数理解析研究所講究録
1197 巻 2001 年 189-195

189

Self-similar solutions to a parabolic system modelling chemotaxis
Yuki Naito(内藤雄基)(神戸大工)

Department of Applied Mathematics, Kobe University

Takashi Suzuki(鈴木貴)(大阪大理)

Department of Mathematics, Osaka University

Kiyoshi Yoshida$(_{\square }^{\pm} \mathrm{f}\mathrm{f}\mathrm{l} \grave{7}\ovalbox{\tt\small REJECT})$

Faculty of Integrated Arts and Sciences, Hiroshima University(広島大総合)We are concerned with the system of partial differential equations$\frac{\partial u}{\partial t}=\nabla\cdot(\nabla u-u\nabla v)$(1.1)$\{$$\tau\frac{\partial v}{\partial t}=\triangle v-\gamma v+u$

forand $t>0$ , where $\tau>0$ andare constants. This system is a mathematical model of chemotaxis (aggregation of organisms sensitive to the gradient of chemicalsubstance) proposed by Keller and Segel [12] in 1970. The functions $u(x, t)$ and $v(x, t)$ in(1.1) denote the cell density of cellular slime molds and the concentration of the chemicalsubstance at place and time , respectively. It is assumed that and are nonnegative.
We deal with a special class of solutions which are called self-similar solutions. If $\gamma=0$ ,then the system (1.1) is invariant under the similarity transformation$x\in \mathbb{R}^{N}$$\gamma\geq 0$

$x$

$t$

$u$

$u_{\lambda}(x, t)=\lambda^{2}u(\lambda x, \lambda^{2}\mathrm{f})$

and

$v$

$v_{\lambda}(x, t)=v(\lambda X, \lambda^{2}t)$

for $\lambda>0$ , that is, if $(u, v)$ is a solution of (1.1) globally in time, then so is. A solution$(u, v)$ is said to be self-similar, when the solution is invariant under this transformation,that is,$(u_{\lambda}, v_{\lambda})$(1.2)

Letting(1.3)$u(x, t)=u_{\lambda}(x, t)$$\lambda=1/\sqrt{t}$and$v(x, t)=v_{\lambda}(x, t)$

for allin (1.2), we see that $(u, v)$ has the form$u(x, t)= \frac{1}{t}\emptyset(\frac{x}{\sqrt{t}})$and$v(x, t)= \psi(\frac{x}{\sqrt{t}})$$\lambda>0$.

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for
only if$x\in \mathbb{R}^{N}$

and $t>0$ . By a direct computation it is shown that $(u, v)$ satisfies (1.1) if andsatisfies$(\phi, \psi)$$\nabla\cdot(\nabla\phi-\emptyset\nabla\psi)+\frac{1}{2}x\cdot\nabla\phi+\emptyset=0$

(1.4)

$\{$

$\triangle\psi+\frac{\tau}{2}X\cdot\nabla\psi+\phi=0$

for$x\in \mathbb{R}^{N}$. It follows that$\int_{\mathbb{R}^{N}}u(x, t)dX=t^{(N-2)}/2\int_{\mathbb{R}^{N}}\phi(y)dy$. Therefore self-similar solution $(u, v)$ preserves the massforif
and only if $N=2$ . Henceforth we study the case $N=2$ . We are concerned with theclassical solutionsof (1.4) satisfying$\phi\in L^{1}(\mathbb{R}^{N})$$||u(\cdot, t)||_{L^{1}(\mathbb{R}^{N})}$$(\phi, \psi)\in C^{2}(\mathbb{R}^{2})\cross C^{2}(\mathbb{R}^{2})$(1.5)$\psi\geq 0$$\phi,$

Define the solution set(1.6)$S=${$(\phi,$$S$

inand$\mathbb{R}^{2}$$\phi(x),$ $\psi(x)arrow \mathrm{O}$as$|x|arrow\infty$.

of (1.4) as$\psi)\in C^{2}(\mathbb{R}^{2})\cross C^{2}(\mathbb{R}^{2}):(\phi,$ $\psi)$

is a solution of (1.4) with (1.5)}.

The problem of existence of self-similar solutions has been studied extensively. The
existence of radial solutionsof (1.4) has been obtained by Mizutani and Nagai[15]. It has shown by Biler [1] that there is an upper bound on the mass of self-similarsolutions. More precisely, the system (1.4) with $\tau=1$ has no radial solutionssatisfying. Furthermore, for every $M\in(0,8\pi)$ , there exists aradial solution
satisfying. In this paper we investigate the structureof the solution set defined by (1.6) more precisely.
First we show the system (1.4) is reduced to a single ordinary differential equation. Put$(\phi, \psi)$$(\phi, \psi)$$||\emptyset||_{L(\mathbb{R}^{2})}1/2\pi\geq 7.82\ldots$$(\phi, \psi)$$||\emptyset||_{L^{1}}(\mathbb{R}^{2})=M$

