Hiroshima Math. J.
30 (2000), 463±497
Chemotactic collapse in a parabolic system of
mathematical biology
Toshitaka Nagai, Takasi Senba, and Takashi Suzuki
(Received June 11, 1999)
(Revised January 21, 2000)
Abstract. In 1970, Keller and Segel proposed a parabolic system describing the
chemotactic feature of cellular slime molds and recently, several mathematical works
have been devoted to it. In the present paper, we study its blowup mechamism and
prove the following. First, chemotactic collapse occurs at each isolated blowup
point. Next, any blowup point is isolated, provided that the Lyapunov function is
bounded from below. Finally, only the origin can be a blowup point of radially
symmetric solutions.
1.
Introduction
A system of parabolic partial di¨erential equations of mathematical biology is attracting interest. It was proposed by Nanjundiah [22] in 1973, as a
simpli®ed model of the Keller and Segel system [16] describing a chemotactic
feature, the aggregation of some organisms (cellular slime molds) sensitive to
gradient of a chemical substance. Precisely, with u…x; t† and v…x; t† standing
for the density of the organism and the concentration of the chemical substance
at the position x A W and the time t A …0; T†, respectively, it is given as
8
qu
>
>
in W …0; T†
>
> qt ˆ ` …`u ÿ wu`v†
>
>
>
>
>
< t qv ˆ Dv ÿ gv ‡ au
in W …0; T†
qt
…KS†
>
>
>
qu qv
>
>
ˆ
ˆ0
on qW …0; T†
>
>
qn qn
>
>
:
v… ; 0† ˆ v0
on W;
u… ; 0† ˆ u0 ;
where
(A1) t, a, g and w are positive constants
(A2) W is a bounded domain in R 2 with smooth boundary qW
(A3) n denotes the unit outer normal vector
2000 Mathematics Subject Classi®cation. 35K57, 92C15
Key words and phrases. system of parabolic equations, chemotactic collapse, blowup mechanism
464
Toshitaka Nagai et al.
(A4) u0 and v0 are smooth, nonnegative, and nontrivial initial values on W.
The ®rst equation describes the conservation of mass. Flux of u is given by
F ˆ ÿ`u ‡ wu`v
so that the e¨ect of di¨usion ` `u and that of chemotaxis ` wu`v are
competing for u to vary. The second equation is linear, and v is produced
proportionarily to u, di¨uses, and is destroyed by a certain rate.
The phenomenon of the blowup in a ®nite time of the solution is important from both mathematical and biological points of view. There are
conjectures by Nanjundiah [22], Childress [5], and Childress and Percus [6];
c ˆ c ˆ c ˆ 8p=…aw† is the threshold number in the following sense: if
ku0 kL 1 …W† < c then the solution exists globally in time and if ku0 kL 1 …W† > c then u…x; t† can form a delta function singularity in a ®nite time. The latter
case is referred to as the chemotactic collapse. The arguments were heuristic,
making use of numerical computations for the stationary problem, while recent
studies are supporting their validity rigorously ([12], [14], [19], [20] and [21]).
First, the existence of such numbers c and c was proven by JaÈger and
Luckhaus [14] for a simpli®ed system. Later, Nagai [19] treated another
system, (KS) with t ˆ 0, referred to as N model in the present paper; as [6]
conjectured, 8p=…aw† is actually the threshold number in the above sense for
radially symmetric solutions. Then, several works were devoted to the full
system, (KS) with t > 0. Particularly, Herrero and VelaÂzquez [12] constructed
a radially symmetric solution with u collapsing at the origin in a ®nite time,
having the concentrated mass equal to 8p=…aw†. Its counter part was shown by
Nagai, Senba, and Yoshida [21]; radial solutions exist globally in time with
uniformly bounded, provided that ku0 kL 1 …W† < 8p=…aw†. In this way, conjecture
[5] has been almost settled down in the a½rmative for radially symmetric
solutions.
