\label{thm;global}
For $(u_0,v_0)\in L_+^1(\mathbb R^2) \times L^1_+(\mathbb R^2)\cap \dot{H}^1(\mathbb R^2)$,
let $(u,v)$ be the solution to \eqref{eqn;KS} on $(0,T)\times\mathbb R^2$.
Suppose that $\|u_0\|_1=8\pi$.
Then, the solution to \eqref{eqn;KS} exists globally in time.

\begin{aligned}
&\pt_t u =\Delta u- \N\cd \left( u \N v \right),
& t>0,\,~ &x\in\R^2,
\\
&\pt_t v = \Delta v - \lambda v +u,
&t>0,\, ~&x\in\R^2,
\\
&(u,v)(0,x)=(u_{0},v_0)(x),
&\, &x\in\R^2
\end{aligned}
\right.
\label{eqn;KS}
\end{equation}
with a constant $\lambda\ge0$, where $u_0,v_0\ge 0$ on $\R^2$ and
$u_0,v_0\not\equiv0$.
The Keller--Segel system~\eqref{eqn;KS} is a fundamental mathematical model
of chemotaxis, describing chemotactic aggregation in the
cellular slime mold {\it Dictyostelium discoideum} during its life cycle \cite{KeSe,Pa}.
In this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and
the chemoattractant, respectively.
The parameter $\lambda$ represents the degradation rate of the chemical.
Cells migrate toward regions of higher concentrations of a chemical substance
secreted by the cells themselves.
From a biological viewpoint, it is natural to assume that the initial data
are nonnegative functions.
From a mathematical viewpoint, cell aggregation is interpreted as the
blowup of solutions at $t=T$ in the sense that
$\lim_{t\to T}\|u(t)\|_\infty=\infty.$

One of the central mathematical features of such systems is the phenomenon of critical
mass.
For positive sufficiently regular solutions $(u,v)$ to~\eqref{eqn;KS}, the first
component $u$ satisfies the mass conservation law
$$\|u(t)\|_1=\|u_0\|_1\quad \text{for}~~~t>0.$$
Moreover, in two space dimensions, the global behavior of solutions is governed
by the size of the initial mass of $u$.
In particular, there exists a threshold value such that solutions with initial
mass below this threshold exist globally in time, whereas solutions with
initial mass above the threshold may blow up in finite time.
This critical-mass phenomenon has attracted considerable attention in the
literature.

The aim of this paper is to establish the global behavior of solutions to
\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.

\vspace{3mm}
The second equation in \eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves
while this is diffusing into the environment.
Since the chemoattractant attains its equilibrium on a time scale much quicker
than that of the cells, the simplified parabolic-elliptic system has also been
investigated \cite{JaLu,Na95}:
\begin{align*}
\left\{
\begin{aligned}
\partial_t u &= \Delta u - \nabla\cdot (u\nabla v),
& t>0,\ & x\in\R^2,\\
0 &= \Delta v - \lambda v + u,
& t>0,\ & x\in\R^2,
\\
&u(0,x)=u_{0}(x),
&\, &x\in\R^2.
\end{aligned}

\begin{prop}[local-in-time solution]\label{prop;LWP}
	Let $4/3<p<2$ and $q=p/(p-1)$.
	For  $(u_0,v_0)\in L^1(\R^2)\times  \dot{H}^{1}(\R^2)$,
	there exist $T\in (0,\infty]$ and a unique mild solution $(u,v)$ to~\eqref{eqn;KS} on $[0,T)$ in the sense of the definition.
	Besides, the solution $(u,v)$ has higher regularity on $(0,T)$ and satisfies the system~\eqref{eqn;KS} in a classical sense on $(0,T)\times\R^2$.
	In addition,
	\begin{equation*}
	\text{either $T=\infty$ or $T<\infty$ and } \lim_{t\to T} \left\{ \|u(t)\|_{r} + \|\nabla v(t)\|_{r+1}\right\} = \infty,~~ 1<r\le \infty.
	\eqntag
	\label{eqn;bucrit}
	\end{equation*}
	Moreover, if $u_0,v_0 \ge 0$ then $u,v> 0$ in $(0,T)\times\R^2$ and
	\begin{equation*}
	\|u(t)\|_1=\|u_0\|_1, \qquad t\in [0,T).
	\end{equation*}
\end{prop}

\begin{aligned}
\partial_t u &= \Delta u - \nabla\cdot (u\nabla v),
& t>0,\ & x\in\R^2,\\
0 &= \Delta v - \lambda v + u,
& t>0,\ & x\in\R^2,
\\
&u(0,x)=u_{0}(x),
&\, &x\in\R^2.
\end{aligned}
\right.
\eqntag
\label{eqn;PKS}
\end{align*}
The system \eqref{eqn;PKS} is also related to models of gravitational interaction
of particles \cite{Bi-Na94,Wo}.
In~\eqref{eqn;PKS}, the second equation can be written as
$v = (-\Delta+\lambda)^{-1}u.$
As a consequence, the system~\eqref{eqn;PKS} can be reduced to a single equation
for~$u$.
Owing to this elliptic structure, the parabolic-elliptic system~\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.
In particular, numerous works have been devoted to the critical-mass
phenomenon.
Indeed, the solution with $\|u_0\|_1\le 8\pi$ exists globally in time~\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},
while the solution may blow up in finite time if $\|u_0\|_1>8\pi$~\cite{DoPe,Ko-Su,Ku-Og03,Wei}.
For the corresponding Cauchy--Neumann problem in bounded domains,
supplemented with homogeneous Neumann boundary conditions, see \cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.

\vspace{3mm}
Unlike the parabolic-elliptic system \eqref{eqn;PKS}, the fully parabolic system
\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches
developed for \eqref{eqn;PKS} are no longer applicable.
The global existence of solutions with sub-critical mass
$\|u_0\|_{1}<8\pi$ has nevertheless been established by combining
Lyapunov functionals with the Trudinger--Moser type inequality and its optimal
constant.
More precisely, the following results are known:
\begin{enumerate}
	\item[(i)]
	If $\|u_0\|_{1}<8\pi$, then the corresponding solution to
	\eqref{eqn;KS} exists globally in time \cite{CaCo08,Mi13,NaOg11}.

	\item[(ii)]
	If $\|u_0\|_{1}>8\pi$ and $(u_0,v_0)$ is radially symmetric, then there
	exists a radially symmetric solution that blows up in finite time
	\cite{Mi20,Mi20-SIAM}.

	\item[(iii)]
	If $\|u_0\|_{1}=8\pi$ and $(u_0,v_0)$ is either radially symmetric or
	satisfies the additional moment condition
	$u_0\ln(1+|x|^2)\in L^1(\R^2)$, then the corresponding solution exists globally
	in time.
	In contrast, for general initial data, the global behavior in the critical case
	remains delicate, and the solution either exists globally or blows up on the plane \cite{CaCo08,Mi13}.
\end{enumerate}
As for positive forward self-similar solutions to \eqref{eqn;KS}, refer to \cite{BiCoDo}.
For the corresponding Cauchy--Neumann problem in bounded domains, we refer to
\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.

As mentioned above, although several partial results are available, no complete global existence result has been obtained
for solutions with critical mass and general initial data without any
symmetry or moment assumptions.
One of the main difficulties lies in controlling the behavior of solutions at~$|x|\to\infty$.
Moreover, in the critical mass case, no global existence result is currently
known  for the corresponding Cauchy--Neumann problem to~\eqref{eqn;KS} in bounded domains.

In this paper, we establish the global existence of solutions with critical mass
for general initial data $(u_0,v_0)$.
To this end, we first recall the definition of solutions.

\vspace{2mm}
\begin{def*}
Let $4/3<p<2$ and $q=p/(p-1)$.
Given $T\in (0,\infty]$ and  $(u_0,v_0) \in L^1(\R^2)\times \dot{H}^{1}(\R^2)$,
we say that a pair of functions $(u,v)$ defined on $[0,T)\times \R^2$ is a mild solution to~\eqref{eqn;KS} on $[0,T)$ if
\begin{enumerate}
	\item $u\in C\big([0,T); L^1(\R^2)\big)\cap C\left( (0,T); L^p(\mathbb{R}^2) \right)$;
	\item $v\in C\big([0,T); \dot{H}^{1}(\R^2)\big)\cap C\left( (0,T); \dot{W}^{1,q}(\mathbb{R}^2) \right)$;
	\item $\dsp\sup_{0<t<T} t^{1-\frac1p}\|u(t)\|_{p}+\dsp\sup_{0<t<T}t^{\frac12-\frac1q}\|\nabla v(t)\|_{q}$;
	\item $(u,v)$ satisfies the integral formulations
	\begin{align*}
	&u(t)=e^{t\Del}u_0-\int_0^t \N\cd e^{(t-s)\Del} \big(u(s)\N v(s)\big) \,\d s,
	\\
	&v(t)= e^{t(\Del-\lm)} v_0+\int_0^t e^{(t-s)(\Del-\lm)} u(s) \,\d s
	\end{align*}
	for $0<t<T$, where $e^{t\Del}$ is the heat semigroup given by
	\begin{align*}
	e^{t\Del} f(x) := G_t*f(x)=\int_{ \R^2}G_t(x-y) f(y)\,\d y,
	\quad
	G_t(x)=\frac{1}{4\pi t} \exp\left(-\frac{|x|^2}{4 t}\right)
	\end{align*}
	for $x\in\R^2$ and $f \in L^p(\R^2)$.
\end{enumerate}
\end{def*}
The Lebesgue spaces $L^p(\R^2)$ are equipped with the norm $\|\cdot\|_{L^p(\R^2)}$,
which is abbreviated as $\|\cdot\|_p$ when there is no ambiguity.
We define the homogeneous Sobolev spaces by
\[
\dot{W}^{1,p}(\R^2)
:= \overline{C_0^\infty(\R^2)}^{\|\nabla (\cdot)\|_{L^p(\R^2)}}
\]
and $\dot{H}^1(\R^2):=\dot{W}^{1,2}(\R^2)$ when $p=2$.
$L_+^1(\R^2):=\{f \in L^1(\R^2); f\ge0~~ \text{and}~~ f\not\equiv0~~ \text{on $\R^2$} \}$.

\vspace{5mm}
The starting point for our analysis is the local well-posedness for \eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\ref{prop;LWP} in Section~\ref{sect;LWP} below.

We are now in a position to state our main result in this paper.
Main result in this paper reads as follows.
\vspace{2mm}
\begin{thm}\label{thm;global}
For $(u_0,v_0)\in L_+^1(\R^2) \times L^1_+(\R^2)\cap \dot{H}^1(\R^2)$,
let $(u,v)$ be the solution to \eqref{eqn;KS} on $(0,T)\times\R^2$.
Suppose that $\|u_0\|_1=8\pi$.
Then, the solution to \eqref{eqn;KS} exists globally in time.
\end{thm}
\vspace{2mm}

\begin{rem}
In Theorem~\ref{thm;global}, the initial data are only required to satisfy
\[
(u_0,v_0)\in L_+^1(\R^2)\times\bigl(L_+^1(\R^2)\cap \dot H^1(\R^2)\bigr)\quad \text{with}~~\|u_0\|_1=8\pi.
\]
In particular, no additional symmetry or moment assumptions are imposed.
\end{rem}
\vspace{2mm}
As mentioned above, the global existence of solutions with critical mass
without any symmetry or moment assumptions remains delicate.
In~\cite[Theorem~1.2]{Mi13}, it is shown that, for general initial data with
critical mass, solutions either exist globally in time or blow up on the plane,
by means of a contradiction argument.
Indeed, although in the sub-critical case regularity estimates can be obtained
by combining (modified) Lyapunov functionals with the Trudinger--Moser
inequality, such estimates are no longer directly available in the critical case due to the lack of regularity of solutions,
see~\eqref{eqn;afford} below.
This difficulty prevents the direct extension of classical entropy methods.

To overcome this difficulty, we introduce a reconstructed Lyapunov functional.
The main novelty of this work lies in the construction of a refined Lyapunov
functional specifically adapted to the whole space setting, which allows us
to control the behavior of solutions at $|x|\to\infty$ and to derive
regularity estimates even in the critical mass regime, without
imposing any symmetry or moment assumptions on the initial data.
As a consequence, we establish global-in-time existence for general initial
data at the critical mass, a result that was previously out of reach by existing
methods, see Subsection~\ref{sect;modified-Lyapunov} for details.
Moreover, the present approach is expected to be applicable to a broad class
of chemotaxis  systems in the whole space setting.