$S$

(1.7)

$\phi(x)=\sigma e^{-|x|^{2}/}e4\psi(x)$where a is a positive constant. Thena solutions of$\phi$,

satisfies the first equation of (1.4), and so if we find$\psi$(1.8)$\triangle\psi+\frac{\tau}{2}x\cdot\nabla\psi+\sigma e^{-|x|^{2}/}4\psi e=0$in$\mathbb{R}^{2}$,

we can obtain the solutionof (1.4). In $[15, 16]$ they have shown the existence ofradial solutions of (1.8) satisfying$(\phi, \psi)$$\psi$(1.9)$\psi(x)arrow 0$

as$|x|arrow\infty$

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by investigating the corresponding ordinary differential equation. Conversely, we havefollowing:
Theorem 1. Assume thatis a nonnegative solution of (1.4) satisfying. Then and are positive and satisfy (1.7), where $\sigma>0$ is a constant. Assume
furthermore that (1.9) holds. Then and are radially symmetric about the origin, andsatisfy $\phi_{r}<0$ and $\psi_{r}<0$ for $r>0$ , and$(\phi, \psi)$$L^{\infty}(\mathbb{R}^{2})$$\phi,$$\psi\in$$\psi$$\phi$$\psi$$\phi$

and$\phi(x)=O(e^{-|x|/4})2$$\psi(x)=O(e^{-\min\{\mathcal{T}}’)1\}|x|2/4$

as$|x|arrow\infty$.

The proof of Theorem 1 consists of two steps. First we reduce the system (1.4) to theequation (1.8) by employing the Liouville type result essensially due to Meyers and Serrin[14]. Then we show the radial symmetry of solutions by the method of moving planes. This
device was first developed by Serrin [23] in PDE theory, and later extended and generalizedby Gidas, Ni, and Nirenberg $[6, 7]$ . With a change of variables we are still able to obtaina symmetry result for the equation (1.8) as in [18].
Next we investigate the structure of the solution set defined by (1.6). F.r om T.heorem 1,
the set $S$ contains nonnegative solutions
satisfyingand (1.9). For$(\phi, \psi)\in S,$
and are radially symmetric about the origin, and satisfy.
$S$$(\phi, \psi)$$\phi\in L^{\infty}(\mathbb{R}^{2})$$\psi$$\phi$$\phi,$$\psi\in L^{1}(\mathbb{R}^{2})$Theorem 2. The solution set is written by one parameter families $(\phi(s), \psi(S))$ on $s\in$that $is_{f}S=\{(\phi(s), \psi(S)) : s\in \mathbb{R}\}$ . The solutions $(\phi(s), \psi(S))$ andthe following properties:(i)andare continuous;(ii) $(\phi(g), \psi(S))arrow(\mathrm{O}, 0)$ in
and
as $sarrow-\infty$ ;(iii)
as $sarrow\infty$ , and$S$$\mathbb{R}_{f}$$\lambda(s)=||\phi(s)||_{L^{1}(\mathbb{R})}2$$satisf\dot{y}$$s-\succ(\phi(s), \psi(S))\in C^{2}(\mathbb{R}^{2})\cross C^{2}(\mathbb{R}^{2})$$srightarrow\lambda(s)\in \mathbb{R}$$C^{2}(\mathbb{R}^{2})\cross C^{2}(\mathbb{R}^{2})$$\lambda(s)arrow \mathrm{O}$$||\psi(s)||_{L^{\infty}(\mathbb{R})}2arrow\infty$

and$\lambda(s)arrow 8m\pi$

for some integer
ger not exceeding
Moreover,$m$$a$$\phi(s)dXarrow 8m\pi\delta_{0}(dX)$as$sarrow\infty$

in the sense of measuresatisfying $1 \leq m\leq\max\{1, [\pi^{2}\tau^{2}/6]\}$ , where $[a]$ is the greatest inteand
denotes Dirac delta function with the support in origin.
$\delta_{0}(dx)$$fS$$\int_{\mathbb{R}^{2}}e^{-1}edyy|^{2}/4\psi(_{S})arrow\infty$as$sarrow\infty$;(iv) Let. Then $8m \pi\leq\lambda^{*}\leq\max\{4\pi^{3}/3,4\pi^{3}\tau^{2}/3\})$(v) If $0<\tau\leq 1/2$ then $0<\lambda(s)<8\pi$ for.
$\lambda^{*}=\sup_{s\in \mathbb{R}}\lambda(s)$$s\in \mathbb{R}$As a consequence of Theorem 2 we obtain the following:

