数理解析研究所講究録
第 1984 巻 2016 年 1-8

1

Global existence of solutions to aparabolic-parabolic chemotaxis system with
subquadratic growth(Kwansei Gakuin Univ.) 1(Tokyo Medical and Dental University) 2

Koichi Osaki
Etsushi Nakaguchi

$*$

$*$

1. Chemotaxis-Growth System
In a study of the chemotactic bacterial pattern formation, Mimura and Tsujikawa [10]analyzed a parabolic-parabolic chemotaxis system with bacterial proliferation, which
is of the following simplified form:(E)$\{\begin{array}{ll}\frac{\partial u}{\partial t}=\triangle u-\chi\nabla\cdot(u\nabla v)+f(u) in \Omega\cross(0, \infty) ,\tau\frac{\partial v}{\partial t}=\triangle v-v+g(u) in \Omega\cross(0, \infty) ,\partial u \partial v \overline{\partial v}=\overline{\partial\nu}=0 on \partial\Omega\cross(0, \infty) ,u(x, 0)=u_{0}(x)\} v(x, 0)=v_{0}(x) in \Omega.\end{array}$$(n=2$ or 3 is a bounded domain with smooth boundaryHere,, and thecoeﬃcients and are positive constants. The unknown functions $u(x, t)$ and $v(x, t)$
are the population density of biological individuals and the concentration of chemical
substance in position at time , respectively. We here note that the limitof thetime scale indicates the parabolic-elliptic simplification of [21] (cf. [12]). We assume
that the function $f(u)$ is a real smooth function of $u\in[0, \infty$ ) such that $f(0)=0$ and$\Omega\subset \mathbb{R}^{n}$$\chi$$\partial\Omega$$)$$\tau$$\tauarrow 0$

$t$

$x$

$f(u)=u-\mu u^{\alpha}$

for suﬃciently large ;$u$and the function $g(u)$ is given by$g(u)=u(1+u)^{\beta-1}$

for $u\geq 0,$

where the exponents and satisfy the relations $1<\alpha\leq 2$ and $0<\beta\leq 1$ , andis a positive constant. The function $f(u)$ models the proliferation and the reduction
in numbers due to death of bacteria as following a logistic process (we refer to theproliferation and reduction in numbers together simply as growth). When $\alpha=2$ , thatis, quadratic degradation is assumed, the function $f(u)$ gives usual logistic growth.
Subquadratic degradation then refers to the case of $\alpha<2$ . The function $g(u)$ models
the nonlinear secretion of the chemical substance from the bacteria, whose increasing, 11, 23
order isIn the context of global existence and blow-up of solutions, the degradation of thegrowth can be considered as an inhibitory eﬀect from the increase of . In fact, if we
suppose that the growth is absent $(f(u)\equiv 0)$ and the secretion is linear $(\beta=1)$ , then$\alpha$$\beta$$\mu$$\beta[5$

$].$

$u$

The research is supported by KAKENHI Nos.23540125 and 26400180.
$*lD$ epartment of Mathematical Sciences, School of Science and Technology, Kwansei Gakuin University, Sanda 669-1337, Japan
College of Liberal Arts and Sciences, Tokyo Medical and Dental University, Ichikawa, Chiba 2720827, Japan$*2$

