# The aggregation-diffusion equation with energy critical exponent

Shen Bian111Corresponding author. Beijing University of Chemical Technology,
100029, Beijing. Email<EMAIL_ADDRESS>
###### Abstract

We consider a Keller-Segel model with non-linear porous medium type diffusion
and nonlocal attractive power law interaction, focusing on potentials that are
less singular than Newtonian interaction. Here, the nonlinear diffusion is
chosen to be $m=\frac{2d}{d+2s}$ in such a way that the associated free energy
is conformal invariant and there is a family of stationary solutions
$U(x)=c\left(\frac{\lambda}{\lambda^{2}+|x-x_{0}|^{2}}\right)^{\frac{d+2s}{2}}$
for any constant $c$ and some $\lambda>0,x_{0}\in{\mathbb{R}}^{d}.$ We analyze
under which conditions on the initial data the regime that attractive forces
are stronger than diffusion occurs and classify the global existence and
finite time blow-up of dynamical solutions by virtue of stationary solutions.
Precisely, solutions exist globally in time if the $L^{m}$ norm of the initial
data $\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}$ is less than the $L^{m}$ norm of
stationary solutions $\|U(x)\|_{L^{m}({\mathbb{R}}^{d})}$. Whereas there are
blowing-up solutions for
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}>\|U(x)\|_{L^{m}({\mathbb{R}}^{d})}$.

Key words. Global existence; Finite time blow-up; Steady states; Critical
exponent

##  1 Introduction

In the present paper, we analyse qualitative properties of nonnegative
solutions for the aggregation-diffusion equation

$\displaystyle\left\\{\begin{array}[]{ll}u_{t}=\Delta
u^{m}-\nabla\cdot\left(u\nabla c\right),{}{}&x\in{\mathbb{R}}^{d},~{}t>0,\\\
u(x,0)=u_{0}(x)\geq 0,{}{}&x\in{\mathbb{R}}^{d}.\end{array}\right.$ (3)

Here the diffusion exponent is taken to be $m=\frac{2d}{d+2s}$ for $2<2s<d$
with $d\geq 3.$ $u$ represents the density of cells, the chemoattractant $c$
is governed by a diffusion process and can be expressed by the convolution of
Riesz potential and $u(x,t)$, that is

$\displaystyle
c(x,t)=\frac{1}{d-2s}\int_{\mathbb{R}^{d}}\frac{u(y,t)}{|x-y|^{d-2s}}dy.$ (4)

The parabolic equation on $u$ gives nonnegative solutions $u(x,t)\geq
0,c(x,t)\geq 0$. Another property we will use is the conservation of the total
number of cells

$\displaystyle
M:=\int_{{\mathbb{R}}^{d}}u_{0}(x)dx=\int_{{\mathbb{R}}^{d}}u(x,t)dx.$ (5)

Initial data will be assumed throughout this paper

$\displaystyle u_{0}\in L_{+}^{1}\cap
L^{\infty}({\mathbb{R}}^{d}),\quad\int_{{\mathbb{R}}^{d}}|x|^{2}u_{0}(x)dx<\infty,\quad\nabla
u_{0}^{m}\in L^{2}({\mathbb{R}}^{d}).$ (6)

The nonlinear diffusion $m>1$ models the local repulsion of cells while the
attractive nonlocal term models cell movement toward chemotactic sources or
attractive interaction between cells due to cell adhesion [7, 21, 23, 24, 25,
26]. The main characteristic of equation (3) is the competition between the
diffusion and the nonlocal aggregation. This is well presented by the free
energy

$\displaystyle
F(u)=\frac{1}{m-1}\int_{{\mathbb{R}}^{d}}u^{m}dx-\frac{1}{2(d-2s)}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u(x,t)u(y,t)}{|x-y|^{d-2s}}dxdy.$
(7)

The competition between these two terms leads to finite-time blow-up and
infinite time existence. As a matter of fact, (3) can be recast as

$\displaystyle u_{t}=\nabla\cdot\left(u\nabla\mu\right),$ (8)

where $\mu$ is the chemical potential

$\displaystyle\mu=\frac{m}{m-1}u^{m-1}-c.$ (9)

Multiplying (8) by $\mu$ and integrating it in space one obtains

$\displaystyle\frac{dF(u)}{dt}+\int_{{\mathbb{R}}^{d}}u|\nabla\mu|^{2}dx=0.$
(10)

This implies that $F[u(\cdot,t)]$ is non-increasing with respect to time.

Notice that equation (3) possesses a scaling invariance which leaves the
$L^{p}$ norm invariant and produces a balance between the diffusion and the
aggregation terms where

$\displaystyle p=\frac{d(2-m)}{2s}=m=\frac{2d}{d+2s}.$ (11)

Indeed, if $u(x,t)$ is a solution to (3), then $u_{\lambda}(x,t)=\lambda
u\left(\lambda^{\frac{2-m}{2s}}x,\lambda^{m-1+\frac{2-m}{s}}t\right)$ is also
a solution and this scaling preserves the $L^{p}$ norm
$\|u_{\lambda}\|_{L^{p}({\mathbb{R}}^{d})}=\|u\|_{L^{p}({\mathbb{R}}^{d})}.$
For the critical case $m_{c}=2-2s/d$, the above scaling becomes the mass
invariant scaling $u_{\lambda}(x,t)=\lambda
u\left(\lambda^{1/d}x,\lambda^{1+\frac{2-2s}{d}}t\right)$. For the subcritical
case $m>2-2s/d$, the aggregation dominates the diffusion for low density and
prevents spreading. While for high density, the diffusion dominates the
aggregation thus blow-up is precluded. On the contrary, for the supercritical
case $1\leq m<2-2s/d,$ the aggregation dominates the diffusion for high
density (large $\lambda$) and the density may have finite-time blow-up. While
for low density (small $\lambda$), the diffusion dominates the aggregation and
the density has infinite-time spreading. These behaviors also appear in many
other physical systems such as thin film, Hele-Shaw, Stellar collapse as well
as the nonlinear Schrödinger equation [8, 16, 31, 32].

Our main goal in this paper is to classify the global existence and finite
time blow-up of solutions by the initial data. The main tool for the analysis
of (3) is the steady state which can be interpreted in the distributional
sense

$\displaystyle\left\\{\begin{array}[]{ll}\frac{m}{m-1}U_{s}^{m-1}-C_{s}=\overline{C},~{}~{}\mbox{in}~{}~{}\Omega,\\\
U_{s}=0\mbox{~{} in ~{}}{\mathbb{R}}^{d}\setminus\Omega,\quad U_{s}>0\mbox{ in
}\Omega,\\\
C_{s}=\frac{1}{d-2s}\int_{\mathbb{R}^{d}}\frac{U_{s}(x)}{|x-y|^{d-2s}}dy,~{}~{}\mbox{in}~{}~{}{\mathbb{R}}^{d}.\end{array}\right.$
(15)

Here $\overline{C}$ is any constant chemical potential and
$\Omega=\\{x\in{\mathbb{R}}^{d}~{}\big{|}~{}U_{s}(x)>0\\}$ is a connected open
set in ${\mathbb{R}}^{d}$. Therefore, let us begin with the analysis of
stationary states. We firstly stress that a lot of efforts have been put to
investigate the steady states of (3) [9, 10, 12, 13, 15].

For $m>2-2s/d,$ this is the diffusion-dominated regime. There exists a unique
up to translations compactly supported steady state to (3). Moreover, such
steady state is radially decreasing and it is the global minimiser of the
energy functional $F(u).$ We refer to [13] for details. In the local case
$s=1,$ the Riesz kernel $\frac{1}{|x|^{d-2s}}$ is replaced by Newtonian
interaction for $d\geq 3$ and log interaction for $d=2.$ Analogous results for
stationary states are provided in [12, 22] for $m>2-2/d~{}(d\geq 3)$ and [11]
for $m>1~{}(d=2).$

In the case $m=2-2s/d,$ this is the fair competition regime where the
nonlinear diffusion and the nonlocal aggregation balance. The existence of the
stationary states can only happen for a critical mass $M_{c}$ which is
characterised by the optimal constant of a variant of the HLS inequality.
$M_{c}$ turns out to be the only value of the mass for which the free energy
$F(u)$ has minimizers and these minimizers are shown to be compactly
supported, radially symmetric and non-increasing stationary solutions of
equation (3). See for instance [9] and the references therein. In the local
case $s=1$, a clear dichotomy arises, that’s, $M_{c}$ is the sharp critical
mass such that if the initial mass $m_{0}<M_{c},$ solutions of (3) exist
globally in time, whereas there are blowing-up solutions for $m_{0}>M_{c}$.
See [3] for the case $m=2-2/d~{}(d\geq 3)$ and [4, 5] for the case
$m=1~{}(d=2)$. Other similar results for stationary states are given in the
above two references as well.

For $1<m<2-2s/d,$ this is the aggregation-dominated regime. We will here focus
on the case $m=\frac{2d}{d+2s}<2-\frac{2s}{d}$ and there are two reasons for
us to concentrate on the special case $m=\frac{2d}{d+2s}$. On one hand, there
are three subcases for stationary states in terms of $m$. For
$\frac{2d}{d+2s}<m<2-\frac{2s}{d},$ all the stationary states are compactly
supported. While they are not compactly supported for $1<m\leq\frac{2d}{d+2s}$
and there is an explicit formula of one parameter family of stationary states
in the case $m=\frac{2d}{d+2s}$ [17]. See Section 3 for more details.
Consequently, $m=\frac{2d}{d+2s}$ is the critical exponent separating steady
states that have compactly support from those that do not. On the other hand,
the free energy $F(u)$ is invariant under translations for
$m=\frac{2d}{d+2s}.$ To be honest, plugging the invariant scaling
$u_{\lambda}(x)=\lambda u\left(\lambda^{\frac{2-m}{2s}}x\right)$ into the free
energy $F(u)$ we obtain

$\displaystyle F(u_{\lambda})=\lambda^{\frac{(d+2s)m-2d}{2s}}F(u),$ (16)

the free energy is invariant when $m=\frac{2d}{d+2s}$. For this reason $m$ is
endowed with the energy critical exponent and denoted as
$m_{ec}:=\frac{2d}{d+2s}.$

Indeed, for $m=\frac{2d}{d+2s},$ we can deduce that $\overline{C}=0$ and
$\Omega={\mathbb{R}}^{d}$ in (15), see Proposition 9. Hence, (15) can be
written as the following integral equation

$\displaystyle C_{s}(x)$
$\displaystyle=\frac{1}{d-2s}\left(\frac{m-1}{m}\right)^{\frac{1}{m-1}}\int_{{\mathbb{R}}^{d}}\frac{C_{s}(y)^{\frac{1}{m-1}}}{|x-y|^{d-2s}}dy$
$\displaystyle=\frac{1}{d-2s}\left(\frac{d-2s}{2d}\right)^{\frac{d+2s}{d-2s}}\int_{{\mathbb{R}}^{d}}\frac{C_{s}(y)^{\frac{d+2s}{d-2s}}}{|x-y|^{d-2s}}dy.$
(17)

It is proved by Chen and Li [17, 18, 19] that every positive solution $C_{s}$
of the integral equation (1) is radially symmetric and decreasing about some
point $x_{0}$ and therefore assumes the form

$\displaystyle
C_{s}(x)=A\left(\frac{\lambda}{\lambda^{2}+|x-x_{0}|^{2}}\right)^{\frac{d-2s}{2}}$
(18)

with some positive constants $A$ and $\lambda.$ Thus, we conclude that (3)
uniquely admits a one-parameter family of stationary states

$\displaystyle
U_{s}(\lambda)=B\left(\frac{\lambda}{\lambda^{2}+|x-x_{0}|^{2}}\right)^{\frac{d+2s}{2}}$
(19)

with some point $x_{0}\in{\mathbb{R}}^{d}$ and positive constants $B,\lambda$.

