# Blow up analysis for Keller-Segel system

Hua Chen, Jian-Meng Li and Kelei Wang

###### Abstract.

In this paper we develop a blow up theory for the parabolic-elliptic Keller-
Segel system, which can be viewed as a parabolic counterpart to the Liouville
equation. This theory is applied to the study of first time singularities,
ancient solutions and entire solutions, leading to a description of the blow
up limit in the first problem, and the large scale structure in the other two
problems.

###### Key words and phrases:

Keller-Segel system; blow up analysis; first time singularity; entire
solution.

###### 2020 Mathematics Subject Classification:

35K58, 35B44, 35B33.

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China.

Email<EMAIL_ADDRESS>(H. Chen<EMAIL_ADDRESS>(J. Li),
<EMAIL_ADDRESS>(K. Wang).

This work is supported by National Natural Science Foundation of China (Grants
No. 12131017 and 11871381).

##  1\. Introduction

###  1.1. Blow up analysis

In this paper we generalize the blow up analysis of Brezis-Merle [1] and Li-
Shafrir [8] for the Liouville equation

(1.1) $-\Delta u=e^{u}$

to a parabolic setting, that is, for the Keller-Segel system (see Keller-Segel
[7])

(1.2) $\left\\{\begin{array}[]{ll}u_{t}=\Delta u-\mbox{div}(u\nabla v),\\\
-\Delta v=u.\end{array}\right.$

We will also apply this theory to the study of first time singularities in
(1.2) and the large scale structure of ancient and entire solutions of (1.2).

Throughout this paper the spatial dimension is $2$. Our first main result is
about the convergence and blow up behavior for sequences of solutions to
(1.2).

######  Theorem 1.1.

Suppose $u_{i}$ is a sequence of smooth, positive solutions of the Keller-
Segel system (1.2) in the unit parabolic cylinder
$Q_{1}:=B_{1}\times(-1,1)\subset\mathbb{R}^{2}\times\mathbb{R}$, satisfying

$\sup_{-1<t<1}\int_{B_{1}}u_{i}(x,t)dx\leq M\quad\mbox{for some
constant}~{}~{}M.$

Then the followings hold.

  1. (1)

There exists a family of Radon measures $\mu_{t}$ on $B_{1}$, $t\in(-1,1)$,
such that after passing to a subsequence of $i$, for any $t\in(-1,1)$,

$u_{i}(x,t)dx\rightharpoonup\mu_{t}\quad\text{weakly as Radon measures};$

  2. (2)

$\mu_{t}$ is continuous in $t$ with respect to the weak topology;

  3. (3)

for any $t\in(-1,1)$, there exist $N(t)$ points $q_{j}(t)$, where $0\leq
N(t)\leq M/(8\pi)$, and $0\leq\rho(t)\in L^{1}(B_{1})$ such that

(1.3) $\mu_{t}=\sum_{j=1}^{N(t)}8\pi\delta_{q_{j}(t)}+\rho(x,t)dx.$

  4. (4)

the blow up locus
$\Sigma:=\cup_{t\in(-1,1)}\cup_{j=1}^{N(t)}\\{(q_{j}(t),t)\\}$ is relatively
closed in $Q_{1}$;

  5. (5)

$\rho\in C^{\infty}(Q_{1}\setminus\Sigma)$ and satisfies (1.2) in this open
set, where

$\nabla
v(x,t)=-4\sum_{j=1}^{N(t)}\frac{x-q_{j}(t)}{|x-q_{j}(t)|^{2}}-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}\rho(y,t)dy;$

  6. (6)

if $\rho$ is smooth in $Q_{1}$, then $N(t)\equiv N$ for some $N\in\mathbb{N}$,
$q_{j}(t)\in C^{1}(-1,1)$ for each $j=1$, $\dots$, $N$ and

(1.4) $q_{j}^{\prime}(t)=4\sum\limits_{k\neq
j}\dfrac{q_{k}(t)-q_{j}(t)}{|q_{k}(t)-q_{j}(t)|^{2}}+\frac{1}{2\pi}\int_{B_{1}}\frac{y-q_{j}(t)}{|y-q_{j}(t)|^{2}}\rho(y,t)dy.$

######  Remark 1.2.

  1. (1)

Eqn. (1.3) says that each Dirac measure in the singular limit has mass $8\pi$.
Near this blow up point, $u_{i}$ should look like a scaled bubble, that is,

$u_{i}(x,t)\sim\frac{8\lambda_{j,i}(t)^{2}}{(\lambda_{j,i}(t)^{2}+|x-q_{j,i}(t)|^{2})^{2}},\quad\text{where}~{}~{}\lambda_{j,i}(t)\to
0,~{}q_{j,i}(t)\to q_{j}(t),$

see Theorem 9.1 and Conjecture 1.11 below. This essentially means that there
is only one bubble at each blow up point, or using the terminology of bubbling
analysis, the blow up is isolated and simple.

It is also worth noticing that although these Dirac measures are separated for
each $t<1$, it is not claimed that they cannot converge to the same one as
$t\to 1$. In fact, this is the picture for the first time singularity in
Keller-Segel system as constructed in Seki-Sugiyama-Velázquez [13]. In this
sense, this result corresponds to the quantization phenomena for blow ups of
Liouville equations, where the mass of Dirac measures in the singular limit is
$8\pi N$ for some $N\in\mathbb{N}$ (Li-Shafrir [8]) and $N$ could be larger
than $1$ (Chen [3]).

  2. (2)

The first term in the right hand side of (1.4) describes the interaction
between different Dirac measures. If $N=1$, (1.4) should be understood as
without this term.

  3. (3)

If the diffusion part $\rho(x,t)dx$ does not appear, then (1.4) is the
gradient flow of the renomarlized energy

$W(q_{1},\cdots,q_{N}):=4\sum_{j\neq k}\log|q_{j}-q_{k}|.$

It arises as a renomalization of the free energy

$\mathcal{F}(u):=\int u(x)\log
u(x)dx+\frac{1}{4\pi}\int\int\log|x-y|u(x)u(y)dxdy.$

Roughly speaking, if $u$ is close to $8\pi\sum_{j=1}^{N}\delta_{q_{j}}$, then
after subtracting the self-interaction terms from $\mathcal{F}$ (which could
be very large), only those terms describing interaction between different
Dirac measures are left, which is $W$, see related discussions in [14, Chapter
2].

For the singular limit of a modified Keller-Segel system, a similar point
dynamics was also established in Velázquez [17, 18].

The proof of Theorem 1.1 uses several tools such as symmetrization of test
functions, $\varepsilon$-regularity theorems. The use of
$\varepsilon$-regularity theorems is standard, just as in the study of blow up
phenomena for many other PDE problems. The symmetrization of test function
technique is mainly used to calculate the equation for the first and second
momentum, which have also been used by many people in the study of Keller-
Segel system. However, because our setting is local in nature, which is
different from most literature on Keller-Segel system, a suitable
_localization of these calculations_ is necessary, see Remark 2.2 below for
the definition of this localization procedure. In particular, the proof of
(1.3) strongly relies on the localized calculation on the second momentum,
which is also perfomed in a limiting form, see Section 5 for details. The
derivation of (1.4) also uses a localized calculation on the first momentum,
see Section 7 .

In the literature, there are various notions of weak solutions about (1.2),
see e.g. Suzuki [14, Chapter 13], which is related to the dynamical law (1.4),
but we will not use them in this paper. Instead, to derive (1.4), we rely
solely on explicit formulas derived from the convergence of sequences of
smooth solutions.

###  1.2. First time singularity

Theorem 1.1 provides a convenient setting for the analysis of blow up behavior
in Keller-Segel system. Our first application of Theorem 1.1 is on the
analysis of first time singularities in the Keller-Segel system (1.2).

By standard parabolic theory, under suitable assumptions, there exists a local
solution to the Cauchy problem of (1.2) on $\mathbb{R}^{2}$. The solution may
not exist globally in time. For example, when the total mass is larger than
the critical one, $8\pi$, the solution must blow up in finite time, see
Dolbeault-Perthame [4]. Then it is natural to analyse the blow up behavior
when the solution blows up at the first time. For the Keller-Segel system
(1.2), blow up of solutions is caused by _aggregation_ , that is,
_concentration of mass_. The following is [14, Theorem 1.1] or [15, Theorem
1.1]. (Similar results also hold for initial-boundary value problems.)

######  Theorem 1.3.

Suppose $u$ blows up at finite time $T$. Then as $t\to T$,

$u(x,t)dx\rightharpoonup u_{T}(x)dx+\sum_{a_{i}}m_{i}\delta_{a_{i}}$

weakly as Radon measures, where $\\{a_{i}\\}$, the set of blow up points, is a
finite set of $\mathbb{R}^{2}$ and $m_{i}\geq 8\pi$, $u_{T}\in
L^{1}(\mathbb{R}^{2})\cap C(\mathbb{R}^{2}\setminus\\{a_{i}\\})$ is a
nonnegative function.

For the construction of finite time blow up solutions, see Herrero-Velázquez
[6], Velázquez [16] and Raphaël-Schweyer [12].

By Theorem 1.3, to study first time singularites, we can work in the following
local setting:

(H1):

$u\in C^{\infty}(\overline{Q_{1}^{-}}\setminus\\{(0,0)\\})$ (here
$Q_{1}^{-}:=B_{1}\times(-1,0)$ is the unit backward parabolic cylinder), $u>0$
and

(1.5) $\sup_{t\in(-1,0)}\int_{B_{1}}u(x,t)dx\leq M;$

(H2):

$u$ satisfies (1.2) in $Q_{1}^{-}$, where $\nabla v$ is given by

(1.6) $\nabla
v(x,t)=-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}u(y,t)dt,\quad\forall(x,t)\in
Q_{1}^{-};$

(H3):

there exists a nonnegative function $u_{0}\in L^{1}(B_{1})$ and a positive
constant $m$ such that as $t\to 0^{-}$,

$u(x,t)dx\rightharpoonup u_{0}(x)dx+m\delta_{0}$

weakly as Radon measures.

Under these hypothesis, we will examine the behavior of $u$ near the blow up
point $(0,0)$. For this purpose, observe that the Keller-Segel system (1.2) is
invariant under the scaling

(1.7) $u^{\lambda}(x,t):=\lambda^{2}u(\lambda x,\lambda^{2}t),\quad\nabla
v^{\lambda}(x,t)=\lambda\nabla v(\lambda
x,\lambda^{2}t),\quad\forall\lambda>0.$

Moreover, because the spatial dimension is $2$, the $L^{1}$ norm of $u$ is
invariant under this scaling. The small scale structure of $u$ near $(0,0)$
can be revealed by examining the convergence of $u^{\lambda}$ as $\lambda\to
0$. This is the blow up procedure.

By applying Theorem 1.1, we get the following result about the blowing up
sequences $u^{\lambda}$.

######  Theorem 1.4.

Under hypothesis (H1)-(H3), the followings hold.

(i) Quantization:

There exists an $N\in\mathbb{N}$ such that $m=8\pi N$.

(ii) Blow-up limit and Self-similarity:

For any sequence $\lambda_{i}\to 0$, there exists a subsequence (not
relabelling) and $N$ distinct points $p_{j}\in\mathbb{R}^{2}$ such that, for
any $t<0$,

$u^{\lambda_{i}}(x,t)dx\rightharpoonup
8\pi\sum_{j=1}^{N}\delta_{\sqrt{-t}p_{j}}\quad\mbox{weakly as Radon
measures}.$

(iii) Renormalized energy:

If $N=1$, then $p_{1}=0$. If $N\geq 2$, then the $N$-tuple
$(p_{1},\cdots,p_{N})$ is a critical point of the renormalized energy

(1.8)
$\mathcal{W}(p_{1},\cdots,p_{N}):=-\frac{1}{4}\sum_{j=1}^{N}|p_{j}|^{2}+4\sum_{1\leq
j\neq k\leq N}\log|p_{j}-p_{k}|.$

######  Remark 1.5.

  1. (1)

In the converse direction, finite time blow up solutions with asymptotic
behavior described as in this theorem have been constructed in Seki-Sugiyama-
Velázquez [13] by using the matched asymptotics method.

  2. (2)

If $N=2$, critical points of $\mathcal{W}$ must be a symmetric pair, that is,
$(-p,p)$ for some $p\in\mathbb{R}^{2}$.

  3. (3)

In many parabolic equations, self-similarity of blow up limits is established
with the help of a monotonicity formula. However, in the above theorem, this
is proved by using the facts that, the ODE system (1.4) (if there is no
diffusion part) is the gradient flow of $W$, and the critical energy levels of
$\mathcal{W}$ are discrete.

  4. (4)

In the renomarlized energy $\mathcal{W}$, the first term comes from a self-
similar transformation. As explained in Remark 1.2, the second term comes from
a renormalization of the free energy $\mathcal{F}$.

  5. (5)

Because blow up limits are obtained by a compactness argument, we do not know
if the $N$-tuple $(p_{1},\cdots,p_{N})$ is unique.

Most of these results are not new, cf. [14, Theorem 14.2] or [15, Chapter 1].
One main difference is the method of the proof. In [14] and [15], Suzuki uses
the self-similar transformation, i.e. by considering similarity variables

$y=\frac{x}{\sqrt{-t}},\quad s=-\log(-t)$

and then taking the transformation

$z(y,s):=|t|u(x,t),\quad w(y,s):=v(x,t),$

one gets the system

(1.9) $\left\\{\begin{array}[]{ll}z_{s}-\Delta
z=-\mbox{div}\left[z\left(\nabla w+\frac{y}{2}\right)\right],\\\ -\Delta
w=z.\end{array}\right.$

Then the proof is reduced to analyse the large time behavior of $(z,w)$. Our
proof uses instead the blow up method, by considering the rescalings in (1.7)
and then analyse its convergence as $\lambda\to 0$. These two convergence
analysis are essentially equivalent, but the later one has an advantage: by
examining the scalings $(u^{\lambda},v^{\lambda})$ at $t=0$, we see that the
trace of $u$ at $t=0$, $u_{0}$, does not enter the singularity formation
mechanism on the mass level. More precisely, Theorem 1.4 implies that

######  Corollary 1.6.

In $Q_{1/2}^{-}$,

$u(x,t)=o\left(\frac{1}{(|x|+\sqrt{-t})^{2}}\right).$

In particular,

$u_{0}(x)=o\left(\frac{1}{|x|^{2}}\right).$

When the solution is radially symmetric, a rather precise asymptotic expansion
of $u_{0}$ near the origin has been given in Herrero-Velázquez [5]. But at
present it is still not known what further regularity on $u_{0}$ can be
obtained in the above general setting. Such a knowledge will be helpful in
defining the continuation after the blow up time.