As for the general case, contrarily to the conjecture, [21] gave only
ku0 kL 1 …W† < 4p=…aw†
as a criterion for the existence of global solutions. (The same result is
obtained by Biler [3] and Gajewski and Zacharias [7] independently.) But this
number 4p=…aw† is also realized to be best possible and the reason for the
discrepancy between radial and non-radial cases has been clari®ed by Nagai,
Senba and Suzuki [20] and Senba and Suzuki [23]. Namely, the former
studied N model and showed, among others, that if 4p=…aw† U ku0 kL 1 …W† <
8p=…aw† and the solution blows up in a ®nite time then the concentration
toward qW occurs to u. (This phenomenon is proven also for the full system
recently by Senba and Suzuki [25] and Harada, Nagai, Senba, and Suzuki
[10].) On the other hand the latter studied the stationary problem in details;
Chemotactic collapse in a parabolic system
465
the underlying variational structure and its e¨ects to the dynamics. In particular, it asserts that many non-radial stationary solutions, missed by [6], exist
and take roles in non-stationary problems even in the case that W is a disc.
Through those studies we are led to the following conjecture:
Component u forms a delta function singularity at each blowup point x0 A W
with the concentrated mass equal to 8p=…aw† and 4p=…aw† according to x0 A W
and x0 A qW, respectively.
Actually Senba and Suzuki [24] studied N model and proved the above
phenomenon with the mass greater than or equal to the expected values. The
present paper studies the full system and proves the following; if the solution
…u; v† blows-up in a ®nite time, then u forms a delta function singularity at each
isolated blowup point, and any blowup point is isolated, provided that the
Lyapunov function described below is bounded. Finally, only the origin can
be a blowup point of radially symmetric solutions.
2.
Summary
Let us put that
tˆaˆgˆwˆ1
for simplicity. The following facts are known.
1. ([27], [3]) Given smooth nonnegative initial data u0 2 0 and v0 , we have a
unique classical solution …u… ; t†; v… ; t†† to (KS) de®ned on the maximal time
interval ‰0; Tmax †. The solution is smooth and positive on W …0; Tmax †. If
Tmax < ‡y, then it holds that
lim sup ku… ; t†kLy …W† ˆ ‡y:
t"Tmax
2.
([21], [3], [7]) Putting
W …t† ˆ
we have
d
W …t† ‡
dt
… W
…
W
1
2
2
u log u ÿ uv ‡ …j`vj ‡ v † dx;
2
vt2 dx ‡
…
W
uj` …log u ÿ v†j 2 dx ˆ 0:
In particular, W …t† is a Lyapunov function.
that either
inf
0Ut<Tmax
W …t† > ÿy
It is monotone decreasing, so
…1†
466
Toshitaka Nagai et al.
or
lim W …t† ˆ ÿy
t"Tmax
holds.
We prepare several notations and de®nitions.
Notation
B…x0 ; R† ˆ fx A R 2 j jx ÿ x0 j < Rg, where x0 A R 2 and R > 0
A…x0 ; r; R† ˆ B…x0 ; R†nB…x0 ; r†
M…S† ˆ fRadon measures on Sg, where S denotes a compact
Hausdor¨ space
(iv) w ÿ lim ˆ weak star limit in M…S†
( v ) d…† ˆ Dirac's delta function concentrated at x ˆ 0 in R 2 and
dx0 …† ˆ d… ÿ x0 † for x0 A R 2
(vi) jWj ˆ the Lebegue measure of W H R 2
(i)
( ii )
(iii)
Definition
In the case of Tmax < ‡y, we say that x0 A W is a blowup point of u if
y
there exist ftk gy
kˆ1 H ‰0; Tmax † and fxk gkˆ1 H W satisfying u…xk ; tk † !
‡y, tk ! Tmax , and xk ! x0 as k ! y. The set of blowup points of u
is denoted by B.
