# Refined blowup analysis and nonexistence of Type II blowups for an energy
critical nonlinear heat equation

Kelei Wang† †School of Mathematics and Statistics
Wuhan University
Wuhan 430072, China<EMAIL_ADDRESS>and  Juncheng Wei∗ ∗Department of
Mathematics
University of British Columbia
Vancouver, B.C., Canada, V6T 1Z2<EMAIL_ADDRESS>
###### Abstract.

We consider the energy critical semilinear heat equation

$\left\\{\begin{aligned} &\partial_{t}u-\Delta u=|u|^{\frac{4}{n-2}}u&\mbox{in
}\mathbb{R}^{n}\times(0,T),\\\ &u(x,0)=u_{0}(x),\end{aligned}\right.$

where $n\geq 3$, $u_{0}\in L^{\infty}(\mathbb{R}^{n})$, and
$T\in\mathbb{R}^{+}$ is the first blow up time. We prove that if $n\geq 7$ and
$u_{0}\geq 0$, then any blowup must be of Type I, i.e.,

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq C(T-t)^{-\frac{1}{p-1}}.$

A similar result holds for bounded convex domains. The proof relies on a
reverse inner-outer gluing mechanism and delicate analysis of bubbling
behavior (bubbling tower/cluster).

###### Key words and phrases:

Nonlinear heat equation; critical exponent; blow up; bubbling analysis.

###### 2020 Mathematics Subject Classification:

35B33; 35B44; 35K58.

K. Wang’s research was supported by the National Natural Science Foundation of
China (No. 11871381).J. Wei’s research is partially supported by NSERC of
Canada.

###### Contents

  1. 1 Introduction
    1. 1.1 Cauchy problem
    2. 1.2 Outline of proof
    3. 1.3 Cauchy-Dirichlet problem
    4. 1.4 List of notations and conventions used throughout the paper.
  2. I Energy concentration behavior
    1. 2 Setting
      1. 2.1 List of notations and conventions used in this part.
    2. 3 Monotonicity formula and $\varepsilon$-regularity
    3. 4 Defect measures
      1. 4.1 Definition and basic properties
      2. 4.2 Definition of the mean curvature
      3. 4.3 Limiting monotonicity formula
    4. 5 Tangent flow analysis, I
    5. 6 The case $m$ is not an integer
    6. 7 Partial regularity of suitable weak solutions
    7. 8 The case $m$ is an integer
    8. 9 The case $p=(n+2)/(n-2)$
      1. 9.1 A local quantization result
      2. 9.2 Proof of Theorem 9.1
      3. 9.3 Further properties of defect measures
  3. II Energy concentration with only one bubble
    1. 10 Setting
      1. 10.1  List of notations and conventions used in this part
    2. 11 Blow up profile
    3. 12 Orthogonal and inner-outer decompositions
      1. 12.1 Orthogonal decomposition
      2. 12.2 Inner-outer decomposition
    4. 13 Inner problem
    5. 14 Outer problem
      1. 14.1 Decomposition of the outer equation
      2. 14.2 Estimates for the outer equation
      3. 14.3 Estimate of $\mathcal{O}(t)$
    6. 15 A Harnack inequality for $\lambda$
    7. 16 Inner problem and outer problem again
    8. 17 Improved estimates on $\phi$
      1. 17.1 Uniform $L^{\infty}$ bound
      2. 17.2 Gradient estimates
      3. 17.3 Estimate of time derivative
    9. 18 Linearization of Pohozaev identity
      1. 18.1 The left hand side
      2. 18.2 The right hand side
    10. 19 A weak form of Schoen’s Harnack inequality
    11. 20 A conditional exclusion of bubble clustering
      1. 20.1 Construction of Green function
      2. 20.2 Local Pohozaev invariants
    12. A Linearization estimates around bubbles
    13. B Estimates on some integrals
  4. III Energy concentration in the general case
    1. 3 Setting
    2. 4 Preliminaries
    3. 5 Bubble tree construction
    4. 6 A remark on the Lipschitz hypothesis in Part II
    5. 7 Exclusion of bubble towering
  5. IV Analysis of first time singularity
    1. 8 Setting
    2. 9 Some integral estimates
    3. 10 Tangent flow analysis, II
    4. 11 No coexistence of Type I and Type II blow ups
    5. 12 Exclusion of bubble clustering
    6. 13 Exclusion of blow up with only one bubble
    7. 14 Proof of main results
      1. 14.1 Proof of Theorem 8.1
      2. 14.2 Cauchy-Dirichlet problems
      3. 14.3 Cauchy problem
      4. 14.4 Energy collapsing and the blow up of $L^{p+1}$ norm

###  1\. Introduction

In this paper we consider the blowup problem for the nonlinear heat equation

$\partial_{t}u-\Delta u=|u|^{p-1}u$ (1.1)

where $p>1$.

####  1.1. Cauchy problem

We first consider the Cauchy problem

$\left\\{\begin{aligned} &\partial_{t}u-\Delta u=|u|^{p-1}u&\mbox{in
}\mathbb{R}^{n}\times(0,T),\\\ &u(x,0)=u_{0}(x).\end{aligned}\right.$ (1.2)

Here $u_{0}\in L^{\infty}(\mathbb{R}^{n})$, and $T\in\mathbb{R}^{+}$ is the
first blow up time, that is, $u\in L^{\infty}(\mathbb{R}^{n}\times[0,t])$ for
any $t<T$, while

$\lim_{t\to T}\|u(t)\|_{L^{\infty}(\mathbb{R}^{n})}=+\infty.$

Problem (1.2) is one of classical nonlinear parabolic equations which has been
studied extensively in recent decades. See the monograph by Quitter and
Souplet [75] for backgrounds and references therein. Following the seminal
work of Fujita [33], it is well-known that finite time blow-up must occur when
the initial datum is sufficiently large (in some suitable sense). An important
and fundamental question is the classification of the blowup near the first
blowup time $T$. Since the equation (1.1) is invariant under the scaling

$u^{\lambda}(x,t):=\lambda^{\frac{2}{p-1}}u(\lambda x,\lambda^{2}t),$ (1.3)

the blow up is divided into two types: it is Type I if there exists a constant
$C$ such that for any $t<T$,

$\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq C(T-t)^{-\frac{1}{p-1}},$
(1.4)

otherwise it is called Type II.

When $p$ is subcritical (i.e. $1<p<+\infty$ for $n=1,2$, or
$1<p<\frac{n+2}{n-2}$ for $n\geq 3$), in classical works of Giga-Kohn [37] (in
the case of $u_{0}\geq 0$) and Giga-Matsui-Sasayama [39] (in the case of sign-
changing $u_{0}$), it is shown that any blow up is of Type I.

In contrast, less is known about the energy critical case, i.e. $n\geq 3$ and
$p=\frac{n+2}{n-2}$, see e.g. [75, Open Problem 2.8 in Appendix I]. In this
paper we establish the following

######  Theorem 1.1.

If $n\geq 7$, $p=\frac{n+2}{n-2}$ and $u_{0}\geq 0$, then any blow up to (1.2)
is of Type I.

Some remarks on Theorem 1.1 are in order. First we remark that the dimension
restriction in Theorem 1.1 is optimal: when $n\leq 6$ and $p=\frac{n+2}{n-2}$,
Type II blowup solutions to (1.2) do exist. This was first predicted and
proved formally in the pioneering work of Filippas-Herrero-Velázquez in [30]
via method of matched asymptotics. The first rigorous proof of Type II blowup
solutions in the radial setting is given by Schweyer in [79] for $n=4$. (For
nonradial setting and multiply blowups in dimension $n=4$ we refer to [23].)
For the remaining dimensions rigorous construction of Type II blowups are
established recently: for $n=3$, see [22], and for $n=5$, we refer to [19,
41]. For $n=6$, see Harada [42]. We should mention that when
$p=\frac{n+2}{n-2}$ all the type II blowup solutions to (1.2) constructed so
far are sign-changing. It is an open question if there are Type II blowups for
positive solutions in low dimensions $n=3,4,5,6$.

Second we remark that the exponent restriction in Theorem 1.1 is also optimal:
when $p>\frac{n+2}{n-2}$ many types of Type II blowup solutions have been
found. The first example was discovered in the radial setting by Herrero and
Velazquez in [43] for $p>p_{JL}$ where $p_{JL}$ is the Joseph-Lundgreen
exponent,

$p_{JL}=\begin{cases}1+\frac{4}{n-4-2\sqrt{n-1}}&\text{if $n\geq 11$}\\\
+\infty,&\text{if $n\leq 10$}.\\\ \end{cases}$

See also Mizoguchi [69] for the case of a ball, Seki [80] for the case of
$p=p_{JL}$, and Collot [10] for the case of general domains with the
restriction that $p>p_{JL}$ and $p$ is an odd integer. A new and interesting
anisotropic Type II blow-up solutions is also constructed recently for
$p>p_{JL}(n-1),n\geq 14$ by Collot, Merle and Raphael in [13]. In the
intermediate exponent regime $\frac{n+2}{n-2}<p<p_{JL}$, Matano and Merle [63,
64] proved that no type II blow-up is present for radial solutions (under some
extra technical conditions). However type II blow-up also exists in this
intermediate regime. In [21], the authors successfully constructed non-radial
type II blow-up solution in the Matano-Merle range
$p=\frac{n+1}{n-3}\in(\frac{n+2}{n-2},p_{JL}(n))$. Another type of non-radial
Type blow-up with shrinking spheres is recently found for $n\geq 5,p=4$ in
[18].

Theorem 1.1 is the first instance of classification in the critical exponent
case for general initial datum. In the pioneering work of Filippas-Herrero-
Velázquez [30] it is shown that blowup is type I if the initial data is
positive and radially decreasing, when $n\geq 3$. For initial datum with low
energy, we mention that in [11], Collot, Merle and Raphael proved a result
similar to Theorem 1.1 under the condition that
$\|u_{0}-W\|_{\dot{H}^{1}(\mathbb{R}^{n})}\ll 1$, where $W$ is a positive
Aubin-Talenti solution satisfying (1.9), i.e. positive steady state of (1.2).
More precisely assume that $\|u_{0}-W\|_{\dot{H}^{1}(\mathbb{R}^{n})}\ll 1$
and the dimension $n\geq 7$, then there is a trichotomy to solution to (1.2):
it either dissipates to zero, or approaches to a rescaled Aubin-Talenti
solution, or blows up in finite time and the blow-up is of Type I. Note that
in Theorem 1.1, we have assumed neither decaying nor energy condition.

####  1.2. Outline of proof

The proof of Theorem 1.1 consists of the following five steps.

  1. (1)

First we perform tangent flow analysis at a possible blow up point, see Part
IV. This is in the same spirit of Giga-Kohn [36, 37, 38], but we rewrite it in
the form which is more familiar in geometric measure theory such as the
tangent cone analysis/ blowing up analysis/tangent flow analysis used in the
study of minimal surfaces, mean curvature flows and many other geometric
variational problems.

  2. (2)

Bubbles may appear during this blow up procedure. A general theory on the
energy concentration phenomena in (1.1) is then developed in Part I, where we
mainly follow the treatment (on harmonic map heat flows) in Lin-Wang [55, 56,
57], see also their monograph [58].

  3. (3)

We then perform a refined analysis of this bubbling phenomena, first in the
case of one bubble (_multiplicity one case_), see Part II. Here we mainly use
_the inner-outer gluing mechanism_ developed by the second author with Davila,
del Pino and Musso in [15], [16], [19], [20], [73], [22] (see also [17] for a
survey);

  4. (4)

In Part III, we combine the analysis in Part I and Part II to establish the
refined analysis in the general case (_higher multiplicity_), where in
particular we will exclude the bubble towering formation in this energy
concentration phenomena. Arguments used here are motivated by those used in
the study of Yamabe problems through pioneering work of Schoen [78], including
secondary blow ups, construction of Green functions, next order expansion of
Pohozaev identities, see for example Schoen [77, 78], Kuhri-Marques-Schoen
[46], Li-Zhu [50], Li-Zhang [48].

  5. (5)

Finally, the refined analysis in Part III and Part II are applied to the first
time singularity problem. Results in Part III are used to exclude blow ups
with bubble clustering, while results in Part II are used to exclude blow ups
with only one bubble.

####  1.3. Cauchy-Dirichlet problem

Our method can also be applied to the initial-boundary value problem

$\left\\{\begin{aligned} &\partial_{t}u-\Delta u=|u|^{p-1}u&\mbox{in
}\Omega^{T}:=\Omega\times(0,T),\\\ &u=0&\mbox{on
}\partial\Omega\times(0,T),\\\ &u(x,0)=u_{0}(x).\end{aligned}\right.$ (1.5)

Here $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with $C^{2}$ boundary,
$u_{0}\in L^{q}(\Omega)$ for some $q\geq n(p-1)/2$. (The exponent
$\frac{n(p-1)}{2}$ is optimal. See [70].) By [75, Theorem 15.2] or [7] and
[88], there exists a $T>0$ such that $u\in
L^{\infty}(\overline{\Omega}\times(0,t])$ for any $t<T$. Here we assume $u$
blows up in finite time, and $T$ is the first blow up time, that is,

$\lim_{t\to T}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}=+\infty.$

For this Cauchy-Dirichlet problem we prove

######  Theorem 1.2.

If $n\geq 7$, $p=\frac{n+2}{n-2}$ and $u_{0}\geq 0$, the first time
singularity in the interior must be Type I, that is, for any
$\Omega^{\prime}\Subset\Omega$, there exists a constant $C$ such that

$\|u(\cdot,t)\|_{L^{\infty}(\Omega^{\prime})}\leq
C(T-t)^{-\frac{1}{p-1}},\quad\forall 0<t<T.$

The proof of this theorem is similar to the one for Theorem 1.1. It is also
generally believed that there is no boundary blow up. This, however, is only
known in some special cases, e.g. when $\Omega$ is convex (cf. [38, Theorem
5.3], [40]).

Once we have this Type I blow up bound, it will be interesting to know if the
set of blow up points enjoy further regularities, and if the blow up profiles
satisfy the uniform, refined estimates as in the subcritical case, see for
example Liu [60], Filippas-Kohn [31], Filippas-Liu [32], Velázquez [83, 84,
85], Merle-Zaag [65, 66, 67, 68], Zaag [92, 93, 94] and Kammerer-Merle-Zaag
[29].

Our proof also implies that the energy collapses and the $L^{p+1}$ norm blows
up at $T$, cf. Giga [35], Zaag [91] and Du [27].

######  Corollary 1.3.

Under the assumptions of Theorem 1.2, if there exists an interior blow up
point, then

$\lim_{t\to T}\int_{\Omega}\left[\frac{1}{2}|\nabla
u(x,t)|^{2}-\frac{1}{p+1}u(x,t)^{p+1}\right]dx=-\infty,$ (1.6)

and

$\lim_{t\to T}\int_{\Omega}u(x,t)^{p+1}dx=+\infty.$ (1.7)

In fact, we prove that

$\lim_{t\to T}\int_{0}^{t}\int_{\Omega}|\partial_{t}u(x,t)|^{2}dx=+\infty.$
(1.8)

Then (1.6) follows from the standard energy identity for $u$. A local version
of (1.8) and (1.6) also hold for the solution of the Cauchy problem (1.2).

We do not know if the $L^{2}(\Omega)$ norm of $\nabla u(t)$ blows up as $t\to
T$. We conjecture that the blow up must be complete, i.e. the solution cannot
be extended beyond $T$ as a weak solution, cf. Baras-Cohen [3], Galaktinov-
Vazquez [34], Martel [62] and [75, Section 27].

####  1.4. List of notations and conventions used throughout the paper.

  * •

The open ball in $\mathbb{R}^{n}$ is denoted by $B_{r}(x)$, and by $B_{r}$ if
the center is the origin $0$.

  * •

The parabolic cylinder is $Q_{r}(x,t):=B_{r}(x)\times(t-r^{2},t+r^{2})$, the
forward parabolic cylinder is $Q_{r}^{+}(x,t):=B_{r}(x)\times(t,t+r^{2})$, and
the backward parabolic cylinder is
$Q_{r}^{-}(x,t):=B_{r}(x)\times(t-r^{2},t)$. If the center is the origin
$(0,0)$, it will not be written down explicitly.

  * •

The parabolic distance is

$\mbox{dist}_{P}\left((x,t),(y,s)\right):=\max\\{|x-y|,\sqrt{|t-s|}\\}.$

Hölder, Lipschitz continuous functions with respect to this distance is
defined as usual.

  * •

Given a domain $\Omega\subset\mathbb{R}^{n}$, $H^{1}(\Omega)$ is the Sobolev
space endowed with the norm

$\left[\int_{\Omega}\left(|\nabla u|^{2}+u^{2}\right)dx\right]^{1/2}.$

Given an interval $\mathcal{I}\subset\mathbb{R}$, we use $L^{2}_{t}H^{1}_{x}$
to denote the space $L^{2}(\mathcal{I};H^{1})$.

  * •

A bubble is an entire solution of the stationary equation

$-\Delta W=|W|^{\frac{4}{n-2}}W\quad\mbox{in }\mathbb{R}^{n}$ (1.9)

with finite energy

$\int_{\mathbb{R}^{n}}|\nabla
W|^{2}=\int_{\mathbb{R}^{n}}|W|^{\frac{2n}{n-2}}<+\infty.$

By the translational and scaling invariance, if $W$ is a bubble, so is

$W_{\xi,\lambda}(x):=\lambda^{\frac{n-2}{2}}W\left(\frac{x-\xi}{\lambda}\right),\quad\xi\in\mathbb{R}^{n},~{}~{}\lambda\in\mathbb{R}^{+}.$

  * •

By Caffarelli-Gidas-Spruck [8], all entire positive solutions to the
stationary equation (1.9) are given by Aubin-Talenti bubbles

$W_{\xi,\lambda}(x):=\left(\frac{\lambda}{\lambda^{2}+\frac{|x-\xi|^{2}}{n(n-2)}}\right)^{\frac{n-2}{2}},\quad\quad\quad\lambda>0,\quad\xi\in\mathbb{R}^{n}.$
(1.10)

They have finite energy, which are always equal to

$\Lambda:=\int_{\mathbb{R}^{n}}|\nabla
W_{\xi,\lambda}|^{2}=\int_{\mathbb{R}^{n}}W_{\xi,\lambda}^{p+1}.$ (1.11)

For simplicity, denote $W:=W_{0,1}$.

  * •

We use $C$ (large) and $c$ (small) to denote universal constants. They could
be different from line to line. Given two quantities $A$ and $B$, if $A\leq
CB$ for some universal constant $C$, we simply write $A\lesssim B$. If the
constant $C$ depends on a quantity $D$, it will be written as $C(D)$ or
$\lesssim_{D}$.

##  Part I Energy concentration behavior

###  2\. Setting

In this part we assume that $p>1$ (which may not necessarily be the critical
exponent), and that the solutions could be sign-changing. Here we study the
energy concentration behavior of the nonlinear heat equation (1.1).

First we need to define what we call as a solution.

######  Definition 2.1 (Suitable weak solution).

A function $u$ is a suitable weak solution of (1.1) in $Q_{1}$, if
$\partial_{t}u,\nabla u\in L^{2}(Q_{1})$, $u\in L^{p+1}(Q_{1})$, and

  * •

$u$ satisfies (1.1) in the weak sense, that is, for any $\eta\in
C_{0}^{\infty}(Q_{1})$,

$\int_{Q_{1}}\left[\partial_{t}u\eta+\nabla
u\cdot\nabla\eta-|u|^{p-1}u\eta\right]=0;$ (2.1)

  * •

$u$ satisfies the localized energy inequality: for any $\eta\in
C_{0}^{\infty}(Q_{1})$,

$\int_{Q_{1}}\left[\left(\frac{|\nabla
u|^{2}}{2}-\frac{|u|^{p+1}}{p+1}\right)\partial_{t}\eta^{2}-|\partial_{t}u|^{2}\eta^{2}-2\eta\partial_{t}u\nabla
u\cdot\nabla\eta\right]\geq 0;$ (2.2)

  * •

$u$ satisfies the stationary condition: for any $Y\in
C_{0}^{\infty}(Q_{1},\mathbb{R}^{n})$,

$\int_{Q_{1}}\left[\left(\frac{|\nabla
u|^{2}}{2}-\frac{|u|^{p+1}}{p+1}\right)\mbox{div}Y-DY(\nabla u,\nabla
u)+\partial_{t}u\nabla u\cdot Y\right]=0.$ (2.3)

A smooth solution satisfies all of these conditions. (In fact, (2.2) will
become an equality, which is just the standard energy identity.) But a
suitable weak solution need not to be smooth everywhere.

######  Definition 2.2.

Given a weak solution $u$ of (1.1), a point $(x,t)$ is a regular point of $u$
if there exists an $r>0$ such that $u\in C^{\infty}(Q_{r}(x,t))$, otherwise it
is a singular point of $u$. The corresponding sets are denoted by
$\mathcal{R}(u)$ and $\mathcal{S}(u)$.

By definition, $\mathcal{R}(u)$ is open and $\mathcal{S}(u)$ is closed.

In this part, $u_{i}$ denotes a sequence of suitable weak solutions to (1.1)
in $Q_{1}$, satisfying a uniform energy bound

$\limsup_{i\to+\infty}\int_{Q_{1}}\left[|\nabla
u_{i}|^{2}+|u_{i}|^{p+1}+|\partial_{t}u_{i}|^{2}\right]dxdt<+\infty.$ (2.4)

The integral bound on $\partial_{t}u_{i}$ in (2.4) can be deduced by
substituting the bounds on the first two integrands into (2.2), if we shrink
the domain $Q_{1}$ a little.

We will state the results about the energy concentration behavior after
introducing some necessary notations. The main results in this part are

  1. (1)

Theorem 4.2, where we establish basic properties about this energy
concentration behavior;

  2. (2)

Theorem 6.1 about the case $2\frac{p+1}{p-1}$ is not an integer, where we
prove a strong convergence result;

  3. (3)

Theorem 8.1 about the case $2\frac{p+1}{p-1}$ is an integer, where we show
that the limiting problem is a generalized Brakke’s flow;

  4. (4)

Theorem 9.1, which is about the energy quantization result in the critical
case $p=\frac{n+2}{n-2}$.

Our treatment in this part mainly follows the work of Lin and Wang [55], [56],
[57]. See also their monograph [58].

There are still many problems remaining open about this energy concentration
behavior, such as the energy quantization result in the general case (cf. Lin-
Riviere [53] for the corresponding results for harmonic maps, and Naber-
Valtorta [74] for Yang-Mills fields), Brakke type regularity result for the
limiting problem (cf. Brakke [6], Kasai-Tonegawa [45]). But we will content
with such a preliminary analysis of this energy concentration phenomena,
because the main goal of this part is to providing a setting for later parts.