###  1.3. Ancient solutions

######  Definition 1.7.

If $u$, $\nabla v$ are smooth in $\mathbb{R}^{2}\times(-\infty,0]$ and satisfy
(1.2), where the second equation is understood as

(1.10) $\nabla
v(x,t)=-\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\frac{x-y}{|x-y|^{2}}u(y,t)dt,\quad\forall(x,t)\in\mathbb{R}^{2}\times(-\infty,0],$

then it is called an ancient solution of (1.2).

Ancient solutions play an important role in the analysis of blow up phenomena
for many nonlinear parabolic equations.

Similar to the blow up analysis used in the proof of Theorem 1.4, we can
analyse the large scale structure of ancient solutions by the blowing down
analysis, that is, to consider the convergence of the sequence defined in
(1.7), but now with $\lambda\to+\infty$.

######  Theorem 1.8.

Assume that $(u,v)$ is an ancient solution, satisfying

$\sup_{t\leq 0}\int_{\mathbb{R}^{2}}u(x,t)dx<+\infty.$

Then the followings hold.

(i) Quantization:

There exists an $N\in\mathbb{N}$ such that

$\int_{\mathbb{R}^{2}}u(x,t)dx\equiv 8\pi N.$

(ii) Blow-down limit:

For any sequence $\lambda_{i}\to+\infty$, there exists a subsequence (not
relabelling) and $N$ distinct points $p_{j}\in\mathbb{R}^{2}$ such that, for
any $t<0$,

$u^{\lambda_{i}}(x,t)dx\rightharpoonup
8\pi\sum_{j=1}^{N}\delta_{\sqrt{-t}p_{j}}\quad\mbox{weakly as Radon
measures}.$

(iii) Renormalized energy:

If $N=1$, then $p_{1}=0$. If $N\geq 2$, then the $N$-tuple
$(p_{1},\cdots,p_{N})$ is a critical point of the renormalized energy
$\mathcal{W}$.

As can be seen, the statement is almost the same with Theorem 1.4. In fact,
the proof is also the same, which is still mainly an application of Theorem
1.1.

###  1.4. Entire solutions

Finally, we study the large scale structure of entire solutions.

######  Definition 1.9.

If $u$,$\nabla v$ are smooth in $\mathbb{R}^{2}\times\mathbb{R}$ and satisfy
(1.2), where the second equation is understood as (1.10), then it is called an
entire solution of (1.2).

Entire solutions appear as a suitable rescalings around blow up points, see
Theorem 9.1. Therefore it is the micro-model of singularity formations at blow
up points.

The large scale structure of entire solutions is described in the following
theorem.

######  Theorem 1.10.

Assume that $(u,v)$ is an entire solution, satisfying the finite mass
condition

(1.11) $\sup_{t\in\mathbb{R}}\int_{\mathbb{R}^{2}}u(x,t)dx<+\infty.$

Then as $\lambda\to+\infty$, for any $t$,

$u^{\lambda}(x,t)dx\rightharpoonup 8\pi\delta_{0}\quad\mbox{weakly as Radon
measures}.$

For a related property on weak solutions, called weak Liouville property, see
[15, Lemma 1.4].

Because the blow-down limit for entire solutions is time-independent, it is
natural to conjecture that the original entire solution is also time-
independent. Then the parabolic equation (1.2) is reduced to

$\left\\{\begin{aligned} &\Delta u-\mbox{div}\left(u\nabla v\right)=0,\\\
&-\Delta v=u.\end{aligned}\right.$

This is essentially the Liouville equation (1.1). By the Liouville theorem of
Chen-Li [2], we thus have

######  Conjecture 1.11.

Any nontrivial entire solution of (1.2) satisfying the finite mass condition
(1.11) has the form

$\left\\{\begin{aligned}
&u(x,t)\equiv\frac{8\lambda^{2}}{(\lambda^{2}+|x-\xi|^{2})^{2}},\\\
&v(x,t)\equiv-2\log\left(\lambda^{2}+|x-\xi|^{2}\right)+C,\end{aligned}\right.$

for some $\lambda>0$, $\xi\in\mathbb{R}^{2}$ and $C\in\mathbb{R}$.

Notations: Throughout this paper, we use the following notations.

  * •

A standard cut-off function $\psi$ subject to two bounded smooth domains
$\Omega_{1}\subset\Omega_{2}$ is a function satisfying $\psi\in
C_{0}^{\infty}(\Omega_{2})$, $0\leq\psi\leq 1$, $\psi\equiv 1$ in $\Omega_{1}$
and for any $k\geq 1$,

$|\nabla^{k}\psi|\leq C_{k}\mbox{dist}(\Omega_{1},\Omega_{2}^{c})^{-k}.$

  * •

A parabolic cylinder is $Q_{r}(x,t):=B_{r}(x)\times(t-r^{2},t+r^{2})$.

  * •

We use standard notation of parabolic Hölder spaces, e.g.
$C^{\alpha,\alpha/2}$ denotes a function which is $\alpha$-Hölder in $x$ and
$\alpha/2$-Hölder in $t$.

The organization of this paper will be as follows. Section 2-Section 7 is
devoted to the blow up analysis, in which the proof of Theorem 1.1 will be
given. We study first time singularities and prove Theorem 1.4 in Section 8.
In Section 9 we prove Theorem 1.10 on entire solutions and explain how to
construct entire solutions by rescalings. Finally, we give a brief remark on
boundary blow up points in Section 10.

##  2\. Setting

We will work in the following local setting, which is a little more general
than (1.2): $u,v\in C^{\infty}(Q_{1})$, $u>0$, and they satisfy

(2.1) $\left\\{\begin{array}[]{ll}u_{t}=\Delta
u-\mbox{div}\left[u\nabla(v+f)\right]+g,\\\ -\Delta v=u\end{array}\right.$

Here we assume

  * •

there exists a constant $M>0$ such that

(2.2) $\sup_{t\in(-1,1)}\int_{B_{1}}u(x,t)dx\leq M;$

  * •

$\nabla f$ and $g$ are two given functions, where for some $\alpha\in(0,1)$,
$\nabla f\in C^{\alpha,\alpha/2}(Q_{1})$, while $g\in L^{\infty}(Q_{1})$;

  * •

the second equation is understood as

(2.3) $\nabla
v(x,t)=-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}u(y,t)dy,\quad\forall
t\in(-1,1).$

The assumption on $g$ can be weakened, for example, by denoting
$p:=2/(1-\alpha)$, $g\in L^{p}(Q_{1})$ is sufficient for most arguments in
this paper. But that will lead to some technical issues, and we will not
pursue it here.

######  Remark 2.1.

The equation (1.9) in self-similar coordinates is a special case of (2.1).

######  Remark 2.2 (Localization).

The form of Eq. (2.1) is invariant under localization in the following sense.
Take an $\eta\in C_{0}^{\infty}(B_{1})$, $\eta\geq 0$ and define
$\widetilde{u}:=u\eta$. Then $\widetilde{u}$ still satisfies (2.1), with
$\nabla v$, $f$ and $g$ replaced respectively by

$\left\\{\begin{aligned}
&\nabla\widetilde{v}(x,t):=-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}\widetilde{u}(y,t)dy,\\\
&\nabla\widetilde{f}(x,t):=\nabla f(x,t)+\nabla v-\nabla\widetilde{v},\\\
&\widetilde{g}(x,t):=g(x,t)\eta(x)+u(x,t)\left[\nabla v(x,t)+\nabla
f(x,t)\right]\cdot\nabla\eta(x)\\\ &\qquad\quad-2\nabla
u(x,t)\cdot\nabla\eta(x)-u(x,t)\Delta\eta(x).\end{aligned}\right.$

As will be seen below, it is very convenient that solutions can be localized,
especially when studying concentration phenomena in (1.2).

##  3\. Two basic tools

In this section we recall two tools, symmetrization of test functions and
$\varepsilon$-regularity theorem. These two tools will be used a lot in this
paper

The first tool is symmetrization of test functions. For any $\psi\in
C^{2}(\mathbb{R}^{2})$, define

(3.1)
$\Theta_{\psi}\left(x,y\right):=\frac{x-y}{|x-y|^{2}}\cdot\left[\nabla\psi(x)-\nabla\psi(y)\right].$

It belongs to $L^{\infty}(\mathbb{R}^{2}\times\mathbb{R}^{2})$, with the
estimate

(3.2) $\|\Theta_{\psi}\|_{L^{\infty}(\mathbb{R}^{2}\times\mathbb{R}^{2})}\leq
C\|\psi\|_{C^{2}(\mathbb{R}^{2})}.$

This symmetrized function $\Theta_{\psi}$ appears in the following estimate.

######  Lemma 3.1.

For any $\psi\in C_{0}^{2}(B_{1})$,

$\displaystyle\frac{d}{dt}\int_{B_{1}}u\psi$ $\displaystyle=$
$\displaystyle\int_{B_{1}}u\Delta\psi+\int_{B_{1}}u\nabla
f\cdot\nabla\psi+\int_{B_{1}}g\psi$ $\displaystyle-$
$\displaystyle\frac{1}{4\pi}\int_{B_{1}}\int_{B_{1}}\Theta_{\psi}(x,y)u(x,t)u(y,t)dxdy.$

As a consequence, there exists a universal constant $C$ such that

$\displaystyle\left|\frac{d}{dt}\int_{B_{1}}u(x,t)\psi(x)dx\right|$
$\displaystyle\leq$ $\displaystyle
C\left(M^{2}\|\nabla^{2}\psi\|_{L^{\infty}(B_{1})}+M\|\Delta\psi\|_{L^{\infty}(B_{1})}\right)$
$\displaystyle+CM\|\nabla
f\|_{L^{\infty}(Q_{1})}\|\nabla\psi\|_{L^{\infty}(B_{1})}$
$\displaystyle+C\|g\|_{L^{\infty}(Q_{1})}\|\psi\|_{L^{\infty}(B_{1})},$

where $M$ is the mass bound in (2.2).

###### Proof.

The proof is similar to the standard Keller-Segel system case, see e.g. [14,
Lemma 5.1]. ∎

The second tool is $\varepsilon$-regularity theorem.

######  Theorem 3.2.

There exist two small constants $0<\varepsilon_{\ast}\ll\theta_{\ast}\ll 1$
(depending only on the constant $M$ in (2.2), $\|\nabla
f\|_{L^{\infty}(Q_{1})}$ and $\|g\|_{L^{\infty}(Q_{1})}$) such that, if
$(u,v)$ is a classical solution of (1.2) in $Q_{1}$, satisfying (2.2) and

(3.5) $\int_{B_{1}(0)}u(x,0)dx\leq\varepsilon_{\ast},$

then

(3.6) $\|u\|_{C^{1+\alpha,(1+\alpha)/2}(Q_{\theta_{\ast}})}\leq C.$

The proof is still similar to the one for the standard Keller-Segel system. It
will be given in Appendix A.

##  4\. Convergence of solutions

In this section we study the convergence of sequences of smooth solutions to
(2.1). We assume $u_{i}$ is a sequence of smooth solutions of (2.1) in
$Q_{1}$, satisfying:

  1. (1)

there exists a constant $M$ such that

(4.1) $\sup_{-1<t<1}\int_{B_{1}}u_{i}(x,t)dx\leq M;$

  2. (2)

$\nabla f_{i}$ are uniformly bounded in $C^{\alpha,\alpha/2}(Q_{1})$, and
$\nabla f_{i}\rightarrow\nabla f$ uniformly in $Q_{1}$;

  3. (3)

$g_{i}$ are uniformly bounded in $L^{\infty}(Q_{1})$, and $g_{i}$ converges to
$g$ $\ast$-weakly in $L^{\infty}(Q_{1})$.

Under these assumptions, we have

######  Theorem 4.1.

  1. (1)

There exists a family of Radon measures $\mu_{t}$ on $B_{1}$, $t\in(-1,1)$,
such that after passing to a subsequence of $i$,

(4.2) $u_{i}(x,t)dx\rightharpoonup\mu_{t}\quad\text{weakly as Radon
measures},\quad\forall t\in(-1,1);$

  2. (2)

$\mu_{t}$ is continuous in $t$ with respect to the weak topology;

  3. (3)

for any $t\in(-1,1)$, there exist $N(t)$ points $q_{j}(t)$,
$\varepsilon_{\ast}\leq m_{i}(t)\leq M$ and $0\leq\rho(t)\in L^{1}(B_{1})$
such that

$\mu_{t}=\sum_{j=1}^{N(t)}m_{j}(t)\delta_{q_{j}(t)}+\rho(x,t)dx.$

  4. (4)

the blow up locus
$\Sigma:=\cup_{t\in(-1,1)}\cup_{j=1}^{N(t)}\\{(q_{j}(t),t)\\}$ is relatively
closed in $Q_{1}$;

  5. (5)

$\rho\in C^{1+\alpha,(1+\alpha)/2}(Q_{1}\setminus\Sigma)$.

This covers most contents of Theorem 1.1, except that $m_{j}(t)=8\pi$ (the
$8\pi$ phenomena) and the dynamical law (1.4). These two properties will be
established below in Proposition 5.1 and Theorem 7.2 respectively.

###### Proof.

(1) Choose a countable, dense subset $\mathcal{D}\subset(-1,1)$. By the mass
bound (4.1) and a diagonal argument, we can choose a subsequence of $i$ so
that for each $t\in\mathcal{D}$, $u_{i}(t)dx$ converges weakly as Radon
measures to a limit Radon measure $\mu_{t}$. In the following, such a
subsequence of $i$ will be fixed, and by abusing notations, it will still be
written as $i$. We will prove that the convergence in (4.2) holds for this
subsequence.

For any $\psi\in C_{0}^{2}(B_{1})$, by Lemma 3.1, the linear functionals

(4.3) $\mathcal{L}_{i,t}(\psi):=\int_{B_{1}}u_{i}(x,t)\psi(x)dx$

are uniformly Lipschitz continuous on $(-1,1)$. After passing to a
subsequence, they converge to a limit function $\mathcal{L}_{t}(\psi)$
uniformly on $(-1,1)$, which is still Lipschitz continuous on $(-1,1)$. Of
course, if $t\in\mathcal{D}$, because $u_{i}(x,t)dx\rightharpoonup\mu_{t}$, we
must have

$\mathcal{L}_{t}(\psi)=\int_{B_{1}}\psi(x)d\mu_{t}.$

Therefore, because $\mathcal{D}$ is dense in $(-1,1)$, the above convergence
of $\mathcal{L}_{i,t}$ holds for the entire sequence $i$ and we do not need to
pass to a further subsequence of $i$.

For any $t\in(-1,1)$ (not necessarily in $\mathcal{D}$), for any further
subsequence $i^{\prime}$ with $u_{i^{\prime}}(x,t)dx$ converging weakly to a
Radon measure $\mu_{t}$, because $C_{0}^{2}(B_{1})$ is dense in
$C_{0}(B_{1})$, we must have

$\mu_{t}=\mathcal{L}_{t},$

where we have used Riesz representation theorem to identify Radon measures
with positive linear functionals on $C_{0}(B_{1})$. As a consequence,
$\mu_{t}$ is independent of the choice of subsequences, that is, for any $t$
and the entire sequence $i$,

$u_{i}(x,t)dx\rightharpoonup\mu_{t}.$

(2) By the Lipschitz continuity of $\mathcal{L}_{t}(\psi)$ (when $\psi\in
C_{0}^{2}(B_{1})$), $\mu_{t}$ is continuous in $t$ with respect to the weak
topology.

