5.26)

Adding the third item in (5.25) and (5.26), and referring to the cut-off
function estimates (5.23) and (5.3) we get

$\displaystyle\psi(\mathscr{L}_{\phi}-\partial_{t})G$
$\displaystyle\leq(2|\nabla\psi|^{2}/\psi-\mathscr{L}_{\phi}\psi)G$
$\displaystyle\leq\frac{1}{R^{2}}\Big{(}2C_{1}^{2}+C_{2}+C_{1}[(m-1)kR+l]R\Big{)}G=:\Phi
G,$ (5.27)

where $\Phi=\Phi(C_{1},C_{2},n,m,k,R,l)$. At this juncture, Lemma 5.5 can now
be applied by taking the evolution of the quantity $G=tF$:

$\displaystyle\psi(\mathscr{L}_{\phi}-\partial_{t})G=$ $\displaystyle
t\psi(\mathscr{L}_{\phi}-\partial_{t})F-G/t$ $\displaystyle\geq$
$\displaystyle\frac{2(1-\epsilon)}{m\delta^{2}p}\frac{G^{2}}{t}+\Big{[}\frac{2(1-\epsilon)}{m}\frac{(\delta
p+\delta-1)^{2}}{\delta^{2}p^{2}}-2\Big{(}\frac{1}{\epsilon}-1\Big{)}\Big{]}\frac{|\nabla
w|^{4}}{w^{4}}t$ $\displaystyle+\frac{4(1-\epsilon)}{m}\frac{(\delta
p+\delta-1)}{\delta^{2}p^{2}}\frac{|\nabla w|^{2}}{w^{2}}G+\frac{2}{p}\nabla
G\nabla\log w$ $\displaystyle-2(m-1)k\frac{|\nabla
w|^{2}}{w^{2}}t+\Omega^{H}|\nabla w|^{2}/w^{2}t-H_{u}G-G/t,$ (5.28)

where $\Omega^{H}:=-[H_{u}-(\alpha/p^{2})uH^{\prime\prime}(u)]$ and
$H_{u}:=H^{\prime}(u)-H(u)/u$.

By the Cauchy-Schwarz and Young’s inequalities we estimates

$\displaystyle 2(m-1)k\frac{|\nabla
w|^{2}}{w^{2}}t\leq\frac{2(1-\epsilon)(\delta-1)^{2}}{m\delta^{2}p^{2}}\frac{|\nabla
w|^{4}}{w^{4}}t+\frac{m\delta^{2}p^{2}k^{2}}{2(1-\epsilon)(\delta-1)^{2}}t,$
(5.29) $\displaystyle\Omega_{H}|\nabla
w|^{2}/w^{2}t\geq-\frac{(\delta-1)^{2}}{2}\frac{|\nabla
w|^{4}}{w^{4}}t-\frac{1}{(\delta-1)^{2}}\sup_{\mathcal{Q}_{2R,T}}\\{[\Omega^{H}]_{+}\\}^{2}t.$
(5.30)

In the next we denote by $\mu^{H}$ and $\gamma^{H}$ the constants

$\mu^{H}=\sup_{\mathcal{Q}_{2R,T}}\\{[H_{u}]_{+}\\}\ \ \text{and}\ \
\gamma^{H}=\sup_{\mathcal{Q}_{2R,T}}\\{[\Omega^{H}]_{+}\\}.$

Choosing $\delta>1$ and $p>0$ such that
$1/p>\Big{[}(m\delta)/[(1-\epsilon)(\delta-1)]\Big{]}\Big{[}1/\epsilon-1+(\delta-1)^{2}/4\Big{]}$
satisfies the hypothesis (5.2) and again makes

$\displaystyle\Big{[}\frac{2(1-\epsilon)}{m}\frac{(\delta-1)}{\delta
p}-2\Big{(}\frac{1}{\epsilon}-1\Big{)}-\frac{(\delta-1)^{2}}{2}\Big{]}\frac{|\nabla
w|^{4}}{w^{4}}t\geq 0.$ (5.31)

Now plugging in (5.29) - (5.31) into (5.4), noting some cancellation and
multiplying through by $\psi$ gives

$\displaystyle\psi(\mathscr{L}_{\phi}-\partial_{t})G\geq$
$\displaystyle\frac{2(1-\epsilon)}{m\delta^{2}p}\frac{G^{2}}{t}\psi+\frac{4(1-\epsilon)}{m}\frac{(\delta
p+\delta-1)}{\delta^{2}p^{2}}\frac{|\nabla w|^{2}}{w^{2}}G\psi$
$\displaystyle+\frac{2}{p}\psi\nabla G\nabla\log
w-\frac{m\delta^{2}p^{2}k^{2}}{2(1-\epsilon)(\delta-1)^{2}}t$
$\displaystyle-\frac{1}{(\delta-1)^{2}}{\gamma^{H}}^{2}\psi
t-\mu^{H}G\psi-\psi G/t.$ (5.32)

By Young’s inequality again while using the identity $\psi\nabla
G=-G\nabla\psi$ derived from (5.25) we have

$\displaystyle\frac{2}{p}\psi\nabla G\nabla\log w$
$\displaystyle=-\frac{2}{p}G\nabla\psi\frac{\nabla w}{w}$
$\displaystyle\geq-\frac{4(1-\epsilon)}{m}\frac{(\delta
p+\delta-1)}{\delta^{2}p^{2}}\frac{|\nabla
w|^{2}}{w^{2}}G\psi-\frac{m\delta^{2}}{4(1-\epsilon)(\delta
p+\delta-1)}\frac{|\nabla\psi|^{2}}{\psi}G.$ (5.33)