\vspace{3mm}
The remainder of this paper is organized as follows.
In Section~\ref{sect;preliminary}, we recall several preliminary lemmas needed
to prove the main result.
Section~\ref{sect;LWP} is devoted to the local well-posedness of~\eqref{eqn;KS}, based on the above definition of solutions.
Finally, in Section~\ref{sect;global}, we present the proof of
Theorem~\ref{thm;global}.

\vspace{5mm}
\section{Preliminary}\label{sect;preliminary}
In this section, we collect some tools to show our main result.
\begin{lem}\label{lem;L2andL3}
	For any $f\in H^1(\R^2)$ and nonnegative cut-off function $\phi\in C^\infty_0(\R^2)$,
	\begin{align*}
	\int_{\R^2}|f|^2\phi\dx\le\,&2\left(\int_{ \{|f|>1\}\cap \{\supp\phi\} }|f|\dx\right)\left(\int_{ \{|f|>1\} }\frac{|\nabla f|^2}{1+|f|} \phi\dx\right)
	\\
	&+4\left(\int_{\R^2}|f \nabla \phi^{\frac12}|\dx\right)^2+4\left(\int_{\R^2}|f|\phi\dx\right)
	\end{align*}
	and for any $\ep>0$
	\begin{align*}
	\int_{\R^2}|f|^3\phi\dx
	\le\,&\ep \left(\int_{  \{\supp\phi\} }(1+|f|)\ln(1+|f|)\dx\right)\left(\int_{\R^2 }|\nabla f|^2\phi\dx\right)
	\\
	&+C\left(\int_{\R^2}|f^{\frac32} \nabla \phi^{\frac12}|\dx\right)^2+C_\ep\left(\int_{\R^2}|f|\phi\dx\right),
	\end{align*}
	where the constant $C_\ep\to \infty$ as $\ep\to0$.
\end{lem}
The proof of Lemma \ref{lem;L2andL3} can be found in \cite[Lemma 2.2]{NaOg16}.

\begin{lem}\label{lem;Lp_Lq_heat}
	For $1 \le q \le p \le \infty$,
	let $f\in L^q (\R^n)$ and let $\al$ be a multi-index.
	Then it follows that
	\begin{equation*}
	\| \pt_x^{\al} e^{t\Del} f \|_p
	\le C t^{-\frac{n}{2} \left(\frac{1}{q} - \frac{1}{p} \right) - \frac{|\al|}{2}}   \|f\|_q
	\end{equation*}
	for all $t>0$.
\end{lem}

The proof is immediately obtained by use of Young's inequality and the
convolution expression of the heat evolution
by the heat kernel, see for instance \cite{GiGiSa}.
\begin{lem}\label{lem;v-Lpbound}
For $v_0\in L^1(\R^2)\cap \dot H^1(\R^2)$ and $f\in L^\infty \left(0,\infty;  L^1(\R^2) \right)$,
let $v$ be the solution to $\partial_t v =\Delta v -\lambda v +f$  on $(0,\infty)\times\R^2$ with the initial data $v_0$ and $\lambda\ge0$. Then, for any $1\le p<\infty$,
\begin{align*}
\| v(t)\|_p
\le\,&
\left\{
\begin{aligned}
&\|v_0\|_p+\frac1p\|f\|_{L^\infty\left(0,\infty; L^1(\R^2)\right)}  t^{\frac1p} &\text{if $\lambda=0$},
\\
&\|v_0\|_p+\lambda^{-\frac1p} \|f\|_{L^\infty\left(0,\infty; L^1(\R^2)\right)}  \Gamma\left(\frac1p\right)&\text{if $\lambda>0$}.
\end{aligned}

\begin{pr}{Proposition \ref{prop;LWP}}
By virtue of Lemmas~\ref{lem;fixed-point} and~\ref{lem;continuous-depend},
we have established the existence of a unique mild solution to~\eqref{eqn;KS} defined on a maximal time interval $[0,T)$ with~$T\le\infty$. As for the regularity of the solutions, we use the standard iteration argument with respect to the derivative. Define $|\N|^\al f(x):=\F^{-1}[ |\xi|^\al \F f(\xi)\,](x)$  for $x\in\R^2$ and $\al>0$, where $\F$ denotes the Fourier transform. Let $\tau\in (0,T)$ and $t\in (0,\tau)$. Recalling that, for $4/3<p<2$,
\begin{equation*}
U_{p,0}(\tau) := \sup_{0< s<\tau}s^{1-\frac1p}\|u(s)\|_{p}<+\infty, \quad V_{q,0}(\tau) := \sup_{0< s<\tau}s^{\frac12-\frac1q}\|\nabla v(s)\|_{q}<+\infty,
\end{equation*}
by Lemma~\ref{lem;fixed-point}, we may argue as in the proof of~\eqref{eqn;L-4/3} to obtain
\begin{align*}
&t^{1-\frac1p+\frac{\al}{2}} \| |\N|^{\al} u(t) \|_{p}
\\
\le\,& t^{1-\frac1p+\frac{\al}{2}} \| |\N|^\al e^{t\Del}u_0\|_{p} + C U_{p,0}(\tau) V_{q,0}(\tau) t^{1-\frac1p+\frac{\al}{2}}  \int_0^t  (t-s)^{-1+\frac1{p}-\frac{1+\al}2} s^{-1+\frac{1}p} s^{-\frac12+\frac1q}\,\d s
\\
\le\,&C\|u_0\|_1 + C  U_{p,0}(\tau) V_{q,0}(\tau) \mathsf{B}\left(\frac1p-\frac{1+\al}2,\frac12\right),
\end{align*}
where $4/3<p<2$ and
\begin{align*}
\al <\frac2p-1<\frac12,
\end{align*}
which implies that
\begin{align*}
U_{p.\al}(\tau):=\sup_{0< t<\tau} t^{1-\frac1p+\frac{\al}{2}} \| |\N|^{\al} u(t)\|_{p} <+\infty,
\quad
\al<\frac2p-1.
\end{align*}
This property yields additional regularity estimates for $v$. Indeed, for $\bt\ge \al$,
\begin{align*}
\| |\N|^{1+\beta} v(t) \|_{q}
\le\,&C t^{-\frac12+\frac1q-\frac\bt2} \| \nabla v_0\|_{2} + C U_{p,\al}(\tau) \int_0^t (t-s)^{-\left(\frac1p-\frac1q\right)-\frac{1+\bt-\al}{2} } s^{-1+\frac1p-\frac\al2}\,\d s
\\
=\,&C  t^{-\frac12+\frac1q-\frac\bt2} \| \nabla v_0\|_{2}
+C U_{p,\al}(\tau)  t^{-\frac12+\frac1q-\frac\bt2} \mathsf{B}\left(\frac12-\frac1p+\frac1q-\frac{\bt-\al}{2}, \frac1p-\frac{\al}{2}\right),
\end{align*}
provided that
\begin{align*}
\al\le \bt<\al+3-\frac4p<2-\frac2p<1,
\end{align*}
from which it follows that
\begin{align*}
V_{q,\bt}(\tau) := \sup_{0< t<\tau} t^{\frac12-\frac1q+\frac\bt2} \| |\N|^{1+\bt} v(t) \|_{q}<+\infty,
\quad  \bt<\al+3-\frac4p\in (\al,1).
\end{align*}
The previous analysis leads us to a higher regularity estimate for $u$. Indeed, let $\th\ge\al$. Since $\al\le\bt\le\gm$,
\begin{align*}
& \| |\N|^{\th}u(t)\|_{p} \\
\le\,&C t^{-\left(1-\frac1p\right)-\frac\th2}\|u_0\|_1
\\
&+C\int_0^t(t-s)^{-\left(1-\frac1p\right)-\frac{1+\th-\al}{2}} \left(\||\N|^{\al} u(s)\|_{p}\|\nabla v(s)\|_q+\|u(s)\|_{p}\| |\N|^{1+\al} v(s)\|_{q}\right)\,\d s
\\
\le\,&Ct^{-\left(1-\frac1p\right)-\frac\th2}\|u_0\|_1
\\
&+ C \big(U_{p,\alpha}(\tau) V_{q,0}(\tau) + U_{p,0}(\tau) V_{q,\al}(\tau)\big)
\int_0^t(t-s)^{-\left(1-\frac1p\right)-\frac{1+\th-\al}{2}} s^{-\frac32+\frac1p+\frac1q-\frac{\al}{2}}\,\d s
\\
=\,&Ct^{-\left(1-\frac1p\right)-\frac\th2}\|u_0\|_1
\\
& + C \big(U_{p,\al}(\tau) V_{q,0}(\tau) + U_{p,\al}(\tau) V_{q,0}(\tau)\big) t^{-\left(1-\frac1p\right)-\frac\th2} \mathsf{B}\left(\frac1p-\frac{1+\th-\al}{2},\frac12-\frac\al2\right),
\end{align*}
where
\begin{align*}
\al \le \th<\al+\frac2p-1<1,\quad
\al<\frac2p-1.
\end{align*}
Hence,
\begin{align*}
U_{p,\th}(\tau) = \sup_{0< t< \tau}t^{1-\frac1p+\frac{\th}{2}}\||\N|^\th u(t)\|_{p}<+\infty,
\quad
\al \le \th<\al+\frac2p-1<1,\quad
\al<\frac2p-1.
\end{align*}
Iterating further the above arguments, we obtain that the solution possesses higher regularity. In fact, using the uniqueness result established in Lemma~\ref{lem;continuous-depend}, we may solve the problem~\eqref{eqn;KS} with the initial data $(u(t_0),v(t_0))$ for some $t_0\in (0,T)$. We thus eventually show that for $s>0$, $u(t)\in L^1(\R^2)\cap W^{s,p}(\R^2)$,
$v(t)\in  \dot{H}^{1}(\R^2)\cap W^{s,q}(\R^2)$
for $t_0<t<T$,
and satisfies the system~\eqref{eqn;KS} in a  classical sense on $(0,T)\times\R^2$.
Also, classical arguments based on the proof of Lemma~\ref{lem;fixed-point} lead to the blowup criterion~\eqref{eqn;bucrit}.

Let  $(u_0,v_0)$ be nonnegative initial conditions with $u_0,v_0\not\equiv0$ and
$\{u_{0,k}\}_{k\in\Nt }$ and
$\{v_{0,k}\}_{k\in\Nt }$ be sequences
of nonnegative functions in $C_0^{\infty}(\R^2)$ converging to $u_0$ in $L^{1}(\R^2)$ and $v_0$ in $\dot{H}^{1}(\R^2)$  as $k\to\infty$, respectively. Then, for each $k\in\Nt$, we can construct the unique, smooth and integrable solution
to~\eqref{eqn;KS} corresponding to the initial data $(u_{0,k},v_{0,k})$.
By the parabolic regularity theory,
the solutions $(u_k,v_k)$ belong to $C^{\infty}( [0,T)\times \R^2)$
and are positive on $(0,T)\times\R^2$. In addition, by the continuous dependence on the initial data established in Lemma~\ref{lem;continuous-depend}, we have the convergences $u_k\to u$ in $C\big([0,T); L^1(\R^2)\big)$ and $v_k\to v$ in $C\big([0,T); \dot{H}^{1}(\R^2)\big)$ as $ k\to \infty$,
where $(u,v)$ is the solution to~\eqref{eqn;KS} with initial data $(u_0,v_0)$. Hence, one can find subsequences of the approximated solutions
which converge to $(u,v)$ almost everywhere in $(0,T)\times\R^2$, which implies that $u$ and $v$ are positive on $(0,T)\times\R^2$.
Moreover the mass conservation law also holds by integrating $u$ over $\R^2$.
\end{pr}

\vspace{5mm}
\section{Global existence of solutions with critical mass}\label{sect;global}

In this section, we prove the global existence of solutions to \eqref{eqn;KS}
with critical mass, as stated in Theorem~\ref{thm;global}.
To this end, we divide the proof into several steps.