######  Remark 1.

  1. 1.

The integral equation (1) is also closely related to the following family of
semi-linear partial differential equations

$\displaystyle(-\Delta)^{s}w=w^{\frac{d+2s}{d-2s}}.$ (20)

Apparently, equation (20) is also of practical interest and importance. For
instance, it arises as the Euler-Lagrange equation of the functional [17]

$\displaystyle
I(w)=\frac{\int_{{\mathbb{R}}^{d}}\left|(-\Delta)^{\frac{s}{2}}w\right|^{2}dx}{\left(\int_{{\mathbb{R}}^{d}}|w|^{\frac{2d}{d-2s}}dx\right)^{\frac{d-2s}{d}}}.$
(21)

The classification of the solutions $C_{s}$ provides the best constant in the
inequality of the critical sobolev embedding from $H^{s}({\mathbb{R}}^{d})$ to
$L^{\frac{2d}{d-2s}}({\mathbb{R}}^{d})$:

$\displaystyle\left(\int_{{\mathbb{R}}^{d}}|w|^{\frac{2d}{d-2s}}dx\right)^{\frac{d-2s}{d}}\leq\bar{K}\int_{{\mathbb{R}}^{d}}\left|(-\Delta)^{\frac{s}{2}}w\right|^{2}dx$
(22)

by
$\bar{K}=\frac{\|(-\Delta)^{s/2}C_{s}\|_{L^{2}({\mathbb{R}}^{d})}^{2}}{\|C_{s}\|_{L^{2d/(d-2s)}({\mathbb{R}}^{d})}^{2}}.$

  2. 2.

It is shown that all the solutions of partial differential equation (20)
satisfy the integral equation (1) and vise versa. Therefore, to classify the
solutions of the partial differential equations, we only need to work on the
integral equations (1).

  3. 3.

Note that the second moment of $U_{s}$ is bounded for $s>1.$

We remark that the stationary solutions $U_{s}(\lambda)$ (19) is the extremum
for the HLS inequality [27] which states as follows.

######  Lemma 2.

(HLS inequality) Let $q,q_{1}>1,d\geq 3$ and $0<\beta<d$ with
$1/q+1/q_{1}+\beta/d=2.$ Assume $f\in L^{q}({\mathbb{R}}^{d})$ and $h\in
L^{q_{1}}({\mathbb{R}}^{d})$. Then there exists a sharp constant
$C(d,\beta,q)$ independent of $f$ and $h$ such that

$\displaystyle\left|\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{f(x)h(y)}{|x-y|^{\beta}}dxdy\right|\leq
C(d,\beta,q)\|f\|_{L^{q}({\mathbb{R}}^{d})}\|h\|_{L^{q_{1}}({\mathbb{R}}^{d})}.$
(23)

If $q=q_{1}=\frac{2d}{2d-\beta},$ then

$\displaystyle
C(d,\beta,q)=C(d,\beta)=\pi^{\beta/2}\frac{\Gamma\left(d/2-\beta/2\right)}{\Gamma\left(d-\beta/2\right)}\left(\frac{\Gamma(d/2)}{\Gamma(d)}\right)^{-1+\beta/d}.$
(24)

In this case there is equality in (23) if and only if $h\equiv(const.)f$ and

$\displaystyle f(x)=A\left(\gamma^{2}+|x-x_{0}|^{2}\right)^{-(2d-\beta)/2}$
(25)

for some $A>0,\gamma\in{\mathbb{R}}$ and $x_{0}\in{\mathbb{R}}^{d}.$
Particularly, for $\beta=d-2s,$

$\displaystyle f=U_{s}(\lambda)~{}~{}\mbox{for some}~{}~{}\lambda>0.$ (26)

Recalling (15) which reads as

$\displaystyle C_{s}(x)=\frac{m}{m-1}U_{s}^{m-1}(x),$ (27)

the following equality from $F(u)$ holds:

$\displaystyle F(U_{s}(x))$
$\displaystyle=\frac{1}{m-1}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}-\frac{1}{2}\int_{{\mathbb{R}}^{d}}U_{s}C_{s}dx$
$\displaystyle=\frac{1}{m-1}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}-\frac{m}{2(m-1)}\int_{{\mathbb{R}}^{d}}U_{s}^{m}dx$
$\displaystyle=\frac{2-m}{2(m-1)}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}=\frac{2s}{d-2s}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}.$
(28)

Here we address that $\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}$ is a constant
independent of $\lambda$ and $x_{0}.$ That is

$\displaystyle\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}$
$\displaystyle=B^{m}\int_{{\mathbb{R}}^{d}}\left(\frac{\lambda}{\lambda^{2}+|x-x_{0}|^{2}}\right)^{d}dx$
$\displaystyle=B^{m}\pi^{\frac{d+1}{2}}2^{1-d}\frac{1}{\Gamma\left(\frac{d+1}{2}\right)}.$
(29)

Before proceeding further to show the dynamical behaviors of solutions, we
define the weak solution which we will deal with in this paper.

######  Definition 3.

(Weak and free energy solution) Let $u_{0}$ be an initial condition satisfying
(6) and $T\in(0,\infty].$

  1. (i)

A weak solution to (3) with initial data $u_{0}$ is a non-negative function
$u\in L^{\infty}\left(0,T;L_{+}^{1}\cap L^{\infty}({\mathbb{R}}^{d})\right)$
and for $\forall~{}\psi\in C_{0}^{\infty}({\mathbb{R}}^{d})$ and $0<t<T$

$\displaystyle\int_{{\mathbb{R}}^{d}}\psi
u(\cdot,t)dx-\int_{{\mathbb{R}}^{d}}\psi
u_{0}(x)dx=\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\Delta\psi u^{m}dxds$
$\displaystyle-\frac{1}{2}\int_{0}^{t}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{[\nabla\psi(x)-\nabla\psi(y)]\cdot(x-y)}{|x-y|^{2}}\frac{u(x,s)u(y,s)}{|x-y|^{d-2s}}dxdyds.$
(30)

  2. (ii)

The weak solution $u$ is also a weak free energy solution to (3) if $u$
satisfies additional regularities that

$\displaystyle\nabla u^{m-1/2}\in
L^{2}\left(0,T;L^{2}({\mathbb{R}}^{d})\right),$ (31) $\displaystyle u\in
L^{3}\left(0,T;L^{\frac{3d}{d-2(1-2s)}}({\mathbb{R}}^{d})\right).$ (32)

Moreover, $F(u(\cdot,t))$ is a non-increasing function and satisfies

$\displaystyle
F[u(\cdot,t)]+\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\left|\frac{2m}{2m-1}\nabla
u^{m-\frac{1}{2}}-\sqrt{u}\nabla c\right|^{2}dxds\leq F(u_{0})$ (33)

for all $t\in(0,T)$ with
$c=\frac{1}{d-2s}\int_{{\mathbb{R}}^{d}}\frac{u(y)}{|x-y|^{d-2s}}dy.$

We emphasize that regularities (31) and (32) are enough to make sense of each
term in (33).

######  Lemma 4.

If $u\in L^{\frac{3d}{d-2(1-2s)}}({\mathbb{R}}^{d})$ and $c$ is expressed by
(4), then

$\displaystyle\left\|u|\nabla c|^{2}\right\|_{L^{1}({\mathbb{R}}^{d})}\leq
C\|u\|_{L^{q}({\mathbb{R}}^{d})}^{3}<\infty,~{}~{}q=\frac{3d}{d-2(1-2s)}\,.$
(34)

Proof.  By the Hölder inequality we have

$\displaystyle\left\|u|\nabla
c|^{2}\right\|_{L^{1}({\mathbb{R}}^{d})}\leq\|u\|_{L^{q}({\mathbb{R}}^{d})}\left\||\nabla
c|^{2}\right\|_{L^{q^{\prime}}({\mathbb{R}}^{d})},~{}~{}\frac{1}{q}+\frac{1}{q^{\prime}}=1.$
(35)

Then by the weak Young inequality [27, formula (9), pp. 107], it follows that

$\displaystyle\left\||\nabla
c|^{2}\right\|_{L^{q^{\prime}}({\mathbb{R}}^{d})}$ $\displaystyle=\|\nabla
c\|_{L^{2q^{\prime}}({\mathbb{R}}^{d})}^{2}=C\left\|u(x)*\frac{x}{|x|^{d+2-2s}}\right\|_{L^{2q^{\prime}}({\mathbb{R}}^{d})}^{2}$
$\displaystyle\leq
C\|u\|_{L^{q}({\mathbb{R}}^{d})}^{2}\left\|\frac{x}{|x|^{d+2-2s}}\right\|_{L_{w}^{\frac{d}{d+1-2s}}({\mathbb{R}}^{d})}^{2}\leq
C\|u\|_{L^{q}({\mathbb{R}}^{d})}^{2},$ (36)

where $1+\frac{1}{2q^{\prime}}=\frac{1}{q}+\frac{d+1-2s}{d}.$ Combining with
(35) follows $q=\frac{3d}{d-2(1-2s)}$ and completes the proof. $\Box$

Consequently, from (32) one has that

$\displaystyle\int_{0}^{T}\left\|\sqrt{u}\nabla
c\right\|_{L^{2}({\mathbb{R}}^{d})}^{2}dt\leq
C\int_{0}^{T}\left\|u\right\|_{L^{3d/(d-2(1-2s))}({\mathbb{R}}^{d})}^{3}dt<\infty.$
(37)

So the term $\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\left|\sqrt{u}\nabla
c\right|^{2}dxds$ in (33) makes sense.