Recall that the quantity appearing on the left hand side of (5.4) has been
estimated in (5.4). Therefore combining (5.4), (5.4) and (5.4) and rearranging
terms leads to

$\displaystyle\frac{2(1-\epsilon)}{m\delta^{2}p}\frac{G^{2}}{t}\psi-\frac{m\delta^{2}}{4(1-\epsilon)(\delta
p+\delta-1)}$
$\displaystyle\frac{|\nabla\psi|^{2}}{\psi}G-\frac{m\delta^{2}p^{2}k^{2}}{2(1-\epsilon)(\delta-1)^{2}}t$
$\displaystyle-\frac{1}{(\delta-1)^{2}}{\gamma^{H}}^{2}\psi
t-\mu^{H}G\psi-\psi G/t-\Phi G\leq 0.$

To obtain a more compact expression from the last inequality, multiply through
it by $\psi t$ again and rearrange terms we obtain

$\displaystyle\frac{2(1-\epsilon)}{m\delta^{2}p}G^{2}\psi^{2}-(1+\mu^{H}t+\widetilde{\Phi}t)G\psi$
$\displaystyle-t^{2}\Big{[}\frac{m\delta^{2}p^{2}k^{2}}{2(1-\epsilon)(\delta-1)^{2}}+\frac{1}{(\delta-1)^{2}}{\gamma^{H}}^{2}\Big{]}\psi^{2}\leq
0,$ (5.34)

where
$\widetilde{\Phi}=\frac{C_{1}^{2}}{R^{2}}\frac{m\delta^{2}}{4(1-\epsilon)(\delta
p+\delta-1)}+\Phi$ with $\Phi$ being as defined in (5.4).

Observe that the last inequality is obviously quadratic in $\psi G$ and can
therefore be compared with the following quadratic inequality $ay^{2}-by-c\leq
0$ with $a>0$, $b,c\geq 0$ being real coefficients, which can be solved using
the formula $y=b/a+\sqrt{c/a}$. So setting $y=\psi G$,
$a=2(1-\epsilon)/(m\delta^{2}p^{2})$, $b=(1+\mu^{H}t+\widetilde{\Phi}t)$ and

$c=\frac{m\delta^{2}p^{2}k^{2}}{2(1-\epsilon)(\delta-1)^{2}}+\frac{1}{(\delta-1)^{2}}{\gamma^{H}}^{2}$

we solve (5.34) for $\psi G$ at $(x_{0},t_{0})$ and obtain

$\displaystyle\psi G$
$\displaystyle\leq\frac{m\delta^{2}p^{2}}{2(1-\epsilon)}(1+\mu^{H}t_{0}+\widetilde{\Phi}t_{0})+t_{0}\sqrt{\frac{m\delta^{2}p^{2}}{2(1-\epsilon)}}\sqrt{\Big{(}\frac{m\delta^{2}p^{2}k^{2}}{2(1-\epsilon)(\delta-1)^{2}}+\frac{1}{(\delta-1)^{2}}{\gamma^{H}}^{2}\Big{)}\psi^{2}}$
$\displaystyle\leq\frac{m\delta^{2}p^{2}}{2(1-\epsilon)}(1+\mu^{H}T+\widetilde{\Phi}T)+T\sqrt{\frac{m\delta^{2}p^{2}}{2(1-\epsilon)}}\left(\sqrt{\frac{m\delta^{2}p^{2}k^{2}}{2(1-\epsilon)(\delta-1)^{2}}}+\frac{1}{(\delta-1)}\gamma^{H}\right)\psi.$

Taking into consideration the properties of $\psi$ (see Lemma 5.6) we set
$\psi\equiv 1$ in $B_{2R}$ and we finally write

$\displaystyle\sup G(x,t)\leq\sup\\{\psi(x_{0})G(x_{0},t_{0})\\}$
$\displaystyle\leq\frac{m\delta^{2}p^{2}}{2(1-\epsilon)}(1+\mu^{H}T+\widetilde{\Phi}T)+T\sqrt{\frac{m\delta^{2}p^{2}}{2(1-\epsilon)}}\left(\sqrt{\frac{m\delta^{2}p^{2}k^{2}}{2(1-\epsilon)(\delta-1)^{2}}}+\frac{1}{(\delta-1)}\gamma^{H}\right)\psi.$

at $(x_{0},t_{0})\in\mathcal{Q}_{2R,T}(\partial M)$.

Noting that $G=tF$ with $F$ defined by $F=|\nabla
w|^{2}/w^{2}+\alpha\widehat{H}(w)+\beta w_{t}/w$, $\alpha,\beta>0$ and by
choice $\beta=\alpha/p$ and $\alpha=\delta p^{2}$. Thus, referring to
$w=u^{-p}$ and $\widehat{H}(w)=H(u)/u$ we can write

$G/t=F=p^{2}|\nabla u|^{2}/u^{2}+\delta p^{2}H(u)/u-\delta p^{2}u_{t}/u.$

Therefore

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}$
$\displaystyle+\delta\frac{H(u)}{u}-\delta\frac{u_{t}}{u}$
$\displaystyle\leq\frac{m\delta^{2}}{2(1-\epsilon)}\Big{(}\frac{1}{T}+\mu^{H}+\widetilde{\Phi}\Big{)}+\frac{m\delta^{2}k}{2(1-\epsilon)(1-\delta)}+\frac{1}{p^{2}(\delta-1)}\sqrt{\frac{m\delta^{2}p^{2}}{2(1-\epsilon)}}\gamma^{H}.$

at $(x,t)\in\mathcal{Q}_{R,T}(\partial M)$. Therefore (5.1) follows since
$T>0$ was arbitrarily fixed and the theorem is proved for the case
$x\not\in\partial M$.

