数理解析研究所講究録
1009 巻 1997 年 22-28

22

Global existence of solutions to the parabolic systemsof chemotaxis
Toshitaka NAGAI(永井敏隆)
Takasi SENBA(仙葉 隆)
Kiyoshi YOSHIDA(吉田清)Department of Mathematics, Kyushu Institute of Technology

Department of Applied Mathematics, Miyazaki University
Faculty of Integrated Arts and Sciences, Hiroshima University

1. Introduction
We consider time.global existence of solutions of some parabolic systems related to chemotaxis. We consider the following system which is called Keller-Segel model.

$u_{t}=\nabla\cdot(\nabla u-\chi u\nabla\phi(v))$$\epsilon v_{t}=\Delta v-v+u$,,$x\in\Omega,$$t>0$ ,$x\in\Omega,$$t>0$ ,$\{$$\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0$,$x\in\partial\Omega,$$u(\cdot,0)=u_{0},$ $v(\cdot, 0)=v_{0}$$x\in\Omega$$t>0$ ,,

withis a bounded and connected domain ofwhere and are postive constants,$\phi’>0,$
are smooth,andwithsmooth boundary
is a smooth function on
nonnegative and non-trivial on .
Keller-Segel model was introduced by Keller and Segel [6] to describe the initiation ofchemotactic aggregation of cellular slime molds. $u(x, t)$ represents the cell density at placeand time . $v(x, t)$ represents the concentration of chemical substance at place and time$\mathrm{R}^{2}$$\Omega$$\epsilon$$\chi$$\partial\Omega,$$(0, \infty)$$\phi$$u_{0}$$v_{0}$$\overline{\Omega}$

$t$

$x$

$t$

$x$

.

Let me explain Keller-Segel model.
The first equation means change of cell density. The term $(-\nabla u+\chi\nabla\phi(v))$ means the, then theflow of cells. The
means the flow due to diffusion. Astermmeans the chemotactic flow due to response to chemical attractant. Namely,cells sense the gradient of chemical concentration. This phenomenon is called chemotaxis.
And chemical substance is an attractant, then the positivity of is neccessary. Then the. is called
function means the relation between the intensity of chemotactic flux and. Several forms of are suggested insensitivity function. Cells measure the gradient ofbiology.
$\nabla\phi(v)=\phi’\nabla v$$\mathrm{t}\mathrm{e}\mathrm{m}-\nabla u$$\chi u\phi’\nabla v$$\phi’$$v,$$\phi$$\phi(v)$$\phi$$\nabla v$$\phi$

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The second equation means change of concentration of chemical substance. The termmeans the flow due to diffusion. The term ofmeans the degradation byreactions. The termmeans the production by cells. Then the degradation and theproduction are proportional to chemical concentration and cell density, respectively.
Those phnomenon suggests the posibility of aggregation. Namely, first, cells move towardhigher concentration. Then cells aggregate at the place and product much attractant. Thencell and chemical substance aggregate at the place.
Then we consider the following problem:$(-1/\epsilon)\nabla v$$v/\epsilon$$u/\epsilon$

Investigate whether solutions can exist globally in time or not for several foms of the 8ensitivity function .
$\phi$

In particular,$\phi$

is specified as the following two cases :(A1) $\phi(v)=v$ ,(A2) $\phi(v)=\log v$ .

First, we describe a result in one dimensional case. In the following theorem,function with $\phi’>0$ .

$\phi$

is a smooth

Theorem 1 Assume that $\Omega=(0, L),$is a nonnegative smooth function on $[0, L]$ andis a positive smooth function on $[0, L]$ . Then the solution is globally bounded in time.
Namely, $T_{\max}=\infty$ and$u_{0}$$v_{0}$$\sup_{t\geq 0}(||u(\cdot, t)||_{L}\infty+||v(\cdot, t)||L^{\infty})<\infty$where$T_{\max}$,

is the maximal time of existence.

Then, in two dimensional case, we expect one dimensional blow-up can not occur.

Theorem 2 Assume $\phi(v)=v$ .
(i) $If||u_{0}||_{L}1<4\pi/\chi$, then the solution is globally bounded in time.
(ii) Let $\Omega=\{x\in \mathrm{R}^{2};|x|<L\}$ andbe radial in . $If||u_{0}||_{L}1<8\pi/\chi$ , then thesolution is globally bounded in time.
$(u_{0}, v\mathrm{o})$We expect that the restriction ofconjecture and resuluts.

