\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 32, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/32\hfil Existence of multiple solutions]
{Existence of multiple solutions for a nonlinearly
 perturbed elliptic parabolic \\ system in $\mathbb{R}^2$}

\author[M. Ishiwata, T. Ogawa, F. Takahashi\hfil EJDE-2009/32\hfilneg]
{Michinori Ishiwata, Takayoshi Ogawa, Futoshi Takahashi}% in alphabetical order

\address{Michinori Ishiwata \newline
Common Subject Division\\
 Muroran Institute of Technology,
 Muroran 050-8585, Japan}
\email{ishiwata@mmm.muroran-it.ac.jp}

\address{Takayoshi Ogawa \newline
 Mathematical Institute,  Tohoku  University \\
 Sendai  980-8578, Japan}
\email{ogawa@math.tohoku.ac.jp}

\address{Futoshi Takahashi \newline
 Graduate School of Science,  Osaka City University \\
 Osaka  558-8585, Japan}
\email{futoshi@sci.osaka-cu.ac.jp}

\thanks{Submitted August 22, 2008. Published February 16, 2009.}
\subjclass[2000]{35K15, 35K55, 35Q60, 78A35}
\keywords{Multiple existence; elliptic-parabolic system; \hfill\break\indent
unconditional uniqueness}

\begin{abstract}
 We consider the following nonlinearly perturbed version
 of the elliptic-parabolic system of Keller-Segel type:
 \begin{gather*}
 \partial_tu -  \Delta  u+ \nabla \cdot(u \nabla v)=0,\quad
 t>0,\; x\in\mathbb{R}^2, \\
 -\Delta v+v-v^p=u,\quad t>0,\; x\in\mathbb{R}^2,\\
 u(0,x) =u_0(x)\ge 0,\quad x\in\mathbb{R}^2,
 \end{gather*} 
 where $1<p<\infty$.
 It has already been shown that the system admits a positive solution
 for a small nonnegative initial data in
 $L^1(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)$
 which corresponds to the local minimum of the associated energy
 functional to the elliptic part of the system.
 In this paper, we show that for a radially symmetric nonnegative
 initial data, there exists {\it another} positive solution which
 corresponds to the critical point of mountain-pass type.
 The $v$-component of the solution bifurcates from the unique 
positive radially symmetric solution of $-\Delta w + w = w^p$ in
 $\mathbb{R}^2$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction} \label{sec01}
 
In this paper, we consider the nonlinearly perturbed version of the
elliptic-parabolic system modeling chemotaxis:
\begin{equation}
 \begin{gathered}
    \partial_tu -  \Delta  u+ \nabla\cdot (u \nabla v)=0,\quad
 t>0,\; x\in\mathbb{R}^2, \\
    -\Delta v+v-v^p=u,\quad t>0,\; x\in\mathbb{R}^2,\\
    u(0,x) =u_0(x)\ge 0, \quad x\in\mathbb{R}^2.
 \end{gathered}   \label{eqn;pKS}
\end{equation}

In the context of mathematical biology, Keller and Segel
\cite{K-S} introduced a parabolic system, called  the Keller-Segel
system, as a mathematical model of 
chemotactic 
collapse (see also
Herrero-Vel\'azquez \cite{H-V1}, \cite{H-V2}, Nagai \cite{Ng},
\cite{Ng3}, Biler \cite{Bl}, Nagai-Senba-Yoshida \cite{N-S-Y},
Nagai-Senba-Suzuki \cite{N-S-S} and Senba-Suzuki \cite{S-S}). When
the diffusion of the chemical 
substance 
is much slower than that
of chemotaxis ameba, 
then the dynamics of chemotaxis is described
by the following simplified system:
\begin{equation}
\begin{gathered}
\partial_tu -  \Delta  u+ \nabla\cdot (u \nabla v)=0,\quad
t>0,\; x\in\mathbb{R}^2, \\
-\Delta v+v=u, \quad t>0,\; x\in\mathbb{R}^2,\\
u(0,x) =u_0(x)\ge 0,\quad x\in\mathbb{R}^2.
\end{gathered} \label{eqn;KS-N}
\end{equation}

It is well known that the existence of the finite time blow up of 
the 
solution for (\ref{eqn;KS-N}) which corresponds to the
concentration of ameba (Herrero-Vel\'azquez \cite{H-V1},
\cite{H-V2}, Nagai \cite{Ng}).

Chen-Zhong \cite{C-Zn} introduced a perturbed system of (\ref{eqn;KS-N}):
For $p>1$,
\begin{equation}
\begin{gathered}
\partial_tu -  \Delta  u+ \nabla\cdot (u \nabla v)=0,\quad
 t>0,\; x\in\mathbb{R}^2, \\
-\Delta v+v+v^p=u,\quad t>0,\; x\in\mathbb{R}^2,\\
u(0,x) =u_0(x)\ge 0,\quad x\in\mathbb{R}^2.
\end{gathered}\label{eqn;KS-p}
\end{equation}
This system is considered as a model of the chemotaxis
with a nonlinear diffusion for the chemical substance.
It has been proved that the solution of (\ref{eqn;KS-p}) has a similar behavior
to the original system (\ref{eqn;KS-N}).
In fact, one can show the local existence theory and finite time blow up with
mass concentration phenomena as is shown for (\ref{eqn;KS-N}),
see Chen-Zhong  \cite{C-Zn} and  Kurokiba-Suzuki \cite{K-S2}.