Case 2: If $x_{0}\in\partial M$, we will show that the gradient estimates
(5.1) holds on $\partial M\times[0,T]$. As in the discussion ensued in
previous sections, at the maximum point $(x_{0},t_{0})$ over $\partial
M\times[0,T]$ we have

$\displaystyle F_{\nu}=(|\nabla w|^{2}/w^{2})_{\nu}\geq 0$ (5.35)

since $w_{t}=0$ and $\alpha\widehat{H}+\beta w_{t}/w\geq 0$ on $\partial
M\times[0,T]$ as informed by the theorem assumption $u_{t}\leq H(u)$,
$u_{t}=0$ on $\partial M\times[0,T]$. This can be seen by setting $w=u^{-p}$
with $\beta=\alpha/p\geq 0$.

Noting also that $|\nabla w|=w_{v}\geq 0$ since $u>0$ and $u_{v}\geq 0$. By
the Reilly-type formula (2.1.3) we have

$\displaystyle(|\nabla
w|^{2}/w^{2})_{\nu}=2(w_{\nu}/w^{2})[\mathscr{L}_{\phi}w-\mathscr{H}_{\phi}w_{\nu}]-2w_{\nu}^{2}w_{\nu}/w^{3}.$
(5.36)

From Lemma 5.2 we have
$\mathscr{L}_{\phi}w=((p+1)/p)(w_{\nu}^{2}/w)+w_{t}+pw\widehat{H}$. It then
follows by combining (5.35) and (5.36) that

$\displaystyle
0\leq(w_{\nu}/w^{2})[(1/p)(w_{\nu}^{2}/w)+w_{t}+pw\widehat{H}-\mathscr{H}_{\phi}w_{\nu}]$

at $(x_{0},t_{0})$. Obviously the condition $u_{t}\leq H(u)$ implies
$w_{t}+pw\widehat{H}\geq 0$ over $\partial M\times[0,T]$. Hence

$\displaystyle-(1/p)(w_{\nu}/w)+\mathscr{H}_{\phi}\leq 0.$

Using the fact that $w_{\nu}/w=-pu_{\nu}/u$ and $p>0$ we obtain

$u_{v}+\mathscr{H}_{\phi}\leq 0$

at $(x_{0},t_{0})$. This implies that

$F(x_{0},t_{0})\leq l^{2}$

which holds for all $(x,t)\in\mathcal{Q}_{R,T}(\partial M)$ by noting that
$\sup F(x,t)\leq\sup F(x_{0},t_{0})$. This therefore completes the proof.

∎

##  6\. Applications of Li-Yau type gradient estimates (Theorem 5.1)

###  6.1. Global Li-Yau gradient estimates

Theorem 5.1 which is local in nature implies the following global gradient
estimates under appropriate global bounds as seen in the following theorem.

######  Theorem 6.1.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary whose mean curvature satisfies the bound
$\mathscr{H}_{\phi}(\partial M)\geq-l$, $l\geq 0$. Let $u=u(x,t)$ be a smooth
positive solution to (1.1) such that $u$ satisfies the Dirichlet boundary
condition (i.e., $u(\cdot,t)|_{\partial M}$ is constant for each time $t$),
$u_{\nu}\geq 0$ and $u_{t}\leq H(u)$ on $\partial M\times(-\infty,+\infty)$.
Denote

$\mu^{H}:=\sup_{M\times(-\infty,+\infty)}\\{[H_{u}]_{+}\\}\ \ \text{and}\ \
\gamma^{H}:=\sup_{M\times(-\infty,+\infty)}\\{[\Omega^{H}]_{+}\\},$

where the quantities $[H_{u}]_{+}$ and $[\Omega^{H}]_{+}$ are as defined in
Theorem 5.1.

  1. (1)

Suppose $\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k,\ k\geq 0$, then for all $x\in M$
and $t\neq 0$, we have

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}$
$\displaystyle+\delta\frac{H(u)}{u}-\delta\frac{u_{t}}{u}\leq\frac{m\delta^{2}}{2}\Big{(}\frac{1}{t}+\frac{k}{\delta-1}+\mu^{H}\Big{)}+\frac{\delta}{p(\delta-1)}\sqrt{\frac{m}{2}}\gamma^{H},$
(6.1)

where $\delta>1$ and $p>0$.

  2. (2)

Suppose $\mathscr{R}ic_{\phi}^{m}\geq 0$, then for all $x\in M$ and $t\neq 0$,
we have

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}$
$\displaystyle+\delta\frac{H(u)}{u}-\delta\frac{u_{t}}{u}\leq\frac{m\delta^{2}}{2t}+\frac{m\delta^{2}}{2}\mu^{H}+\frac{\delta}{p(\delta-1)}\frac{\sqrt{m}}{\sqrt{2}}\gamma^{H},$
(6.2)

where $\delta>1$ and $p>0$.

###### Proof.

The proof is a direct consequence of Theorem 5.1 once the bounds on
$\mathscr{R}ic_{\phi}^{m}$ and $\mathscr{H}_{\phi}$, and all the supremums are
taken in global sense. Estimates (6.1) therefore follows from (5.1) by sending
$\epsilon\searrow 0$, whilts (6.2) follows also by setting $k\equiv 0$ since
$\mathscr{R}ic_{\phi}^{m}\geq 0$ for all $x\in M$. ∎

###  6.2. Li-Yau Harnack inequalities

The gradient estimates obtained in Theorem 5.1 can be utilized to derive
parabolic Harnack inequality for the positive solution of (1.1) on
$M\times(-\infty,\infty)$ as seen in the following theorem.