2
the system reduces to the classical parabolic-parabolic Keller-Segel system [7], for which
several mathematicians showed the blow-up of solutions: Herrero and Vel\’azquez [4]$(n=2)$ , Horstmann and Wang [6] $(n=2)$ and Winkler $[26](n\geq 3)$ . Therefore, wehave known that if $f(u)\equiv 0$ or $\alpha=1$ with a special choice of $\mu=1$ and $\beta=1,$
then no inhibitory eﬀect can cause a chemotactic collapse in the -dimensional domain$(n\geq 2)$ . In contrast, for the parabolic-parabolic chemotaxis-growth system with $\alpha=2$
and $\beta=1$ , the global existence of solutions is assured even if the initial total massand the chemotactic coeﬃcient are suﬃciently large when $n=2$ by one of
the authors et al. [19] and $n\geq 3$ by Winkler [25]. From these results, we find that if
degradation is quadratic $(\alpha=2)$ and secretion is linear $(\beta=1)$ , then the blow-up ofsolutions is prevented independently of the space dimension. We can then conjecturethat the critical degradation order is in the interval [1, 2] when $\beta=1$ ; however, thecorresponding result for the parabolic-parabolic system (E) has yet to be established.
We then introduced sub-linear secretion $\beta<1$ , and showed a suﬃcient condition
for the existence of global and bounded solutions under certain relations between
and when $n=2$ or $n=3[14]$ . In fact, we can obtain the following:$n$$\Vert u_{0}\Vert_{L_{1}}$$\chi$$\alpha$$\alpha$$\beta$Theorem 1.1. Let be an arbitrarily fixed exponent satisfying $0<\epsilon<1/4$ . For the
exponents and , assume the relation$\epsilon$$\alpha$$\beta$$\frac{2(n+4)}{n+6}<\alpha\leq 2, 0<\beta<\frac{n+6}{2(n+2)}(\alpha-1)$.

(1)Then, for each initial function $0\leq u_{0}\in H^{(n/2)-1}(\Omega)\subset L_{n}(\Omega)$ andthe problem (E) admits a unique global solution $(u, v)$ in the junction space$0\leq v_{0}\in W\subset C(\overline{\Omega})$,$\{\begin{array}{l}0\leq u\in C([0, \infty);H^{(n/2)-1}(\Omega))\cap C((O, \infty);H_{N}^{3}(\Omega))\cap C^{1}((0, \infty);H^{1}(\Omega)) ,0\leq?1\in C([0, \infty);W)\cap C((O, \infty);H_{N^{2}}^{4+\epsilon}(\Omega))\cap C^{1}((0, \infty);H_{N}^{2+\epsilon}(\Omega)) .\end{array}$Here, the function spaces$H_{N}^{s}(\Omega)$,$H_{N^{2}}^{s}(\Omega)$

and $W$ are defined by$H_{N}^{s}( \Omega)=\{w\in H^{s}(\Omega);\frac{\partial w}{\partial n}=0$on$\partial\Omega\}$$H_{N^{2}}^{s}(\Omega)=\{w\in H_{N}^{2}(\Omega);\triangle w\in H_{N}^{s-2}(\Omega)\}$for$s> \frac{3}{2},$for$s> \frac{7}{2},$$W=\{\begin{array}{ll}H^{1+\epsilon}(\Omega) when n=2,H_{N}^{(3/2)+\epsilon}(\Omega) when n=3,\end{array}$for some fixed exponent $0<\epsilon<1/4.$
In this report, we show a sketch of a proof of the theorem and also the asymptotic
behavior of the solutions. In the paper [14], the global attractors and the exponential-uniform
attractors also were constructed. Indeed, we derived the higher order
estimates of the solutions $(u, v)$ , and then proved not only the boundedness of the
topology in which any solutions
solutions but also the existence of a ball in thethat start from a bounded set of the universal space shall eventually be contained
within a finite time. We call the set an absorbing set. The omega-limit set of theabsorbing set is the global attractor (e.g. [22]). Hence, the dynamics including many
complex pattern formations reduces to the restricted region in the function space andall the orbits traced out by the solutions converge to the global attractor as $tarrow\infty.$$H^{2}\cross H^{3}$$H^{2}\cross H^{3}$

3

plane [13, 14, 15]. The X mark denotes an
Figure 1: Region of global existence inoccurrence of a blow-up of solutions in the Keller-Segel system $(\alpha=\mu=\beta=1)$ . The
critical degradation order between global existence and blow-up has not been foundfor (E).
$\alpha-\beta$$\alpha$Figure 2: Dot and hexagonal pattern formation of solutions to the system (E). A dot
pattern (left) and a hexagonal pattern (right) [9, 16]. In actual phenomena, some dot
patterns have been observed; on the other hand, hexagonal patterns have not been
observed, as far as the authors know.