In Subsection~\ref{sect;modified-Lyapunov}, we introduce a suitably reconstructed
Lyapunov functional and derive regularity estimates for the associated
dissipative terms.
In Subsection~\ref{sect;exterior}, we establish various regularity estimates
in exterior domains by using the estimates obtained in
Subsection~\ref{sect;modified-Lyapunov}.
Thanks to the results in the previous subsections, we then derive interior
regularity estimates in Subsection~\ref{sect;interior}.
Finally, in Subsection~\ref{sect;main}, we complete the proof of
Theorem~\ref{thm;global} by combining the exterior and interior estimates.

Throughout this section, we denote by $(u,v)$ the solution to~\eqref{eqn;KS}
with maximal existence time $T\le\infty$.
In the following, we derive a priori estimates on $[t_0,\tau]\cap(0,T)$ for arbitrary $t_0,\tau$ with $t_0<\tau$, which will allow us to conclude global existence by a contradiction argument.
Moreover, $C$ denotes a generic positive constant depending only on $\lambda$
and the initial data~$(u_0,v_0)$, which may vary from line to line.
The dependence of $C$ on additional parameters will be indicated explicitly.

\subsection{Refined Lyapunov  functional}\label{sect;modified-Lyapunov}

We here introduce a suitably reconstructed Lyapunov functional which is non-increasing along the evolution of solutions, and derive regularity estimates associated with the corresponding dissipation mechanism. These estimates further yield refined regularity properties for solutions to \eqref{eqn;KS}.

The existence of a Lyapunov functional is known to play a fundamental role in the analysis of the long-time behavior of solutions, including global existence and finite-time blowup phenomena. In the whole space setting, however, the use of Lyapunov-based methods in~\eqref{eqn;KS} faces intrinsic difficulties due to
 the necessity of controlling the behavior as $|x|\to\infty$. This issue becomes particularly delicate at the critical mass.

First of all, it is worth mentioning that the system \eqref{eqn;KS} has a usual Lyapunov functional~$\L(t)$ as follows:
\begin{align*}
\frac{\d}{\d t}\L(t)+ \D(t) =0,
\eqntag
\label{eqn;Lyapunov}
\end{align*}
where
\begin{align*}
\L(t):=\int_{\R^2}u\ln u\dx-\int_{\R^2}uv\dx+\frac12\|\nabla v\|_2^2+\frac\lambda2\|v\|_2^2
\end{align*}
and the dissipative term $\D(t)$ is given by
\begin{align*}
\D(t):=\int_{\R^2} u \left|\nabla(\ln u -v)\right|^2\dx+\|\partial_t v\|_2^2.
\end{align*}
The upper bound for $\L(t)$ follows from the monotonicity \eqref{eqn;Lyapunov} and
the lower bound for that can be obtained by the optimal constraint of the initial mass $\|u_0\|_1$ deriving from the Trudinger--Moser type inequality and its best possible constant, see \cite{Na-Se-Yo} and also \cite{CaCo08,Mi13,NaOg11} for instance.
On the other hand,
in order to ensure the well-definedness of a usual entropy $\int_{\R^2}u\ln u \dx$ on the whole space,
at least the logarithmic moment assumption for the initial data $u_0\ln(1+|x|^2)\in L^1(\R^2)$ is required to control the behavior of solutions as $|x|\to\infty$, that is,
\begin{align*}
-\int_{\R^2}u(\ln u)_-\dx \le\,C\int_{\R^2}u\ln(1+|x|^2)\dx,
\end{align*}
where $(\ln u)_-:=\min\{\ln u,0\}$, according to \cite[Lemma 2.4]{CaCo08}.
Later, to get rid of any moment assumptions,
the modified entropy is adapted by Nagai \cite{Na11};
\begin{align*}
\int_{\R^2}(1+u)\ln(1+u)\dx,
\end{align*}
as well as the following identity for the modified Lyapunov functional $\L_m(t)$ is introduced:
\begin{align*}
\frac{\d}{\d t} \L_m(t) + \D_m(t)=\frac14\|\nabla v(t)\|_2^2,
\eqntag
\label{eqn;usualmodifiedfunctiona}
\end{align*}
where
\begin{align*}
\L_m(t):=\int_{\R^2}(1+u)\ln(1+u)\dx-\int_{\R^2}uv\dx+\frac12\|\nabla v\|_2^2+\frac\lambda2\|v\|_2^2
\end{align*}
and
\begin{align*}
\D_m(t):=\int_{\R^2}u\left|\nabla(\ln(1+u)-v)\right|^2\dx+\int_{\R^2}\left|\nabla\left(\ln(1+u)-\frac12v\right)\right|^2\dx+\|\partial_tv\|_2^2.
\end{align*}
The functional $\L_m(t)$ is no longer the Lyapunov functional, however, it is still useful for proving
the global existence of the solution to \eqref{eqn;KS} with sub-critical mass. Indeed, it follows
from the lower bound for $\L_m(t)$ that
for any $\al\in[0,1)$
\begin{align*}
\al\int_{\R^2}(1+u)\ln(1+u)\dx+\frac12\left[1-\frac{\|u_0\|_1}{8\pi(1-\al)}\right]&\|\nabla v\|_2^2
+\int_0^t \D_m(s)\,\d s
\\
\le\,&\L_m(0)+C(\tau,\alpha)+\frac14\int_0^t\|\nabla v\|_2^2\,\d s
\eqntag
\label{eqn;afford}
\end{align*}
for $t\in[0,\tau]\cap(0,T)$ with any $\tau>0$ and some $C(\tau,\alpha)>0$ (cf.~\cite{CaCo08,Mi13}). Hence, if $\|u_0\|<8\pi$, then all terms on the left hand side of \eqref{eqn;afford} are positive by choosing $\al>0$ sufficiently small which depends on $\|u_0\|_1$, and Gronwall's inequality implies that
$\nabla v \in L^2((0,T)\times \R^2)$,
so that we also obtain the bound for the modified entropy $\int_{\R^2}(1+u)\ln (1+u)\dx$.
Nevertheless, for the critical mass $\|u_0\|_1=8\pi$, this necessarily leads to $\alpha=0$, as a result,
\begin{align*}
\int_0^t \D_m(s)\,\d s\le\,&\L_m(0)+C(\tau)+\frac14\int_0^t\|\nabla v\|_2^2\,\d s,
\eqntag
\label{eqn;lack-lyapunov}
\end{align*}
which is useless and fails to obtain even the estimates for $\D_m(t)$ due to the lack of the regularity of solutions.
Therefore, we introduce a reconstructed Lyapunov functional $\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\F_m(t)$ be the functional for solutions to \eqref{eqn;KS} defined as
\begin{align*}
\F_m(t):=\L_m(t)
+\int_{\R^2}\ln(1+u)\,\mathrm{d}x-\int_{\R^2}v\,\d x.
\eqntag
\label{eqn;modifiedLyapunov}
\end{align*}
Then, the following functional differential inequality holds true:
\begin{prop}\label{prop;energy-est}
Let $(u,v)$ be the solution to \eqref{eqn;KS}. Then, the functional $\F_m(t)$ defined in~\eqref{eqn;modifiedLyapunov} satisfies the following identity:
	\begin{align*}
	\frac{\d}{\d t}\F_m(t) +\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\frac12\|\partial_tv\|_2^2
	\le\,&
	\lambda\int_{\R^2}v\,\d x.
	\end{align*}
	Therefore, if $\lambda=0$ then
	\begin{align*}
	\frac{\d}{\d t}\F_m(t) +\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\frac12\|\partial_tv\|_2^2 \le\,0.
	\end{align*}
	If $\lambda>0$ then
	\begin{align*}
	\frac{\d}{\d t}\F_m(t) +\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\frac12\|\partial_tv\|_2^2 \le\,\|v_0\|_1+\|u_0\|_1.
	\end{align*}
\end{prop}
\vspace{5mm}
It is worth emphasizing that $\F_m(t)$ is non-increasing in time when $\lambda=0$ as well as
unlike the identity \eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\ref{prop;energy-est} can be easily controlled by initial data.
This allows us to show the regularity estimates for the dissipative term associated with $\F_m(t)$, see Proposition~\ref{prop;bound-energy} below.
\begin{lem}\label{lem;ModifiedLF}
	Suppose assumptions as in Theorem \ref{thm;global}. Then,
\begin{align*}
\frac{\d}{\d t}\F_m(t)+ \widetilde{\D}(t)=-\int_{\R^2}\partial_t v \frac{u}{1+u}\,\d x-\int_{\R^2}\frac{u}{1+u}\,\mathrm{d}x+\lambda\int_{\R^2}\frac{v}{1+u}\,\d x,
\end{align*}
where
$\F_m(t)$ is the modified functional defined in \eqref{eqn;modifiedLyapunov} and the dissipative term $\widetilde{\D}(t)$ is given by
\begin{align*}
\widetilde{\D}(t):=\int_{\R^2}u\left|\nabla \left(\ln(1+u)-v\right)\right|^2\,\d x+\|\partial_t v\|_2^2.
\end{align*}
\end{lem}

\begin{pr}{Lemma \ref{lem;ModifiedLF}}
The following computations are already well-known:
\begin{align*}
\frac{\d}{\d t} \int_{\R^2}(1+u)\ln (1+u)\,\d x=\,&
-\int_{\R^2}(1+u)|\nabla\ln(1+u)|^2\,\d x
+\int_{\R^2}u\nabla\ln(1+u)\cdot\nabla v\,\d x
\end{align*}
as well as
\begin{align*}
-\frac{\d}{\d t}\int_{\R^2}uv\,\d x=\,&\int_{\R^2}(1+u)\nabla\ln(1+u)\cdot\nabla v\,\d x-\int_{\R^2}u|\nabla v|^2\,\d x
\\
&-\|\partial_tv\|_2^2-\frac12\frac{\d}{\d t}\|\nabla v\|_2^2-\frac\lambda2\frac{\d}{\d t}\|v\|_2^2,
\end{align*}
so that
\begin{align*}
&\frac{\d}{\d t}\left[\int_{\R^2}(1+u)\ln(1+u)\,\d x-\int_{\R^2}uv\,\d x+\frac12\left(\|\nabla v\|_2^2+\lambda\|v\|_2^2\right)\right]
\\
&+\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\|\partial_tv\|_2^2
\\
=\,&-\|\nabla\ln(1+u)\|_2^2+\int_{\R^2}\nabla\ln(1+u)\cdot\nabla v\,\d x.
\end{align*}
Next,
\begin{align*}
\frac{\d}{\d t}\int_{\R^2}\ln(1+u)\,\d x=\,&\int_{\R^2}\frac{1}{1+u} \left[\Delta u -\nabla\cdot( u\nabla v)\right]\,\mathrm{d}x
\\
=\,&-\int_{\R^2}\nabla\left(\frac{1}{1+u}\right)\cdot \nabla u\,\d x+\int_{\R^2}u\nabla\left(\frac{1}{1+u}\right)\cdot \nabla v\,\d x
\\
=\,&\int_{\R^2}|\nabla \ln(1+u)|^2\,\d x-\int_{\R^2}\frac{u}{1+u}\nabla\ln(1+u)\cdot\nabla v\,\d x
\\
=\,&\int_{\R^2}|\nabla \ln(1+u)|^2\,\d x-\int_{\R^2}\nabla\ln(1+u)\cdot\nabla v\,\d x
\\
&+\int_{\R^2}\frac{1}{1+u}\nabla\ln(1+u)\cdot\nabla v \,\d x.
\end{align*}
Hence, combining the above computations implies that
\begin{align*}
&\frac{\d}{\d t}\left[\int_{\R^2}(1+u)\ln(1+u)\,\d x-\int_{\R^2}uv\,\d x+\frac12\left(\|\nabla v\|_2^2+\lambda\|v\|_2^2\right)+\int_{\R^2}\ln(1+u)\,\d x\right]
\\
&+\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\|\partial_tv\|_2^2
\\
=\,&\int_{\R^2}\frac{1}{(1+u)^2}\nabla u\cdot\nabla v \,\d x.
\end{align*}
Since
\begin{align*}
\int_{\R^2}\frac{1}{(1+u)^2}\nabla u\cdot\nabla v \,\d x=\,&-\int_{\R^2}\nabla\left(\frac{1}{1+u}\right)\cdot \nabla v\,\d x
\\
=\,&\int_{\R^2}\frac{1}{1+u}\Delta v\,\d x
\\
=\,&\int_{\R^2}\frac{\partial_t v}{1+u}\,\d x-\int_{\R^2}\frac{u}{1+u}\,\d x+\lambda\int_{\R^2}\frac{v}{1+u}\,\d x
\\
=\,&\frac{\d}{\d t}\int_{\R^2}v\,\d x-\int_{\R^2}\partial_t v \frac{u}{1+u}\,\d x-\int_{\R^2}\frac{u}{1+u}\,\d x+\lambda\int_{\R^2}\frac{v}{1+u}\,\d x,
\end{align*}
we end up with
\begin{align*}
&\frac{\d}{\d t}\F_m(t)+\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\|\partial_tv\|_2^2
\\
=\,&-\int_{\R^2}\partial_t v \frac{u}{1+u}\,\d x-\int_{\R^2}\frac{u}{1+u}\,\d x+\lambda\int_{\R^2}\frac{v}{1+u}\,\d x,
\end{align*}
as desired.
\end{pr}