######  Theorem 6.2.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary and let $u$ be a smooth positive solution to (1.1). Under the
assumptions of Theorem 5.1 we have for all $(x,t)\in\mathcal{Q}_{R,T}(\partial
M)$ with $x_{1},x_{2}\in B_{R}(\partial M)$, $-\infty<t_{1}<t_{2}<+\infty$,
$t_{1},t_{2}\neq 0$ and $\delta>1$ that

$\displaystyle u(x_{1},t_{1})\leq
u(x_{2},t_{2})\left(\frac{t_{2}}{t_{1}}\right)^{\frac{m\delta}{2(1-\epsilon)}}\exp\left\\{\frac{\delta
d^{2}(x_{1},x_{2})}{4(t_{2}-t_{1})}+A(t_{2}-t_{1})\right\\},$ (6.3)

where $d(x_{1},x_{2})$ is the geodesic distance between $x_{1}$ and $x_{2}$,
and

$\displaystyle A$
$\displaystyle=A(m,p,\delta,\epsilon,k,\mu^{H},\gamma^{H},\sigma^{H})$
$\displaystyle=\frac{m\delta}{2(1-\epsilon)}\Big{(}\frac{k}{\delta-1}+\widetilde{\Phi}+\mu^{H}\Big{)}+\frac{1}{p(\delta-1)}\sqrt{\frac{m}{2(1-\epsilon)}}\gamma^{H}-\sigma^{H}$

with $\displaystyle\sigma^{H}=\inf_{\mathcal{Q}_{2R,T}(\partial
M)}\left\\{[H(u)/u]_{-}\right\\}$, $\delta>1$ and $p>0$.

###### Proof.

For the purpose of the proof of Li-Yau Harnack inequality, (5.1) can be
written as

$\displaystyle-\frac{u_{t}}{u}\leq\frac{m\delta}{2(1-\epsilon)t}-\frac{|\nabla
u|^{2}}{\delta u^{2}}+A,$ (6.4)

where

$\displaystyle A$
$\displaystyle=\frac{m\delta}{2(1-\epsilon)}\Big{(}\frac{k}{\delta-1}+\widetilde{\Phi}+\mu^{H}\Big{)}+\frac{1}{p(\delta-1)}\sqrt{\frac{m}{2(1-\epsilon)}}\gamma^{H}-\sigma^{H},$
$\displaystyle\sigma^{H}$ $\displaystyle=\inf_{\mathcal{Q}_{2R,T}(\partial
M)}\left\\{[H(u)/u]_{-}\right\\}.$

Let $\gamma$ be the shortest geodesic joining $x_{1}$ and $x_{2}$ in
$\mathcal{Q}_{R,T}(\partial M)$ such that $\gamma(t_{1})=x_{1}$ and
$\gamma(t_{2})=x_{2}$. By a straightforward computation we have

$\displaystyle\log u(\gamma(t),t)\Big{|}_{t_{1}}^{t_{2}}$
$\displaystyle=\int_{t_{1}}^{t_{2}}\frac{d}{dt}\log u(\gamma(t),t)dt$
$\displaystyle=\int_{t_{1}}^{t_{2}}\left[\dot{\gamma}\nabla
u/u+u_{t}/u\right]dt$
$\displaystyle\geq\int_{t_{1}}^{t_{2}}\left[\dot{\gamma}\nabla u/u+|\nabla
u|^{2}/(\delta u^{2})-A-m\delta/2(1-\epsilon)t\right]dt$

by using (6.4). Applying the completing the square method to rewrite the
expression $\dot{\gamma}\nabla u/u+|\nabla u|^{2}/(\delta u^{2})$ and using
the fact that $(1/\delta)\left|(\delta/2)\dot{\gamma}+\nabla
u/u\right|^{2}\geq 0$, we have

$\displaystyle\log u(x_{2},t_{2})-\log
u(x_{1},t_{1})\geq\int_{t_{1}}^{t_{2}}\left[-\frac{\delta}{4}|\dot{\gamma}|^{2}-A-\frac{m\delta}{2(1-\epsilon)t}\right]dt.$

Therefore

$\displaystyle\log\frac{u(x_{2},t_{2})}{u(x_{1},t_{1})}\geq-\int_{t_{1}}^{t_{2}}\frac{\delta}{4}|\dot{\gamma}|^{2}dt-A(t_{2}-t_{1})-\frac{m\delta}{2(1-\epsilon)}\log\left(\frac{t_{2}}{t_{1}}\right).$
(6.5)

By exponentiating both sides of (6.5) we arrive at

$\displaystyle
u(x_{2},t_{2})/u(x_{1},t_{1})\geq\left(t_{2}/t_{1}\right)^{-\frac{m\delta}{2(1-\epsilon)}}\exp\left\\{\delta/4\int_{t_{1}}^{t_{2}}|\dot{\gamma}|^{2}dt-A(t_{2}-t_{1})\right\\},$

which when rearranging yields

$\displaystyle u(x_{1},t_{1})\leq
u(x_{2},t_{2})\left(t_{2}/t_{1}\right)^{\frac{m\delta}{2(1-\epsilon)}}\exp\left\\{\delta\rho+A(t_{2}-t_{1})\right\\},$

where
$\displaystyle\rho=\rho(x_{1},x_{2},t_{2}-t_{1})=\inf_{\gamma}\left\\{\frac{1}{4(t_{2}-t_{2})}\int_{0}^{1}|\dot{\gamma}|^{2}dt\right\\}.$
We arrive at the desired inequality by using the fact that
$d(x_{1},x_{2})=\inf_{\gamma\in\Gamma}\int_{0}^{1}|\dot{\gamma}|dt$ since
every pair of points in $M$ can be joined by the minimizing geodesic, with

$\Gamma:=\left\\{\gamma:[0,1]\to M\ |\
\gamma(0)=x_{1},\gamma(1)=x_{2}\right\\}$

being a smooth curve in $\mathcal{Q}_{2R,T}(\partial M)$.