Note that the nonlinear term $v^p$ in the second equation in
\eqref{eqn;pKS} has a different sign compared to (\ref{eqn;KS-p}).
According to this difference, the behavior of the solution for
\eqref{eqn;pKS} is much different from  the one 
for 
(\ref{eqn;KS-p}). Indeed, the nonhomogeneous elliptic problem
corresponding to the second equation of \eqref{eqn;pKS}:
\begin{equation}
     -\Delta v+v-v^p =f,\quad x\in\mathbb{R}^2
\label{eqn;p-NE}
\end{equation}
admits at least two positive solutions when $f$ is a sufficiently
small nonnegative nontrivial function in $H^{-1}(\mathbb{R}^2)$,
while
\begin{equation*}
     -\Delta v+v+v^p =f,\quad x\in\mathbb{R}^2
\end{equation*}
has only one solution. Moreover, it is also known that if the
external force $f$ is large in $H^{-1}$ sense, then there is no
positive solution for the equation (\ref{eqn;p-NE}). Hence it is
an interesting question whether the finite time blow up of the
solution may occur in the case \eqref{eqn;pKS}, or more 
primitively, whether the time local solution exists properly and
the system is well posed in some sense or not. In this point, the
structure of the time dependent positive solutions of
\eqref{eqn;pKS} seems to be very much different from that of the
original system (\ref{eqn;KS-N}) or the perturbed system
(\ref{eqn;KS-p}).

In this paper, we shall consider solutions of \eqref{eqn;pKS} in the
following sense:
\begin{align*}
 &u \in C([0,T);L^2(\mathbb{R}^2))\cap C^1((0,T);L^2(\mathbb{R}^2))
     \cap C((0,T);\dot{H}^2(\mathbb{R}^2)), \\
 & v \in C((0,T);H^1(\mathbb{R}^2))\cap C((0,T);W^{2,2}(\mathbb{R}^2))
\end{align*}
for some $T >0$.

Recently Kurokiba-Ogawa-Takahashi \cite{K-Og-T} proved that, for a
small nonnegative initial data, there exists a solution for
\eqref{eqn;pKS} which is, in  a sense, ``small'' one. On the other
hand, as is mentioned above, the perturbed nonlinear elliptic
equation (\ref{eqn;p-NE}) 
admits at least two positive solutions for small and nonnegative $f \not\equiv 0$.
Therefore it is natural to ask whether the time dependent equation \eqref{eqn;pKS}
also has a second positive solution.
The main issue of this paper is to show the existence of
{\it  two positive time dependent solutions of \eqref{eqn;pKS}}
under the radially symmetric setting.

\begin{theorem}[Multiple existence] \label{thm;second-sol}
Let $1<p<\infty$. Then there exists a constant $C_{**}>0$ such that,
if the radially symmetric nonnegative initial data
$u_0\in L^1 \cap L^2(\mathbb{R}^2)$  satisfies
$$
   \|u_0\|_2\le C_{**},
$$
then there exist two positive radial pair of
solutions $(u_1(t),v_1(t))$ and $(u_2(t),v_2(t))$ for  \eqref{eqn;pKS}.
One of them is different from the solution obtained in \cite{K-Og-T}.
\end{theorem}

Note that the solution obtained in \cite{K-Og-T} exists globally in time
if in addition $\| u_0 \|_1$ is sufficiently small.

The main idea to construct the time dependent solutions heavily relies
on the variational structure of
the elliptic part of the system.
The $v$-component of the solution obtained in \cite{K-Og-T} corresponds
to the
solution of (\ref{eqn;p-NE}) 
bifurcating 
from the trivial solution with $f=0$.
On the other hand, it has been known that
the problem (\ref{eqn;p-NE}) with $f=0$ has a unique positive
solution $w$ (see Berestycki-Lions \cite{BeLi}, Gidas-Ni-Nirenberg \cite{GNN} and Kwong \cite{Kw01}).
This solution is obtained as a mountain pass critical point of the energy functional
\[
I_0(v)=\frac{1}{2}\int_{\mathbb{R}^2}|\nabla
 v|^2dx+\frac{1}{2}\int_{\mathbb{R}^2}|v|^2dx
-\frac{1}{p+1}\int_{\mathbb{R}^2} |v|^{p+1}dx.
\]
If the second variation of $I_0$ at $w$
is nondegenerate and if $f$ is small,
then we may construct the solution $v$ of (\ref{eqn;p-NE})
bifurcating 
from the mountain pass solution $w$.
This is not always possible,
since the kernel of the Hessian of $I_0$ at $w$ is nontrivial.
If we restrict the
class of initial data, however,
there is a possibility of constructing
the second local-in-time solution of \eqref{eqn;pKS}.
In this paper, we shall show that this is indeed possible under
the radially symmetric setting.