####  2.1. List of notations and conventions used in this part.

  * •

For any $\lambda>0$ and each set $A\subset\mathbb{R}^{n}\times\mathbb{R}$, we
use $\lambda A$ to denote the parabolic scaling of $A$, which is the set

$\left\\{(\lambda x,\lambda^{2}t):(x,t)\in A\right\\}.$

  * •

Given a set $A\subset\mathbb{R}^{n}\times\mathbb{R}$, its time slices are
$A_{t}:=A\cap\left(\mathbb{R}^{n}\times\\{t\\}\right)$.

  * •

Denote $m:=\frac{2(p+1)}{p-1}$. Note that $m>2$, and $p=\frac{m+2}{m-2}$. If
$m$ is an integer, then $p$ is the Sobolev critical exponent in dimension $m$.

  * •

For $s\geq 0$, $\mathcal{P}^{s}$ denotes the $s$-dimensional parabolic
Hausdorff measure. The dimension of a subset of
$\mathbb{R}^{n}\times\mathbb{R}$ is always understood to be the parabolic
Hausdorff dimension.

###  3\. Monotonicity formula and $\varepsilon$-regularity

In this section, $u$ denotes a fixed suitable weak solution of (1.1) in
$Q_{1}$. Here we recall the monotonicity formula and $\varepsilon$-regularity
theorems. We also derive a Morrey space estimate, which will be used below in
Section 5 to preform the tangent flow analysis. Our use of Morrey space
estimates follows closely Chou-Du-Zheng [9] and Du [25, 27], which is
different from the ones used in Blatt-Struwe [5] and Souplet [81].

Fix a function $\psi\in C^{\infty}_{0}(B_{1})$ such that $0\leq\psi\leq 1$,
$\psi\equiv 1$ in $B_{3/4}$ and $|\nabla\psi|+|\nabla^{2}\psi|\leq 100$. For
$t>0$ and $x\in\mathbb{R}^{n}$, let

$G(x,t):=\left(4\pi t\right)^{-\frac{n}{2}}e^{-\frac{|x|^{2}}{4t}}$

be the standard heat kernel on $\mathbb{R}^{n}$.

Take a large constant $C$ and a small constant $c$. For each $(x,t)\in
B_{3/4}\times(-3/4,1]$ and $s\in(0,1/4)$, define

$\displaystyle\Theta_{s}(x,t;u)$ $\displaystyle:=$ $\displaystyle
s^{\frac{p+1}{p-1}}\int_{B_{1}}\left[\frac{|\nabla
u(y,t-s)|^{2}}{2}-\frac{|u(y,t-s)|^{p+1}}{p+1}\right]G(y-x,s)\psi(y)^{2}dy$
$\displaystyle+$
$\displaystyle\frac{1}{2(p-1)}s^{\frac{2}{p-1}}\int_{B_{1}}u(y,t-s)^{2}G(y-x,s)\psi(y)^{2}dy+Ce^{-cs^{-1}}.$

######  Remark 3.1.

Because $u$ and $\nabla u$ are only integrable in space-time, rigorously we
should integrate one more time in $s$, e.g. to consider the quantity

$\overline{\Theta}_{s}(x,t;u):=s^{-1}\int_{s}^{2s}\Theta_{\tau}(x,t;u)d\tau.$

However, to simplify notations we will not use this quantity.

The following is a localized version of the monotonicity formula of Giga and
Kohn, see [36, 38] and [39].

######  Proposition 3.2 (Localized monotonicity formula).

If $C$ is universally large and $c$ is universally small, then for any
$(x,t)\in B_{3/4}\times(-3/4,1]$ and $0<s_{1}<s_{2}<1/4$,

$\displaystyle\Theta_{s_{2}}(x,t)-\Theta_{s_{1}}(x,t)$ $\displaystyle\geq$
$\displaystyle\int_{s_{1}}^{s_{2}}\tau^{\frac{2}{p-1}-1}\int_{B_{1}}\Big{|}(t-\tau)\partial_{t}u(y,t-\tau)+\frac{u(y,t-\tau)}{p-1}+\frac{y}{2}\cdot\nabla
u(y,t-\tau)\Big{|}^{2}$ $\displaystyle\quad\quad\quad\quad\quad\quad\times
G(y-x,\tau)\psi(y)^{2}dyd\tau.$

This almost monotonicity allows us to define

$\Theta(x,t):=\lim_{s\to 0^{+}}\Theta_{s}(x,t).$

For each $s>0$, from the definition we see
$s^{-1}\int_{s}^{2s}\Theta_{\tau}(x,t)d\tau$ is a continuous function of
$(x,t)$. Combining this fact with the monotonicity formula, we get

######  Lemma 3.3.

$\Theta(x,t)$ is upper semi-continuous in $(x,t)$.

The following Morrey space bound is essentially Giga-Kohn’s [37, Proposition
2.2] or [38, Proposition 3.1].

######  Proposition 3.4.

For any $(x,t)\in Q_{1/2}$ and $r\in(0,1/4)$,

$\displaystyle r^{m-2-n}\int_{Q_{r}^{-}(x,t-r^{2})}\left(|\nabla
u|^{2}+|u|^{p+1}\right)+r^{m-n}\int_{Q_{r}^{-}(x,t-r^{2})}|\partial_{t}u|^{2}$
(3.1) $\displaystyle\leq$ $\displaystyle
C\max\left\\{\Theta_{4r^{2}}(x,t),\Theta_{4r^{2}}(x,t)^{\frac{2}{p+1}}\right\\}.$

Combining this estimate with the monotonicity formula, we obtain a Morrey
space estimate of $u$.

######  Corollary 3.5 (Morrey space bound).

There exists a universal constant $M$ depending on $\int_{Q_{1}}\left[|\nabla
u|^{2}+|u|^{p+1}\right]$ such that for any $(x,t)\in Q_{1/2}$ and
$r\in(0,1/4)$,

$r^{m-2-n}\int_{Q_{r}(x,t)}\left(|\nabla
u|^{2}+|u|^{p+1}\right)+r^{m-n}\int_{Q_{r}(x,t)}|\partial_{t}u|^{2}\leq M.$
(3.2)

These Morrey space estimates are invariant under the scaling (1.3).

Combining (3.1) and (3.2), we also get

######  Corollary 3.6 (Change of base point).

For any $\varepsilon>0$, there exist
$0<\delta(\varepsilon)\ll\theta(\varepsilon)\ll 1$ and $C(\varepsilon)$ so
that the following holds. For any $(x,t)\in Q_{1/2}$, $r<1/4$ and $(y,s)\in
Q_{\delta r}(x,t)$,

$\Theta_{\theta r^{2}}(y,s)\leq
C\max\left\\{\Theta_{r^{2}}(x,t),\Theta_{r^{2}}(x,t)^{\frac{2}{p+1}}\right\\}+\varepsilon.$

Next we recall two standard $\varepsilon$-regularity theorems. The first one
is a reformulation of Du [27, Theorem 3.1].

######  Theorem 3.7 ($\varepsilon$-regularity I).

There exist three universal constants $\varepsilon_{\ast}$, $\delta_{\ast}$
and $c_{\ast}$ so that the following holds. If

$\Theta_{r^{2}}(x,t)\leq\varepsilon_{\ast},$

then

$\sup_{Q_{\delta_{\ast}r}(x,t)}|u|\leq c_{\ast}r^{-\frac{2}{p-1}}.$

Once we have an $L^{\infty}$ bound on $u$, higher order regularity follows by
applying standard parabolic estimates to (1.1). As a consequence, we get

######  Corollary 3.8.

$\mathcal{R}(u)=\\{\Theta=0\\}$ and
$\mathcal{S}(u)=\\{\Theta\geq\varepsilon_{\ast}\\}$.

The second $\varepsilon$-regularity theorem is Chou-Du-Zheng [9, Theorem 2]
(see also Du [27, Proposition 4.1]).

######  Theorem 3.9 ($\varepsilon$-regularity II).

After decreasing $\varepsilon_{\ast}$, $\delta_{\ast}$, and enlarging
$c_{\ast}$ further, the following holds. There exists a constant
$\theta_{\ast}>0$ such that, if

$\int_{Q_{r}^{-}(x,t-\theta_{\ast}r^{2})}\left(|\nabla
u|^{2}+|u|^{p+1}\right)\leq\varepsilon_{\ast}r^{n+2-m},$

then

$\sup_{Q_{\delta_{\ast}r}(x,t)}|u|\leq c_{\ast}r^{-\frac{2}{p-1}}.$

A standard covering argument then gives

######  Corollary 3.10.

If $u$ is a suitable weak solution of (1.1) in $Q_{1}$, then

$\mathcal{P}^{n+2-m}\left(\mathcal{S}(u)\right)=0.$

In particular, $\mbox{dim}\left(\mathcal{S}(u)\right)\leq n+2-m$.

###  4\. Defect measures

From now on until Section 9, we will be concerned with the energy
concentration behavior in (1.1). In this section we define the defect measure
and prove some basic properties about it.

####  4.1. Definition and basic properties

By (2.4), after passing to a subsequence, we may assume $u_{i}$ converges
weakly to a limit $u_{\infty}$ in $L^{p+1}(Q_{1})$, $\nabla u_{i}$ and
$\partial_{t}u_{i}$ converges weakly to $\nabla u_{\infty}$ and
$\partial_{t}u_{\infty}$ respectively in $L^{2}(Q_{1})$.

By Sobolev embedding theorem and an interpolation argument, for any $1\leq
q<p+1$, $u_{i}$ converges to $u_{\infty}$ strongly in $L^{q}_{loc}(Q_{1})$. As
a consequence, $u_{\infty}$ is a weak solution of (1.1).

There exist three Radon measures $\mu_{1},\mu_{2}$ and $\nu$ such that

$\left\\{\begin{aligned}
&|u_{i}|^{p+1}dxdt\rightharpoonup|u_{\infty}|^{p+1}dxdt+\mu_{1},\\\ &|\nabla
u_{i}|^{2}dxdt\rightharpoonup|\nabla u_{\infty}|^{2}dxdt+\mu_{2},\\\
&|\partial_{t}u_{i}|^{2}dxdt\rightharpoonup|\partial_{t}u_{\infty}|^{2}dxdt+\nu.\end{aligned}\right.$

These measures, called _defect measures_ , characterize the failure of
corresponding strong convergence.

We also let

$\nabla u_{i}\otimes\nabla u_{i}dxdt\rightharpoonup\nabla
u_{\infty}\otimes\nabla u_{\infty}dxdt+\mathcal{T}d\mu_{2},$

where $\mathcal{T}$ is a matrix valued $\mu_{2}$-measurable functions.
Furthermore, $\mathcal{T}$ is symmetric, semi-positive definite and its trace
equals $1$ $\mu_{2}$-a.e.

######  Lemma 4.1 (Energy partition).

$\mu_{1}=\mu_{2}$.

###### Proof.

For any $\eta\in C_{0}^{\infty}(Q_{1})$, multiplying the equation (1.1) by
$u_{i}\eta^{2}$ and integrating by parts, we obtain

$\int_{Q_{1}}\left[-u_{i}^{2}\eta\partial_{t}\eta+|\nabla
u_{i}^{2}|\eta^{2}-|u_{i}|^{p+1}\eta^{2}+2\eta u_{i}\nabla
u_{i}\nabla\eta\right]=0.$ (4.1)

Letting $i\to\infty$, and noting that $u_{\infty}$ also satisfies (4.1), we
obtain

$\int_{Q_{1}}\eta^{2}d\left(\mu_{1}-\mu_{2}\right)=0.$

Since this holds for any $\eta\in C_{0}^{\infty}(Q_{1})$, $\mu_{1}=\mu_{2}$ as
Radon measures. ∎

By this lemma, we can write $\mu_{1}$ and $\mu_{2}$ just as $\mu$. Denote
$\Sigma:=\mbox{spt}(\mu)$, and define

$\Sigma^{\ast}:=\left\\{(x,t):~{}~{}\forall
r>0,~{}~{}\limsup_{i\to\infty}r^{m-2-n}\int_{Q_{r}^{-}(x,t)}\left(|\nabla
u_{i}|^{2}+|u_{i}|^{p+1}\right)\geq\varepsilon_{\ast}/2\right\\}$

to be _the blow up locus_.

Now we state some basic properties on this energy concentration phenomena.

######  Theorem 4.2.

Suppose $u_{i}$ is a sequence of suitable weak solutions of (1.1) in $Q_{1}$,
satisfying (2.4). Define $\mu,\nu$, $\Sigma$ and $\Sigma^{\ast}$ as above.
Then the following holds.

  1. (1)

(Morrey space bound) For any $(x,t)\in Q_{1/2}$ and $r\in(0,1/4)$,

$\left\\{\begin{aligned} &\mu(Q_{r}(x,t))+\int_{Q_{r}(x,t)}\left(|\nabla
u_{\infty}|^{2}+|u_{\infty}|^{p+1}\right)\leq Mr^{n+2-m},\\\
&\nu(Q_{r}(x,t))+\int_{Q_{r}(x,t)}|\partial_{t}u_{\infty}|^{2}\leq
Mr^{n-m}.\end{aligned}\right.$ (4.2)

Here $M$ is the constant in Corollary 3.5.

  2. (2)

The blow up locus $\Sigma^{\ast}$ is closed.

  3. (3)

(Smooth convergence) $u_{i}$ converges to $u_{\infty}$ in
$C_{loc}^{\infty}(Q_{1}\setminus\Sigma^{\ast})$. As a consequence,

     * •

$u_{\infty}\in C^{\infty}(Q_{1}\setminus\Sigma^{\ast})$, that is,
$\mathcal{S}(u_{\infty})\subset\Sigma^{\ast}$.

     * •

$\Sigma\subset\Sigma^{\ast}$ and $\mbox{spt}(\nu)\subset\Sigma^{\ast}$.

  4. (4)

(Measure estimate of the blow up locus) For any $(x,t)\in\Sigma^{\ast}\cap
Q_{1/2}$ and $0<r<1/2$,

$\frac{\varepsilon_{\ast}}{C}r^{n+2-m}\leq\mathcal{P}^{n+2-m}(\Sigma^{\ast}\cap
Q_{r}(x,t))\leq\frac{C}{\varepsilon_{\ast}}r^{n+2-m}.$

  5. (5)

(Lower density bound) For $\mu$-a.e. $(x,t)\in\Sigma\cap Q_{1/2}$ and
$r\in(0,1/2)$,

$\mu(Q_{r}(x,t))\geq\mu(Q_{r/2}^{-}(x,t-\theta_{\ast}r^{2}))\geq\varepsilon_{\ast}r^{n+2-m}.$
(4.3)

  6. (6)

$\mathcal{P}^{n+2-m}\left(\Sigma^{\ast}\setminus\Sigma\right)=0$.

  7. (7)

There exists a measurable function $\theta$ on $\Sigma$ such that

$\mu=\theta(x,t)\mathcal{P}^{n+2-m}\lfloor_{\Sigma}.$

Moreover,

$\frac{\varepsilon_{\ast}}{C}\leq\theta\leq
C\quad\quad\mathcal{P}^{n+2-m}-\mbox{a.e. in}~{}~{}\Sigma.$

Before presenting the proof, we first note the following Federer-Ziemer type
result [28]. It can be proved by a Vitali covering argument.

######  Lemma 4.3.

  1. (1)

For $\mathcal{P}^{n+2-m}$ a.e. $(x,t)\in Q_{1/2}$,

$\lim_{r\to 0}r^{m-2-n}\int_{Q_{r}(x,t)}\left(|\nabla
u_{\infty}|^{2}+|u_{\infty}|^{p+1}\right)=0.$

  2. (2)

For $\mathcal{P}^{n-m}$ a.e. $(x,t)\in Q_{1/2}$,

$\lim_{r\to 0}r^{m-n}\int_{Q_{r}(x,t)}|\partial_{t}u_{\infty}|^{2}=0.$

###### Proof of Theorem 4.2.

  1. (1)

This Morrey space bound follows directly by passing to the limit in (3.1) (for
$u_{i}$).

  2. (2)

By definition, for any $(x,t)\notin\Sigma^{\ast}$, there exists an $r>0$ such
that for all $i$ large,

$\int_{Q_{r}^{-}(x,t)}\left(|\nabla
u_{i}|^{2}+|u_{i}|^{p+1}\right)<\varepsilon_{\ast}r^{n+2-m}.$

By Theorem 3.9 and standard parabolic regularity theory, $u_{i}$ are uniformly
bound in $C^{k}(Q_{\delta_{\ast}r}(x,t))$ for each $k\in\mathbb{N}$. By the
weak convergence of $u_{i}$ and Arzela-Ascolli theorem, they converge to
$u_{\infty}$ in $C^{\infty}(Q_{\delta_{\ast}r}(x,t))$.

  3. (3)

This follows directly from the previous point.

  4. (4)

This follows from a standard Vitali type covering argument, by utilizing the
upper density bound in (4.2) and the lower density bound coming from the
definition of $\Sigma^{\ast}$, that is, for any $(x,t)\in\Sigma^{\ast}\cap
Q_{1/2}$ and $0<r<1/2$,

$\mu(Q_{r}(x,t))+\int_{Q_{r}(x,t)}\left(|\nabla
u_{\infty}|^{2}+|u_{\infty}|^{p+1}\right)\geq\frac{\varepsilon_{\ast}}{2}r^{n+2-m}.$
(4.4)

  5. (5)

This follows by combining (4.4) and Lemma 4.3.

  6. (6)

This follows from a Vitali type covering argument, by using (4.4) and Lemma
4.3.

  7. (7)

This follows from differentiation theorem for measures, and an application of
(4.2) and (4.3). ∎

####  4.2. Definition of the mean curvature

In this subsection we define a mean curvature type term for defect measures.

######  Lemma 4.4.

There exists an $\mathbb{R}^{n}$ valued function $\mathbf{H}\in
L^{2}(Q_{1},d\mu)$ such that

$\partial_{t}u_{i}\nabla u_{i}dxdt\rightharpoonup\partial_{t}u_{\infty}\nabla
u_{\infty}dxdt+\frac{1}{m}\mathbf{H}d\mu$

weakly as Radon measures.

###### Proof.

By Cauchy-Schwarz inequality,

$\int_{Q_{1}}|\partial_{t}u_{i}||\nabla
u_{i}|\leq\left(\int_{Q_{1}}|\partial_{t}u_{i}|^{2}\right)^{1/2}\left(\int_{Q_{1}}|\nabla
u_{i}|^{2}\right)^{1/2}$ (4.5)

are bounded as $i\to+\infty$. Therefore we may assume

$\partial_{t}u_{i}\nabla u_{i}dxdt\rightharpoonup\xi\quad\mbox{weakly as
vector valued Radon measures},$

where $\xi$ is an $\mathbb{R}^{n}$-valued Radon measure.

Take the Radon-Nikodym decomposition of $\xi$ with respect to the Lebesgue
measure, $\xi=\xi^{a}+\xi^{s}$, where $\xi^{a}$ is the absolute continuous
part, and $\xi^{s}$ is the singular part.

By Point (3) in Theorem 4.2,

$\xi=\partial_{t}u_{\infty}\nabla
u_{\infty}dxdt\quad\mbox{outside}~{}~{}\Sigma^{\ast}.$

In view of Point (4) in Theorem 4.2, this is just the absolutely continuous
part $\xi^{a}$. In other words,

$\xi^{a}=\partial_{t}u_{\infty}\nabla u_{\infty}dxdt.$ (4.6)

On the other hand, we can also define

$\mathbf{H}_{i}:=\begin{cases}\frac{\partial_{t}u_{i}\nabla u_{i}}{|\nabla
u_{i}|^{2}},&\mbox{if }|\nabla u_{i}|\neq 0\\\
0,&\mbox{otherwise}.\end{cases}$

It satisfies

$\int_{Q_{1}}|\mathbf{H}_{i}|^{2}|\nabla
u_{i}|^{2}dxdt\leq\int_{Q_{1}}|\partial_{t}u_{i}|^{2}dxdt.$

By Hutchinson [44], we may assume $(\mathbf{H}_{i},|\nabla u_{i}|^{2}dxdt)$
converges to $(\widetilde{\mathbf{H}},|\nabla u_{\infty}|^{2}dxdt+\mu)$ weakly
as measure-functions pairs. By Fatou lemma,

$\int_{Q_{1}}|\widetilde{\mathbf{H}}|^{2}|\nabla
u_{\infty}|^{2}dxdt+\int_{Q_{1}}|\widetilde{\mathbf{H}}|^{2}d\mu<+\infty.$
(4.7)

Since $\xi$ is the weak limit of $\partial_{t}u_{i}\nabla u_{i}dxdt$, we have

$\xi=\widetilde{\mathbf{H}}\left(|\nabla u_{\infty}|^{2}dxdt+\mu\right).$
(4.8)

By (4.6),

$\widetilde{\mathbf{H}}=\begin{cases}\frac{\partial_{t}u_{\infty}\nabla
u_{\infty}}{|\nabla u_{\infty}|^{2}},&\mbox{if }|\nabla u_{\infty}|\neq 0\\\
0,&\mbox{otherwise},\end{cases}$

a.e. with respect to the Lebesgue measure in $Q_{1}$. (In fact, this holds
everywhere in $Q_{1}\setminus\Sigma^{\ast}$.) Hence in view of (4.6), we get

$\xi^{s}=\widetilde{\mathbf{H}}d\mu.$

In particular, $\widetilde{\mathbf{H}}$ is the Radon-Nikodym derivative
$d\xi^{s}/d\mu$. The proof is complete by defining
$\mathbf{H}:=m\widetilde{\mathbf{H}}$. ∎

Similar to (4.5), we obtain

######  Corollary 4.5.

For each $Q_{r}(x,t)\subset Q_{1}$,

$\frac{1}{m^{2}}\int_{Q_{r}(x,t)}|\mathbf{H}|^{2}d\mu\leq\nu(Q_{r}(x,t))\mu(Q_{r}(x,t)).$

A more precise estimate will be given in Lemma 8.3 below.