$L^{1}$$x$-norm is necessary. Because, there are the followingChildress [2] and Childress and Percus [3] have given a conjecture such that if$8\pi/\chi$$\int_{\Omega}u_{0}(x)dx<$

then the solution exists globally in time, and if $\int_{\Omega}u_{0}(x)dx>8\pi/\chi$ then the solution

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can blow up in finite time, in the case of $\phi(v)=v$ and radial initial functions$\Omega=\{X\in \mathrm{R}^{2};|X|<L\}$$(u_{0},v\mathrm{o})$on.

. He has given a result such that if
T. Nagai [7] deal with the limiting system as$\int_{\Omega}u_{0}(x)dx<8\pi/\chi$ then the solution is globally bounded in time, and if$\epsilonarrow 0$$\int_{\Omega}u\mathrm{o}(X)dx>8\pi/\chi$, then the solution blows up in finite time, in the case of $\phi(v)=v$
on $\Omega=\{x\in \mathrm{R}^{2};|x|<L\}$ .
and radial initial functions
and$\int_{\Omega}u\mathrm{o}(X)|x|2d_{X}\ll 1$$(u_{0}, v\mathrm{o})$Theorem 3 $A_{\mathit{8}}sume\phi(v)=\log v$ and is positive in .
(i) If $\chi<1$ , ffien the solution globally exists in time. Namdy, $T_{\max}=\infty$ and$\overline{\Omega}$$v_{0}$$\sup_{0\leq t\leq\tau}(||u(\cdot,t)||_{L}\infty+||v(\cdot, t)||_{L}\infty)=c\tau<\infty$

for $T>0$ .

(ii) Let $\Omega=\{x\in \mathrm{R}^{2};|x|<L\}$ andglobally exists in time.

$(u_{0},v\mathrm{o})$be radial in . If $\chi<5/2$ , then the solution$x$–norm is not necessary. Because, T. Senba [8] deal
We expect that the restriction of. I have given a result such that the solution is globwith the limiting system ason
ally bounded in time, in the case of $\phi(v)=\log v$ and radial initial functions$L^{1}$$\epsilonarrow 0$$(u_{0},v_{0})$$\Omega=\{X\in \mathrm{R}^{2};|X|<L\}$.

2. Proof of Theorem 2.
Lemma 2.1 Put.

$W(t)=I_{\Omega} \{u\log u-\chi uv+\frac{\chi}{2}(|\nabla v|2+v^{2})\}dx$

Then we have$\frac{dW}{dt}(t)+\chi\epsilon\int\Omega d_{X}(v_{t})^{2}+\int_{\Omega}u|\nabla\cdot(\log u-xv)|^{2}d_{X}--0$.

Proof. Multiplying $\log u-\chi v$ by the first equation and using Green’s formula and thesecond equation, we have this lemma.
Lemma 2.2(i) Letbe a bounded and connected domain in$\Omega$$\exists C_{\Omega}>0\mathrm{s}.\mathrm{t}$$\mathrm{R}^{2}$

with smooth boundary. Then,.
$\int_{\Omega}\exp|u|dx\leq C_{\Omega}\exp\{\frac{1}{8\pi}||\nabla u||_{2}^{2}+\frac{2}{|\Omega|}||u||_{1\}}$

for$u\in H^{1}(\Omega)$. (S.Y.A. Chang and$\mathrm{P}.\mathrm{C}$. Yang [1])

25(ii) Let $\Omega=\{x\in \mathrm{R}^{2};|x|<L\}$ . Then for$\forall\delta>0,$$\exists C=C_{\delta}>0\mathrm{s}.\mathrm{t}$.

$\int_{\Omega}\exp|u|dx\leq C_{\delta}\exp\{(\frac{1}{16\pi}+\delta)||\nabla u||^{2}2+\frac{2}{|\Omega|}||u||_{1\}}$for$u\in H^{1}(\Omega)$with $u(x)=u(|x|)$ .

Lemma 2.3 If$C$.

$||u_{0}||_{1}<\pi^{*}/\chi,$$\exists C$(independent of )$t$$>0\mathrm{s}.\mathrm{t}$.

$\int_{\Omega}$$uvdx\leq C$ and $|W(t)|\leq$,$8\pi$

where $\pi^{*}=\{$

Proof. Let $a>0$ . For fix$4\pi$$t\in(\mathrm{O}, T)$, in radially symmetric case,,otherwise.

, put$\psi(x, t)=\frac{M}{\mu}e^{av(t)}x,$$M= \int_{\Omega}u(x, t)dX$

By Lemma 2.2 for$\forall\delta>0,$$\exists C_{\delta}>0\mathrm{s}.\mathrm{t}$and, where$\mu=\int_{\Omega}e^{av(x,t)}dx$.

