数理解析研究所講究録
第 1720 巻 2010 年 37-67

37

Smoluchowski-Poisson equation and harmonic heatflow quantization observed in nonlinear analysis
and diffusion geometry-

鈴木 貴 (阪大基礎工) TAKASHI SUZUKI
Division of Mathematical Science, Department of Systems Innovation,
Graduate School of Engineering Science, Osaka University

1

Introduction

The purpose of the present paper is to describe similarity and difference
between harmonic heat flow and Smoluchowski-Poisson equation definedon two-dimensional domain. Starting with stationary states, the former iscalled the harmonic map, while the latter arises as the point vortex mean
field equation.
First, equilibrium statistical mechanics of Gibbs is achieved with Hamiltonian denoted by $H$ . Classical gas molecular dynamics is described by the
Hamilton system$\frac{dq_{i}}{dt}=\frac{\partial H}{\partial p_{i}}$,$\frac{dp_{i}}{dt}=-\frac{\partial H}{\partial q_{i}}$,$1\leq i\leq N$

where $H$ casts total energy. Then micro-canonical ensemble indicates thecoset in $R^{6N}/\{H\}$ using the phase variable $x=(q_{1}, \ldots, q_{N},p_{1}, \ldots,p_{N})\in$. Micro-canonical measure$R^{N}$$d \mu^{H,N}=\frac{1}{\Omega(H)}\cdot\frac{d\Sigma(H)}{|\nabla H|}$,$\Omega(H)=\int_{\{H(x)=H\}}\frac{d\Sigma(H)}{|\nabla H|}$is thus defined with $d\Sigma(H)$ on $\{x\in R^{6N}|H(x)=H\}$ which satisfies$dx=dH \cdot\frac{d\Sigma(H)}{|\nabla H|}$by the principle of equal a priori probabilities. Canoncial mechanics, on
the other hand, is concerned with iso-thermal system, or closed system in38physical chemistry. Canonical ensemble is the coset in $R^{6N}/\{T\}$ and thenmicro-canonical measure is formulated by$d \mu^{\beta,N}=\frac{e^{-\beta H}dx}{Z(\beta,N)}$,$Z( \beta, N)=\int_{R^{6N}}e^{-\beta H}dx$

where $T,$ $\beta=1/(kT)$ , and denote temperature, inverse temperature, andthe Boltzmann constant, respectively. These two measures are equivalent inthermal equilibrium through the thermodynamical relation$k$$\beta=\frac{\partial}{\partial H}\log\Omega(H)$.

Onsager [26] introduced point vortex mean field equation using canonicalmeasure from the point vortex system$\alpha_{i}\frac{dx_{i}}{dt}=\nabla_{x_{i}}^{\perp}H_{N}$,$\nabla^{\perp}=(\begin{array}{l}\partial/\partial x_{2}-\partial/\partial x_{1}\end{array})$,$x=(x_{1}, x_{2})$$H_{N}(x_{1}, \ldots, x_{N})=\sum_{i}\frac{\alpha_{i}^{2}}{2}R(x_{j})+\sum_{i<j}\alpha_{i}\alpha_{j}G(x_{i}, x_{j})$defined on simply-connected bounded domain
with smooth boundary
where $G=G(x, x’)$ is the Green’s function ofprovided withand$\Omega\subset R^{2}$$\partial\Omega$$-\triangle$$|_{\partial\Omega}=0$$R(x)=[G(x, x’)+ \frac{1}{2\pi}\log|x-x’|]_{x=x}$

is the Robin function. The above point vortex system is derived from theEuler equation$v_{t}+(v\cdot\nabla)v=-\nabla p,$that is$\omega=\nabla\cross v,$

intensity$\alpha_{i}=\alpha$$\nabla\cdot v=0$

$N\uparrow+\infty$$\Omega\cross(O, T)$$\omega(dx, t)=\sum_{i=1}^{N}\alpha_{i}\delta_{x_{i}(t)}(dx)$,$\nu\cdot v=0$

on$\partial\Omega\cross(O, T)$. Thus, in the case of equal

there arises the point vortex mean field equation$\rho=\frac{e^{-\beta\psi}}{\int_{\Omega}e^{-\beta\psi}}$

as

in

,

$\psi=\int_{\Omega}G(\cdot, x’)\rho(x’)dx’$(1). Mathematical justification is done by [6, 7, 17] under the

relative Boltzmann factors $\{Z\}$ bound and unique existence of the solution
to (1), re-formulated by$- \Delta v=\frac{\lambda e^{v}}{\int_{\Omega}e^{v}}$in$\Omega$

,

$v=0$

on

$\partial\Omega$

(2)

39
using
and
. More precisely, weak convergence of canonicalmeasures, equivalence between canonical and micro-canonical ensembles in
the mean field limit level, and propagation of caos formulated by$v=e^{-\beta\psi}$$\lambda=-\beta$$\rho_{k}^{n}arrow\rho^{\otimes k}=\Pi_{i=1}^{k}\rho(x_{i})$wheredenotes the k-th pdf, see [42]. This property is actually the caseforwhich is proven by [39]. The caseis also exciting as thefollowing theorem [22] indicates.
$\rho_{k}^{n}$$\lambda<8\pi$$\lambda\geq 8\pi$

Theorem 1 Let

be a solution sequence to (2) satisfying
then it holds that
N. Passing to a
subsequence we obtain$\# S=l$ such that
locally uniformly inwhere $v_{0}(x)=8 \pi\sum_{x_{0}\in S}G(x, x_{0})$ andare the singular$\lambda_{0}\in(0, \infty),$$\{(\lambda_{k}, v_{k})\}$$\lambda_{k}arrow$$\Vert v_{k}\Vert_{\infty}arrow\infty$$\lambda_{0}=S\pi P,$$S\subset\Omega,$$\ell\in$$v_{k}arrow v_{0}$$\overline{\Omega}\backslash S$$S=\{x_{1}^{*}, \ldots, x_{\ell}^{*}\}$

limit and the blowup set prescribed by $\nabla_{x_{i}}H_{\ell}|_{(x1}=0,1\leq i\leq$, respectively, where$\ell$$H_{l}(x_{I}, \ldots, x_{l})=\frac{1}{2}\sum_{i}R(x_{i})+\sum_{i<j}G(x_{i}, x_{j})$denotes the original Hamiltonian.
We thus obtain quantized blowup mechanism in the total mass and also
recursivity of Hamiltonian.
Geometric background of the above theorem is described by the Liouvilleintegral of (2), see [40]. Thus we obtain a meromorphic function $F=F(z)$ ,$z\in\Omega\subset R^{2}\cong C$
satisfying$\lambda$$\rho(F)=(\frac{\sigma}{8})^{1/2}e^{v/2}$,$\lambda=\sigma\int_{\Omega}e^{v}$

where $\rho(F)=\frac{|F’|}{1+|F|^{2}}$ stands for the spherical derivative. Then (2) is trans-

formed to$\rho(F)|_{\partial\Omega}=(\frac{\sigma}{8})^{1/2}$, that is to find conformal immesion$\sqrt{8}F$:$d\Sigma$$\Omegaarrow S^{2}$such that–$=\sigma^{1/2}$

where$(S^{2}, d\Sigma)$denotes the sphere with

$ds\partial\Omega$$|S^{2}|=8\pi$. Since$\int_{\Omega}(\frac{d\Sigma}{ds})^{2}dx=8\int_{\Omega}\rho(F)^{2}dx=\int_{\Omega}\sigma e^{v}$

40stands for the immersed area of$\sqrt{8}F(\Omega)$, the conclusion$\lambda=\int_{\Omega}\sigma e^{v}arrow 8\pi l$indicates -covering of the sphere by this mapping.
At this moment it will be natural to suspect similarity between harand the target
monic map. Given the domain of m-compact manifold$Narrow*R^{n}$compact manifold
without boundary thus we put$\ell$$(\Omega, g)$$H^{1}(\Omega, N)=\{u\in H^{1}(\Omega, R^{n})|u\in N a.e.

on \Omega\}$,$E(u)= \frac{1}{2}\int_{\Omega}|\nabla u|^{2}$

to define the harmonic map $u\in H^{1}(\Omega, N)$ by$\frac{d}{d\epsilon}E(\Pi(u+\epsilon\phi))_{\epsilon=0}=0$,$\forall_{\phi\in C^{\infty}(\Omega,R^{n})}$is a tubular neighwhere $\Pi:Uarrow N$ is the geodesic projection andbourhood of $N$ . There is energy quantization for $m=2$ . The difference,
however, is the collision of bubbles classified as bubble on bubble and separated bubble which forms bubble tree.
$U$

2

Duality-Symmetry

The Smoluchowski-Poisson equation$u_{t}=\nabla\cdot(\nabla u-u\nabla v)$$\frac{\partial u}{\partial\nu}-u\frac{\partial v}{\partial\nu}=\frac{\partial v}{\partial\nu}=0$,$- \Delta v=u-\frac{1}{|\Omega|}\int_{\Omega}u$on$\partial\Omega\cross(0, T)$,in$\Omega\cross(0, T)$(3)$\int_{\Omega}v=0$

is a fundamental equation in transport theory which arises in the context
of semi-conductor physics, high molecular chemistry, astrophysics, and cellbiology [41, 42] where
denotes bounded domain with smooth boundary. Fundamental properties are the positivity of the solution $u\geq 0$ under, total mass conservationthat of the initial value$\Omega\subset R^{n}$$u_{0}\geq 0$(4)$\frac{d}{dt}\Vert u\Vert_{1}=0$

and decrease of the free energy$\frac{d}{dt}\mathcal{F}(u)=-\int_{\Omega}u|\nabla(\log u-v)|^{2}\leq 0$$\mathcal{F}(u)=\int_{\Omega}u(\log u-1)-\frac{1}{2}\iint_{\Omega\cross\Omega}G(x, x’)u\otimes u$

.