An exponential attractor contains the global attractor and attracts the orbits at an
exponential rate. Moreover, its fractal dimension is finite [3]. We thus find that the
dynamics of solutions exponentially converges to the restricted compact region, of
which the degree of freedom is finite. It is known that such characteristics have someadvantages for numerical computations (e.g. [22]). Of the results including numericalcomputations of chemotaxis-growth systems, we cite here only a selection of the articles
and books: the famous book on mathematical biology by Murray [11]; one-dimensional
pattern formations by Kurata et al. [8], Okuda and Osaki [18], Painter and Hillen [20]and Uemichi and Osaki [24]; two-dimensional pattern formations by Aida et al. [1],Okuda and Osaki [17] and Kuto et al. [9] (see Fig. 2); and three-dimensional patternformations by Narumi and Osaki [16] (see Figs. 2 and 3).

2. Global Existence of Solutions
After showing the local unique existence of solutions, we show the global existence ofsolutions by obtaining several a priori estimates.
Stepl ( -uniform estimates of ). By integrating the first equation of (E) in$L_{1}$$u$$\Omega,$4

Figure 3: Three-dimensional face centered cubic (FCC) pattern formation in the
chemotaxis-growth system (E) [16].

we obtain$\frac{d}{dt}f_{\iota}udx=\int_{l}f(u)dx\leq\int_{1}(a-\frac{\mu}{2}u^{\alpha})dx\leq\int_{Jl}(a_{1}-u)dx.$

From Gronwall’s inequality, we obtaina constant. At the same time, we have$\Vert u\Vert_{L_{1}}\leq e^{-t}\Vert u_{0}\Vert_{L_{1}}+a_{1}|\Omega|(1-e^{-t})$, where$\int_{0}^{t}e^{-\omega(t-s)}\int_{tl}u^{\alpha}dxds\leq\frac{2\omega}{\mu}\{(\frac{a}{\omega}+a_{1})|\Omega|+\Vert u_{0}\Vert_{L_{1}}\}\leq C(1+\Vert u_{0}\Vert_{L_{1}})$

is$a_{1}$.

Step 2 ( -uniform estimate of : degradation vs. secretion). By multiplying theand integrating the result over , under the assumption
second equation of (E) by$0<2\beta\leq\alpha$ , we obtain$H^{1}$$v$$-\Delta v$$\Omega$$\frac{\tau}{2}\frac{d}{dt}\int_{\Omega}|\nabla v|^{2}dx\leq-\frac{1}{2}\int_{\Omega}(\Delta v)^{2}dx-\int_{\Omega}|\nabla v|^{2}dx+\frac{1}{2}\int_{\Omega}(1+u)^{2\beta}dx$$\leq-\frac{1}{2}\int_{l}(\Delta v)^{2}dx-\int_{l}|\nabla v|^{2}dx+Cl_{\iota}(1+u^{\alpha})dx.$

Therefore, we have$\int_{\zeta\}}|\nabla v|^{2}dx\leq e^{-2t/\tau}\int_{tl}|\nabla v_{0}|^{2}dx+C\int_{0}^{t}e^{-2(t-s)/\tau}\int_{tt}(1+u^{\alpha})dxds$$\leq e^{-2t/\tau}\int_{l}|\nabla v_{0}|^{2}dx+C(1+\Vert u_{0}\Vert_{L_{1}})$.

Here, we note that the assumptions $0<\beta\leq\alpha/2$ and $\alpha<2$ imply sub-linear secretion$\beta<1.$

Step 3 (-uniform estimate: chemotaxis vs. degradation). By multiplying
the first equation of (E) by $(1+u)^{\theta-1}$ and integrating the result over , we have
$L_{\theta}\cross H^{2}$$\Omega$$\frac{1}{\theta}\frac{d}{dt}\int_{\Omega}(1+u)^{\theta}dx$$=-( \theta-1)l_{l}(1+u)^{\theta-2}|\nabla u|^{2}dx+\chi(\theta-1)\int_{Il}u(1+u)^{\theta-2}\nabla u\cdot\nabla vdx$$+ \int_{1}(1+u)^{\theta-1}f(u)dx$$\leq-(\theta-1)\int_{l}(1+u)^{\theta-2}|\nabla u|^{2}dx+\chi\int_{l}(1+u)^{\theta}|\Delta v|dx-\int_{ti}u^{\alpha+\theta-1}dx+C.$