Also it should be noted that our problem is related to the
unconditional uniqueness problem in the general nonlinear evolution equations.
Let $X$ be a Banach space.
If an initial value problem admits the unique solution in the class $C([0,T);X)$
with initial data in $X$,
then we call the {\it unconditional uniqueness} holds for this problem.
If the class of the solution is reasonably restricted,
the unconditional uniqueness is expected to hold for the well-posed problem.
For our problem \eqref{eqn;pKS}, however,
there is no possibility to have the unconditional uniqueness
by restricting the regularity.
Namely, 
no matter how the class of the solution is restricted from the regularity point of view,
at least two solutions for \eqref{eqn;pKS} do exist.
Only the variational characterization of the second component $v$
distinguishes two solutions
and the uniqueness class is not definable by means of function spaces.
In this sense, the unconditional uniqueness never holds for \eqref{eqn;pKS}.
This kind of phenomena may occur for a general nonlinear problem.
In our particular setting, there exists at least two time dependent solutions
and are uniquely continued in time
each other under the variational restriction.


We use the following notation.  The Lebesgue space $L^p(\mathbb{R}^2)$ is
denoted by $L^p$ with $1\le p\le \infty$ with the norm $\|\cdot \|_p$.
For $k=1,2,\cdots$ 
and for 
$1\le p\le \infty$, let
 $W^{k,p}=W^{k,p}(\mathbb{R}^2)$ be the Sobolev space with the
norm $\|f\|_p+\|\nabla f\|_p$.  We frequently use $H^1=W^{1,2}(\mathbb{R}^2)$, and
$L^2_r$ and
$H^1_r$ denote the radially symmetric subspaces of $L^2$ and $H^1$,
respectively.
$(H^1_r)^*$ denotes 
the dual space of $H^1_r$.  
For a Banach  space $X$,  $B_{\delta,X}$  stands for the open ball in $X$ with the
radius $\delta>0$ and 
the center $0$. 
The constant $C$ may vary from line to line.

\section{Variational Structure of the Lagrangian Functional}
\label{sec03}

The existence of multiple positive solutions for the semilinear
elliptic equation
\begin{equation}
-\Delta v+v=v^p+f , \quad x\in \mathbb{R}^2 \label{eqn;Pf}
\end{equation}
is known for small nonnegative external 
forces 
$f \not\equiv 0$ in $H^{-1}$,
see e.g. Zhu \cite{Zh} and Cao-Zhou \cite{C-Z}.
According to their results,
there exists a solution of (\ref{eqn;Pf}) for small $f$
(in the $H^{-1}$ sense) 
which is not a local minimizer of the functional
$I_f$ defined by
\begin{equation*} 
 I_f(v)=\frac{1}{2}\int_{\mathbb{R}^2} |\nabla v|^2dx
 +\frac{1}{2}\int_{\mathbb{R}^2}
|v|^2dx-\frac{1}{p+1}\int_{\mathbb{R}^2} |v|^{p+1}
      -\int_{\mathbb{R}^2}f vdx,\quad v \in H^1(\mathbb{R}^2).
\end{equation*}

In this section, we give some analysis on the dependence of
this non-minimal solution with respect to $f$, 
namely, we show some refined results 
compared to those 
of
Zhu \cite{Zh} and Cao-Zhou \cite{C-Z} from a bifurcation
theoretical point of view.

As is mentioned in the introduction,
the nonlinear elliptic problem  (\ref{eqn;Pf})
with $f\equiv 0$,
\begin{equation}
-\Delta v+v=v^p, \quad x\in \mathbb{R}^2, \label{eqn;P0}
\end{equation}
has 
a radially symmetric positive unique solution $w$ \cite{BeLi,GNN,Kw01}. 
This solution is obtained as a critical point of the
variational functional $I_0$
by the well known mountain pass lemma in $H^1$.
Note that the Hessian operator
of $I_0$ at $u\in H^1$ is realized by $L_u:=-\Delta+1-p|u|^{p-1}$,
which is an operator from $H^1$ to $H^{-1}$.
As for the kernel of the linearized operator $L_w$ at $w$,
the following is well-known (see e.g. \cite{CdP01, D01, Fei01, NiTa01}).

\begin{proposition}[Kernel of the linearized operator]\label{prop:main}
For the radially symmetric positive unique 
solution $w$ to {\rm (\ref{eqn;P0})}, 
$\ker L_w$, the kernel of the operator
$L_w = -\Delta+1-p|w|^{p-1}$, is spanned by $\partial_{x_1} w$
and $\partial_{x_2} w$.
In particular, $\ker L_w \cap H^1_r = \{ 0 \}$.
\end{proposition}

According to Proposition \ref{prop:main},
we may construct a solution branch of
the nonminimal solution of  (\ref{eqn;Pf}) with the aid of
the implicit function theorem if we restrict our problem 
to the class of
radially symmetric functions.