(5)

41
Then we obtain formation of collapse with quantized mass. Henceforth$T=T_{\max}\in(0, +\infty]$ denotes the existence time of the solution.
Theorem 2 If $n=2$ and $T=T_{\max}<+\infty$ in (3) it holds that$u(x, t)dx arrow\sum_{x_{0}\in S}m(x_{0})\delta_{x0}(dx)+f(x)dx$in$\mathcal{M}$(St) as$t\uparrow T$

(6)

where

$0\leq f=f(x)\in L^{1}(\Omega)\cap C(\overline{\Omega}\backslash S)$$S=\{x0\in$St$|^{\text{ョ}}(x_{k},$$t_{k})arrow(x0,$ $T)such$that $u(x_{k},$ $t_{k})arrow+\infty\}$(7)$m(x_{0})=m_{*}(x_{0})\equiv\{\begin{array}{l}8\pi, x_{0}\in\Omega 4\pi, x_{0}\in\partial\Omega\end{array}$

Since$\Vert u(t)\Vert_{1}=\Vert u_{0}\Vert_{1}$we obtain$2\#(S\cap\Omega)+\#(S\cap\partial\Omega)\leq\Vert u_{0}\Vert_{1}/(4\pi)$,(8)

and, in particular, the blowup set $S$ is finite. The strict inequality is actuallytrue in (8).
The above described quantized blowup mechanism was suspected fromthat of stationary states. Since (4) $-(5)$ this stationary state is defined by$\log u-v=$ constant,$\Vert u\Vert_{1}=\lambda$

with the prescribed total mass . This property implies$\lambda$(9)$u= \frac{\lambda e^{v}}{\int_{\Omega}e^{v}}$and hence$- \triangle v=\lambda(\frac{e^{v}}{\int_{\Omega}e^{v}}-\frac{1}{|\Omega|})$in$\Omega$,$\frac{\partial v}{\partial\nu}=0$on$\partial\Omega$,$\int_{\Omega}v=0$(10)which is reduced to the point vortex mean field equation defined on Riemannsurface where the quantized blowup mechanis is observed. This profile thatquantization of stationary states implies that of non-stationary states may
be called nonlinear spectml theory.

42

.
and
Toland duality [48, 49] is observed in (1) between
In the context of (9) $-(10)$ , the variational structure is provided with bothand , that is the free energy$v=e^{-\beta\psi}$$u=\lambda\rho$

$u$

$v$

$\mathcal{F}(u)=\int_{\Omega}u(\log u-1)-\frac{1}{2}\langle(-\Delta)^{-1}u,$ $u\rangle$,,$u\geq 0$$\Vert u\Vert_{1}=\lambda$

and the field functional,$v\in H^{1}(\Omega),$on$\partial\Omega$$\mathcal{J}_{\lambda}(v)=\frac{1}{2}\Vert\nabla v\Vert_{2}^{2}-\lambda\log\int_{\Omega}e^{v}+\lambda(\log\lambda-1)$$\int_{\Omega}v=0$

where $v=(-\triangle)^{-1}u$ if and only if$- \Delta v=u-\frac{1}{|\Omega|}\int_{\Omega}u$in$\Omega$,$\frac{\partial v}{\partial\nu}=0$

In fact (3) is the model (B) equation derived from$u_{t}=\nabla u\cdot\nabla\delta \mathcal{F}(u)$,,$\mathcal{F}(u)$$\int_{\Omega}v=0$.

, that is$u \frac{\partial}{\partial\nu}\delta \mathcal{F}(u)\partial\Omega^{=0}$.

Such duality between field and particles is observed in many models in mathematical physics, see [42]. The other variational structure is based on thesymmetry of the Green function $G(x, x’)=G(x’, x)$ coming from actionreaction law, that is the weak formulation of (3),$s$$\frac{d}{dt}\int_{\Omega}\varphi(x)u(x, t)dx=\int_{\Omega}\Delta\varphi(x)\cdot u(x, t)dx$$+ \frac{1}{2}\iint_{\Omega\cross\Omega}\rho_{\varphi}(x, x’)u(x, t)u(x’, t)dxdx’$$\rho_{\varphi}(x, x’)=\nabla_{x}G(x, x’)\cdot\nabla\varphi(x)+\nabla_{x’}G(x, x’)\cdot\nabla\varphi(x’)$(11)

valid to$\varphi\in C^{2}(\overline{\Omega})$3,$\frac{\partial\varphi}{\partial\nu}\partial\Omega^{=0}$

.

(12)

Blowup Analysis-Partial Regularity

The choice $n=2$ of Theorem 2 comes from scaling invariance of the equation
defined on whole space$u_{t}=\triangle u-\nabla\cdot(u\nabla\Gamma*u)$,$u\geq 0$,$-\Delta\Gamma=\delta$

in $R^{n}\cross(0, T)$

(13)

43
that is$u_{\mu}(x, t)=\mu^{2}u(\mu x, \mu^{2}t)$,$\mu>0$.

More presicely, this transformation is consistent to the total mass conservation 1
if and only if $n=2$ . The critical massof thisdimension is then detected by scaling of free energy, that is$u_{\mu}\Vert_{1}=\Vert u\Vert_{1}$$\lambda_{*}=8\pi$$\Vert u_{\mu}\Vert_{1}=\Vert u\Vert_{1}\equiv\lambda$$u_{\mu}(x)=\mu^{2}u(\mu x)$,$\mathcal{F}(u_{\mu})=(2\lambda-\frac{\lambda^{2}}{4\pi})\log\mu+\mathcal{F}(u)$,$\mu>0$ ,where $0\leq u=u(x)\in L^{1}(R^{2})$ . The above critical dimension and thetotal mass are associated with the stationary state of (13),with, that is$\delta \mathcal{F}(u)=0$$\Vert u\Vert_{1}=\lambda$$\log u-\Gamma*u=$ constant,

in

$u>0$

$R^{2}$,$\Vert u\Vert_{1}=\lambda$

.

(14)

In fact, (14) is equivalent to$-\triangle v=e^{v}$

in$R^{2}$,$\int_{R^{2}}e^{v}=\lambda<+\infty$(15)

in terms of $v=\Gamma*u+$ constant, see [41, 42]. Then (15) admits a family of solutions all of which take the value, see [10]. These critical dimension
and threshold mass were already noticed heuristically to examine collapse
formation and blowup threshold [11]. Mass quantization in non-compact stationary solution sequence on bounded domains was actually clarified later[22, 19]. Then two conjectues made in the context of cell biology, collapse
formation [24] and blowup threshold [11], are combined and solved in theaffirmative as the quantized blowup mechanism [41, 42, 32].
The above described scaling invariance is observed in the fundamental equations of mathematical physics which induces hierarchical argument
called blowup analysis. We take a simple example$\lambda=8\pi$$-\triangle v=v^{p}$,$v\geq 0$,$1<p<\infty$

which admits the self-similar transformation$v_{\mu}(x)=\mu^{2/(p-1)}v(\mu x)$,$\mu>0$.

Then we can use blowup analysis to show the following theorem [14].

44
Theorem 3 Given a bounded domain
with smooth boundary$1<p<\frac{n+2}{n-2}$, there is $C>0$ such that any solution $v=v(x)$ to
and$\Omega\subset R^{n}$$\partial\Omega$$-\Delta v=v^{p},$admits the estimate$v>0in\Omega$ ,$\Vert v\Vert_{\infty}\leq C$$v=0on\partial\Omega$.

To prove the above theorem we assume the contrary, existence ofsatisfying$-\triangle v_{k}=\uparrow \mathscr{S}_{k},$$v_{k}>0$in$\Omega$,$v_{k}=0$$m_{k}=v_{k}(x_{k})=\Vert v_{k}\Vert_{\infty}arrow+\infty$Then$\tilde{v}_{k}(x)=\mu^{\frac{2}{kp-1}}v_{k}(\mu_{k}x+x_{k})$on$\{v_{k}\}$$\partial\Omega$.

satisfies the same equation on the rescaled

domain where
and hence. Elliptic
regularity now guarantees the scaled limit $v=v(x)$ satisfying either$\mu_{k}=m_{k}^{-\frac{2}{p-1}}\downarrow 0$,$\Vert\tilde{v}_{k}\Vert_{\infty}=\tilde{v}_{k}(0)=1$

in$-\Delta v=v^{p},$ $0\leq v\leq v(0)=1$ in$-\triangle v=v^{p}$$0\leq v\leq v(0)=1$$R^{n}$$R_{+}^{n}$

,

$v=0$

on

$\partial R_{+}^{n}$(16)

accoding to the approaching speed of. The Liouville property,
towardhowever, guarantees the non-existence of the solution to (16) in case $1<$
, a contradiction. Ingredients of blowup analysis are thus scaling
invariance of the model, control of rescaled solution at infinity, hierarchicalargument, and classification of scaling limit.
Finiteness of blowup points is the simplest partial regularity. It is now
well-known that this property is achieved by -regularity and monotonicity
formula. We take a model harmonic heat flow defined on the torus,$\partial\Omega$$x_{k}$$p< \frac{n+2}{n-2}$$\epsilon$$u_{t}-\Delta u=u|\nabla u|^{2}$,$|u|=1$in$\Omega\cross(0, T)$$u=u(x, t)$ : $\Omega\cross[0, T)arrow S^{n-1}\subset R^{n}$(17)

where $\Omega=R^{2}/aZ\cross bZ$ . It holds that$\Vert u_{t}\Vert_{2}^{2}+\frac{1}{2}\frac{d}{dt}\Vert\nabla u\Vert_{2}^{2}=\int_{\Omega}u\cdot u_{t}|\nabla u|^{2}=\frac{1}{2}\int_{\Omega}(\frac{\partial}{\partial t}|u|^{2})|\nabla u|^{2}=0$(18)which implies the fundamental property, decrease of total energy$\frac{dE}{dt}=-\Vert u_{t}\Vert_{2}^{2}\leq 0$,$E= \frac{1}{2}\Vert\nabla u\Vert_{2}^{2}$

.

(19)

45
There is a self-similar transformation to $u_{\mu}(x, t)=u(\mu x, \mu^{2}t)$ of (17) consistent to $E$ and the classification of the scaled stationary state which is theorigin of energy quantization in the level of stationary states described in
the previous section.
We have -regularity so that there issuch that $u=u(x, t)$ issmooth in $B_{R/2}\cross[0, T]$ provided that$\epsilon_{0}>0$$\epsilon$(20)$\sup E(u(\cdot, t), B_{R})<\epsilon_{0}$$t\in[0,T]$

where$E(u, R)= \frac{1}{2}\int_{B_{R}}|\nabla u|^{2},$$B_{R}=B(0, R)$ .

Inequality (20) means small-ness of local energy in global-in-time which is reduced to the initial state by
the monotonicity formula$E(u(\cdot, T), B_{R})\leq E(u_{0}, B_{2R})+CE_{0}T/R^{2}$,$E_{0}= \frac{1}{2}\Vert\nabla u_{0}\Vert_{2}^{2}$

.

(21)

Theory of partial regularity is thus composed of -regularity derived fromstandard local parabolic theory and monotonicity formula which trades-off$\epsilon$

space and time variables.

4

$2D$

Smoluchowski-Poisson Equation

Global-in-time existence of the solution to (3) is related to the fundamental
properties (4) $-(5)$ , that is the Thrudinger-Moser inequality. This inequality is
associated with the scaling property described above, but the critical massis reduced to a half because the boundary blowup is involved, that is$\inf\{\mathcal{F}(u)|u\geq 0, \Vert u\Vert_{1}=4\pi\}>-\infty$(22)

valid to $n=2$ . Then we obtain the following theorem.
Theorem 4 ([2, 13, 21]) If$\lambda=\Vert u_{0}\Vert_{1}<4\pi$, then $T=+\infty$ in (13).