5The chemotaxis term can be estimated as$\int_{tt}(1+u)^{\theta}|\Delta v|dx\leq\Vert(1+u)^{\theta}\Vert_{L_{\frac{2(n+4)}{n+6}}}\Vert\Delta v\Vert_{L_{\frac{2(n+4)}{n+2}}}$$\leq C_{n}\Vert(1+u)^{\theta}\Vert_{L_{\frac{2(n+4)}{n+b}}}\Vert\Delta v\Vert_{H^{\frac{n}{n+4}}}\leq C_{n}\Vert(1+u)^{\theta}\Vert_{L_{\frac{2(n+4)}{n+6}}}\Vert v\Vert^{\frac{2}{H^{1}n+4}}\Vert v\Vert^{\frac{n+2}{H^{3}n+4}}$$\leq\eta\Vert v\Vert_{H^{3}}^{2}+C_{\eta}\Vert(1+u)^{\theta}\Vert_{\frac{n+62(n+4)+4)}{n+6}}^{\frac{2(n}{L}}\Vert v\Vert^{\frac{4}{H^{1}n+6}}$

We here adopt the assumption (chemotaxis$<$

degradationthat is,$\frac{2(n+4)}{n+6}\theta<\alpha+\theta-1 \Leftrightarrow \theta<\frac{n+6}{n+2}(\alpha-1)$.

Then, we obtain$\frac{1}{\theta}\frac{d}{dt}l_{l}(1+u)^{\theta}dx\leq-(\theta-1)\int_{\iota}(1+u)^{\theta-2}|\nabla u|^{2}dx+\eta\int_{l}|\nabla\Delta v|^{2}dx$$- \frac{\mu}{4}\int_{t1}u^{\alpha+\theta-1}dx+\psi(\Vert v\Vert_{H^{1}}+\eta^{-1})$

Meanwhile, by applying operator to the second equation of (E), multiplying by
and integrating the result over , we have$\nabla$.

$\nabla\Delta v,$$\Omega$$\frac{\tau}{2}\frac{d}{dt}\int_{\zeta f}|\Delta v|^{2}dx=-\int_{)}|\nabla\Delta v|^{2}dx-\int_{\zeta 1}|\Delta v|^{2}dx-\int_{t}\nabla\Delta v\cdot g’(u)\nabla udx$$\leq-\frac{1}{2}\int_{1}|\nabla\Delta v|^{2}dx-\int_{\zeta)}|\Delta v|^{2}dx+\frac{\beta^{2}}{2}\int_{l}(u+1)^{2\beta-2}|\nabla u|^{2}dx.$, we obtain an
With an additional assumption-uniform estimate.
Step 4 ( -uniform estimate of ). From the first equation, we have
$L_{\theta}\cross H^{2}$$2\beta\leq\theta$$L_{2\theta}$$u$$\frac{1}{2\theta}\frac{d}{dt}\int_{t1}(1+u)^{2\theta}dx\leq-(2\theta-1)\int_{1l}(1+u)^{2\theta-2}|\nabla u|^{2}dx$$+ \chi\int_{l}(1+u)^{2\theta}|\Delta v|dx+Cl_{l}(1-u^{\alpha+2\theta-1})dx.$

The chemotaxis term can be estimated as$l_{1}(1+u)^{2\theta}|\Delta v|dx\leq\Vert(1+u)^{\theta}\Vert_{L_{4}}^{2}\Vert\Delta v\Vert_{L_{2}}$$\leq C(\Vert(1+u)^{\theta}\Vert\frac{3n}{H^{1}2n+4}\Vert(1+u)^{\theta}\Vert\frac{4-n}{L_{1}2n+4})^{2}\Vert\Delta v\Vert_{L_{2}}$$\leq\eta\Vert(1+u)^{\theta}\Vert_{H^{1}}^{2}+C_{\eta}\Vert 1+u\Vert_{L_{\theta}}^{2\theta}\Vert v\Vert^{\frac{2n+4}{H^{2}4-n}}$Then, we have a recurrence relationof , which allows -uniform estimateswith arbitrary (cf. [2]).
By similar arguments to the above, we can also obtain the higher order uniform
estimate [14].
$\theta_{j+1}=2\theta_{j}$$\theta$$\square$$\theta$$L_{\theta}$

6
3. Dynamical System and Attractors
By the above results, we can show the existence of attractors in the dynamical systemof the solutions. In fact: let us define the universal space $H$ as$H=L_{2}(\Omega)\cross H^{1}(\Omega)$.