Thanks to (1), we are now ready to state our main result by employing the
steady states of (3).

######  Theorem 5.

Let $d\geq 3$ and $m=\frac{2d}{d+2s}.$ We assume that $u_{0}(x)\in
L^{1}_{+}\cap L^{\infty}({\mathbb{R}}^{d})$ and
$\int_{{\mathbb{R}}^{d}}|x|^{2}u_{0}(x)dx<\infty,$ $U_{s}$ is the steady state
of (3). Suppose $F(u_{0})<F(U_{s}).$

  1. (i)

If $\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}<\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})},$
then there exists a global weak solution to (3) satisfying that for any
$0<t<\infty,$

$\displaystyle\|u(\cdot,t)\|_{L^{1}\cap L^{\infty}({\mathbb{R}}^{d})}\leq
C\left(\|u_{0}\|_{L^{1}\cap L^{\infty}({\mathbb{R}}^{d})}\right).$ (38)

Furthermore, the weak solution is also a weak free energy solution satisfying
the energy inequality (33).

  2. (ii)

If $\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})},$
then the weak solution of (3) blows up in finite time $T$ in the sense that

$\displaystyle\displaystyle\limsup_{t\to
T}\|u(\cdot,t)\|_{L^{q}({\mathbb{R}}^{d})}=\infty,~{}~{}\mbox{for
any}~{}~{}1<q\leq\infty.$ (39)

Let us mention that the above results also hold true in the local setting
$s=1$ for the lower dimensional case $d=2$, where Riesz potential is replaced
by the log interaction and the energy critical exponent $\frac{2d}{d+2}$ and
mass critical exponent $2-\frac{2}{d}$ coincides, i.e.
$\frac{2d}{d+2}=2-\frac{2}{d}=1.$ In this case, the $L^{1}$ norm of the steady
states is $8\pi$ and the global behaviors of solutions can be classified by
the $L^{1}$ norm of the initial data. Actually, if
$\|u_{0}\|_{L^{1}({\mathbb{R}}^{2})}<8\pi,$ the solution of (3) exists
globally in time, while the solution blows up in finite time for
$\|u_{0}\|_{L^{1}({\mathbb{R}}^{2})}>8\pi$ [4, 5]. Notice that in the nonlocal
case $s>1,$ it is not straightforward to extend the above results to $d=2.$ In
fact, there is a gap between $\frac{2d}{d+2s}$ and $2-\frac{2s}{d}$, i.e.
$2-\frac{2s}{d}<\frac{2d}{d+2s}$ and the two dimensional degenerate problem is
essentially new aspect.

The results are organised as follows. This work is entirely devoted to
establish an exact criteria on the initial data for the global existence and
finite time blow-up of dynamical solutions to (3). With this aim we firstly
construct the existence criterion which shows a key maximal existence time for
the free energy solution of (3) in Section 2. Section 3 mainly explores
stationary solutions to (3) and shows that they are global minimizers of the
free energy. Based on the above analysis, Section 4 makes use of the steady
states to prove the main theorem concerning the dichotomy, the global
existence for
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}<\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$ and
finite time blow-up for
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$.
Finally, Section 5 concludes the main work of this paper, some open questions
for equation (3) are also addressed.

##  2 Existence criterion

In this section, we will establish the existence criterion of a bounded weak
solution to (3). As in [1, 2, 30], we consider the regularized problem

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}u_{\varepsilon}=\Delta
u^{m}_{\varepsilon}+\varepsilon\Delta
u_{\varepsilon}-\nabla\cdot\left(u_{\varepsilon}\nabla
c_{\varepsilon}\right),~{}~{}x\in{\mathbb{R}}^{d},~{}t>0,\\\
u_{\varepsilon}(x,0)=u_{0\varepsilon}\geq 0,\end{array}\right.$ (42)

where $c_{\varepsilon}$ is defined as

$\displaystyle c_{\varepsilon}=R_{\varepsilon}\ast u_{\varepsilon}$ (43)

with the regularized Riesz potential

$\displaystyle
R_{\varepsilon}(x)=\frac{1}{(d-2s)\left(|x|^{2}+\varepsilon^{2}\right)^{\frac{d-2s}{2}}}.$
(44)

Here $u_{0\varepsilon}$ is a sequence of approximation for $u_{0}$ and can be
constructed to satisfy that there exists $\varepsilon_{0}>0$ such that for any
$0<\varepsilon<\varepsilon_{0},$

$\displaystyle\left\\{\begin{array}[]{ll}u_{0\varepsilon}\geq
0,~{}~{}\|u_{\varepsilon}(x,0)\|_{L^{1}({\mathbb{R}}^{d})}=\|u_{0}\|_{L^{1}({\mathbb{R}}^{d})},\\\
\int_{{\mathbb{R}}^{d}}|x|^{2}u_{0\varepsilon}dx\to\int_{{\mathbb{R}}^{d}}|x|^{2}u_{0}(x)dx,~{}~{}\mbox{as}~{}~{}\varepsilon\to
0,\\\ u_{0\varepsilon}(x)\to
u_{0}(x)~{}~{}\mbox{in}~{}~{}L^{q}({\mathbb{R}}^{d}),~{}~{}\mbox{for}~{}~{}1\leq
q<\infty,~{}~{}\mbox{as}~{}~{}\varepsilon\to 0.\end{array}\right.$ (48)

This regularized problem has global in time smooth solutions for any
$\varepsilon>0$. This approximation has been proved to be convergent. More
precisely, following the arguments in [2, Theorem 4.2], [12, Lemma 4.8] and
[30, Section 4] we assert that if

$\displaystyle\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}({\mathbb{R}}^{d})}<C_{0},$
(49)

where $C_{0}$ is independent of $\varepsilon>0,$ then there exists a
subsequence $\varepsilon_{n}\to 0$ such that

$\displaystyle\begin{array}[]{ll}u_{\varepsilon_{n}}\rightarrow
u~{}~{}\mbox{in}~{}~{}L^{r}(0,T;L^{r}({\mathbb{R}}^{d})),~{}~{}1\leq
r<\infty\end{array}$ (51)

and $u$ is a weak solution to (3) on $[0,T).$

According to the above analysis, a weak solution to (3) on $[0,T)$ exists when
(49) is fulfilled. So we shall focus on establishing the availability of the
$L^{\infty}$-bound. As we will see in the following theorem where the local in
time existence and blow-up criteria are constructed, such a bound follows from
the $L^{r}$ norm for $r>m=\frac{2d}{d+2s}$ which additionally provides a
characterisation of the maximal existence time.

######  Theorem 6.

(Local in time existence and blow-up criteria) Under assumption (6) on the
initial condition, there are a maximal existence time $T_{w}\in(0,\infty]$ and
a weak solution $u$ to (3) on $[0,T_{w})$. If $T_{w}<\infty$, then

$\displaystyle\|u(\cdot,t)\|_{L^{m}({\mathbb{R}}^{d})}\to\infty~{}~{}\mbox{as}~{}~{}t\to
T_{w}.$ (52)

Proof.  To prove this result we need to refine the arguments used in the proof
of [1, Theorem 2.11]. We follow a procedure analogous to the ones in [3, Lemma
2.3].

Step 1 ($L^{r}$-estimates, $r\in(m,\infty)$) It’s obtained by multiplying the
equation (42) with $ru_{\varepsilon}^{r-1}$ that

$\displaystyle\frac{d}{dt}\int_{{\mathbb{R}}^{d}}u_{\varepsilon}^{r}dx+\frac{4mr(r-1)}{(m+r-1)^{2}}\int_{{\mathbb{R}}^{d}}\left|\nabla
u_{\varepsilon}^{\frac{m+r-1}{2}}\right|^{2}dx+\varepsilon\frac{4(r-1)}{r}\int_{{\mathbb{R}}^{d}}\left|\nabla
u_{\varepsilon}^{r/2}\right|^{2}dx$ $\displaystyle=$
$\displaystyle(2s-2)(r-1)\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u_{\varepsilon}^{r}(x)u_{\varepsilon}(y)}{\left(|x-y|^{2}+\varepsilon^{2}\right)^{\frac{d+2-2s}{2}}}dxdy.$
(53)

Due to $2s>2,$ we treat the right hand side of (2) by applying the HLS
inequality (23) with $f=u_{\varepsilon}^{r}(x)$ and $h=u_{\varepsilon}(y)$
such that

$\displaystyle\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u_{\varepsilon}^{r}(x)u_{\varepsilon}(y)}{\left(|x-y|^{2}+\varepsilon^{2}\right)^{\frac{d+2-2s}{2}}}dxdy$
$\displaystyle\leq\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u_{\varepsilon}^{r}(x)u_{\varepsilon}(y)}{|x-y|^{d+2-2s}}dxdy$
$\displaystyle\leq
C_{dsr}\|u_{\varepsilon}^{r}\|_{L^{q}({\mathbb{R}}^{d})}\|u_{\varepsilon}\|_{L^{q_{1}}({\mathbb{R}}^{d})}=C_{dsr}\left\|u_{\varepsilon}\right\|_{L^{\frac{r+1}{1+(2s-2)/d}}({\mathbb{R}}^{d})}^{r+1},$
(54)

where $\frac{1}{q}+\frac{1}{q_{1}}+\frac{d+2-2s}{d}=2$ and we have used
$q_{1}=rq=\frac{r+1}{1+(2s-2)/d}$, $C_{dsr}$ is a bounded constant depending
on $d,s,r$. Furthermore, for

$1\leq q_{0}<r_{0}<\frac{2d}{d-2},$

we have the following GNS inequality [28]: there exists a positive constant
$C$ such that