Formation of collapse with mass estimate from below is done by localizing
the above theorem.

1. Using nice cut-off functions, we show the formation of collapse at eachisolated blowup point $x_{0}\in S$ in the form of$\lim_{R\downarrow 0}\lim_{t\uparrow T}\inf\Vert u(t)\Vert_{L^{1}(\Omega\cap B(x0,R))}\geq m_{*}(x_{0})$provided that $T<+\infty$ .

(23)

46
2. The Gagliardo-Nirenberg inequality guarantees -regularity. There isthus an absolute constantsuch that$\epsilon$$\epsilon_{0}>0$$\Rightarrow$$\lim_{R\downarrow 0}\lim_{t\uparrow T}\sup\Vert u(\cdot, t)\Vert_{L^{1}(\Omega\cap B(x_{0},R))}<\epsilon_{0}$$x_{0}\not\in S$or equivalently$x_{0} \in S\Rightarrow\lim_{t\uparrow T}\sup\Vert u(\cdot, t)\Vert_{L^{1}(\Omega\cap B(x0,R))}\geq\epsilon_{0}$,$\forall_{R>0}$

.

(24)

3. If we can replace limbyin (24), we obtain $\# S<+\infty$by the total mass conservation. Then (23) arises$x_{0}\inS$
for any
because any blowup point is now isolated.
$supt\uparrow T$$\lim\inf_{t\uparrow T}$$\Vert u(t)\Vert_{1}=\Vert u_{0}\Vert_{1}=\lambda$

4. Above replacement is justified by the weak formulation (11)-(12). In
more details, we have(25)$| \frac{d}{dt}\int_{\Omega}u(\cdot, t)\varphi|\leq C_{\varphi}(\lambda+\lambda^{2})$by$\rho_{\varphi}\in L^{\infty}(\Omega\cross\Omega)$

,

(26)

recall $n=2$ , which takes place of the monotonicity formula. Thus$u(x, t)dx=\mu(dx, t)$ is extended
up to$t=T$ . Then (23) implies$\mu(dx, t)\in C_{*}([0, T], \mathcal{M}(\overline{\Omega}))$$\mu(dx, T)=\sum_{xo\in S}m(x_{0})\delta_{x0}(dx)+f(x)dx$(27)

with $m(x_{0})\geq m_{*}(x_{0})$ and $0\leq f=f(x)\in L^{1}(\Omega)$ .
The reverse inequality$m(x_{0})\leq m_{*}(x_{0})$(28)regarded as a localization of blowup threshold. Global blowup criterion
follows from the weak formulation (26) and the second moment of which
plot-type argument is founded in [3]. In the case of (13), thus, the equality$\frac{d}{dt}\int_{R^{2}}|x|^{2}u(\cdot, t)=4\lambda-\frac{\lambda^{2}}{2\pi}$holds under $u_{0}\in L^{1}(R^{2}, (1+|x|^{2})dx)$ which implies $T<+\infty$ in case.
Employing the method of localization to the above described second moment
argument, we obtain the following theorem [30].
$\lambda>8\pi$47Theorem 5 There is a constant $\eta=\eta(m)>0$ such that if we have
and $0<R\ll 1$ satisfying$x0\in R^{2}$$\frac{1}{R^{2}}\int_{\Omega\cap B(x_{0},2R)}|x-x_{0}|^{2}u_{0}<\eta$,$m= \int_{\Omega\cap B(x_{0},R)}u_{0}>m_{*}(x_{0})$(29)then $T=T_{\max}<+$ oo.

The following are the proof of (28).
1. We take the backward self-similar variables defined by$y=(x-x_{0})/(T-t)^{1/2}$ ,$s=-\log(T-t)$ ,$t<T$ .

(30)

Then the transformation consistent to the ODE blowup rate$z(y, s)=(T-t)u(x, t)$ , $w(y, s)=v(x, t)$(31)

inducesin$z_{s}=\nabla\cdot(\nabla z-z\nabla(w+|y|^{2}/4))$$- \triangle w=z-\frac{\lambda}{|\Omega_{s}|}$

where

in

$Q$

,

$\frac{\partial w}{\partial\nu}=0$

$Q$

on ,
$\Gamma$$\int_{\Omega_{s}}w=0$(32), and $\Omega_{s}=(T-$$t)^{-1/2}(\Omega-\{x_{0}\})$ .
Similarly to the pre-scaled case there is genemtion
admitsand $\zeta(dy, s)$of weak solution so that any
such that$z(y, s+s_{k}’)dyarrow\zeta(dy, s)$(33)$Q= \bigcup_{s>-\log T}\Omega_{s}\cross\{s\},$$\Gamma=\bigcup_{s>-\log T}\partial\Omega_{s}\cross\{s\}$$s_{k}\uparrow+\infty$$\{s_{k}’\}\subset\{s_{k}\}$$(-oo, +\infty;\mathcal{M}_{0}(R^{2}))$ . Here,in
0-extension of $z(y, s)$ is taken where
it is not defined, $\mathcal{M}_{0}(R^{2})=C_{0}(R^{2})’$ , anddenotes the set ofcontinuous functions on
taking at . This $\zeta(dy, s)$ is a(finite) Radon measure on
for each . Contrary to the pre-scaled
case, $y=\infty$ is excluded in convergence (33) to derive collapse massestimate above.
$C_{*}$$C_{0}(R^{2})$$R^{2}\cup\{\infty\}$$R^{2}$

2. Taking even extension in caseweak solution to$0$$\infty$$s$$x_{0}\in\partial\Omega$$z_{s}=\nabla\cdot(\nabla z-z\nabla(\Gamma*z+|y|^{2}/4))$, the above $\zeta(dy, s)$ becomes ain $R^{2}\cross(-\infty, +\infty)$ .

(34)

48

There is thus

$0\leq \mathcal{K}=\mathcal{K}(\cdot, s)\in \mathcal{E}’,$$\Vert \mathcal{K}(\cdot, s)\Vert_{\mathcal{E}’}\leq\lambda^{2}$, a.e.

$s$such that$\mathcal{K}(\cdot, s)|_{X}=\zeta(dy, s)\otimes\zeta(dy’, s)$$\frac{d}{ds}\langle\varphi,$$\zeta(dy, s)\rangle=\langle\Delta\varphi+\frac{1}{2}y\cdot\nabla\varphi,$$\zeta(dy, s)\rangle+\frac{1}{2}\langle\rho_{\varphi}^{0},$

for
where
lutely continuous. Here$\varphi\in C_{0}^{2}(R^{2})$$s\in(-\infty, +\infty)\mapsto\langle\varphi,$ $\zeta(dy, s)\rangle$$\mathcal{K}(\cdot, s)\rangle$

a.e.

$s$

is locally abso-$\rho_{\varphi}^{0}=\rho_{\varphi}^{0}(y, y’)=-\frac{y-y’}{2\pi|y-y’|^{2}}\cdot(\nabla\varphi(y)-\nabla\varphi(y’))$and$\mathcal{E}$is the closure of the linear hull of$\mathcal{E}_{0}$defined by$\mathcal{E}_{0}=\{\rho_{\varphi}+\psi|\varphi\in C_{0}^{2}(R^{2}), \psi\in X\}$

$X=C_{0}(R^{2}\cross R^{2})\oplus[(C_{0}(R^{2})\oplus R)\otimes R]\oplus[R\otimes(C_{0}(R^{2})\oplus R)]$with

$C_{0}(R^{2}\cross R^{2})$standing for the set of continuous functions onvanishing at$(R^{2}\cup\{\infty\})\cross(R^{2}\cup\{\infty\})$$(R^{2}\cup\{\infty\})]$.

$[(R^{2}\cup\{\infty\})\cross\{\infty\}]\cup[\{\infty\}\cross$

3. We use pambolic envelope, infinitely large parabolic region. Henceforth
is a function satisfying (12) and$\varphi=\varphi_{x0,R}$$\overline{\Omega}\cap\overline{B(x_{0},R/2)}$$\varphi_{x_{0},R}(x)=\{\begin{array}{l}1,0,\end{array}$$\overline{\Omega}\backslash B(x_{0}, R)$.

First, we refine (25) as(35)$| \frac{d}{dt}\int_{\Omega}u(\cdot, t)\varphi_{x_{0},R1}\leq C(\lambda+\lambda^{2})R^{-2}$with a constant $C>0$ independent of $0<R\leq 1$ which implies$| \int_{\Omega}\varphi_{x0,R}u(\cdot, t)-\langle\varphi_{x_{0},R},$$\mu(dx, T)\rangle|\leq C_{\lambda}R^{-2}(T-t)$$\lim_{b\uparrow+\infty}\lim_{t\uparrow}\sup_{T}|\langle\varphi_{x0,b(T-t)^{1/2}},$$\mu(dx, t)\rangle-m(x_{0})|=0$

.

(36)

It says that infinitely wide parabolic region in $xt$ space associated withbackward self-similar variables contains the whole blowup mechanism.

We thus obtain$\hat{m}(x_{0})=\zeta(R^{2}, s)$,$-\infty<s<+\infty$

(37)

49
for
$\hat{m}(x_{0})=\{\begin{array}{ll}m(x_{0}), x_{0}\in\Omega 2m(x_{0}) x_{0}\in\partial\Omega\end{array}$which means that collapse mass is equal to total mass of rescaled weaklimit with $y=\infty$ excluded in the limit process.

4. We take the scale-back $\zeta(dy, s)=e^{-s}A(dy’, s’),$ $y’=e^{-s/2}y,$ $s’=-e^{-s}$to (34) and obtain weak solution to$A_{s}=\nabla\cdot(\nabla A-A\nabla\Gamma*A)$,$A\geq 0$

in $R^{2}\cross(-$ oo, $0)$$-\infty<s<0$ .

$A(R^{2}, s)=\hat{m}(x_{0})$Here we use$| \frac{d}{ds}\langle\varphi,$$A(dy, s)\rangle|\leq C_{\varphi}$,$A(dy, s)\geq 0$ ,$A(R^{2}, s)=m(xo)$to take the translation weak limit where
R. Anythus admits
such that $A(dy, s-s_{k}’)arrow a(dy, s)$ in$(-oo, +\infty;M(R^{2}))$ where $a=a(dy, s)$ is a weak solution to$\varphi\in C_{0}^{2}(R^{2})\oplus$$+\infty$$s_{k}\uparrow$$\{s_{k}’\}\subset\{s_{k}\}$$C_{*}$$a_{s}=\nabla\cdot(\nabla a-a\nabla\Gamma*a)$,$a\geq 0$$a(R^{2}, s)=\hat{m}(x_{0})$in $R^{2}\cross(-oo, +\infty)$$-\infty<s<+\infty$(38)

and $\mathcal{M}(R^{2})=[C_{0}(R^{2})\oplus R]’$ . Here we use
to envelope the$a(R^{2},s)=\hat{m}(x_{0})$rescaled total mass which gurantees. Then (28) is
derived similarly to the classical solution done by [18], see below.
$\mathcal{M}(R^{2})$5. First we use the local second moment. We obtain$\frac{d}{ds}\langle c(|y|^{2})+1,$ $a(dy, s)\rangle\leq

a.e.