(3) For any $t\in(-1,1)$, define

$\Sigma_{t}:=\\{x:~{}~{}\mu_{t}(\\{x\\})\geq\varepsilon_{\ast}/2\\}.$

Hence there is an atom of $\mu_{t}$ at each point in $\Sigma_{t}$, whose mass
is at least $\varepsilon_{\ast}/2$. Because $\mu_{t}(B_{1})\leq M$,
$\Sigma_{t}$ is a finite set.

If $x\notin\Sigma_{t}$, then there exists an $r<1-|x|$ such that
$\mu_{t}(B_{r}(x))<\varepsilon_{\ast}/2$. By the convergence of
$u_{i}(x,t)dx$, for all $i$ large,

$\int_{B_{r}(x)}u_{i}(y,t)dy<\varepsilon_{\ast}.$

By Theorem 3.2, $u_{i}$ are uniformly bounded in
$C^{1+\alpha,(1+\alpha)/2}(Q_{\theta_{\ast}r}(x,t))$. Hence they converge in
$C(Q_{\theta_{\ast}r}(x,t))$. As a consequence, $\mu_{t}=\rho(x,t)dx$ in
$B_{\theta_{\ast}r}(x)$ for some function $\rho\in
C^{1+\alpha,(1+\alpha)/2}(Q_{\theta_{\ast}r}(x,t))$. In conclusion, $\mu_{t}$
is absolutely continuous with respect to Lebesgue measure outside
$\Sigma_{t}$.

(4) Since each $\Sigma_{t}$ is a finite set, we need only to show that
Claim. If $t_{i}\to t$, $x_{i}\in\Sigma_{t_{i}}$ and $x_{i}\to x$, then
$x\in\Sigma_{t}$.

Indeed, assume by the contrary that $x\notin\Sigma_{t}$, then there exists an
$r>0$ such that

$\mu_{t}(B_{r}(x))<\varepsilon_{\ast}/2.$

Take a cut-off function $\psi$ subject to $B_{r/2}(x)\subset B_{r}(x)$. By the
Lipschitz continuity of $\mathcal{L}_{t}(\psi)$, there exists a small
$\delta>0$ such that for any $s\in(t-\delta,t+\delta)$,

$\mu_{s}(B_{r/2}(x))<\varepsilon_{\ast}/2.$

This implies that for all of these $s$, $B_{r/2}(x)$ is disjoint from
$\Sigma_{s}$. This is a contradiction with the fact that
$x_{i}\in\Sigma_{t_{i}}$ and $t_{i}\to t$, and the claim follows.

(5) The regularity of $\rho$ in $Q_{1}\setminus\Sigma$ has already been
established in the proof of (3). ∎

Because

$\nabla
v_{i}(x,t)=-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}u_{i}(y,t)dy,$

by the convergence of $u_{i}(x,t)dx$ established in this theorem, we see
$\nabla v_{i}$ converges to

(4.4) $\nabla
v_{\infty}(x,t):=-\sum_{j=1}^{N(t)}\frac{m_{j}(t)}{2\pi}\frac{x-q_{j}(t)}{|x-q_{j}(t)|^{2}}-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}\rho(y,t)dy.$

By standard estimates on Newtonian potential, this convergence is strong in
$L^{q}(B_{1})$ for any $q<2$, and uniform in any compact set of
$B_{1}\setminus\Sigma_{t}$.

##  5\. The $8\pi$ phenomena

This section is devoted to the proof of the following result, which says that
the mass of each atom in $\mu_{t}$ is exactly $8\pi$. We keep the notations
used in Theorem 4.1.

######  Proposition 5.1.

For any $t\in(-1,1)$ and each $q_{j}(t)\in\Sigma_{t}$, $m_{j}(t)=8\pi$.

###### Proof.

Take an arbitrary $t\in(-1,1)$ and a point $q_{j}(t)\in\Sigma_{t}$. Denote
$m:=m_{j}(t)$. The remaining proof is divided into three steps.

Step 1. Reduction to a simple setting. For any $r>0$ and
$s\in((t+1)/r^{2},(1-t)/r^{2})$, define the Radon measure

$\widehat{\mu}_{s}^{r}(A):=\mu_{t+r^{2}s}\left(q_{j}(t)+rA\right),\quad\forall
A\subset\mathbb{R}^{2}~{}~{}\text{Borel}.$

Correspondingly, define

$\left\\{\begin{aligned}
&u_{i}^{r}(y,s):=r^{2}u_{i}\left(q_{j}(t)+ry,t+r^{2}s\right),\\\ &\nabla
v_{i}^{r}(y,s):=r\nabla v_{i}\left(q_{j}(t)+ry,t+r^{2}s\right),\\\ &\nabla
f_{i}^{r}(y,s):=r\nabla f_{i}\left(q_{j}(t)+ry,t+r^{2}s\right),\\\
&g_{i}^{r}(y,s):=r^{2}g_{i}\left(q_{j}(t)+ry,t+r^{2}s\right).\end{aligned}\right.$

Then $u_{i}^{r}(y,s)dy$ converges to $\widehat{\mu}_{s}^{r}$ in the sense of
Theorem 4.1.

As in the proof of Theorem 4.1, we can take a subsequence $r_{i}\to 0$ so that
for any $s\in\mathbb{R}$, $\widehat{\mu}_{s}^{r_{i}}$ converges weakly to
$\widehat{\mu}_{s}$ as finite Radon measures on $\mathbb{R}^{2}$. Then by a
diagonal argument, we can take a further subsequence of $i$ (not relabelling)
so that $u_{i}^{r_{i}}(y,s)dy$ converges to $\widehat{\mu}_{s}$ as finite
Radon measures on $\mathbb{R}^{2}$. In the following we denote
$\widehat{u}_{i}:=u_{i}^{r_{i}}$.

Since $\Sigma_{t}$ is a finite set, there exists an $r>0$ such that
$\Sigma_{t}\cap B_{r}(q_{j}(t))=\\{q_{j}(t)\\}$. Then by the structure of
$\mu_{t}$ in Theorem 4.1 (3), we get

$\widehat{\mu}^{r}_{0}=m\delta_{0}+r^{2}\rho(rx,0)dx.$

Because $\rho\in L^{1}(B_{1})$, sending $r\to 0$ we obtain

(5.1) $\widehat{\mu}_{0}=m\delta_{0}.$

Step 2. Clearing out. Take a cut-off function $\psi_{1}$ subject to
$\left(B_{7/8}\setminus B_{1/4}\right)\subset\left(B_{1}\setminus
B_{1/8}\right)$. By Lemma 3.1 and our assumption on the uniform bound on
$\nabla f_{i}$ and $g_{i}$,

$\left|\frac{d}{dt}\int_{B_{1}}\widehat{u}_{i}(x,t)\psi_{1}(x)dx\right|\leq
C\left(M+M^{2}\right)+o_{i}(1).$

Combining this inequality with (5.1), we deduce that for any $t$ satisfying

$|t|\leq t_{\ast}:=\frac{\varepsilon_{\ast}}{4C\left(M+M^{2}\right)},$

it holds that

$\int_{B_{7/8}\setminus
B_{1/4}}\widehat{u}_{i}(x,t)dx\leq\frac{\varepsilon_{\ast}}{3}.$

By Theorem 3.2 and Arzela-Ascoli theorem, $\widehat{u}_{i}$ converges
uniformly to a function $\widehat{\rho}$ in $\left(B_{3/4}\setminus
B_{3/8}\right)\times(-t_{\ast},t_{\ast})$, that is, $\widehat{u}_{i}$ does not
blow up in this annular cylinder.

Step 3. Second momentum calculation. Take a cut-off function $\psi_{2}$
subject to $B_{1/2}\subset B_{5/8}$. Consider the second momentum

$M(s):=\int_{B_{1}}|x|^{2}\psi_{2}(x)d\widehat{\mu}_{s}.$

We claim that
Claim. $M(s)$ is differentiable in $(-t_{\ast},t_{\ast})$, and

(5.2) $M^{\prime}(0)=4m-\frac{1}{2\pi}m^{2}.$

By definition $M(s)\geq 0$, while by (5.1), $M(0)=0$. The differentiability
then implies that

$M^{\prime}(0)=0.$

This identity combined with (5.2) implies that $m=8\pi$. Hence the proof of
this proposition is complete, provided this claim is true.

Proof of the claim. A direct calculation shows that

$\Theta_{|x|^{2}\psi_{2}(x)}(x,y)=\frac{1}{2\pi}\psi_{2}(x)\psi_{2}(y)+\Phi(x,y),$

where

(5.3) $\displaystyle\Phi(x,y)$ $\displaystyle:=$
$\displaystyle\frac{1}{2\pi}\frac{x-y}{|x-y|^{2}}\cdot\left[x\psi_{2}(x)\left(1-\psi_{2}(y)\right)-y\psi_{2}(y)\left(1-\psi_{2}(x)\right)\right]$
$\displaystyle+$
$\displaystyle\frac{1}{4\pi}\frac{x-y}{|x-y|^{2}}\cdot\left[|x|^{2}\nabla\psi_{2}(x)-|y|^{2}\nabla\psi_{2}(y)\right].$

Denote

$M_{i}(s):=\int_{B_{1}}|x|^{2}\psi_{2}(x)\widehat{u}_{i}(x,s)dx.$

By (3.1), we obtain

$\displaystyle M_{i}^{\prime}(s)$ $\displaystyle=$ $\displaystyle
4\int_{B_{1}}\psi_{2}(x)\widehat{u}_{i}(x,s)dx-\frac{1}{2\pi}\left[\int_{B_{1}}\psi_{2}(x)\widehat{u}_{i}(x,s)dx\right]^{2}$
$\displaystyle+$
$\displaystyle\int_{B_{1}}\left[|x|^{2}\Delta\psi_{2}(x)+4x\cdot\nabla\psi_{2}(x)\right]\widehat{u}_{i}(x,s)dx$
$\displaystyle+$
$\displaystyle\int_{B_{1}}\nabla\widehat{f}_{i}(x,s)\cdot\nabla\left(|x|^{2}\psi_{2}(x)\right)\widehat{u}_{i}(x,s)dx+\int_{B_{1}}\widehat{g}_{i}(x,t)|x|^{2}\psi_{2}(x)dx$
$\displaystyle+$
$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\Phi(x,y)\widehat{u}_{i}(x,s)\widehat{u}_{i}(y,s)dxdy$
$\displaystyle=:$
$\displaystyle\mathrm{I}_{i}+\mathrm{II}_{i}+\mathrm{III}_{i}+\mathrm{IV}_{i}+\mathrm{V}_{i}+\mathrm{VI}_{i}.$

Let us analyse the convergence of these six terms one by one.

  1. (1)

By the convergence of $\widehat{u}_{i}(x,s)dx$, $\mathrm{I}_{i}$ converges
uniformly in $(-t_{\ast},t_{\ast})$ to

$4\int_{B_{1}}\psi_{2}(x)d\widehat{\mu}_{s}(x).$

  2. (2)

In the same way, $\mathrm{II}_{i}$ converges uniformly in
$(-t_{\ast},t_{\ast})$ to

$-\frac{1}{2\pi}\left[\int_{B_{1}}\psi_{2}(x)d\widehat{\mu}_{s}(x)\right]^{2}.$

  3. (3)

Similarly, $\mathrm{III}_{i}$ converges uniformly in $(-t_{\ast},t_{\ast})$ to

$\int_{B_{1}}\left[|x|^{2}\Delta\psi_{2}(x)+4x\cdot\nabla\psi_{2}(x)\right]d\widehat{\mu}_{s}.$

  4. (4)

Because $\|\nabla\widehat{f}_{i}\|_{L^{\infty}}\leq Cr_{i}$, $\mathrm{IV}_{i}$
converges uniformly to $0$.

  5. (5)

Because $\|\widehat{g}_{i}\|_{L^{\infty}(Q_{1})}\leq Cr_{i}^{2}$,
$\mathrm{V}_{i}$ converges uniformly to $0$.

  6. (6)

Concerning $\mathrm{VI}_{i}$, first because we have established the uniform
convergence of $\widehat{u}_{i}$ in $\left(B_{3/4}\setminus
B_{3/8}\right)\times(-t_{\ast},t_{\ast})$ in Step 2, an application of the
dominated convergence theorem gives us the uniform in
$s\in(-t_{\ast},t_{\ast})$ convergence

$\displaystyle\int_{B_{3/4}\setminus B_{3/8}}\int_{B_{3/4}\setminus
B_{3/8}}\Phi(x,y)\widehat{u}_{i}(x,s)\widehat{u}_{i}(y,s)dxdy$
$\displaystyle\to\int_{B_{3/4}\setminus B_{3/8}}\int_{B_{3/4}\setminus
B_{3/8}}\Phi(x,y)d\widehat{\mu}_{s}(x)d\widehat{\mu}_{s}(y).$

Next, by the form of $\Phi$ in (5.3) and using the facts that $\psi_{2}\equiv
0$ in $B_{1/2}$, $\psi_{2}\equiv 1$ outside $B_{5/8}$, we see $\Phi$ is
continuous outside $\left(B_{3/4}\setminus
B_{3/8}\right)\times\left(B_{3/4}\setminus B_{3/8}\right)$.111In fact, the
only trouble appears on the diagonal $\\{x=y\\}$, but when $(x,y)$ belongs to
this domain, it is directly checked that $\Phi(x,y)=0$ if $|x-y|\leq 1/16$.
Then by the weak convergence of $\widehat{u}_{i}$, the remaining part in
$\mathrm{VI}_{i}$ converges uniformly in $(-t_{\ast},t_{\ast})$ to

$\int\int_{\left(\mathbb{R}^{2}\times\mathbb{R}^{2}\right)\setminus\left[\left(B_{3/4}\setminus
B_{3/8}\right)\times\left(B_{3/4}\setminus
B_{3/8}\right)\right]}\Phi(x,y)d\widehat{\mu}_{s}(x)d\widehat{\mu}_{s}(y).$

In conclusion, $\mathrm{VI}_{i}$ converges uniformly to

$\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\Phi(x,y)d\widehat{\mu}_{s}(x)d\widehat{\mu}_{s}(y).$

Putting these six facts together, we deduce that $M_{i}(s)$ converges to
$M(s)$ in $C^{1}(-t_{\ast},t_{\ast})$. Moreover,

$\displaystyle M^{\prime}(s)$ $\displaystyle=$ $\displaystyle
4\int_{B_{1}}\psi_{2}(x)d\widehat{\mu}_{s}(x)-\frac{1}{2\pi}\left[\int_{B_{1}}\psi_{2}(x)d\widehat{\mu}_{s}(x)\right]^{2}$
$\displaystyle+$
$\displaystyle\int_{B_{1}}\left[|x|^{2}\Delta\psi_{2}(x)+4x\cdot\nabla\psi_{2}(x)\right]d\widehat{\mu}_{s}$
$\displaystyle+$
$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\Phi(x,y)d\widehat{\mu}_{s}(x)d\widehat{\mu}_{s}(y).$

Evaluating this identity at $s=0$, and then using (5.1) and the definition of
$\psi_{2}$, we get (5.2). The proof of the claim is thus complete. ∎

######  Remark 5.2.

By the clearing out property established in the above proof, in the following
we will assume without loss generality that $\Sigma\subset
B_{1/2}\times(-1,1)$.