.

$\log\mu\leq\log C_{\delta}+\frac{2a}{|\Omega|}||v||_{1}+\{^{\frac{1}{2\pi^{*}}+\delta}\}a^{2}||\nabla v||_{2}2$

By.

and Jensen’s inequality,$\int_{\Omega}\frac{\psi}{u}\frac{u}{M}dx=1$

$0$

$=$

$- \log\int_{\Omega}\frac{\psi}{u}\frac{u}{M}d_{X}$$\leq$$\int_{\Omega}\{-\log\frac{\psi}{u}\}\frac{u}{M}dX=\frac{1}{M}\int\Omega \mathrm{o}u\mathrm{l}\mathrm{g}\frac{u}{\psi}dX$.

Then$\{\frac{\chi}{2}-M(\frac{1}{2\pi^{*}}+\delta)a^{2}\}||\nabla v||_{2}2+(a-\chi)\int_{\Omega}$$\leq$

Lemma 2.4$\exists C$

uvdx$M \{\log C_{\delta}+\frac{2a}{|\Omega|}||v||1^{-\mathrm{l}M\}}\mathrm{o}\mathrm{g}+W(t)$(independent of )$t$$>0\mathrm{s}.\mathrm{t}$.

$||u(\cdot, t)||_{2}\leq C$

Proof. For simplicity we put $\chi=\epsilon=1$ . Multiply$\frac{1}{2}\frac{d}{dt}\int_{\Omega}u^{2}dx+\int_{\Omega}|\nabla u|^{2}dX=$$=- \frac{1}{2}\int_{\Omega}u^{2}\Delta vdX=$

$u$.

.

by the first equation, we have$- \int_{\Omega}\nabla\cdot(u\nabla v)ud_{X}$$- \frac{1}{2}\int_{\Omega}u^{2}(v_{t}+v-u)dX$.

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We can show that$||u||_{3}$$\leq$$\delta||\nabla u||_{2}/3|2|u\log u||_{1^{/3}}1$$+C_{\delta}\{||u\log u||_{1}+||u||^{1/}1\}3$

By H\"order and$\mathrm{G}\mathrm{a}\mathrm{g}\mathrm{l}\mathrm{i}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{c}\succ \mathrm{N}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}$,inequality, we have$\int_{\Omega}|u^{2}v_{t}|dx\leq\delta||\nabla u||_{2}2+C\{||v_{t}||_{2}^{2}+||v_{t}||_{2\}}||u||_{2}^{2}$,

which together above formulas leads to$\frac{d}{dt}||u||_{2}^{2}+2||\nabla u||_{2}2\leq\delta||\nabla u||_{2}2+C\{||v_{t}||_{2}2+||v_{t}||_{2}\}||u||_{2}^{2}$$+\delta^{\mathrm{s}}||u\log u||1||\nabla u||22+C\{||u\log u||_{1}\mathrm{s}+||u||_{1}\}$.

By the above inequality and Gronwall’s inequality, we have this lemma.
By applying the estimes ofand standard argments to the second equation, we havethe boundedness of, which yields the bondedness ofandby applying
Moser’s technique to the first equation.
$||u||_{2}$$||\nabla v||_{\infty}$$||v||_{\infty}$$||u||_{\infty}$

3. Proof of Theorem 3. Since the proof of (\"u) is similar to one of (i), we shall prove
only (ii).
Lemma 3.1 Let$a$be a positive constant. Then we have$\frac{d}{dt}\int_{\Omega}$( $u\log$ u–au $\log v$ )$dX+ \frac{a}{\epsilon}\int_{\Omega}\frac{u^{2}}{v}dx$$+ \int_{\Omega}u\{|\nabla\log u|2-(\chi+\frac{2a}{\epsilon})\nabla\log u\cdot\nabla\log v$$+ \frac{a}{2}(\chi+1)|\nabla\log v|^{2\}u_{0}}d_{X}=\frac{a}{\epsilon}||_{1}$Proof. Multiplying $\log u-a\log v$ by the first equation and using Green’s formula and thesecond equation, we have this lemma.

Lemma 3.2 For$\forall p\geq 1,$$\exists C_{p}$(independent of )$t$$>0\mathrm{s}.\mathrm{t}$.

$||v(\cdot, t)||_{\mathrm{P}}\leq c(\mathrm{P}||u0||_{1}+||v_{0}||_{p})$Proof. Using the following estimet of Green’s function G.
$|G(x,y,t)| \leq\frac{\epsilon C}{t}\exp(-C\mathcal{E}\frac{|x-y|^{2}}{t}-\frac{t}{\epsilon})$.