$s$

C\langle c(|y|^{2})+1,$ $a(dy, s) \rangle+\delta\hat{m}(x_{0})(4-\frac{\hat{m}(x_{0})}{2\pi})$with $C>0,$ $\delta>0$ where$0\leq c’(s)\leq 1,$$s\geq 0$,$-1\leq c(s)\leq 0,$ $s\geq 0$$c(s)=\{\begin{array}{ll}s-1, 0\leq s\leq 1/40, s\geq 4.\end{array}$

In case $\hat{m}(x_{0})>8\pi$ , therefore, the condition$\langle c(|y|^{2})+1,$ $a(dy, 0) \rangle<\eta=\frac{\delta\hat{m}(x_{0})}{2\pi C}(\hat{m}(x_{0})-8\pi)$50
implies$\langle c(|y|^{2})+1,$ $a(dy, s)\rangle<0$,$s\gg 1$a contradiction. Hence it must hold that$\langle c(|y|^{2})+1,$ $a(dy, 0)\rangle\geq\eta$

(39)

.

6. Problem (38) is invariant under the transformation$a^{\mu}(dy’, s’)=\mu^{2}a(dy, s)$

and so is true for$\eta$,$y’=\mu y,$$s’=\mu s$,$\mu>0$in (39). In case $\hat{m}(x_{0})>8\pi$ , therefore, we obtain$\langle c(|y|^{2})+1,$ $a^{\mu}(dy, 0)\rangle\geq\eta$$\langle c(\mu^{-2}|y|^{2})+1,$ $a(dy, 0)\rangle\geq\eta$,$\forall_{\mu>0}$.

Then we apply the dominated convergence theorem using$0\leq c(\mu^{-2}|y|^{2})+1\leq 1$

as$\mu\uparrow+\infty$

which implies,$\eta\leq 0$$c(\mu^{-2}|y|^{2})+1arrow 0$,$\forall_{y\in R^{2}}$, a contradiction.

Having proven (6) $-(7)$ , now we can show the following theorem.
Theorem 6 ([28, 23]) Every $x_{0}\in S$ is of type

. More strongly, $we$

$\Pi$

have

$\lim_{t\uparrow T}(T-t)\Vert u(\cdot, t)\Vert_{L(B(x_{0},b(T-t)^{1/2}))}\infty=+\infty$

for any $b>0$ and$z(y, s+s’)dyarrow 8\pi\delta_{0}(dy)$in$C_{*}(-\infty, +\infty;\mathcal{M}(R^{2}))$

as$s’\uparrow+\infty$

(40)
(41)

.

Relation (41) is called formation of sub-collapse which says that total blowup mechanism is enclosed in hyper-pambola, infinitesimally small
parabolic region. In fact it is easy to see that (41) implies (40), while thelatter is proven as follows.

1. Similarly to (35) we obtain$| \frac{d}{dt}\int_{\Omega}|x-x_{0}|^{2}u(\cdot, t)\varphi_{x0,R1}\leq C(\lambda+\lambda^{2})$51which implies$\int_{\Omega}|x-x0|^{2}u(\cdot, t)\varphi_{x0,R}\leq C_{\lambda}(T-t)+\int_{\Omega}|x-x_{0}|^{2}\varphi_{x0,R}f(x)dx$

and hence$\int_{\Omega}|\frac{x-x_{0}}{R(t)}|^{2}\varphi_{x_{0},bR(t)}\cdot u(\cdot, t)\leq C+4b^{2}\langle\varphi_{x0,bR(t)},$ $f\rangle$$b>0$ .

$R(t)=(T-t)^{1/2}$ ,2. Given, we take. Inequality (42) implies$t_{k}\uparrow T$$\{s_{k}’\}\subset\{s_{k}\}$$t_{k})$$\langle|y|^{2},$with$b\uparrow+\infty$(42)

satisfying (33) for $s_{k}=-\log(T-$and hence$\langle|y|^{2}\varphi_{x0,b},$$\zeta(dy, s)\rangle\leq C$,$\zeta(dy, s)\rangle\leq C$(43)$-\infty<s<\infty$.

3. Putting $I(s)=\langle|y|^{2},$$\zeta(dy, s)\rangle$, now we obtain$\frac{dI}{ds}=4\hat{m}(x_{0})-\frac{\hat{m}(x_{0})^{2}}{2\pi}+I=I$

by (32) and (37) which implies(43). It follows that$\langle|y|^{2},$a.e. $s\in R$$\zeta(dy, s)\rangle=0,$

$-$

(44)

oo $<s<\infty$ by(45)$\zeta(dy, s)=8\pi\delta_{0}(dy)$and hence (40).

Here we mention the problem of quantization of blowup in infinite time ofthe solution to (13) which says that $T=+\infty$ withwill imply. This expected property indicates the nonstationary redisual vanishing which is assured for non-compact sequence ofstationary states [19]. We also expect that the movement of collapse formed
in infinite time is subject to a Hamiltonian. These profiles are actually thecases of radially symmetric solutions and that of the first critical mass, respectively, see [25]. The fundamental property of $\sup+$ inf inequality valid
to the stationary problem, however, will not hold to the non-stationary problem. In [1], the repulsive self-interaction is studied where (32) is replacedby$v_{t}=\nabla\cdot(\nabla v+v\nabla(\Gamma*z+|x|^{2}/4))$
in $R^{n}\cross(0, +\infty)$ .

$\lim_{t\uparrow+\infty}\Vert u(t)\Vert_{\infty}=+\infty$$\lambda=\Vert u_{0}\Vert_{1}\in 8\pi N$

In this case there is exponential decay of a relative entropy.

52

5

Harmonic Heat Flow

The following theorem is obtained by the above described -regularity andmonotonicity formula for the model harmonic heat flow [34].
$\epsilon$

Theorem 7 There is a global-in-time-solution to (17) with finite
singular points inwhere $\Omega=R^{2}/aZ\cross bZ$ .
$H^{1}$$\Omega\cross[0, +\infty)$

We show the key ingredient, monoticity formula (21). This property is
regarded as bounded variation in time of local energy. For the harmonicheat flow, this property is derived from the definite sign of the density of $E$ ,, more precisely,that is$|\nabla u|^{2}$(46)$\frac{1}{2}\sup_{t\in[0,T)}\Vert\nabla u(\cdot, t)\Vert_{2}^{2}+\int_{0}^{T}\Vert u_{t}(\cdot, s)\Vert_{2}^{2}ds\leq\frac{1}{2}\Vert\nabla u_{0}\Vert_{2}^{2}=E_{0}$by (18). We have also$\int_{\Omega}|u_{t}|^{2}\varphi^{2}+\int_{\Omega}\nabla u\cdot\nabla(u_{t}\varphi^{2})=\frac{1}{2}\int_{\Omega}[\frac{\partial}{\partial t}|u|^{2}]|\nabla u|^{2}\varphi^{2}=0$

with$\int_{\Omega}\nabla u\cdot\nabla(u_{t}\varphi^{2})$$=$$\int_{\Omega}(\nabla u\cdot\nabla u_{t})\varphi^{2}+\int_{\Omega}[(u_{t}\cdot\nabla)u]\cdot\nabla\varphi^{2}$$=$$\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla u|^{2}\varphi^{2}+\int_{\Omega}[(u_{t}\cdot\nabla)u]\cdot\nabla\varphi^{2}$,

and, therefore,$\int_{\Omega}|u_{t}|^{2}\varphi^{2}+\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla u|^{2}\varphi^{2}+\int_{\Omega}[(u_{t}\cdot\nabla)u]\cdot\nabla\varphi^{2}=0$

for each$\varphi\in C^{1}(\Omega)$. It holds that$| \frac{1}{2}\frac{d}{dt}\int_{\Omega}|\nabla u|^{2}\varphi^{2}|$$\leq$$\leq$$\Vert u_{t}\Vert_{2}^{2}\Vert\varphi\Vert_{\infty}^{2}+\Vert u_{t}\Vert_{2}\cdot\{\int_{\Omega}|\nabla u|^{2}|\nabla\varphi^{2}|^{2}\}^{1/2}$$\Vert u_{t}\Vert_{2}^{2}\Vert\varphi||_{\infty}^{2}+\sqrt{2}\Vert u_{t}\Vert_{2}\cdot E_{0}^{1/2}\Vert\nabla\varphi^{2}\Vert_{\infty}$(47)

and hence$\int_{0}^{T}|\frac{d}{dt}\int_{\Omega}|\nabla u|^{2}\varphi^{2}|dt<+\infty$by (46). Then (21) follows from (46)-(47).

(48)

53
Inequality (48) assures collapse formation with finite blowup points,(49)$| \nabla u(x, t)|^{2}dxarrow\sum_{xo\in S}m(x_{0})\delta_{x_{0}}(dx)+f(x)dx$

as$t\uparrow T$

in$\mathcal{M}(\Omega)$with $m(x_{0})\geq\epsilon_{0},0\leq f=f(x)\in L^{1}(\Omega)$ where$\mathcal{B}=\{x_{0}\in\Omega|\lim_{t\uparrow}\sup_{T}\Vert\nabla v(\cdot, t)\Vert_{L^{2}(B(x0,R))}<+\infty$,$S=\Omega\backslash \mathcal{B}$$\text{ョ_{}R>0\}}$,.