##  6\. Local structure of blow up locus

In this section we study the local structure of the blow up locus, $\Sigma$.
Since we are only interested in interior case, by Remark 5.2, we will assume
$\Sigma\subset B_{1/2}\times(-1,1)$. The main result of this section is

######  Proposition 6.1.

For any $r<1$, there exists an $N\in\mathbb{N}$ and $N$ functions
$\xi_{i}:[-r,r]\mapsto B_{1/2}$ which are $1/2$-Hölder, such that
$\Sigma\cap\\{|t|\leq r\\}\subset\cup_{i=1}^{N}\\{(\xi_{i}(t),t)\\}$.

First we need two technical lemmas, which are global version of Proposition
6.1. Below they will be used to study blow up limits of $\mu_{t}$. The first
lemma is about a backward problem.

######  Lemma 6.2.

Suppose a family of Radon measures on $\mathbb{R}^{2}$, $\mu_{t}$,
$t\in(-\infty,0]$ is a weak limit given in Theorem 4.1, from a sequence of
solutions $u_{i}$, where in the equation of $u_{i}$, the two terms $\nabla
f_{i}$ and $g_{i}$ converge to $0$ uniformly. Assume there exists a constant
$M$ such that

$\mu_{t}(\mathbb{R}^{2})\leq M,\quad\forall t\in(-\infty,0].$

If $\mu_{0}=8\pi\delta_{0}$, then for any $t\leq 0$, $\mu_{t}=8\pi\delta_{0}$.

###### Proof.

Step 1. Let $\rho$ be the regular part in the Lebesgue decomposition of
$\mu_{t}$ as in Theorem 4.1. We claim that $\rho\equiv 0$.

In fact, in the open set
$\Omega:=\left(\mathbb{R}^{2}\times(-\infty,0)\right)\setminus\Sigma$, $\rho$
is a classical solution of a linear parabolic equation (i.e. the first
equation in (1.2)). By Theorem 4.1, $\Omega$ is connected. By our assumption,
$\rho(x,0)=0$ in $\mathbb{R}^{2}\setminus\\{0\\}$. Hence we can apply the
strong maximum principle, which implies that $\rho\equiv 0$ in $\Omega$.
Because for each $t$, $\rho(\cdot,t)\in L^{1}(\mathbb{R}^{2})$ and
$\Sigma_{t}$ is a finite set, we get the claim.

Step 2. For each $R>0$, take a cut-off function $\psi_{R}$ subject to
$B_{R}\subset B_{2R}$. By Lemma 3.1 and Theorem 4.1,

$\left|\frac{d}{dt}\int_{\mathbb{R}^{2}}\psi_{R}(x)d\mu_{t}(x)\right|\leq\frac{C(M+M^{2})}{R^{2}}.$

Integrating this inequality in $t$, we obtain

$\left|\int_{\mathbb{R}^{2}}\psi_{R}(x)d\mu_{t}(x)-8\pi\right|\leq\frac{C(M+M^{2})}{R^{2}}|t|.$

Letting $R\to+\infty$, we deduce that

(6.1) $\mu_{t}(\mathbb{R}^{2})=8\pi,\quad\forall t\in(-\infty,0].$

Combining this fact with Proposition 5.1 and the result in Step 1, we deduce
that for each $t$, there exists a $q(t)\in\mathbb{R}^{2}$ such that
$\mu_{t}=8\pi\delta_{q(t)}$.

Step 3. Take the cut-off function $\psi_{R}$ as before and define

$M_{R}(t):=\int_{\mathbb{R}^{2}}|x|^{2}\psi_{R}(x)d\mu_{t}(x)=8\pi|q(t)|^{2}\psi_{R}(q(t)).$

By the same derivation of (5) as in the proof of Proposition 5.1, we get

$\displaystyle M_{R}^{\prime}(t)$ $\displaystyle=$ $\displaystyle
4\int_{\mathbb{R}^{2}}\psi_{R}(x)d\mu_{t}(x)-\frac{1}{2\pi}\left[\int_{\mathbb{R}^{2}}\psi_{R}(x)d\mu_{t}(x)\right]^{2}$
$\displaystyle+$
$\displaystyle\int_{\mathbb{R}^{2}}\left[|x|^{2}\Delta\psi_{R}(x)+4x\cdot\nabla\psi_{R}(x)\right]d\mu_{t}(x)$
$\displaystyle+$
$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\Phi(x,y)d\mu_{t}(x)d\mu_{t}(y),$

where $\Phi(x,y)$ is defined in (5.3).

For any $T>0$, if $R$ has been chosen to be large enough so that for any
$t\in[-T,0]$, $q(t)\in B_{R}$, by using the fact that
$\mu_{t}=8\pi\delta_{q(t)}$ and the form of $\Phi$ etc., we deduce that

$M_{R}^{\prime}(t)\equiv 0,\quad\forall t\in[-T,0].$

By the assumption that $q(0)=0$, we also have $M_{R}(0)=0$, so $M_{R}(t)\equiv
0$. This then implies that $q(t)\equiv 0$. ∎

The second lemma is about a forward problem.

######  Lemma 6.3.

Suppose a family of Radon measures on $\mathbb{R}^{2}$, $\mu_{t}$,
$t\in[0,+\infty)$ is a weak limit given in Theorem 4.1, from a sequence of
solutions $u_{i}$, where in the equation of $u_{i}$, the two terms $\nabla
f_{i}$ and $g_{i}$ converge to $0$ uniformly. Assume there exists a constant
$M$ such that

$\mu_{t}(\mathbb{R}^{2})\leq M,\quad\forall t\in[0,+\infty).$

If $\mu_{0}=8\pi\delta_{0}$, then for any $t\geq 0$, $\mu_{t}=8\pi\delta_{0}$.

###### Proof.

First as in Step 2 in the proof of the previous lemma, we still have (6.1)
(for any $t\in[0,+\infty)$). Combining this fact with Theorem 4.1, we see for
any $t>0$,

  * •

Alternative I: either there exists a $q(t)\in\mathbb{R}^{2}$ such that
$\mu_{t}=8\pi\delta_{q(t)}$ ;

  * •

Alternative II: or there exists a positive smooth function $\rho(t)$ on
$\mathbb{R}^{2}$ such that $\mu_{t}=\rho(x,t)dx$.

Define

$T_{\ast}:=\sup\\{t:~{}\mbox{Alternative I holds at }t\\}.$

Similar to Step 1 in the proof of the previous lemma, we deduce that if
$T_{\ast}>0$, then for any $t\in[0,T_{\ast}]$, Alternative I holds.

To finish the proof, we need to show that

(6.2) $T_{\ast}=+\infty.$

Proof of (6.2) Assume by the contrary that $T_{\ast}<+\infty$. Then in
$\mathbb{R}^{2}\times(T_{\ast},+\infty)$, $\mu_{t}=\rho(x,t)dx$, where $\rho$
is a smooth solution of (1.2). For any $t>T_{\ast}$ and $R>1$, take a cut-off
function $\psi_{R}$ subject to $B_{R}\subset B_{2R}$ and define

$M_{R}(t):=\int_{\mathbb{R}^{2}}|x|^{2}\psi_{R}(x)\rho(x,t)dx,\quad\mathcal{E}_{R}(t):=\int_{B_{R}^{c}}\rho(x,t)dx.$

By (3.1), we have

$\displaystyle M_{R}^{\prime}(t)$ $\displaystyle=$ $\displaystyle
4\int_{\mathbb{R}^{2}}\psi_{R}(x)\rho(x,t)dx-\frac{1}{2\pi}\left[\int_{\mathbb{R}^{2}}\psi_{R}(x)\rho(x,t)dx\right]^{2}$
$\displaystyle+$
$\displaystyle\int_{\mathbb{R}^{2}}\left[4x\cdot\nabla\psi_{R}(x)+|x|^{2}\Delta\psi_{R}(x)\right]\rho(x,t)dx$
$\displaystyle+$
$\displaystyle\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\frac{(x-y)\cdot
x}{|x-y|^{2}}\psi_{R}(x)\left[1-\psi_{R}(y)\right]\rho(x,t)\rho(y,t)dxdy$
$\displaystyle-$
$\displaystyle\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\frac{(x-y)\cdot
y}{|x-y|^{2}}\psi_{R}(y)\left[1-\psi_{R}(x)\right]\rho(x,t)\rho(y,t)dxdy$
$\displaystyle-$
$\displaystyle\frac{1}{4\pi}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\frac{x-y}{|x-y|^{2}}\cdot\left[|x|^{2}\nabla\psi_{R}(x)-|y|^{2}\nabla\psi_{R}(y)\right]\rho(x,t)\rho(y,t)dxdy.$

By (6.1),

$\int_{\mathbb{R}^{2}}\rho(x,t)dx=8\pi.$

Plugging this identity into (6), we obtain

$|M_{R}^{\prime}(t)|\leq C\mathcal{E}_{R}(t).$

Because $M_{R}(T_{\ast})=0$, this inequality implies that for any fixed
$t>T_{\ast}$,

$M_{R}(t)\leq C(t-T_{\ast})\max_{T_{\ast}\leq s\leq t}\mathcal{E}_{R}(s).$

By the definition of $\psi_{R}$, $M_{R}(t)$ is non-decreasing in $R$. On the
other hand, we have

$\lim_{R\to+\infty}\max_{T_{\ast}\leq s\leq t}\mathcal{E}_{R}(s)=0.$

Thus

$\displaystyle M_{R}(t)$ $\displaystyle\leq$
$\displaystyle\lim_{R\to+\infty}M_{R}(t)$ $\displaystyle\leq$ $\displaystyle
C(t-T_{\ast})\lim_{R\to+\infty}\max_{T_{\ast}\leq s\leq t}\mathcal{E}_{R}(s)$
$\displaystyle=$ $\displaystyle 0.$

This is possible only if $\mu_{t}$ is a Dirac measure at $0$, which is a
contradiction with our assumption. In other words, (6.2) must hold.

Finally, once we have shown that $\mu_{t}=8\pi\delta_{q(t)}$ for any $t>0$,
following the argument in Step 3 in the proof of the previous lemma, we deduce
that for any $t>0$, $M_{R}(t)=0$ for all $R$ large enough. This implies that
$q(t)\equiv 0$. ∎

######  Remark 6.4.

The above proof can be used to show that, there does not exist forward self-
similar solutions of the Keller-Segel system (1.2) with mass not smaller than
$8\pi$. More precisely, there does not exist solutions satisfying the
following three conditions:

  1. (1)

$u\in C^{2,1}(\mathbb{R}^{n}\times(0,+\infty))$ is a solution of (1.2);

  2. (2)

it is forward self-similar in the sense that

$u(\lambda x,\lambda^{2}t)=\lambda^{2}u(x,t),\quad\forall\lambda>0;$

  3. (3)

the total mass satisfies

$\int_{\mathbb{R}^{2}}u(x,t)dx\equiv m\geq 8\pi.$

Indeed, under these assumptions, the initial condition is $u(0)=m\delta_{0}$.
The above calculation in the proof of (6.2) then leads to a contradiction. On
the other hand, if $m<8\pi$, such forward self-similar solutions do exist, see
Naito-Suzuki [11].

For any $(q_{j}(t),t)\in\Sigma$, define $\widehat{\mu}^{r}_{s}$ as in the
first step in the proof of Proposition 5.1, that is,

$\widehat{\mu}_{s}^{r}(A):=\mu_{t+r^{2}s}(q_{j}(t)+rA),\quad\forall
A\subset\mathbb{R}^{2}\quad\text{Borel}.$

We have the following charaterization of its limit as $r\to 0$.

######  Lemma 6.5.

For any $s\in\mathbb{R}$, $\widehat{\mu}^{r}_{s}\rightharpoonup
8\pi\delta_{0}$ as $r\to 0$,

###### Proof.

First, as in the proof of Theorem 4.1, we can assume (after passing to a
subsequence of $r\to 0$) that, for any $s\in\mathbb{R}$,
$\widehat{\mu}^{r}_{s}$ converges weakly to $\mu^{0}_{s}$ as Radon measures.
Moreover, $\mu^{0}_{s}$, as a family of Radon measures, is a weak limit of a
sequence of solutions $\widehat{u}_{i}$, where in the equation of
$\widehat{u}_{i}$, the two terms $\nabla\widehat{f}_{i}$ and $\widehat{g}_{i}$
converge to $0$ uniformly.

By Theorem 4.1 and Proposition 5.1, for any $R>0$,

$\displaystyle\widehat{\mu}_{0}^{r}(B_{R})$ $\displaystyle=$
$\displaystyle\mu_{0}\left(B_{Rr}(q_{j}(t))\right)$ $\displaystyle=$
$\displaystyle 8\pi+\int_{B_{Rr}(q_{j}(t))}\rho(x,t)dx\to
8\pi\quad\text{as}~{}r\to 0.$

Thus $\mu^{0}_{0}=8\pi\delta_{0}$.

Then by Lemma 6.2, for any $t\leq 0$, $\mu_{t}^{0}=8\pi\delta_{0}$, and by
Lemma 6.3, for any $t\geq 0$, $\mu_{t}^{0}=8\pi\delta_{0}$. ∎

######  Corollary 6.6.

For any $(q_{j}(t),t)\in\Sigma$, as $r\to 0$, $\Sigma\cap Q_{r}(q_{j}(t),t)$
belongs to an $o(r)$ neighborhood of $\\{q_{j}(t)\\}\times(t-r^{2},t+r^{2})$.

Finally, let us prove Proposition 6.1. Recall that we always assume that
$\Sigma\subset B_{1/2}\times(-1,1)$.

###### Proof of Proposition 6.1.

We need a fact about the extension of Hölder functions: If $f$ is an
$\alpha$-Hölder continuous function (for any $\alpha\in(0,1)$) on a closed
subset $D\subset\mathbb{R}^{n}$, then it can be extended to the whole
$\mathbb{R}^{n}$. Indeed, if $f$ is scalar, the extension can be defined as

$\widetilde{f}(x):=\inf_{y\in D}\left[f(y)+L|x-y|^{\alpha}\right],$

where $L$ is the Hölder constant of $f$, while if $f$ is vector valued, we can
take this extension for each component.