$\displaystyle\|v\|_{L^{r_{0}}({\mathbb{R}}^{d})}\leq C\|\nabla
v\|_{L^{2}({\mathbb{R}}^{d})}^{\theta}\|v\|_{L^{q_{0}}({\mathbb{R}}^{d})}^{1-\theta},\quad\theta=\frac{\frac{1}{q_{0}}-\frac{1}{r_{0}}}{\frac{1}{q_{0}}-\frac{d-2}{2d}},$
(55)

which we apply with $v=u_{\varepsilon}^{\frac{m+r-1}{2}}$ to obtain

$\displaystyle\left\|u_{\varepsilon}\right\|_{L^{\frac{r+1}{1+(2s-2)/d}}({\mathbb{R}}^{d})}^{r+1}\leq
C\left\|\nabla
u_{\varepsilon}^{\frac{m+r-1}{2}}\right\|_{L^{2}({\mathbb{R}}^{d})}^{a\theta}\|u_{\varepsilon}\|_{L^{q_{0}(m+r-1)/2}({\mathbb{R}}^{d})}^{a(1-\theta)\frac{m+r-1}{2}},$
(56)

where the parameters satisfy

$\displaystyle a\frac{m+r-1}{2}=r+1,\quad
r_{0}\frac{m+r-1}{2}=\frac{r+1}{1+(2s-2)/d}.$

Here we pick

$\displaystyle m=\frac{d(2-m)}{2s}<q_{0}\frac{m+r-1}{2}$

and simple computations show that

$\displaystyle
a\theta=2\left(1+\frac{\frac{2-m}{(m+r-1)q_{0}}-\frac{2s}{2d}}{\frac{1}{q_{0}}-\frac{d-2}{2d}}\right)<2$

in case of $1<m<2-2s/d$. By Young’s inequality, from (56) we continue to
obtain

$\displaystyle\left\|u_{\varepsilon}\right\|_{L^{\frac{r+1}{1+(2s-2)/d}}({\mathbb{R}}^{d})}^{r+1}\leq\frac{2mr}{C_{dsr}(2s-2)(m+r-1)^{2}}\left\|\nabla
u_{\varepsilon}^{\frac{m+r-1}{2}}\right\|_{L^{2}({\mathbb{R}}^{d})}^{2}+C\|u_{\varepsilon}\|_{L^{q_{0}(m+r-1)/2}({\mathbb{R}}^{d})}^{(r+1)(1-\theta)/(1-a\theta/2)},$
(57)

where the constant $C_{dsr}$ is defined as in (2) and

$\displaystyle\frac{(r+1)(1-\theta)}{1-a\theta/2}=\frac{2s-(2-m)(2s-2)+2sr-d(2-m)}{2s-\frac{2d(2-m)}{q_{0}(m+r-1)}}>q_{0}\frac{m+r-1}{2}$
(58)

in case of

$\frac{d(2-m)}{2s}<q_{0}\frac{m+r-1}{2}\leq r.$

In particular, taking

$q_{0}\frac{m+r-1}{2}=r$

we write

$\displaystyle(2s-2)C_{dsr}(r-1)\|u_{\varepsilon}\|_{L^{\frac{r+1}{1+(2s-2)/d}}({\mathbb{R}}^{d})}^{r+1}\leq$
$\displaystyle\frac{2mr(r-1)}{(m+r-1)^{2}}\left\|\nabla
u_{\varepsilon}^{\frac{m+r-1}{2}}\right\|_{L^{2}({\mathbb{R}}^{d})}^{2}$
$\displaystyle+C\left(\|u_{\varepsilon}\|_{L^{r}({\mathbb{R}}^{d})}^{r}\right)^{1+\frac{\frac{2s}{d}-(2-m)\frac{2s-2}{d}}{\frac{2s}{d}r-(2-m)}}$
(59)

for $r>\frac{d(2-m)}{2s}=m$ due to $m=\frac{2d}{d+2s}.$ Therefore,
substituting (2) and (2) into (2) we thus end up with

$\displaystyle\frac{d}{dt}\|u_{\varepsilon}\|_{L^{r}({\mathbb{R}}^{d})}^{r}+\frac{2mr(r-1)}{(m+r-1)^{2}}\int_{{\mathbb{R}}^{d}}\left|\nabla
u_{\varepsilon}^{\frac{m+r-1}{2}}\right|^{2}dx+\varepsilon\frac{4(r-1)}{r}\int_{{\mathbb{R}}^{d}}\left|\nabla
u_{\varepsilon}^{r/2}\right|^{2}dx$ $\displaystyle\leq$ $\displaystyle
C\left(\|u_{\varepsilon}\|_{L^{r}({\mathbb{R}}^{d})}^{r}\right)^{1+\frac{\frac{2s}{d}-(2-m)\frac{2s-2}{d}}{\frac{2s}{d}r-(2-m)}}.$
(60)

Hence the local in time $L^{r}$-estimates are followed

$\displaystyle\|u_{\varepsilon}(\cdot,t)\|_{L^{r}({\mathbb{R}}^{d})}^{r}$
$\displaystyle\leq\frac{1}{\left(\|u_{\varepsilon}(0)\|_{L^{r}({\mathbb{R}}^{d})}^{-r\delta}-\delta
C(d,s,r)t\right)^{\frac{1}{\delta}}}$
$\displaystyle=\left(\frac{C(m,s,r)}{T_{0}-t}\right)^{\frac{1}{\delta}},\qquad
T_{0}=\frac{\|u_{\varepsilon}(0)\|_{L^{r}({\mathbb{R}}^{d})}^{-r\delta}}{\delta
C(d,s,r)},$ (61)

where $\delta=\frac{1-\frac{(2-m)(2s-2)}{2s}}{r-\frac{d(2-m)}{2s}}>0$ for
$r>\frac{d(2-m)}{2s}$ because of $1<m<2-\frac{2s}{d}.$ Moreover, coming back
to (2), we further deduce that

$\displaystyle\left\|\nabla
u_{\varepsilon}^{\frac{m+r-1}{2}}\right\|_{L^{2}(0,T_{0};L^{2}({\mathbb{R}}^{d}))}\leq
C\left(\|u_{\varepsilon}(0)\|_{L^{r}({\mathbb{R}}^{d})}\right).$ (62)

Step 2 ($L^{m}$-estimates) The local in time $L^{m}$-estimate is derived from
(2) by the interpolation inequality

$\displaystyle\|u_{\varepsilon}\|_{L^{m}({\mathbb{R}}^{d})}$
$\displaystyle\leq\|u_{\varepsilon}\|_{L^{r}({\mathbb{R}}^{d})}^{\eta}\|u_{\varepsilon}\|_{L^{1}({\mathbb{R}}^{d})}^{1-\eta}$
$\displaystyle\leq
M^{1-\eta}\left(\frac{C(m,s,r)}{T_{0}-t}\right)^{\frac{\eta}{r\delta}},\qquad\eta=\frac{1-\frac{1}{m}}{1-\frac{1}{r}}.$
(63)

Step 3 ($L^{\infty}$-estimates) The fact (2) allows us to imitate the proof of
[14, Theorem 2.3] and [2, Theorem 4.2] to conclude that

$\displaystyle\displaystyle\sup_{0<t<T_{0}}\|u_{\varepsilon}\|_{L^{\infty}({\mathbb{R}}^{d})}\leq
C\left(\|u_{\varepsilon}(0)\|_{L^{1}({\mathbb{R}}^{d})},\|u_{\varepsilon}(0)\|_{L^{\infty}({\mathbb{R}}^{d})}\right)$
(64)

by virtue of Moser iterative method. Thus following similar arguments of [30,
Section 4] with the aid of regularities (62) and (64), we deduce that there
exists a weak solution $u$ of (3) by passing to the limit $u_{\varepsilon}\to
u$ as $\varepsilon\to 0$ (without relabeling). Finally, a direct consequence
of [3, Theorem 2.4] characterises the maximal existence time of the weak
solution and completes the proof. $\Box$

Furthermore, a detailed analysis of the proof of [1, Theorem 2.11] shows that
the weak solution is also a free energy solution satisfying $F(u(\cdot,t))\leq
F(u_{0}).$

######  Proposition 7.

(Existence of the free energy solution) Under assumption (6) on the initial
data and $u_{\varepsilon}(\cdot,t)\in
L^{\infty}\left(0,T_{w};L^{\infty}({\mathbb{R}}^{d})\right)$ on the
approximated sequence, there exists a free energy solution to (3) on
$[0,T_{w}).$

Proof.  Multiplying
$\mu_{\varepsilon}=\frac{m}{m-1}u_{\varepsilon}^{m-1}-c_{\varepsilon}$ to the
regularized equation (42) yields

$\displaystyle\frac{d}{dt}F\left(u_{\varepsilon}(\cdot,t)\right)+\int_{{\mathbb{R}}^{d}}\left|\frac{2m}{2m-1}\nabla
u_{\varepsilon}^{m-\frac{1}{2}}-\sqrt{u_{\varepsilon}}\nabla
c_{\varepsilon}\right|^{2}dx+\frac{4\varepsilon}{m}\int_{{\mathbb{R}}^{d}}\left|\nabla
u_{\varepsilon}^{m/2}\right|^{2}dx$ $\displaystyle=$
$\displaystyle\varepsilon(2s-2)\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u_{\varepsilon}(x)u_{\varepsilon}(y)}{\left(|x-y|^{2}+\varepsilon^{2}\right)^{\frac{d+2-2s}{2}}}dxdy$
(65)

for any $t\in[0,T_{w}).$ Integrating (2) in time from $0$ to $t$ arrives at

$\displaystyle
F\left[u_{\varepsilon}(\cdot,t)\right]+\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\left|\frac{2m}{2m-1}\nabla
u_{\varepsilon}^{m-\frac{1}{2}}-\sqrt{u_{\varepsilon}}\nabla
c_{\varepsilon}\right|^{2}dxds$ $\displaystyle\leq$ $\displaystyle
F\left[u_{\varepsilon}(0)\right]+\varepsilon(2s-2)\int_{0}^{t}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u_{\varepsilon}(x)u_{\varepsilon}(y)}{\left(|x-y|^{2}+\varepsilon^{2}\right)^{\frac{d+2-2s}{2}}}dxdyds$
(66)

From (62) and (64) we know that for any $t\in[0,T_{w})$, the following basic
estimates hold true:

$\displaystyle\|u_{\varepsilon}\|_{L^{\infty}\left(0,t;L_{+}^{1}\cap
L^{\infty}({\mathbb{R}}^{d})\right)}\leq C,$ (67) $\displaystyle\left\|\nabla
u_{\varepsilon}^{\frac{m+r-1}{2}}\right\|_{L^{2}(0,t;L^{2}({\mathbb{R}}^{d}))}\leq
C,~{}~{}\mbox{for}~{}~{}1\leq r<\infty.$ (68)

Here $C$ represent constants depending only on
$\|u_{\varepsilon}(0)\|_{L_{+}^{1}\cap L^{\infty}({\mathbb{R}}^{d})}$. We
shall use (67) and (68) to pass to the limit $\varepsilon\to 0$ in (2).