( ii ) We say that x0 A B is isolated if there exists R > 0 such that
(i)
sup
0Ut<Tmax
ku… ; t†kLy …A…x0 ; r; R†VW† < ‡y
for any r A …0; R†. The set of isolated blowup points of u is denoted by
BI .
(iii) System (KS) is called radially symmetric if W ˆ fx A R 2 j jxj < 1g and
u0 ˆ u0 …jxj†, v0 ˆ v0 …jxj†.
Our results are stated as follows.
Theorem 1.
Given x0 A BI , we have 0 < R f 1, m V m , and
f A L 1 …B…x0 ; R† V W† V C…B…x0 ; R† V Wnfx0 g†;
satisfying f V 0 and
w ÿ lim u… ; t†dx ˆ mdx0 …dx† ‡ f dx
t"Tmax
in M…B…x0 ; R† V W†, where
m ˆ
8p
4p
…x0 A W†
…x0 A qW†
…2†
Chemotactic collapse in a parabolic system
Theorem 2.
467
If (1) occurs, then B ˆ BI .
Theorem 3. If (KS) is radially symmetric and Tmax < ‡y, then B ˆ f0g.
In our notation, the delta function dx0 …dx† A M…W† acts as
hh; dx0 …dx†i ˆ h…x0 †
for x0 A W and h A C…W†. It is easy to see that L 1 norm of u… ; t† is preserved
(see section 3). Therefore, Theorem 1 implies that the number of isolated
blowup points is ®nite. More precisely,
2 ]…BI V W† ‡ ]…BI V qW† U ku0 kL 1 …W† =4p:
Condition (1) actually holds for the blowup solution constructed by [12].
However, except for this example any criteria for (1) have not been known. In
this connection, it may be worth mentioning about the semilinear heat equation
ut ˆ Du ‡ juj pÿ1 u
in W …0; T†
with ujqW ˆ 0
…3†
n‡2
, it is
nÿ2
known that blowup occurs if and only if limt"Tmax J…u…t†† ˆ ÿy, where
on a bounded domain W H R n .
For the subcritical case 1 < p <
1
1
p‡1
kukp‡1
J…u† ˆ k`uk22 ÿ
2
p‡1
stands for the Lyapunov function ([8], [13], e.g.). To our knowledge, it has not
been clari®ed whether limt"Tmax J…u…t†† > ÿy and Tmax < ‡y can occur for
the other cases of (3). But those relations between Lyapunov functions and
blowup mechanisms may suggest that (KS) with two space dimension obeys
some features of (3); in the former case the boundedness of the Lyapunov
function implies the ®niteness of blowup points.
Our theorems are proven through localized energy estimates, particularly
the localized Lyapunov function. Concluding the section, we describe it in
short.
The localized Lyapunov function is de®ned by
… 1
2
2
u log u ÿ uv ‡ …j`vj ‡ v † j dx;
Wj …t† ˆ
2
W
where j is a nonnegative C y function. If j 1 1, Wj …t† is equal to W …t†, but
usually j is a cut-o¨ function satisfying
0UjU1
in R 2 ;
Actually it is taken in the following way.
qj
ˆ0
qn
on qW:
…4†
468
Toshitaka Nagai et al.
Given x0 A W, we have 0 < R 0 < R with B…x0 ; 2R† H W. Then we take j
satisfying
1
…x A B…x0 ; R 0 ††
j…x† ˆ
…5†
0
…x A R 2 nB…x0 ; R††:
Given x0 A qW, we ®rst prepare z A C0y …R 2 † satisfying z ˆ z…j yj†, 0 U z U 1 in
R 2 , and
1
…y A B…0; 1=2††
z…y† ˆ
0
…y A R 2 nB…0; 1††:
Next, we take a smooth conformal mapping X : B…x0 ; 2R† ! O H R 2 satisfying
x0 7! 0 and
X …B…x0 ; R† V W† H f…x1 ; x2 †jx2 > 0g
X …B…x0 ; R† V qW† H f…x1 ; x2 †jx2 ˆ 0g
X …B…x0 ; R 0 †† H B…0; 1=2†
X …R 2 nB…x0 ; R†† H R 2 nB…0; 1†
for 0 < R 0 < R f 1.