Take an arbitrary $t\in[-r,r]$ and $q_{j}(t)\in\Sigma_{t}$. By Theorem 4.1 and
Proposition 5.1, there exists an $r>0$ such that

$\left\\{\begin{aligned} &\Sigma_{t}\cap B_{r}(q_{j}(t))=\\{q_{j}(t)\\},\\\
&8\pi\leq\mu_{t}(B_{r}(q_{j}(t)))<8\pi+\varepsilon_{\ast}/4.\end{aligned}\right.$

By the weak continuity of $\mu_{t}$, there exists a $\delta>0$ such that for
any $s\in(t-\delta,t+\delta)$,

$8\pi-\varepsilon_{\ast}/2<\mu_{s}(B_{r/2}(q_{j}(t)))<8\pi+\varepsilon_{\ast}/2.$

By Proposition 5.1, for these $s$, there exists at most one atom of $\mu_{s}$
in $B_{r/2}(q_{j}(t))$.

Let $I\subset(t-\delta,t+\delta)$ be the set containing those $s$ such that
$\mu_{s}$ consists of an atom inside $B_{r/2}(q_{j}(t))$. Because $\Sigma$ is
a closed set, $I$ is relatively closed in $(t-\delta,t+\delta)$. Then

$\Sigma\cap\left(B_{r}(q_{j}(t))\times(t-\delta,t+\delta)\right)=\\{(q_{j}(s),s):~{}~{}s\in
I\\}.$

By Corollary 6.6, $q_{j}$ is $1/2$-Hölder on $I$. We extend it to
$(t-\delta,t+\delta)$ by the above remark about extension of Hölder continuous
functions. Because $\Sigma_{t}$ consists of $N(t)$ points, the number of these
curves is $N(t)$. Moreover, these curves do not intersect (perhaps after
shrinking $\delta$).

Because $\Sigma$ is a closed set, the above discussion implies that for any
$t\in[-r,r]$, there exists an interval $(t-\delta(t),t+\delta(t))$ such that
$\Sigma\cap(t-\delta(t),t+\delta(t))$ is contained in these $1/2$-Hölder
curves emanating from $\Sigma_{t}$. These open intervals form a covering of
$[-r,r]$. Take a finite sub-covering from it, denoted by
$\\{(t_{j}^{-},t_{j}^{+})\\}$. Let $N_{j}:=N((t_{j-1}+t_{j})/2)$, so
$\Sigma\cap(t_{j-1},t_{j})$ is contained in $N_{j}$ curves.

In the intersection of two neighboring intervals, that is,
$(t_{j}^{+},t_{j+1}^{-})$, define a continuous, nonnegative function
$\eta_{j}$ such that $\eta_{j}(t_{j}^{+})=1$, $\eta_{j}(t_{j+1}^{-})=0$. In
$(t_{j}^{-},t_{j}^{+})$, $\Sigma$ is contained in $N_{j}$ curves
$\gamma_{k}^{1}:=\\{(q_{k}^{1}(t),t)\\}$, $k=1$, …, $N_{j}$, while in
$(t_{j+1}^{-},t_{j+1}^{+})$, $\Sigma$ is contained in $N_{j+1}$ curves
$\gamma_{\ell}^{2}:=\\{(q_{\ell}^{2}(t),t)\\}$, $\ell=1$, …, $N_{j+1}$. In
each group, the distance between any curve and other curves has a positive
lower bound. Because these curves are graphs over the $t$-axis, each
$\Gamma_{k}^{1}$ has nonempty intersection with at most one
$\Gamma_{\ell}^{2}$, and vice versa. If two such curves, say $\gamma_{k}^{1}$
and $\gamma_{\ell}^{2}$, do intersect, define their joint as

$q_{k\ell}^{12}(t)=\eta_{j}(t)q_{k}^{1}(t)+\left[1-\eta_{j}(t)\right]q_{\ell}^{2}(t).$

If there is no curves with such nonempty intersection, we stop here (either in
the left or the right end). In this way, we glue all of those local
$1/2$-Hölder curves to be defined on a maximal interval. Some of these maximal
curves are not defined on the entire interval $[-r,r]$, but we can extend it
suitably to $[-r,r]$ so that after extension, they still do not intersect with
each other. These curves are the desired one. Moreover, from the construction,
we see that the number of these curves is at most $\sum_{j}N_{j}$. ∎

######  Remark 6.7.

It is possible that $N(t)$ (the number of points in $\Sigma_{t}$) is not
constant in time. If finite time blow up solutions can be continued past the
singular time, then it implies that $N(t)$ will jump up at the singular time.
On the other hand, it seems highly impossible that a singularity can disappear
once it has been created.222In a global setting, this is exactly what Lemma
6.3 says, which claims that if the initial value is $8\pi\delta_{0}$, then the
solution will always be $8\pi\delta_{0}$, that is, it cannot be smoothed.
Hence it is natural to conjecture that $N(t)$ is non-decreasing and right
continuous in $t$, but we do not know how to prove this.

##  7\. Limit equations

By the smooth convergence of $u_{i}$ in $Q_{1}\setminus\Sigma$, we see $\rho$
satisfies

(7.1) $\rho_{t}-\Delta\rho=-\mbox{div}\left[\rho\left(\nabla v_{\infty}+\nabla
f\right)\right]+g\quad\mbox{in}~{}~{}Q_{1}\setminus\Sigma.$

Here by Proposition 5.1, the definition of $\nabla v_{\infty}$ (cf. (4.4)) is

(7.2) $\nabla
v_{\infty}(x,t)=-4\sum_{j=1}^{N(t)}\frac{x-q_{j}(t)}{|x-q_{j}(t)|^{2}}-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}\rho(y,t)dy.$

In this section, we will establish the dynamical law for $\mu_{t}$ only in the
case that $\rho$ is smooth enough and it satisfies (7.1) in the entire
$Q_{1}$. The reason to add this assumption is that, only under this
assumption, we can show that the blow up locus $\Sigma$ is regular in the
following sense.

######  Lemma 7.1.

If $\rho\in C(Q_{1})$, then there exists an $N\in\mathbb{N}$ such that
$N(t)\equiv N$, that is, for any $t\in(-1,1)$, $\Sigma_{t}$ consists of
exactly $N$ distinct points, $q_{1}(t)$, $\dots$, $q_{N}(t)$. Moreover, each
$q_{j}(t)$ is continuous.

###### Proof.

Because $\rho$ is continuous, $\rho(x,t)dx$, viewed as Radon measures, is
continuous in the weak topology. By Theorem 4.1 (2), $\mu_{t}$ is also
continuous in the weak topology. Hence by Theorem 4.1 (3) and Proposition 5.1,

$\mu_{t}-\rho(x,t)dx=\sum_{j=1}^{N(t)}8\pi\delta_{q_{j}(t)}$

is continuous in the weak topology. By testing this weak continuity with
suitable compactly supported, continuous functions, we deduce that its total
mass, which is $8\pi N(t)$, is constant in time. In other words, there exists
an $N\in\mathbb{N}$ such that for any $t\in(-1,1)$, $\Sigma_{t}$ consists of
exactly $N$ distinct points, $q_{1}(t)$, $\dots$, $q_{N}(t)$. The continuity
of $q_{j}(t)$ also follows from this weak continuity. ∎

Next we establish the dynamical law for these $q_{j}(t)$ under this
assumption.

######  Theorem 7.2.

For each $j=1$, …, $N$, $q_{j}$ is differentiable in $(-1,1)$. Moreover, it
satisfies

(7.3) $q_{j}^{\prime}(t)=4\sum_{k\neq
j}\dfrac{q_{k}(t)-q_{j}(t)}{|q_{k}(t)-q_{j}(t)|^{2}}+\frac{1}{2\pi}\int_{B_{1}}\frac{y-q_{j}(t)}{|y-q_{j}(t)|^{2}}\rho(y,t)dy+\nabla
f(q_{j}(t),t).$

If $f=0$, this is (1.4) in Theorem 1.1. The proof of Theorem 1.1 is thus
complete.

Before going into the details of the proof, we explain briefly the idea. The
main task is to single out the interaction between different atoms and the
effect of the diffusion part $\rho$ to these atoms. In the following proof we
achieve this goal by taking a localization (as in Remark 2.2) around a
selected atom. We can also use Lemma 3.1 directly, by calculating a local
first momentum as in the proof of Proposition 5.1. However, this will need a
detailed knowledge of $\Theta_{x\eta}$ (with $\eta$ a suitable cut-off
function). Nevertherless, by the localization argument, the self-interaction
term, the interaction terms between different atoms as well as the interaction
between the selected atom and the diffusion part can be revealed in a clear
way, so we will use this approach in the following proof.

###### Proof.

Take an arbitrary $t_{0}\in(-1,1)$ and a point
$q_{j}(t_{0})\in\Sigma_{t_{0}}$. By Lemma 7.1, there exists an $r>0$ such that

(7.4) $\Sigma\cap Q_{2r}(q_{j}(t_{0}),t_{0})=\\{(q_{j}(t),t)\\},$

By Lemma 6.5, after shrinking $r$ further, we may assume for any
$t\in(t_{0}-r^{2},t_{0}+r^{2})$, $q_{j}(t)\in B_{r/4}(q_{j}(t_{0}))$.

Take a cut-off function $\eta$ subject to $B_{r/2}(q_{j}(t_{0}))\subset
B_{r}(q_{j}(t_{0}))$. Following the localization procedure in Remark 2.2, set
$\widetilde{u}_{i}:=u_{i}\eta$, and then define $\nabla\widetilde{v}_{i}$,
$\nabla\widetilde{f}_{i}$, $\widetilde{g}_{i}$ accordingly. By the convergence
of $u_{i}$ etc., we have the following convergence results.

  1. (1)

For any $t\in(t_{0}-r^{2},t_{0}+r^{2})$,
$\widetilde{u}_{i}(x,t)dx\rightharpoonup
8\pi\delta_{q_{j}(t)}+\widetilde{\rho}(x,t)dx$, where
$\widetilde{\rho}(x,t):=\rho(x,t)\eta(x)$.

  2. (2)

By the convergence of $\widetilde{u}_{i}$, $\nabla\widetilde{v}_{i}$ converges
to

$\nabla\widetilde{v}_{\infty}(x,t)-4\frac{x-q_{j}(t)}{|x-q_{j}(t)|^{2}}$

in $L^{q}(B_{1})$ for any $q<2$, and uniformly in any compact set of
$Q_{r}(q_{j}(t_{0}),t_{0})\setminus\\{(q_{j}(t_{0}),t)\\}$. Here

$\nabla\tilde{v}_{\infty}(x,t)=-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}\widetilde{\rho}(y,t)dy.$

  3. (3)

By definition,

$\nabla
v_{i}(x,t)-\nabla\widetilde{v}_{i}(x,t)=-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}u_{i}(y,t)[1-\eta(y)]dy.$

Because

$u_{i}(y,t)[1-\eta(y)]dy\rightharpoonup 8\pi\sum_{k\neq
j}\delta_{q_{k}(t)}+\rho(y,t)\left[1-\eta(y)\right]dy,$

and in particular, $u_{i}$ converges uniformly to $\rho$ in
$(B_{2r}(q_{j}(t_{0}))\setminus
B_{r/4}(q_{j}(t_{0})))\times(t_{0}-r^{2},t_{0}+r^{2})$, by standard estimates
on Newtonian potential, $\nabla v_{i}(x,t)-\nabla\widetilde{v}_{i}(x,t)$
converges uniformly in $Q_{r}(q_{j}(t_{0}),t_{0})$ to

$-4\sum_{k\neq
j}\frac{x-q_{k}(t)}{|x-q_{k}(t)|^{2}}-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}\rho(y,t)[1-\eta(y)]dy.$

Then by the convergence of $\nabla f_{i}$, we deduce that
$\nabla\widetilde{f}_{i}$ converges uniformly in $Q_{r}(q_{j}(t_{0}),t_{0})$
to

(7.5) $\nabla\widetilde{f}(x,t):=\nabla f(x,t)-4\sum_{k\neq
j}\frac{x-q_{k}(t)}{|x-q_{k}(t)|^{2}}-\frac{1}{2\pi}\int_{B_{1}}\frac{x-y}{|x-y|^{2}}\rho(y,t)\left[1-\eta(y)\right]dy.$

  4. (4)

In the support of $\nabla\eta$, $u_{i}$, $\nabla u_{i}$ and $\nabla v_{i}$
converge uniformly to $\rho$, $\nabla\rho$ and $\nabla v_{\infty}$
respectively, so $\widetilde{g}_{i}(x,t)$ converges uniformly to

$\displaystyle\widetilde{g}(x,t)$ $\displaystyle:=$ $\displaystyle
g(x,t)\eta(x)+\rho(x,t)\left[\nabla v_{\infty}(x,t)+\nabla
f(x,t)\right]\cdot\nabla\eta(x)$
$\displaystyle-2\nabla\rho(x,t)\cdot\nabla\eta(x)-\rho(x,t)\Delta\eta(x).$

By definition, for any $t\in(t_{0}-r^{2},t_{0}+r^{2})$,
$\widetilde{u}_{i}(x,t)$ is compactly supported in $B_{r}(q_{j}(t_{0}))$.
Plugging $\psi(x)=x$ as test function into (3.1), by noting that
$\Delta\psi\equiv 0$ and $\Theta_{\psi}\equiv 0$, we obtain

$\frac{d}{dt}\int_{B_{1}}\psi(x)\widetilde{u}_{i}(x,t)dx=\int_{B_{1}}\left[\nabla\widetilde{f}_{i}(x,t)\cdot\nabla\psi(x)\widetilde{u}_{i}(x,t)+\widetilde{g}_{i}(x,t)\psi(x)\right]dx.$

Sending $i\to+\infty$, by the weak convergence of $\widetilde{u}_{i}$ and the
uniform convergence of $\nabla\widetilde{f}_{i}$ and $\widetilde{g}_{i}$, we
get

$\displaystyle\frac{d}{dt}\left[\int_{B_{1}}\psi(x)\widetilde{\rho}(x,t)dx+8\pi
q_{j}(t)\right]$ $\displaystyle=$
$\displaystyle\int_{B_{1}}\nabla\widetilde{f}(x,t)\cdot\nabla\psi(x,t)\widetilde{\rho}(x,t)dx+\int_{B_{1}}\widetilde{g}(x,t)\psi(x)dx+8\pi\nabla\widetilde{f}(q_{j}(t),t).$

Note that this differential identity should be understood in the
distributional sense.