\begin{proposition}\label{prop:conti}
There exists $\delta>0$ and $h\in C(B_{\delta,(H^1_r)^*};H^1_r)$
such that
$h(f)$ is a critical point of $I_f$
which is not a local minimum
for $f \in B_{\delta,(H^1_r)^*}$
with $h(0)=w$.
Moreover, $h$ is a Lipschitz continuous
 mapping in $B_{\delta,(H^1_r)^*}$, namely, there exists $C>0$ such that
\begin{equation}
\|h(f_1)-h(f_2)\|_{H^1}<C\|f_1-f_2\|_{(H_r^1)^*},\quad\forall f_1,\ f_2\in
 B_{\delta,(H^1_r)^*}.
\label{eq:con04}
\end{equation}
If $f\geq 0$, then $h(f)\geq 0$ holds.
\end{proposition}

\begin{proof}
We employ the implicit function theorem for
$$
g: (H^1_r)^* \times  H^1_r \ni (f,u) \mapsto g(f,u)
:= (dI_{f})_u\in (H^1_r)^*
$$
around $(0,w) \in (H^1_r)^* \times  H^1_r$.
Hereafter the functional derivatives of $g$
with respect to $f\in (H^1_r)^*$ and $u\in H^1_r$ are denoted by
$D_1g$ and $D_2g$, respectively.

Let $\varphi\in H^1_r$, $\eta\in (H^1_r)^*$ and $v\in H^1_r$.
Then it is easy to see that
\begin{equation*}
(D_1g)_{(f,u)}(\eta)\varphi =
 \frac{d}{dt}g(f+t\eta,u)\varphi\Big|_{t=0}
  =\int_{\mathbb{R}^2} \eta\varphi.
\end{equation*}
Hence $(D_1g)_{(f,u)}$: $(H^1_r)^* \to (H^1_r)^*$ is an identity mapping.
Therefore
\[
D_1g:\,(H^1_r)^*\times H^1_r\ni(f,u)\mapsto (D_1g)_{(f,u)}\in
 L((H_r^1)^*;(H_r^1)^*)
\]
is a constant map, especially continuous,
where $L(X,Y)$ denotes the space of bounded linear operators
between Banach spaces $X$ and $Y$. Similarly,
\[
    (D_2g)_{(f,u)}(v)\varphi
=\frac{d}{dt}g(f,u+tv)\varphi\Big|_{t=0}
=\int_{\mathbb{R}^2}L_uv\varphi,
\]
i.e.,
$(D_2g)_{(f,u)}: H^1_r \to (H^1_r)^*$ is given by
\begin{equation}
(D_2g)_{(f,u)}(v)=L_uv, \quad v \in H^1_r.
\label{eqpr:con02}
\end{equation}
In particular, 
\begin{align*}
D_2g:\,(H^1_r)^*\times H^1_r\ni(f,u)\mapsto (D_2g)_{(f,u)}
\in L(H_r^1;(H_r^1)^*)
\end{align*}
is also a continuous map; thus $g\in C^1((H^1_r)^*\times H^1_r;(H^1_r)^*)$.
Note that (\ref{eqpr:con02}) implies
$(D_2g)_{(0,w)}=L_w$
and $L_w$ restricted to $H^1_r$ should have a trivial kernel
by virtue of Proposition \ref{prop:main}.
Therefore by the implicit function theorem (see e.g. (\cite[Theorem 5.9]{Lang1})),
there exist $\delta>0$ and $h$:
 $B_{\delta,(H_r^1)^*}\to H_r^1$ such that $g(f,h(f))=0$ in $(H_r^1)^*$
for any $f\in
 B_{\delta,(H_r^1)^*}$ and $h(0)=w$. 
The latter implies that $h(f)$ is a critical
 point which is not a local minimum of $I_f$. 
Moreover, by the symmetric criticality principle of Palais \cite{Pa},
$h(f)$ is a critical point of $I_f$ not only on $H_r^1$ but also on
$H^1$. Therefore the first part of the 
proposition follows.

Also by the implicit function theorem,
we have $h\in C^1(B_{\delta,(H_r^1)^*};H_r^1)$,
thus there exists a constant $C>0$ such that
\[
\|(dh)_f\|_{L((H_r^1)^*;H_r^1)}<C\ \text{ for\ } f\in B_{\delta,(H_r^1)^*}
\]
if $\delta>0$ is sufficiently small.
Then for $f_1$, $f_2\in B_{\delta,(H_r^1)^*}$,
\begin{align*}
\|h(f_1)-h(f_2)\|_{H^1}
&\leq
\Big\|\int_0^1dt\Big(
\frac{d}{dt}h(tf_2+(1-t)f_1)\Big)\Big\|_{H^1} \\
&\leq
\int_0^1dt\|(dh)_{tf_2+(1-t)f_1}\|_{L((H^1_r)^*;H^1_r)}
\|f_2-f_1\|_{(H^1_r)^*} \\
&\leq C\|f_2-f_1\|_{(H_r^1)^*},
\end{align*}
hence (\ref{eq:con04}) follows.
The nonnegativity of $h(f)$ for $f\geq 0$ follows from the standard
argument 
as in e.g. \cite{C-Z}.
\end{proof}