Firstly, the term in the right hand side of (2) is bounded with the help of
(67) and the HLS inequality (23) that

$\displaystyle\int_{0}^{t}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u_{\varepsilon}(x)u_{\varepsilon}(y)}{\left(|x-y|^{2}+\varepsilon^{2}\right)^{\frac{d+2-2s}{2}}}dxdyds\leq
C\int_{0}^{t}\|u_{\varepsilon}\|_{L^{2d/(d-2+2s)}({\mathbb{R}}^{d})}^{2}ds\leq
C$ (69)

for any $t\in[0,T_{w}).$

Secondly, we note that the dissipation term is uniformly bounded

$\displaystyle\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\left|\frac{2m}{2m-1}\nabla
u_{\varepsilon}^{m-\frac{1}{2}}-\sqrt{u_{\varepsilon}}\nabla
c_{\varepsilon}\right|^{2}dxds\leq C.$ (70)

Actually, taking $r=m$ in (68) yields $\left\|\nabla
u_{\varepsilon}^{m-1/2}\right\|_{L^{2}(0,t;L^{2}({\mathbb{R}}^{d}))}\leq C$.
Besides, by Lemma 4 and using (67) we find

$\displaystyle\int_{0}^{t}\left\|u_{\varepsilon}|\nabla
c_{\varepsilon}|^{2}\right\|_{L^{1}({\mathbb{R}}^{d})}ds\leq
C\int_{0}^{t}\|u_{\varepsilon}\|_{L^{\frac{3d}{d-2(1-2s)}}({\mathbb{R}}^{d})}^{3}ds\leq
C.$ (71)

Now by going through similar steps as the proof of [1, Theorem 2.11], it is
straightforward from the convergence properties (51) to pass to the limit
$\varepsilon\to 0$ (without relabeling) and thus obtain the lower semi-
continuity of the dissipation term

$\displaystyle\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\left|\frac{2m}{2m-1}\nabla
u^{m-1/2}-\sqrt{u}\nabla c\right|^{2}dxds$ $\displaystyle\leq$
$\displaystyle\displaystyle\liminf_{\varepsilon\to
0}\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\left|\frac{2m}{2m-1}\nabla
u_{\varepsilon}^{m-1/2}-\sqrt{u_{\varepsilon}}\nabla
c_{\varepsilon}\right|^{2}dxds.$ (72)

In addition, the convergence of the free energy can be directly deduced from
(51) that

$\displaystyle
F(u_{\varepsilon}(\cdot,t))=\frac{1}{m-1}\int_{{\mathbb{R}}^{d}}u_{\varepsilon}^{m}dx-\frac{1}{2(d-2s)}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u_{\varepsilon}(x,t)u_{\varepsilon}(y,t)}{|x-y|^{d-2s}}dxdy\to$
$\displaystyle\frac{1}{m-1}\int_{{\mathbb{R}}^{d}}u^{m}dx-\frac{1}{2(d-2s)}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u(x,t)u(y,t)}{|x-y|^{d-2s}}dxdy=F(u(\cdot,t))~{}~{}\mbox{a.e.~{}in}~{}(0,T_{w}).$
(73)

Hence combining (69), (2) and (2), letting $\varepsilon\to 0$ in (2) follows
that

$\displaystyle
F\left[u(\cdot,t)\right]+\int_{0}^{t}\int_{{\mathbb{R}}^{d}}\left|\frac{2m}{2m-1}\nabla
u^{m-\frac{1}{2}}-\sqrt{u}\nabla c\right|^{2}dxds\leq F[u_{0}(x)],\quad
a.e.~{}~{}t\in(0,T_{w}).$

Thus $u$ is a free energy solution satisfying the free energy inequality (33).
$\Box$

##  3 Steady states

As we have just seen, the existence time of the free energy solution to (3)
heavily depends on the behavior of its $L^{m}$-norm. Moreover, the first term
of the free energy involves the $L^{m}$-norm and (26) tells us that the second
term of the free energy is related to the stationary solution $U_{s}$ to (3).
Hence, in order to prove the existence of a global weak solution by the energy
functional, the information given by steady states will be of paramount
importance. Therefore, in this section, we will give a detailed analysis of
steady states to (3).

The steady equation of (3) is followed in the sense of distribution

$\displaystyle\Delta U_{s}^{m}-\nabla\cdot\left(U_{s}(x)\nabla
C_{s}(x)\right)=0,~{}~{}x\in{\mathbb{R}}^{d},$ (74)

where $C_{s}(x)$ is given by

$\displaystyle
C_{s}(x)=\frac{1}{d-2s}\int_{{\mathbb{R}}^{d}}\frac{U_{s}(y)}{|x-y|^{d-2s}}dy.$
(75)

Starting from a Pohozaev type identity, we will show a full characterization
of minimizers of $F(u)$ to relate to the steady states.

######  Lemma 8.

(A Pohozaev identity for steady solutions) Assume $U_{s}\in
L^{m}({\mathbb{R}}^{d})$ satisfying (74), then the steady solution satisfies
the following identity in the sense of distribution

$\displaystyle\int_{{\mathbb{R}}^{d}}U_{s}^{m}dx=\frac{d-2s}{2d}\int_{{\mathbb{R}}^{d}}C_{s}U_{s}dx.$
(76)

Proof.  We follow a similar argument as done in [5, 28]. Consider a cut-off
function $\psi_{R}(x)\in C_{0}^{\infty}({\mathbb{R}}^{d})$ and
$\psi_{R}(x)=|x|^{2}$ for $|x|<R,~{}\psi_{R}(x)=0$ for $|x|\geq 2R.$ Then we
compute

$\displaystyle\int_{{\mathbb{R}}^{d}}\Delta
U_{s}^{m}\psi_{R}(x)dx=\int_{{\mathbb{R}}^{d}}U_{s}^{m}\Delta\psi_{R}(x)dx$
(77)

and

$\displaystyle\int_{{\mathbb{R}}^{d}}\psi_{R}(x)\nabla\cdot(U_{s}\nabla
C_{s})dx=-\int_{{\mathbb{R}}^{d}}\nabla\psi_{R}(x)\cdot U_{s}\nabla C_{s}dx$
$\displaystyle=$
$\displaystyle\frac{1}{2}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{[\nabla\psi_{R}(x)-\nabla\psi_{R}(y)]\cdot(x-y)}{|x-y|^{2}}\frac{U_{s}(x)U_{s}(y)}{|x-y|^{d-2s}}dxdy.$
(78)

Both terms in the right-hand side of (77) and (3) are bounded (because
$\Delta\psi_{R}(x)$ and
$\frac{[\nabla\psi_{R}(x)-\nabla\psi_{R}(y)]\cdot(x-y)}{|x-y|^{2}}$ are
bounded and $U_{s}\in L^{m}({\mathbb{R}}^{d})$). Therefore, as $R\to\infty,$
we may pass to the limit in each term using the Lebesgue monotone convergence
theorem and obtain the identity. $\Box$

With this identity in hand, we are now ready to make use of the variational
structure of the free energy to deduce the following results.

######  Proposition 9.

(Critical point of $F(u)$) Let $\bar{u}\in L_{+}^{1}\cap
L^{\infty}({\mathbb{R}}^{d})$ with $\|\bar{u}\|_{L^{1}({\mathbb{R}}^{d})}=M$
be a global minimizer of $F(u)$. Then $\bar{u}$ satisfies the Euler-Lagrange
condition

$\displaystyle\frac{m}{m-1}\bar{u}^{m-1}-\frac{1}{d-2s}\bar{u}\ast\frac{1}{|x|^{d-2s}}=\overline{C},~{}~{}\forall~{}x\in
supp(\bar{u}),$ (79)

where the constant

$\overline{C}=\frac{1}{M}\left(\frac{m}{m-1}\|\bar{u}\|_{L^{m}({\mathbb{R}}^{d})}^{m}-\frac{1}{d-2s}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{\bar{u}(x)\bar{u}(y)}{|x-y|^{d-2s}}dxdy\right).$

Furthermore, such minimizer is the steady state $U_{s}$ of (3) in the
distributional sense (74). Therefore, for $m=\frac{2d}{d+2s},$
$supp(U_{s})={\mathbb{R}}^{d}.$

Proof.  We use some ideas from [1, 11, 13]. Set $Y_{M}=\left\\{u\in
L_{+}^{1}\cap
L^{\infty}({\mathbb{R}}^{d}),\|u\|_{L^{1}({\mathbb{R}}^{d})}=M\right\\}$. Let
$\bar{u}\in Y_{M}$ be a global minimizer of $F(u)$ with
$\Omega=supp(\bar{u}).$ For any $\psi\in C_{0}^{\infty}(\Omega),$ denoting

$\displaystyle\varphi=\left(\psi-\frac{1}{M}\int_{\Omega}\psi\bar{u}dx\right)\bar{u}(x),$
(80)

then $supp(\varphi)\subseteq\Omega$ and $\int_{\Omega}\varphi dx=0.$ Besides,
there exists

$\displaystyle\varepsilon_{0}:=\frac{\displaystyle\min_{y\in
supp\varphi}\bar{u}(y)}{\displaystyle\max_{y\in
supp\varphi}\left|\varphi(y)\right|}>0$

such that $\bar{u}+\varepsilon\varphi\geq 0$ in $\Omega$ for
$0<\varepsilon<\varepsilon_{0}.$ Now $\bar{u}$ is a global minimizer of $F(u)$
in $\Omega$ if and only if

$\displaystyle\frac{d}{d\varepsilon}\Big{|}_{\varepsilon=0}F(\bar{u}+\varepsilon\varphi)=0.$
(81)

The above definition implies that for any $\int_{\Omega}\varphi(x)dx=0$

$\displaystyle\int_{\Omega}\left(\frac{m}{m-1}\bar{u}^{m-1}-\frac{1}{d-2s}\int_{{\mathbb{R}}^{d}}\frac{\bar{u}(y)}{|x-y|^{d-2s}}dy\right)\varphi(x)dx=0.$
(82)

Then by the definition (80) of $\varphi,$ (82) can be read as

$\displaystyle\int_{\Omega}\left(\frac{m}{m-1}\bar{u}^{m-1}-\frac{1}{d-2s}\int_{{\mathbb{R}}^{d}}\frac{\bar{u}(y)}{|x-y|^{d-2s}}dy-\overline{C}\right)\bar{u}\psi
dx=0,~{}~{}\mbox{for any}~{}\psi\in C_{0}^{\infty}(\Omega).$ (83)

This gives

$\displaystyle\frac{m}{m-1}\bar{u}^{m-1}-\frac{1}{d-2s}\int_{{\mathbb{R}}^{d}}\frac{\bar{u}(y)}{|x-y|^{d-2s}}dy=\overline{C},~{}~{}a.e.~{}~{}\mbox{in}~{}\Omega.$
(84)

Since $\bar{u}\in L_{+}^{1}\cap L^{\infty}({\mathbb{R}}^{d}),$ we have from
similar arguments of [12, Lemma 2.3] that $\bar{u}$ is continuous in $\Omega$.
Owning the regularity we arrive at

$\displaystyle\nabla\cdot\left(\nabla\bar{u}^{m}-\frac{1}{d-2s}\bar{u}\nabla\bar{u}\ast\frac{1}{|x|^{d-2s}}\right)=\nabla\cdot\left(\bar{u}\nabla\left(\frac{m}{m-1}\bar{u}^{m-1}-\frac{1}{d-2s}\bar{u}\ast\frac{1}{|x|^{d-2s}}\right)\right)=0$
(85)

in the sense of distribution. Therefore, all global minimizers of the free
energy are stationary solutions of (3) in the distributional sense
characterized by (85). Then by employing Theorem 2.2 of [12], the steady
states must be radially decreasing up to translations.