Because $\rho$ is a solution of (7.1) in $Q_{1}$, $\widetilde{\rho}=\rho\eta$
satisfies

$\widetilde{\rho}_{t}-\Delta\widetilde{\rho}=-\text{div}\left[\widetilde{\rho}\left(\nabla\widetilde{v}_{\infty}-4\frac{x-q_{j}(t)}{|x-q_{j}(t)|^{2}}+\nabla\widetilde{f}\right)\right]+\widetilde{g}.$

Then by Lemma 3.1, we have

$\displaystyle\frac{d}{dt}\int_{B_{1}}\psi(x)\widetilde{\rho}(x,t)dx$
$\displaystyle=$
$\displaystyle\int_{B_{1}}\left[-4\frac{x-q_{j}(t)}{|x-q_{j}(t)|^{2}}+\nabla\widetilde{f}(x,t)\right]\cdot\nabla\psi(x,t)\widetilde{\rho}(x,t)dx+\int_{B_{1}}\widetilde{g}(x,t)\psi(x)dx.$

In view of the form of $\nabla\widetilde{f}$ in (7.5), subtracting this
identity from (7) gives (7.3).

Finally, although by now (7.3) is only understood in the distributional sense,
by Lemma 7.1 and our assumption on $\rho$, the right hand side of (7.3) is
continuous in $t$. Then by standard regularity theory for ODEs, we deduce that
$q_{j}$ is $C^{1}$ in $t$ and it satisfies (7.3) in the classical sense. ∎

##  8\. Blow-up limits: Proof of Theorem 1.4

Now we turn to the study of first time singularities and the proof of Theorem
1.4.

Recall that $u$ satisfies (H1-H3) and the blowing up sequence is defined as

$u^{\lambda}(x,t):=\lambda^{2}u(\lambda x,\lambda^{2}t),\quad\nabla
v^{\lambda}(x,t)=\lambda\nabla v(\lambda x,\lambda^{2}t).$

These functions are defined on $Q_{\lambda^{-1}}^{-}$, which tends to
$\mathbb{R}^{2}\times(-\infty,0]$ as $\lambda\to 0$. By Theorem 4.1, we may
assume that for a subsequence (not relabelling) $\lambda\to 0$,

$u^{\lambda}(x,t)dx\rightharpoonup\mu_{t},\quad\forall t\leq 0.$

First we notice the following two facts. Recall that Hypothesis (H3) says that
$u(x,t)$ forms a Dirac measure of mass $m$ at the origin as $t\to 0^{-}$.

######  Lemma 8.1.

$\mu_{0}=m\delta_{0}$.

###### Proof.

For any $\lambda>0$, by (H3), as $t\to 0^{-}$,

$u^{\lambda}(x,t)dx\rightharpoonup
m\delta_{0}+u_{0}^{\lambda}(x)dx,\quad\mbox{where}\quad
u_{0}^{\lambda}(x):=\lambda^{2}u_{0}(\lambda x).$

Because $u_{0}\in L^{1}(B_{1})$, as $\lambda\to 0$,

$u_{0}^{\lambda}(x)dx\rightharpoonup 0.$

Then by the uniform Lipschitz continuity of
$\mathcal{L}_{t}^{\lambda_{i}}(\psi)$ (for $\psi\in
C_{0}^{2}(\mathbb{R}^{2})$, see (4.3)), we deduce that

$\lim_{t\to 0}\lim_{\lambda\to 0}u^{\lambda}(x,t)dx=m\delta_{0}.\qed$

######  Lemma 8.2.

$\rho\equiv 0$ in
$\left(\mathbb{R}^{2}\times(-\infty,0]\right)\setminus\Sigma$.

###### Proof.

In the open set
$\Omega:=\left(\mathbb{R}^{2}\times(-\infty,0)\right)\setminus\Sigma$, $\rho$
is a classical solution of a linear parabolic equation (i.e. the first
equation in (1.2)). By Theorem 4.1, $\Omega$ is connected. By Lemma 8.1,
$\rho(x,0)=0$ in $\mathbb{R}^{2}\setminus\\{0\\}$. The desired claim follows
by applying the strong maximum principle. ∎

By this lemma and Lemma 7.1, there exists an $N\in\mathbb{N}$ such that for
each $t<0$, $\Sigma_{t}$ consists of exactly $N$ distinct points, $q_{1}(t)$,
$\dots$, $q_{N}(t)$, and

$\left\\{\begin{array}[]{ll}\mu_{t}=8\pi\sum_{j=1}^{N}\delta_{q_{j}(t)},\\\
\nabla
v_{\infty}(x,t)=-4\sum_{j=1}^{N}\frac{x-q_{j}(t)}{|x-q_{j}(t)|^{2}}.\end{array}\right.$

Furthermore, each $q_{j}(t)$ is $1/2$-Hölder in $t$ and $q_{j}(0)=0$.

As a consequence we obtain the quantization of mass, $m=8N\pi$. This also
implies that, even though the blow-up limit may be not unique, the number of
Dirac measures in every blow-up limit is $N$.

Because $\rho\equiv 0$, an application of Theorem 7.2 gives

######  Proposition 8.3.

For each $j$, $q_{j}(t)\in C((-\infty,0],\mathbb{R}^{2})\cap
C^{1}((-\infty,0),\mathbb{R}^{2})$. Moreover,

  1. (1)

if $N=1$, then $q_{1}(t)\equiv 0$;

  2. (2)

if $N\geq 2$, then the vector valued function $(q_{j}(t))$ satisfies

(8.1) $q_{j}^{\prime}(t)=4\sum\limits_{k\neq
j}\dfrac{q_{k}(t)-q_{j}(t)}{|q_{k}(t)-q_{j}(t)|^{2}},\quad q_{j}(0)=0.$

To finish the proof of Theorem 1.4, it remains to show that

######  Proposition 8.4.

There exist $N$ distinct points $p_{1}$,$\dots$, $p_{N}\in\mathbb{R}^{2}$ such
that

$q_{j}(t)\equiv\sqrt{-t}p_{j},\quad\forall t\leq 0.$

Moreover, $(p_{j})$ is a critical point of the renormalized energy
$\mathcal{W}$.

If $N=1$, it has been established in Propositon 8.3 that $q_{1}(t)\equiv 0$,
so we will assume $N\geq 2$ in the remaining part of this section.

We first prove several technical estimates on $q_{j}(t)$.

######  Lemma 8.5.

For any $t<0$,

(8.2) $\sum_{j=1}^{N}q_{j}(t)=0,$ (8.3)
$\sum_{j=1}^{N}|q_{j}(t)|^{2}=2N(N-1)|t|.$

###### Proof.

By (8.1), we have

$\left\\{\begin{array}[]{ll}\frac{d}{dt}\sum_{j=1}^{N}q_{j}(t)=4\sum_{j=1}^{N}\sum_{k\neq
j}\frac{q_{k}(t)-q_{j}(t)}{|q_{k}(t)-q_{j}(t)|^{2}}=0,\\\
\sum_{j=1}^{N}q_{j}(0)=0.\end{array}\right.\\\ $

Then (8.2) follows by continuity.

In the same way, we obtain

$\left\\{\begin{array}[]{ll}\frac{d}{dt}\sum_{j=1}^{N}|q_{j}(t)|^{2}=8\sum_{j=1}^{N}q_{j}(t)\cdot\sum_{k\neq
j}\frac{q_{k}(t)-q_{j}(t)}{|q_{k}(t)-q_{j}(t)|^{2}}=2N(N-1),\\\
\sum_{j=1}^{N}|q_{j}(0)|^{2}=0.\end{array}\right.\\\ $

Then (8.3) follows by an integration in $t$. ∎

######  Corollary 8.6.

For any $t<0$,

$\max_{1\leq j\leq N}|q_{j}(t)|\leq\sqrt{2N(N-1)}\sqrt{-t}.$

######  Lemma 8.7.

There exists a constant $c_{\ast}>0$ such that for any $t<0$,

$\min_{1\leq j\neq k\leq N}|q_{j}(t)-q_{k}(t)|\geq c_{\ast}\sqrt{-t}.$

###### Proof.

Assume by the contrary, there exists a sequence of $t_{i}<0$, and two indicies
$j\neq k$ such that

(8.4) $\lim_{i\to+\infty}\frac{|q_{j}(t_{i})-q_{k}(t_{i})|}{\sqrt{-t_{i}}}=0.$

Set

$\mu^{i}_{t}(A):=\mu_{|t_{i}|t}\left(\frac{A}{\sqrt{-t_{i}}}\right),\quad\forall
A\subset\mathbb{R}^{2}~{}~{}\text{Borel}.$

As in the first step in the proof of Theorem 4.1, after passing to a
subsequence, we may assume $\mu^{i}_{t}\rightharpoonup\mu^{\infty}_{t}$ for
any $t\leq 0$. By a diagonal argument as in the proof of Proposition 5.1 (see
Step 1 therein), we deduce that $\mu^{\infty}_{t}$ is also a blow up limit of
$u$ at $(0,0)$.

By the definition of $\mu^{i}_{t}$,

(8.5) $\mu_{-1}^{i}\geq
8\pi\delta_{q_{j}(t_{i})/\sqrt{-t_{i}}}+8\pi\delta_{q_{k}(t_{i})/\sqrt{-t_{i}}}.$

By Corollary 8.6, we may take a subsequence of $i$ so that both
$q_{j}(t_{i})/\sqrt{-t_{i}}$ and $q_{k}(t_{i})/\sqrt{-t_{i}}$ converge. By
(8.4), their limit points coincide, which is denoted by $q_{\infty}$. Passing
to the limit in (8.5), we obtain

$\mu_{-1}^{\infty}\geq 16\pi\delta_{q_{\infty}}.$

This is a contradiction with Proposition 5.1. ∎

In view of these estimates, it is better to normalize the blow-up limits
$(q_{j}(t))$.

######  Definition 8.8 (Renormalized blow-up limits).

By setting $s:=-\log(-t)$, $s\in\mathbb{R}$, we define the renormalized blow-
up limit to be

$(p_{j}(s)):=\left(\frac{1}{\sqrt{-t}}q_{j}(t)\right).$

The following lemma gives a characterization of the energy level of
remormialized blow-up limits at infinite time, by using the renormalized
energy $\mathcal{W}$ (see (1.8)).

######  Lemma 8.9.

  1. (1)

For any renormalized blow up limit $(p_{j}(s))$ constructed from a blowing up
sequence at scales $\lambda_{i}\to 0$, it is a solution to the ODE system

(8.6) $p_{j}^{\prime}(s)=\frac{1}{2}p_{j}(s)+4\sum_{k\neq
j}\frac{p_{k}(s)-p_{j}(s)}{|p_{k}(s)-p_{j}(s)|^{2}},\quad s\in\mathbb{R}.$

  2. (2)

The two limits

$\mathcal{W}_{\pm}(\\{\lambda_{i}\\}):=\lim_{s\to\pm\infty}\mathcal{W}\left(p_{1}(s),\cdots,p_{N}(s)\right)$

exist, and they are critical energy levels of $\mathcal{W}$.

  3. (3)

There exists a constant $C(N)$ depending on $N$ only such that

(8.7)
$-C(N)\leq\mathcal{W}_{-}(\\{\lambda_{i}\\})\leq\mathcal{W}_{+}(\\{\lambda_{i}\\})\leq
C(N).$

###### Proof.

The ODE (8.6) follows by taking a change of variables in (8.1). This system of
ODEs is the gradient flow of the function $\mathcal{W}$. Hence
$\mathcal{W}(p_{1}(s),\cdots,p_{N}(s))$ is non-increasing in $s$.

By Corollary 8.6,

(8.8) $\max_{1\leq j\leq N}\sup_{s\in\mathbb{R}}|p_{j}(s)|\leq\sqrt{2N(N-1)}.$

By Lemma 8.7,

(8.9) $\min_{1\leq j\neq k\leq N}\inf_{s\in\mathbb{R}}|p_{j}(s)-p_{k}(s)|\geq
c_{\ast}.$

Hence

$\mathcal{W}_{\pm}(\\{\lambda_{i}\\}):=\lim_{s\to\pm\infty}\mathcal{W}(p_{1}(s),\cdots,p_{N}(s))$

are two well-defined, finite constants. This also implies that the
$\omega$-limit points and $\alpha$-limit points of $(p_{j}(s))$ are critical
points of $\mathcal{W}$. They may not be unique, but the corresponding energy
levels are $\mathcal{W}_{\pm}(\\{\lambda_{i}\\})$ respectively.

Finally, (8.7) is a consequence of (8.8) and (8.9). ∎

In the following we denote, for any $s\in\mathbb{R}$,

$\overline{u}^{s}(y):=e^{-s}u\left(e^{-s/2}y,-e^{-s}\right),$

which is just the slice of the blowing up sequence $u^{e^{-s/2}}$ at time
$-1$. By the blow up analysis developed so far, we have

######  Lemma 8.10.

There exists a $T_{\sharp}>0$ such that for any $s>T_{\sharp}$, there exist
$N$ points $\widetilde{p}_{1}(s)$, $\dots$, $\widetilde{p}_{N}(s)$ satisfying

(8.10) $\min_{j\neq k}|\widetilde{p}_{j}(s)-\widetilde{p}_{k}(s)|\geq
c_{\ast}/2\quad\mbox{and}\quad\sum_{j=1}^{N}|\widetilde{p}_{j}(s)|^{2}\leq
2N^{2},$

such that $\overline{u}^{s}(y)dy$ is close to
$8\pi\sum_{j=1}^{N}\delta_{\widetilde{p}_{j}(s)}$ weakly as Radon measures.

Next we give a canonical construction of the Dirac measures by using an
optimal approximation procedure. For any
$(p_{1},\cdots,p_{N})\in\mathbb{R}^{2N}$ satisfying ($c_{\ast}$ as in Lemma
8.7)

$\min_{j\neq k}|p_{j}-p_{k}|\geq
c_{\ast}/4\quad\text{and}\quad\sum_{j=1}^{N}|p_{j}|^{2}\leq
2N^{2}+c_{\ast}^{2},$

define a positive smooth function $\omega(x;p_{1},\cdots,p_{N})$ such that it
equals $|x-p_{j}|^{2}$ in $B_{c_{\ast}/8}(p_{j})$, it is bounded below by
$c_{\ast}^{2}/64$ outside $\cup_{j=1}^{N}B_{c_{\ast}/4}(p_{j})$ and it is
equal to a positive constant outside $B_{2N+2c_{\ast}}$.

######  Lemma 8.11.

There exists a $T_{\ast}>0$ such that for any $s>T_{\ast}$, the following
minimization problem

$\min_{(p_{1},\dots,p_{N})\in\mathbb{R}^{2N}}\int_{B_{2N+3c_{\ast}}}\omega(x;p_{1},\cdots,p_{N})\overline{u}^{s}(x)dx$

has a unique minimizer $(p_{j,\ast}(s))$. Moreover,

  1. (1)

for any $s>T_{\ast}$,

(8.11) $\min_{j\neq k}|p_{j,\ast}(s)-p_{k,\ast}(s)|\geq
c_{\ast}/8\quad\text{and}\quad\sum_{j=1}^{N}|p_{j,\ast}(s)|^{2}\leq
2N^{2}+2c_{\ast}^{2};$

  2. (2)

for each $j$, $p_{j,\ast}\in C^{1}(T_{\ast},+\infty)$.