The following corollary follows immediately  from Proposition
\ref{prop:conti}.

\begin{corollary}\label{cor:conti}
There exists $\rho>0$ such that the conclusion of Proposition
{\rm\ref{prop:conti}} holds
when $B_{\delta;(H_r^1)^*}$ and $(H_r^1)^*$ are replaced
by $B_{\rho,L^2_r}$ and $L_r^2$, respectively.
\end{corollary}

\section{Proof of Main Theorem}
\label{sec04}

In this section, we give the proof of Theorem
\ref{thm;second-sol}.
Let $1<p<\infty$.
We choose $M$ with $M<\rho$ where $\rho$ is the number 
obtained in
Corollary \ref{cor:conti}.
We shall construct a solution of \eqref{eqn;pKS} in the complete
metric space
$$
   X_{T,M}=\big \{  \phi\in
                C([0,T);L_r^{2})
                    \cap L^{2}(0,T;\dot H_r^1)
             ; \ \phi\ge 0, \
              |\!|\!| \phi|\!|\!| _X\le M\big \}
$$
with the metric $d(\phi, \psi)\equiv
\sup_{t\in [0,T]}|\!|\!| \phi-\psi|\!|\!| _X$,
where
$$
   |\!|\!| \phi|\!|\!| _X
    \equiv\Big( \sup_{\tau\in[0,T)} \|\phi(\tau)\|_{2}^2
         +\int_0^T \|\nabla\phi(\tau)\|_2^{2} d\tau\Big)^{1/2}
$$
and $T>0$ is chosen to be small later.

For a nonnegative function
$a\in L^2_r$, we define a map
\begin{equation*}\label{eqn;sol-op}
  \Phi_a:X_{T,M} \ni f \mapsto u\in X_{T,M},
\end{equation*}
where $u$ is the solution of the following system:
\begin{equation}
    \begin{gathered}
    \partial_tu -  \Delta  u+ \nabla\cdot (u \nabla v)=0,\quad t>0,\; x\in\mathbb{R}^2, \\
    -\Delta v+v=v^p+f,\quad t>0,\; x\in\mathbb{R}^2,\\
    u(0,x) =a(x), \quad x\in\mathbb{R}^2.
    \end{gathered} \label{eqn;DS3}
\end{equation}
Here  we choose the solution  $v(t)$ of the elliptic part of the above
system as $h(f(t))$ where $h$ is a map constructed in Corollary
 \ref{cor:conti}. Note that $\Phi_a$ is well defined by virtue of 
Corollary \ref{cor:conti},
since $\sup_{\tau\in [0,T)}\|f(\tau)\|_2\le |\!|\!| f|\!|\!| _X\leq M<\rho$.

It is also easy to see that Corollary \ref{cor:conti} yields
\begin{equation}
\begin{split}
\sup_{\tau\in [0,T)}\|h(f(\tau))\|_{H^1}
    \leq &
    \sup_{\tau\in[0,T)}\|h(f(\tau))-h(0)\|_{H^1}+\|h(0)\|_{H^1}\\
    \leq &
     C\sup_{\tau\in [0,T)}\|f(\tau)-0\|_2+\|w\|_{H^1} \\
    \leq & CM+\|w\|_{H^1}=:\sigma,
\end{split}
\label{eq:hbd02}
\end{equation}
where $w$ is the unique, radially symmetric positive 
function satisfying $-\Delta w+w=w^p$ in $\mathbb{R}^2$.
Hereafter for $f$ and $\bar{f}\in X_{T,M}$, we denote $h(f(\tau))$
and $h(\bar{f}(\tau))$ by
$v(\tau)$ and $\bar{v}(\tau)$ (or simply $v$ and $\bar{v}$), respectively.

Our first lemma is as follows.

\begin{lemma}\label{lem:wb}
For any $q \ge 2$, there exists a constant $C_q>0$ such that
\begin{gather}
\|v(\tau)\|_{W^{1,q}}<C_q , \label{eq:wb02}
\\
\|v(\tau)-\bar{v}(\tau)\|_{W^{1,q}}<C_q\|f(\tau)-\bar{f}(\tau)\|_{2}
\label{eq:wb04}
\end{gather}
holds for any $f$, $\bar{f}\in X_{T,M}$ and for any $\tau\in [0,T)$.
\end{lemma}

\begin{proof}
Recall that $v$ satisfies $-\Delta v+v=v^p+f$ in $\mathbb{R}^2$.
Now for a given  $g\in L^2$,
the unique solution of $-\Delta \tilde{v}+\tilde{v}=g$ in
$\mathbb{R}^2$ satisfies
\begin{equation}
\|\tilde{v}\|_{W^{1,q}}\leq A_q\|g\|_2
\label{eqpr:pre102}
\end{equation}
for some constant $A_q>0$ when $q\geq 2$.
Thus the Sobolev embedding $H^1\hookrightarrow L^{2p}$
and (\ref{eq:hbd02}) yields
\begin{equation*}
\|v\|_{W^{1,q}}\leq A_q\|v^p+f\|_2\leq A_q(C\|v\|_{H^1}^{p}+\|f\|_2)\leq
A_q(C \sigma^p+M)=:C_q,
\end{equation*}
hence (\ref{eq:wb02}).