Now we declare that the steady state $U_{s}$ of (3) satisfies

$\displaystyle\frac{m}{m-1}U_{s}^{m-1}-\frac{1}{d-2s}\int_{{\mathbb{R}}^{d}}\frac{U_{s}(y)}{|x-y|^{d-2s}}dy=\overline{C},~{}~{}\forall~{}x\in\Omega=supp(U_{s}).$
(86)

Keeping (76) in mind, taking inner product to (86) by $U_{s}$ in
${\mathbb{R}}^{d}$ yields

$\displaystyle\overline{C}=\frac{1}{M}\left(\frac{1}{m-1}-\frac{d+2s}{d-2s}\right)\int_{{\mathbb{R}}^{d}}U_{s}^{m}dx.$
(87)

When $m=\frac{2d}{d+2s},\overline{C}=0,$ thus from (86) we claim that
$\Omega={\mathbb{R}}^{d}.$ If not, choose a point $x_{0}\in\partial\Omega,$ as
we take a sequence of point $x_{n}\to x_{0}$ with $x_{n}\in\Omega,$ we have
that $\frac{m}{m-1}U_{s}^{m-1}(x_{n})\to 0,$ whereas the sequence
$\frac{1}{d-2s}\int_{{\mathbb{R}}^{d}}\frac{U_{s}(y)}{|x_{n}-y|^{d-2s}}dy>0$,
a contradiction. $\Box$

Now we are ready to assert that the steady states of (3) can be expressed
explicitly. By Proposition 9 one has that

$\displaystyle\left\\{\begin{array}[]{ll}\frac{m}{m-1}U_{s}^{m-1}-C_{s}=0,~{}~{}\mbox{in}~{}~{}{\mathbb{R}}^{d},\\\
C_{s}=\frac{1}{d-2s}U_{s}\ast\frac{1}{|x|^{d-2s}}.\end{array}\right.$ (90)

Hence $C_{s}$ is endowed with the following integral equation

$\displaystyle C_{s}(x)$
$\displaystyle=\frac{1}{d-2s}\left(\frac{m-1}{m}\right)^{\frac{1}{m-1}}\int_{{\mathbb{R}}^{d}}\frac{C_{s}(y)^{\frac{1}{m-1}}}{|x-y|^{d-2s}}dy$
$\displaystyle=\frac{1}{d-2s}\left(\frac{d-2s}{2d}\right)^{\frac{d+2s}{d-2s}}\int_{{\mathbb{R}}^{d}}\frac{C_{s}(y)^{\frac{d+2s}{d-2s}}}{|x-y|^{d-2s}}dy.$
(91)

It has been proved in [17] that every positive solution of (3) is radially
symmetric and decreasing about some point $x_{0}\in{\mathbb{R}}^{d}$ and
therefore assumes the form

$\displaystyle
C_{s}(x)=c\left(\frac{\lambda}{\lambda^{2}+|x-x_{0}|^{2}}\right)^{\frac{d-2s}{2}}$
(92)

with some positive constants $c$ and $\lambda.$ It gives our final estimates
for this part.

######  Proposition 10.

For $m=\frac{2d}{d+2s},$ any steady state to (3) is radially symmetric up to
translation and given by a one-parameter family for some $\lambda>0$ and $c>0$

$\displaystyle
U_{s}(x)=c\left(\frac{\lambda}{\lambda^{2}+|x-x_{0}|^{2}}\right)^{\frac{d+2s}{2}}$
(93)

with $\int_{{\mathbb{R}}^{d}}U_{s}^{m}(x)dx=C(d).$

##  4 Proof of Theorem 5

With the aid of the characterization of steady states and the free energy, we
can proceed to distinguish the two cases
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}<\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$ and
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$ to
show $\|u(\cdot,t)\|_{L^{m}({\mathbb{R}}^{d})}$ can be bounded from below or
above separately, then we establish the global existence and finite time blow-
up of solutions to (3).

###  4.1 Proof of the finite time blow-up (ii) of Theorem 5

In this subsection, we start with the case
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$ and
show the finite time blow-up of the weak solution to (3). Firstly we prepare a
priori bound.

######  Lemma 11.

(A priori bound) Under assumption (6), if the initial free energy
$F(u_{0})<F(U_{s})$ and
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$,
then there exists a constant $\mu>1$ such that the corresponding free energy
solution $u$ satisfies that for any $t>0$

$\displaystyle\|u(\cdot,t)\|_{L^{m}({\mathbb{R}}^{d})}>\mu\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}.$
(94)

Proof.  Firstly letting

$\displaystyle f=u(x),h=u(y),\beta=d-2s,q=q_{1}=\frac{2d}{2d-\beta}$

in the HLS inequality (23), we can infer from the expression of $F(u)$ that

$\displaystyle F(u)$
$\displaystyle=\frac{1}{m-1}\|u\|_{L^{m}({\mathbb{R}}^{d})}^{m}-\frac{1}{2(d-2s)}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u(x)u(y)}{|x-y|^{d-2s}}dxdy$
$\displaystyle\geq\frac{1}{m-1}\|u\|_{L^{m}({\mathbb{R}}^{d})}^{m}-\frac{C(d,\beta)}{2(d-2s)}\|u\|_{L^{m}({\mathbb{R}}^{d})}^{2}.$
(95)

While (26) also tells us that

$\displaystyle
F(U_{s})=\frac{1}{m-1}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}-\frac{C(d,\beta)}{2(d-2s)}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{2}.$
(96)

We define an auxiliary function

$g(x)=\frac{1}{m-1}x-\frac{C(d,\beta)}{2(d-2s)}x^{2/m}.$

It can be seen from (4.1) and (96) that

$g\left(\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}\right)=F(U_{s})\geq
g\left(\|u\|_{L^{m}({\mathbb{R}}^{d})}^{m}\right)$

for all $u\in L^{m}({\mathbb{R}}^{d}).$ Namely, the maximum point $x_{0}$ of
$g(x)$ is attained by

$\displaystyle
x_{0}=\left(\frac{m(d-2s)}{(m-1)C(d,\beta)}\right)^{\frac{m}{2-m}}=\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}.$
(97)

In addition, in the case that $F(u_{0})<F(U_{s}),$ there is a $0<\delta<1$
such that

$\displaystyle F(u_{0})<\delta F(U_{s}).$ (98)

Thus, by the monotonicity of $F(u)$ with respect to time, it follows that

$\displaystyle g\left(\|u\|_{L^{m}({\mathbb{R}}^{d})}^{m}\right)\leq F(u)\leq
F(u_{0})<\delta F(U_{s})=\delta
g\left(\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}\right).$ (99)

Therefore, for any $x>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m},g(x)$ is
strictly decreasing and it has a strictly decreasing inverse function
$g^{-1}.$ So if
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}^{m}>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m},$
one has that for some $\mu>1$ depending on $\delta,$

$\displaystyle\|u(\cdot,t)\|_{L^{m}({\mathbb{R}}^{d})}^{m}>\mu\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}.$
(100)

This completes this lemma. $\Box$

When
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}<\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$,
following the lines (4.1)-(99) in the proof of Lemma 11, we can deduce the
following reverse result.

######  Corollary 12.

Under assumption (6), if the initial free energy $F(u_{0})<F(U_{s})$ and
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}<\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$,
then there exists a constant $\mu_{1}<1$ such that the corresponding free
energy solution $u$ satisfies that for any $t>0$

$\displaystyle\|u(\cdot,t)\|_{L^{m}({\mathbb{R}}^{d})}<\mu_{1}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}.$
(101)

Now we can utilize the standard argument relying on the evolution of the
second moment of solutions as originally done in [20] to prove the finite time
blow-up result in Theorem 5.

######  Proposition 13.

(Finite time blow-up) Assume
$\int_{{\mathbb{R}}^{d}}|x|^{2}u_{0}(x)dx<\infty,F(u_{0})<F(U_{s})$ and
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$. Let
$u(x,t)$ be a weak free energy solution to (3) on $[0,T_{w}),$ then it
satisfies

$\displaystyle\frac{d}{dt}\int_{{\mathbb{R}}^{d}}|x|^{2}u(x,t)dx=-4s\int_{{\mathbb{R}}^{d}}u^{m}dx+2(d-2s)F(u),~{}0<t<T_{w}.$
(102)

Here $T_{w}<\infty$ and $\|u\|_{L^{m}({\mathbb{R}}^{d})}$ blows up in finite
time.