Then we set j…x† ˆ z…X …x††.
It holds that
q
qX
zX ˆ
…`z X † ˆ 0
on qW
qn
qn
because …qX †=…qn† is proportional to …0; ÿ1† on qW, and such j satis®es (4)
and (5).
We have the following.
Lemma 2.1. It holds that
…
…
…
d
d
2
2
Wj …t† ‡ vt j dx ‡ uj`…log u ÿ v†j j dx ˆ
uj dx ‡ R1 …u; v; j†;
dt
dt W
W
W
where
…
R1 …u; v; j† ˆ
…6†
…
W
‰…1 ÿ v†`u ÿ …u log u ÿ uv ‡ vt †`vŠ `j dx ‡
W
…u log u†hj dx:
Proof. Multiplying …log u ÿ v†j by the ®rst equation of (KS) and using
Green's formula, we have
…
ut …log u ÿ v†j dx
W
…
ˆ
W
` …`u ÿ u`v†…log u ÿ v†j dx
…
ˆÿ
W
2
uj`…log u ÿ v†j j dx ÿ
…
W
…log u ÿ v†…`u ÿ u`v† `j dx:
…7†
Chemotactic collapse in a parabolic system
469
Here, it holds that
…
…
…
…
d
d
ut …log u ÿ v†j dx ˆ
…u log u ÿ uv†j dx ÿ
uj dx ‡ uvt j dx
dt W
dt W
W
W
and
…
…
W
…log u†`u `j dx ˆ ÿ
W
…
u` …log u`j†dx ‡
…
ˆÿ
W
qW
…u log u†
…8†
qj
dx
qn
f…u log u†hj ‡ `u `jgdx:
…9†
In use of the second equation of (KS), we have
…
…
uvt j dx ˆ …vt ÿ hv ‡ v†vt j dx
W
W
… ˆ
W
vt2
…
1 q
2
2
…j`vj ‡ v † j dx ‡ vt `v `j dx;
‡
2 qt
W
which, together with (7), (8) and (9), leads to
…
…
d
Wj ‡ vt2 j dx ‡ uj`…log u ÿ v†j 2 j dx
dt
W
W
…
…
d
uj dx ‡ …u log u†hj dx
ˆ
dt W
W
…
‡ ‰…1 ÿ v†`u ÿ …u log u ÿ uv ‡ vt †`vŠ `j dx:
W
The proof is complete.
r
We sometimes write j ˆ jx0 ; R 0 ; R .
Now we describe the way of proof and some technical di½culties.
Theorem 1 is proven by the method of [20], localizing estimates of [21]. The
crucial point for the proof of Theorem 2 is showing ®niteness of blowup
points. As is described in [24], it follows if local L 1 norms of u have bounded
variations in time, and this actually holds if the Lyapunov function is
bounded. (In N model, it can be shown that the local L 1 norms have always
bounded variation in time thanks to remarkable properties of the Green's
function. See [24].) Finally, Theorem 3 is a consequence of those arguments.
3.
Preliminaries
Regard ÿh ‡ 1 as a closed operator in L p …W† …1 < p < y†, denoted by
Ap , by
470
Toshitaka Nagai et al.
D…Ap † ˆ
qv
ˆ 0 on qW :
v A W 2; p …W† qn
It is sectorial so that ÿAp generates an analytic semigroup denoted by fTp …t†g
(see [15]). The spectrum s…Ap † is independent of p and satis®es s…Ap † H
fz A C j Re…z† V 1g. We have the following because W H R 2 is bounded and
qW is smooth (see [26]):
Tp …t† is an operator of integration with the symmetric kernel G…x; y; t†
independent of p, satisfying
!