We study  the Cauchy problem for the parabolic-parabolic Keller--Segel system in~$\mathbb R^2$
\begin{equation}
\left\{
\begin{aligned}
&\pt_t u =\Delta u- \nabla\cdot \left( u \nabla v \right),
& t>0,\,~ &x\in\mathbb R^2,
\\
&\pt_t v = \Delta v - \lambda v +u,
&t>0,\, ~&x\in\mathbb R^2,
\\
&(u,v)(0,x)=(u_{0},v_0)(x),
&\, &x\in\mathbb R^2
\end{aligned}
\right.
\label{eqn;KS}
\end{equation}
with a constant $\lambda\ge0$, where $u_0,v_0\ge 0$ on $\mathbb R^2$ and
$u_0,v_0\not\equiv0$.
The Keller--Segel system~\eqref{eqn;KS} is a fundamental mathematical model
of chemotaxis, describing chemotactic aggregation in the
cellular slime mold {\it Dictyostelium discoideum} during its life cycle \cite{KeSe,Pa}.
In this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and
the chemoattractant, respectively.
The parameter $\lambda$ represents the degradation rate of the chemical.
Cells migrate toward regions of higher concentrations of a chemical substance
secreted by the cells themselves.
From a biological viewpoint, it is natural to assume that the initial data
are nonnegative functions.
From a mathematical viewpoint, cell aggregation is interpreted as the
blowup of solutions at $t=T$ in the sense that
$\lim_{t\to T}\|u(t)\|_\infty=\infty.$

One of the central mathematical features of such systems is the phenomenon of critical
mass.
For positive sufficiently regular solutions $(u,v)$ to~\eqref{eqn;KS}, the first
component $u$ satisfies the mass conservation law
$$\|u(t)\|_1=\|u_0\|_1\quad \text{for}~~~t>0.$$
Moreover, in two space dimensions, the global behavior of solutions is governed
by the size of the initial mass of $u$.
In particular, there exists a threshold value such that solutions with initial
mass below this threshold exist globally in time, whereas solutions with
initial mass above the threshold may blow up in finite time.
This critical-mass phenomenon has attracted considerable attention in the
literature.

The aim of this paper is to establish the global behavior of solutions to
\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.

\vspace{3mm}
The second equation in \eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves
while this is diffusing into the environment.
Since the chemoattractant attains its equilibrium on a time scale much quicker
than that of the cells, the simplified parabolic-elliptic system has also been
investigated \cite{JaLu,Na95}:
\begin{align*}
\left\{
\begin{aligned}
\partial_t u &= \Delta u - \nabla\cdot (u\nabla v),
& t>0,\ & x\in\mathbb R^2,\\
0 &= \Delta v - \lambda v + u,
& t>0,\ & x\in\mathbb R^2,
\\
&u(0,x)=u_{0}(x),
&\, &x\in\mathbb R^2.
\end{aligned}
\right.
\addtocounter{equation}{1}\tag{\theequation}
\label{eqn;PKS}
\end{align*}
The system \eqref{eqn;PKS} is also related to models of gravitational interaction
of particles \cite{Bi-Na94,Wo}.
In~\eqref{eqn;PKS}, the second equation can be written as
$v = (-\Delta+\lambda)^{-1}u.$
As a consequence, the system~\eqref{eqn;PKS} can be reduced to a single equation
for~$u$.
Owing to this elliptic structure, the parabolic-elliptic system~\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.
In particular, numerous works have been devoted to the critical-mass
phenomenon.
Indeed, the solution with $\|u_0\|_1\le 8\pi$ exists globally in time~\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},
while the solution may blow up in finite time if $\|u_0\|_1>8\pi$~\cite{DoPe,Ko-Su,Ku-Og03,Wei}.
For the corresponding Cauchy--Neumann problem in bounded domains,
supplemented with homogeneous Neumann boundary conditions, see \cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.

\vspace{3mm}
Unlike the parabolic-elliptic system \eqref{eqn;PKS}, the fully parabolic system
\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches
developed for \eqref{eqn;PKS} are no longer applicable.
The global existence of solutions with sub-critical mass
$\|u_0\|_{1}<8\pi$ has nevertheless been established by combining
Lyapunov functionals with the Trudinger--Moser type inequality and its optimal
constant.
More precisely, the following results are known:
\begin{enumerate}
	\item[(i)]
	If $\|u_0\|_{1}<8\pi$, then the corresponding solution to
	\eqref{eqn;KS} exists globally in time \cite{CaCo08,Mi13,NaOg11}.

	\item[(ii)]
	If $\|u_0\|_{1}>8\pi$ and $(u_0,v_0)$ is radially symmetric, then there
	exists a radially symmetric solution that blows up in finite time
	\cite{Mi20,Mi20-SIAM}.

	\item[(iii)]
	If $\|u_0\|_{1}=8\pi$ and $(u_0,v_0)$ is either radially symmetric or
	satisfies the additional moment condition
	$u_0\ln(1+|x|^2)\in L^1(\mathbb R^2)$, then the corresponding solution exists globally
	in time.
	In contrast, for general initial data, the global behavior in the critical case
	remains delicate, and the solution either exists globally or blows up on the plane \cite{CaCo08,Mi13}.
\end{enumerate}
As for positive forward self-similar solutions to \eqref{eqn;KS}, refer to \cite{BiCoDo}.
For the corresponding Cauchy--Neumann problem in bounded domains, we refer to
\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.

As mentioned above, although several partial results are available, no complete global existence result has been obtained
for solutions with critical mass and general initial data without any
symmetry or moment assumptions.
One of the main difficulties lies in controlling the behavior of solutions at~$|x|\to\infty$.
Moreover, in the critical mass case, no global existence result is currently
known  for the corresponding Cauchy--Neumann problem to~\eqref{eqn;KS} in bounded domains.

In this paper, we establish the global existence of solutions with critical mass
for general initial data $(u_0,v_0)$.
To this end, we first recall the definition of solutions.

\vspace{2mm}
\begin{def*}
Let $4/3<p<2$ and $q=p/(p-1)$.
Given $T\in (0,\infty]$ and  $(u_0,v_0) \in L^1(\mathbb R^2)\times \dot{H}^{1}(\mathbb R^2)$,
we say that a pair of functions $(u,v)$ defined on $[0,T)\times \mathbb R^2$ is a mild solution to~\eqref{eqn;KS} on $[0,T)$ if
\begin{enumerate}
	\item $u\in C\big([0,T); L^1(\mathbb R^2)\big)\cap C\left( (0,T); L^p(\mathbb{R}^2) \right)$;
	\item $v\in C\big([0,T); \dot{H}^{1}(\mathbb R^2)\big)\cap C\left( (0,T); \dot{W}^{1,q}(\mathbb{R}^2) \right)$;
	\item $\displaystyle\sup_{0<t<T} t^{1-\frac1p}\|u(t)\|_{p}+\displaystyle\sup_{0<t<T}t^{\frac12-\frac1q}\|\nabla v(t)\|_{q}$;
	\item $(u,v)$ satisfies the integral formulations
	\begin{align*}
	&u(t)=e^{t\Delta}u_0-\int_0^t \nabla\cdot e^{(t-s)\Delta} \big(u(s)\nabla v(s)\big) \,\mathrm{d} s,
	\\
	&v(t)= e^{t(\Delta-\lambda)} v_0+\int_0^t e^{(t-s)(\Delta-\lambda)} u(s) \,\mathrm{d} s
	\end{align*}
	for $0<t<T$, where $e^{t\Delta}$ is the heat semigroup given by
	\begin{align*}
	e^{t\Delta} f(x) := G_t*f(x)=\int_{ \mathbb R^2}G_t(x-y) f(y)\,\mathrm{d} y,
	\quad
	G_t(x)=\frac{1}{4\pi t} \exp\left(-\frac{|x|^2}{4 t}\right)
	\end{align*}
	for $x\in\mathbb R^2$ and $f \in L^p(\mathbb R^2)$.
\end{enumerate}
\end{def*}
The Lebesgue spaces $L^p(\mathbb R^2)$ are equipped with the norm $\|\cdot\|_{L^p(\mathbb R^2)}$,
which is abbreviated as $\|\cdot\|_p$ when there is no ambiguity.
We define the homogeneous Sobolev spaces by
\[
\dot{W}^{1,p}(\mathbb R^2)
:= \overline{C_0^\infty(\mathbb R^2)}^{\|\nabla (\cdot)\|_{L^p(\mathbb R^2)}}
\]
and $\dot{H}^1(\mathbb R^2):=\dot{W}^{1,2}(\mathbb R^2)$ when $p=2$.
$L_+^1(\mathbb R^2):=\{f \in L^1(\mathbb R^2); f\ge0~~ \text{and}~~ f\not\equiv0~~ \text{on $\mathbb R^2$} \}$.

\vspace{5mm}
The starting point for our analysis is the local well-posedness for \eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\ref{prop;LWP} in Section~\ref{sect;LWP} below.

We are now in a position to state our main result in this paper.
Main result in this paper reads as follows.
\vspace{2mm}

\vspace{3mm}
The second equation in \eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves
while this is diffusing into the environment.
Since the chemoattractant attains its equilibrium on a time scale much quicker
than that of the cells, the simplified parabolic-elliptic system has also been
investigated \cite{JaLu,Na95}:
\begin{align*}
\left\{
\begin{aligned}
\partial_t u &= \Delta u - \nabla\cdot (u\nabla v),
& t>0,\ & x\in\mathbb R^2,\\
0 &= \Delta v - \lambda v + u,
& t>0,\ & x\in\mathbb R^2,
\\
&u(0,x)=u_{0}(x),
&\, &x\in\mathbb R^2.
\end{aligned}
\right.
\addtocounter{equation}{1}\tag{\theequation}
\label{eqn;PKS}
\end{align*}
The system \eqref{eqn;PKS} is also related to models of gravitational interaction
of particles \cite{Bi-Na94,Wo}.
In~\eqref{eqn;PKS}, the second equation can be written as
$v = (-\Delta+\lambda)^{-1}u.$
As a consequence, the system~\eqref{eqn;PKS} can be reduced to a single equation
for~$u$.
Owing to this elliptic structure, the parabolic-elliptic system~\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.
In particular, numerous works have been devoted to the critical-mass
phenomenon.
Indeed, the solution with $\|u_0\|_1\le 8\pi$ exists globally in time~\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},
while the solution may blow up in finite time if $\|u_0\|_1>8\pi$~\cite{DoPe,Ko-Su,Ku-Og03,Wei}.
For the corresponding Cauchy--Neumann problem in bounded domains,
supplemented with homogeneous Neumann boundary conditions, see \cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.

\vspace{3mm}
Unlike the parabolic-elliptic system \eqref{eqn;PKS}, the fully parabolic system
\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches
developed for \eqref{eqn;PKS} are no longer applicable.
The global existence of solutions with sub-critical mass
$\|u_0\|_{1}<8\pi$ has nevertheless been established by combining
Lyapunov functionals with the Trudinger--Moser type inequality and its optimal
constant.
More precisely, the following results are known:
\begin{enumerate}
 \item[(i)]
 If $\|u_0\|_{1}<8\pi$, then the corresponding solution to
 \eqref{eqn;KS} exists globally in time \cite{CaCo08,Mi13,NaOg11}.