Proof.  Here we show the formal computation, the passing to the limit from the
approximated problem (42) can be done on account of [1, Theorem 2.11] and [29,
Lemma 6.2] without any further complication. By integrating by parts in (3)
and symmetrising the second term, the time derivative of the second moment of
the weak solution is endowed with

$\displaystyle\frac{d}{dt}m_{2}(t)$
$\displaystyle=\frac{d}{dt}\int_{{\mathbb{R}}^{d}}|x|^{2}u(x,t)dx=2d\int_{{\mathbb{R}}^{d}}u^{m}dx-\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\frac{u(x,t)u(y,t)}{|x-y|^{d-2s}}dxdy$
$\displaystyle=\left(2d-\frac{2}{m-1}(d-2s)\right)\int_{{\mathbb{R}}^{d}}u^{m}dx+2(d-2s)F(u)$
$\displaystyle=-4s\int_{{\mathbb{R}}^{d}}u^{m}dx+2(d-2s)F(u).$ (103)

Then applying Lemma 11 with some $\mu>1$ and the decreasing of $F(u)$ in time
indicates

$\displaystyle\frac{d}{dt}\int_{{\mathbb{R}}^{d}}|x|^{2}u(x,t)dx$
$\displaystyle\leq-4s\mu\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}+2(d-2s)F(u_{0})$
$\displaystyle<-4s\mu\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}+2(d-2s)F(U_{s})$
$\displaystyle=4s(1-\mu)\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}<0,$ (104)

where we have used the identity
$F(U_{s})=\frac{2s}{d-2s}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}$ which can be
derived by

$\displaystyle F(U_{s})$
$\displaystyle=\frac{1}{m-1}\int_{{\mathbb{R}}^{d}}U_{s}^{m}dx-\frac{1}{2}\int_{{\mathbb{R}}^{d}}U_{s}C_{s}dx$
$\displaystyle=\frac{1}{m-1}\int_{{\mathbb{R}}^{d}}U_{s}^{m}dx-\frac{m}{2(m-1)}\int_{{\mathbb{R}}^{d}}U_{s}^{m}dx$
$\displaystyle=\frac{2-m}{2(m-1)}\int_{{\mathbb{R}}^{d}}U_{s}^{m}dx=\frac{2s}{d-2s}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}^{m}$
(105)

due to $C_{s}=\frac{m}{m-1}U_{s}^{m-1}.$ Thus there exists a $T>0$ such that
$\displaystyle\lim_{t\to T}m_{2}(t)=0$.

On the other hand, it follows from the Hölder inequality that

$\displaystyle\int_{{\mathbb{R}}^{d}}u(x)dx=\int_{|x|\leq
R}u(x)dx+\int_{|x|>R}u(x)dx$ $\displaystyle\leq$ $\displaystyle
CR^{(m-1)d/m}\|u\|_{L^{m}({\mathbb{R}}^{d})}+\frac{1}{R^{2}}m_{2}(t).$ (106)

Choosing
$R=\left(\frac{Cm_{2}(t)}{\|u\|_{L^{m}({\mathbb{R}}^{d})}}\right)^{\frac{1}{a+2}}$
with $a=\frac{(m-1)d}{m}$ gives

$\displaystyle\|u_{0}\|_{L^{1}({\mathbb{R}}^{d})}=\|u\|_{L^{1}({\mathbb{R}}^{d})}\leq
c\|u\|_{L^{m}({\mathbb{R}}^{d})}^{\frac{2}{a+2}}m_{2}(t)^{\frac{a}{a+2}}.$
(107)

It follows that

$\displaystyle\displaystyle\limsup_{t\to
T}\|u(\cdot,t)\|_{L^{m}({\mathbb{R}}^{d})}\geq\displaystyle\lim_{t\to
T}\frac{\|u_{0}\|_{L^{1}({\mathbb{R}}^{d})}^{(a+2)/2}}{cm_{2}(t)^{a/2}}=\infty.$
(108)

Thus the proof is completed. $\Box$

###  4.2 Proof of the global existence (i) of Theorem 5

A direct consequence of Theorem 6 and Corollary 12 is the following global
existence result.

######  Proposition 14.

(Global existence) Under assumption (6), assume $F(u_{0})<F(U_{s})$ and
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}<\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$,
then there exists a free energy solution to (3) on $[0,\infty).$

Proof.  By Theorem 6, there are $T_{w}$ and a free energy solution to (3) in
$[0,T_{w})$ with initial data $u_{0}.$ We then infer from Corollary 12 that
there exists a $\mu_{1}<1$, for all $t\in[0,T_{w})$,

$\|u(\cdot,t)\|_{L^{m}({\mathbb{R}}^{d})}<\mu_{1}\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}.$

Therefore, we can deduce that $u\in
L^{\infty}\left(0,\min(T,T_{w});L^{m}({\mathbb{R}}^{d})\right)$ for every
$T>0$ which implies that $T_{w}=\infty$ by Theorem 6. $\Box$

##  5 Conclusions

This paper concerns equation (3) for the energy critical exponent
$m=\frac{2d}{d+2s}$. It is proved that any steady states $U_{s}(x)$ of (3) is
radially symmetric and decreasing about some point $x_{0}\in{\mathbb{R}}^{d}$
and assumes the from
$U_{s}(x)=c\left(\frac{\lambda}{\lambda^{2}+|x-x_{0}|^{2}}\right)^{\frac{d+2s}{2}}$
for some positive constants $c$ and $\lambda$. Our results show that the
$L^{p}$ norm of the stationary solutions plays a critical role on determining
the global existence and finite time blow-up of solutions to (3). Here
$p=\frac{d(2-m)}{2s}=m$ and the $L^{p}$ norm produces a balance between the
diffusion and the aggregation terms under the invariant scaling for equation
(3). Actually, $\|U_{s}\|_{L^{p}({\mathbb{R}}^{d})}$ is a constant only
depending on $d,m$ and we prove that the $L^{p}$ norm of the stationary
solutions $\|U_{s}\|_{L^{p}({\mathbb{R}}^{d})}$ is the sharp condition on the
initial data separating infinite-time existence from finite-time blow-up.
Precisely, in the case that the initial free energy $F(u_{0})$ is less than
$F(U_{s})$ which is the free energy of the stationary solutions, there exists
a global weak solution satisfying the free energy inequality when the initial
data satisfies
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}<\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$ and
the solution blows up at finite time provided by the condition
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}>\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})}$. For
the critical case
$\|u_{0}\|_{L^{m}({\mathbb{R}}^{d})}=\|U_{s}\|_{L^{m}({\mathbb{R}}^{d})},$
whether the solution exists globally in time or blows up in finite time is
still unknown.

## References

  * [1] S. Bian, J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$. Comm. Math. Phys., 323 (2013), 1017-1070.
  * [2] S. Bian, J.-G. Liu and C. Zou, Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$. Kinetic and Related Models, 7(1) (2014), 9-28.
  * [3] A. Blanchet, J.A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions. Calc. Var. Parital Differential Equations, 35 (2009), 133-168.
  * [4] A. Blanchet, J.A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller- Segel model in ${\mathbb{R}}^{2}$. Comm. Pure Appl. Math., 61 (2008), 1449-1481.
  * [5] A. Blanchet, J. Dolbeault and B. Perthame, Two- dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions. Eletron. J. Differential Equations, 44 (2006), 1-33.
  * [6] A.L. Bertozzi, C.M. Topaz, M.A. Lewis, A nonlocal continuum model for biological aggregation. Bull. Math. Biol., 68 (2006), 1601-1623.
  * [7] M.P. Brenner, L.S. Levitov, E.O. Budrene, Physical mechanisms for chemotactic pattern formation by bacteria. Biophysical Journal, 74 (1998), 1677-1693.
  * [8] M.P. Brenner, P. Constantin, L.P. Kadanoff, A. Schenkel and S.C. Venkataramani, Diffusion, attraction and collapse. Nonlinearity, 12 (1999), 1071-1098.
  * [9] V. Calvez, J.A. Carrillo, F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime. Nonlinear Anal., 159 (2017), 85-128.
  * [10] V. Calvez, J.A. Carrillo, F. Hoffmann, Uniqueness of stationary states for singular Keller-Segel type models. Nonlinear Anal., 205 (2021), 1-24.
  * [11] J.A. Carrillo, D. Castorina, B. Volzone, Ground states for diffusion dominated free energies with logarithmic interaction. SIAM J. Math. Anal., 47(1) (2015), 1-25.
  * [12] J.A. Carrillo, S. Hittmeir, B. Volzone and Y. Yao, Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics. Invent. Math., 218(3) (2019), 889-977.
  * [13] J.A. Carrillo, F. Hoffmann, E. Mainini, B. Volzone, Ground states in the diffusion-dominated regime. Calc. Var. Partial Differential Equations, 57 (2018), doi: 10.1007/s00526-018-1402-2.
  * [14] J.A. Carrillo, J. Wang, Uniform in time $L^{\infty}$-estimates for nonlinear aggregation diffusion equations. Acta Applicandae Mathematica, 164 (2019), 1-19.
  * [15] H. Chan, M.D.M. González, Y. Huang, E. Mainini and B. Volzone, Uniqueness of entire ground states for the fractional plasma problem. Calc. Var. Partial Differential Equations, 59(195) (2020), https://doi.org/10.1007/s00526-020-01845-y.
  * [16] S. Chandrasekhar, An Introduction to the Study of Stellar Structure. Dover, New York, 1967.
  * [17] W. Chen, C. Li, B. Ou, Classification of solutions for an integral equation. Comm. Pure Appl. Math., 59(3) (2006), 330-343.
  * [18] W. Chen, C. Li, B. Ou, Qualitative properties of solutions for an integral equation. Disc. Cont. Dyn. Sys., 12 (2005), 347-354.
  * [19] W. Chen, C. Li, B. Ou, Classification of solutions for a system of integral equations. Comm. in PDE, 30 (2005), 59-65.
  * [20] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis. Trans. Amer. Math. Soc., 239(2) (1992), 819-821.
  * [21] E.F. Keller, L.A. Segel, Initiation of slide mold aggregation viewed as an instability. J. Theor. Biol., 26 (1970), 399-415.
  * [22] I. Kim, Y. Yao, The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle. SIAM J. Math. Anal., 44 (2012), 568-602.
  * [23] K.J. Painter, J.M. Bloomfield, J.A. Sherratt, A. Gerisch, A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bull. Math. Biol., 77(6) (2015), 1132-1165.
  * [24] P. Friedl, K. Wolf, Tumor-cell invasion and migration: diversity and escape mechanisms. Nature reviews (cancer), 3 (2003), 362-374.
  * [25] H. Murakawa, H. Togashi, Continuous models for cell-cell adhesion. J. Theor. Biol., 374 (2015), 1-12.
  * [26] A. Gerisch, M.A.J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: local and nonlocal models and the effect of adhesion. J. Theor. Biol., 250(4) (2008), 684-704.
  * [27] E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics. V. 14, American Mathematical Society Providence, Rhode Island, 2nd edition, 2001.
  * [28] B. Perthame, Transport Equations in Biology. Birkhaeuser Verlag, Basel-Boston-Berlin, 2007.
  * [29] Y. Sugiyama, Global existence in subcritical cases and finite time blow-up in supercritical cases to degenerate Keller-Segel systems. Differ. Integral Equ., 19 (2006), 841-876.
  * [30] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate keller-segel model with a power factor in drift term. J. Differential Equations, 227 (2006), 333-364.
  * [31] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87 (1983), 567-576.
  * [32] T.P. Witelski, A.J. Bernoff and A.L. Bertozzi, Blowup and dissipation in critical-case unstable thin film equaion. Eur. J. Appl. Math., 15 (2004), 223-256.