∎

Theorem 6.2 gives the local Li-Yau inequality, which also implies some global
form if the bound are taken in global sense with the same argument. For
instance we obtain the following corollary.

######  Corollary 6.3.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq 0$ and
$\mathscr{H}_{\phi}\geq 0$. Let $u$ be a smooth positive solution to (1.1)
satisfying the assumption of Theorem 6.1. Then we have for all $x_{1},x_{2}\in
M$ $-\infty<t_{1}<t_{2}<+\infty$, $t_{1},t_{2}\neq 0$ that

$\displaystyle u(x_{1},t_{1})\leq
u(x_{2},t_{2})\left(\frac{t_{2}}{t_{1}}\right)^{\frac{m}{2}}\exp\left\\{\frac{\delta
d^{2}(x_{1},x_{2})}{4(t_{2}-t_{1})}+\widetilde{A}(t_{2}-t_{1})\right\\},$
(6.6)
$A=A(m,p,\delta,\mu^{H},\gamma^{H},\sigma^{H})=\frac{m}{2}\mu^{H}+\frac{1}{p(\delta-1)}\frac{\sqrt{m}}{\sqrt{2}}\gamma^{H}-\sigma^{H}$

with
$\displaystyle\sigma^{H}=\inf_{M\times(-\infty,+\infty)}\left\\{[H(u)/u]_{-}\right\\}$,
$\delta>1$ and $p>0$.

###### Proof.

The proof of Corollary 6.3 follows the same argument as in the proof of
Theorem 6.2 with the use of global bounds and global gradient estimates in
Theorem 6.1. Since $\mathscr{R}ic_{\phi}^{m}\geq 0$ we just set $k\equiv 0$
and $\epsilon\searrow 0$, and we obtain (6.6) at once.

∎

######  Remark 6.4.

If we set $\epsilon=1/2$ we still obtain (6.6) with power of $(t_{2}/t_{1})$
becoming $m$, while the quantity $A$ becomes
$A=m\mu^{H}+(\sqrt{m}/p)\gamma^{H}-\sigma^{H}$.

###  6.3. Li-Yau type gradient estimates for elliptic equations and Liouville-
type theorems

Here we discuss Li-Yau type gradient estimates for elliptic form (steady
state) of (1.1). The first result here is the local gradient estimates on the
positive smooth solution of the following weighted elliptic equation

$\displaystyle\begin{cases}\displaystyle\mathscr{L}_{\phi}u+H(u)&=0,\\\
\displaystyle\hskip 42.67912ptu|_{\partial M}&=\text{const.},\end{cases}$
(6.7)

where $H=H(s)$ is a sufficiently smooth nonlinear function for $s>0$.

######  Theorem 6.5.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k$ and
$\mathscr{H}_{\phi}\geq-l$, $k,l\geq 0$. Let $u=u(x)$ be a smooth positive
solution to (6.7) on $B_{R}(\partial M)$ such that $u_{\nu}\geq 0$ over
$\partial M$. Then for some $\epsilon\in(0,1)$ and $\delta>1$, we have

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}$
$\displaystyle+\delta\frac{H(u)}{u}\leq\frac{m\delta^{2}}{2(1-\epsilon)}\Big{(}\frac{k}{\delta-1}+\widetilde{\Phi}+\widetilde{\mu}^{H}\Big{)}+\frac{\delta}{p(\delta-1)}\sqrt{\frac{m}{2(1-\epsilon)}}\widetilde{\gamma}^{H},$
(6.8)

for all $x\in B_{R}(\partial M)$ with
$0<p\leq[(1-\epsilon)(\delta-1)]/\left[m\delta\left(\frac{1}{\epsilon}-1+\frac{(\delta-1)^{2}}{4}\right)\right]$,
where

$\widetilde{\mu}^{H}:=\sup_{B_{R}(\partial M)}\\{[H_{u}]_{+}\\}\ \ \text{and}\
\ \widetilde{\gamma}^{H}:=\sup_{B_{R}(\partial M)}\\{[\Omega^{H}]_{+}\\},$

$[H_{u}]_{+}$ and $[\Omega^{H}]_{+}$ are as defined in Theorem 5.1.

Globally we have the following Li-Yau type gradient estimates.

######  Theorem 6.6.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq 0$ and
$\mathscr{H}_{\phi}\geq 0$. Let $u=u(x)$ be a smooth positive solution to
(6.7) such that $u_{\nu}\geq 0$ over $\partial M$. Then for some $\delta>1$,
we have

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}$
$\displaystyle+\delta\frac{H(u)}{u}\leq\frac{m\delta^{2}}{2(1-\epsilon)}\widetilde{\mu}_{M}^{H}+\frac{\delta}{p(\delta-1)}\sqrt{\frac{m}{2(1-\epsilon)}}\widetilde{\gamma}_{M}^{H},$
(6.9)

for all $x\in M$, where

$\widetilde{\mu}_{M}^{H}:=\sup_{M}\\{[H_{u}]_{+}\\}\ \ \text{and}\ \
\widetilde{\gamma}_{M}^{H}:=\sup_{M}\\{[\Omega^{H}]_{+}\\}.$

In particular, choosing
$p=[(1-\epsilon)(\delta-1)]/\left[m\delta\left(\frac{1}{\epsilon}-1+\frac{(\delta-1)^{2}}{4}\right)\right]$,
$\delta=2$ and $\epsilon=\frac{1}{2}$,

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}$ $\displaystyle+2\frac{H(u)}{u}\leq
4m\ \widetilde{\mu}_{M}^{H}+10m\sqrt{m}\ \widetilde{\gamma}_{M}^{H}.$ (6.10)

As a consequence of the above Li-Yau type gradient estimates we can prove the
following Liouville-type results on nonnegative weighted Ricci curvature and
mean curvature conditions.