Inequality (47) implies also$| \int_{\Omega}|\nabla u(\cdot, t)|^{2}\varphi_{x_{0},R}^{2}-\int_{\Omega}f(x)\varphi_{x_{0},R}^{2}-m(x_{0})|\leq 2Cl^{T}\Vert u_{t}(\cdot, s)\Vert_{2}^{2}ds$$+2\sqrt{2}E_{0}^{1/2}\{l^{T}\Vert u_{t}(\cdot, s)\Vert_{2}^{2}ds\}^{1/2}C(T-t)^{1/2}/R$,$x_{0}\in S$

with $C>0$ independent of $0<R\ll 1$ . Then it follows that$\lim_{t\uparrow T}\int_{\Omega}|\nabla u(\cdot, t)|^{2}\varphi_{x,bR(t)}^{2_{0}}=m(x_{0})$from (46) again, where $b>0$ is arbitrary and $R(t)=(T-t)^{1/2}$ . Hyperparabola thus arises and we obtain type (II) blowup rate at each, see Theorem 12 below concerning (51) for the proof. Inequality (46),furthermore, implies$x_{0}\in$

$S$

(50)

$\lim_{t\uparrow T}\inf(T-t)\Vert u_{t}(t)\Vert_{2}^{2}arrow 0$

which assures a stationary profile of, see [50].
as
The convergence (49) is compatible to $u(\cdot, t)arrow u(\cdot, T),$ $|\nabla u(\cdot, T)|^{2}=f$asinbecause this convergence is valid also in.
Then the solution is extended beyond $t=T$ with the updated initial value$u(\cdot, T)\in H^{1}(\Omega)$ .
This process ends after finitely many times. The solutionin Theorem 7 is thus constructed with weak continuity in.
The concentration lemma [31] guarantees to generalize the initial value$u_{0}=u_{0}(x)\geq 0$ tofunction in (3). This solution is defined as the limitof regular solutions for regularized initial values and becomes regular for$0<t\ll 1$ . If we take $0\leq f=f(x)\in L^{1}(\Omega)$ in (6), we can extend the
blowup solution beyond $t=T$ and this process ends with finitely manytimes. This solution may be comparable to Struwe’s solution for harmonicheat fiow in Theorem 7, although its continuity is not certain in standard
function spaces.
$u(\cdot, t)$$t\uparrow T$$H^{1}(\Omega)$$t\uparrow T$$C_{loc}^{1}(\Omega\backslash S)$$H^{1}(\Omega)$$L^{1}$

54

6Mass Quantization in Higher Dimension
Degenerate parabolic equation is introduced by [8] to describe the motionof the mean field of many self-interacting particles, that is$u_{t}= \frac{m-1}{m}\Delta u^{m}-\nabla\cdot(u\nabla\Gamma*u)$,$u\geq 0$

in $R^{n}\cross(0, T)$(51)$\Gamma(x)=\frac{1}{\omega_{n-1}(n-2)|x|^{n-2}}$

denoting the area of the boundary of the unit ballinso that. First, the particle density at $(x, t)\in R^{n}\cross(0, T)$with the velocity $v\in R^{n}$ is denoted by $0\leq f=f(x, v, t)$ which satisfies the
kinetic equation
with $n\geq 3$ , and$R^{n}$$\omega_{n-1}$$-\Delta\Gamma=\delta$$f_{t}+v\cdot\nabla_{x}f-\nabla\varphi\cdot\nabla_{v}f=-\nabla_{v}\cdot j$provided with the general dissipation flux $term-\nabla_{v}\cdot j$ , where is the gravitational potential generated by $f$ . We have the density-pressure relationand the Poisson equation$\varphi$$p=p(\mu, \theta)$

,

(52)

$\triangle\varphi=\mu$

where and stand for the pressure and the temperature, respectively. The
dissipation flux
, next, is determined by the maximum entropy
production principle, so that maximize the local entropy$\theta$$p$$term-\nabla_{v}\cdot j$$f$$S= \int_{R^{n}}s(f(x, v, t))dv$

under the constraint$\mu(x, t)=\int_{R^{n}}f(x, v, t)dv,$$p(x, t)= \frac{1}{n}\int_{R^{n}}|v|^{2}f(x, v, t)dv$.

Averaging over the velocities $v\in R^{n}$ and the passage to the limit of large
friction or large times lead to$f$$\mu_{t}=\nabla[D_{*}\cdot(\nabla p+\mu\nabla\varphi)]$

,

(53)

that is a hydrodynamical limit of self-gravitating particles whereby the total

mass$\lambda=\int_{R^{n}}\mu(x, t)dx$

is conserved during the evolution. We have, thus,

several mean field equations according to the entropy function $s(f)$ subject

55
to the law of partition of macroscopic states of particles into mezoscopic
states, that is the entropies of Boltzmann, Fermi-Dirac, Bose-Einstein, andso forth. System (52)-(53) is still under-determined, and there are several
theories to prescribe the temperature . In the canonical statistics one takes
the iso-thermal setting, and hence the temperature is a constant. In the
micro-canonical statistics, on the other hand, is a function of while thetime-independent total energy
is prescribed in$\theta$$\theta$$\theta$$t$$E= \frac{n}{2}\int_{\Omega}pdx+\frac{1}{2}\int_{\Omega}\mu\varphi dx$

advance. R\’enyi-Tsallis’ entropy $S= \frac{-1}{q-1}\int_{R^{n}}(f^{q}-f)dv$ is q-analogue ofthe Boltzmann entropy. Adopting this entropy, (52) takes
where $\kappa>0$ is a constant and, see [9, 4]. Normalizing physicalconstants and taking the iso-thermal setting, we can reduce (52)-(53) to thedegenerate parabolic equation (51) where the new unknown is a positive
constant times and. When $n=3$ and, the case$m=2- \frac{2}{n}=\frac{4}{3}$ arises to (51).
Equation (51) with $m=2- \frac{2}{n}$ shares,
actually, similar variational and scaling properties to the SmoluchowskiPoisson equation (13) with $n=2$ .
The solution to (51) which we handle with is the weak solution formulated by [43]. Given the initial value$p=\kappa\theta^{1-\frac{\gamma n}{2}}\mu^{1+\gamma}$$\frac{1}{\gamma}=\frac{1}{q-1}+\frac{n}{2}$$u$$\frac{1}{m-1}=\frac{1}{q-1}+\frac{n}{2}$$\mu$$q= \frac{5}{3}$$0\leq u_{0}\in L^{1}(R^{n})\cap L^{\infty}(R^{n})$we thus take the approximate solution,$u_{0}^{m}\in H^{1}(R^{n})$$u_{\epsilon}=u_{\epsilon}(x, t)$$u_{\epsilon t}= \frac{m-1}{m}\triangle(u_{\epsilon}+\epsilon)^{m}-\nabla\cdot(u_{\epsilon}\nabla\Gamma*u_{\epsilon})$

,

(54)

satisfying
in $R^{n}\cross(0, T)$

for $0<\epsilon\ll 1$ with sufficiently regular initial value
and takethe limit process. If the existence time of the weak solution denoted$T=T_{\max}\in(0, +\infty]$ is finite, then$u_{0\epsilon}=u_{0\epsilon}(x)$$\epsilon\downarrow 0$$u$$\lim_{t\uparrow T}\Vert u(t)\Vert_{\infty}=+\infty$

For the proof, first,
while$u_{\epsilon}=u_{\epsilon}(x, t)$

.

(55)

is extended as far as 1

$B \equiv\sup_{t\in(0,T)}\Vert u_{\epsilon}(t)\Vert_{\infty}\leq\frac{\Vert u_{0}||_{\infty}}{1-T\Vert u0\Vert_{\infty}}$,$u_{\epsilon}(t)\Vert_{\infty}$

is bounded,$0<T<\Vert u_{0}\Vert_{\infty}^{-1}$holds by the-energy method [27]. Then we derive several estimates of
uniform in $0<\epsilon\ll 1$ and $0\leq t\leq T$ to take the limit. Then the$L^{\infty}$$u_{\epsilon}(\cdot, t)$

56
existence time of the weak solution $u=u(\cdot, t)$ is bounded from below using, and, consequently, it extends in as far as
is bounded.
$T=T_{\max}<+\infty$
In particular, (55) arises in the case
. The scheme toconstruct weak solution is due to [38, 36, 35] where the Besse potential isused instead of . We compensate the lack of decay at the infinity of by
the decomposition $\nabla\Gamma*u=g_{1}+g_{2}$ ,$t$$\Vert u_{0}\Vert_{\infty}$$\Vert u(t)\Vert_{\infty}$$\Gamma$$\Gamma$$g_{1}=[\nabla\Gamma\cdot\chi_{R^{n}\backslash B(0,1)}]*u$

,

(56)

$g_{2}=[\nabla\Gamma\cdot\chi_{B(0,1)}]*u$

and the $Calder6n$-Zygmund estimate$\Vert D^{2}u\Vert_{p}\leq C(n,p)\Vert\Delta u\Vert_{p}$,$u\in W^{2,p}(R^{n}),$$1<p<\infty$ .

We emphasize that the blowup criterion (55) is concerned with the weaksolution. In [20], for instance, the unique existence of the classical solution
local in time is proven for $n=1,$ $m=3$ , and, provided with$u_{0}=u_{0}(x)$the regular non-negative initial value. If
has compact
support, furthermore, this classical solution breaks down in finite time exposing the profile,. Thus we have the continuation after $t=T_{c}$ of this classical solution
as a weak solution. Several arguments described below are formal becauseof the lack of sufficient regularity of the solution althoug are justified by the$(-\triangle+1)\Gamma=\delta$$u_{0}\not\equiv 0$$T_{c}$$\lim_{t\uparrow T_{c}}\Vert\partial_{x}u(t)\Vert_{\infty}=+\infty,$$\lim\sup_{t\uparrow T_{c}}\Vert u(t)\Vert_{p}<+\infty$$1\leq$$p\leq\infty$

approximate solution.
First, (51) is a model $B$ equation, see [42], associated with the free energy$\mathcal{F}(u)=\int_{R^{n}}\frac{u^{m}}{m}dx-\frac{1}{2}\langle\Gamma*u,$$u\rangle$,$u\geq 0$.

In fact we have$\langle v,$$\delta \mathcal{F}(u)\rangle=\frac{d}{ds}\mathcal{F}(u+sv)_{s=0}=(v, u^{m-1}-\Gamma*u)$-inner product , . Under this identification $\delta \mathcal{F}(u)=u^{m-1}-$
we can write (51) as
in $R^{n}\cross(0, T)$ which implies
the the total mass conservation and the decrease of the free energy
using the$\Gamma*u$$L^{2}$$($$)$$u_{t}=\nabla\cdot u\nabla\delta \mathcal{F}(u)$$\Vert u(t)\Vert_{1}=\Vert u_{0}\Vert_{1}=\lambda$,$\frac{d}{dt}\mathcal{F}(u)=-\int_{R^{n}}u|\nabla\delta \mathcal{F}(u)|^{2}\leq 0$

.

(57)

Next we examine scaling invariance compatible to total mass conservation,that is$u_{\mu}(x, t)=\mu^{n}u(\mu x, \mu^{n}t)$ ,$\mu>0$(58)

57
valid to $m=2- \frac{2}{n}$ . The Trudinger-Moser inequality(59)$\inf\{\mathcal{F}(u)|u\geq 0, suppu\subset B(0, R), \int_{R^{n}}u=\lambda_{*}\}>-\infty$

actually arises for each $R>0$ to this exponent $m=2- \frac{2}{n}$ , see [52, 51, 33],
and the threshold value
is detected by the stationary state of (51),$\lambda_{*}$$u^{m-1}-\Gamma*u=$ constantin $\{u>0\}$ ,$\int_{R^{n}}u=\lambda$

.