The initial functions are taken in the following set:
$K= \{(u, v)\in H^{\frac{n}{2}-1}(\Omega)\cross W;u\geq 0, v\geq 0\}, 0<\epsilon<\frac{1}{4}.$Then, the global unique solutions belong to, which shows that a$(u_{0},v_{0})\mapsto(u(t),v(t))\inK\cap \mathcal{D}$ can be$Karrow
K$$S(t)$
semigroup
:
such
that
continuous, we have an absorbing set
defined. From the higher order uniform estimate in; that is, for every bounded set $B\subset K$ , there exists a timethat may depend on. More precisely, we can show the following:such that$\mathcal{D}=H_{N}^{2}(\Omega)\cross H_{N}^{3}(\Omega)$$\mathcal{D}[14]$$\mathcal{B}$$t_{B}$$B$$\bigcup_{t\geq t_{B}}S(t)B\subset \mathcal{B}$

Proposition 3.1. A bounded ball$\mathcal{B}$

of

$K$

$\mathcal{B}=\{(u, v)\in H_{N}^{2}(\Omega)\cross H_{N}^{3}(\Omega);\Vert u\Vert_{H^{2}}+\Vert v\Vert_{H^{3}}\leq C, u\geq 0, v\geq 0\}\subset K$

is an absorbing set of the dynamical system $(S(t), K, H)$ . Here, the constantuniversal constant, which is suitably determined from the a priori estimates.

From the existence of the absorbing set
set$\mathcal{H}$$\mathcal{B}$

$C$

is a

, we can construct a positively invariantas$\mathcal{H}=\bigcup_{t\geq t_{B}}S(t)\mathcal{B},$whose topology of the closure is of $K$ . Therefore, the asymptotic behavior of thesolutions is reduced to the eventual dynamical system $(S(t), \mathcal{H}, H)$ . In the dynamicalsystem, the global attractor , which is a compact and invariant set in $H$ and attracts
every bounded subset of , is obtained as the -limit set of;if
A subsetis called the exponential attractor for
has finite fractal dimension
is a compact subset of $H$ and is invariant for$t\geq0$$c_{1}>$ O.
; andwith some constantsforHere, $h(B_{0}, B_{1})= \sup_{U\in B_{0}}\inf_{V\in B_{1}}\Vert U-V\Vert_{H}$ denotes the Hausdorﬀ pseudodistance of. Eden et al. [3, Proposition 3.1 and Theorem 3.1] showed that
two sets
andunder the above setting, if some Lipschitz conditions hold for the equations, then an. We can verify
exponential attractor
exists for the dynamical system
the Lipschitz conditions by arguments quite similar to those in [19, Section 5]. Then
we obtain the existence theorem of the global attractor and an exponential attractor
as follows:
Theorem 3.2. There exist the global attractor and an exponential attractorfor(E).
systemchemotaxis-growththethe dynamical systemof$\mathcal{A}$$\mathcal{H}$$\mathcal{B}:\mathcal{A}=\bigcap_{t>0}\bigcup_{s>t}S(t)\mathcal{B}.$$\omega$$(S(t), \mathcal{H}, H)$$\mathcal{M}\subset \mathcal{H}$$\overline{\mathcal{A}}\subset\overline{\mathcal{M}}\subset \mathcal{H}$$S(t);\mathcal{M}$$\mathcal{M}$$h(S(t)\mathcal{H}, \mathcal{M})\leq c_{0}\exp(-c_{1}t)$$d_{F}(\mathcal{M})$$B_{0}$$c_{0},$$B_{1}$$(S(t), \mathcal{H}, H)$$\mathcal{M}$$\mathcal{A}$$\mathcal{M}$$(S(t), \mathcal{H}, H)$

References[1]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of theattractor dimension for a chemotaxis growth system, J. London Math. Soc. 74 (2006),