Passing to the limit in (2.2) gives the limiting energy inequality:

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\int_{Q_{1}}\left[\left(\frac{|\nabla
u_{\infty}|^{2}}{2}-\frac{|u_{\infty}|^{p+1}}{p+1}\right)\partial_{t}\eta^{2}-|\partial_{t}u_{\infty}|^{2}\eta^{2}-2\eta\partial_{t}u_{\infty}\nabla
u_{\infty}\cdot\nabla\eta\right]$ (4.9) $\displaystyle+$
$\displaystyle\frac{1}{m}\int_{Q_{1}}\partial_{t}\eta^{2}d\mu-\int_{Q_{1}}\eta^{2}d\nu-\frac{1}{m}\int_{Q_{1}}\nabla\eta^{2}\cdot\mathbf{H}d\mu.$

Passing to the limit in (2.3) gives the limiting stationary condition: for any
$Y\in C_{0}^{\infty}(Q_{1},\mathbb{R}^{n})$,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{Q_{1}}\left[\left(\frac{|\nabla
u_{\infty}|^{2}}{2}-\frac{|u_{\infty}|^{p+1}}{p+1}\right)\mbox{div}Y-DY(\nabla
u_{\infty},\nabla u_{\infty})+\partial_{t}u_{\infty}\nabla u_{\infty}\cdot
Y\right]$ (4.10) $\displaystyle+$
$\displaystyle\frac{1}{m}\int_{Q_{1}}\left[\left(I-m\mathcal{T}\right)\cdot
DY+\mathbf{H}\cdot Y\right]d\mu.$

####  4.3. Limiting monotonicity formula

In this section we establish the monotonicity formula for the limit
$(u_{\infty},\mu)$. For this purpose, we need first to define the time slices
of $\mu$.

######  Lemma 4.6.

There exists a family of Radon measures$\mu_{t}$ on $B_{1}$ (defined for a.e.
$t\in(-1,1)$) such that

$\mu=\mu_{t}dt.$ (4.11)

###### Proof.

Denote the projection onto the time axis by $\pi$. For each $r<1$, let
$\mu^{r}$ be the restriction of $\mu$ to $B_{r}\times(-1,1)$.

Take a function $\eta_{r}\in C_{0}^{\infty}(B_{1})$, $0\leq\eta_{r}\leq 1$ and
$\eta_{r}\equiv 1$ in $B_{r}$. The limiting energy inequality (4.9) implies
that

$\int_{B_{1}}\left(\frac{1}{2}|\nabla
u_{\infty}(x,t)|^{2}-\frac{1}{p+1}|u_{\infty}(x,t)|^{p+1}\right)\eta_{r}(x)^{2}dx+\frac{1}{m}\pi_{\sharp}\left(\eta_{r}\mu\right)$

is a BV function on $(-1,1)$. Hence there exists a function
$\overline{e}_{r}(t)\in L^{1}(-1,1)$ such that

$\pi_{\sharp}\left(\eta_{r}\mu\right)=\overline{e}_{r}(t)dt,$

Because

$0\leq\pi_{\sharp}\mu^{r}\leq\pi_{\sharp}\left(\eta_{r}\mu\right),$

we find another function $e_{r}(t)\in L^{1}(-1,1)$ such that

$\pi_{\sharp}\mu^{r}=e^{r}(t)dt.$

By the disintegration theorem, there exists a family of probability measure
$\overline{\mu}_{t}$ on $(-1,1)$ such that

$\mu^{r}=\overline{\mu}_{t}d\left(\pi_{\sharp}\mu^{r}\right).$

By defining

$\mu_{t}:=e_{r}(t)\overline{\mu}_{t},$

we get (4.11). ∎

######  Remark 4.7.

Unlike harmonic map heat flows, because the energy density for (1.1) is sign-
changing, we do not know if

$\left[\frac{1}{2}|\nabla
u_{\infty}(x,t)|^{2}-\frac{1}{p+1}|u_{\infty}(x,t)|^{p+1}\right]dx+\mu_{t}$

is a well-defined measure _for all $t$_. Similarly, we also do not know if
there is an estimate on the Hausdorff measure of $\Sigma^{\ast}_{t}$.

Define $\Theta_{s}(x,t;u_{\infty},\mu)$ to be

$\displaystyle s^{\frac{p+1}{p-1}}\int_{B_{1}}\left[\frac{|\nabla
u_{\infty}(y,t-s)|^{2}}{2}-\frac{|u_{\infty}(y,t-s)|^{p+1}}{p+1}\right]G(y-x,s)\psi(y)^{2}dy$
$\displaystyle+$
$\displaystyle\frac{1}{2(p-1)}s^{\frac{2}{p-1}}\int_{B_{1}}u_{\infty}(y,t-s)^{2}G(y-x,s)\psi(y)^{2}dy$
$\displaystyle+$
$\displaystyle\frac{1}{m}s^{\frac{p+1}{p-1}}\int_{\mathbb{R}^{n}}G(y-x,s)\psi(y)^{2}d\mu_{t-s}(y)+Ce^{-cs^{-1}}.$

As in Remark 3.1, rigourously we should integrate one more time in $s$.

Passing to the limit in the monotonicity formula for $u_{i}$, we obtain

######  Proposition 4.8 (Limiting, localized monotonicity formula).

For any $(x,t)\in Q_{1/2}$ and a.a. $0<s_{1}<s_{2}<1/4$,

$\displaystyle\Theta_{s_{2}}(x,t;u_{\infty},\mu)-\Theta_{s_{1}}(x,t;u_{\infty},\mu)$
$\displaystyle\geq$
$\displaystyle\int_{s_{1}}^{s_{2}}\tau^{\frac{2}{p-1}-1}\int_{B_{1}}\Big{|}(t-\tau)\partial_{t}u_{\infty}(y,t-\tau)+\frac{u_{\infty}(y,t-\tau)}{p-1}+\frac{y}{2}\cdot\nabla
u_{\infty}(y,t-\tau)\Big{|}^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\times
G(y-x,\tau)\psi(y)^{2}dyd\tau$ $\displaystyle+$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{B_{1}}\tau^{\frac{2}{p-1}-1}(t-\tau)^{2}G(y-x,\tau)\psi(y)^{2}d\nu(y,\tau)$
$\displaystyle+$
$\displaystyle\int_{s_{1}}^{s_{2}}\int_{B_{1}}\tau^{\frac{2}{p-1}-1}\frac{|y|^{2}}{4}G(y-x,\tau)\psi(y)^{2}d\mu(y,\tau)$
$\displaystyle+$
$\displaystyle\frac{1}{m}\int_{s_{1}}^{s_{2}}\int_{B_{1}}\tau^{\frac{2}{p-1}-1}(t-\tau)y\cdot\mathbf{H}(y,\tau)G(y-x,\tau)\psi(y)^{2}d\mu(y,\tau).$

###  5\. Tangent flow analysis, I

In this section we perform the tangent flow analysis for $(u_{\infty},\mu)$.

For any $(x,t)\in\Sigma^{\ast}$ and a sequence $\lambda_{i}\to 0$, define the
blowing up sequence

$\left\\{\begin{aligned}
&u_{\infty}^{\lambda_{i}}(y,s):=\lambda_{i}^{\frac{2}{p-1}}u_{\infty}(x+\lambda_{i}y,t+\lambda_{i}^{2}s),\\\
&\mu^{\lambda_{i}}(A):=\lambda_{i}^{m-2-n}\mu(\lambda_{i}A),\quad\nu^{\lambda_{i}}(A):=\lambda_{i}^{m-n}\nu(\lambda_{i}A),\quad\mbox{for
any}~{}~{}A\subset\mathbb{R}^{n}\times\mathbb{R}.\end{aligned}\right.$

By scaling (4.2), we see that with the same constant $M$ in Corollary 3.5, for
any $Q_{R}\subset\mathbb{R}^{n}\times\mathbb{R}$, we have

$\left\\{\begin{aligned} &\mu^{\lambda_{i}}(Q_{R})+\int_{Q_{R}}\left(|\nabla
u^{\lambda_{i}}_{\infty}|^{2}+|u_{\infty}^{\lambda_{i}}|^{p+1}\right)\leq
MR^{n+2-m},\\\
&\nu^{\lambda_{i}}(Q_{R})+\int_{Q_{R}}|\partial_{t}u_{\infty}^{\lambda_{i}}|^{2}\leq
MR^{n-m}.\end{aligned}\right.$ (5.1)

Therefore there exists a subsequence (still denoted by $\lambda_{i}$) such
that $u_{\infty}^{\lambda_{i}}\rightharpoonup u_{\infty}^{0}$ weakly in
$L^{2}_{t}H^{1}_{x,loc}(\mathbb{R}^{n}\times\mathbb{R})\cap
L^{p+1}_{loc}(\mathbb{R}^{n}\times\mathbb{R})$, and

$\left\\{\begin{aligned} &|\nabla
u_{\infty}^{\lambda_{i}}|^{2}dxdt+\mu^{\lambda_{i}}\rightharpoonup|\nabla
u_{\infty}^{0}|^{2}dxdt+\mu^{0},\\\
&|\partial_{t}u_{\infty}^{\lambda_{i}}|^{2}dxdt+\nu^{\lambda_{i}}\rightharpoonup|\partial_{t}u_{\infty}^{0}|^{2}dxdt+\nu^{0}\end{aligned}\right.$

weakly as Radon measures on any compact set of
$\mathbb{R}^{n}\times\mathbb{R}$.

######  Remark 5.1.

By Lemma 4.3, we can avoid a set of zero $\mathcal{P}^{n+2-m}$ measure so that

$\lim_{r\to 0}\left[r^{m-2-n}\int_{Q_{r}(x,t)}\left(|\nabla
u_{\infty}|^{2}+|u_{\infty}|^{p+1}\right)+r^{m-n}\int_{Q_{r}(x,t)}|\partial_{t}u_{\infty}|^{2}\right]=0.$
(5.2)

Under this assumption, $u_{\infty}^{\lambda_{i}}$ converges to $0$ strongly in
$L^{2}_{t}H^{1}_{x,loc}(\mathbb{R}^{n}\times\mathbb{R})\cap
L^{p+1}_{loc}(\mathbb{R}^{n}\times\mathbb{R})$,
$\partial_{t}u_{\infty}^{\lambda_{i}}$ converges to $0$ strongly in
$L^{2}_{loc}(\mathbb{R}^{n}\times\mathbb{R})$, so

$\mu^{\lambda_{i}}\rightharpoonup\mu^{0},\quad\nu^{\lambda_{i}}\rightharpoonup\nu^{0}.$

This special choice will be used in Section 6.

Because $\mu$ is the defect measure coming from $u_{i}$, we can take a further
subsequence of $u_{i}$ so that, by defining

$u_{i}^{\lambda_{i}}(y,s):=\lambda_{i}^{\frac{2}{p-1}}u_{i}\left(x+\lambda_{i}y,t+\lambda_{i}^{2}s\right),$

we have

$\left\\{\begin{aligned} &|\nabla
u_{i}^{\lambda_{i}}|^{2}dyds\rightharpoonup|\nabla
u_{\infty}^{0}|^{2}dxdt+\mu^{0},\\\
&|\partial_{t}u_{i}^{\lambda_{i}}|^{2}dyds\rightharpoonup|\partial_{t}u_{\infty}^{0}|^{2}dxdt+\nu^{0}.\end{aligned}\right.$

As a consequence, all results in Section 4 hold for $u_{\infty}^{0}$,
$\mu^{0}$ and $\nu^{0}$. In particular,

  1. (1)

by Lemma 4.6, there exists a family of Radon measures $\mu^{0}_{t}$ on
$\mathbb{R}^{n}$ (for a.a. $t\in\mathbb{R}$) such that

$\mu^{0}=\mu^{0}_{t}dt;$ (5.3)

  2. (2)

there exists an $\mathbf{H}^{0}\in
L^{2}_{loc}(\mathbb{R}^{n}\times\mathbb{R},d\mu^{0})$ such that

$\partial_{t}u_{i}^{\lambda_{i}}\nabla
u_{i}^{\lambda_{i}}dxdt\rightharpoonup\partial_{t}u_{\infty}^{0}\nabla
u_{\infty}^{0}dxdt+\frac{1}{m}\mathbf{H}^{0}d\mu^{0};$

  3. (3)

for any $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}$ and $s>0$, we define
$\Theta_{s}(x,t;u_{\infty}^{0},\mu^{0})$ to be

$\displaystyle s^{\frac{p+1}{p-1}}\int_{\mathbb{R}^{n}}\left[\frac{|\nabla
u_{\infty}^{0}(y,t-s)|^{2}}{2}-\frac{|u_{\infty}^{0}(y,t-s)|^{p+1}}{p+1}\right]G(y-x,s)dy$
$\displaystyle+$
$\displaystyle\frac{1}{2(p-1)}s^{\frac{2}{p-1}}\int_{\mathbb{R}^{n}}u_{\infty}^{0}(y,t-s)^{2}G(y-x,s)dy+\frac{1}{m}s^{\frac{p+1}{p-1}}\int_{\mathbb{R}^{n}}G(y-x,s)d\mu_{t-s}^{0}(y),$

which is still non-decreasing in $s>0$.

For any $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}$, we still define

$\Theta(x,t;u_{\infty}^{0},\mu^{0}):=\lim_{s\to
0}\Theta_{s}(x,t;u_{\infty}^{0},\mu^{0}).$

By the scaling invariance of $\Theta$, we obtain

$\Theta_{s}(0,0;u_{\infty}^{0},\mu^{0})=\Theta(x,t;u_{\infty},\mu),\quad\forall
s>0.$ (5.4)

Then an application of Proposition 4.8 gives

######  Lemma 5.2.

  1. (1)

The function $u_{\infty}^{0}$ is backwardly self-similar in the sense that

$u_{\infty}^{0}(\lambda
x,\lambda^{2}t)=\lambda^{-\frac{2}{p-1}}u_{\infty}(x,t),\quad\forall(x,t)\in\mathbb{R}^{n}\times\mathbb{R}^{-},~{}~{}\lambda>0.$

  2. (2)

The measure $\mu^{0}$ is backwardly self-similar in the sense that

$\mu^{0}(\lambda
A)=\lambda^{n+2-m}\mu^{0}(A),\quad\forall\lambda>0,~{}~{}~{}A\subset\mathbb{R}^{n}\times\mathbb{R}^{-}.$

By (5.4), following the proof of Lin-Wang [58, Lemma 8.3.3], we obtain

######  Lemma 5.3.

For any $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}$,

$\Theta(x,t;u_{\infty}^{0},\mu^{0})\leq\Theta(0,0;u_{\infty}^{0},\mu^{0}).$

Moreover, if the equality is attained at $(x,t)$, then $u_{\infty}^{0}$ and
$\mu^{0}$ are translational invariant in the $(x,t)$-direction.

By this lemma,

$\mathcal{L}(u_{\infty}^{0},\mu^{0}):=\left\\{(x,t):~{}~{}\Theta(x,t;u_{\infty}^{0},\mu^{0})=\Theta(0,0;u_{\infty}^{0},\mu^{0})\right\\}$

is a linear subspace of $\mathbb{R}^{n}\times\mathbb{R}$, which is called the
invariant subspace of $(u_{\infty}^{0},\mu^{0})$.

######  Definition 5.4.

The invariant dimension of $(u_{\infty}^{0},\mu^{0})$ is

$\begin{cases}k+2,&\mbox{if
}\mathcal{L}_{\mu_{0}}=\mathbb{R}^{k}\times\mathbb{R}\\\
k,&\mbox{otherwise}.\end{cases}$

Using these notations we can define a stratification of $\Sigma^{\ast}$, and
give a dimension estimate on these stratifications as in White [89], see also
[58, Section 8.3].

###  6\. The case $m$ is not an integer

In this section, we use Marstrand theorem ([61], see also [59, Theorem
1.3.12]) to study the case when $m$ is not an integer. This is similar to the
elliptic case studied in Du [26] and the authors [86]. The main result in this
case is

######  Theorem 6.1.

Suppose $m$ is not an integer. Then under the assumptions of Theorem 4.2,
$u_{i}$ converges strongly in $L^{p+1}_{loc}(Q_{1})$ and $\nabla u_{i}$
converges strongly to $\nabla u_{\infty}$ in $L^{2}_{loc}(Q_{1})$. In other
words, $\mu=0$. As a consequence, $u_{\infty}$ is a suitable weak solution of
(1.1) in $Q_{1}$.

Here we do not claim the strong convergence of $\partial_{t}u_{i}$.

To prove this theorem, we use the tangent flow analysis in the previous
section, but only performed at a point $(x,t)$ satisfying the condition (5.2).
Hence by Remark 5.1, $u_{\infty}^{0}=0$ and

$\mu^{\lambda}\rightharpoonup\mu^{0},\quad\nu^{\lambda}\rightharpoonup\nu^{0}.$

By (4.3) in Theorem 4.2, $\mu^{0}$ is not identically zero in
$\mathbb{R}^{n}\times\mathbb{R}^{-}$. Then by the self-similarity of
$\mu^{0}$, there exists a point $(x_{0},-1)\in\mbox{spt}(\mu^{0})$. We blow up
$\mu^{0}$ again at $(x_{0},-1)$ as in Section 5, producing a tangent measure
$\mu^{1}$ to $\mu^{0}$ at this point.

######  Lemma 6.2.

The measure $\mu^{1}$ is static, that is,

$\partial_{t}\mu^{1}=0\quad\mbox{in the distributional sense}.$

###### Proof.

By Lemma 5.2 and the scaling invariance of $\Theta_{s}$, for any $\lambda>0$,

$\Theta_{\lambda^{2}s}(\lambda
x_{1},-\lambda^{2};\mu^{0})=\Theta_{s}(x_{1},-1;\mu^{0}).$

Letting $s\to 0$, we obtain

$\Theta(\lambda x_{1},-\lambda^{2};\mu^{0})=\Theta(x_{1},-1;\mu^{0}).$

After blowing up to $\mu^{1}$, this equality implies that

$\Theta(0,t;\mu^{1})=\Theta(0,0;\mu^{1}),\quad\forall t\in\mathbb{R}.$

By Lemma 5.3, $\mu^{1}$ is invariant under translations in the time direction.
∎

######  Corollary 6.3.

$\nu^{1}=0$ and $\mathbf{H}^{1}=0$ $\mu^{1}$-a.e..

###### Proof.

This follows from the energy identity for $\mu^{1}$,

$\frac{d}{dt}\int_{\mathbb{R}^{n}}\eta d\mu^{1}_{t}=-m\int\eta
d\nu^{1}-\int_{\mathbb{R}^{n}}\nabla\eta\cdot\mathbf{H}^{1}d\mu^{1}_{t}.$

Note that $\mathbf{H}^{1}$ can be controlled by $\nu^{1}$ as in Corollary 4.5.
∎

By the previous lemma, we can view $\mu^{1}$ as a Radon measure on
$\mathbb{R}^{n}$. Now the stationary condition (4.10) reads as

$\int_{\mathbb{R}^{n}}\left(I-m\mathcal{T}\right)\cdot
DYd\mu^{1}=0,\quad\forall Y\in C_{0}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{n}).$
(6.1)

Following Moser [72], this is called a stationary measure. Similar to [72,
Lemma 2.1], we have

######  Lemma 6.4.

For any $x\in\mathbb{R}^{n}$,

$\Theta_{r}(x;\mu^{1}):=r^{m-n}\mu_{1}(B_{r}(x))$

is non-decreasing in $r>0$.

As a consequence,

$\Theta(x;\mu^{1}):=\lim_{r\to 0}r^{m-n}\mu_{1}(B_{r}(x))$

exists. By (4.2), (4.3) and Lemma 4.3,

$\frac{\varepsilon_{\ast}}{2}\leq\Theta(x;\mu^{1})\leq
C,\quad\mu^{1}-\mbox{a.e. in}~{}~{}\mathbb{R}^{n}.$

By Marstrand theorem, $m$ must be an integer. In other words, if $m$ is not an
integer, then $\mu^{1}$, and consequently, $\mu^{0}$ and $\mu$, must be
trivial. This finishes the proof of Theorem 6.1.

###  7\. Partial regularity of suitable weak solutions

In this section, as an application of Theorem 6.1, we prove the following
partial regularity result for a fixed, suitable weak solution. This is almost
the same with the one obtained in [9], with a small improvement.

######  Theorem 7.1.

Suppose $u$ is a suitable weak solution of (1.1) in $Q_{1}$. Then

  * •

If $1<p<\frac{n+2}{n-2}$, $u$ is smooth, that is, $\mathcal{S}(u)=\emptyset$.

  * •

If there exists an integer $k$ with $3\leq k<n$ such that
$\frac{k+3}{k-1}<p<\frac{k+2}{k-2}$, then
$\mbox{dim}_{\mathcal{P}}(\mathcal{S}(u))\leq n-k+2$.

  * •

If $m$ is an integer, then $\mathcal{P}^{n-m+2}(\mathcal{S}(u))=0$.

We have already obtained a dimension bound on the singular set
$\mathcal{S}(u)$ in Corollary 3.10. Here we need only to improve this bound
when $m$ is not an integer. This follows from a standard dimension reduction
argument, cf. [54], [86].

Suppose $(x,t)\in\mathcal{S}(u)\cap Q_{1/2}$. By Theorem 3.9, for any
$r\in(0,1/2)$,

$\int_{Q_{r}^{-}(x,t)}\left(|\nabla
u|^{2}+|u|^{p+1}\right)\geq\varepsilon_{\ast}r^{n+2-m}.$ (7.1)

For $\lambda\to 0$, define the blowing up sequence

$u^{\lambda}(y,s):=\lambda^{\frac{2}{p-1}}u(x+\lambda y,t+\lambda^{2}s).$

As in Section 5, we can take a subsequence so that $u^{\lambda}\rightharpoonup
u^{0}$ weakly in $L^{2}_{t}H^{1}_{x,loc}(\mathbb{R}^{n}\times\mathbb{R})\cap
L^{p+1}_{loc}(\mathbb{R}^{n}\times\mathbb{R})$, and

$\left\\{\begin{aligned} &|\nabla u^{\lambda}|^{2}dxdt\rightharpoonup|\nabla
u^{0}|^{2}dxdt+\mu^{0},\\\
&|\partial_{t}u^{\lambda}|^{2}dxdt\rightharpoonup|\partial_{t}u^{0}|^{2}dxdt+\nu^{0}\end{aligned}\right.$

weakly as Radon measures on any compact set of
$\mathbb{R}^{n}\times\mathbb{R}$.

Furthermore, because $m$ is not an integer, by Theorem 6.1, we deduce that
$\mu^{0}=0$ and $u^{\lambda}$ in fact converges strongly. Because tangent
flows are always nontrivial, $u^{0}\neq 0$.

By Lemma 5.2, $u^{0}$ is backwardly self-similar. For these solutions, the
following Liouville theorem was established by Giga-Kohn in [36].

######  Theorem 7.2.

When $p\leq(n+2)/(n-2)$, if $u$ is a backward self-similar solution of (1.1)
in $\mathbb{R}^{n}\times\mathbb{R}^{-}$, satisfying

$\sup_{x\in\mathbb{R}^{n}}\int_{Q_{1}^{-}(x,-1)}\left(|\nabla
u|^{2}+|u|^{p+1}\right)<+\infty,$ (7.2)

then either $u\equiv 0$, or
$u\equiv\pm(p-1)^{-\frac{1}{p-1}}(-t)^{-\frac{1}{p-1}}$.