(60)

Then $v=\Gamma*u+$ constant satisfies$-\triangle v=v_{+}^{q}$in$R^{n}$,(61)$\int_{R^{n}}v_{+}^{q}=\lambda$

where $m=1+ \frac{1}{q}$ and hence. Problem (61) for this exponent isinvariant under the scaling transformation $v_{\mu}(x)=\mu^{n-2}v(\mu x),$ $\mu>0$ .
This stationary problem admits a family of solutions each of which isnecessarily radially symmetric, and determines the threshold value
uniquely. There is also mass quantization to the non-compact stationary
solution sequence on bounded domains [51, 47]. Threshold value
is alsoprescribed by the best constant $C(n)$ of the Hardy-Littlewood-Sobolev inequality$q= \frac{n}{n-2}$$\lambda=\lambda_{*}$$\lambda_{*}$$|\langle f,$$\Gamma*f\rangle|\leq C(n)\Vert f\Vert_{m}^{m}\Vert f\Vert_{1}^{2/n}$,$m=2- \frac{2}{n}$,see [5]. There is a difference between (61) and the two-dimensional problem(15), however, thathas a compact support. The other difference is scaling
propery of free energy
which refines (59) as$v_{+}^{q}$$\mathcal{F}(u_{\mu})=\mu^{n-2}\mathcal{F}(u)$$j_{*}= \inf\{\mathcal{F}(u)|0\leq u\in L^{m}(R^{n}), \int_{R^{n}}u=\lambda_{*}\}=0$

Ifis the case we obtainand (62). Moser’s iteration scheme guarantees
and hence $T=+\infty$ . Assuming$\Vert u_{0}\Vert_{1}=\lambda<\lambda_{*}$.

$\lim\sup_{t\uparrow T}\Vert u(t)\Vert_{m}<+$$|x|^{2}u_{0}\in L^{1}(R^{n})$(62)oo by (57)$\lim\sup_{t\uparrow T}\Vert u(t)\Vert_{\infty}<+\infty$

,

(63)

on the other hand, we have$\frac{d}{dt}\int_{R^{n}}|x|^{2}u(\cdot, t)$$=$$\frac{m-1}{m}\cdot 2n\int_{R^{n}}u^{m}(\cdot, t)-(n-2)\langle\Gamma*u,$$=$$2(n-2)\mathcal{F}(u)$$u\rangle$

(64)

58
which implies $T<+\infty$ in case$\mathcal{F}(u_{0})<0$. Since (62) is sharp and$\inf\{\mathcal{F}(u)|0\leq u\in L^{m}(R^{n}), \int_{R^{n}}u=\lambda\}=-\infty$

each
and 1

$\lambda>\lambda_{*}$$0$

admits $u_{0}=u_{0}(x)\geq 0$ with compact support such that$u_{0}\Vert_{1}=\lambda$$\mathcal{F}(u_{0})<$.

Theorem 8 ([5, 44]) For
determined by the dimension $n\geq 3$
it holds that if $u_{0}=u_{0}(x)$ is the initial value satisfying (54), (63), and, then $T=+\infty$ in (51) for $m=2- \frac{2}{n}$ . Each, on theother hand, takes $u_{0}=u_{0}(x)$ such that (54), (63),, and $T<+\infty$ .
$\lambda_{*}>0$$\Vert u_{0}\Vert_{1}<\lambda_{*}$$\lambda>\lambda_{*}$$\Vert u_{0}\Vert_{1}=\lambda$

Localization of the above criteria, particularly, deriving monotonicity formula, however, has not been achieved. This difficulty is due to the above
described different relation from $2D$ Smolchowski-Poisson equation, between

total mass, free energy, and second moment of . In fact we have
and $x\cdot\nabla\Gamma=-(n-2)\Gamma$ for
defined by (13) and (51), respectively.
The blowup set is defined by,$x\cdot\nabla\Gamma=$$u$$\Gamma(x)$$- \frac{1}{2\pi}$$S=R^{n}\backslash \mathcal{B}$$\mathcal{B}=\{x0\in R^{n}|^{\text{ョ}}r>$such that$\lim\sup_{t\uparrow T}\Vert u(t)\Vert_{L(B(x_{0},r))}\infty<+\infty\}$which is non-empty because weak solution $u=u(x, t)$ satisfies the blowup
criterion (55) for $T<+\infty$ . To confirm the blowup rate, next, we write (51)as$u_{t}= \frac{m-1}{m}\triangle u^{m}-\nabla u\cdot\nabla\Gamma*u+u^{2}$and take the ODE part
. It follows that $((t)=(T-t)^{-1}$ which
defines type I blowup rate of $u=u(x, t)$ , that is $\Vert u(t)\Vert_{\infty}=O((T-t)^{-1})$ .
Then we say that $x_{0}\in S$ is type I if$\dot{\zeta}=(^{2}$$\lim_{t\uparrow T}\inf(T-t)\Vert u(t)\Vert_{L^{\infty}(B(x_{0},ro))}<+\infty$

for some $r_{0}>0$ and type II in the other case. Then we obtain finitenessof type II blowup points, recall, in $2D$ Smoluchowski-Poisson equation (13)any $x_{0}\in S$ is type II.

Theorem 9 ([46]) Let $u_{0}=u_{0}(x)$ be the initial value satisfying (54)and (63), and assume $T<+\infty$ for the above described weak solution $u=$$u(x, t)$ to (51) with $m=2- \frac{2}{n}$ . Thenis bounded andis finite where$S$$S_{II}$$S_{II}= \{x_{0}\in S|\lim_{t\uparrow T}(T-t)||u(t)\Vert_{L\infty(B(x0,r_{0}))}=+\infty,$$\forall_{r_{0}}>0\}$.

59
The first step to prove Theorem 9 is the -regularity stated below.
$\epsilon$

Theorem 10 ([45]) We have
$0<R\ll 1$ such that$\sup\Vert u(t)\Vert_{L^{1}(B(x0,R))}<\epsilon_{0}$$\epsilon_{0}>0,$$\Rightarrow$$t\in(0,T)$$C>0$ independent of $x_{0}\in R^{n}$ ,(65)$\sup\Vert u(t)\Vert_{L\infty(B(x_{0},R/2))}\leq C$$t\in(0,T)$

where $u=u(x, t)$ is a weak solution to (51).
For the proof of Theorem 10 we use the decomposition (56),$v=\Gamma*u=v_{1}+v_{2}$,$v_{1}=[\Gamma\cdot\chi_{R^{n}\backslash B(0,1)}]*u$,$v_{2}=[\Gamma\cdot\chi_{B(0,1)}]*u$combined with the relation $v=G*v_{I}+G*v_{2}+G*u$ where $G=G(x)$ is
the Bessel potential satisfying $(-\triangle+1)G=\delta$ . This potential, studied by[37], decays exponentially at infinity. Direct consequence of Theorem 10 isthe boundedness of the blowup set derived from (64). In fact we have$S$$\int_{R^{n}}|x|^{2}u(\cdot, t)\leq C(T, u_{0})=2(n-2)T\mathcal{F}(u_{0})+\int_{R^{n}}|x|^{2}u_{0}$and hence$\lim_{t\uparrow}\sup_{T}\int_{|x|>R}u(\cdot, t)\leq R^{-2}C(T, u_{0})$. We obtain,$S\subset R^{n}\backslash B(0, R)$taking $R\gg 1$ as $C(T, u_{0})R^{-2}<\epsilon_{0}$ ,The constantin (65) is involved by the initial value. This propery,
however, is compensated by the parabolic regularity concerning local normsof the solution noticed by [31] to (13).
$C$

Theorem 11 ([46]) Given $r\in[2, \infty)$ and $R>0$ , we have $0<\epsilon_{r}\ll 1$such that$\sup\Vert u(t)\Vert_{L^{1}(B(x2R))}0,<\epsilon_{r}$$\Rightarrow$$\Vert u(t)\Vert_{L^{r}(B(x0,R))}\leq t^{-1}$,$0<t\leq 1$.

$t\in(0,1)$

Given $x_{0}\in S$ and $0<R\ll 1$ , we take $0\leq\varphi=\varphi_{x0,R}(x)\in C_{0}^{\infty}(R^{n})$satisfying$supp\varphi\subset\overline{B(x_{0},R)},$ $\varphi=1$

on $B(x_{0}, R/2)$ . Put$A(t)= \int_{R^{n}}\varphi u(\cdot, t)$.

First, it holds that$| \frac{d}{dt}\int_{R^{n}}\varphi u|2 =| \int_{R^{n}}u\nabla(u^{m-1}-\Gamma*u)\cdot\nabla\varphi|^{2}\leq\int_{R^{n}}u|\nabla\varphi|^{2}$.

$\int_{R^{n}}u|\nabla(u^{m-1}-\Gamma*u)|^{2}\cdot\leq-\lambda\Vert\nabla\varphi\Vert_{\infty}^{2}\frac{d}{dt}\mathcal{F}(u)$

(66)

60
which means$(A’)^{2} \leq-\frac{||\nabla\varphi\Vert_{\infty}^{2}\lambda}{2(n-2)}H’’$,$H(t)= \int_{R^{n}}|x|^{2}u(\cdot, t)$

.

(67)

If
(68)

$\lim_{t\uparrow T}\mathcal{F}(u(t))>-\infty$

is the case, therefore, it follows that$\int_{0}^{T}|\frac{d}{dt}\int_{R^{n}}\varphi u|dt\leq T^{1/2}\{\int_{0}^{T}|\frac{d}{dt}\int_{R^{n}}\varphi u|^{2}dt\}^{1/2}<+\infty$

and hence$\lim_{t\uparrow T}A(t)=\lim_{t\uparrow T}\int_{R^{n}}\varphi u(\cdot, t)$exists. Since Theorem 10 guarantees$\lim_{t\uparrow T}\inf A(t)=\lim_{t\uparrow T}\sup A(t)\geq\lim_{t\uparrow T}\sup\Vert u(t)\Vert_{L^{1}(B(x0,R))}\geq\epsilon_{0}$,for any $x_{0}\in S$ , and hence
we obtain
the finiteness of by the total mass conservation.
oo, we havefor some
In the other case of$t_{0}\in[0, T)$ . We may assume $t_{0}=0$ without loss of generality. Inequality$\lim_{R\downarrow 0}\lim\inf_{t\uparrow T}\Vert u(t)\Vert_{L^{1}(B(x0,2R))}\geq\epsilon_{0}$$S$$\mathcal{F}(u(t_{0}))<0$$\lim_{t\uparrow T}\mathcal{F}(u(t))=-$(64) then implies$\frac{dH}{dt}<0$

and the existence of $H(T)= \lim_{t\uparrow T}H(t)\geq 0$ .