######  Theorem 6.7.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq 0$ and
$\mathscr{H}_{\phi}\geq 0$. Let $u=u(x)$ be a smooth bounded positive solution
to (6.7) such that $u_{\nu}\geq 0$ over $\partial M$, where $H(u)\geq 0$,
$H^{\prime}(u)-H(u)/u\leq 0$ and $H^{\prime}(u)-H(u)/u-\delta
uH^{\prime\prime}(u)\geq 0$ for $\delta>1$ and all $x\in M$. Then $u$ must be
a constant.

###  6.4. Some special examples of $H(u)$

Lastly in this subsection, we demonstrate that the global estimates in Theorem
6.6 can be applied to investigate Liouville-type results for more special
cases of $H(u)$ with a power like growth or logarithmic-type or exponential-
type nonlinearities, such as cases of $H(u)=au^{\lambda}(\log u)^{\alpha}$,
$H(u)=au^{q}+bu^{s}$, $H(u)=Ae^{au}$ or a mixture of these nonlinearities.

### 1\. The case $H(u)=au^{\lambda}\log u$ with $a<0$, $\lambda\geq 1$

Consider $H(u)=au^{\lambda}(\log u)^{\alpha}$, but for simplicity we keep
$\alpha=1$, $a\leq 0$ and $\lambda\geq 1$. Thus we consider the specific case
$H(u)=au^{\lambda}\log u$ for bounded positive function $u$. Here we see that

$H_{u}=H^{\prime}(u)-H(u)/u=a\left(1+(\lambda-1)\log u\right)u^{\lambda-1}$

and

$H^{\prime}(u)-H(u)/u-\delta
uH^{\prime\prime}(u)=a\left([1-\delta(2\lambda-1)]+[(\lambda-1)(1-\delta\lambda)]\log
u\right)u^{\lambda-1}$

which satisfy the assumptions that $H^{\prime}(u)-H(u)/u\leq 0$ and
$H^{\prime}(u)-H(u)/u-\delta uH^{\prime\prime}(u)\geq 0$ for $0<u(x)<c\leq 1$
since $\delta>1$, $a\leq 0$ and $\lambda\geq 1$. Clearly

$\widetilde{\mu}^{H}_{M}=\sup_{M}\\{[H_{u}]_{+}\\}=[a(1+(\lambda-1)\log
u)]_{+}\sup_{M}\\{u^{\lambda-1}\\}=0$

and

$\widetilde{\gamma}_{M}^{H}=\sup_{M}\\{[\Omega^{H}]_{+}\\}=\sup_{M}\\{[-[H^{\prime}(u)-H(u)/u-\delta
uH^{\prime\prime}(u)]]_{+}\\}=0.$

By the above discussion we have the following Liouville-type results

######  Proposition 6.8.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq 0$ and
$\mathscr{H}_{\phi}\geq 0$. Let $u$ be a bounded positive solution to

$\displaystyle\begin{cases}\displaystyle\mathscr{L}_{\phi}u+au^{\lambda}\log
u&=0,\hskip 14.22636pt\lambda\geq 1\\\ \displaystyle\hskip
56.9055ptu|_{\partial M}&=\text{const.}\end{cases}$ (6.11)

such that $u_{\nu}\geq 0$ over $\partial M$. If $a\leq 0$, then $u$ must be a
constant. If a is further restricted to $a<0$, then
$u\geq\exp\\{2/(1-\lambda)\\}$.

###### Proof.

Combining the above discussion (which gives $\widetilde{\mu}^{H}_{M}=0$,
$\widetilde{\gamma}_{M}^{H}=0$) together with (6.10) (Theorem 6.6), it then
follows that $|\nabla u|\equiv 0$. Referring to the assumption that $H(u)\geq
0$ and $H^{\prime}(u)-H(u)/u-\delta uH^{\prime\prime}(u)\geq 0$, we have $\log
u\geq 2/(1-\lambda)$.

∎

### 2\. The case $H(u)=au^{q}+bu^{s}$ with real coefficients $a,b$ and real
exponents $s,q$

Consider $H(u)=au+bu^{q}$ for bounded positive function $u$. Here we compute
that

$H_{u}=H^{\prime}(u)-H(u)/u=b(q-1)u^{q-1}$

and

$H^{\prime}(u)-H(u)/u-\delta uH^{\prime\prime}(u)=b(q-1)(1-\delta q)u^{q-1}$

which satisfy the assumptions that $H^{\prime}(u)-H(u)/u\leq 0$ and
$H^{\prime}(u)-H(u)/u-\delta uH^{\prime\prime}(u)\geq 0$ for $\delta>1$ and
for either $b\geq 0$ and $q<1$ or $b\leq 0$ and $q>1$. Clearly

$\widetilde{\mu}^{H}_{M}=\sup_{M}\\{[H_{u}]_{+}\\}=[b(q-1)]_{+}\sup_{M}\\{u^{q-1}\\}=0$

and

$\widetilde{\gamma}_{M}^{H}=\sup_{M}\\{[\Omega^{H}]_{+}\\}=[b(q-1)(\delta
q-1)]_{+}\sup_{M}\\{u^{q-1}\\}=0.$

The following Liouville-type results can now be proved in a similar way.