Since $v$ and $\bar{v}$ satisfy
\begin{equation*}
-\Delta (v-\bar{v})+(v-\bar{v})=v^p-\bar{v}^p+f-\bar{f},
\end{equation*}
we have again from (\ref{eqpr:pre102}),
\begin{equation}
\|v-\bar{v}\|_{W^{1,q}}
 \leq A_q\big(\|v^p-\bar{v}^p\|_2+\|f-\bar{f}\|_2\big).
\label{eqpr:wb02}
\end{equation}
Here we note that,
by the Sobolev embedding $H^1\hookrightarrow L^{2p}$, (\ref{eq:hbd02}) and
Corollary \ref{cor:conti},
\begin{equation*}
\|v^p-\bar{v}^p\|_2^2
\leq
C(\|v\|_{2p}^{2(p-1)}+\|\bar{v}\|_{2p}^{2(p-1)})\|v-\bar{v}\|_{2p}^2
\leq C\|v-\bar{v}\|_{H^1}^2\leq C\|f-\bar{f}\|_2^2
\end{equation*}
holds for suitable $C>0$.
Hence this fact together with
(\ref{eqpr:wb02}) yields (\ref{eq:wb04}).
\end{proof}

\begin{lemma}\label{lem:bge}
There exists $C>0$ such that
\begin{eqnarray*}
&&\|\nabla v(\tau)\|_\infty^2\leq C(1+\|\nabla f(\tau)\|_2)
\end{eqnarray*}
for $\tau\in [0,T)$.
\end{lemma}

\begin{proof}
The second equation of (\ref{eqn;DS3}),
the Sobolev embedding $H^1\hookrightarrow L^{2p}$ and
(\ref{eq:hbd02}) lead
\begin{equation*}
\|\Delta v\|_2 \leq \|v\|_2+\|v^p\|_2+\|f\|_2<C
\end{equation*}
for some $C>0$.
Hence by using a version of the Brezis-Gallouet inequality
\cite{B-G}:
\begin{equation*}
\|h\|_\infty^2\leq C\left(
\|h\|_{H^1}^2(1+\|\Delta h\|_2^{1/2})+\|\Delta
h\|_2\right)
\end{equation*}
for all $h\in H^2(\mathbb{R}^2)$,
we have
\begin{equation}
\begin{split}
\|\nabla v\|_\infty^2
&\leq C((\|\Delta v\|_2^2+\|\nabla v\|_2^2)(1+\|\nabla\Delta
 v\|_2^{1/2})+\|\nabla\Delta v\|_2)\\
&\leq C(1+\|\nabla\Delta v\|_2).
\end{split}
\label{eqpr:pre02}
\end{equation}
Note that by Lemma \ref{lem:wb},
the Sobolev embedding $H^1\hookrightarrow L^{2p}$ and (\ref{eq:hbd02}),
\begin{equation}
\|\nabla v^p\|_2^2
=p^2\int_{\mathbb{R}^2}
|v|^{2(p-1)}|\nabla v|^2\leq p^2\|v\|_{2p}^{2(p-1)}\|\nabla
v\|_{2p}^2
<C
\label{eqpr:pre04}
\end{equation}
holds. Then the second equation of (\ref{eqn;DS3}) together with
(\ref{eq:hbd02}) and (\ref{eqpr:pre04}) yields
\[
\|\nabla\Delta v\|_2\leq \|\nabla v\|_2+\|\nabla v^p\|_2+\|\nabla f\|_2
\leq C(1+\|\nabla f\|_2).
\]
Hence combining this relation with (\ref{eqpr:pre02}),
we have the conclusion.
\end{proof}

Using the estimate for $v$ obtained above, we can verify the
following key proposition for the verification of
Theorem \ref{thm;second-sol}.