Lemma 1 It holds that$t’ \in[t,\frac{t+Tp}{2}]suA(t’)\leq A(t)+C(H(t)-H(T))^{1/2}$.

Proof: Inequality (67) implies$\int^{t’}(t’-s)A’(s)^{2}ds\leq\frac{||\nabla\varphi\Vert_{\infty}^{2}\lambda}{2(n-2)}(H(t)-H(t’))$for $0\leq t\leq t’<T$ by $H’(t)\leq 0$ , and, therefore, it holds that$|A( \frac{t+t’}{2})-A(t)|^{2}=|\int^{\frac{t+t’}{2}}A’(s)ds|^{2}\leq l^{\frac{t+t’}{2}}(t’-s)^{-1}ds$.

$l^{t’}(t^{f}-s)A’(s)^{2}ds \leq\frac{\log 2}{2}\cdot\frac{\Vert\nabla\varphi\Vert_{\infty}^{2}}{n-2}$$\leq\frac{\log 2}{2}\cdot\frac{\Vert\nabla\varphi\Vert_{\infty}^{2}}{n-2}\cdot\lambda\cdot(H(t)-H(T))$. $\lambda\cdot(H(t)-H(t’))$,$t’\in[t, T)$

(69)

61
which implies$A( \frac{t+t’}{2})\leq A(t)+C(H(t)-H(T))^{I/2}$,$t’\in[t, T)$and hence (69).

1

Now we use the scaling property (58).

Lemma 2 Each $r_{0}>0$ admits $t_{0}\in[0, T)$ and $C>0$ such that$\Vert u(\cdot, t_{1})\Vert_{L^{1}(B(x0_{)}ro))}<\epsilon_{0}/2$$\Rightarrow$$\sup_{t\in(t_{1}+\frac{1}{8}(T-t_{1}),t_{1}+\frac{3}{8}(T-t_{1}))}(T-t)\Vert u(t)\Vert_{L^{\infty}(B(x0,(T-t_{1})^{1/n}))}\leq C$

where$x_{0}\in R^{n}$(70)

and $t_{1}\in[t_{0}, T)$ .

Proof: We have$A(t_{1})<\epsilon 0/2,$ $A(t)= \int_{R^{n}}\varphi_{x0,r0}u(\cdot, t)$

by the assumption(70) and hence$0<T-t_{0}\ll 1,$$t_{1}\in[t_{0}, T)$(71)$A(t’)<\epsilon_{0}$$\sup$$\Rightarrow$$t’\in[t_{1^{\frac{T+t}{2}}},]$by Lemma 1. Here we use the scaling property (58) and take $\mu>0$ andby$\tilde{u}(x, t)$$\tilde{u}(x, t)=\mu^{n}u(\mu x+x_{0}, \mu^{n}t+t_{1})$,$\mu^{n}+t_{1}=\frac{T+t_{1}}{2}$.

It holds that$\tilde{u}_{t}=\frac{m-1}{m}\triangle\tilde{u}^{m}-\nabla\cdot(\tilde{u}\nabla\Gamma*\tilde{u})$$\mu^{n}=\frac{T-t_{1}}{2}$,,$\tilde{u}\geq 0$

in $R^{n}\cross(0,1)$$\sup\Vert\tilde{u}(t)\Vert_{L^{1}(B(0,r0\mu^{-1}))}<\epsilon_{0}$$t\in(0,1)$

by (71). Now we use Theorem 11 and then Moser’s iteration scheme applied
to the proof of Theorem 10. We obtain$\sup$$\Vert\tilde{u}(t)\Vert_{L^{\infty}(B(0,1))}\leq C_{1}$$t\in[1/4,3/4]$

similarly, because$r_{0}\mu^{-1}\geq 2$

holds for $0<T-t_{I}\ll 1$ which means$\sup_{t\in(t_{1}+\frac{1}{8}(T-t_{1}),t_{1}+\frac{3}{8}(T-t_{1}))}(T-t_{1})\Vert u(t)\Vert_{L^{\infty}(B(x_{0},(T-t_{1})^{1/n}))}\leq C_{1}$.

62Then (70) follows for$C= \frac{3}{4}C_{1}$.

IThe proof of Theorem 2 is complete by showing$inf\Vert u(t)\Vert_{L^{1}(B(x0,r))}\geq\epsilon_{0}/2$$\inf_{xo\in S_{II}}\lim_{r\downarrow 0}\lim_{t\uparrow}$.

In fact, then $\# S_{II}<+\infty$ follows from the total mass conservation. Assuming
such thatthe contrary, we have $x0\in S_{II},$ $r_{0}>0$ , and$t_{j}\uparrow T$$\Vert u(t_{j})\Vert_{L^{1}(B(x_{0},2r_{0}))}<\epsilon_{0}/2$

for $j=1,2,$$\cdots$. Then we obtain, and,$y \in B(xr)\sup_{0,0}\Vert u(t_{j})\Vert_{L^{1}(B(y,ro))}<\epsilon_{0}/2$

therefore,$\sup$$(T-t)\Vert u(t)\Vert_{L^{\infty}(B(y,(T-t_{j})^{1/n}))}\leq C$$t \in(t_{j}+\frac{1}{8}(T-t_{j}),t_{j}+\frac{3}{8}(T-t_{j}))$by Lemma 2, where $y\in B(x_{0}, r_{0})$ is arbitrary. This inequality implies$\sup_{t\in(t_{j}+\frac{1}{8}(T-t_{j}),t_{j}+\frac{3}{8}(T-t_{j}))}(T-t)\Vert u(t)\Vert_{L^{\infty}(B(x0,ro))}\leq C$$\lim_{t\uparrow T}\inf(T-t)\Vert u(t)\Vert_{L^{\infty}(B(x0,ro))}<+\infty$that is$x_{0}\in S_{I}$, a contradiction.

1

The proof of the next theorem is valid to Theorem 6.
Theorem 12 If (68) holds, then each $x_{0}\in S$ is typestrongly,. We have, more$\Pi$$\lim_{t\uparrow T}(T-t)\Vert u(t)\Vert_{L^{\infty}(B(x_{0)}b(T-t)^{1/n})}=+\infty$(72)

for any $b>0$ .
Proof. We have shown that the formation of collapse arises in this case. By$\cdot$(66), furthermore, it holds that$\int_{0}^{T}|\frac{d}{dt}\int_{\Omega}\varphi u|dt\leq C(T\lambda)^{1/2}\Vert\nabla\varphi\Vert_{\infty}$

Putting$\varphi=\varphi_{x0,R}$.

, we obtain$|\langle\varphi_{x0,R},$$u(t)\rangle-\langle\varphi_{x0,R},$ $\mu(dx,T)\rangle|\leq C(T\lambda)^{1/2}R^{-1}(T-t)$$\mu(dx, T)=\sum_{xo\in S}m(x_{0})\delta_{x0}(dx)+f(x)dx$

.

(73)

63
Given $b>0$ , we can take $R=b(T-t)$ for $0<T-t\ll 1$ in (73), and again(74)$\lim_{b\uparrow+\infty}\lim_{t\uparrow}\sup_{T}|\int_{B(x0,b(T-t))}u(\cdot, t)-m(x_{0})|=0$

follows for any $b>0$ . Using $v(y, s)=(T-t)u(x, t),$ $y=(x-x_{0})/(T-t)^{1/n}$ ,$s=-\log(T-t)$ , inequality (74) reads;$\lim_{b\uparrow+\infty}\lim_{s\uparrow+}\sup_{\infty}|\int_{B(0,be^{-\frac{n-1}{n}s})}v(\cdot, s)-m(x_{0})|=0$

Since$\int_{R^{n}}v(\cdot, s)=\lambda,$$s>-\log T$ , any

admits$t_{k}\uparrow T$.

$\{s_{k}’\}\subset\{s_{k}\}$(75)

for $s_{k}=$$-\log(T-t_{k})$ such that$v(y, s_{k}’)dyarrow\zeta(dy)$

,

(76)

$\zeta(dy)\geq m(x_{0})\delta_{0}(dy)$inby (75). Relations (76) imply
any $b>0$ and hence (72).
$\mathcal{M}(R^{n})$$\lim_{karrow\infty}\Vert v(s_{k}’)\Vert_{L^{\infty}(B(0,b))}=+\infty$

for1

Scaling property combined with the compactness of solution sequence
derives other aspects of type II blowup points. The following theorem may
be comparable to non-degeneracy of the blowup point concerning the semilinear parabolic equation with sub-critical nonlinearity [15].

Theorem 13 ([46]) We have $\# S_{*,\xi}<+\infty,$ $\xi>0$ where$S_{*,\xi}=\{x0\in R^{n}|^{\text{ョ}}y(t)arrow x0,$$\text{ョ_{}b}>0,$$|y(t)-x0|=O((T-t)^{1/n})$$\lim_{t\uparrow T}\inf(T-t)\Vert u(t)\Vert_{L^{\infty}(B(y(t),b(T-t)^{1/n}))}\geq\xi\}$

Proof: If$S_{*},$$\inf_{xo\in S_{*,\xi}}\lim_{r\downarrow 0}\lim_{t\uparrow T}\inf\Vert u(t)\Vert_{L^{1}(B(x0,r))}>0$$r_{k}>0,0<T-t_{jk}< \frac{1}{jk},$ $j,$ $k=1,2,$ $\cdots$

is not the case we have$k$be fixed. Since$x_{k}\in S_{*}$$x_{k}\in$such that$\int_{B(x4r_{k})}k,u(x, t_{jk})dx<\min\{\frac{\epsilon_{0}}{2},$Let.

$\frac{1}{2k}\}$.

, there is $y_{k}(t)arrow x_{k},$ $b_{k}>0,$ $L_{k}>0$ such that$|y_{k}(t)-y_{k}(s)|\leq L_{k}|t-s|^{1/n}$,$0<T-t,$ $T-s\ll 1$$\lim_{t\uparrow T}\inf(T-t)\Vert u(t)\Vert_{L^{\infty}(B(y_{k}(t),b_{k}(T-t)^{1/n}))}\geq\xi$.

64
We have $j_{k}\gg 1$ such that $B(y_{k}(t_{jk}), 3r_{k})\subset B(x_{k}, 4r_{k}),$$j\geq j_{k}$$\Vert u(t_{jk})\Vert_{L^{1}(B(y,2r_{k}))}<\min\{\frac{\epsilon_{0}}{2},$$\sup$$\frac{1}{2k}\}$and hence.