\item[(iii)]
 If $\|u_0\|_{1}=8\pi$ and $(u_0,v_0)$ is either radially symmetric or
 satisfies the additional moment condition
 $u_0\ln(1+|x|^2)\in L^1(\mathbb R^2)$, then the corresponding solution exists globally
 in time.
 In contrast, for general initial data, the global behavior in the critical case
 remains delicate, and the solution either exists globally or blows up on the plane \cite{CaCo08,Mi13}.
\end{enumerate}
As for positive forward self-similar solutions to \eqref{eqn;KS}, refer to \cite{BiCoDo}.
For the corresponding Cauchy--Neumann problem in bounded domains, we refer to
\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.

\vspace{2mm}
\begin{def*}
Let $4/3<p<2$ and $q=p/(p-1)$.
Given $T\in (0,\infty]$ and $(u_0,v_0) \in L^1(\mathbb R^2)\times \dot{H}^{1}(\mathbb R^2)$,
we say that a pair of functions $(u,v)$ defined on $[0,T)\times \mathbb R^2$ is a mild solution to~\eqref{eqn;KS} on $[0,T)$ if
\begin{enumerate}
 \item $u\in C\big([0,T); L^1(\mathbb R^2)\big)\cap C\left( (0,T); L^p(\mathbb{R}^2) \right)$;
 \item $v\in C\big([0,T); \dot{H}^{1}(\mathbb R^2)\big)\cap C\left( (0,T); \dot{W}^{1,q}(\mathbb{R}^2) \right)$;
 \item $\displaystyle\sup_{0<t<T} t^{1-\frac1p}\|u(t)\|_{p}+\displaystyle\sup_{0<t<T}t^{\frac12-\frac1q}\|\nabla v(t)\|_{q}$;
 \item $(u,v)$ satisfies the integral formulations
 \begin{align*}
 &u(t)=e^{t\Delta}u_0-\int_0^t \nabla\cdot e^{(t-s)\Delta} \big(u(s)\nabla v(s)\big) \,\mathrm{d} s,
 \\
 &v(t)= e^{t(\Delta-\lambda)} v_0+\int_0^t e^{(t-s)(\Delta-\lambda)} u(s) \,\mathrm{d} s
 \end{align*}
 for $0<t<T$, where $e^{t\Delta}$ is the heat semigroup given by
 \begin{align*}
 e^{t\Delta} f(x) := G_t*f(x)=\int_{ \mathbb R^2}G_t(x-y) f(y)\,\mathrm{d} y,
 \quad
 G_t(x)=\frac{1}{4\pi t} \exp\left(-\frac{|x|^2}{4 t}\right)
 \end{align*}
 for $x\in\mathbb R^2$ and $f \in L^p(\mathbb R^2)$.
\end{enumerate}
\end{def*}
The Lebesgue spaces $L^p(\mathbb R^2)$ are equipped with the norm $\|\cdot\|_{L^p(\mathbb R^2)}$,
which is abbreviated as $\|\cdot\|_p$ when there is no ambiguity.
We define the homogeneous Sobolev spaces by
\[
\dot{W}^{1,p}(\mathbb R^2)
:= \overline{C_0^\infty(\mathbb R^2)}^{\|\nabla (\cdot)\|_{L^p(\mathbb R^2)}}
\]
and $\dot{H}^1(\mathbb R^2):=\dot{W}^{1,2}(\mathbb R^2)$ when $p=2$.
$L_+^1(\mathbb R^2):=\{f \in L^1(\mathbb R^2); f\ge0~~ \text{and}~~ f\not\equiv0~~ \text{on $\mathbb R^2$} \}$.

\vspace{5mm}
The starting point for our analysis is the local well-posedness for \eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\ref{prop;LWP} in Section~\ref{sect;LWP} below.

We are now in a position to state our main result in this paper.
Main result in this paper reads as follows.
\vspace{2mm}

\vspace{5mm}
The starting point for our analysis is the local well-posedness for \eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\ref{prop;LWP} in Section~\ref{sect;LWP} below.

First of all, it is worth mentioning that the system \eqref{eqn;KS} has a usual Lyapunov functional~$\L(t)$ as follows:
\begin{align*}
\frac{\d}{\d t}\L(t)+ \D(t) =0,
\eqntag
\label{eqn;Lyapunov}
\end{align*}
where
\begin{align*}
\L(t):=\int_{\R^2}u\ln u\dx-\int_{\R^2}uv\dx+\frac12\|\nabla v\|_2^2+\frac\lambda2\|v\|_2^2
\end{align*}
and the dissipative term $\D(t)$ is given by
\begin{align*}
\D(t):=\int_{\R^2} u \left|\nabla(\ln u -v)\right|^2\dx+\|\partial_t v\|_2^2.
\end{align*}
The upper bound for $\L(t)$ follows from the monotonicity \eqref{eqn;Lyapunov} and
the lower bound for that can be obtained by the optimal constraint of the initial mass $\|u_0\|_1$ deriving from the Trudinger--Moser type inequality and its best possible constant, see \cite{Na-Se-Yo} and also \cite{CaCo08,Mi13,NaOg11} for instance.
On the other hand,
in order to ensure the well-definedness of a usual entropy $\int_{\R^2}u\ln u \dx$ on the whole space,
at least the logarithmic moment assumption for the initial data $u_0\ln(1+|x|^2)\in L^1(\R^2)$ is required to control the behavior of solutions as $|x|\to\infty$, that is,
\begin{align*}
-\int_{\R^2}u(\ln u)_-\dx \le\,C\int_{\R^2}u\ln(1+|x|^2)\dx,
\end{align*}
where $(\ln u)_-:=\min\{\ln u,0\}$, according to \cite[Lemma 2.4]{CaCo08}.
Later, to get rid of any moment assumptions,
the modified entropy is adapted by Nagai \cite{Na11};
\begin{align*}
\int_{\R^2}(1+u)\ln(1+u)\dx,
\end{align*}
as well as the following identity for the modified Lyapunov functional $\L_m(t)$ is introduced:
\begin{align*}
\frac{\d}{\d t} \L_m(t) + \D_m(t)=\frac14\|\nabla v(t)\|_2^2,
\eqntag
\label{eqn;usualmodifiedfunctiona}
\end{align*}
where
\begin{align*}
\L_m(t):=\int_{\R^2}(1+u)\ln(1+u)\dx-\int_{\R^2}uv\dx+\frac12\|\nabla v\|_2^2+\frac\lambda2\|v\|_2^2
\end{align*}
and
\begin{align*}
\D_m(t):=\int_{\R^2}u\left|\nabla(\ln(1+u)-v)\right|^2\dx+\int_{\R^2}\left|\nabla\left(\ln(1+u)-\frac12v\right)\right|^2\dx+\|\partial_tv\|_2^2.
\end{align*}
The functional $\L_m(t)$ is no longer the Lyapunov functional, however, it is still useful for proving
the global existence of the solution to \eqref{eqn;KS} with sub-critical mass. Indeed, it follows
from the lower bound for $\L_m(t)$ that
for any $\al\in[0,1)$
\begin{align*}
\al\int_{\R^2}(1+u)\ln(1+u)\dx+\frac12\left[1-\frac{\|u_0\|_1}{8\pi(1-\al)}\right]&\|\nabla v\|_2^2
+\int_0^t \D_m(s)\,\d s
\\
\le\,&\L_m(0)+C(\tau,\alpha)+\frac14\int_0^t\|\nabla v\|_2^2\,\d s
\eqntag
\label{eqn;afford}
\end{align*}
for $t\in[0,\tau]\cap(0,T)$ with any $\tau>0$ and some $C(\tau,\alpha)>0$ (cf.~\cite{CaCo08,Mi13}). Hence, if $\|u_0\|<8\pi$, then all terms on the left hand side of \eqref{eqn;afford} are positive by choosing $\al>0$ sufficiently small which depends on $\|u_0\|_1$, and Gronwall's inequality implies that
$\nabla v \in L^2((0,T)\times \R^2)$,
so that we also obtain the bound for the modified entropy $\int_{\R^2}(1+u)\ln (1+u)\dx$.
Nevertheless, for the critical mass $\|u_0\|_1=8\pi$, this necessarily leads to $\alpha=0$, as a result,
\begin{align*}
\int_0^t \D_m(s)\,\d s\le\,&\L_m(0)+C(\tau)+\frac14\int_0^t\|\nabla v\|_2^2\,\d s,
\eqntag
\label{eqn;lack-lyapunov}
\end{align*}
which is useless and fails to obtain even the estimates for $\D_m(t)$ due to the lack of the regularity of solutions.
Therefore, we introduce a reconstructed Lyapunov functional $\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\F_m(t)$ be the functional for solutions to \eqref{eqn;KS} defined as
\begin{align*}
\F_m(t):=\L_m(t)
+\int_{\R^2}\ln(1+u)\,\mathrm{d}x-\int_{\R^2}v\,\d x.
\eqntag
\label{eqn;modifiedLyapunov}
\end{align*}
Then, the following functional differential inequality holds true:
\begin{prop}\label{prop;energy-est}
Let $(u,v)$ be the solution to \eqref{eqn;KS}. Then, the functional $\F_m(t)$ defined in~\eqref{eqn;modifiedLyapunov} satisfies the following identity:
 \begin{align*}
 \frac{\d}{\d t}\F_m(t) +\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\frac12\|\partial_tv\|_2^2
 \le\,&
 \lambda\int_{\R^2}v\,\d x.
 \end{align*}
 Therefore, if $\lambda=0$ then
 \begin{align*}
 \frac{\d}{\d t}\F_m(t) +\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\frac12\|\partial_tv\|_2^2 \le\,0.
 \end{align*}
 If $\lambda>0$ then
 \begin{align*}
 \frac{\d}{\d t}\F_m(t) +\int_{\R^2}u\left|\nabla\left(\ln(1+u)-v\right)\right|^2\,\mathrm{d}x+\frac12\|\partial_tv\|_2^2 \le\,\|v_0\|_1+\|u_0\|_1.
 \end{align*}
\end{prop}
\vspace{5mm}
It is worth emphasizing that $\F_m(t)$ is non-increasing in time when $\lambda=0$ as well as
unlike the identity \eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\ref{prop;energy-est} can be easily controlled by initial data.
This allows us to show the regularity estimates for the dissipative term associated with $\F_m(t)$, see Proposition~\ref{prop;bound-energy} below.
\begin{lem}\label{lem;ModifiedLF}
 Suppose assumptions as in Theorem \ref{thm;global}. Then,
\begin{align*}
\frac{\d}{\d t}\F_m(t)+ \widetilde{\D}(t)=-\int_{\R^2}\partial_t v \frac{u}{1+u}\,\d x-\int_{\R^2}\frac{u}{1+u}\,\mathrm{d}x+\lambda\int_{\R^2}\frac{v}{1+u}\,\d x,
\end{align*}
where
$\F_m(t)$ is the modified functional defined in \eqref{eqn;modifiedLyapunov} and the dissipative term $\widetilde{\D}(t)$ is given by
\begin{align*}
\widetilde{\D}(t):=\int_{\R^2}u\left|\nabla \left(\ln(1+u)-v\right)\right|^2\,\d x+\|\partial_t v\|_2^2.
\end{align*}
\end{lem}