\begin{proposition}\label{prop:ien}
Let $a$, $\overline{a}\in L^2_r$ be smooth
nonnegative radial functions. Then for some
 $C>0$, we have for the solution operator $\Phi_a$ defined by
\eqref{eqn;sol-op},
\begin{align}
  \big(1-CT^{1/2}(T^{1/2}+M)\big)&|\!|\!| \Phi_a(f)|\!|\!| _{X}^2
      \leq  \|a\|_2^2,   \label{eq:ien02} \\
  \big(1-CT^{1/2}(T^{1/2}+M)\big)&|\!|\!| \Phi_a(f)-\Phi_{\overline{a}}  
   (\bar{f})|\!|\!| _{X}^2 \notag\\
  &\leq  \|a-\overline{a}\|_2^2
     +C|\!|\!| \Phi_a(f)|\!|\!| _{X}^2T^{1/2}
       |\!|\!| f-\bar{f}|\!|\!| _{X}^2 \label{eq:ien04}
\end{align}
for $f$, $\bar{f}\in X_{T,M}$.
\end{proposition}

\begin{proof}
The existence of a smooth solution for the system (\ref{eqn;DS3}) with
a smooth initial
data follows from  the standard theory of evolution equations.
Under the assumption of the proposition, we denote solutions
$\Phi_a(f(\tau))$ and
 $\Phi_{\overline{a}}(\bar{f}(\tau))$ of
 (\ref{eqn;DS3}) by
$u(\tau)$ and $\bar{u}(\tau)$ (or simply $u$
 and $\bar{u}$), respectively. We also denote $h(f(\tau))$
and $h(\bar{f}(\tau))$ by $v(\tau)$ and $\bar{v}(\tau)$ (or simply $v$ and
 $\bar{v}$), respectively.
Now multiplying the first equation of (\ref{eqn;DS3}) by $u=u(\tau)$ and
integrating by parts,
we have
\begin{equation}
\frac{1}{2}\frac{d}{d\tau}\| u(\tau)\|_2^2 +\|\nabla u(\tau)\|_2^2
\le \frac12 \| u(\tau)\|_2^2 \|\nabla  v(\tau) \|_{\infty}^2
                     + \frac12 \|\nabla  u(\tau)\|_2^2.
\label{eqn;engy1}
\end{equation}
Then, the integration of (\ref{eqn;engy1})
from $0$ to $t$ in $\tau$ leads
\begin{align*}
  \| u(t)\|_2^2  +  \int_0^t \|\nabla u(\tau)\|_2^2 d\tau
    \le &  \|a\|_2^2
          +\int_0^t  \|u(\tau)\|_2^2\|\nabla v(\tau)\|_\infty^2d\tau\\
   \le &  \|a\|_2^2+ \sup_{\tau\in [0,T)}\|u(\tau)\|_2^2C
            \int_0^t (1+\|\nabla f(\tau)\|_2)d\tau\\
   \le &  \|a\|_2^2+\sup_{\tau\in [0,T)}\|u(\tau)\|_2^2CT^{1/2}(T^{1/2}+M)
\end{align*}
for $t\in [0,T)$,
here we have used
Lemma \ref{lem:bge} and $\sqrt{\int_0^t\|\nabla f\|_2^2d\tau}\leq M$,
thus (\ref{eq:ien02}).