(77)$y\in B(y_{k}(t_{jk}),r_{k})$In particular, it holds that$\sup_{y\in B(y_{k}(t_{jk}),r_{k})}\Vert u(t_{jk})\Vert_{L^{1}(B(y,4(T-t_{jk})^{1/n}))}<\frac{\epsilon_{0}}{2}$with$j_{k}$

replaced larger if necessary which implies$\sup_{t\in(t_{jk}+\frac{1}{8}(T-t_{jk}),t_{jk}+\frac{3}{8}(T-t_{jk}))}(T-t)\Vert u(t)$

I

(78)

$L^{\infty}(B(y_{k}(t_{jk}),r_{k}))\leq C$

by Lemma 2. We obtain, also,(79)$\sup_{t\in(t_{jk},\frac{1}{2}(T+t_{jk}))}\Vert u(t)\Vert_{L^{1}(B(y_{k}(t_{jk}),2r_{k}))}<\frac{1}{k}$by (77) and Lemma 1 under the same agreement. Inequalities (78)-(79)imply,$\sup$$\sup\Vert u_{jk}(t)\Vert_{L^{1}(B(0,2\mu_{jk}^{-1}r_{k}))}<\frac{1}{k}$$\Vert u_{jk}(t)\Vert_{L^{\infty}(B(0,\mu_{jk}^{-1}r_{k}))}\leq C$$t\in[0,1]$$t \in(\frac{1}{4},\frac{3}{4})$$\mu_{jk}=\frac{1}{2}(T-t_{jk})$,$u_{jk}(x, t)=\mu_{jk}^{n}u(\mu_{jk}x+y_{k}(t_{jk}), \mu_{jk}^{n}t+t_{jk})$.

Since the parabolic regularity does not apply to the family $u_{jk}=u_{jk}(x, t)$
we come back to the approximate solution. Then passing to a subsequence
locally uniformlyofdenoted by the same symbol, we haveby diagonal argument wherein
as $jarrow\infty$ for $k=1,2,$$u_{k}=u_{k}(x, t)$ is a solution to (51) satisfying$\{j\}$$u_{jk}arrow u_{k}$$\cdots$$R^{n}\cross[\frac{3}{8}, \frac{5}{8}]$$\sup_{t\in[\frac{3}{8},\frac{5}{8}]}\Vert u_{k}(t)\Vert_{L(R^{n})}\infty\leq C$,$\sup_{t\in[\frac{3}{8},\frac{5}{8}]}\Vert u_{k}(t)\Vert_{L^{1}(R^{n})}\leq\frac{1}{k}$locally uniformly in
Then it holds that$0<\eta<\xi$ , therefore, we havesuch thatlarge, and, then, there arises. This inequality impliesany$u_{k}arrow 0$$R^{n}\cross[\frac{3}{8}, \frac{5}{8}]$$\Vert u_{k}\Vert_{L^{\infty}(B(0,2b)\cross[\frac{3}{8},\frac{5}{8}])}<\frac{\eta}{2}$$j_{b,\eta,k}$as.

$karrow\infty$

for a

$k$

. Givensufficiently
for$\Vert u_{jk}\Vert_{L^{\infty}(B(0,2b)\cross[\frac{3}{8},\frac{5}{8}])}<\eta$$j\geq j_{b,\eta,k}$$\sup_{t\in(t_{jk}+\frac{3}{16}(T-t_{jk}),t_{jk}+\frac{5}{16}(T-t_{jk}))}(T-t)\Vert u(t)\Vert_{L^{\infty}(B(y_{k}(t_{jk}),\mu_{jk}b))}<\eta$$\lim_{t\uparrow T}\inf(T-t)\Vert u(t)\Vert_{L^{\infty}(B(y_{k}(t),b(T-t)^{1/n}))}\leq\eta$65
sending$jarrow\infty$, a contradiction.

1

The following theorem suggests that
may be a positive dimensional$\#
S=+\infty$
set if. Since the lower dimensionally reduced solution does not
blowup because of the total mass conservation, one may suspect the validity
of $\# S<+\infty$ . Here we say that $x(t)\in R^{n}$ attains a positive local maximum
if
is positive in a neighborhood of $x(t)$ in x-space and $x=x(t)$ takes
a local maximum of.
$S$$u(\cdot, t)$$u(\cdot, t)$

that

Theorem 14 If $\# S=+\infty$ , there are infinite number of $x_{0}\in S$ such,

(80)$\lim_{t\uparrow T}\sup\frac{dist(x(t),x_{0})}{(T-t)^{1/n}}=+\infty$

pmvided that $x(t)$ attains a positive local maxmimum of$\lim_{t\uparrow}\sup_{T}u(x(t), t)=+\infty$$u(\cdot, t)$such that.

(81)

Proof: If $\# S=+\infty$ is the case, there are infinite number of$x_{0}\in S\backslash S_{*,1}$by Theorem 13. Since $x(t)$ attains a positive local maximum offollows thatfor $m(t)=u(x(t), t)$ , see [12], and hence$u(\cdot, t)$it$\dot{m}\leq m^{2}$$m(t)=u(x(t), t)\geq(T-t)^{-1}$ ,$0\leq t<T$holds by (81). In particular we have$\lim_{t\uparrow T}\inf(T-t)\Vert u(t)\Vert_{L^{\infty}(B(x(t),b(T-t)^{1/n})}\geq 1$

Hence$x_{0}\in S_{*,1}$,$\forall_{b}>0$.

if (80) is not the case, a contradiction.

1

References[1] A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter, Comm. Partial
Differential Equations 26 (2001) 43-100.

[2] P. Biler, Adv. Math. Sci. Appl. 8 (1998) 715-743.
[3] P. Biler, D. Hilhorst, and T. Nadzieja, Colloq. Math. 67 (1994) 297-308.

[4] P. Biler and R. Sta\’{n}czy, Folia Matematica 13 (2006) 3-19.
[5] A. Blanchet, J.A. Carrillo, and P. Lauren\caot, Calc. Var. 35 (2009) 133-168.

66[6] E. Caglioti, P.-L. Lions, C. Marchioro, and M. Pulvirenti, Comm. Math. Phys.
143 (1992) 501-525,[7] E. Caglioti, P.-L. Lions, C. Marchioro, and M. Pulvirenti, Comm. Math. Phys.
174 (1995) 229-260.

[8] P.-H. Chavanis, Banach Center Publications 66 (2004) 79-101.
[9] P.-H. Chavanis and C. Sire, Phys. Rev. E69 (2004) 016116.
[10] W. Chen and C. Li, Duke Math. J. 63 (1991) 615-622.

[ 垣] S. Childress and J.K. Percus, Math. Biosci. 56 (1981) 217-237.
[12] A. Friedman and B. McLeod, Indiana Univ. Math. J. 34 (1985) 425-447.
[13] H. Gajewski and K. Zacharias, Math. Nachr. 195 (1998) 77-114.
[14] B. Gidas and J. Spruck, Comm. Pure Appl. Math. 34 (1981) 525-598.

[15] Y. Giga and R.V. Kohn, Comm. Pure Appl. Math. 47 (1989) 845-884.

[16] M.A. Herrero, and J.J.L. Vel\’azquez, Math. Ann., Vol. 306 (1996) 583-623.
[17] M.K.H. Kiessling, Comm. Pure Appl. Math. 46 (1993) 27-56.
[18] M. Kurokiba and T. Ogawa, Differential and Integral Equations 16 (2003)

427-452.
[19] Y.Y. Li and I. Shafrir, Indiana Univ. Math. J. 43 (1994) 1255-1270.
[20] D. Li and X. Zhang, preprint.
[21] T. Nagai, T. Senba, and K. Yoshida, Funkcial. Ekvac. 40 (1997) 411-433.

[22] K. Nagasaki and T. Suzuki, Asymptotic Analysis 3 (1990) 173-188.
[23] Y. Naito and T. Suzuki, Colloq. Math. 111 (2008) 11-34.
[24] V. Nanjundiah, J. Theor. Biol. 42 (1973) 63-105.

[25] H. Ohtsuka, T. Senba, and T. Suzuki, Adv. Math. Sci. Appl. 17 (2007) 445-472.
[26] L. Onsager, Suppl. Nuovo Cimento 6 (1949) 279-287.
[27] M. Otani, in; Nonlinear Partial Differential Equations and their Applications,pp. 505-516, Gakkotosho, Tokyo, 2004.
[28] T. Senba, Nonlinear Analysis, TMA 66 (2007) 1817-1839.

[29] T. Senba and T. Suzuki, Adv. Dffierential Equations 6 (2001)$21arrow 50$.

[30] T. Senba and T. Suzuki, Meth. Appl. Anal. 8 (2001) 349-368.
[31] T. Senba and T. Suzuki, Asymptotic Analysis 32 (2001) 349-368.

[32] T. Senba and T. Suzuki, Applied Analysis-Mathematical Methods in Natuml
Science, second edition, Imperior College Press, London, 2010.

67[33] I. Shafrir and G. Wolansky, J. Euro. Math. Soc. 7 (2005) 413-448.
[34] M. Struwe, Comment. Math. Helv. 60 (1985) 558-581.
[35] Y. Sugiyama, SIAM J. Math. Anal. 41 (2009) 1664-1692.
[36] Y. Sugiyama, J. Math. Anal. Appl. 364 (2010) 51-70.

[37] Y. Sugiyama, In; Noncommutativity and sigularities, pp. 137-160, Adv. Stud.
Pure Math. 55 Math. Soc. Japan, Tokyo 2009.
[38] Y. Sugiyama and H. Kunii, J. Differential Equations 227 (2006) 333-364.
[39] T. Suzuki, Ann. Inst. Henri Poincar\’e AN, 9 (1992) 367-398.

[40] T. Suzuki, Semilinear Elliptic Equations, Gakkotosho, Tokyo, 1994.
[41] T. Suzuki, Free Energy and Self-Intemcting Particles, Birkh\"auser, Boston,2005.

[42] T. Suzuki, Mean Field Theories and Dual Variation, Atlantis Press, Amsterdam-Paris, 2008.

[43] T. Suzuki and R. Takahashi, Adv. Differential Equations 14 (2009) 433-476.
[44] T. Suzuki and R. Takahashi, Differential and Integral Equations, 22 (2009)1153-1172.

[45] T. Suzuki and R. Takahashi, Differential and Integral Equations, to appear.

[46] T. Suzuki and R. Takahashi, Adv. Differential Equations 15 (2010) 853-892.
[47] T. Suzuki and R. Takahashi, preprint.

[48] J.F. Toland, J. Math. Anal. Appl. 66 (1978) 399-415.
[49] J.F. Toland, Arch. Rational Mech. Anal. 71 (1979) 41-61.

[50] P. Topping, Math. Z. 247 (2004) 279-302.
[51] G. Wang and D. Ye, Math. Z. 244 (2003) 531-548.
[52] G. Wolansky, Nonlinear Anal. 26 (1996) $971arrow 995$ .