453-474.
[2] N. D. Alikakos,
bounds of solutions of reaction-diﬀusion equations, Comm. Partial
4 (1979), 827-868.
EquationsDiﬀerential$L^{p}$

7[3] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative
Evolution Equations, Research in Applied Mathematics, vol. 37, John-Wiley and Sons,
Chichester, 1994.
[4] M. A. Herrero and J. J. L. Vel\’azquez, A blow-up mechanism for a chemotaxis model, Ann.
Scoula Norm. Sup. Pisa Cl. Sci. IV 24 (1997), 633-683.
[5] T. Hillen and K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol.
58 (2009), 183-217.
[6] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math. 12 (2001), 159-177.
[7] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,
J. Theor. Biol. 26 (1970), 399-415.
[8] N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena ofpattern solution to Mimura-Tsujikawa model in one dimension, GAKUTO Internat. Ser.
Math. Sci. Appl. 29 (2008), 265-278.
[9] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in achemotaxis-diﬀusion-growth model, Physica D241 (2012), 1629-1639.
[10] M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model
including growth, Physica A 230 (1996), 499-543.
[11] J. D. Murray, Mathematical Biology, II: Spatial Models and Biomedical Applications, 3rdedition, Springer-Verlag, New York, 2003.
[12] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domain, J. Inequal. Appl. 6 (2001), 37-55.
[13] E. Nakaguchi and K. Osaki,-Estimates of Solutions to -Dimensional ParabolicParabolic System for Chemotaxis with Subquadratic Degradation, Funkcial. Ekvac., inpress.
[14] E. Nakaguchi and K. Osaki, Global Solutions and Exponential Attractors of a ParabolicParabolic System for Chemotaxis with SubQuadratic Degradation, DCDS-B 18 (2013),$L^{p}$$n$2627-2646.
[15] E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic systemfor chemotaxis with weak degradation, Nonlinear Anal. 74 (2011), 286-297.
[16] T. Narumi and K. Osaki, Three-Dimensional Pattern Formations in a Biological Modelof Chemotaxis and Growth,
1917 (2014), 86-93.
[17] T. Okuda and K. Osaki, Bifurcation of Hexagonal Patterns in a Chemotaxis-DiﬀusionGrowth System, Nonlinear Anal. Real World Appl. 12 (2011), 3294-3305.
[18] T. Okuda and K. Osaki, Application of Normal Form Theory to a Chemotaxis System
in One Dimension,
1742 (2011), 49-61.
[19] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal. 51 (2002), 119-144.
[20] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica
240 (2011), 363-375.
[21] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial
Diﬀerential Equations 32 (2007), 849-877.
[22] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2ndedition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.
[23] M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical
approaches used to model bacterial chemotaxis II: Bacterial populations, Bull. Math.
Biol. 70 (2008), 1570-1607.
[24] K. Uemichi and K. Osaki, Hopf Bifurcation of Oscillatory Solutions to One-Dimensional$\overline{F_{\backslash }}g\beta\star\not\cong \mathscr{X}\ddagger\Sigma\Phi ffi\Re_{J\iota\overline{|^{1}}]i\ovalbox{\tt\small REJECT}-y_{\iota\ovalbox{\tt\small REJECT}}^{t}}^{pt}$$\overline{p_{\backslash }}ffi;\lambda\ovalbox{\tt\small REJECT} \mathscr{X}\ovalbox{\tt\small REJECT}\emptyset F7\hslash\Re^{p}j_{L}^{D}\overline{p}fi-\ovalbox{\tt\small REJECT}_{J\iota}^{p セ}\ovalbox{\tt\small REJECT}$$D$8
Chemotaxis-Growth System, Proceedings of the International Conference Functional
Analysis andEvolution Equations and Control
honor of Prof.
Shin-ichi Nakagiri on the occasion of his retirement edited by G. Akagi (2014), 207-210.
[25] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis
system with logistic source, Comm. Partial Diﬀerential Equations 35 (2010), 1516-1537.
[26] M. Winkler, Aggregation vs. global diﬀusive behavior in the higher-dimensional KellerSegel model, J. Diﬀerential Equations 248 (2010), 2889-2905.
$Application\fbox{Error::0x0000}$$Theor\fbox{Error::0x0000}in$