###### Proof.

Denote

$\mathcal{J}^{s}(p_{1},\cdots,p_{N}):=\int_{B_{2N+3c_{\ast}}}\omega(x;p_{1},\cdots,p_{N})\overline{u}^{s}(x)dx.$

It is smooth in $p_{1}$, …, $p_{N}$ and positive everywhere.

By Lemma 8.10, we have

  1. (1)

$\lim_{s\to+\infty}\mathcal{J}^{s}(\widetilde{p}_{1}(s),\cdots,\widetilde{p}_{N}(s))=0$;

  2. (2)

for any $\varepsilon>0$, there exist $T(\varepsilon)>0$ and
$\delta(\varepsilon)>0$ such that, for any $s>T(\varepsilon)$, if $(p_{j})$
satisfies $\sum_{j=1}^{N}|p_{j}-\widetilde{p}_{j}(s)|>\delta(\varepsilon)$,
then

$\mathcal{J}^{s}(p_{1},\cdots,p_{N})>\varepsilon.$

Thus there exists a $T_{\ast}>0$ such that for any $s\geq T_{\ast}$, the
minima of $\mathcal{J}^{s}$ is attained at some point. Denote it by
$(p_{j,\ast}(s))$. By Property 1, we have

$\lim_{s\to+\infty}\sum_{j=1}^{N}|p_{j,\ast}(s)-\widetilde{p}_{j}(s)|=0.$

Combining this relation with (8.10), we obtain (8.11), perhaps after taking a
larger $T_{\ast}$.

By Lemma 8.10, there exists a small constant $\delta>0$ such that for any
$s>T_{\ast}$ and $(p_{j})$ satisfying
$\sum_{j=1}^{N}|p_{j}-\widetilde{p}_{j}(t)|<\delta$, it holds that

(8.12)
$\sum_{j,k=1}^{N}\left|\int_{B_{2N+3c_{\ast}}}\frac{\partial^{2}\omega}{\partial
p_{j}\partial
p_{k}}(x;p_{1},\cdots,p_{N})\overline{u}^{s}(x)dx-\delta_{jk}\right|\leq\delta,$

where $\delta_{jk}$ is the Kronecker delta symbol. This implies that
$\mathcal{J}^{s}$ is strictly convex in a fixed neighborhood of
$(\widetilde{p}_{j}(s))$. Therefore its minima point is unique. By the
implicit function theorem, $p_{j,\ast}(s)$ is continuously differentiable in
$s$. ∎

######  Lemma 8.12.

For any sequence $T_{i}\to+\infty$, there exists a subsequence (not
relabelling) so that $(p_{j,\ast}(s+T_{i}))$ converges to a renormalized blow-
up limit in $C^{1}_{loc}(\mathbb{R})$.

###### Proof.

Let $\lambda_{i}:=e^{-T_{i}/2}$. Take a subsequence of $\lambda_{i}$ so that
the blowing up sequence $u^{\lambda_{i}}$ converges to a blow-up limit
$8\pi\sum_{j=1}^{N}\delta_{q_{j}(t)}$ (in the sense of Theorem 4.1). Let
$(p_{j}(s))$ be the corresponding renormalized blow up limit. By the
definition of $\mathcal{J}^{s}$, we have

$\lim_{T_{i}\to+\infty}\mathcal{J}^{s+T_{i}}\left(p_{1}(s),\cdots,p_{N}(s)\right)=0.$

Then by the minimality of $(p_{j,\ast}(s))$ and the strict convexity of
$\mathcal{J}^{s+T_{i}}$ near $\left(p_{j}(s)\right)$, we deduce that
$(p_{j,\ast}(s+T_{i}))$ converges to $(p_{j}(s))$ uniformly on any compact set
of $\mathbb{R}$.

Next we prove the uniform convergence of $(p_{j,\ast}^{\prime}(s+T_{i}))$. The
proof of the previous lemma implies that for all $t<0$, the minimization
problem

$\int_{B_{2N+3c_{\ast}}}\omega(x;p_{1},\cdots,p_{N})u^{\lambda_{i}}(x,t)dx$

has a unique minimizer $(p_{j}^{\lambda_{i}}(t))$. In fact, the uniqueness
implies that

$\left(p_{j}^{\lambda_{i}}(t)\right)=\left(p_{j,\ast}(-\log(-t)+T_{i})\right).$

Hence it suffices to establish the uniform convergence for the derivative of
$p_{j}^{\lambda_{i}}$.

By the minimization condition for $(p_{j}^{\lambda_{i}}(t))$, we have

$\int_{B_{2N+3c_{\ast}}}\frac{\partial\omega}{\partial
p_{j}}(x;p_{1}^{\lambda_{i}}(t),\cdots,p_{N}^{\lambda_{i}}(t))u^{\lambda_{i}}(x,t)dx=0,\quad\forall
j=1,\cdots,N.$

Differentiating this identity in $t$, we obtain

$\displaystyle\sum_{k=1}^{N}\left(\int_{B_{2N+3c_{\ast}}}\frac{\partial^{2}\omega}{\partial
p_{j}\partial
p_{k}}(x;p_{1}^{\lambda_{i}}(t),\cdots,p_{N}^{\lambda_{i}}(t))u^{\lambda_{i}}(x,t)dx\right)\frac{dp_{k}^{\lambda_{i}}}{dt}(t)$
$\displaystyle=$
$\displaystyle-\int_{B_{2N+3c_{\ast}}}\frac{\partial\omega}{\partial
p_{j}}(x;p_{1}^{\lambda_{i}}(t),\cdots,p_{N}^{\lambda_{i}}(t))\partial_{t}u^{\lambda_{i}}(x,t)dx$
$\displaystyle=$
$\displaystyle-\int_{B_{2N+3c_{\ast}}}\Delta_{x}\frac{\partial\omega}{\partial
p_{j}}(x;p_{1}^{\lambda_{i}}(t),\cdots,p_{N}^{\lambda_{i}}(t))u^{\lambda_{i}}(x,t)dx$
$\displaystyle+\frac{1}{4\pi}\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\Theta_{\frac{\partial\omega}{\partial
p_{j}}}(x,y)u^{\lambda_{i}}(x,t)u^{\lambda_{i}}(y,t)dxdy.$

In the above, we have used Lemma 3.1 and the fact that
$\frac{\partial\omega}{\partial
p_{j}}(x;p_{1}^{\lambda_{i}}(t),\cdots,p_{N}^{\lambda_{i}}(t))$ is compactly
supported in $x$. In view of (8.12), we can solve $dp_{k}^{\lambda_{i}}/dt$
from this equation. By the convergence of $u^{\lambda_{i}}(x,t)dx$, the last
two integrals converge uniformly on any compact set of $(-\infty,0)$. This
then gives the uniform convergence of $dp_{k}^{\lambda_{i}}/dt$ on any compact
set of $(-\infty,0)$, and consequently, the uniform convergence of
$(p_{j,\ast}^{\prime}(s+T_{i}))$ on any compact set of $\mathbb{R}$. ∎

Let

$E(s):=\mathcal{W}\left(p_{1,\ast}(s),\cdots,p_{N,\ast}(s)\right).$

######  Lemma 8.13.

The limit $\lim_{s\to+\infty}E(s)$ exists, and it is a critical energy level
of $\mathcal{W}$ in $[-C(N),C(N)]$.

###### Proof.

First, by (8.11) and the definition of $\mathcal{W}$, $E(s)$ is a continuous
function on $[T_{\ast},+\infty)$. Moreover, for any $s\geq T_{\ast}$,

(8.14) $-C(N)\leq E(s)\leq C(N).$

Next, let us recall a fact about the critical energy levels of $\mathcal{W}$.
Because critical points of $\mathcal{W}$ are identical to those of
$e^{-\mathcal{W}}$, and $e^{-\mathcal{W}}$ is a real analytic function, by the
Lojasiewicz inequality for real analytic functions (see Lojasiewicz [10]), its
critical energy levels are discrete. Hence there are only finitely many
critical energy levels of $\mathcal{W}$ in the interval $[-C(N),C(N)]$. Denote
them by $E_{1},\cdots,E_{K}$ for some $K\in\mathbb{N}$. Let

(8.15) $\delta:=\min_{1\leq i\neq j\leq K}|E_{i}-E_{j}|>0$

be the gap between these energy levels. Then for any non-static gradient flow
$(p_{j}(s))$ of $\mathcal{W}$, if
$\mathcal{W}(p_{1}(s),\cdots,p_{N}(s))\in[-C(N),C(N)]$, we must have

(8.16)
$\lim_{s\to-\infty}\mathcal{W}(p_{1}(s),\cdots,p_{N}(s))-\lim_{s\to+\infty}\mathcal{W}(p_{1}(s),\cdots,p_{N}(s))\geq\delta.$

For any $\varepsilon\in(0,\delta/10)$, set

$\mathcal{I}_{\varepsilon}:=\\{s:~{}s>T_{\ast},~{}\min_{j=1,\cdots,K}|E(s)-E_{j}|>\varepsilon\\}.$

It is an open subset of $(T_{\ast},+\infty)$. Assume

$\mathcal{I}_{\varepsilon}=\cup_{\alpha}\mathcal{I}_{\varepsilon,\alpha},$

where
$\mathcal{I}_{\varepsilon,\alpha}=(t_{\varepsilon,\alpha}^{-},t_{\varepsilon,\alpha}^{+})$
is an open interval. We claim that
Claim. The number of these intervals is finite.

By Lemma 8.12, for any $T>0$ and any $s\geq T_{\ast}+T$, $(p_{j,\ast})$ is
close to a gradient flow of $\mathcal{W}$ in $[s-T,s+T]$. Because in each
$\mathcal{I}_{\varepsilon,\alpha}$, $E(s)$ is not close to any critical energy
level $E_{j}$, in every $\mathcal{I}_{\varepsilon,\alpha}$ (with at most
finitely many exceptions), $(p_{j,\ast})$ cannot be close to any static
solution of (8.6) (i.e. a critical point of $\mathcal{W}$). In view of (8.16),
this is possible only if (at least when $t_{\varepsilon,\alpha}^{-}$ is
sufficiently large)

$E(t_{\varepsilon,\alpha}^{-})>E(t_{\varepsilon,\alpha}^{+})+\delta-4\varepsilon.$

Then because

$\sup_{s_{1},s_{2}\in[t_{\varepsilon,\alpha-1}^{+},t_{\varepsilon,\alpha}^{-}]}|E(s_{1})-E(s_{2})|\leq
2\varepsilon,$

we get a discrete monotonicity relation

$E(t_{\varepsilon,\alpha-1}^{+})-E(t_{\varepsilon,\alpha}^{+})\geq\delta-6\varepsilon\geq\frac{\delta}{8}.$

Combining this inequality with (8.14), we deduce that there are only finitely
many open intervals in $\mathcal{I}_{\varepsilon}$. This finishes the proof of
this claim.

This claim can be reformulated as the statement that, for any $\varepsilon>0$,
there exists a $T(\varepsilon)>0$ such that for any $s\geq T(\varepsilon)$,

$\min_{j=1,\cdots,K}|E(s)-E_{j}|\leq\varepsilon.$

By the gap between different critical energy levels in (8.15) and the
continuity of $E(s)$, we then deduce that there exists a fixed $j$ such that
for any $\varepsilon\in(0,\delta/10)$ and $s\geq T(\varepsilon)$,

$|E(s)-E_{j}|\leq\varepsilon.\qed$

###### Completion of the proof of Theorem 1.4.

By the previous lemma, there exists an energy critical level $E$ of
$\mathcal{W}$ such that, for any renormalized blow-up limit $(p_{j}(s))$,

$\mathcal{W}(p_{1}(s),\cdots,p_{N}(s))\equiv E.$

Because $(p_{j}(s))$ is the gradient flow of $\mathcal{W}$, this is possible
only if it does not depend on $s$. Hence it must be a critical point of
$\mathcal{W}$. ∎

######  Remark 8.14.

Because blow up limits are obtained by a compactness argument and critical
points of $\mathcal{W}$ are not discrete, it is not clear if the renormalized
blow-up limit is unique.

##  9\. Entire solutions

In this section we prove Theorem 1.10. We will also explain, in the setting of
Section 4, how to obtain entire solutions from suitable rescalings around a
blow up point.

###### Proof of Theorem 1.10.

If $u$ is an entire solution, by Theorem 1.1, the blowing down sequence
converges to a limit $\mu_{t}$ in $\mathbb{R}^{2}\times\mathbb{R}$. First by
Theorem 1.8, there exists an $N\in\mathbb{N}$ such that $\mu_{0}=8\pi N$. On
the other hand, because the blowing down sequence converges in $Q_{1}$, an
application of Proposition 5.1 at $t=0$ shows that $N=1$. Then by Lemma 6.2
and Lemma 6.3, we conclude the proof of Theorem 1.10. ∎

Next, we use the convergence theory in Section 4 to show how entire solutions
arise as micro-models of singularity formations.

We work under the same assumptions of Theorem 4.1. After localization (see
Remark 2.2), we may assume that

(9.1) $u_{i}(x,0)dx\rightharpoonup
8\pi\delta_{0}+\rho(x,0)dx\quad\mbox{in}~{}~{}B_{1}.$

This implies that

$\max_{B_{1}}u_{i}(x,0)\to+\infty,$

and it is attained at an interior point, say $x_{i}$. Because $u_{i}$
converges to $\rho$ in $C_{loc}(B_{1}\setminus\\{0\\})$,

$|x_{i}|\to 0.$

Denote $R_{i}:=u_{i}(x_{i},0)^{1/2}$. Set
$\widetilde{u}_{i}(x,t):=R_{i}^{-2}u_{i}(x_{i}+R_{i}^{-1}x,R_{i}^{-2}t)$ and
define $\nabla\widetilde{v}_{i}$ as in (1.6) by using $\widetilde{u}_{i}$.
Then we have

######  Theorem 9.1.

As $i\to+\infty$, $\widetilde{u}_{i}$ converges to a limit
$\widetilde{u}_{\infty}$ in $C^{2,1}_{loc}(\mathbb{R}^{2}\times\mathbb{R})$,
where $\widetilde{u}_{\infty}$ is an entire solution of (1.2).