######  Proposition 6.9.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq 0$ and
$\mathscr{H}_{\phi}\geq 0$. Let $u$ be a bounded positive solution to

$\displaystyle\begin{cases}\displaystyle\mathscr{L}_{\phi}u+au+bu^{q}&=0,\\\
\displaystyle\hskip 56.9055ptu|_{\partial M}&=\text{const}.\end{cases}$ (6.12)

on $M$ such that $u_{\nu}\geq 0$ over $\partial M$, where $a,b,q\in\mathbb{R}$
and $q\neq 1$. If either $b\geq 0$ and $q<1$ or $b\leq 0$ and $q>1$, then $u$
must be a constant. Moreover, if $a>0$, $b<0$ and $q>1$, then $u$ is the
constant $u=\sqrt[q-1]{a/(-b)}$.

######  Remark 6.10.

The last proposition includes Liouville-type results for bounded solution of
the following specific equation of mathematical physics, namely;

  1. (1)

Weighted Allen-Cahn equation: $\mathscr{L}_{\phi}u+u(1-u^{3})=0$, $0<u\leq 1$.
That is, setting $a=1$, $b=-1$ and $q=3$ in (6.13).

  2. (2)

Weighted elliptic Fisher KPP equation: $\mathscr{L}_{\phi}u+cu(1-u)=0$, $c>0$,
$0<u\leq 1$. That is, setting $a=c>0$, $b=-c$ and $q=2$ in (6.13).

  3. (3)

Weighted semilinear elliptic equation: $\mathscr{L}_{\phi}u+bu^{q}=0$, $b,q$
are real. That is, setting $a=0$ in (6.13).

  4. (4)

In general, a similar Liouville type result holds for the case $\displaystyle
H(u)=\sum_{l=1}^{r}b_{l}u^{\lambda_{l}}$, $1\leq l\leq r$. That is, any
bounded positive solution to the weighted boundary value problem

$\displaystyle\begin{cases}\displaystyle\mathscr{L}_{\phi}u+\sum_{l=1}^{r}b_{l}u^{\lambda_{l}}&=0,\\\
\displaystyle\hskip 56.9055ptu|_{\partial M}&=\text{const}.\end{cases}$ (6.13)

on $M$ such that $u_{\nu}\geq 0$ over $\partial M$ with
$\mathscr{R}ic_{\phi}^{m}\geq 0$ and $\mathscr{H}_{\phi}\geq 0$ must be
constant if either $b_{l}^{\prime}s\geq 0$ and $\lambda_{l}\leq 1$ or
$b_{l}^{\prime}s\leq 0$ and $\lambda_{l}\geq 1$.

##  7\. Applications of elliptic gradient estimates

First in this section we mention that local Hamilton-Souplet-Zhang type
gradient estimates in Theorem 3.1 (resp. Theorem 2.3) and Theorem 4.1 (resp.
Theorem 2.4) have global counterparts when ensuring that all bounds and
supremums are globally defined on the whole of $M$. The resulting global
gradient estimates can be proved by allowing $R$ to pass to the limit
$R\to\infty$ in the respective local estimates. To this end and for the
purpose of other applications of elliptic gradient estimates, we focus on
Theorem 4.1, while noting that Theorem 3.1 can be applied in similar contexts.

###  7.1. Global elliptic gradient estimates

Theorem 4.1 yields the following global estimates by passing to the limit
$R\to\infty$.

######  Theorem 7.1.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k$ and
$\mathscr{H}_{\phi}\geq-l$ for some $k,l\geq 0$. Let $u$ be a smooth positive
solution to (1.1) such that $u$ satisfies the Dirichlet boundary condition,
$u_{\nu}\geq 0$ and $u_{t}\leq H(u)$ on $\partial M\times(-\infty,+\infty)$.
Then there exists a constant $C_{\varepsilon}>0$ depending only on
$n,m,\varepsilon$ with $\varepsilon\in(0,1/2)$ such that

$\displaystyle\frac{|\nabla u|}{u^{1-3\varepsilon/2}}$ $\displaystyle\leq
C_{\varepsilon}\Bigg{\\{}\frac{1}{\sqrt{T}}+\sqrt{L}+\Gamma^{\varepsilon}_{+}[u]\Bigg{\\}}\Big{(}\sup_{M\times(-\infty,\infty)}\\{\sqrt{u^{3\varepsilon}}\\}\Big{)},$
(7.1)

where $L$ and $\Gamma^{\varepsilon}_{+}[u]$ are as defined in Theorem 4.1, but
with the supremums taking all over the whole of $M$.

An immediate consequence of the above result is the following global gradient
estimates for weighted nonlinear elliptic equation on manifold with boundary

$\displaystyle\displaystyle\mathscr{L}_{\phi}u(x)+H(u(x))$ $\displaystyle=0\
\text{in}\ M,\ \ \ u(x)=\text{const.}\ \text{on}\ \ \partial M.$ (7.2)

######  Theorem 7.2.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq-(m-1)k$ and
$\mathscr{H}_{\phi}\geq-l$, $k,l\geq 0$. Let $u=u(x)$ be a smooth positive
bounded solution to (7.2) such that $u_{\nu}\geq 0$ over $\partial M$. Then
for $\varepsilon\in(0,1/2]$ we have

$\displaystyle\frac{|\nabla u|}{u^{1-3\varepsilon/2}}$ $\displaystyle\leq
C_{\varepsilon}\Big{(}\sqrt{L}+\Gamma^{\varepsilon}_{+}[u]\Big{)}\Big{(}\sup_{M}\\{\sqrt{u^{3\varepsilon}}\\}\Big{)},$
(7.3)

where the constant $C_{\varepsilon}$ and the quantities $L$ and
$\Gamma^{\varepsilon}_{+}[u]$ are as defined in Theorem 7.1.

###### Proof.

The proof is an immediate consequence of Theorem 7.1 by letting $T\to\infty$
since the solution of (7.2) is time-independent. ∎

###  7.2. Liouville-type results

Another important implication of these estimates is the following Liouville-
type theorem.