S3b | 100 | solid | gold | 10 | 675 | 0.5pi | 0 | 0 | 0
S3c | 100 | solid | gold | 10 | 675 | 0 | 0 | 0.5pi | 0
S4a-f | 100 | solid | gold | 10 | 675 | 0.5pi | 0 | 0 | 0
S4g-l | 100 | solid | gold | 10 | 675 | 0.5pi | 0 | 0.5pi | 0
S5 | 100 | solid | gold | 10 | 675 | 0 | 0 | 0.25pi | 0
S6 | 100 | solid | gold | 10 | 675 | 0.5pi | 0 | 0 | 0
S14a | 50 | solid | gold | 5 | 680 | 0 | 0 | 0.25pi | 0
| 50 | solid | gold | 5 | 680 | 0 | pi | 0.75pi | 0
S17 | 50 | solid | gold | 5 | 680 | 0 | 0 | 0.25pi | 0
S19 | 100 | solid | gold | 10 | 675 | 0 | 0 | 0.25pi | 0
S21 | 50 | solid | gold | 5 | 680 | 0:0.5pi | 0 | 0:pi | 0:0.5pi
S26a/b | 100 | solid | gold | 10 | 675 | 0.5pi | 0 | 0 | 0
| microscope
---|---
Figure | focus | dist_glass | l_max | m_max | N | r | M | num_px | px_size
| [nm] | [nm] |  |  |  | [m] |  |  | [nm]
2d(1)/S7a | 120 | 20 | 10 | 3 | 100 | 1 | 363.63 | 50 | 15
2d(2)/S7b | 120 | 20 | 10 | 3 | 100 | 1 | 363.63 | 50 | 15
2d(3)/S7c | 70 | 20 | 10 | 3 | 100 | 1 | 363.63 | 50 | 15
S3b/c | 120 | 20 | 10 | 3 | 100 | 1 | 363.63 | 50 | 15
S4a-l | 120 | 20 | 1:20 | l_max | 100 | 1 | 363.63 | 50 | 15
S5 | 120 | 20 | 2:30 | 3 | 4:100 | 1 | 363.63 | 2:32 | 30
S6 | 120 | 20 | 10 | 3 | 100 | 1 | 363.63 | 50 | 15
S14a | 56 | 6 | 10 | 3 | 100 | 1 | 100 | 65 | 10
S17 | 56 | 6 | 10 | 3 | 100 | 1 | 100 | 50 | 15
S19 | 120 | 20 | 10 | 3 | 100 | 1 | 363.63 | 20 | 30
S21 | 56 | 6 | 10 | 3 | 100 | 1 | 100 | 50 | 15
S26a | 120 | 20 | 10 | 1 | 1:100 | 1 | 363.63 | 24 | 30
S26b | 120 | 20 | 10 | 1 | 100 | 1e-9:1 | 363.63 | 24 | 30
Table S1: Parameters for Analytically calculated PSFs.

## Supplementary Table 2:
Numerical PSF parameters

Table S2 below shows the numerical parameters used for all the numerically
calculated PSFs throughout the paper. The numerical parameters that are the
same for all calculations are not mentioned in the table, and are:

  * •

gap between NP and glass = 20 nm

  * •

gap between NP and dipole = 10 nm

  * •

dipole orientation azimuthal angle = 0 deg

  * •

FDTD simulation time = 40 fs

  * •

detector z-position = -20 nm (20 nm below water-glass interface)

  * •

refractive index water = 1.333

  * •

refractive index glass = 1.52

  * •

refractive index gold = taken from Johnson and Christy [Johnson and Christy,
1972]

| NP | Dipole emitter | FDTD
---|---|---|---
Figure | radius | lambda | orientation | positon | size | accuracy | fine mesh
| [nm] | [nm] | polar [deg] |  | [$\mu$m] |  | nm
2c | 100 | 675 | 90 | top | 8 | 1-8 | 3
2d (1)/S7a | 100 | 675 | 90 | top | 8 | 6 | 3
2d (2)/S7b | 100 | 400 | 45 |  | 45 deg
---
8 | 6 | 5
2d (3)/S7c | 50 | 675 | 90 | side | 8 | 6 | 0.6
S3b | 100 | 675 | 90 | top | 8 | 6 | 3
S3c | 100 | 675 | 90 | side | 8 | 6 | 3
S6a | 100 | 675 | 90 | top | 0.25-16 | 6 | 3
S6b | 100 | 675 | 90 | top | 8 | 1-8 | 3
Table S2: Parameters for numerically calculated PSFs. This able shows the
numerical parameters used for all numerically calculated PSFs throughout this
paper. Explanation of the parameters can be found in the Methods section of
the main text.

## Bibliography for SI

  * [Abramowitz et al., 1964] Abramowitz, M., Stegun, I. A., et al. (1964).  Handbook of mathematical functions, volume 55.  Dover New York.
  * [Bloksma and Zijlstra, 2021] Bloksma, F. and Zijlstra, P. (2021).  Imaging and localization of single emitters near plasmonic particles of different size, shape, and material.  The Journal of Physical Chemistry C, 125(40):22084–22092.
  * [Bohren and Huffman, 1983] Bohren, C. F. and Huffman, D. R. (1983).  Absorption and scattering of light by small particles.  Wiley.
  * [Boström et al., 1991] Boström, A., Kristensson, G., and Ström, S. (1991).  Transformation properties of plane, spherical and cylindrical scalar and vector wave functions.  In Acoustic, Electromagnetic and Elastic Wave Scattering, Field Representations and Introduction to Scattering, pages 165–210. Elsevier.
  * [Chew, 1988] Chew, W. C. (1988).  A quick way to approximate a sommerfeld-weyl-type integral.  IEEE Transactions on Antennas and Propagation, 36:1654–1657.
  * [Doicu et al., 2006] Doicu, Y. A., Wriedt, A., and Eremin, T. (2006).  Light scattering by systems of particles: null-field method with discrete sources: theory and programs.  Springer.
  * [Dyakov et al., 2009] Dyakov, S. A., Tolmachev, V. A., Astrova, E. V., Tikhodeev, S. G., Timoshenko, V. Y., and Perova, T. S. (2009).  Numerical methods for calculation of optical properties of layered structures.  volume 7521, page 75210G. SPIE.
  * [Egel, 2019] Egel, A. (2019).  Accurate optical simulation of disordered scattering layers for light extraction from organic light emitting diodes.  PhD thesis, Karlsruher Institut fur Technologie (KIT).
  * [Egel et al., 2021] Egel, A., Czajkowski, K. M., Theobald, D., Ladutenko, K., Kuznetsov, A. S., and Pattelli, L. (2021).  Smuthi: A python package for the simulation of light scattering by multiple particles near or between planar interfaces.  Journal of Quantitative Spectroscopy and Radiative Transfer, 273:107846.
  * [Enderlein, 2000] Enderlein, J. (2000).  Theoretical study of detection of a dipole emitter through an objective with high numerical aperture.  Optics letters, 25:634–636.
  * [Enderlein and Böhmer, 2003] Enderlein, J. and Böhmer, M. (2003).  Influence of interface–dipole interactions on the efficiency of fluorescence light collection near surfaces.  Optics letters, 28:941–943.
  * [Johnson and Christy, 1972] Johnson, P. B. and Christy, R.-W. (1972).  Optical constants of the noble metals.  Physical review B, 6(12):4370.
  * [Jungmann et al., 2010] Jungmann, R., Steinhauer, C., Scheible, M., Kuzyk, A., Tinnefeld, P., and Simmel, F. C. (2010).  Single-molecule kinetics and super-resolution microscopy by fluorescence imaging of transient binding on dna origami.  Nano letters, 10(11):4756–4761.
  * [Ko and Sambles, 1988] Ko, D. Y. K. and Sambles, J. (1988).  Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals.  JOSA A, 5(11):1863–1866.
  * [Mie, 1908] Mie, G. (1908).  Beiträge zur optik trüber medien, speziell kolloidaler metallösungen.  Annalen der Physik, 330:377–445.
  * [Mishchenko et al., 2017] Mishchenko, M. I., Zakharova, N. T., Khlebtsov, N. G., Videen, G., and Wriedt, T. (2017).  Comprehensive thematic t-matrix reference database: A 2015–2017 update.  Journal of Quantitative Spectroscopy and Radiative Transfer, 202:240–246.
  * [Mortensen et al., 2010] Mortensen, K. I., Churchman, L. S., Spudich, J. A., and Flyvbjerg, H. (2010).  Optimized localization analysis for single-molecule tracking and super-resolution microscopy.  Nature methods, 7:377–381.
  * [Novotny and Hecht, 2006] Novotny, L. and Hecht, B. (2006).  Principles of Nano-Optics.  Cambridge University Press.
  * [Ru and Etchegoin, 2008] Ru, E. L. and Etchegoin, P. (2008).  Principles of Surface-Enhanced Raman Spectroscopy: and related plasmonic effects.  Elsevier.
  * [Shafiqa et al., 2018] Shafiqa, A., Abdul Aziz, A., and Mehrdel, B. (2018).  Nanoparticle optical properties: size dependence of a single gold spherical nanoparticle.  In Journal of Physics: Conference Series, volume 1083, page 012040\. IOP Publishing.
  * [Taflove et al., 2005] Taflove, A., Hagness, S. C., and Piket-May, M. (2005).  Computational electromagnetics: the finite-difference time-domain method.  The Electrical Engineering Handbook, 3:629–670.
  * [Wigner, 1959] Wigner, E. (1959).  Group theory: and its application to the quantum mechanics of atomic spectra, volume 5.  Elsevier.
  * [Yee, 1966] Yee, K. (1966).  Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media.  IEEE Transactions on antennas and propagation, 14(3):302–307.