###### Proof.

By Theorem 4.1, for each $t\in\mathbb{R}$, there exist finitely many points
$q_{j}(t)$, $j=1$, …, $N(t)$, such that as Radon mearures on $\mathbb{R}^{2}$,

(9.2) $\widetilde{u}_{i}(x,t)dx\rightharpoonup
8\pi\sum_{j=1}^{N(t)}\delta_{q_{j}(t)}+\widetilde{u}_{\infty}(x,t)dx.$

By (9.1) and Lemma 3.1, for any $\varepsilon>0$, there exists a $\delta>0$
such that for any $t\in(-\delta,\delta)$,

$\limsup_{i\to+\infty}\int_{B_{\delta}}u_{i}(x,t)dx<8\pi+\varepsilon.$

After a scaling, this is transformed into

(9.3) $\limsup_{i\to+\infty}\int_{B_{\delta
R_{i}}}\widetilde{u}_{i}(x,t)dx\leq 8\pi+\varepsilon,\quad\forall t\in(-\delta
R_{i}^{2},\delta R_{i}^{2}).$

Combining this upper bound with (9.2), we see $N(t)\leq 1$. Moreover, by
Theorem 4.1 and Proposition 5.1,

  * •

Alternative I: either $N(t)=1$ and $\widetilde{u}_{\infty}(t)\equiv 0$ ;

  * •

Alternative II: or $N(t)=0$ and $\widetilde{u}_{\infty}(t)\in
C^{2}(\mathbb{R}^{2})$ .

Let $\mathcal{I}$ be the set of those $t$ satisfying Alternative I. By the
same strong maximum principle argument used in the proof of Lemma 8.2, we
deduce that there exists a $T\leq+\infty$ such that

$\mathcal{I}=(-\infty,T].$

On the other hand, by definition, we have

$\max_{B_{R_{i}}}\widetilde{u}_{i}(x,0)=\widetilde{u}_{i}(0,0)=1.$

Then by Theorem 3.2 (applied to the cylinder $Q_{r}(x,0)$, for any
$x\in\mathbb{R}^{2}$ and a fixed, sufficiently small $r$), there exists a
$\delta>0$ such that $\widetilde{u}_{i}\in
C^{2,1}(\mathbb{R}^{2}\times(-\delta,\delta))$. As a consequence,
$0\notin\mathcal{I}$, or equivalently, $T<0$.

If $T>-\infty$, $\widetilde{u}_{\infty}$ is smooth in
$\mathbb{R}^{2}\times(T,+\infty)$ and at time $T$ it is a Dirac measure with
mass $8\pi$. Then the same proof of Lemma 6.3 (more precisely, the proof of
(6.2)) leads to a contradiction. This contradiction implies that $T=-\infty$,
that is, Alternative II holds for all $t$. Then by Theorem 4.1,
$\widetilde{u}_{i}$ converges uniformly to $\widetilde{u}_{\infty}$ on any
compact set of $\mathbb{R}^{2}\times\mathbb{R}$. By standard parabolic
regularity theory, we deduce that $\widetilde{u}_{\infty}$, with the
corresponding $\nabla\widetilde{v}_{\infty}$, is a classical solution of (1.2)
on $\mathbb{R}^{2}\times\mathbb{R}$, i.e. it is an entire solution. Moreover,
by passing to the limit in (9.3), we see $\widetilde{u}_{\infty}$ satisfies
the finite mass condition (1.11). ∎

##  10\. Boundary blow up points

Finally, we give a remark on boundary blow up points. Let
$\Omega\subset\mathbb{R}^{2}$ be a smooth domain. Consider the initial-
boundary value problem

$\left\\{\begin{aligned} &u_{t}=\Delta u-\mbox{div}(u\nabla
v),\quad&\mbox{in}~{}~{}\Omega\times(0,T),\\\ &-\Delta
v+v=u,\quad&\mbox{in}~{}~{}\Omega\times(0,T),\\\
&\partial_{\nu}u=\partial_{\nu}v=0,\quad&\mbox{on}~{}~{}\partial\Omega\times(0,T),\end{aligned}\right.$

where $\nu$ is the outward unit normal vector of $\partial\Omega$.

Solutions to this problem could also blow up in finite time, which is still
caused by the concentration of $u$, see [14, Chapter 11]. The blow up points
could lie on the boundary. The blow up analysis can also be used to study
boundary blow up points. More precisely, similar to Theorem 1.4, we have

######  Theorem 10.1.

Assume $T$ is the first blow up time, and $a\in\partial\Omega$ is a boundary
blow up point. Let

$u^{\lambda}(x,t):=\lambda^{2}u(a+\lambda x,T+\lambda^{2}t),\quad\lambda\to
0.$

For any sequence $\lambda_{i}\to 0$, there exists a subsequence (not
relabelling), an half space $H$ and $N$ distinct points $p_{j}\in H$ such
that, for any $t<0$,

$u^{\lambda_{i}}(x,t)dx\rightharpoonup
8\pi\sum_{j=1}^{N}\delta_{\sqrt{-t}p_{j}}\quad\mbox{weakly as Radon
measures}.$

We only briefly explain the proof of Theorem 10.1. We can still work in the
local setting as in Theorem 1.4: first take a small ball $B_{r}(a)$ around
$a$, then take a diffeomorphism to flatten $\partial\Omega\cap B_{r}(a)$.
After scaling the radius of this ball to be $1$, extend $u$ and $v$ evenly to
the whole $B_{1}$. Denote by $g$ the Riemannian metric obtained by pushing
forward the original Euclidean metric through these transformations. Then we
are in the following local setting:

  * •

$u\in C^{\infty}(\overline{Q_{1}^{-}}\setminus\\{(0,0)\\})$, $u>0$ and

$\sup_{t\in(-1,0)}\int_{B_{1}}u(x,t)dx\leq M;$

  * •

$u$ satisfies

$u_{t}-\Delta_{g}u=-\mbox{div}_{g}\left(u\nabla_{g}v+\nabla_{g}f\right)\quad\mbox{in}~{}~{}Q_{1}^{-},$

where $\Delta_{g}$ is the Beltrami-Laplace operator with respect to the
Rimannian metric $g$, $\text{div}_{g}$ is the corresponding divergence
operator, $f$ is a smooth function in $Q_{1}$, $\nabla v$ is given by

$\nabla v(x,t)=\int_{B_{1}}\left[-\frac{1}{2\pi}\frac{x-y}{|x-y|^{2}}+\nabla
R(x,y)\right]u(y,t)dt,\quad\forall(x,t)\in Q_{1}^{-},$

with $R$ a smooth function of $(x,y)$ (the regular part of the Green
function);

  * •

there exists a nonnegative function $u_{0}\in L^{1}(B_{1})$ and a positive
constant $m$ such that as $t\to 0^{-}$,

$u(x,t)dx\rightharpoonup u_{0}(x)dx+m\delta_{0}\quad\text{weakly as Radon
measures}.$

After blowing up, the inhomogeneity caused by the Riemannian metric $g$ will
disappear. The remaining proof is exactly the same as the one of Theorem 1.4.

##  Appendix A Proof of Theorem 3.2

In this appendix we prove Theorem 3.2. It is a consequence of the following
theorem.

######  Theorem A.1.

Suppose $(u,v)$ is a classical solution of (2.1) in $Q_{1}$, satisfying

(A.1) $\sup_{|t|<1}\int_{B_{1}}u(x,t)dx<2\varepsilon_{\ast},$

then

(A.2) $\|u\|_{C^{1+\alpha,(1+\alpha)/2}(Q_{1/2})}\leq C.$

The only difference with Theorem 3.2 is that now we make a stronger assumption
(A.1). However, (A.1) follows by combining (3.5) with Lemma 3.1, perhaps after
restricting to a smaller cylinder and then scaling this cylinder to the unit
one.

Before proving this theorem, we recall several inequalities. The first two
inequalities are taken from [14, Lemma 4.2 and Lemma 11.1].

######  Lemma A.2.

For any $\psi\in C_{0}^{\infty}(B_{1})$, the following inequalities hold for
any $s>1$, where $C>0$ is a constant determined by $\psi$ only:

(A.3) $\int_{B_{1}}u^{2}\psi dx\leq C\|u\|_{1}\left(\int_{B_{1}}u^{-1}|\nabla
u|^{2}\psi dx\right)+C\|u\|_{1}^{2},$ (A.4) $\int_{B_{1}}u^{3}\psi
dx\leq\dfrac{C}{\log s}\left(\int_{B_{1}}(u\log
u+e^{-1})dx\right)\left(\int_{B_{1}}|\nabla u|^{2}\psi
dx\right)+C\|u\|_{1}^{3}+10s^{3}.$

The next two inequalities are similar to the corresponding ones for (1.2) (see
[14, Eqns. (11.15) and (11.16)]). Although there are additional terms $\nabla
f$ and $g$ in (2.1), due to their lower order nature, they only produce terms
which can be incorporated into the constant term.

######  Lemma A.3.

For any $\psi\in C_{0}^{2}(B_{1})$,

(A.5) $\dfrac{d}{dt}\int_{B_{1}}(u\log u)\psi
dx\leq-\dfrac{1}{2}\int_{B_{1}}u^{-1}|\nabla u|^{2}\psi
dx+2\int_{B_{1}}u^{2}\psi dx+C.$

(A.6) $\frac{d}{dt}\int_{B_{1}}u^{2}\psi dx\leq-\int_{B_{1}}|\nabla u|^{2}\psi
dx+3\int_{B_{1}}u^{3}\psi dx+C.$

###### Proof of Theorem A.1.

Take two functions $\psi_{1}\in C_{0}^{\infty}(B_{1}),\psi_{2}\in
C_{0}^{\infty}(B_{3/4})$ such that $0\leq\psi_{1},\psi_{2}\leq 1$,
$\psi_{1}\equiv 1$ in $B_{3/4}$ and $\psi_{2}\equiv 1$ in $B_{2/3}$. Denote

$I_{1}(t):=\int_{B_{1}}\left(u\log u+e^{-1}\right)\psi_{1}dx,\quad
J_{1}(t):=\int_{B_{1}}u^{2}\psi_{1}dx$

and

$I_{2}(t):=\int_{B_{1}}u^{2}\psi_{2}dx,\quad
J_{2}(t):=\int_{B_{1}}u^{3}\psi_{2}dx.$

Step 1. In view of (A.1), combining (A.5) and (A.3) gives

$I_{1}^{\prime}(t)\leq-cJ_{1}(t)+C.$

By the Hölder inequality, we have

$I_{1}(t)\leq CJ_{1}(t)^{\frac{1}{3}}+C.$

Hence we get

$I_{1}^{\prime}(t)\leq-cI_{1}(t)^{3}+C.$

Because $I_{1}(t)\geq 0$ for any $t\in(-1,1)$, this differential inequality
implies that

(A.7) $I_{1}(t)\leq C,\quad\forall t\in[-7/8,7/8].$

Step 2. In view of (A.7), combining (A.4) and (A.6) (where we choose a
sufficiently large $s$, determined by the upper bound of $J_{1}$ in (A.7))
gives

$I_{2}^{\prime}(t)\leq-cJ_{2}(t)+C.$

By the Hölder inequality, we have

$I_{2}(t)\leq CJ_{2}(t)^{\frac{2}{3}}+C.$

Hence we get

$I_{2}^{\prime}(t)\leq-cI_{2}(t)^{\frac{3}{2}}+C.$

Because $I_{2}(t)\geq 0$ for any $t\in(-7/8,7/8)$, this differential
inequality implies that

(A.8) $I_{2}(t)\leq C,\quad\forall t\in[-3/4,3/4].$

Step 3. We have shown that $u\in L^{\infty}(-3/4,3/4;L^{2}(B_{3/4}))$. Then by
standard $W^{2,p}$ estimate and Sobolev embedding theorem, $\nabla v\in
L^{\infty}(-3/4,3/4;L^{3}(B_{2/3}))$. This implies that $u\nabla v\in
L^{\infty}(-3/4,3/4;L^{6/5}(B_{3/4}))$. Then we can lift the regularity of $u$
and $v$ by bootstrapping standard $W^{2,p}$ estimate and Schauder estimate
(see [9, Theorem 4.8]). ∎

## References

  * [1] Haïm Brezis and Frank Merle.  Uniform estimates and blow-up behavior for solutions of $-\Delta u=V(x)e^{u}$ in two dimensions.  Comm. Partial Differential Equations, 16(8-9):1223–1253, 1991.
  * [2] Wen Xiong Chen and Congming Li.  Classification of solutions of some nonlinear elliptic equations.  Duke Math. J., 63(3):615–622, 1991.
  * [3] Xiuxiong Chen.  Remarks on the existence of branch bubbles on the blowup analysis of equation $-\Delta u=e^{2u}$ in dimension two.  Comm. Anal. Geom., 7(2):295–302, 1999.
  * [4] Jean Dolbeault and Benoît Perthame.  Optimal critical mass in the two-dimensional Keller-Segel model in $\mathbb{R}^{2}$.  C. R. Math. Acad. Sci. Paris, 339(9):611–616, 2004.
  * [5] Miguel A. Herrero and Juan J. L. Velázquez.  Singularity patterns in a chemotaxis model.  Math. Ann., 306(3):583–623, 1996.
  * [6] Miguel A. Herrero and Juan J. L. Velázquez.  A blow-up mechanism for a chemotaxis model.  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24(4):633–683 (1998), 1997.
  * [7] Evelyn F Keller and Lee A Segel.  Model for chemotaxis.  Journal of theoretical biology, 30(2):225–234, 1971.
  * [8] Yan Yan Li and Itai Shafrir.  Blow-up analysis for solutions of $-\Delta u=Ve^{u}$ in dimension two.  Indiana Univ. Math. J., 43(4):1255–1270, 1994.
  * [9] Gary M. Lieberman.  Second order parabolic differential equations.  World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
  * [10] Stanisław Lojasiewicz.  Ensembles semi-analytiques.  IHES notes, 1965.
  * [11] Yūki Naito and Takashi Suzuki.  Self-similar solutions to a nonlinear parabolic-elliptic system.  In Proceedings of Third East Asia Partial Differential Equation Conference, volume 8, pages 43–55, 2004.
  * [12] Pierre Raphaël and Rémi Schweyer.  On the stability of critical chemotactic aggregation.  Math. Ann., 359(1-2):267–377, 2014.
  * [13] Yukihiro Seki, Yoshie Sugiyama, and Juan J. L. Velázquez.  Multiple peak aggregations for the Keller-Segel system.  Nonlinearity, 26(2):319–352, 2013.
  * [14] Takashi Suzuki.  Free energy and self-interacting particles, volume 62 of Progress in Nonlinear Differential Equations and their Applications.  Birkhäuser Boston, Inc., Boston, MA, 2005.
  * [15] Takashi Suzuki.  Chemotaxis, reaction, network.  World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018.
  * [16] J. J. L. Velázquez.  Stability of some mechanisms of chemotactic aggregation.  SIAM J. Appl. Math., 62(5):1581–1633, 2002.
  * [17] J. J. L. Velázquez.  Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions.  SIAM J. Appl. Math., 64(4):1198–1223, 2004.
  * [18] J. J. L. Velázquez.  Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions.  SIAM J. Appl. Math., 64(4):1224–1248, 2004.