Next, we consider equations
\begin{gather*}
\partial_tu -  \Delta  u+ \nabla \cdot (u \nabla v) =0,\quad u(0)=a, \\
\partial_t\bar{u} -  \Delta \bar{u}+ \nabla \cdot (\bar{u} \nabla\bar{v})
=0, \quad \bar{u}(0)=\bar{a}.
\end{gather*}
Multiplying $u-\bar{u}$ to the
difference of these equations and integrating by parts,
we see
\begin{equation}
\begin{split}
 \frac12\frac{d }{d\tau} &\| u(\tau)-\bar{u}(\tau) \|_2^2
     +\|\nabla ( u(\tau)-\bar{u}(\tau) )\|_2^2 \\
     = &
       \int_{\mathbb{R}^2} u\nabla (v - \bar{v} )\cdot \nabla (u-\bar{u} )dx
     + \int_{\mathbb{R}^2}  (u  -\bar{u})\nabla\bar{v} \cdot \nabla (u-\bar{u} )dx.
\label{eqpr:m00}
\end{split}
\end{equation}
Then
\begin{equation}
\begin{split}
  \left|\int_{\mathbb{R}^2} u\nabla (v-\bar{v})\nabla (u-\bar{u})dx\right|
   &\leq
    \|u\|_4\|\nabla (v-\overline{v})\|_4\|\nabla (u-\overline{u})\|_2 \\
   &\leq
    \|u\|_4^2\|\nabla (v-\overline{v})\|_4^2
        +\frac{1}{4}\|\nabla (u-\overline{u})\|_2^2\\
   &\leq
    \|u\|_4^2C
       \sup_{\tau\in [0,T)}\|f(\tau)-\overline{f}(\tau)\|_2^2
      +\frac{1}{4}\|\nabla (u-\overline{u})\|_2^2,
\end{split}
\label{eqpr:m02}
\end{equation}
where we have used Lemma \ref{lem:wb}.
Also Lemma \ref{lem:bge} gives
\begin{equation}
\begin{split}
&\left|\int_{\mathbb{R}^2}  (u  -\bar{u})\nabla\bar{v} \cdot
\nabla (u-\bar{u})dx\right|\\
&\leq \|u-\bar{u}\|_2\|\nabla\bar{v}\|_\infty\|\nabla(u-\bar{u})\|_2  \\
&\leq \|u-\bar{u}\|_2^2\|\nabla\bar{v}\|_\infty^2
+\frac{1}{4}\|\nabla(u-\bar{u})\|_2^2\\
&\leq C(1+\|\nabla f\|_2)\sup_{\tau\in [0,T)}\|u(\tau)-\bar{u}(\tau)\|_2^2
+\frac{1}{4}\|\nabla (u-\bar{u})\|_2^2.
\end{split}
\label{eqpr:m04}
\end{equation}
Then plugging (\ref{eqpr:m02}) and (\ref{eqpr:m04}) into
 (\ref{eqpr:m00}) and integrating from $0$ to $t$ in $\tau$, we have
\begin{equation}
\begin{split}
  &\|u(t)-\bar{u}(t)\|_2^2+\int_0^t\|\nabla (u(\tau)-\bar{u}(\tau))\|_2^2d\tau \\
  &\leq
     \|a-\bar{a}\|_2^2
     +2C\int_0^T\|u(\tau)\|_4^2d\tau
      \sup_{\tau\in [0,T)}\|f(\tau)-\overline{f}(\tau)\|_2^2\\
   &\quad +2C\sup_{\tau\in [0,T)}\|u(\tau)-\bar{u}(\tau)\|_2^2\int_0^T
     (1+\|\nabla f(\tau)\|_2)d\tau.
\end{split}
\label{eqpr:m06}
\end{equation}
Here, we recall the Ladyzhenskaya inequality (see e.g., \cite{FT}):
\[
\Big(\int_0^T\|\varphi(\tau)\|_4^4d\tau\Big)^{1/2}
\leq
C \Big( \sup_{\tau\in [0,T)}\|\varphi(\tau)\|_2^2+\int_0^T\|\nabla
\varphi(\tau)\|_2^2d\tau \Big)
\]
for $\varphi\in C([0,T);L^2)\cap L^2(0,T;\dot{H}^1)$.
Then we obtain
\begin{equation}
\int_0^T\|u(\tau)\|_4^2d\tau\leq
 \Big(\int_0^T\|u(\tau)\|_4^4d\tau\Big)^{1/2}
T^{1/2}
\leq
C |\!|\!| u|\!|\!| _X^2T^{1/2}.
\label{eqpr:m08}
\end{equation}
Hence plugging (\ref{eqpr:m08}) 
into (\ref{eqpr:m06}) and noting
$\int_0^T(1+\|\nabla f(\tau)\|_2)d\tau\leq T^{1/2}(T^{1/2}+M)$
again, 
we have 
\begin{equation*}
\begin{split}
\left(1-CT^{1/2}(T^{1/2}+M)\right)
&\sup_{\tau\in
[0,T)}\|u(\tau)-\bar{u}(\tau)\|_2^2
+\int_0^t\|\nabla
(u(\tau)-\bar{u}(\tau))\|_2^2d\tau\\
&\leq
\|a-\bar{a}\|_2^2+
C|\!|\!| u|\!|\!| _X^2T^{1/2}\sup_{\tau\in
[0,T)}\|f(\tau)-\bar{f}(\tau)\|_2^2
\end{split}
\label{eqpr:c08}
\end{equation*}
and (\ref{eq:ien04}) follows.
\end{proof}

Now we are in the position to give the proof of Theorem
\ref{thm;second-sol}.

\begin{proof}

Take any $M<\rho$ where $\rho$ is the
number which 
is obtained in Corollary
\ref{cor:conti}. Then choose $T>0$ so
small that
$\frac{1}{2}\leq 1-CT^{1/2}(T^{1/2}+M)$ and
$CM^2T^{1/2}\leq
\frac{1}{4}$ hold,
where $C$ is the constant in Proposition
\ref{prop:ien}. Let
$u_0\in L_r^2(\mathbb{R}^2)$ be a nonnegative initial data
with
$\|u_0\|_2^2<M^2/2=:C_{**}^2$. Then by using the approximation of
$u_0$
 by a sequence of smooth functions and Proposition \ref{prop:ien},
we can easily verify that
$\Phi_{u_0}$ is a contraction mapping from $X_{T,M}$ to $X_{T,M}$.
Therefore, the Banach fixed point theorem
implies that there
exists a unique solution of $u=\Phi_{u_0}(u)$.
It is obvious that
$(u,v)=(u,h(u))$ gives a solution of \eqref{eqn;pKS}.
The standard
parabolic regularity argument gives that the solution
becomes regular
immediately after $t>0$. The continuous dependence of
the solution on the initial data also follows from
(\ref{eq:ien04}).
\end{proof}

\subsection*{Acknowledgments}
Work of M. Ishiwata is partially supported by Grant-in-Aid for 
Young Scientists (B) \#19740081, 
The Ministry of Education, Culture, Sports,
Science and Technology, Japan. 
Work of T. Ogawa is partially supported by JSPS Grant-in-Aid for
Scientific Research (A) \#20244009.  
Work of F. Takahashi is partially
supported by JSPS Grant-in aid
for Scientific Research (C) \#20540216.�@


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\end{document}