C
jx ÿ yj 2 ÿdt
a b
…10†
e
jDx Dy G…x; y; t†j U …2‡jaj‡j bj†=2 exp ÿ
Ct
t
for jaj U 2, jbj U 2, and …x; y; t† A W W …0; ‡y†, where C > 0 is a constant
and 0 < d < 1.
An immediate consequence is
kTp …t†kL…L p …W†; L p …W†† U Cp ;
where Cp > 0 is a constant determined by p A …1; y†.
For b A ‰0; 1Š the fractional powers Apb of Ap is de®ned, and the domain
b
Xp ˆ D…Apb † is a Banach space under the norm kukX b ˆ kApb ukL p …W† . We have
p
the following ([11]):
2
2
Xpb H W k; q …W† and Xpb H C m …W†, provided that k ÿ < 2b ÿ , q V p and
q
p
2
0 U m < 2b ÿ , respectively.
p
Making use of those estimates instead of the elliptic estimate for the
second equation, we get the following similarly to N model (see [20]). We
have v…t0 † A D…Ap † for 0 < t0 < Tmax and henceforth suppose that v0 A D…Ap †.
Proposition 3.1. The following relations hold for the solution …u; v† to
(KS), where Cq; e > 0 is a constant determined by q A …1; 2† and e A …0; 1=2†:
ku… ; t†kL 1 …W† ˆ ku0 kL 1 …W†
kv… ; t†kW 1; q …W† U Cq; e …kAq1=2‡e v0 kL q …W† ‡ ku0 kL 1 …W† †
Proof. Integrating the equations of (KS) over W, we have
…
d
u…x; t†dt ˆ 0
dt W
…11†
…12†
…13†
Chemotactic collapse in a parabolic system
and
d
dt
…
…
W
v…x; t†dt ˆ ÿ
471
…
W
v…x; t†dt ‡
Equality (13) implies (11) because u > 0.
W
u…x; t†dt:
…14†
Then,
kv… ; t†kL 1 …W† ˆ eÿt kv0 kL 1 …W† ‡ …1 ÿ eÿt †ku0 kL 1 …W†
follows from (14) and v > 0.
PoincareÂ-Wirtinger's inequality assures the equivalence
kvkW 1; q …W† A k`vkL q …W† ‡ kvkL 1 …W† ;
so that (12) is reduced to
k`v… ; t†kL q …W† U Cq; e …kAq1=2‡e v0 kL q …W† ‡ ku0 kL 1 …W† †:
Rewrite the second equation of (KS) as
…t
v… ; t† ˆ Tq …t†v0 ‡ Tq …t ÿ s†u… ; s†ds:
0
Inequality (15) will follow from
…t
` Tq …t ÿ s†u… ; t†ds
0
W 1; q …W†
U Cq ku0 kL 1 …W†
and
kTq …t†v0 kW 1; q …W† U Cq; e kAq1=2‡e v0 kW 1; q …W† :
In fact, we have
…t
q
` Tq …t ÿ s†u… ; t†ds
0
L q …W†
q
… … t …
ˆ `x G…x; y; t ÿ s†u… y; s†dyds dx
W
…
U
W
0 W
I …t† qÿ1 II …x; t†dx
with
Iˆ
…t …
0 W
…t ÿ s†…6ÿ5q†=…4qÿ4† u…y; s†e dq…sÿt†=…2qÿ2† dyds
and
II ˆ
…t …
0 W
…t ÿ s†…5qÿ6†=4 j`x G…x; y; t ÿ s†j q u…y; s†e dq…tÿs†=2 dyds:
…15†
472
Toshitaka Nagai et al.