Here the original $L^{\infty}(\mathbb{R}^{n})$ assumption in [36, Theorem 1]
is replaced by an integral one (7.2). With this condition, we are still able
to do the same computation in [36] to deduce the Pohozaev identity [36,
Proposition 1]. For $u^{0}$, the condition (7.2) follows by scaling (3.2) and
passing to the limit.

By this theorem, if $p<(n+2)/(n-2)$ is subcritical, because $u^{0}$ also
satisfies

$\int_{Q_{1}}|u^{0}|^{p+1}<+\infty,$

we must have $u^{0}=0$. This contradiction implies that there is no singular
point of $u$. In other words, $u$ is smooth.

If $p>(n+2)/(n-2)$ is supcritical, by Point (3) in Theorem 4.2, the strong
convergence of $u^{\lambda}$ implies that if $\lambda$ is small enough, then
$\lambda^{-1}\left(\mathcal{S}(u)-(x,t)\right)$ is contained in a small
neighborhood of $\mathcal{S}(u^{0})$. Since $u^{0}$ is backwardly self-similar
(see Lemma 5.2), applying White’s stratification theorem in [89], we conclude
the proof of Theorem 7.1.

###  8\. The case $m$ is an integer

In this section we continue the analysis of energy concentration behavior, now
under the assumption that $m$ is an integer. For this case we prove

######  Theorem 8.1.

If $m$ is an integer, then $(u_{\infty},\mu)$ is a generalized Brakke’s flow
in the following sense: For any $\varphi\in
C_{0}^{\infty}(B_{1};\mathbb{R}^{+})$ and $-1<t_{1}<t_{2}<1$,

$\displaystyle\left[\int_{B_{1}}\left(\frac{1}{2}|\nabla
u_{\infty}(t_{2})|^{2}-\frac{1}{p+1}|u_{\infty}(t_{2})|^{p+1}\right)\varphi
dx+\frac{1}{m}\int_{B_{1}}\varphi d\mu_{t_{2}}\right]$
$\displaystyle-\left[\int_{B_{1}}\left(\frac{1}{2}|\nabla
u_{\infty}(t_{1})|^{2}-\frac{1}{p+1}|u_{\infty}(t_{1})|^{p+1}\right)\varphi
dx+\frac{1}{m}\int_{B_{1}}\varphi d\mu_{t_{1}}\right]$ $\displaystyle\leq$
$\displaystyle-\int_{t_{1}}^{t_{2}}\int_{B_{1}}\left[|\partial_{t}u_{\infty}|^{2}\varphi+\partial_{t}u_{\infty}\nabla
u_{\infty}\cdot\nabla\varphi\right]dxdt$
$\displaystyle-\frac{1}{m}\int_{t_{1}}^{t_{2}}\int_{\Sigma_{t}}\left[|\mathbf{H}_{t}|^{2}\varphi-\nabla\varphi\cdot\mathbf{H}_{t}\right]d\mu_{t}dt.$

By Lemma 4.6, for a.e. $t\in(-1,1)$ and any $Y\in
C_{0}^{\infty}(B_{1},\mathbb{R}^{n})$,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{B_{1}}\left[\left(\frac{|\nabla
u_{\infty}(x,t)|^{2}}{2}-\frac{|u_{\infty}(x,t)|^{p+1}}{p+1}\right)\mbox{div}Y-DY\left(\nabla
u_{\infty}(x,t),\nabla u_{\infty}(x,t)\right)\right]dx$ $\displaystyle+$
$\displaystyle\int_{B_{1}}\partial_{t}u_{\infty}(x,t)\nabla
u_{\infty}(x,t)\cdot Ydx$ $\displaystyle+$
$\displaystyle\int_{B_{1}}\left[\left(\frac{1}{m}I-\mathcal{T}(x,t)\right)\cdot
DY+\frac{1}{m}\mathbf{H}(x,t)\cdot Y\right]d\mu_{t}(x).$

In view of the lower density bound in (4.3), as in [1] or [52] (see also [72]
or [2, Proposition 3.1]), we deduce that $\mu_{t}$ is countably
$(n-m)$-rectifiable. In other words, $\Sigma_{t}$ is countably
$(n-m)$-rectifiable, and

$I-m\mathcal{T}=T_{x}\Sigma_{t},\quad\mathcal{H}^{n-m}~{}\mbox{a.e.
on}~{}\Sigma_{t},$ (8.3)

where $T_{x}\Sigma_{t}$ is the weak tangent space (identified with the
projection map onto it) of $\Sigma_{t}$ at $x$.

Similar to Lin-Wang [58, Lemma 9.2.2], we get

######  Lemma 8.2.

For a.e. $t\in(-1,1)$,

$\mathbf{H}(x,t)\bot
T_{x}\Sigma_{t},\quad\mbox{for}~{}\mathcal{H}^{n-m}~{}\mbox{a.e.
}~{}x\in\Sigma_{t}.$

Similar to Lin-Wang [58, Lemma 9.2.7], we also get

######  Lemma 8.3.

For any $\eta\in C_{0}^{\infty}(Q_{1})$,

$\int_{Q_{1}}\eta^{2}|\mathbf{H}(x,t)|^{2}d\mu_{t}(x)dt\leq\frac{1}{m}\int_{Q_{1}}\eta^{2}d\nu.$

Plugging this estimate into (4.9), we obtain (8.1). This finishes the proof of
Theorem 8.1.

###  9\. The case $p=(n+2)/(n-2)$

In this section we assume $p=(n+2)/(n-2)$ is the Sobolev critical exponent,
and all of the solutions $u_{i}$ are _smooth_. Note that now
$m=2(p+1)/(p-1)=n$. The main result of this section is about the quantization
of energy.

######  Theorem 9.1.

For any $(x,t)\in\Sigma$, there exist finitely many bubbles $W^{k}$,
$k=1,\cdots,N$, such that

$\Theta(x,t)=\frac{1}{n\left(4\pi\right)^{n/2}}\sum_{k=1}^{N}\int_{\mathbb{R}^{n}}|\nabla
W^{k}|^{2}.$

Furthermore, if all solutions are positive, then there exists an
$N\in\mathbb{N}$ such that

$\Theta(x,t)=N\frac{\Lambda}{n\left(4\pi\right)^{n/2}}.$

We first prove a local version of this proposition in Subsection 9.1, then
prove this theorem in Subsection 9.2. During this course, some further
properties of defect measures will also be established in Subsection 9.2 and
Subsection 9.3, in particular, for applications in Part II, Part III and Part
IV, the special case of positive solutions will be discussed.

####  9.1. A local quantization result

In this subsection we prove the following

######  Lemma 9.2.

Given a constant $M>0$, suppose a sequence of smooth solutions $u_{i}$ to
(1.1) in $Q_{1}$ satisfies

$\left\\{\begin{aligned} &|\nabla u_{i}|^{2}dxdt\rightharpoonup
M\delta_{0}\otimes dt,\\\ &|u_{i}|^{p+1}dxdt\rightharpoonup M\delta_{0}\otimes
dt,\\\ &\int_{Q_{1}}|\partial_{t}u_{i}|^{2}dxdt\to 0.\end{aligned}\right.$

Then there exist finitely many bubbles $W^{k}$ such that

$M=\sum_{k}\int_{\mathbb{R}^{n}}|\nabla W^{k}|^{2}dx.$

First let us present some immediate consequences of the assumptions in this
lemma.

  * •

An application of the $\varepsilon$-regularity theorem (Theorem 3.9) implies
that

$u_{i}\to
0\quad\mbox{in}~{}C^{\infty}_{loc}\left((B_{1}\setminus\\{0\\})\times(-1,1)\right).$
(9.1)

  * •

By the $L^{2}(Q_{1})$ bound on $\partial_{t}u_{i}$, we deduce that $u_{i}\to
0$ in $C_{loc}(-1,1;L^{2}_{loc}(B_{1}))$.

  * •

By Fatou lemma,

$\int_{-1}^{1}\lim_{i\to+\infty}\left(\int_{B_{1}}|\partial_{t}u_{i}(x,t)|^{2}dx\right)dt\leq\lim_{i\to+\infty}\int_{Q_{1}}|\partial_{t}u_{i}|^{2}dxdt=0.$

Hence for a.e. $t\in(-1,1)$,

$\lim_{i\to+\infty}\int_{B_{1}}|\partial_{t}u_{i}(x,t)|^{2}dx=0.$ (9.2)

The following lemma describes the energy concentration behavior for a.e. time
slice of $u_{i}$, under the assumptions in Lemma 9.2.

######  Lemma 9.3.

For a.e. $t\in(-1,1)$,

$\left\\{\begin{aligned} &|\nabla u_{i}(x,t)|^{2}dx\rightharpoonup
M\delta_{0},\\\ &|u_{i}(x,t)|^{p+1}dx\rightharpoonup
M\delta_{0}.\end{aligned}\right.$

###### Proof.

For any $\eta\in C_{0}^{\infty}(B_{1})$, by the energy identity for $u_{i}$,
the function

$E_{\eta,i}(t):=\int_{B_{1}}\left(\frac{|\nabla
u_{i}(x,t)|^{2}}{2}-\frac{|u_{i}(x,t)|^{p+1}}{p+1}\right)\eta(x)^{2}dx$

is uniformly bounded in $BV_{loc}(-1,1)$. After passing to a subsequence, they
converge in $L^{1}_{loc}(-1,1)$ and a.e. in $(-1,1)$ to the limit
$(M/n)\eta(0)^{2}$.

Another consequence of this uniform BV bound is, $E_{\eta,i}$ are uniformly
bounded in any compact set of $(-1,1)$. If (9.2) holds at $t$, then

$\int_{B_{1}}\left(|\nabla
u_{i}(t)|^{2}-|u_{i}(t)|^{p+1}\right)\eta^{2}=-\int_{B_{1}}\left[\partial_{t}u_{i}(t)u_{i}(t)\eta^{2}+2\eta
u_{i}(t)\nabla u_{i}(t)\cdot\nabla\eta\right]$

are also bounded as $i\to+\infty$. Therefore

$\limsup_{i\to+\infty}\int_{B_{1}}\left[|\nabla
u_{i}(t)|^{2}+|u_{i}(t)|^{p+1}\right]dx<+\infty.$ (9.3)

Because $u_{i}(t)\to 0$ in $L^{2}(B_{1})$, with the help of (9.1), we find a
nonnegative constant $M(t)$ such that

$|\nabla u_{i}(x,t)|^{2}dx\rightharpoonup
M(t)\delta_{0},\quad|u_{i}(x,t)|^{p+1}dx\rightharpoonup M(t)\delta_{0}.$

Then using the above a.e. convergence of $E_{\eta,i}(t)$, and letting $\eta$
vary in $C_{0}^{\infty}(B_{1})$, we deduce that $M(t)=M$. ∎

For those $t$ satisfying (9.2), we have the uniform $H^{1}(B_{1})$ bound
(9.3), hence by Struwe’s global compactness theorem ([82]), the following
bubble tree convergence holds for $u_{i}(t)$.

######  Proposition 9.4 (Bubble tree convergence).

There exist $N(t)$ points $\xi_{ik}^{\ast}(t)$, positive constants
$\lambda_{ik}^{\ast}(t)$, $k=1,\cdots,N(t)$, all converging to $0$ as
$i\to+\infty$, and $N(t)$ bubbles $W^{k}$, such that

$u_{i}(x,t)=\sum_{k=1}^{N(t)}W^{k}_{\xi_{ik}^{\ast}(t),\lambda_{ik}^{\ast}(t)}(x)+o_{i}(1),$

where $o_{i}(1)$ are measured in $H^{1}(B_{1})$.

As a consequence,

$\int_{B_{1}}|\nabla
u_{i}(x,t)|^{2}dx=\sum_{k=1}^{N(t)}\int_{\mathbb{R}^{n}}|\nabla
W^{k}|^{2}+o_{i}(1).$ (9.4)

######  Remark 9.5.

If $u_{i}$ are positive, then all bubbles constructed in this proposition are
positive, see [24, Section 3.2]. In view of the Liouville theorem of
Caffarelli-Gidas-Spruck [8], we can take all of these $W^{k}$ to be the
standard Aubin-Talenti bubble $W$. As a consequence, (9.4) reads as

$\int_{B_{1}}|\nabla u_{i}(x,t)|^{2}dx=N(t)\Lambda+o_{i}(1).$ (9.5)

######  Remark 9.6.

About bubble tree convergence, there are two phenomena that we will
investigate more closely latter.

Bubble towering:

Two bubbles at $\xi_{ik}^{\ast}(t)$ and $\xi_{i\ell}^{\ast}(t)$ are towering
if

$\limsup_{i\to+\infty}\frac{|\xi_{ik}^{\ast}(t)-\xi_{i\ell}^{\ast}(t)|}{\max\\{\lambda_{ik}^{\ast}(t),\lambda_{i\ell}^{\ast}(t)\\}}<+\infty.$

In this case, either
$\frac{\lambda_{ik}^{\ast}(t)}{\lambda_{i\ell}^{\ast}(t)}\to+\infty$ or
$\frac{\lambda_{i\ell}^{\ast}(t)}{\lambda_{ik}^{\ast}(t)}\to+\infty$. Hence
these two bubbles are located at almost the same point (with respect to the
bubble scales), but the height of one bubble is far larger than the other
one’s.

Bubble clustering:

If for some $k\neq\ell$,

$\lim_{i\to+\infty}|\xi_{ik}^{\ast}(t)-\xi_{i\ell}^{\ast}(t)|=0$

but

$\lim_{i\to+\infty}\frac{|\xi_{ik}^{\ast}(t)-\xi_{i\ell}^{\ast}(t)|}{\max\\{\lambda_{ik}^{\ast}(t),\lambda_{i\ell}^{\ast}(t)\\}}=+\infty,$

we say these bubbles are clustering.

Using the terminology introduced in the study of Yamabe problem (see Schoen
[77]), if there is no bubble clustering and towering, the blow up is called
_isolated and simple_.

Combining Lemma 9.3 with Proposition 9.4, we conclude the proof of Lemma 9.2.
Furthermore, if all solutions are positive, by Remark 9.5, there exists an
$N\in\mathbb{N}$ such that

$M=N\Lambda,\quad\mbox{and}\quad N(t)=N\quad\mbox{a.e. in }~{}~{}(-1,1).$
(9.6)

####  9.2. Proof of Theorem 9.1

Under the critical exponent assumption, the tangent flow analysis in Section 5
can give more information.

First, by noting that $m=n$, in the definition of the blowing up sequence, we
have

$\nu^{\lambda}(A):=\nu\left((x,t)+\lambda A\right),\quad\forall
A\subset\mathbb{R}^{n}\times\mathbb{R}.$

In the same way, for any $R>0$,

$\int_{Q_{R}}|\partial_{t}u_{\infty}^{\lambda}|^{2}=\int_{Q_{\lambda
R}}|\partial_{t}u_{\infty}|^{2}\to 0\quad\mbox{as}~{}\lambda\to 0.$ (9.7)

Therefore by defining

$Q:=\lim_{\lambda\to 0}\nu\left(Q_{\lambda}(x,t)\right),$ (9.8)

we get

$\nu^{0}=Q\delta_{(0,0)}.$ (9.9)

There is an atom of $\nu$ at $(x,t)$ if and only if $Q>0$.

By Corollary 4.5, we find that

$\mathbf{H}^{0}d\mu=\mathbf{H}^{0}(0,0)\delta_{(0,0)}.$ (9.10)

Recall that $u_{\infty}$ is backwardly self-similar (Lemma 5.2). By combining
Theorem 7.2 with (9.7), we deduce that

$u_{\infty}^{0}\equiv 0\quad\mbox{in}~{}~{}\mathbb{R}^{n}\times\mathbb{R}.$

Hence in view of (9.9) and (9.10), the energy identity (4.9) for
$(u_{\infty}^{0},\mu^{0})$ reads as

$\frac{1}{m}\int_{Q_{1}}\partial_{t}\eta
d\mu^{0}=Q\eta(0,0)+\frac{1}{m}\nabla\eta(0,0)\cdot\mathbf{H}^{0}(0,0),\quad\forall\eta\in
C_{0}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}).$ (9.11)

In particular,

$\partial_{t}\mu^{0}=0\quad\mbox{in the distributional sense in
}~{}\left(\mathbb{R}^{n}\times\mathbb{R}\right)\setminus\\{(0,0)\\}.$

Combining this fact with the backward self-similarity of $\mu^{0}$ (see Lemma
5.2), we deduce that there exists a constant $M\geq Q$ such that

$\mu^{0}=M\delta_{0}\otimes
dt\lfloor_{\mathbb{R}^{-}}+\left(M-Q\right)\delta_{0}\otimes
dt\lfloor_{\mathbb{R}^{+}}.$ (9.12)

By choosing $\eta(x,t)=\varphi(x)\psi(t)x$ in (9.11), where $\varphi\in
C_{0}^{\infty}(\mathbb{R}^{n})$ and $\psi\in C_{0}^{\infty}(\mathbb{R})$, we
also deduce that

$\mathbf{H}^{0}(0,0)=0.$ (9.13)

Now we come to the proof of Theorem 9.1.

###### Proof.

As in Section 5, there exist two sequences of solutions to (1.1) (with
$p=(n+2)/(n-2)$) satisfying the assumptions in Lemma 9.2 in $Q_{1}(0,-2)$ and
$Q_{1}(0,2)$. Therefore there exist two groups of finitely many bubbles,
$\\{W^{k}\\}$ and $\\{W^{\ell}\\}$, such that

$M=\sum_{k}\int_{\mathbb{R}^{n}}|\nabla W^{k}|^{2}dx,\quad
M-Q=\sum_{\ell}\int_{\mathbb{R}^{n}}|\nabla W^{\ell}|^{2}dx.$ (9.14)

Furthermore, if all solutions are positive, then there exist
$N_{1},N_{2}\in\mathbb{N}$ such that

$M=N_{1}\Lambda,\quad M-Q=N_{2}\Lambda.$ (9.15)

By the weak convergence of $|\nabla
u_{\infty}^{\lambda}|^{2}dxdt+\mu^{\lambda}$ etc., we get

$\displaystyle\Theta(x,t;u_{\infty},\mu)$ $\displaystyle=$
$\displaystyle\lim_{\lambda\to
0}\int_{1}^{2}\Theta_{\lambda^{2}s}(x,t;u_{\infty},\mu)ds$ $\displaystyle=$
$\displaystyle\lim_{\lambda\to
0}\int_{1}^{2}\Theta_{s}(0,0;u_{\infty}^{\lambda},\mu^{\lambda})ds$
$\displaystyle=$
$\displaystyle\frac{1}{n}\int_{1}^{2}s^{\frac{n}{2}}\left[\int_{\mathbb{R}^{n}}G(y,1)d\mu^{0}_{-s}(y)dy\right]ds$
$\displaystyle=$ $\displaystyle\frac{M}{n\left(4\pi\right)^{n/2}}.$

Substituting (9.14) into this equality, we conclude the proof. ∎

A consequence of Theorem 9.1 is the following relation between $\Theta$ and
$\theta$.

######  Corollary 9.7.

For $\mu$-a.e. $(x,t)$,

$\Theta(x,t;u_{\infty},\mu)=\frac{1}{n\left(4\pi\right)^{\frac{n}{2}}}\theta(x,t),$

and consequently, there exist finitely many bubbles $W^{k}$, $k=1,\cdots,N$,
such that

$\theta(x,t)=\sum_{k=1}^{N}\int_{\mathbb{R}^{n}}|\nabla W^{k}|^{2}.$

Moreover, if all solutions are positive, there exist an $N(x,t)\in\mathbb{N}$
such that

$\theta(x,t)=N(x,t)\Lambda\quad\mu-a.e..$

###### Proof.

For those $(x,t)$ satisfying the condition (5.2), we have

$\theta(x,t)=\lim_{\lambda\to
0}\frac{\mu\left(Q_{\lambda}(x,t)\right)}{2\lambda^{2}}=\frac{1}{2}\mu^{0}(Q_{1}).$

We can also assume there is no atom for $\nu$ at $(x,t)$ by avoiding a at most
countable set. Then by the above structure theory of tangent flows and Theorem
9.1, we get

$\mu^{0}(Q_{1})=2M=2n\left(4\pi\right)^{\frac{n}{2}}\Theta(x,t).\qed$

####  9.3. Further properties of defect measures

Some consequences follow from the above structure result on tangent flows. The
first one is

######  Lemma 9.8.

For each $t$, $\Sigma_{t}^{\ast}$ is isolated.

###### Proof.

Assume by the contrary, there exists a sequence of points
$(x_{j},t)\in\Sigma_{t}^{\ast}$ converging to a limit point $(x,t)$. Because
$\Sigma^{\ast}$ is closed, $(x,t)\in\Sigma^{\ast}$. Denote

$\lambda_{j}:=|x_{j}-x|\to 0.$

Define the blow up sequence $(u_{\infty}^{\lambda_{j}},\mu^{\lambda_{j}})$
with respect to the bast point $(x,t)$ as before. Assume without loss of
generality that

$\frac{x_{j}-x}{\lambda_{j}}\to x_{\infty}\in\partial B_{1}.$

By the upper semi-continuity of $\Theta$, we get

$\Theta(x_{\infty},0;\mu^{0})\geq\limsup_{j\to+\infty}\Theta(x_{j},t;u_{\infty},\mu)\geq\varepsilon_{\ast}.$
(9.16)

On the other hand,the tangent flow analysis shows that
$\mbox{spt}(\mu^{0})=\\{0\\}\times\mathbb{R}$. This fact, together with the
$\varepsilon$-regularity theorem and the fact that $u_{\infty}^{0}=0$, implies
that

  * •

$u_{\infty}^{\lambda_{j}}\to 0$ in
$C^{\infty}_{loc}\left((\mathbb{R}^{n}\setminus\\{0\\})\times\mathbb{R}\right)$,

  * •

and for all $\lambda_{j}$ small, $\mbox{spt}(\mu^{\lambda_{j}})$ is contained
in $B_{1/2}\times\mathbb{R}$.

Therefore,

$\Theta(x_{\infty},0;\mu^{0})=\lim_{j\to+\infty}\Theta\left(\frac{x_{j}-x}{\lambda_{j}},0;u_{\infty}^{\lambda_{j}},\mu^{\lambda_{j}}\right)=0.$

This is a contradiction with (9.16). In other words, there does not exist
converging sequences in $\Sigma_{t}^{\ast}$. ∎

The next one is about the form of the stress-energy tensor $\mathcal{T}$.

######  Lemma 9.9.

$\mathcal{T}=I/n$ $\mu$-a.e..