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Lemma 3.3 If $\chi<1,$$\exists C$(independent of )$t$$\int_{\Omega}u\log udx$

Proof. By using Lemma 3.1 with

and$>0\mathrm{s}.\mathrm{t}$.

.

$\int_{0}^{t}\int_{\Omega}u|\nabla\log u|d_{X}dS\leq Ct$$a=\epsilon/2$,$\frac{d}{dt}\int_{\Omega}(u\log u-\frac{\epsilon}{2}u\log v)d_{X}+\frac{1-\chi}{2}\int_{\Omega}u|\nabla\log u|^{2}d_{X}$$\leq\frac{1}{2}||u_{0}||_{1}$Put$\psi(x)=\frac{M}{\mu}v^{p}$$M=||u||_{1}$and.

, where$\mu=||v||_{p}^{p}$. By

and Jensen’s inequality,$\int_{\Omega}\frac{\psi}{u}\frac{u}{M}dx=1$$0=- \log\int\Omega\frac{\psi}{u}\frac{u}{M}dX\leq\int_{\Omega}\{-\log\frac{\psi}{u}\}\frac{u}{M}dx$

Then$p \int_{\Omega}u\log vdx\leq\int_{\Omega}u\log udX+M\log\frac{\mu}{M}$

Combining the first eq. and the$1\mathrm{a}s\mathrm{t}$.

eq. implies this lemma.

Proof of Theorem 3 By Gagliardo-Nirenberg inequality, we have.

:$||u||_{2}2=||\sqrt{u}||_{4}4\leq’(||\nabla$$\sqrt{u}||_{2}^{2}+||\sqrt{u}||^{2}2)||\sqrt{u}||_{2}^{2}$.

The above inequality and Lemma 3.3 implies that$\int_{0}^{t}\int_{\Omega}u^{2}dxds\leq Ct$$\mathrm{M}\mathrm{u}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{p}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{g}-\triangle v$.

(1)by the second equation, we have.

$\frac{d}{dt}||\nabla v||_{2}2+||\Delta v||_{2}2+||\nabla v||2^{2}\leq||u||2||\triangle v||_{2}$Combining the above inequality and (1) implies that$\int_{0}^{t}||\triangle v||_{2}2$

Mulitiply

$u$

and

$||\nabla v||_{2}2\leq Ct$

by the first equation and using Gagliard($\frac{d}{dt}||u||_{2}^{2}+||\nabla u||_{2}2$$=$$= \frac{\chi}{2}\int_{\Omega}u^{2}\Delta\log vdX=$$\leq$.

$\succ \mathrm{N}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}$(2)inequality, we have$- \chi\int_{\Omega}u\nabla u\cdot\nabla\log vdx$$\frac{\chi}{2}\int_{\Omega}\frac{u^{2}}{v}(\triangle v-|\nabla\log v|^{2})d_{X}$$\frac{\chi}{2V_{m}}||\Delta v||_{2}||\nabla u||_{2}||u||_{2}$,28,where $V_{m}(t)= \min_{\Omega}v(\cdot,t)\geq V_{m}(0)e^{-t}$ , which together (2) implies$||u||_{2}^{2}\leq C\exp(cte^{2t})$.

and standard argments to the second equation, we have theBy applying the estimes ofby using Moser’sof, which yields theandboundedness oftechnique for the first equation.
$||u||_{2}$$||\nabla v||_{\infty}$$||v||_{\infty}$

bond.edness$||u||_{\infty}$

References[1] S. Y. A. Chang and P. C. Yang, Conformal deformation of metrics onGeom., 27 (1988), 259-296.

$S^{2}$, J. Differential[2] S. Childress, Chemotactic collapse in two dimensions, Lecture Notes in Biomath., vol.
-New York, 1984, 61-66.
55, Springer,$\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{i}\mathrm{n}-\mathrm{H}\mathrm{e}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}$[3] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56(1981), 217-237.

[4] M. A. Herrero and J. J. L. Vel\’azquez, Singularity patterns in a chemotaxis model,preprint.
[5] W. J\"ager and S. Luckhaus, On explosions of solutions to a system of partial differential
equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
[6] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
[7] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math.
Sci. Appl., 5 (1995), 581-601.
[8] T. Senba, Blow-up of radially symmetric solutions to some systems of partial differential equations modelling chemotaxis, Adv. Math. Sci. Appl., to appear.

[9] A. Yagi, Norm behavior of solutions to the parabolic system of chemotaxis, preprint.