If q A …1; 2† then …6 ÿ 5q†=…4q ÿ 4† > ÿ1, so that we have
I …t† U Cq ku0 kL 1 …W† :
On the other hand, inequality (10) gives that
…
W
II …x; t†dx U C
UC
… …t …
W 0 R2
…y
0
…t ÿ s†
ÿ…q‡6†=4
!
qjx ÿ yj 2
exp ÿ
u…y; s†e dq…sÿt†=2 dxdsdy
C…t ÿ s†
tÿ…q‡2†=4 eÿdqt=2 dtku0 kL 1 …W† ˆ Cq ku0 kL 1 …W† :
Therefore, we get
…t
q
` Tq …t ÿ s†u… ; s†ds
q
0
L …W†
U Cq ku0 kLq 1 …W† :
Finally,
kTq …t†v0 kW 1; q …W† U Cq; e kAq1=2‡e Tq …t†v0 kL q …W†
ˆ Cq; e kTq …t†Aq1=2‡e v0 kL q …W† U Cq;0 e kAq1=2‡e v0 kL q …W†
and (15) follows.
The proof is complete.
r
We note that inequality (122) with 1 U q < 2 implies
kv… ; t†kp U Cp
…16†
for 1 < p < y.
Lemma 5.10 of Adams [1] reads;
kwkL2 2 …W† U K 2 …kwkL2 1 …W† ‡ k`wkL2 1 …W† †
…17†
for w A W 1; 1 …W†, where K > 0 is a constant determined by W. Inequality (17)
implies some estimates on u.
Recall the cut-o¨ function jx0 ; R 0 ; R intoduced at the end of section 2.
Then, c ˆ …jx0 ; R 0 ; R † 6 satis®es
1
…x A B…x0 ; R 0 ††
c…x† ˆ
0
…x A R 2 nB…x0 ; R††
0UcU1
j`cj U Ac 5=6 ;
qc
ˆ0
qn
on qW
jhcj U Bc 2=3
in R 2 ;
in R 2 ;
where A > 0 and B > 0 are constants determined by 0 < R 0 < R f 1.
Chemotactic collapse in a parabolic system
Lemma 3.2
constant:
…
2
W
…
W
u c dx U 2K
u 2 c dx U
473
The following inequalities hold for any s > 1, where C > 0 is a
2
…
…
4K 2
log s
B…x0 ; R†VW
u dx …
B…x0 ; R†VW
W
2
ÿ1
u j`uj c dx ‡ K
…u log u ‡ eÿ1 †dx …
W
2
A2
‡ 1 kukL2 1 …W†
2
uÿ1 j`uj 2 c dx
‡ CkukL2 1 …W† ‡ 3s 2 jWj
…
W
u 3 c dx U
72K 2
log s
…
B…x0 ; R†VW
…18†
…19†
…u log u ‡ eÿ1 †dx …
W
j`uj 2 c dx
‡ CkukL3 1 …B…x0 ; R†VW† ‡ 10jWjs 3
…20†
Proof. Putting w ˆ uc 1=2 , we have
…
W
j`wjdx
2
…
U2
W
…
U2
…
U2
W
j`ujc 1=2 dx
j`ujc 1=2 dx
2
2
…
B…x0 ; R†VW
u dx W
…
‡2
‡
W
uj`c 1=2 jdx
2
A2
kukL2 1 …W†
2
uÿ1 j`uj 2 c dx ‡
A2
kukL2 1 …W† :
2
Hence (18) follows from (17) and kwkL 1 …W† U kukL 1 …W† .
We turn to (19). Take w ˆ …u ÿ s†‡ c 1=2 with a‡ ˆ maxfa; 0g.
kwkL2 2 …W† ˆ
…
fu>sg
…u ÿ s† 2 c dx
1 2
u ÿ s 2 c dx
fu>sg 2
…
…
…
1 2
1 2
u c dx ÿ
u c dx ÿ s 2 c dx
ˆ
W2
fuUsg 2
W
…
1
3
u 2 c dx ÿ s 2 jWj:
V
2 W
2
…
V
On the other hand we have kwkL2 1 …W† U kukL2 1 …W† and
We have
474
Toshitaka Nagai et al.