###### Proof.

This is (8.3) in this special case. Here we explain briefly a direct proof.
First, because $\mathcal{T}$ is $\mu$-measurable, it is approximately
continuous $\mu$-a.e.. Choose a point $(x,t)$ so that $\mathcal{T}$ is
approximate continuous and Lemma 4.3 holds at this point. Take a sequence
$\lambda\to 0$ and define the tangent flow at this point as before, denoted by
$\mu^{0}$. Let

$\mathcal{T}^{\lambda}(y,s):=\mathcal{T}(x+\lambda y,t+\lambda^{2}s).$

By the approximate continuity of $\mathcal{T}$ at $(x,t)$, we see

$\mathcal{T}^{\lambda}d\mu^{\lambda}\rightharpoonup\mathcal{T}(x,t)d\mu^{0}.$

Then we obtain the stationary condition for the tangent flow,

$\int_{\mathbb{R}^{n}\times\mathbb{R}}\left[I-n\mathcal{T}(x,t)\right]\cdot
DYd\mu^{0}=0,\quad\forall Y\in
C_{0}^{\infty}(\mathbb{R}^{n}\times\mathbb{R},\mathbb{R}^{n}).$

In view of the form of $\mu^{0}$ in (9.12), substituting suitable $Y$ as test
functions into this identity we deduce that $\mathcal{T}(x,t)=I/n$. ∎

The following lemma is a rigourous statement that zero dimensional objects
should have zero mean curvatures.

######  Lemma 9.10.

$\mathbf{H}=0$ $\mu$-a.e..

###### Proof.

We will show that, for a.e. $t\in(-,1,1)$,

$\mathbf{H}=0\quad\mu_{t}-\mbox{everywhere}.$ (9.17)

For a.e. $t$,

$\int_{B_{1}}\left[|\nabla
u_{\infty}(x,t)|^{2}+|u_{\infty}(x,t)|^{p+1}+|\partial_{t}u_{\infty}(x,t)|^{2}\right]dx<+\infty,$
(9.18)

and by Lemma 4.6, $\mu_{t}$ is a well defined Radon measure.

By the form of $\mathcal{T}$, now the stationary condition (4.10) reads as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{Q_{1}}\left[\left(\frac{|\nabla
u_{\infty}|^{2}}{2}-\frac{|u_{\infty}|^{p+1}}{p+1}\right)\mbox{div}Y-DY(\nabla
u_{\infty},\nabla u_{\infty})+\partial_{t}u_{\infty}\nabla u_{\infty}\cdot
Y\right]$ (9.19) $\displaystyle+$ $\displaystyle\frac{1}{m}\int_{Q_{1}}\
\mathbf{H}\cdot Yd\mu,\quad\quad\forall Y\in
C_{0}^{\infty}(Q_{1},\mathbb{R}^{n}).$

Hence for a.e. $t$, we have the stationary condition

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{B_{1}}\left[\left(\frac{|\nabla
u_{\infty}|^{2}}{2}-\frac{|u_{\infty}|^{p+1}}{p+1}\right)\mbox{div}X-DX(\nabla
u_{\infty},\nabla u_{\infty})+\partial_{t}u_{\infty}\nabla u_{\infty}\cdot
X\right]$ (9.20) $\displaystyle+$ $\displaystyle\frac{1}{m}\int_{B_{1}}\
\mathbf{H}\cdot Xd\mu_{t},\quad\quad\forall X\in
C_{0}^{\infty}(B_{1},\mathbb{R}^{n}).$

Now we claim that
Claim. $u_{\infty}(t)\in W^{2,2}_{loc}(B_{1})\cap L^{2p}_{loc}(B_{1})$.

Once we have this claim, by integration by parts, we can show that
$u_{\infty}(t)$ satisfies the stationary condition

$0=\int_{B_{1}}\left[\left(\frac{|\nabla
u_{\infty}|^{2}}{2}-\frac{|u_{\infty}|^{p+1}}{p+1}\right)\mbox{div}X-DX(\nabla
u_{\infty},\nabla u_{\infty})+\partial_{t}u_{\infty}\nabla u_{\infty}\cdot
X\right].$

Combining this identity with (9.20) we get

$\int_{B_{1}}\ \mathbf{H}\cdot Xd\mu_{t}=0,\quad\quad\forall X\in
C_{0}^{\infty}(B_{1},\mathbb{R}^{n}),$

from which (9.17) follows.

Proof of the claim. By Lemma 9.8, $\Sigma_{t}^{\ast}$ is isolated. Since
$u_{\infty}$ is smooth outside $\Sigma^{\ast}$, we only need to show that for
each $\xi\in\Sigma_{t}^{\ast}$, there exists a ball $B_{r}(\xi)$ such that
$u_{\infty}(t)\in W^{2,2}(B_{r}(\xi))\cap L^{2p}(B_{r}(\xi))$.

We take a sufficiently small $\sigma$ and choose this ball so that

$\int_{B_{r}(\xi)}|u_{\infty}(t)|^{\frac{2n}{n-2}}\leq\sigma.$

Take a standard cut-off function $\eta$ in $B_{r}(\xi)$ such that $\eta\equiv
1$ in $B_{r/2}(\xi)$. A direct calculation gives

$-\Delta\left(u_{\infty}(t)\eta\right)=|u_{\infty}(t)|^{\frac{n-2}{2}}u_{\infty}\eta+f_{\eta},$

where $f_{\eta}\in L^{2}(B_{r}(\xi))$. Then by the $W^{2,2}$ estimate for
Laplacian operator and an application of Hölder inequality, if $\sigma$ is
sufficiently small, we obtain

$\|u_{\infty}(t)\eta\|_{W^{2,2}(B_{r}(\xi))}\lesssim\|u(t)\|_{L^{2}(B_{r}(\xi))}+\|f_{\eta}\|_{L^{2}(B_{r}(\xi))}.$

The $L^{2p}$ estimate on $u_{\infty}(t)$ follows by combining this estimate
with the equation for $u_{\infty}(t)$. The proof of the claim is complete. ∎

######  Corollary 9.11.

$u_{\infty}$ satisfies the stationary condition (2.3).

We do not know if the energy inequality (4.9) can be decoupled in the same
way.

Finally, for positive solutions, we note the following fact as a consequence
of (9.15):

  * •

either $Q=0$, which implies that there is no atom at $(x,t)$,

  * •

or $Q\geq\Lambda$, which implies that there is an atom in $\nu$ at $(x,t)$ and
its mass is at least $\Lambda$.

In conclusion, we get

######  Lemma 9.12.

If all solutions are positive, then the mass of each atom in $\nu$ is at least
$\Lambda$. Consequently, there are at most finitely many atoms in $\nu$.

The atoms of $\nu$ correspond to the singular point of $\Sigma$.
Microscopically, such an atom comes from scalings of a connecting orbit (or a
terrace of connecting orbits)

$\left\\{\begin{aligned} &\partial_{t}u-\Delta
u=u^{p},\quad\mbox{in}~{}~{}\mathbb{R}^{n}\times\mathbb{R},\\\
&\int_{\mathbb{R}^{n}}\left[\frac{1}{2}|\nabla
u|^{2}-\frac{1}{p+1}u^{p+1}\right]\Bigg{|}_{+\infty}^{-\infty}=Q.\end{aligned}\right.$

However, we do not know if $\Sigma$ is smooth outside this singular set.

##  Part II Energy concentration with only one bubble

###  10\. Setting

In this part, $p=(n+2)/(n-2)$ is the Sobolev critical exponent, $u_{i}$
denotes a sequence of _smooth, positive solutions_ of (1.1) in $Q_{1}$,
satisfying the following three assumptions:

(II.a) Weak limit:

$u_{i}$ converges weakly to $u_{\infty}$ in $L^{p+1}(Q_{1})$, and $\nabla
u_{i}$ converges weakly to $\nabla u_{\infty}$ in $L^{2}(Q_{1})$. Here
$u_{\infty}$ is a _smooth solution_ of (1.1) in $Q_{1}$.

(II.b) Energy concentration set:

weakly as Radon measures

$\left\\{\begin{aligned} &|\nabla u_{i}|^{2}dxdt\rightharpoonup|\nabla
u_{\infty}|^{2}dxdt+\Lambda\delta_{0}\otimes dt,\\\
&u_{i}^{p+1}dxdt\rightharpoonup u_{\infty}^{p+1}dxdt+\Lambda\delta_{0}\otimes
dt.\end{aligned}\right.$

(II.c) Convergence of time derivatives:

as $i\to\infty$, $\partial_{t}u_{i}$ converges to $\partial_{t}u_{\infty}$
strongly in $L^{2}(Q_{1})$.

The assumption (II.b) says there is only one bubble. From the above
assumptions, it is also seen that $u_{i}$ converges to $u_{\infty}$ in
$C(-1,1;L^{2}(B_{1}))$.

The main result of this part is the following theorem, which can be viewed as
a weak form of Schoen’s Harnack inequality for Yamabe problem (see [77],
[47]). As in Yamabe problem, this will be used to prove that there is no
bubble towering.

######  Theorem 10.1.

Under the above assumptions, we must have $u_{\infty}\equiv 0$.

In the following, we also assume there exists a constant $L$ such that for all
$i$,

$\sup_{-1<t<1}\int_{B_{1}}|\partial_{t}u_{i}(x,t)|^{2}dx\leq L.$ (10.1)

This assumption is in fact a consequence of (II.a-II.c), see Section 6 in Part
III for the proof.

The proof of Theorem 10.1 uses mainly a reverse version of _the inner-outer
gluing mechanism_.

Figure 1. Inner-outer gluing mechanism

  1. (1)

In Section 11, we describe the blow up profile of $u_{i}$, i.e. the form of
bubbles when energy concentration phenomena appears, see Proposition 11.1.
This is the main order term of $u_{i}$, and it provides us the starting point
for the decomposition in the next step.

  2. (2)

In Section 12, we take two decompositions for $u_{i}$: the first one is an
orthogonal decomposition where we decompose $u_{i}$ into a standard bubble
(which is the main order term) and an error function (which is the next order
term); the second one is the inner-outer decomposition, where we divide the
error function into two further parts, the first one (the inner part) is
localized near the bubble, and the second one (the outer part) is on the
original scale.

  3. (3)

In Section 13, we establish an estimate for the inner problem, where we mainly
use the nondegenaracy of the bubble (see Section A). Roughly speaking, this
estimate reads as

$\mathcal{I}\leq A\mathcal{O}+\mbox{higher order terms from scaling parameters
etc.},$

where $\mathcal{I}$ is a quantity measuring the inner component, $\mathcal{O}$
is a quantity measuring the outer component, and $A$ is a constant.

  4. (4)

In Section 14, we establish an estimate for the outer problem. This estimate
reads as

$\displaystyle\mathcal{O}$ $\displaystyle\leq$ $\displaystyle
B\mathcal{I}+\mbox{effect from initial-boundary value}$ $\displaystyle+$
$\displaystyle\mbox{higher order terms from scaling parameters etc.},$

where $B$ is a constant.

The _inner-outer gluing mechanism_ works thanks to the fact that

$AB<1.$

This follows from a fast decay estimate away from the bubble domains, where we
mainly rely on a Gaussian bound on heat kernels associated to a parabolic
operator with small Hardy term, see Moschini-Tesei [71].

  5. (5)

In Section 15, we combine these two estimates on inner and outer problems to
establish an Harnack inequality for the scaling parameter. This gives a
uniform in time control on the height of bubbles.

  6. (6)

In Section 16, by the estimate in Section 15, we improve the estimates on
inner and outer problems.

  7. (7)

In Section 17, we improve further the estimates on the error function to an
optimal one, such as uniform $L^{\infty}$ estimate, first order gradient
Hölder estimate and Schauder estimate.

  8. (8)

In Section 18, with the help of these estimates on the error function, we are
able to linearize the Pohozaev identity. This is because a Pohozaev identity
holds for $u$, and $u$ is approximated by the bubble at main order, which also
satisfies a Pohozaev identity, then the next order term in the Pohozaev
identity of $u$ gives further information.

  9. (9)

In Section 19, we use this linearization of Pohozaev identity to establish a
weak form of Schoen’s Harnack inequality, which also finishes the proof of
Theorem 10.1.

  10. (10)

In Section 20, we use this weak form of Schoen’s Harnack inequality to exclude
a special case of bubble clustering, where we use the same idea of
constructing Green functions in the study of Yamabe problems. This result will
be used in Section 12 of Part IV.

####  10.1.  List of notations and conventions used in this part

  * •

Given $\theta\in[0,1)$, $\alpha\geq 0$ and $k\in\mathbb{N}$, let
$\mathcal{X}_{\alpha}^{k+\theta}$ be the space of functions $\phi\in
C^{k,\theta}(\mathbb{R}^{n})$ with the weighted $C^{k,\theta}$ norm

$\|\phi\|_{\alpha,k+\theta}:=\sup_{x\in\mathbb{R}^{n}}\sum_{\ell=0}^{k-1}\left(1+|x|\right)^{\ell+\alpha}|\nabla^{\ell}\phi(x)|+\sup_{x\in\mathbb{R}^{n}}\left(1+|x|\right)^{k+\alpha}\|\nabla^{k}\phi\|_{C^{\theta}(B_{1}(x))}.$

If $\theta=0$ and $k=0$, this space is written as $\mathcal{X}_{\alpha}$.

  * •

Throughout this part, an even function $\eta\in C_{0}^{\infty}(-2,2)$ will be
fixed, which satisfies $\eta\equiv 1$ in $(-1,1)$ and
$|\eta^{\prime}|+|\eta^{\prime\prime}|\leq 10$. For any $R>0$, denote

$\eta_{R}(y):=\eta\left(\frac{|y|}{R}\right).$

  * •

$\alpha:=\frac{n-2}{2}$.

  * •

$\bar{p}:=\min\\{p,2\\}$.

  * •

Two large constants $K\gg L\gg 1$ will be chosen in Section 12.

  * •

In this part, unless otherwise stated, it is always assumed that $n\geq 7$,
which implies $\alpha>2$.

###  11\. Blow up profile

In this section we only assume $n\geq 3$. Here we prove

######  Proposition 11.1 (Blow up profile).

For any $t\in[-81/100,81/100]$, there exists a unique maxima point of
$u_{i}(\cdot,t)$ in the interior of $B_{1}(0)$. Denote this point by
$\xi_{i}^{\ast}(t)$ and let
$\lambda_{i}^{\ast}(t):=u_{i}(\xi_{i}^{\ast}(t),t)^{-\frac{2}{n-2}}$.

As $i\to+\infty$,

$\lambda_{i}^{\ast}(t)\to 0,\quad\xi_{i}^{\ast}(t)\to 0,\quad\mbox{uniformly
in }C([-81/100,81/100]),$

and the function

$u_{i}^{t}(y,s):=\lambda_{i}^{\ast}(t)^{\frac{n-2}{2}}u_{i}\left(\xi_{i}^{\ast}(t)+\lambda_{i}^{\ast}(t)y,t+\lambda_{i}^{\ast}(t)^{2}s\right),$

converges to $W(y)$ in $C^{\infty}_{loc}(\mathbb{R}^{n}\times\mathbb{R})$.

Let us first present some preliminary results, which are needed for the proof
of this proposition. The first one is a direct consequence of Theorem 3.9
(with the help of (II.b)).

######  Corollary 11.2.

As $i\to+\infty$, $u_{i}$ are uniformly bounded in
$C^{\infty}_{loc}\left(Q_{1}\setminus\Sigma\right)$.

We also note that there is no concentration for $\partial_{t}u_{i}$.

######  Lemma 11.3.

For any $\sigma>0$, there exists an $r(\sigma)>0$ such that for any $(x,t)\in
Q_{9/10}$,

$\limsup_{i\to+\infty}\int_{Q_{r(\sigma)}(x,t)}|\partial_{t}u_{i}|^{2}<\sigma.$

###### Proof.

Assume by the contrary, there exists a $\sigma>0$, a sequence of points
$(x_{i},t_{i})\in Q_{9/10}$ and a sequence of $r_{i}\to 0$ such that

$\int_{Q_{r_{i}}(x_{i},t_{i})}|\partial_{t}u_{i}|^{2}\geq\sigma.$ (11.1)

Without loss of generality, assume $(x_{i},t_{i})\to(x_{\infty},t_{\infty})$.
For any $r>0$ fixed, by (II.c), if $i$ is large enough,

$\int_{Q_{r}(x_{\infty},t_{\infty})}|\partial_{t}u_{\infty}|^{2}\geq\int_{Q_{r_{i}}(x_{i},t_{i})}|\partial_{t}u_{i}|^{2}\geq\sigma.$

This is a contradiction with the fact that $\partial_{t}u_{\infty}\in
L^{2}(Q_{1})$. ∎

Next is a result about the energy concentration behavior of each time slice
$u_{i}(t)$.

######  Lemma 11.4.

For any $\delta>0$, there exists an $r(\delta)\in(0,1)$ such that for any
$t\in[-81/100,81/100]$,

$\limsup_{i\to+\infty}\left(\left|\int_{B_{r(\delta)}}|\nabla
u_{i}(x,t)|^{2}dx-\Lambda\right|+\left|\int_{B_{r(\delta)}}u_{i}(x,t)^{p+1}dx-\Lambda\right|\right)<\delta.$
(11.2)

###### Proof.

Take a sufficiently small $\sigma>0$, and then choose $r(\sigma)$ according to
Lemma 11.3. Without loss of generality, assume $t=0$. By the scaling
invariance of energy, in the following we need only to prove the corresponding
result for the rescaling of $u_{i}$,

$\widetilde{u}_{i}(x,t):=r(\sigma)^{\frac{n-2}{2}}u_{i}\left(r(\sigma)x,r(\sigma)^{2}t\right).$

However, in order not to complicate notations, we still use $u_{i}$ to denote
$\widetilde{u}_{i}$. In particular, now $u_{i}$ also satisfyies

$\int_{Q_{1}}|\partial_{t}u_{i}|^{2}\leq\sigma.$ (11.3)

For each $r\in(0,1)$, recall that $\eta_{r}(x):=\eta(|x|/r)$ is a standard
cut-off function in $B_{2r}$. Because $u_{i}$ is smooth, we have the standard
localized energy identity

$\displaystyle\frac{d}{dt}\int_{B_{1}}\left[\frac{1}{2}|\nabla
u_{i}(x,t)|^{2}-\frac{1}{p+1}u_{i}(x,t)^{p+1}\right]\eta_{r}(x)^{2}dx$
$\displaystyle=$
$\displaystyle-\int_{B_{1}}|\partial_{t}u_{i}(x,t)|^{2}\eta_{r}(x)^{2}dx+2\int_{B_{1}}\eta_{r}(x)\partial_{t}u_{i}(x,t)\nabla
u_{i}(x,t)\cdot\nabla\eta_{r}(x)dx.$

Integrating this identity and applying Cauchy-Schwarz inequality, we get

$\displaystyle\mbox{osc}_{t\in(-1,1)}\int_{B_{1}}\left[\frac{1}{2}|\nabla
u_{i}(x,t)|^{2}-\frac{1}{p+1}u_{i}(x,t)^{p+1}\right]\eta_{r}(x)^{2}$
$\displaystyle\lesssim$
$\displaystyle\left(\int_{Q_{1}}|\partial_{t}u_{i}|^{2}\right)^{1/2}$ (11.5)
$\displaystyle\lesssim$ $\displaystyle\sigma^{1/2},$

where we have used (11.3).

By (11.5), (II.b) and the smoothness of $u_{\infty}$, we get

$\displaystyle\int_{B_{1}}\left[\frac{1}{2}|\nabla
u_{i}(x,t)|^{2}-\frac{1}{p+1}u_{i}(x,t)^{p+1}\right]\eta_{r}(x)^{2}dxdt$
$\displaystyle=$
$\displaystyle\frac{\Lambda}{n}+O\left(\sigma^{1/2}\right)+O\left(r^{n}\right)+o_{i}(1),\quad\quad\quad\forall
t\in(-1,1).$

Next, for each $t\in(-1,1)$, multiplying (1.1) by $u_{i}(t)\eta_{r}^{2}$ and
integrating on $B_{1}$, we get

$\displaystyle\int_{B_{1}}\left[|\nabla
u_{i}(x,t)|^{2}-u_{i}(x,t)^{p+1}\right]\eta_{r}(x)^{2}dx$ $\displaystyle=$
$\displaystyle\int_{B_{1}}\left[\partial_{t}u_{i}(x,t)u_{i}(x,t)\eta_{r}(x)^{2}-2\eta_{r}(x)u_{i}(x,t)\nabla\eta_{r}(x)\cdot\nabla
u_{i}(x,t)\right]dx.$

Concerning the right hand side of this equation, the following estimates hold.

  * •

By Cauchy-Schwarz inequality and (10.1), we have

$\displaystyle\left|\int_{B_{1}}\partial_{t}u_{i}(x,t)u_{i}(x,t)\eta_{r}(x)^{2}dx\right|$
$\displaystyle\lesssim$
$\displaystyle\left(\int_{B_{r}}\partial_{t}u_{i}(x,t)^{2}dx\right)^{\frac{1}{2}}\left(\int_{B_{r}}u_{i}(x,t)^{2}dx\right)^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle
L^{1/2}\left[\left(\int_{B_{r}}u_{\infty}(x,t)^{2}dx\right)^{\frac{1}{2}}+o_{i}(1)\right],$

where in the second inequality we have used the Lipschitz assumption (10.1)
and the uniform convergence of $u_{i}(t)$ in $C_{loc}(-1,1;L^{2}(B_{1}))$.

  * •

Still by the uniform convergence of $u_{i}(t)$ in
$C_{loc}(-1,1;L^{2}(B_{1}))$, we obtain

$\displaystyle\int_{B_{1}}\eta_{r}(x)u_{i}(x,t)\nabla\eta_{r}(x)\cdot\nabla
u_{i}(x,t)dx$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{B_{1}}u_{i}(x,t)^{2}\Delta\eta_{r}(x)^{2}dx$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{B_{1}}u_{\infty}(x,t)^{2}\Delta\eta_{r}(x)^{2}dx+o_{i}(1).$

Substituting these two estimates into (11) and noting the smoothness of
$u_{\infty}$, we get

$\int_{B_{1}}\left[|\nabla
u_{i}(x,t)|^{2}-u_{i}(x,t)^{p+1}\right]\eta_{r}(x)^{2}dx=o_{r}(1)+o_{i}(1).$
(11.8)

Plugging this relation into (11), we obtain

$\left\\{\begin{aligned} &\int_{B_{1}}|\nabla
u_{i}(x,t)|^{2}\eta_{r}(x)^{2}=\Lambda+O\left(\sigma^{1/2}\right)+o_{r}(1)+o_{i}(1),\\\
&\int_{B_{1}}u_{i}(x,t)^{p+1}\eta_{r}(x)^{2}=\Lambda+O\left(\sigma^{1/2}\right)+o_{r}(1)+o_{i}(1).\end{aligned}\right.$
(11.9)

Then (11.2) follows by noting that $\eta_{r}\leq\chi_{B_{r}}\leq\eta_{2r}$ and
choosing a suitable $r$. ∎

Some consequences follow directly from this lemma.