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Corollary 1. There exist an integer $m$ and a constantand$1 \leq m\leq\max\{1, [\pi^{2}\tau^{2}/6]\}$

satisfying$\lambda^{*}$$8m \pi\leq\lambda^{*}\leq\max\{4\pi^{3}/3,4\pi^{3}\tau^{2}/3\}$,

respectively, such that(i) for every, there exists a solution $(\phi, \psi)\in S$ satisfying;, there exists no solution $(\phi, \psi)\in S$ satisfying(ii) for(iii) there exists a sequencesatisfying $\phi_{k}dxarrow 8m\pi\delta 0(dx)$ as $karrow\infty$ in thesense of measure.
Moreover, if $0<\tau\leq 1/2$ then $m=1$ and.
$\lambda\in(0, \lambda^{*})$$||\emptyset||_{L(\mathbb{R}^{2})}1=\lambda$$\lambda>\lambda^{*}$$||\phi||_{L(\mathbb{R}^{2})}1=\lambda_{i}$$(\phi_{k}, \psi_{k})\subset S$$\lambda^{*}=8\pi$

The proof of Theorem 2 is based on the ODE arguments. Furthermore, we employ
the blow-up analysis by Brezis-Merle [2] and Li-Shafrir [13] to investigate the asymptotic
behavior of $(\phi, \psi)\in S$ as. To show the upper bounds of the mass of we)use the techniques due to Bilar [1]. Let. bom (1.7) it follows that$||\psi||_{L^{\infty(}}\mathbb{R}^{2}$$arrow\infty$$\phi$$\lambda=||\phi||_{L(\mathbb{R}^{2})}1$$\lambda=\sigma\int_{\mathbb{R}^{2}}e^{-1y}e^{\psi}d(y)y|^{2}/4$.

Then (1.8) is rewritten as the elliptic equation with a non-local term
(1.10)$\triangle\psi+\frac{\tau}{2}x\cdot\nabla\psi+\lambda e^{-|x|^{2}}e/4\psi/\int_{\mathbb{R}^{2}}e^{-1y}e^{\psi}d(y)y|^{2}/4=0$. This equation plays an important role to investigate the blow-up properties ofS.
Finally, we obtain the result concerning the existence of solutions to (1.8) with (1.9). Thisrefines the previous results [15, Theorem 1], [16, Theorems 1 and 2], and [17, Theorem 1.1].
for$x\in \mathbb{R}^{2}$$(\phi, \psi)\in$

Theorem 3. For any $\tau>0$ there exists
such that(i) if, then (1.8) with (1.9) has no solution;$(1.8)$ with (1.9) has at least one solutionj(ii) if(iii) if $0<\sigma<\sigma^{*}$ , then (1.8) with (1.9) has at least two distinct solutions
satisfying
and.
$\sigma^{*}>0$$\sigma>\sigma^{*}$$\sigma--\sigma^{*},$$\lim_{\sigmaarrow 0}\underline{\psi}_{\sigma}(0)=0$$\overline{\psi}_{\sigma}$$\underline{\psi}_{\sigma},$$\lim_{\sigmaarrow 0}\overline{\psi}_{\sigma}(0)=\infty$

Recently, attentions have been paid to blowup problems for the system (1.1) for $(x, t)\in$
subject to the boundary and initial condition$\Omega\cross(0, T)$$\underline{\partial u}\underline{\partial v}==0$(1.11)$\{$$\partial \mathrm{z}\text{ノ}$on$\partial\Omega\cross(0, T)$,$\partial\nu$$u(x, 0)=u_{0}$and$v(x, 0)=v_{0}$on$\Omega$,

193

whereis a bounded domain with smooth boundary, and l ノ is the outer normal
unit vector. Childress and Percus [4] and Childress [3] have studied the stationary problem
and have conjectured that there exists a thresholdof blowup, that is, ifthen the solution $(u, v)$ exists globally in time, and if
then$u(x, t)$ can form a delta function singularity in finite time. Their arguments were heuristic,
while recent studies are supporting their validity rigorously, see, [21]. We also refer to [9],[19], and [22].
On the other hand, it is asserted that self-similar solutions take an important role for
the Cauchy problem for the semilinear parabolic equations on the whole space, see, e.g.,[5], [10], and [11]. By the definition, self-similar solutions are global in time, and they are
expected to describe large time behavior of general solutions generically. From Corollary
1, we are led to the following conjectures for the Cauchy problem (1.1) with $\gamma=0$ .
$\Omega\subset \mathbb{R}^{2}$$\partial\Omega$$||u_{0}||_{L^{1}()}\Omega=8\pi$$||u_{0}||_{L^{1}(\Omega)}<8\pi$$||u_{0}||_{L()}1\Omega>8\pi$

For $0<\tau\leq 1/2$ , ifthen the solution of the Cauchy problem exists
globally in time, and $if||u_{0}||_{L()}1\Omega>8\pi$ then the solution can blowup in a finite time.
$||u_{0}||_{L(\mathbb{R}^{2})}1<8\pi$

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