k`wkL2 1 …W† U
(…
)2
fu>sg
…j`ujc 1=2 ‡ …u ÿ s†‡ j`c 1=2 j†dx
(…
U2
)2
fu>sg
j`ujc
1=2
j`ujc
1=2
(…
U2
dx
…
‡2
)2
fu>sg
dx
‡
uj`c
W
1=2
jdx
2
A2
kukL2 1 …W† :
2
Here,
(…
fu>sg
)2
j`ujc
1=2
dx
…
U
U
…
B…x0 ; R†Vfu>sg
1
log s
u dx …
B…x0 ; R†VW
fu>sg
uÿ1 j`uj 2 c dx
…u log u ‡ eÿ1 †dx …
W
uÿ1 j`uj 2 c dx
because s log s V ÿeÿ1 for any s > 0. This implies (19).
Finally, take w ˆ …u ÿ s†‡3=2 c 1=2 . We have
kwkL2 2 …W† ˆ
…
fu>sg
…u ÿ s†‡3 c dx
1 3
3
u ÿ s c dx
V
fu>sg 4
…
V
1
4
…
W
5
u 3 c dx ÿ s 3 jWj:
4
Because
j`wj U
3
1
…u ÿ s†‡1=2 j`ujc 1=2 ‡ A…u ÿ s†‡3=2 c 1=3
2
2
we have
k`wkL2 1 …W†
9
U
2
(…
)2
fu>sg
Here, it holds that
…u ÿ s†
1=2
j`ujc
1=2
dx
A2
‡
2
(…
fu>sg
)2
…u ÿ s†
3=2
c
1=3
dx
:
Chemotactic collapse in a parabolic system
(…
)2
fu>sg
…u ÿ s† 1=2 j`ujc 1=2 dx
(…
U
)2
fu>sg
u 1=2 j`ujc 1=2 dx
…
U
475
…
B…x0 ; R†Vfu>sg
u dx …
U
1
log s
)2
(…
B…x0 ; R†VW
fu>sg
j`uj 2 c dx
…u log u ‡ eÿ1 †dx …
W
j`uj 2 c dx
and
(…
fu>sg
…u ÿ s†
3=2
c
1=3
dx
U
)2
fu>sg
…
U
W
…
Ue
W
3
uc
1=3 1=2
u c dx
u
2=3
9
2 log s
‡
Since c
1=2
Uc
A2
e
2
…
B…x0 ; R†VW
…
W
kukL 1 …B…x0 ; R†VW† jWj 1=3
u 3 c dx ‡ Ce jWjkukL3 1 …B…x0 ; R†VW† ;
where Ce > 0 is a constant determined by e > 0.
k`wkL2 1 …W† U
dx
Therefore,
…u log u ‡ eÿ1 †dx u 3 c dx ‡
…
W
j`uj 2 c dx
A2
Ce jWj kukL3 1 …B…x0 ; R†VW† :
2
1=3
, it follows from (21) that
…
kwkL2 1 …W† U e u 3 c dx ‡ Ce jWjkukL3 1 …B…x0 ; R†VW† :
W
We get
2
…
1
2 A
‡1 e
ÿK
u 3 c dx
2
4
W
…
…
9K 2
…u log u ‡ eÿ1 †dx j`uj 2 c dx
U
log s B…x0 ; R†VW
W
‡ K 2 Ce jWj
by (17).
Taking e > 0 as
2
A
5
‡ 1 kukL3 1 …B…x0 ; R†VW† ‡ s 3 jWj
4
2
…21†
476
Toshitaka Nagai et al.
2
1
1
2 A
ÿK
‡1 eˆ ;
4
8
2
we obtain (20).
4.
r
Finiteness of blowup points
This section is devoted to the proof of Therems 2 and 3. First, a
technical estimate is derived for local norms of the solution …u; v† to (KS).