  1. (1)

There exists a constant $C(u_{\infty},L)$ such that

$\int_{B_{4/5}}\left[|\nabla u_{i}(t)|^{2}+u_{i}(t)^{p+1}\right]\leq
C(u_{\infty},L),\quad\forall t\in[-81/100,81/100].$ (11.10)

  2. (2)

By the convergence of $u_{i}(t)$ in $C^{\infty}_{loc}(B_{1}\setminus\\{0\\})$,
we deduce that for any $t\in[-81/100,81/100]$,

$|\nabla u_{i}(t)|^{2}dx\rightharpoonup|\nabla
u_{\infty}|^{2}dx+\Lambda\delta_{0},\quad u_{i}(t)^{p+1}dx\rightharpoonup
u_{\infty}^{p+1}dx+\Lambda\delta_{0}.$ (11.11)

Equation (11.11) implies that for any $t\in[-81/100,81/100]$,

$\sup_{B_{1}}u_{i}(x,t)\to+\infty\quad\mbox{as}\quad i\to+\infty.$

Because $u_{i}(t)$ are bounded in $C_{loc}(B_{1}\setminus\\{0\\})$, this
supremum must be a maxima, and it is attained near $0$. Take one such maximal
point $\xi_{i}^{\ast}(t)$, and define $\lambda_{i}^{\ast}(t)$, $u_{i}^{t}$ as
in the statement of Proposition 11.1. By definition,

$u_{i}^{t}(0,0)=\max_{y\in B_{\lambda_{i}^{\ast}(t)^{-1}/2}}u_{i}^{t}(y,0)=1.$
(11.12)

Scaling (10.1) gives

$\lim_{i\to+\infty}\int_{B_{R}}\partial_{t}u_{i}^{t}(y,s)^{2}dy=0,\quad\mbox{for
any }R>0,~{}~{}~{}s\in\mathbb{R}.$ (11.13)

By scaling (11.2), we see for any $R>0$ and $s\in\mathbb{R}$,

$\left\\{\begin{aligned} &\limsup_{i\to+\infty}\int_{B_{R}}|\nabla
u_{i}^{t}(y,s)|^{2}dy\leq\Lambda,\\\
&\limsup_{i\to+\infty}\int_{B_{R}}u_{i}^{t}(y,s)^{p+1}dy\leq\Lambda.\end{aligned}\right.$
(11.14)

Then by Struwe’s global compactness theorem (see [82]), we deduce that for any
$s\in\mathbb{R}$, $u_{i}^{t}(\cdot,s)$ converges strongly in
$W^{1,2}_{loc}(\mathbb{R}^{n})$ and $L^{p+1}_{loc}(\mathbb{R}^{n})$. Denote
its limit by $u_{\infty}^{t}$. By (11.13) and (11.14), $u_{\infty}^{t}$ is a
nonnegative solution of (1.9). In view of its $H^{1}$ regularity, it must be
smooth. Hence for any $(y,s)\in\mathbb{R}^{n}\times\mathbb{R}$, there exists
an $r>0$ such that

$\int_{Q_{r}^{-}(y,s)}\left[|\nabla
u_{\infty}^{t}|^{2}+\left(u_{\infty}^{t}\right)^{p+1}\right]\leq\frac{\varepsilon_{\ast}}{2}r^{2}.$

By the above strong convergence of $u_{i}^{t}$, we get

$\limsup_{i\to+\infty}\int_{Q_{r}^{-}(y,s)}\left[|\nabla
u_{i}^{t}|^{2}+\left(u_{i}^{t}\right)^{p+1}\right]<\varepsilon_{\ast}r^{2}.$

Therefore Theorem 3.7 is applicable, which implies that $u_{i}^{t}$ are
uniformly bounded in $C^{\infty}(Q_{\delta_{\ast}r}(y,s))$. In conclusion,
$u_{i}^{t}$ are uniformly bounded in
$C^{\infty}_{loc}(\mathbb{R}^{n}\times\mathbb{R})$. Hence it converges to
$u_{\infty}^{t}$ in $C^{\infty}_{loc}(\mathbb{R}^{n}\times\mathbb{R})$. Then
we can let $i\to+\infty$ in (11.12), which leads to

$u_{\infty}^{t}(0)=\max_{y\in\mathbb{R}^{n}}u_{\infty}^{t}(y)=1.$

By the Liouville theorem of Caffarelli-Gidas-Spruck [8], we get

$u_{\infty}^{t}\equiv W\quad\mbox{in }\mathbb{R}^{n}.$

Next we give a scaling invariant estimate for $u_{i}$ away from
$\xi_{i}^{\ast}(t)$.

######  Lemma 11.5.

For any $R$ large enough, there exist two constants $\sigma(R)\ll 1$ and
$C(R)>0$ such that, outside $B_{R\lambda_{i}^{\ast}(t)}(\xi_{i}^{\ast}(t))$ we
have

$\left\\{\begin{aligned}
&u_{i}(x,t)\leq\sigma(R)|x-\xi_{i}^{\ast}(t)|^{-\frac{n-2}{2}}+C(R),\\\
&|\nabla u_{i}(x,t)|\leq\sigma(R)|x-\xi_{i}^{\ast}(t)|^{-\frac{n}{2}}+C(R),\\\
&|\nabla^{2}u_{i}(x,t)|+|\partial_{t}u_{i}(x,t)|\leq\sigma(R)|x-\xi_{i}^{\ast}(t)|^{-\frac{n+2}{2}}+C(R).\end{aligned}\right.$
(11.15)

###### Proof.

For any $\delta>0$, by the smooth convergence of $u_{i}^{t}$ obtained above,
there exists an $R(\delta)\gg 1$ such that for any $t\in[-81/100,81/100]$,

$\left\\{\begin{aligned}
&\int_{B_{R(\delta)\lambda_{i}^{\ast}(t)}(\xi_{i}^{\ast}(t))}|\nabla
u_{i}(x,t)|^{2}\geq\Lambda-\delta/4,\\\
&\int_{B_{R(\delta)\lambda_{i}^{\ast}(t)}(\xi_{i}^{\ast}(t))}u_{i}(x,t)^{p+1}\geq\Lambda-\delta/4.\end{aligned}\right.$
(11.16)

By Lemma 11.4, there exists an $r(\delta)\ll 1$ such that for any
$t\in[-81/100,81/100]$,

$\left\\{\begin{aligned} &\int_{B_{r(\delta)}}|\nabla
u_{i}(x,t)|^{2}\leq\Lambda+\delta/4,\\\
&\int_{B_{r(\delta)}}u_{i}(x,t)^{p+1}\leq\Lambda+\delta/4.\end{aligned}\right.$
(11.17)

Combining (11.16) with (11.17), we get

$\sup_{|t|\leq 81/100}\int_{B_{r(\delta)}\setminus
B_{R(\delta)\lambda_{i}^{\ast}(t)}(\xi_{i}^{\ast}(t))}\left[|\nabla
u_{i}(x,t)|^{2}+u_{i}(x,t)^{p+1}\right]dx\leq\delta.$ (11.18)

By Corollary 11.2, there exists a constant $C(\delta)$ such that

$u_{i}(x,t)+|\nabla
u_{i}(x,t)|+|\nabla^{2}u_{i}(x,t)|+|\partial_{t}u_{i}(x,t)|\leq
C(\delta),\quad\mbox{if}~{}x\notin B_{r(\delta)}(\xi_{i}^{\ast}(t)).$

It remains to prove (11.15) when $x\in B_{r(\delta)}\setminus
B_{R(\delta)\lambda_{i}^{\ast}(t)}(\xi_{i}^{\ast}(t))$. We argue by
contradiction, so assume there exists a sequence $(x_{i},t_{i})$, with
$t_{i}\in[-81/100,81/100]$ and $x_{i}\in B_{r(\delta)}(0)\setminus
B_{R(\delta)\lambda_{i}^{\ast}(t_{i})}(\xi_{i}^{\ast}(t_{i}))$, but for some
$\sigma$ (to be determined below),

$u_{i}(x_{i},t_{i})\geq\sigma|x_{i}-\xi_{i}^{\ast}(t_{i})|^{-\frac{n-2}{2}}.$
(11.19)

Let $r_{i}:=|x_{i}-\xi_{i}^{\ast}(t_{i})|$ and

$\widetilde{u}_{i}(y,s):=r_{i}^{\frac{n-2}{2}}u_{i}\left(\xi_{i}^{\ast}(t_{i})+r_{i}y,t_{i}+r_{i}^{2}s\right).$

Denote

$\widetilde{x}_{i}:=\frac{x_{i}-\xi_{i}^{\ast}(t_{i})}{r_{i}}.$

It lies on $\partial B_{1}$. Assume it converges to a limit point
$\widetilde{x}_{\infty}\in\partial B_{1}$.

Scaling (11.16) and (11.2) leads to

$\left\\{\begin{aligned}
&\int_{B_{R(\delta)^{-1}}}|\nabla\widetilde{u}_{i}(y,0)|^{2}\geq\Lambda-\delta/4,\\\
&\int_{B_{R(\delta)^{-1}}}\widetilde{u}_{i}(y,0)^{p+1}\geq\Lambda-\delta/4\end{aligned}\right.$
(11.20)

and for any $R>0$ and $s\in\mathbb{R}$,

$\left\\{\begin{aligned}
&\int_{B_{R}(0)}|\nabla\widetilde{u}_{i}(y,s)|^{2}\leq\Lambda+\delta/4,\\\
&\int_{B_{R}(0)}\widetilde{u}_{i}(y,s)^{p+1}\leq\Lambda+\delta/4.\end{aligned}\right.$
(11.21)

Then by noting that for any $s$, $\partial_{s}\widetilde{u}_{i}(\cdot,s)$
converges to $0$ strongly in $L^{2}_{loc}(\mathbb{R}^{n})$, we can apply Lemma
11.4 to find an $\rho(\delta)>0$ such that for any $s\in[-1,1]$,

$\left|\int_{B_{\rho(\delta)}}|\nabla\widetilde{u}_{i}(y,s)|^{2}dy-\Lambda\right|+\left|\int_{B_{\rho(\delta)}}\widetilde{u}_{i}(y,s)^{p+1}dy-\Lambda\right|<\delta.$

Combining this estimate with (11.21), we get

$\int_{-1}^{0}\int_{B_{1/2}(\widetilde{x}_{i})}\left(|\nabla\widetilde{u}_{i}|^{2}+\widetilde{u}_{i}^{p+1}\right)\leq
C\delta.$

By choosing $C\delta<\varepsilon_{\ast}$, we can apply Theorem 3.7 to deduce
that

$\widetilde{u}_{i}(\widetilde{x}_{i},0)+|\nabla\widetilde{u}_{i}(\widetilde{x}_{i},0)|+|\nabla^{2}\widetilde{u}_{i}(\widetilde{x}_{i},0)|+|\partial_{s}\widetilde{u}_{i}(\widetilde{x}_{i},0)|\leq
c(\delta),$

where $c(\delta)$ is a small constant depending on $\delta$. Scaling back to
$u_{i}$ we get a contradiction with (11.19). In other words, (11.15) must hold
in $B_{r(\delta)}\setminus
B_{R(\delta)\lambda_{i}^{\ast}(t)}(\xi_{i}^{\ast}(t))$. ∎

Combining Proposition 11.1 and Lemma 11.5. we get

######  Corollary 11.6.

For each $t\in[-81/100,81/100]$, $\xi_{i}^{\ast}(t)$ is the unique maximal
point of $u_{i}(t)$ in $B_{1}$.

This completes the proof of Proposition 11.1.

###  12\. Orthogonal and inner-outer decompositions

For simplicity of notations, from here to Section 16 we are concerned with a
fixed solution $u_{i}$ with sufficiently large index $i$, and denote it by
$u$. In this section we define the orthogonal decomposition and inner-outer
decomposition for $u$.

####  12.1. Orthogonal decomposition

From now on, until Section 19, a constant $K\gg 1$ will be fixed. In the
following we will use the notations about bubbles and kernels to their
linearized equations, $W_{\xi,\lambda}$ and $Z_{i,\xi,\lambda}$, see Appendix
A.

######  Proposition 12.1 (Orthogonal condition).

For any $t\in[-81/100,81/100]$, there exists a unique
$(a(t),\xi(t),\lambda(t))\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{+}$
with

$\frac{|\xi(t)-\xi^{\ast}(t)|}{\lambda(t)}+\Big{|}\frac{\lambda(t)}{\lambda^{\ast}(t)}-1\Big{|}+\Big{|}\frac{a(t)}{\lambda(t)}\Big{|}=o(1),$
(12.1)

such that for each $i=0,\cdots,n+1$,

$\int_{B_{1}}\left[u(x,t)-W_{\xi(t),\lambda(t)}(x)-a(t)Z_{0,\xi(t),\lambda(t)}(x)\right]\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)dx=0.$
(12.2)

###### Proof.

For these $t$, set

$\widetilde{u}(y):=\lambda^{\ast}(t)^{\frac{n-2}{2}}u(\xi^{\ast}(t)+\lambda^{\ast}(t)y,t).$

Define a smooth map $F$ from $B_{1/2}^{n+2}\subset\mathbb{R}^{n+2}$ into
$\mathbb{R}^{n+2}$ as

$F(\xi,\lambda,a):=\left(\int_{\mathbb{R}^{n}}\left[\widetilde{u}(y)-W_{\xi,1+\lambda}(y)-aZ_{0,\xi,1+\lambda}(y)\right]\eta_{K}\left(\frac{y-\xi}{1+\lambda}\right)Z_{i,\xi,1+\lambda}(y)dy\right)_{i=0}^{n+1}.$

The task is reduced to find a solution of $F(\xi,\lambda,a)=0$.

By Proposition 11.1, $|F(0,0,0)|\ll 1$. By a direct calculation and applying
Proposition 11.1 once again, we find a fixed small constant $\rho_{\ast}$ such
that, for any $(\xi,\lambda,a)\in B_{\rho_{\ast}}^{n+2}$, $DF(\xi,\lambda,a)$
is diagonally dominated, and hence invertible. Then the inverse function
theorem implies the existence and uniqueness of the solution $F=0$ in
$B_{\rho_{\ast}}^{n+2}$. ∎

In the above, in order to show that $DF(\xi,\lambda,a)$ is diagonally
dominated, we need $n\geq 5$ to make sure $Z_{n+1}\in L^{2}(\mathbb{R}^{n})$,
and we also need the $n\geq 7$ assumption because at first we only know that
$u$ decays like $|y|^{-(n-2)/2}$. (This term appears in the integral
containing $\nabla\eta$.)

For later purpose, we notice a Lipschitz bound on the parameters
$(a(t),\xi(t),\lambda(t))$.

######  Lemma 12.2.

For any $t\in[-81/100,81/100]$,

$\left|a^{\prime}(t)\right|+\left|\xi^{\prime}(t)\right|+\left|\lambda^{\prime}(t)\right|\lesssim\left(\int_{B_{1}}\partial_{t}u(x,t)^{2}dx\right)^{1/2}.$
(12.3)

###### Proof.

For each $i=1,\cdots,n$, because $Z_{i}\eta_{K}$ is orthogonal to $Z_{0}$ in
$L^{2}(\mathbb{R}^{n})$, the orthogonal condition (12.2) reads as

$\int_{B_{1}}\left[u(x,t)-W_{\xi(t),\lambda(t)}(x)\right]\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)dx=0.$

Differentiating this equality in $t$, by the $L^{2}$ orthogonal relations
between different $Z_{i}$, we obtain

$\displaystyle\left[\int_{B_{1}}\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)^{2}dx\right]\xi_{i}^{\prime}(t)$
$\displaystyle=$
$\displaystyle\int_{B_{1}}\partial_{t}u(x,t)\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)dx$
$\displaystyle+$
$\displaystyle\int_{B_{1}}\left[u(x,t)-W_{\xi(t),\lambda(t)}(x)\right]\frac{\partial}{\partial
t}\left[\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)\right]dx.$

First, by the definition of $Z_{i,\xi(t),\lambda(t)}$, we have

$\int_{B_{1}}\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)^{2}dx=\int_{\mathbb{R}^{n}}\eta_{K}(y)Z_{i}(y)^{2}dy\geq
c.$

Next, by Cauchy-Schwarz inequality and (10.1), we get

$\displaystyle\left|\int_{B_{1}}\partial_{t}u(x,t)\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)dx\right|$
$\displaystyle\leq$
$\displaystyle\left(\int_{B_{1}}\partial_{t}u(x,t)^{2}dx\right)^{1/2}\left(\int_{B_{1}}\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)^{2}Z_{i,\xi(t),\lambda(t)}(x)^{2}dx\right)^{1/2}$
$\displaystyle\lesssim$
$\displaystyle\left(\int_{B_{1}}\partial_{t}u(x,t)^{2}dx\right)^{1/2}.$

Finally, because in $B_{K\lambda(t)}(\xi(t))$,

$\left\\{\begin{aligned}
&\left|u(x,t)-W_{\xi(t),\lambda(t)}(x)\right|=o(1)\lambda(t)^{-\frac{n-2}{2}}\left(1+\frac{|x-\xi(t)|}{\lambda(t)}\right)^{-\frac{n-2}{2}},\\\
&\left|\frac{\partial}{\partial
t}\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)\right|\lesssim\frac{\left|\xi^{\prime}(t)\right|+\left|\lambda^{\prime}(t)\right|}{\lambda(t)^{\frac{n+2}{2}}}\left(1+\frac{|x-\xi(t)|}{\lambda(t)}\right)^{1-n},\\\
&\left|\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)\frac{\partial}{\partial
t}Z_{i,\xi(t),\lambda(t)}(x)\right|\lesssim\frac{\left|\xi^{\prime}(t)\right|+\left|\lambda^{\prime}(t)\right|}{\lambda(t)^{\frac{n+2}{2}}}\left(1+\frac{|x-\xi(t)|}{\lambda(t)}\right)^{1-n},\end{aligned}\right.$

we get

$\displaystyle\left|\int_{B_{1}}\left[u(x,t)-W_{\xi(t),\lambda(t)}(x)\right]\frac{\partial}{\partial
t}\left[\eta_{K}\left(\frac{x-\xi(t)}{\lambda(t)}\right)Z_{i,\xi(t),\lambda(t)}(x)\right]dx\right|$
$\displaystyle\ll$
$\displaystyle\lambda(t)^{-n}\left(\left|\xi^{\prime}(t)\right|+\left|\lambda^{\prime}(t)\right|\right)\int_{B_{1}}\left(1+\frac{|x-\xi(t)|}{\lambda(t)}\right)^{2-\frac{3n}{2}}$
$\displaystyle\lesssim$
$\displaystyle\left|\xi^{\prime}(t)\right|+\left|\lambda^{\prime}(t)\right|.$

Plugging these three estimates into (12.1), we obtain

$|\xi^{\prime}(t)|\lesssim\left(\int_{B_{1}}\partial_{t}u(x,t)^{2}dx\right)^{1/2}+o\left(\left|\xi^{\prime}(t)\right|+\left|\lambda^{\prime}(t)\right|\right).$
(12.5)

In the same way, we get

$|\lambda^{\prime}(t)|\lesssim\left(\int_{B_{1}}\partial_{t}u(x,t)^{2}dx\right)^{1/2}+o\left(\left|\xi^{\prime}(t)\right|+\left|\lambda^{\prime}(t)\right|+\left|a^{\prime}(t)\right|\right)$
(12.6)

and

$|a^{\prime}(t)|\lesssim\left(\int_{B_{1}}\partial_{t}u(x,t)^{2}dx\right)^{1/2}+o\left(\left|\xi^{\prime}(t)\right|+\left|\lambda^{\prime}(t)\right|+\left|a^{\prime}(t)\right|\right).$
(12.7)

Adding these three estimates together we get (12.3). ∎

####  12.2. Inner-outer decomposition

In the following we denote the error function

$\phi(x,t):=u(x,t)-W_{\xi(t),\lambda(t)}(x)-a(t)Z_{0,\xi(t),\lambda(t)}(x),$
$W_{\ast}(x,t):=W_{\xi(t),\lambda(t)}(x),\quad
Z_{i,\ast}(x,t):=Z_{i,\xi(t),\lambda(t)}(x)$

and $Z_{\ast}:=(Z_{0,\ast},Z_{1,\ast},\cdots,Z_{n+1,\ast})$.

Combining Lemma 11.5 and Proposition 12.1, we obtain

######  Proposition 12.3.

In $B_{1}$,

$\left\\{\begin{aligned}
&|\phi(x,t)|=o\left(\left(\lambda(t)+|x-\xi_{\ast}(t)|\right)^{-\frac{n-2}{2}}\right)+O(1),\\\
&|\nabla\phi(x,t)|=o\left(\left(\lambda(t)+|x-\xi_{\ast}(t)|\right)^{-\frac{n}{2}}\right)+O(1),\\\
&|\nabla^{2}\phi(x,t)|+|\phi_{t}(x,t)|=o\left(\left(\lambda(t)+|x-\xi_{\ast}(t)|\right)^{-\frac{n+2}{2}}\right)+O(1).\end{aligned}\right.$

The error function $\phi$ satisfies

$\partial_{t}\phi-\Delta\phi=pW_{\ast}^{p-1}\phi+\mathcal{N}+\left(-a^{\prime}+\mu_{0}\frac{a}{\lambda^{2}},\xi^{\prime},\lambda^{\prime}\right)\cdot
Z_{\ast}-a\partial_{t}Z_{0,\ast},$ (12.8)

where $\\{\\}^{\prime}$ denotes differentiation in $t$, and the nonlinear term
is defined by

$\mathcal{N}:=\left(W_{\ast}+\phi+aZ_{0,\ast}\right)^{p}-W_{\ast}^{p}-pW_{\ast}^{p-1}\left(\phi+aZ_{0,\ast}\right).$

Now take a further decomposition of $\phi$, _the inner-outer decomposition_ ,
as follows. Keep $K$ as the large constant used in Proposition 12.1. Take
another constant $L$ satisfying $1\ll L\ll K$. Denote

$\eta_{in}(x,t):=\eta\left(\frac{x-\xi(t)}{K\lambda(t)}\right),\quad\eta_{out}(x,t):=\eta\left(\frac{x-\xi(t)}{L\lambda(t)}\right).$
(12.9)

Set

$\phi_{in}(x,t):=\phi(x,t)\eta_{K}(x,t),\quad\phi_{out}(x,t):=\phi(x,t)\left[1-\eta_{out}(x,t)\right].$

For later purpose, we introduce two quantities:

$\mathcal{I}(t):=\left[L\lambda(t)\right]^{\alpha}\sup_{B_{2L\lambda(t)}(\xi(t))\setminus
B_{L\lambda(t)}(\xi(t))}|\phi|+\left[L\lambda(t)\right]^{1+\alpha}\sup_{B_{2L\lambda(t)}(\xi(t))\setminus
B_{L\lambda(t)}(\xi(t))}|\nabla\phi|$

and

$\mathcal{O}(t):=\left[K\lambda(t)\right]^{\alpha}\sup_{B_{2K\lambda(t)}(\xi(t))\setminus
B_{K\lambda(t)}(\xi(t))}|\phi|+\left[K\lambda(t)\right]^{1+\alpha}\sup_{B_{2K\lambda(t)}(\xi(t))\setminus
B_{K\lambda(t)}(\xi(t))}|\nabla\phi|.$

By Proposition 12.3, for any $t\in[-81/100,81/100]$,

$\mathcal{I}(t)+\mathcal{O}(t)\ll 1.$

Starting from this smallness, we will improve it to an explicit bound in the
following sections.