\vspace{5mm}
\begin{pr}{Theorem \ref{thm;global}}
Let $R$ be taken as in Lemma \ref{lem;L^2-exterior}.
By Lemma \ref{lem;LlogL-bound-exterior},
\begin{align*}
\int_{ |x|>2R } (1+u) \ln(1+u) \dx\le\, C(t_0,\tau,R).
\end{align*}
On the other hand, Lemma \ref{prop;interior-bdd-entropy} gives
\begin{align*}
\int_{ \R^2}(u\ln u)\psi_R^2\dx\le\, C(t_0,\tau,R)
\end{align*}
for $t\in[t_0,\tau]\cap(0,T)$.
Now, according to \cite[Lemma 2.3]{NaOg16},
\begin{align*}
\int_{ \Omega} (1+u)\ln(1+u)\dx\le\,2\int_{ \Omega} u |\ln u | \dx+(2\ln 2)\int_{ \Omega} u\dx,
\end{align*}
where $\Omega$ is a measurable set in $\R^2$,
 so that since $\supp\psi_R\subset \{x; |x|\le 16 R\}$
 \begin{align*}
 \int_{ |x|<8R }(1+u)\ln(1+u)\dx\le\,&2\int_{ |x|<8R }u|\ln u| \dx+(2\ln 2)\|u_0\|_1
 \\
 \le\,\,&2\int_{ \R^2 }u|\ln u| \psi_R^2\dx+(2\ln 2)\|u_0\|_1
 \\
 =\,&2\int_{ \R^2 }(u\ln u) \psi_R^2\dx-4\int_{ \R^2 }(u\ln u)_-\psi_R^2\dx+(2\ln 2)\|u_0\|_1
 \\
 \le\,&C(t_0,\tau,R) +\frac4e|B_{16R}(0)|
 \end{align*}
 for $t\in[t_0,\tau]\cap(0,T)$, where we use $x\ln x \ge -1/e$ for $x\ge0$.
 This implies that
 \begin{align*}
&\int_{ \R^2 }(1+u)\ln(1+u)\dx
\\=\,&\int_{ |x|>2R } (1+u) \ln(1+u) \dx
+\int_{ |x|\le 2R } (1+u) \ln(1+u) \dx
\\
\le\,&\int_{ |x|>2R } (1+u) \ln(1+u) \dx+\int_{ |x|< 8R } (1+u) \ln(1+u) \dx
\\
\le\,&C(t_0,\tau,R)
 \end{align*}
 for $t\in[t_0,\tau]\cap(0,T)$.
 Hence, we have along with Proposition \ref{prop;bound-energy}
 \begin{align*}
\int_{ \R^2 }(1+u) \ln (1+u)\dx+\int_{t_0}^t \|\partial_tv\|_2^2\,\d s\le\, C(t_0,\tau,R)
 \eqntag
 \label{eqn;uniform-ulogu}
 \end{align*}
 for $t\in[t_0,\tau]\cap(0,T)$.
By the parabolic regularity argument, we obtain a uniform $L^2$-bound for~$u$
on $[t_0,\tau]\cap(0,T)$ from \eqref{eqn;uniform-ulogu} (cf. \cite{Mi13,Na-Se-Yo}), which yields further regularity estimates.
In particular, the solution~$(u,v)$ cannot blow up in finite time.
Therefore, the solution to \eqref{eqn;KS} exists globally in time.
\end{pr}

\vspace{2mm}

\begin{rem}
In Theorem~\ref{thm;global}, the initial data are only required to satisfy
\[
(u_0,v_0)\in L_+^1(\mathbb R^2)\times\bigl(L_+^1(\mathbb R^2)\cap \dot H^1(\mathbb R^2)\bigr)\quad \text{with}~~\|u_0\|_1=8\pi.
\]
In particular, no additional symmetry or moment assumptions are imposed.
\end{rem}
\vspace{2mm}
As mentioned above, the global existence of solutions with critical mass
without any symmetry or moment assumptions remains delicate.
In~\cite[Theorem~1.2]{Mi13}, it is shown that, for general initial data with
critical mass, solutions either exist globally in time or blow up on the plane,
by means of a contradiction argument.
Indeed, although in the sub-critical case regularity estimates can be obtained
by combining (modified) Lyapunov functionals with the Trudinger--Moser
inequality, such estimates are no longer directly available in the critical case due to the lack of regularity of solutions,
see~\eqref{eqn;afford} below.
This difficulty prevents the direct extension of classical entropy methods.

To overcome this difficulty, we introduce a reconstructed Lyapunov functional.
The main novelty of this work lies in the construction of a refined Lyapunov
functional specifically adapted to the whole space setting, which allows us
to control the behavior of solutions at $|x|\to\infty$ and to derive
regularity estimates even in the critical mass regime, without
imposing any symmetry or moment assumptions on the initial data.
As a consequence, we establish global-in-time existence for general initial
data at the critical mass, a result that was previously out of reach by existing
methods, see Subsection~\ref{sect;modified-Lyapunov} for details.
Moreover, the present approach is expected to be applicable to a broad class
of chemotaxis  systems in the whole space setting.

\vspace{3mm}
The remainder of this paper is organized as follows.
In Section~\ref{sect;preliminary}, we recall several preliminary lemmas needed
to prove the main result.
Section~\ref{sect;LWP} is devoted to the local well-posedness of~\eqref{eqn;KS}, based on the above definition of solutions.
Finally, in Section~\ref{sect;global}, we present the proof of
Theorem~\ref{thm;global}.

\vspace{5mm}

The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving Theorem~\ref{thm;global}: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\to\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (Section~\ref{sect;preliminary}), local well-posedness (Section~\ref{sect;LWP}), then the global argument proving Theorem~\ref{thm;global} (Section~\ref{sect;global}).

The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving the main theorem: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\to\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (proved next), local well-posedness (proved after that), then the global argument establishing the main theorem (proved later).

\label{thm;global}
For $(u_0,v_0)\in L_+^1(\mathbb R^2) \times L^1_+(\mathbb R^2)\cap \dot{H}^1(\mathbb R^2)$,
let $(u,v)$ be the solution to
\begin{aligned}
&\pt_t u =\Delta u- \N\cd \left( u \N v \right),
& t>0,\,~ &x\in\R^2,
\\
&\pt_t v = \Delta v - \lambda v +u,
&t>0,\, ~&x\in\R^2,
\\
&(u,v)(0,x)=(u_{0},v_0)(x),
&\, &x\in\R^2
\end{aligned}
\right.
\label{eqn;KS}
\end{equation}
with a constant $\lambda\ge0$, where $u_0,v_0\ge 0$ on $\R^2$ and
$u_0,v_0\not\equiv0$
on $(0,T)\times\mathbb R^2$.
Suppose that $\|u_0\|_1=8\pi$.
Then, in establishing the main theorem, the solution to the system above exists globally in time.

Question:
Consider the two-dimensional fully parabolic Keller--Segel system
\[
\begin{cases}
\partial_t u = \Delta u-\nabla\!\cdot\!\big(u\nabla v\big), & t>0,\ x\in\mathbb R^2,\\
\partial_t v = \Delta v-\lambda v+u, & t>0,\ x\in\mathbb R^2,\\
(u,v)(0,x)=(u_0,v_0)(x), & x\in\mathbb R^2,
\end{cases}
\]
where \(\lambda\ge 0\). Assume
\(u_0\in L_+^1(\mathbb R^2)\) and \(v_0\in L_+^1(\mathbb R^2)\cap \dot H^1(\mathbb R^2)\), with
\[
L_+^1(\mathbb R^2):=\{f\in L^1(\mathbb R^2): f\ge 0\ \text{and}\ f\not\equiv 0\},
\]
so in particular \(u_0,v_0\) are nonnegative and nontrivial, and suppose that the initial cell mass satisfies
\[
\|u_0\|_{L^1(\mathbb R^2)}=8\pi.
\]
Which of the following conclusions about the corresponding solution \((u,v)\) holds?

The solution \((u,v)\) exists globally in time; equivalently, it can be continued for all \(t\ge 0\) and does not blow up in finite time.

Choices:
B. The solution \((u,v)\) exists globally in time provided, in addition, that the initial data are either radially symmetric or satisfy a finite logarithmic moment condition such as \(u_0\ln(1+|x|^2)\in L^1(\mathbb R^2)\); without one of these extra assumptions, finite-time blow-up may occur at mass \(8\pi\).
C. The corresponding solution \((u,v)\) has a local-in-time mild solution on some interval \([0,T)\) for a positive maximal existence time \(T>0\).
D. For every such initial datum with \(\|u_0\|_{L^1(\mathbb R^2)}=8\pi\), the solution \((u,v)\) exists globally in time and moreover remains uniformly bounded in \(L^\infty(\mathbb R^2)\times \dot W^{1,\infty}(\mathbb R^2)\) for all \(t\ge 0\).
E. There exists a time \(T=T(u_0,v_0)>0\) such that the solution \((u,v)\) exists on \([0,T)\), and if it is global then this conclusion requires constants in the a priori estimates to depend on additional decay of the data at spatial infinity; at critical mass \(8\pi\), global existence is therefore not guaranteed for arbitrary \(u_0\in L_+^1(\mathbb R^2)\) and \(v_0\in L_+^1(\mathbb R^2)\cap \dot H^1(\mathbb R^2)\).

The stem does not explicitly reveal the correct conclusion. It states the PDE, hypotheses, and critical-mass condition, but the answer still depends on knowing or reasoning about the exact existence theorem.

The item is close to theorem recall: it essentially asks which existence conclusion matches the stated hypotheses. However, it is not a pure restatement because the options include weaker, stronger, and conditionally modified alternatives.

The question requires some discrimination among nearby claims (local vs global existence, extra decay assumptions, stronger boundedness conclusions), but it mainly tests recognition of the correct theorem rather than substantial generative reasoning.

The distractors are mathematically plausible and target realistic confusions: requiring extra symmetry/decay assumptions, settling for only local existence, or overstrengthening the conclusion to uniform L^infty bounds.

A solid theorem-discrimination MCQ with good distractors and no major answer leakage, though it leans more toward recall of a specific result than deep generative reasoning.


Question:
Consider the two-dimensional fully parabolic Keller--Segel system
\[
\begin{cases}
\partial_t u = \Delta u-\nabla\!\cdot\!\big(u\nabla v\big), & t>0,\ x\in\mathbb R^2,\\
\partial_t v = \Delta v-\lambda v+u, & t>0,\ x\in\mathbb R^2,\\
(u,v)(0,x)=(u_0,v_0)(x), & x\in\mathbb R^2,
\end{cases}
\]
where \(\lambda\ge 0\). Assume
\(u_0\in L_+^1(\mathbb R^2)\) and \(v_0\in L_+^1(\mathbb R^2)\cap \dot H^1(\mathbb R^2)\), with
\[
L_+^1(\mathbb R^2):=\{f\in L^1(\mathbb R^2): f\ge 0\ \text{and}\ f\not\equiv 0\},
\]
so in particular \(u_0,v_0\) are nonnegative and nontrivial, and suppose that the initial cell mass satisfies
\[
\|u_0\|_{L^1(\mathbb R^2)}=8\pi.
\]
Which of the following conclusions about the corresponding solution \((u,v)\) holds?

Choices:
B. The solution \((u,v)\) exists globally in time provided, in addition, that the initial data are either radially symmetric or satisfy a finite logarithmic moment condition such as \(u_0\ln(1+|x|^2)\in L^1(\mathbb R^2)\); without one of these extra assumptions, finite-time blow-up may occur at mass \(8\pi\).
C. The corresponding solution \((u,v)\) has a local-in-time mild solution on some interval \([0,T)\) for a positive maximal existence time \(T>0\).
D. For every such initial datum with \(\|u_0\|_{L^1(\mathbb R^2)}=8\pi\), the solution \((u,v)\) exists globally in time and moreover remains uniformly bounded in \(L^\infty(\mathbb R^2)\times \dot W^{1,\infty}(\mathbb R^2)\) for all \(t\ge 0\).
E. There exists a time \(T=T(u_0,v_0)>0\) such that the solution \((u,v)\) exists on \([0,T)\), and if it is global then this conclusion requires constants in the a priori estimates to depend on additional decay of the data at spatial infinity; at critical mass \(8\pi\), global existence is therefore not guaranteed for arbitrary \(u_0\in L_+^1(\mathbb R^2)\) and \(v_0\in L_+^1(\mathbb R^2)\cap \dot H^1(\mathbb R^2)\).

The stem states hypotheses and asks for the resulting conclusion, but it does not explicitly reveal that global existence is the correct outcome. The critical mass 8π is mathematically relevant, not answer leakage.

The item is close to a theorem-recall question: under the stated assumptions, choose the theorem’s conclusion. It is not a pure verbatim restatement because the options include weaker, stronger, and hypothesis-altered alternatives.

Some reasoning is needed to reject plausible variants such as needing extra symmetry/moment assumptions, accepting only local existence, or inferring uniform L∞ bounds. However, the main task is still recognition of the correct theorem-level conclusion rather than substantial derivation.