######  Theorem 7.3.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq 0$ and
$\mathscr{H}_{\phi}\geq 0$. Let $u=u(x)$ be a smooth positive bounded solution
to (7.2) such that $u_{\nu}\geq 0$ over $\partial M$, where
$2H^{\prime}(u)+(3\varepsilon-2)H(u)/u\leq 0$ for all $x\in M$ and
$\varepsilon\in(0,1/2]$. Then $u$ must be constant. In particular $H(u)\equiv
0$.

###### Proof.

The proof follows from Theorem 7.2. By the assumption
$2H^{\prime}(u)+(3\varepsilon-2)H(u)/u\leq 0$, $\varepsilon\in(0,1/2]$, we see
that

$\Gamma^{\varepsilon}_{+}[u]=\left[2H^{\prime}(u)+(3\varepsilon-2)H(u)/u\right]_{+}=0.$

Considering also the global bounds $\mathscr{R}ic_{\phi}^{m}\geq 0$ and
$\mathscr{H}_{\phi}\geq 0$ we set $L=(k^{2}+l^{4})^{1/2}=0$. Thus the global
estimates in Theorem 7.2 implies

$\displaystyle\frac{|\nabla u|}{u^{1-3\varepsilon/2}}$ $\displaystyle\leq
C_{\varepsilon}\Gamma^{\varepsilon}_{+}[u]\Big{(}\sup_{M}\\{\sqrt{u^{3\varepsilon}}\\}\Big{)}=0$

since $\sup_{M}\\{\sqrt{u^{3\varepsilon}}\\}$ is finite. It therefore follows
that $|\nabla u|\equiv 0$ on $M$, and by implication $u$ is a constant.
Substituting this constant solution into (7.1) yields $H(u)\equiv 0$.

∎

Finally, we remark that the above global gradient estimates can be applied to
establish Liouville-type theorems for several special cases of $H(u)$ as in
the previous section. As examples we want to consider two specific nonlinear
elliptic equations, namely:

$\displaystyle\displaystyle\mathscr{L}_{\phi}u+\sum_{j=1}^{r}q_{j}u^{\lambda_{j}}$
$\displaystyle=0\ \text{in}\ M,\ \ \ u=\text{const.}\ \text{on}\ \ \partial M$
(7.4)

and

$\displaystyle\displaystyle\mathscr{L}_{\phi}u+au^{s}\log u+bu^{r}$
$\displaystyle=0\ \text{in}\ M,\ \ \ u=\text{const.}\ \text{on}\ \ \partial
M.$ (7.5)

Example 1. Consider the case

$H(u)=\sum_{j=1}^{r}q_{j}u^{\lambda_{j}}$

and compute

$2H^{\prime}(u)+(3\varepsilon-2)H(u)/u=\sum_{j=1}^{r}q_{j}[2\lambda_{j}+(3\varepsilon-2)]u^{\lambda_{j}-1}\leq
0$

provided either $q_{j}\leq 0$ and $\lambda_{j}\geq 1-3\varepsilon/2$ or
$q_{j}\geq 0$ and $\lambda_{j}\leq 1-3\varepsilon/2$ for each $j$ in the
interval $1\leq j\leq r$.

Note that the quantity $\omega(\varepsilon)=1-3\varepsilon/2$ which serves as
bound for $\lambda_{j}$ can be computed for each $\varepsilon\in(0,1/2]$. For
instance, if we allow $\varepsilon\searrow 0$ the quantity
$\omega(\varepsilon)$ becomes $\omega(0)=1$ or when picking $\varepsilon=1/3$
the quantity $\omega(\varepsilon)$ yields $\omega(1/3)=1/2$. By the above
discussion we can therefore apply Theorem 7.3 to prove the following result.

######  Proposition 7.4.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq 0$ and
$\mathscr{H}_{\phi}\geq 0$. Let $u=u(x)$ be any smooth positive bounded
solution to (7.4) such that $u_{\nu}\geq 0$ over $\partial M$. Then $u$ must
be constant if either $q_{j}\leq 0$ and $\lambda_{j}\geq 1-3\varepsilon/2$ or
$q_{j}\geq 0$ and $\lambda_{j}\leq 1-3\varepsilon/2$ for each $j$. In
particular $\sum_{j=1}^{r}q_{j}u^{\lambda_{j}}\equiv 0$.

Example 2. Consider the case $H(u)=au^{s}\log u+bu^{r}$, where $a,b,s$ and $r$
are real. A straightforward calculation yields

$2H^{\prime}(u)+(3\varepsilon-2)H(u)/u=a\left[2+[2(s-1)+3\varepsilon]\log
u\right]u^{s-1}+b\left[2(r-1)+3\varepsilon\right]u^{r-1}$

which is nonpositive for $\varepsilon\in(0,1/2]$ if the following assumptions
hold:

$\displaystyle\begin{cases}a\leq 0,\ \ s\geq 1,\ \ \log
u\geq-\frac{2}{2(s-1)+3\varepsilon}\\\ \text{or}\\\ a\geq 0,\ \ s\leq
1-\frac{3\varepsilon}{2},\ \ \log
u\leq-\frac{2}{2(s-1)+3\varepsilon}\end{cases}$ (7.6)

and

$\displaystyle b\leq 0,\ \ r\geq 1\ \ \text{or}\ \ b\geq 0,\ \ \ r\leq
1-\frac{3\varepsilon}{2}.$ (7.7)

Thus an application of Theorem 7.3 yields the following result.

######  Proposition 7.5.

Let $M$ be an $n$-dimensional complete smooth metric measure space with
compact boundary satisfying $\mathscr{R}ic_{\phi}^{m}\geq 0$ and