###  13\. Inner problem

In this section we give an $C^{1,\theta}$ estimate on the inner component
$\phi_{in}$, see Proposition 13.4.

Define a new coordinate system around $\xi(t)$ by

$y:=\frac{x-\xi(t)}{\lambda(t)},\quad\quad\tau=\tau(t).$

Here the new time variable $\tau$ is determined by the relation

$\tau^{\prime}(t)=\lambda(t)^{-2},\quad\quad\tau(0)=0.$

Because there is a one to one correspondence between $\tau$ and $t$, in the
following we will not distinguish between them. For example, we will use
$\mathcal{O}(\tau)$ instead of the notation $\mathcal{O}(t(\tau))$.

It is convenient to write $\phi$ in these new coordinates. By denoting

$\varphi(y,\tau):=\lambda(t)^{\frac{n-2}{2}}\phi(x,t),$

we get

$\displaystyle\partial_{\tau}\varphi-\Delta\varphi$ $\displaystyle=$
$\displaystyle
pW^{p-1}\varphi+\left(-\frac{\dot{a}}{\lambda}+\mu_{0}\frac{a}{\lambda},\frac{\dot{\xi}}{\lambda},\frac{\dot{\lambda}}{\lambda}\right)\cdot
Z$ $\displaystyle+$
$\displaystyle\mathcal{N}+\frac{\dot{\xi}}{\lambda}\cdot\nabla\varphi+\frac{\dot{\lambda}}{\lambda}\left(y\cdot\nabla\varphi+\frac{n-2}{2}\varphi\right)$
$\displaystyle+$
$\displaystyle\frac{a}{\lambda}\left[\frac{\dot{\xi}}{\lambda}\cdot\nabla
Z_{0}+\frac{\dot{\lambda}}{\lambda}\left(y\cdot\nabla
Z_{0}+\frac{n}{2}Z_{0}\right)\right].$

Here $\dot{\\{\\}}$ denotes differentiation in $\tau$, and by abusing
notations,

$\mathcal{N}:=\left(W+\varphi+\frac{a}{\lambda}Z_{0}\right)^{p}-W^{p}-pW^{p-1}\left(\varphi+\frac{a}{\lambda}Z_{0}\right).$

We will need the following pointwise estimate on $\mathcal{N}$. Recall that
$\bar{p}=\min\\{p,2\\}$.

######  Lemma 13.1.

The following point-wise inequality holds:

$|\mathcal{N}|\lesssim|\varphi|^{\bar{p}}+\Big{|}\frac{a}{\lambda}\Big{|}^{\bar{p}}Z_{0}^{\bar{p}}.$

###### Proof.

There exists a $\vartheta\in[0,1]$ such that

$\displaystyle\left(W+\varphi+\frac{a}{\lambda}Z_{0}\right)^{p}-W^{p}-pW^{p-1}\left(\varphi+\frac{a}{\lambda}Z_{0}\right)$
$\displaystyle=$ $\displaystyle
p\left\\{\left[W+\vartheta\left(\varphi+\frac{a}{\lambda}Z_{0}\right)\right]^{p-1}-W^{p-1}\right\\}\left(\varphi+\frac{a}{\lambda}Z_{0}\right).$

Since both $W$ and $W+\varphi+\frac{a}{\lambda}Z_{0}=u$ are bounded by a
universal constant, and $\varphi+\frac{a}{\lambda}Z_{0}$ is small, by
considering the two cases $W\geq|\varphi+\frac{a}{\lambda}Z_{0}|$ and
$W<|\varphi+\frac{a}{\lambda}Z_{0}|$ separately, we get

$|\mathcal{N}|\lesssim\Big{|}\varphi+\frac{a}{\lambda}Z_{0}\Big{|}^{\bar{p}}\lesssim|\varphi|^{\bar{p}}+\Big{|}\frac{a}{\lambda}\Big{|}^{\bar{p}}Z_{0}^{\bar{p}}.\qed$

By scaling the estimates in Proposition 12.3, we obtain

######  Proposition 13.2.

For each $\tau$,

$K^{\alpha}\sup_{y\in
B_{2K}}\left[|\varphi(y,\tau)|+K|\nabla\varphi(y,\tau)|+K^{2}\left(|\nabla^{2}\varphi(y,\tau)|+|\partial_{\tau}\varphi(y,\tau)|\right)\right]=o(1).$

In $(y,\tau)$ coordinates, the inner error function $\phi_{in}$ is

$\varphi_{K}(y,\tau):=\varphi(y,\tau)\eta_{K}(y).$

For each $\tau$, the support of $\varphi_{K}(\cdot,\tau)$ is contained in
$B_{K}$. By (12.2), for each $i=0,\cdots,n+1$,

$\int_{\mathbb{R}^{n}}\varphi_{K}(y,\tau)Z_{i}(y)dy=0.$ (13.2)

The equation for $\varphi_{K}$ is

$\partial_{\tau}\varphi_{K}-\Delta_{y}\varphi_{K}=pW^{p-1}\varphi_{K}+\lambda^{-1}\left(-\dot{a}+\mu_{0}a,\dot{\xi},\dot{\lambda}\right)\cdot
Z+E_{K}.$ (13.3)

Here

$\displaystyle E_{K}$ $\displaystyle:=$
$\displaystyle-2\nabla\varphi\cdot\nabla\eta_{K}-\varphi\Delta\eta_{K}+\mathcal{N}\eta_{K}$
$\displaystyle+$
$\displaystyle\lambda^{-1}\left(-\dot{a}+\mu_{0}a,\dot{\xi},\dot{\lambda}\right)\cdot
Z\left(\eta_{K}-1\right)$ $\displaystyle+$
$\displaystyle\lambda^{-1}\left[\dot{\lambda}\left(\frac{n-2}{2}\varphi+y\cdot\nabla_{y}\varphi\right)+\dot{\xi}\cdot\nabla_{y}\varphi\right]\eta_{K}$
$\displaystyle+$
$\displaystyle\frac{a}{\lambda}\left[\frac{\dot{\xi}}{\lambda}\cdot\nabla
Z_{0}+\frac{\dot{\lambda}}{\lambda}\left(y\cdot\nabla
Z_{0}+\frac{n}{2}Z_{0}\right)\right]\eta_{K}.$

The following estimate holds for $E_{K}$.

######  Lemma 13.3.

For any $\tau$, we have

$\displaystyle\|E_{K}(\tau)\|_{2+\alpha}$ $\displaystyle\lesssim$
$\displaystyle\mathcal{O}(\tau)+K^{2+\alpha-\bar{p}\alpha}\|\varphi_{K}(\tau)\|_{\alpha}^{\bar{p}}+\Big{|}\frac{a(\tau)}{\lambda(\tau)}\Big{|}^{\bar{p}}$
$\displaystyle+$ $\displaystyle
K^{3+\alpha-n}\left|\frac{\dot{\xi}(\tau)}{\lambda(\tau)}\right|+K^{4+\alpha-n}\left|\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\right|+e^{-cK}\Big{|}\frac{\dot{a}(\tau)-\mu_{0}a(\tau)}{\lambda(\tau)}\Big{|}.$

###### Proof.

  1. (1)

Because $\alpha=(n-2)/2$, by the definition of $\mathcal{O}$, we have

$K^{2+\alpha}\sup_{y}\left|2\nabla\varphi(y,\tau)\cdot\nabla\eta_{K}(y)+\varphi(y,\tau)\Delta\eta_{K}(y)\right|\lesssim\mathcal{O}(\tau).$

  2. (2)

By Lemma 13.1, we obtain

$\displaystyle|\mathcal{N}\eta_{K}|$ $\displaystyle\lesssim$
$\displaystyle|\varphi|^{\bar{p}}\eta_{K}+\left|\frac{a}{\lambda}\right|^{\bar{p}}Z_{0}^{\bar{p}}$
$\displaystyle=$
$\displaystyle|\varphi|^{\bar{p}}\left(\eta_{K}-\eta_{K}^{\bar{p}}\right)+|\varphi_{K}|^{\bar{p}}+\left|\frac{a}{\lambda}\right|^{\bar{p}}Z_{0}^{\bar{p}}.$

First, because $\mathcal{O}(\tau)\ll 1$,

$\sup_{y\in\mathbb{R}^{n}}K^{2+\alpha}|\varphi(y,\tau)|^{\bar{p}}\left[\eta_{K}(y)-\eta_{K}(y)^{\bar{p}}\right]\lesssim
K^{2+\alpha}\mathcal{O}(\tau)^{\bar{p}}\ll\mathcal{O}(\tau).$

Secondly, because the support of $\varphi_{K}$ is contained in $B_{K}$,

$\sup_{y\in\mathbb{R}^{n}}\left(1+|y|\right)^{2+\alpha}|\varphi_{K}(y,\tau)|^{\bar{p}}\lesssim
K^{2+\alpha-\bar{p}\alpha}\|\varphi_{K}(\tau)\|_{\alpha}^{\bar{p}}.$

Finally, by the exponential decay of $Z_{0}$ at infinity,

$\sup_{y\in\mathbb{R}^{n}}\left(1+|y|\right)^{2+\alpha}\left|\frac{a(\tau)}{\lambda(\tau)}\right|^{\bar{p}}|Z_{0}(y)|^{\bar{p}}\lesssim\left|\frac{a(\tau)}{\lambda(\tau)}\right|^{\bar{p}}.$

  3. (3)

For $i=1,\cdots,n$, $Z_{i}(y)=O\left(|y|^{1-n}\right)$. Hence

$\sup_{y\in\mathbb{R}^{n}}\left(1+|y|\right)^{2+\alpha}\left|\frac{\dot{\xi}(\tau)}{\lambda(\tau)}\cdot
Z(y)\left[\eta_{K}(y)-1\right]\right|\lesssim
K^{3+\alpha-n}\left|\frac{\dot{\xi}(\tau)}{\lambda(\tau)}\right|.$

  4. (4)

As in the previous case, because $Z_{n+1}(y)=O\left(|y|^{2-n}\right)$, we get

$\sup_{y\in\mathbb{R}^{n}}\left(1+|y|\right)^{2+\alpha}\left|\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}Z_{n+1}(y)\left[\eta_{K}(y)-1\right]\right|\lesssim
K^{4+\alpha-n}\left|\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\right|.$

  5. (5)

By the exponential decay of $Z_{0}$, we get

$\sup_{y\in\mathbb{R}^{n}}\left(1+|y|\right)^{2+\alpha}\left|\frac{\dot{a}(\tau)-\mu_{0}a(\tau)}{\lambda(\tau)}Z_{0}(y)\left[\eta_{K}(y)-1\right]\right|\lesssim
e^{-cK}\frac{\left|\dot{a}(\tau)-\mu_{0}a(\tau)\right|}{\lambda(\tau)}.$

  6. (6)

By Proposition 13.2, we have

$\sup_{y\in\mathbb{R}^{n}}\left(1+|y|\right)^{2+\alpha}\left|\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\left[\frac{n-2}{2}\varphi(y)+y\cdot\nabla_{y}\varphi(y)\right]\eta_{K}(y)\right|\ll
K^{4+\alpha-n}\frac{|\dot{\lambda}(\tau)|}{\lambda(\tau)}.$

  7. (7)

As in the previous case,

$\displaystyle\sup_{y\in\mathbb{R}^{n}}\left(1+|y|\right)^{2+\alpha}\left|\frac{a(\tau)}{\lambda(\tau)}\left[\frac{\dot{\xi}(\tau)}{\lambda(\tau)}\cdot\nabla
Z_{0}(y)+\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\left(y\cdot\nabla
Z_{0}(y)+\frac{n}{2}Z_{0}(y)\right)\right]\eta_{K}(y)\right|$
$\displaystyle\ll$ $\displaystyle
K^{4+\alpha-n}\left(\frac{|\dot{\xi}(\tau)|}{\lambda(\tau)}+\frac{|\dot{\lambda}(\tau)|}{\lambda(\tau)}\right).$

Putting these estimates together we finish the proof. ∎

Because $n\geq 7$ and $\alpha=(n-2)/2$,

$4+\alpha-n<0.$

Furthermore, by noting that those terms like $\|\varphi_{K}(\tau)\|_{\alpha}$
are small, this lemma can be restated as

$\displaystyle\|E_{K}(\tau)\|_{2+\alpha}$ $\displaystyle\lesssim$
$\displaystyle
o\left(\|\varphi_{K}(\tau)\|_{\alpha}+\Big{|}\frac{\dot{\xi}(\tau)}{\lambda(\tau)}\Big{|}+\Big{|}\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\Big{|}+\Big{|}\frac{a(\tau)}{\lambda(\tau)}\Big{|}+\Big{|}\frac{\dot{a}(\tau)-\mu_{0}a(\tau)}{\lambda(\tau)}\Big{|}\right)$
(13.4) $\displaystyle+$ $\displaystyle\mathcal{O}(\tau).$

In particular, the main order term in $E_{K}$ is $\mathcal{O}(\tau)$, which
comes from the outer component.

Now we prove our main estimate in this section.

######  Proposition 13.4 ($C^{1,\theta}$ estimates).

  * •

For any $\tau$,

$\left|\frac{\dot{\xi}(\tau)}{\lambda(\tau)}\right|+\left|\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\right|+\left|\frac{\dot{a}(\tau)-\mu_{0}a(\tau)}{\lambda(\tau)}\right|\lesssim\mathcal{O}(\tau)+K^{2+\alpha-\bar{p}\alpha}\|\varphi_{K}(\tau)\|_{\alpha}^{\bar{p}}+\Big{|}\frac{a(\tau)}{\lambda(\tau)}\Big{|}^{\bar{p}}.$
(13.5)

  * •

For any $\sigma>0$, there exist two constants $C$ (universal) and
$T(\sigma,K)$ such that for any $\tau_{1}<\tau_{2}$,

$\displaystyle\|\varphi_{K}\|_{C^{(1+\theta)/2}(\tau_{1},\tau_{2};\mathcal{X}_{\alpha}^{1+\theta})}$
$\displaystyle\leq$
$\displaystyle\sigma\|\varphi_{K}(\tau_{1}-T(\sigma,K))\|_{\mathcal{X}_{\alpha}}$
$\displaystyle+C\left\|\left(\mathcal{O},\left|\frac{a}{\lambda}\right|^{\bar{p}}\right)\right\|_{L^{\infty}(\tau_{1}-T(\sigma,K),\tau_{2})}.$

###### Proof.

Take a decomposition $\varphi_{K}=\varphi_{K,1}+\varphi_{K,2}$, where
$\varphi_{K,1}$ is the solution of (A.3) with initial value (at
$\tau_{1}-T(\sigma)$, with $T(\sigma,K)$ to be determined below)
$\varphi_{K}$, and $\varphi_{K,2}$ is the solution of (A.8) with non-
homogeneous term $E_{K}$.

To apply Lemma A.4, take an $\alpha^{\prime}>n/2$. Because $\varphi_{K}$ is
supported in $B_{K}$, we have

$\left\|\varphi_{K}(\tau_{1}-T(\sigma,K))\right\|_{\alpha^{\prime}}\leq\left(2K\right)^{\alpha^{\prime}-\alpha}\left\|\varphi_{K}(\tau_{1}-T(\sigma,K))\right\|_{\alpha}.$
(13.7)

By Lemma A.4, there exists a $T(\sigma,K)>0$ such that

$\|\varphi_{K,1}\|_{C^{(1+\theta)/2}(\tau_{1},\tau_{2};C^{1+\theta}(B_{2K}))}\leq\sigma\left(4K\right)^{-1-\alpha}\left(2K\right)^{\alpha-\alpha^{\prime}}.$
(13.8)

For $\varphi_{K,2}$, in view of the estimate in Lemma 13.3, a direct
application of Lemma A.5 gives

$\|\varphi_{K,2}\|_{C^{(1+\theta)/2}(\tau_{1},\tau_{2};\mathcal{X}_{\alpha}^{1+\theta})}\leq
C(\alpha)\left\|\left(\mathcal{O},\left|\frac{a}{\lambda}\right|^{\bar{p}}\right)\right\|_{L^{\infty}(\tau_{1}-T(\sigma,K),\tau_{2})}.$
(13.9)

Adding (13.8) and (13.9), by using again the fact that $\varphi_{K}(\tau)$ is
supported in $B_{K}$, we get (• ‣ 13.4). ∎

For applications in Section 18, we give another estimate on the parameters
$\dot{\lambda}$ etc. in terms of the $L^{\infty}$ norm of $\varphi$.

######  Lemma 13.5.

For each $\tau$,

$\left|\frac{\dot{a}(\tau)-\mu_{0}a(\tau)}{\lambda(\tau)}\right|+\left|\frac{\dot{\xi}(\tau)}{\lambda(\tau)}\right|+\left|\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\right|\lesssim\sup_{B_{2K}\setminus
B_{K}}|\varphi(\tau)|+\sup_{B_{2K}}|\varphi(\tau)|^{\bar{p}}+\left|\frac{a(\tau)}{\lambda(\tau)}\right|^{\bar{p}}.$
(13.10)

###### Proof.

For each $j=0,\cdots,n+1$, multiplying (13.3) by $Z_{j}$, utilizing the
orthogonal condition (13.2), we get

$\left\\{\begin{aligned}
&\frac{\dot{a}(\tau)-\mu_{0}a(\tau)}{\lambda(\tau)}=\int_{\mathbb{R}^{n}}Z_{0}(y)E_{K}(y,\tau)dy,\\\
&\frac{\dot{\xi}_{j}(\tau)}{\lambda(\tau)}=-\int_{\mathbb{R}^{n}}Z_{j}(y)E_{K}(y,\tau)dy,\quad
j=1,\cdots,n,\\\
&\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}=-\int_{\mathbb{R}^{n}}Z_{n+1}(y)E_{K}(y,\tau)dy.\end{aligned}\right.$

An integration by parts gives

$\displaystyle\left|\int_{\mathbb{R}^{n}}\left(2\nabla\varphi\cdot\nabla\eta_{K}+\varphi\Delta\eta_{K}\right)Z_{j}\right|$
$\displaystyle=$
$\displaystyle\left|\int_{\mathbb{R}^{n}}\varphi\left(2\nabla\eta_{K}\cdot\nabla
Z_{j}+\Delta\eta_{K}Z_{j}\right)\right|$ (13.11) $\displaystyle\lesssim$
$\displaystyle\sup_{B_{2K}\setminus B_{K}}|\varphi|.$

By the at most $|y|^{2-n}$ decay of $Z_{j}$, Lemma 13.1 and Proposition 13.2,
we find the contribution from other terms in $E_{K}$ is of the order

$O\left(\sup_{y\in
B_{2K}}|\varphi(y,\tau)|^{\bar{p}}+\left|\frac{a(\tau)}{\lambda(\tau)}\right|^{\bar{p}}\right)+o\left(\left|\frac{\dot{\xi}(\tau)}{\lambda(\tau)}\right|+\left|\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\right|+\left|\frac{\dot{a}(\tau)-\mu_{0}a(\tau)}{\lambda(\tau)}\right|\right).$

Adding these estimates for
$\frac{\dot{a}(\tau)-\mu_{0}a(\tau)}{\lambda(\tau)}$,
$\frac{\dot{\xi}_{j}(\tau)}{\lambda(\tau)}$ and
$\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}$ together, we get (13.10). ∎

Finally, we will need the following backward estimate on $a/\lambda$.

######  Lemma 13.6.

For any $\tau_{1}<\tau_{2}$, we have

$\displaystyle\frac{|a(\tau_{1})|}{\lambda(\tau_{1})}\lesssim
e^{-\frac{\mu_{0}}{2}(\tau_{2}-\tau_{1})}\frac{|a(\tau_{2})|}{\lambda(\tau_{2})}+\int_{\tau_{1}}^{\tau_{2}}e^{-\frac{\mu_{0}}{2}(\tau_{2}-s)}\left[\mathcal{O}(s)+\|\varphi_{K}(s)\|_{\alpha}^{\bar{p}}\right]ds.$
(13.12)

###### Proof.

By a direct differentiation and then applying (13.5), we obtain

$\displaystyle\frac{d}{d\tau}\frac{a(\tau)}{\lambda(\tau)}$ $\displaystyle=$
$\displaystyle\frac{\dot{a}(\tau)}{\lambda(\tau)}-\frac{\dot{\lambda}(\tau)}{\lambda(\tau)}\frac{a(\tau)}{\lambda(\tau)}$
$\displaystyle=$
$\displaystyle\mu_{0}\frac{a(\tau)}{\lambda(\tau)}+O\left(\mathcal{O}(\tau)+\|\varphi_{K}(\tau)\|_{\alpha}^{\bar{p}}+\left|\frac{a(\tau)}{\lambda(\tau)}\right|^{\bar{p}}\right)$
$\displaystyle=$
$\displaystyle\left[\mu_{0}+o(1)\right]\frac{a(\tau)}{\lambda(\tau)}+O\left(\mathcal{O}(\tau)+\|\varphi_{K}(\tau)\|_{\alpha}^{\bar{p}}\right).$

Integrating this differential inequality gives (13.12). ∎

###  14\. Outer problem

In this section we establish an estimate on the outer component $\phi_{out}$.
Our main goal is to obtain an estimate of $\mathcal{O}$ in terms of
$\mathcal{I}$ and the parameters $\lambda^{\prime}$ etc..