The distractors are plausible and distinct: one imports extra assumptions from related results, one gives a weaker true local statement, one overstates regularity, and one casts doubt on critical-mass global existence. These align with realistic mathematical confusions.

A solid MCQ with strong distractors and no real answer leakage, but it is primarily a theorem-identification item rather than a deeply generative reasoning question.


The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory.
A classical result by Goodman~\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound
\[
T(n) =
\begin{cases}
    \dfrac{u(u-1)(u-2)}{3}, & \text{if } n = 2u; \\
    \dfrac{2u(u-1)(4u+1)}{3}, & \text{if } n = 4u+1; \\
    \dfrac{2u(u+1)(4u-1)}{3}, & \text{if } n = 4u+3.
\end{cases}
\]
\ignore{
\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\binom{n}{r}/2^{\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \cite{erdHos1962number}. This conjecture was later disproved by Thomason \cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \approx 2.18$ \cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \cite{csv}.}
\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}
}

While these deterministic settings have attracted considerable attention and yielded several intriguing results,
the same question in the context of random graphs,
such as Erdős–Rényi random graphs, remains far less understood, even for  the special  case of triangles. In this paper, we consider this question of finding the smallest number of monochromatic subgraphs in bipartitions of random graphs. We do so in a general setting,
where the subgraph is only assumed to be
strictly balanced.

To set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$   is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and
$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.
 A random \( F \)-graph \( \Hg_F(n, q) \) is an \( F \)-graph with vertex set \( V=[n]=\cbc{1,2,\ldots,n} \), where each of the
\begin{align}\label{eq:N-trials}
\binom{n}{r} \cdot \frac{r!}{{\rm aut}(F)}
\end{align}
potential copies of \( F \) on vertices in \( [n] \) is included in the edge set $E$ independently with probability \( q \). Here ${\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case
when $F$ is $K_2$ (two nodes
connected by an edge), we use a more common notation $\mathbb{G}(n,q)$.

Given a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \(  V=V_1 \cup V_2, V_1\cap V_2=\emptyset \), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and  whose vertex sets have a non-empty intersection with  \( V_1 \) and \( V_2 \). That is, each hyperedge of the cut is  neither entirely contained in \( V_1 \), nor is it entirely contained in \( V_2 \). The  \emph{cut value} associated with a cut is
the cardinality of this set.
 The \emph{max-cut value}
 denoted by ${\rm MaxC}(G,F)$
 is defined as the largest such value taken over all possible bipartitions of \(V\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$
  associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum
  number of monochromatic $F$-subgraphs is denoted by ${\rm MinM}(G,F)$. Naturally, ${\rm MinM}(G,F)+{\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.

In combinatorial optimization and theoretical computer science, there is a long-standing interest in understanding the    minimum number of monochromatic subgraphs achievable by  varying over all bipartitions of  graphs, both in the special case of max-cut values associated with $F=K_2$, and beyond ~\cite{ckpsty2013,dms2017,dl2018,sen2018optimization,shabanov2021maximum}. This is the question we address in our paper.

\relax

Let $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\rd_1(F')<{\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\neq F$.  Here
    \begin{align*}
        \rd_1(G)=\frac{\abs{E(G)}}{\abs{V(G )}-1}.
    \end{align*}
There are many examples of strictly 1-balanced graphs, including  cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.

Recently a great progress was achieved in understanding the relationship between $\mathbb{G}(n,p)$ and $\Hg_F(n,q)$
with $q$ judiciously chosen
as $q=p^{|E(F)|}$. In particular, as shown in \cite{burghart2024sharp} (see Proposition~\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs
$\mathbb{G}(n,p)$ and $\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\mathbb{G}(n,p)$.

Our first result is as follows.

The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory.
A classical result by Goodman~\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound
\[
T(n) =
\begin{cases}
 \dfrac{u(u-1)(u-2)}{3}, & \text{if } n = 2u; \\
 \dfrac{2u(u-1)(4u+1)}{3}, & \text{if } n = 4u+1; \\
 \dfrac{2u(u+1)(4u-1)}{3}, & \text{if } n = 4u+3.
\end{cases}
\]
\ignore{
\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\binom{n}{r}/2^{\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \cite{erdHos1962number}. This conjecture was later disproved by Thomason \cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \approx 2.18$ \cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \cite{csv}.}
\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}
}

To set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and
$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.
 A random \( F \)-graph \( \Hg_F(n, q) \) is an \( F \)-graph with vertex set \( V=[n]=\cbc{1,2,\ldots,n} \), where each of the
\begin{align}\label{eq:N-trials}
\binom{n}{r} \cdot \frac{r!}{{\rm aut}(F)}
\end{align}
potential copies of \( F \) on vertices in \( [n] \) is included in the edge set $E$ independently with probability \( q \). Here ${\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case
when $F$ is $K_2$ (two nodes
connected by an edge), we use a more common notation $\mathbb{G}(n,q)$.

Given a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \( V=V_1 \cup V_2, V_1\cap V_2=\emptyset \), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \( V_1 \) and \( V_2 \). That is, each hyperedge of the cut is neither entirely contained in \( V_1 \), nor is it entirely contained in \( V_2 \). The \emph{cut value} associated with a cut is
the cardinality of this set.
 The \emph{max-cut value}
 denoted by ${\rm MaxC}(G,F)$
 is defined as the largest such value taken over all possible bipartitions of \(V\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$
 associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum
 number of monochromatic $F$-subgraphs is denoted by ${\rm MinM}(G,F)$. Naturally, ${\rm MinM}(G,F)+{\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.

Let $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\rd_1(F')<{\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\neq F$. Here
 \begin{align*}
 \rd_1(G)=\frac{\abs{E(G)}}{\abs{V(G )}-1}.
 \end{align*}
There are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.

Recently a great progress was achieved in understanding the relationship between $\mathbb{G}(n,p)$ and $\Hg_F(n,q)$
with $q$ judiciously chosen
as $q=p^{|E(F)|}$. In particular, as shown in \cite{burghart2024sharp} (see Proposition~\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs
$\mathbb{G}(n,p)$ and $\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\mathbb{G}(n,p)$.

Our first result is as follows.

\begin{proposition}[{\cite[Theorem 1.6]{burghart2024sharp}}]\label{thm-bur-1.6}\label{prop:coupling}
Let $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \geq 2$ vertices. There exist constants $\delta, \varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \leq n^{-1/\rd_1(F) + \varepsilon}$ and $q \leq (1 - n^{-\delta})p^{s}$. Whp as $n\to\infty$ there exists a permutation $\tau: [n] \to [n]$ such that
 for every $F(v_1, \ldots, v_r) \in E(\Hg_F(n,q))$ also $F(\tau(v_1), \ldots, \tau(v_r)) \subset \G(n,p)$.
\end{proposition}

Our first result is as follows.

\ignore{
\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\dg{good point. Done}
}

The following key result from \cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $\G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $\Hg_F(n,q)$:
\begin{proposition}[{\cite[Theorem 1.6]{burghart2024sharp}}]\label{thm-bur-1.6}\label{prop:coupling}
Let $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \geq 2$ vertices. There exist constants $\delta, \varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \leq n^{-1/\rd_1(F) + \varepsilon}$ and $q \leq (1 - n^{-\delta})p^{s}$. Whp as $n\to\infty$ there exists a permutation $\tau: [n] \to [n]$ such that
 for every $F(v_1, \ldots, v_r) \in E(\Hg_F(n,q))$ also $F(\tau(v_1), \ldots, \tau(v_r)) \subset \G(n,p)$.
\end{proposition}
\ignore{
\yd{I think the order should be "whp as $n\to\infty$, here exists a permutation $\tau: [n] \to [n]$". The permutations aren't fixed beforehand.}
\dg{done}
\yd{Can't $\epsilon, \delta$ be arbitrarily small?}
\dg{We don't want them to be small. In fact the larger is $\epsilon$ the stronger the result. $\epsilon$ controls how high above $n^{-{1\over d_1(F)}}$ we can go and theorem still be valid. Makes sense?}
\yd{}
}
In the above $A\subset B$ means $A$ is a subgraph of $B$ (not necessarily induced one).
\Cref{thm-bur-1.6} implies that if we choose $q = (1 - n^{-\delta})p^{s}$, each $F$-hyperedge in $\Hg_F(n,q)$ corresponds to an ordered copy of $F$ in $\G(n,p)$. If we instead choose $q = p^s$, then, on the one hand, the number of $F$-hyperedges in $\Hg_F(n,q)$ does not increase too much, while on the other hand, it is close to the expected number of ordered copies of $F$ in $\G(n,p)$. In particular,
we claim the following.

\begin{corollary}\label{cor-erg-hg}
Let $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \geq 2$ vertices. Fix $c > 0$, and let $p = c n^{-1/\rd_1(F)}$, $q = p^s = c^s n^{1-r}$. Whp as $n\to\infty$, there exists a permutation
$\tau:[n]\to [n]$ such that the number of distinct ordered copies $F(v_1, \ldots, v_r)$ for which exactly one of the events $F(v_1, \ldots, v_r) \in E(\Hg_F(n,q))$ or $F(\tau(v_1), \ldots, \tau(v_r)) \subset \G(n,p)$ holds, is $o(n)$.
\end{corollary}
\ignore{
\yd{Again the existence of permutation is whp}
\dg{done}
}
The proof crucially uses the fact that
for strictly 1-balanced graph that when $p=O(n^{-{1/d_1(F)}})$, the number of pairs of distinct ordered copies of $F$ in $\G(n,p)$ which share at least one edge is small.
\ignore{
\yd{remove "ensures that"}\dg{done}
}

Throughout this paper, we use $G(n,p)$ to denote the Erdős–Rényi random graph with vertex set $[n]$ and edge probability $p$.
Recent progress in the study of random \( F \)-graphs enables us to extend this relation to more complex substructures, by analyzing the minimum number of monochromatic \( F \)-subgraphs over all bipartitions of Erdős–Rényi random graphs, where \( F \) is a strictly 1-balanced graph~\cite{burghart2024sharp}.
\begin{theorem}\label{thm-convergence}
 Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \geq 2$ vertices. For $c>0$, let $p= cn^{-1/\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. Let $\Min(F,n,p)$ be the minimum number of monochromatic \( F \)-subgraphs over all bipartitions of the Erdős–Rényi random graph $G(n,p)$. Let $\Max(F,n,q)$ be the size of max-cut in the random $F$-graph $R_F(n,q)$. Then, there exists $m(F,p)$ such that
 \begin{align*}
 \frac{\Min(F,n,p)}{n}\xlongrightarrow{\text{p}}m(F,p),
 \end{align*}
 and
 \begin{align*}
 \frac{\Max(F,n,q)}{n}\xlongrightarrow{\text{p}}\frac{c^s}{\aut(F)}-m(F,p).
 \end{align*}
\end{theorem}

We further illustrate how this minimum evolves as the host graph becomes denser, and we estimate its limiting behavior as the number of vertices in $F$ tends to infinity.
For any integer \( k \geq 0 \), any positive function \( f = f(a_1, \ldots, a_k) \), and any function \( g = g(a_1, \ldots, a_k) \), we write \( g = o_{a_1,\ldots,a_k}(f) \) if
\[
\limsup_{a_k \to \infty} \cdots \limsup_{a_1 \to \infty} \left| \frac{g}{f} \right| = 0;
\]
we write \( g = O_{a_1,\ldots,a_k}(f) \) if there exists a positive constant \( M \) such that
\[
\limsup_{a_k \to \infty} \cdots \limsup_{a_1 \to \infty} \left| \frac{g}{f} \right| \leq M.
\]
\begin{theorem}\label{thm-limit}
 With the notion in \Cref{thm-convergence}, for sufficently large $r$,
 \begin{align*}
 m(F,p)=(1+o_r(1))\sqrt{\frac{\log2}{2^{r-2}\aut(F)}c^s }+\frac{c^s}{2^{r-1}\aut(F)}+o_{c^s}(c^{s/2}).
 \end{align*}
\end{theorem}

A key result from \cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $R_F(n,q)$:
\begin{proposition}[{\cite[Theorem 1.6]{burghart2024sharp}}]\label{thm-bur-1.6}
Let $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \geq 2$ vertices. Then there exist constants $\delta, \varepsilon > 0$ such that the following holds. For any sequences satisfying $p \leq n^{-1/\rd_1(F) + \varepsilon}$ and $q \leq (1 - n^{-\delta})p^{s}$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \whp, there exists a permutation $\tau = \tau(G, R) : [n] \to [n]$ for which every $F(v_1, \ldots, v_r) \in E(R)$ satisfies $F(\tau(v_1), \ldots, \tau(v_r)) \subseteq G$.

\begin{corollary}\label{cor-erg-hg}
Let $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \geq 2$ vertices. For $c > 0$, let $p = c n^{-1/\rd_1(F)}$ and $q = p^s = c^s n^{1-r}$. Then there exists $\delta > 0$ such that the following holds. For any $t > 0$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \whp, there exists a permutation $\tau = \tau(G, R) : [n] \to [n]$ for which the number of distinct ordered copies $F(v_1, \ldots, v_r)$ such that exactly one of the events $F(v_1, \ldots, v_r) \in E(R)$ or $F(\tau(v_1), \ldots, \tau(v_r)) \subseteq G$ holds is at most $tn$.
\end{corollary}
It is worth noting that the definition of a strictly 1-balanced graph ensures that, for $G$ defined as in \Cref{cor-erg-hg}, the number of pairs of distinct ordered copies that share at least one edge is limited. Specifically, cases like \Cref{exa_1} occur only rarely.
 This property is essential for the validity of \Cref{thm-bur-1.6} and \Cref{cor-erg-hg}.

\begin{theorem}\label{thm-convergence}
    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that
    \begin{align*}
        \frac{{\rm MinM}(\G(n,p),F)}{n}
        \to m(F,c),
    \end{align*}
\whp~ as $n\to\infty$.
\end{theorem}

\ignore{
\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\dg{good point. Done}
}

Our proof approach is based on similar results
for the contiguous model $\mathbb{H}(n,q), q=p^s$ which are already known in the literature. Specifically, the existence of $m(F,c)$ such that
\begin{align*}
\frac{{\rm MinM}(\mathbb{H}(n,q),F)}{n}
\to m(F,c),
\end{align*}
is already known~\cite{shabanov2021maximum} based on the combinatorial interpolation technique introduced in~\cite{bayati2010combinatorial} and used
for establishing the existence of
such limits.
For similar contiguity reasons
we also have that
$m(F,c)=0$ when $c$ is
sufficiently small positive
constant $c$, and $m(F,c)>0$ when $c$ is sufficiently large.
The former claim  is
obtained by choosing $c$ small enough so that
the  random graph $\mathbb{H}(n,q)$ does not contain
a giant component. In this case the
locally tree-like
structure of the graph allows for cutting nearly
every $F$-edge  of the graph. Conversely, when
$c$ is sufficiently large, the
non-existence of near
perfect bipartition follows from a simple union
bound. We regard these observations as folklore
and will not provide a formal verification of
these claims.

We conjecture that the value
${\rm MinM}(\mathbb{G}(n,p),F){n}$ also undergoes a different type of phase transition. Specifically, we conjecture the existence of $c^*$ which
depends on $F$ only such that this value is $0$ w.h.p. as $n\to\infty$
when $c<c^*$, and is positive  when $c>c^*$ also w.h.p.
The basis for this conjecture is a similar conjecture for random K-SAT (and several other related models such as proper coloring
of a random graph), which was proven for large $K$~\cite{ding2022proof}, but is still open for general $K$.

Our next result concerns obtaining explicit
asymptotic limit
values when the size of the host graph $F$ grows.
Introduce a short-hand notation
\begin{align*}
\kappa(F,c,s,r)={c^s\over 2^{r-1}{\rm aut}(F)}.
\end{align*}

\begin{theorem}\label{thm-limit}
   Under the assumption of \Cref{thm-convergence},
\begin{align*}
       m(F,c)=\kappa(F,c,s,r)+
    (1+o_r(1))\sqrt{(2\log 2)\kappa(F,c,s,r)}
       +o_{c}(c^{s/2}).
   \end{align*}
\end{theorem}
Namely, we obtain explicit limit values when both the number of nodes of the host graph $F$ and the leading coefficient $c$ of the random graph
parameter diverge to infinity. The proof inspiration
is based on large-$p$
approximation technique
for $p$-spin glass models employed recently
in~\cite{gamarnik2025shattering}, and porting these
results to sparse random graphs as was done
in~\cite{dms2017}
and~\cite{sen2018optimization}. The latter two works relate $p$-spin models to sparse graphs using the Lindeberg's interpolation method.
Instead, we develop relevant asymptotics directly for sparse graphs using the second
moment method employed
in~\cite{gamarnik2025shattering}
for the mean field $p$-spin model.

\paragraph{Organization} The remainder of this paper is organized as follows. In \Cref{sec-hypergraph-coupling}, we introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph, and prove \Cref{thm-convergence}. In \Cref{sec-spinglass}, we reduce the study of \( m(F,p) \) to the analysis of the maximum of a family of Gaussian random variables. Section~\ref{sec-maximizer}  provides auxiliary results that en route to deriving this maximum. Finally, we prove \Cref{thm-limit} in \Cref{sec-proof}.

To prove Theorem~\ref{thm-convergence}, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\frac{{\rm MinM}(\mathbb{H}(n,q),F)}{n}\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of~\cite{bayati2010combinatorial}, as used in~\cite{shabanov2021maximum}). Then, in \Cref{sec-hypergraph-coupling}, the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to “prove \Cref{thm-convergence}.”

To prove the main theorem, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\frac{{\rm MinM}(\mathbb{H}(n,q),F)}{n}\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of Bayati et al., \emph{Combinatorial Interpolation and the Cavity Method} (2010), as used in Shabanov, \emph{Maximum matchings in random graphs via interpolation} (2021)). Then, later the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to complete the proof of the main theorem.

\label{thm-convergence}
    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that
    \begin{align*}
        \frac{{\rm MinM}(\mathbb{G}(n,p),F)}{n}
        \to m(F,c),
    \end{align*}
\whp~ as $n\to\infty$.,

Existential–Universal
Asymptotic or Limit

Question: Let \(F\) be a strictly 1-balanced graph with \(s>0\) edges and \(r\ge 2\) vertices, where \(\mathrm d_1(H)=|E(H)|/(|V(H)|-1)\) and “strictly 1-balanced” means \(\mathrm d_1(F')<\mathrm d_1(F)\) for every proper subgraph \(F'\subsetneq F\). Fix \(c>0\), and define \(p=c\,n^{-1/\mathrm d_1(F)}\). Let \(\mathbb G(n,p)\) be the Erdős–Rényi random graph on \([n]\), and let \({\rm MinM}(\mathbb G(n,p),F)\) denote the minimum, over all bipartitions \([n]=V_1\cup V_2\), of the number of copies of \(F\) in \(\mathbb G(n,p)\) whose vertices lie entirely in \(V_1\) or entirely in \(V_2\) (that is, the minimum number of monochromatic copies of \(F\)). Which statement holds?

There exists a constant \(m(F,c)\) such that, with high probability as \(n\to\infty\),
\[
\frac{{\rm MinM}(\mathbb G(n,p),F)}{n}\to m(F,c).
\]

Choices:
B.
There exists a constant \(m(F,c)\) such that, with high probability as \(n\to\infty\),
\[
\frac{{\rm MinM}(\mathbb G(n,p),F)}{n}\to m(F,c)
\]
uniformly for all fixed \(c>0\), with the same limiting constant independent of \(c\).
C.
There exists a constant \(m(F,c)\) and a sequence \(n_k\to\infty\) such that, with high probability along the subsequence \(n_k\),
\[
\frac{{\rm MinM}(\mathbb G(n_k,p),F)}{n_k}\to m(F,c).
\]
D.
There exists a constant \(m(F,c)\) such that, for every fixed \(c>0\), almost surely as \(n\to\infty\),
\[
\frac{{\rm MinM}(\mathbb G(n,p),F)}{n}\to m(F,c).
\]
E.
There exists a constant \(m(F,c)\) such that, with high probability as \(n\to\infty\),
\[
\frac{{\rm MinM}(\mathbb G(n,p),F)}{n^{r-1}}\to m(F,c).
\]

dependence of the limit on the parameter c

full-sequence convergence whp weakened to subsequential whp convergence

mode of convergence strengthened from with high probability to almost sure

linear normalization in n coming from the contiguous model replaced by n^{r-1}

The stem gives definitions and scaling but does not explicitly or implicitly reveal that the correct conclusion is convergence with high probability to a constant. The answer is not leaked by wording in the prompt itself.

The item is close to a theorem-identification question: the correct option appears to match a precise known limiting statement, while other options tweak mode of convergence or hypotheses. It is not a pure restatement, but it is only a mild reformulation rather than a deeply non-tautological problem.

Some reasoning is needed to distinguish high probability from in probability or almost sure convergence, and to judge the necessity of strict 1-balance and the coupling claim. However, the question primarily tests recognition of the exact theorem statement rather than substantial independent derivation.

The distractors are strong: one is a weaker true-looking alternative, one improperly strengthens the convergence mode, one overextends the hypothesis class, and one adds a plausible but overly strong coupling conclusion. These reflect realistic mathematical failure modes.

A solid MCQ with little answer leakage and strong distractors, but it leans more toward theorem recall/statement recognition than genuine generative reasoning.


Question: Let \(F\) be a strictly 1-balanced graph with \(s>0\) edges and \(r\ge 2\) vertices, where \(\mathrm d_1(H)=|E(H)|/(|V(H)|-1)\) and “strictly 1-balanced” means \(\mathrm d_1(F')<\mathrm d_1(F)\) for every proper subgraph \(F'\subsetneq F\). Fix \(c>0\), and define \(p=c\,n^{-1/\mathrm d_1(F)}\). Let \(\mathbb G(n,p)\) be the Erdős–Rényi random graph on \([n]\), and let \({\rm MinM}(\mathbb G(n,p),F)\) denote the minimum, over all bipartitions \([n]=V_1\cup V_2\), of the number of copies of \(F\) in \(\mathbb G(n,p)\) whose vertices lie entirely in \(V_1\) or entirely in \(V_2\) (that is, the minimum number of monochromatic copies of \(F\)). Which statement holds?

Choices:
B.
There exists a constant \(m(F,c)\) such that, with high probability as \(n\to\infty\),
\[
\frac{{\rm MinM}(\mathbb G(n,p),F)}{n}\to m(F,c)
\]
uniformly for all fixed \(c>0\), with the same limiting constant independent of \(c\).
C.
There exists a constant \(m(F,c)\) and a sequence \(n_k\to\infty\) such that, with high probability along the subsequence \(n_k\),
\[
\frac{{\rm MinM}(\mathbb G(n_k,p),F)}{n_k}\to m(F,c).
\]
D.
There exists a constant \(m(F,c)\) such that, for every fixed \(c>0\), almost surely as \(n\to\infty\),
\[
\frac{{\rm MinM}(\mathbb G(n,p),F)}{n}\to m(F,c).
\]
E.
There exists a constant \(m(F,c)\) such that, with high probability as \(n\to\infty\),
\[
\frac{{\rm MinM}(\mathbb G(n,p),F)}{n^{r-1}}\to m(F,c).
\]

The stem defines the setup and notation but does not explicitly state the limiting conclusion. The correct answer is not leaked directly; one must distinguish among several nearby asymptotic formulations.

The item is close to a theorem-identification question: one option is essentially the exact theorem statement, while the others are quantifier/convergence/normalization variants. So it is not fully tautological, but it is still largely a restatement-discrimination task rather than a fresh conclusion derived from premises.

There is some reasoning pressure because the distractors differ in subtle but meaningful ways (dependence on c, subsequence vs full sequence, whp vs almost sure, and scaling by n vs n^{r-1}). However, the problem mainly tests recognition of the correct theorem-strength statement rather than substantial generative mathematical reasoning.

The distractors are strong: they are plausible, mathematically distinct, and target common failure modes such as overstrengthening convergence, weakening to subsequences, removing parameter dependence, or using an incorrect normalization.

A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it leans more toward recall/statement recognition than genuine generative reasoning.