####  14.1. Decomposition of the outer equation

First we need to take a further decomposition of $\phi_{out}$. Recall that
$\phi_{out}$ satisfies

$\displaystyle\partial_{t}\phi_{out}-\Delta\phi_{out}$ $\displaystyle=$
$\displaystyle\left(u^{p}-W_{\ast}^{p}\right)\left(1-\eta_{out}\right)$
$\displaystyle-$
$\displaystyle\phi\left(\partial_{t}\eta_{out}-\Delta\eta_{out}\right)+2\nabla\phi\cdot\nabla\eta_{out}$
$\displaystyle+$
$\displaystyle\left(-a^{\prime}+\mu_{0}\frac{a}{\lambda^{2}},\xi^{\prime},\lambda^{\prime}\right)Z_{\ast}\left(1-\eta_{out}\right)$
$\displaystyle-$ $\displaystyle
a\partial_{t}Z_{0}^{\ast}\left(1-\eta_{out}\right).$

Introducing

$\left\\{\begin{aligned}
&V_{\ast}:=\begin{cases}\frac{u^{p}-W_{\ast}^{p}}{u-W_{\ast}}\chi_{B_{L\lambda(t)}(\xi(t))^{c}},&\mbox{if
}u\neq W_{\ast},\\\
pu^{p-1}\chi_{B_{L\lambda(t)}(\xi(t))^{c}},&\mbox{otherwise},\end{cases}\\\
&F_{2}:=-\phi\left(\partial_{t}\eta_{out}-\Delta\eta_{out}\right)+2\nabla\phi\cdot\nabla\eta_{out},\\\
&F_{3}:=\lambda^{\prime}Z_{n+1,\ast}\left(1-\eta_{out}\right),\\\
&F_{4}:=\sum_{j=1}^{n}\xi_{j}^{\prime}Z_{j,\ast}\left(1-\eta_{out}\right),\\\
&F_{5}:=\left[aV_{\ast}Z_{0}^{\ast}-\left(a^{\prime}-\mu_{0}\frac{a}{\lambda^{2}}\right)Z_{0}^{\ast}-a\partial_{t}Z_{0}^{\ast}\right]\left(1-\eta_{out}\right),\end{aligned}\right.$

then (14.1) is written as

$\partial_{t}\phi_{out}-\Delta\phi_{out}=V_{\ast}\phi_{out}+F_{2}+F_{3}+F_{4}+F_{5}.$
(14.2)

Corresponding to this decomposition of the right hand side, we have
$\phi_{out}=\phi_{1}+\phi_{2}+\phi_{3}+\phi_{4}+\phi_{5}$, with these five
functions solving the following five equations respectively: first,

$\left\\{\begin{aligned}
&\partial_{t}\phi_{1}-\Delta\phi_{1}=V_{\ast}\phi_{1}\quad&\mbox{in
}Q_{9/10},\\\ &\phi_{1}=\phi_{out},\quad&\mbox{on
}\partial^{p}Q_{9/10};\end{aligned}\right.$

next,

$\left\\{\begin{aligned}
&\partial_{t}\phi_{2}-\Delta\phi_{2}=V_{\ast}\phi_{2}+F_{2}\quad&\mbox{in
}Q_{9/10},\\\ &\phi_{2}=0,\quad&\mbox{on
}\partial^{p}Q_{9/10};\end{aligned}\right.$

and $\phi_{3},\phi_{4},\phi_{5}$ solve the same equation with $\phi_{2}$, but
with the right hand side term replaced by $F_{3}$, $F_{4}$ and $F_{5}$
respectively.

The following estimates hold for $V_{\ast}$ and $F_{2}$-$F_{5}$.

  1. (i)

Because $u>0$ and $W_{\ast}>0$, by convexity of the function $x\mapsto x^{p}$
and Lemma 11.5, for any $\delta>0$ and $C>0$, there exists a constant
$R(\delta)>0$ such that, if $L\geq R(\delta)$, then

$|V_{\ast}|\leq
p\left(u^{p-1}+W_{\ast}^{p-1}\right)\leq\frac{\delta}{|x-\xi(t)|^{2}}+C.$
(14.3)

  2. (ii)

By the definition of $\eta_{out}$ in (12.9), we get

$\displaystyle\big{|}\phi\left(\partial_{t}\eta_{out}-\Delta\eta_{out}\right)\big{|}$
$\displaystyle\lesssim$
$\displaystyle|\phi|\left(\frac{|\lambda^{\prime}(t)|}{\lambda(t)}+\frac{|\xi^{\prime}(t)|}{L\lambda(t)}+\frac{1}{L^{2}\lambda(t)^{2}}\right)\chi_{B_{2L\lambda(t)}(\xi(t))\setminus
B_{L\lambda(t)}(\xi(t))}$ $\displaystyle\lesssim$
$\displaystyle\frac{\mathcal{I}(t)}{L^{\alpha}\lambda(t)^{\alpha}}\left(\frac{|\lambda^{\prime}(t)|}{\lambda(t)}+\frac{|\xi^{\prime}(t)|}{L\lambda(t)}+\frac{1}{L^{2}\lambda(t)^{2}}\right)\chi_{B_{2L\lambda(t)}(\xi(t))\setminus
B_{L\lambda(t)}(\xi(t))}.$

In the same way, we get

$\displaystyle 2|\nabla\phi||\nabla\eta_{out}|$ $\displaystyle\lesssim$
$\displaystyle\frac{|\nabla\phi|}{L\lambda(t)}\chi_{B_{2L\lambda(t)}(\xi(t))\setminus
B_{L\lambda(t)}(\xi(t))}$ $\displaystyle\lesssim$
$\displaystyle\frac{\mathcal{I}(t)}{L^{2+\alpha}\lambda(t)^{2+\alpha}}\chi_{B_{2L\lambda(t)}(\xi(t))\setminus
B_{L\lambda(t)}(\xi(t))}.$

Combining ((ii)) and ((ii)), we obtain

$|F_{2}|\lesssim\frac{\mathcal{I}(t)}{L^{\alpha}\lambda(t)^{\alpha}}\left[\frac{|\lambda^{\prime}(t)|}{\lambda(t)}+\frac{|\xi^{\prime}(t)|}{L\lambda(t)}+\frac{1}{L^{2}\lambda(t)^{2}}\right]\chi_{B_{2L\lambda(t)}(\xi(t))\setminus
B_{L\lambda(t)}(\xi(t))}.$ (14.6)

  3. (iii)

By the decay of $Z_{n+1}$ at infinity, we get

$|F_{3}|\lesssim\lambda(t)^{\frac{n-4}{2}}|\lambda^{\prime}(t)||x-\xi(t)|^{2-n}\chi_{B_{L\lambda(t)}(\xi(t))^{c}}.$
(14.7)

  4. (iv)

By the decay of $Z_{1},\cdots,Z_{n}$ at infinity, we get

$|F_{4}|\lesssim\lambda(t)^{\frac{n-2}{2}}|\xi^{\prime}(t)||x-\xi(t)|^{1-n}\chi_{B_{L\lambda(t)}(\xi(t))^{c}}.$
(14.8)

  5. (v)

By the decay of $Z_{0}$ at infinity, we get

$\Big{|}\left(a^{\prime}-\mu_{0}\frac{a}{\lambda^{2}}\right)Z_{0}^{\ast}\left(1-\eta_{out}\right)\Big{|}\lesssim\Big{|}a^{\prime}(t)-\mu_{0}\frac{a(t)}{\lambda(t)^{2}}\Big{|}\lambda(t)^{-\frac{n}{2}}e^{-c\frac{|x-\xi(t)|}{\lambda(t)}}\chi_{B_{L\lambda(t)}(\xi(t))^{c}}.$
(14.9)

Similarly, we have

$\Big{|}a\partial_{t}Z_{0}^{\ast}\left(1-\eta_{out}\right)\Big{|}\lesssim|a(t)|\lambda(t)^{-\frac{n+2}{2}}\left(|\lambda^{\prime}(t)|+|\xi^{\prime}(t)|\right)e^{-c\frac{|x-\xi(t)|}{\lambda(t)}}\chi_{B_{L\lambda(t)}(\xi(t))^{c}}.$
(14.10)

Finally, by (14.3), we obtain

$\displaystyle\left|aV_{\ast}Z_{0}^{\ast}\left(1-\eta_{out}\right)\right|$
$\displaystyle\lesssim$
$\displaystyle\frac{|a(t)|}{|x-\xi(t)|^{2}}\lambda(t)^{-\frac{n}{2}}e^{-c\frac{|x-\xi(t)|}{\lambda(t)}}\chi_{B_{L\lambda(t)}(\xi(t))^{c}}$
$\displaystyle\lesssim$
$\displaystyle\frac{|a(t)|}{\lambda(t)^{2}}\lambda(t)^{-\frac{n}{2}}e^{-c\frac{|x-\xi(t)|}{\lambda(t)}}\chi_{B_{L\lambda(t)}(\xi(t))^{c}}.$

Combining (14.9)-((v)), we get

$\displaystyle|F_{5}|$ $\displaystyle\lesssim$
$\displaystyle\left[\frac{|a(t)|}{\lambda(t)^{2}}+\Big{|}a^{\prime}(t)-\mu_{0}\frac{a(t)}{\lambda(t)^{2}}\Big{|}+\frac{|a(t)|}{\lambda(t)}\left(|\lambda^{\prime}(t)|+|\xi^{\prime}(t)|\right)\right]$
$\displaystyle\quad\quad\times\lambda(t)^{-\frac{n}{2}}e^{-c\frac{|x-\xi(t)|}{\lambda(t)}}\chi_{B_{L\lambda(t)}(\xi(t))^{c}}.$

For each $i=1,\cdots,5$, let

$\overline{\phi}_{i}(x,t):=\left|\phi_{i}\left(x-\xi(t),t\right)\right|.$
(14.13)

By the Kato inequality, we obtain

$\partial_{t}\overline{\phi}_{1}-\Delta\overline{\phi}_{1}+\xi^{\prime}(t)\cdot\nabla\overline{\phi}_{1}\leq|V_{\ast}|\overline{\phi}_{1}$
(14.14)

and for $i=2,3,4,5$,

$\partial_{t}\overline{\phi}_{i}-\Delta\overline{\phi}_{i}+\xi^{\prime}(t)\cdot\nabla\overline{\phi}_{i}\leq|V_{\ast}|\overline{\phi}_{i}+|F_{i}|.$
(14.15)

####  14.2. Estimates for the outer equation

In this subsection, we establish some pointwise estimates on
$\overline{\phi}_{1},\cdots,\overline{\phi}_{5}$.

Denote the Dirichlet heat kernel for the operator
$\mathcal{H}:=\partial_{t}-\Delta-\left(\delta|x|^{-2}+C\right)+\xi^{\prime}(t)\cdot\nabla$
in $Q_{1}$ by $G(x,t;y,s)$ ($t>s$).

Let

$\gamma:=\frac{n-2}{2}-\sqrt{\frac{(n-2)^{2}}{4}-4\delta}.$

Then $w(x):=|x|^{-\gamma}$ is a weak solution of the elliptic equation

$-\Delta w(x)=\frac{\delta}{|x|^{2}}w(x).$

######  Lemma 14.1 (Estimate on $\overline{\phi}_{1}$).

$\overline{\phi}_{1}(x,t)\lesssim|x|^{-\gamma}$ in $Q_{8/9}$.

###### Proof.

Set $\varphi_{1}:=\overline{\phi}_{1}/w$. It is a sub-solution to the
parabolic equation

$\partial_{t}\varphi=w^{-2}\mbox{div}\left(w^{2}\nabla\varphi\right)-\xi^{\prime}(t)\cdot\nabla\varphi+\left[C+\gamma\xi^{\prime}(t)\cdot\frac{x}{|x|^{2}}\right]\varphi.$
(14.16)

We note the following three facts about the coefficients of this equation.

  1. (1)

Because $\gamma\ll 1$, by [71, Theorem 3.1], the volume doubling property and
the scaling invariant Poincare inequality holds on $\mathbb{R}^{n}$ with the
weighted measure $w^{2}dx$. By [76, Theorem 5.2.3], for some $q>2$
(independent of $\gamma$), there is a local Sobolev embedding from
$W^{1,2}(B_{1},w^{2}dx)$ into $L^{q}(B_{1},w^{2}dx)$.

  2. (2)

By the Lipschitz hypothesis (10.1), $\xi^{\prime}(t)\in L^{\infty}(-1,1)$.

  3. (3)

Similarly, the zeroth order term
$C+\gamma\xi^{\prime}(t)\cdot\frac{x}{|x|^{2}}\in
L^{\infty}(-1,1;L^{n-3\gamma}(B_{1},w^{2}dx))$.

Then by standard De Giorgi-Nash-Moser estimate (cf. [71, Section 3] and [76,
Chapter 5]), we deduce that $\varphi_{1}$ is bounded in $Q_{8/9}$. (The lower
order terms in this parabolic operator do not affect this argument by noting
their higher integrability.) ∎

In fact, Moser’s Harnack inequality holds for positive solutions of (14.16).
This is similar to [71, Theorem 3.5]. Then by [76, Section 5.4.7] (see also
[71, Theorem 4.3]), the heat kernel $G$ satisfies a Gaussian bound

$G(x,t;y,s)\leq
C\left(t-s\right)^{-\frac{n}{2}}e^{-c\frac{|x-y|^{2}}{t-s}}\left(1+\frac{\sqrt{t-s}}{|x|}\right)^{\gamma}\left(1+\frac{\sqrt{t-s}}{|y|}\right)^{\gamma}.$
(14.17)

Hereafter we fix two constants $\beta>1$ (but sufficiently close to $1$) and
$\mu\in(0,1)$ (to be determined below). We choose $\mu$ so that

$\left(\frac{n-2}{2}-2\gamma\right)\left(1-\mu\right)>2.$

which is denoted by $2+2\kappa$. This inequality is guaranteed by the
assumption that $n\geq 7$.

######  Lemma 14.2 (Estimate on $\overline{\phi}_{2}$).

If $K\lambda(t)/2\leq|x|\leq 4K\lambda(t)$, then

$\displaystyle\overline{\phi}_{2}(x,t)$ $\displaystyle\lesssim$
$\displaystyle\left[K^{-\left(\frac{n-2}{2}-\gamma\right)(\beta-1)}+\frac{L^{\frac{n-2}{2}-\gamma}}{K^{\frac{n-2}{2}-(2\beta-1)\gamma}}\right]\left[K\lambda(t)\right]^{-\frac{n-2}{2}}\|\mathcal{I}\|_{L^{\infty}(t-\lambda(t)^{2\mu},t)}$
$\displaystyle+$ $\displaystyle\lambda(t)^{-\gamma(1-\mu)-\frac{n-2}{2}\mu}.$

###### Proof.

By the heat kernel representation formula, for any $(x,t)$ we have

$\overline{\phi}_{2}(x,t)\leq\int_{-81/100}^{t}\int_{B_{2L\lambda(s)}\setminus
B_{L\lambda(s)}}G(x,t;y,s)\frac{\mathcal{I}(s)}{L^{\alpha}\lambda(s)^{\alpha}}\left(\frac{|\lambda^{\prime}(s)|}{\lambda(s)}+\frac{|\xi^{\prime}(s)|}{L\lambda(s)}+\frac{1}{L^{2}\lambda(s)^{2}}\right).$

Divide this integral into three parts, the first part $\mathrm{I}$ being on
$(-81/100,t-\lambda(t)^{2\mu})$, the second part $\mathrm{II}$ involving the
integral on $(t-\lambda(t)^{2\mu},t-K^{2\beta}\lambda(t)^{2})$, and the third
part $\mathrm{III}$ involving the integral on
$(t-K^{2\beta}\lambda(t)^{2},t)$.

Estimate of $\mathrm{I}$. By (13.5),

$|\lambda^{\prime}(s)|+|\xi^{\prime}(s)|\ll\lambda(s)^{-1}.$ (14.18)

Hence

$\left\|\frac{\mathcal{I}(s)}{L^{\alpha}\lambda(s)^{\alpha}}\left(\frac{|\lambda^{\prime}(s)|}{\lambda(s)}+\frac{|\xi^{\prime}(s)|}{L\lambda(s)}+\frac{1}{L^{2}\lambda(s)^{2}}\right)\chi_{B_{2L\lambda(s)}\setminus
B_{L\lambda(s)}}\right\|_{L^{\frac{2n}{n+2}}(B_{1})}\lesssim\mathcal{I}(s).$

By the Guassian bound on $G(x,y;t,s)$ in (14.17),

$\left\|G(x,t;\cdot,s)\right\|_{L^{\frac{2n}{n-2}}(B_{1})}\lesssim(t-s)^{-\frac{n+2}{4}}\left(1+\frac{\sqrt{t-s}}{|x|}\right)^{\gamma}.$
(14.19)

Then by Hölder inequality we get

$\displaystyle\mathrm{I}$ $\displaystyle\lesssim$
$\displaystyle|x|^{-\gamma}\int_{-81/100}^{t-\lambda(t)^{2\mu}}(t-s)^{-\frac{n+2}{4}+\frac{\gamma}{2}}\mathcal{I}(s)ds$
$\displaystyle\lesssim$ $\displaystyle
K^{-\gamma}\lambda(t)^{-(1-\mu)\gamma-\frac{n-2}{2}\mu},$

where in the last step we used only the estimate $|\mathcal{I}(s)|\leq C$.

Estimate of $\mathrm{II}$. This case differs from the previous one only in the
last step. Now we have

$\displaystyle\mathrm{II}$ $\displaystyle\lesssim$
$\displaystyle|x|^{-\gamma}\int_{t-\lambda(t)^{2\mu}}^{t-K^{2\beta}\lambda(t)^{2}}(t-s)^{-\frac{n+2}{4}+\frac{\gamma}{2}}\mathcal{I}(s)$
$\displaystyle\lesssim$ $\displaystyle
K^{-\frac{n-2}{2}\beta+(\beta-1)\gamma}\lambda(t)^{-\frac{n-2}{2}}\|\mathcal{I}\|_{L^{\infty}(t-\lambda(t)^{2\mu},t-K^{2\beta}\lambda(t)^{2})}.$

Estimate of $\mathrm{III}$. Still by the heat kernel representation formula,
$\mathrm{III}$ is bounded by

$\displaystyle\int_{t-K^{2\beta}\lambda(t)^{2}}^{t}\int_{B_{2L\lambda(s)}\setminus
B_{L\lambda(s)}}\left(t-s\right)^{-\frac{n}{2}}e^{-c\frac{|x-y|^{2}}{t-s}}\left(1+\frac{\sqrt{t-s}}{|x|}\right)^{\gamma}\left(1+\frac{\sqrt{t-s}}{|y|}\right)^{\gamma}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\frac{\mathcal{I}(s)}{L^{\alpha}\lambda(s)^{\alpha}}\left(\frac{|\lambda^{\prime}(s)|}{\lambda(s)}+\frac{|\xi^{\prime}(s)|}{L\lambda(s)}+\frac{1}{L^{2}\lambda(s)^{2}}\right)dyds$
$\displaystyle=$
$\displaystyle\int_{t-K^{2\beta}\lambda(t)^{2}}^{t}\frac{\mathcal{I}(s)}{L^{\alpha}\lambda(s)^{\alpha}}\left(\frac{|\lambda^{\prime}(s)|}{\lambda(s)}+\frac{|\xi^{\prime}(s)|}{L\lambda(s)}+\frac{1}{L^{2}\lambda(s)^{2}}\right)\left(1+\frac{\sqrt{t-s}}{|x|}\right)^{\gamma}$
$\displaystyle\times\left[\int_{B_{2L\lambda(s)}\setminus
B_{L\lambda(s)}}\left(t-s\right)^{-\frac{n}{2}}e^{-c\frac{|x-y|^{2}}{t-s}}\left(1+\frac{\sqrt{t-s}}{|y|}\right)^{\gamma}dy\right]ds.$

For $s\in[t-K^{2\beta}\lambda(t)^{2},t]$, by Proposition 11.1 we have

$\lambda(s)\sim\lambda(t).$ (14.21)

Thus by noting that $K\lambda(t)/2\leq|x|\leq 4K\lambda(t)$ and $K\gg L$, we
have

$|x|\gg|y|\quad\mbox{for any}~{}~{}y\in B_{2L\lambda(s)}.$

Therefore in the integral above we can replace $e^{-c\frac{|x-y|^{2}}{t-s}}$
by $e^{-c\frac{|x|^{2}}{t-s}}$.

Then in view of (14.18), $\mathrm{III}$ is controlled by

$\displaystyle
K^{(\beta-1)\gamma}\|\mathcal{I}\|_{L^{\infty}(t-K^{2\beta}\lambda(t)^{2},t)}L^{\frac{n-2}{2}}\lambda(t)^{\frac{n-2}{2}}$
$\displaystyle\quad\quad\quad\quad\times\int_{t-K^{2\beta}\lambda(t)^{2}}^{t}\left(t-s\right)^{-\frac{n}{2}}e^{-c\frac{|x|^{2}}{t-s}}\left(1+\frac{\sqrt{t-s}}{L\lambda(t)}\right)^{\gamma}ds$
$\displaystyle\lesssim$ $\displaystyle
K^{(\beta-1)\gamma}\|\mathcal{I}\|_{L^{\infty}(t-K^{2\beta}\lambda(t)^{2},t)}L^{\frac{n-2}{2}}\lambda(t)^{\frac{n-2}{2}}$
$\displaystyle\quad\quad\quad\quad\left(1+\frac{K^{\beta}}{L}\right)^{\gamma}\int_{t-K^{2\beta}\lambda(t)^{2}}^{t}\left(t-s\right)^{-\frac{n}{2}}e^{-c\frac{|x|^{2}}{t-s}}ds$
$\displaystyle\lesssim$ $\displaystyle
K^{(2\beta-1)\gamma}L^{\frac{n-2}{2}-\gamma}\|\mathcal{I}\|_{L^{\infty}(t-K^{2\beta}\lambda(t)^{2},t)}\lambda(t)^{\frac{n-2}{2}}|x|^{2-n}$
$\displaystyle\lesssim$
$\displaystyle\frac{L^{\frac{n-2}{2}-\gamma}}{K^{\frac{n-2}{2}-(2\beta-1)\gamma}}\left[K\lambda(t)\right]^{-\frac{n-2}{2}}\|\mathcal{I}\|_{L^{\infty}(t-K^{2}\lambda(t)^{2},t)}.$

Adding up the estimates for $\mathrm{I}$, $\mathrm{II}$ and $\mathrm{II}$, we
finish the proof. ∎

######  Lemma 14.3 (Estimate on $\overline{\phi}_{3}$).

For $K\lambda(t)/2\leq|x|\leq 4K\lambda(t)$,

$\overline{\phi}_{3}(x,t)\lesssim\lambda(t)^{-\frac{n-2}{2}\mu+2(\mu-1)\gamma}+K^{-\frac{n-2}{2}(\beta-1)+2(\beta-1)\gamma}\left[K\lambda(t)\right]^{-\frac{n-2}{2}}\|\lambda\lambda^{\prime}\|_{L^{\infty}(t-\lambda(t)^{2\mu},t)}.$

###### Proof.

By the heat kernel representation,

$\displaystyle\overline{\phi}_{3}(x,t)$ $\displaystyle\lesssim$