Henceforth, we always asume Tmax < ‡y and a generic positive constant
(possibly changing from line to line) is denoted by C.
Lemma 4.1. It holds that
…
…
…
d
1
…u log u†c dx ‡
uÿ1 j`uj 2 c dx ‡ uvc dx
dt W
2 W
W
…
…
…
1
1 d
vt2 c dx ‡
…j`vj 2 ‡ v 2 †c dx U 2 u 2 c dx ‡ C:
‡
2 W
2 dt W
W
Proof. We show the following equality ®rst:
…
…
…
d
1 d
…u log u†c dx ‡ uÿ1 j`uj 2 c dx ‡ uvc dx ‡
…j`vj 2 ‡ v 2 †c dx
dt W
2 dt W
W
W
…
…
u 2 c dx ÿ vt2 c dx ÿ I ÿ II ÿ III ÿ IV ;
ˆ
…22†
…
W
W
where
…23†
…
Iˆ
W
vt `v `c dx
…
II ˆ
W
…1 ‡ log u†`u `c dx
…
III ˆ
W
v…1 ‡ log u†`u `c dx
…
IV ˆ
W
…uv log u†hc dx:
This can be derived by (6), but here we prove it directly by (KS).
In fact, multiplying …log u†c by the ®rst equation of (KS), we get that
…
…
…
d
…u log u†c dx ˆ
ut …log u†c dx ‡ ut c dx
dt W
W
W
…
ˆ f` …`u ÿ u`v†g…1 ‡ log u†c dx
W
Chemotactic collapse in a parabolic system
…
ˆÿ
477
…
W
`u `f…1 ‡ log u†cgdx ‡
W
u`v `f…1 ‡ log u†cgdx
ˆ ÿV ‡ VI :
Here,
…
u`v …uÿ1 c`u ‡ …1 ‡ log u†`c†dx
…
…
ˆ …`v `u†c dx ‡ u…1 ‡ log u†`v `c dx
W
W
…
…
ˆ ÿ u` …c`v†dx ‡ u…1 ‡ log u†`v `c dx
W
W
…
…
ˆ ÿ uchv dx ‡ …u log u†`v `c dx
W
W
…
…
ˆ ÿ u…vt ‡ v ÿ u†c dx ‡ …u log u†`v `c dx
W
W
…
…
…
ˆ ÿ u…vt ‡ v ÿ u†c dx ÿ v…u log u†Dc dx ÿ v…1 ‡ log u†`u `c dx
VI ˆ
W
W
W
W
by the second equation of (KS). On the other hand,
…
`u fuÿ1 c`u ‡ …1 ‡ log u†`cgdx
Vˆ
W
…
…
2
ÿ1
u j`uj c dx ‡ …1 ‡ log u†`u `c dx:
ˆ
W
W
Therefore, it holds that
…
…
…
d
…u log u†c dx ‡ uÿ1 j`uj 2 c dx ‡ uvc dx
dt W
W
W
…
…
…
ˆ …u 2 ÿ vt u†c dx ÿ …1 ‡ log u†`u `c dx ‡ …u log u†`v `c dx
W
W
W
…
2
ˆ …u ÿ vt u†c dx ÿ II ÿ III ÿ IV :
W
On the other hand we have
…
…
1 d
…j`vj 2 ‡ v 2 †c dx ˆ …`vt `v ‡ vt v†c dx
2 dt W
W
…
…
vt …ÿDv ‡ v†c dx ÿ vt `v `c dx
ˆ
W
…
ˆ
Equality (23) has been proven.
W
W
…ÿvt2 ‡ vt u†c dx ÿ I :
478
Toshitaka Nagai et al.
Now we proceed to the proof of (22). First, in use of (18), we get that
…
jII j U C u 1=6 u 1=3 j1 ‡ log ujc 1=3 uÿ1=2 j`ujc 1=2 dx
W
1=6
U CkukL 1 …W†
…
1=3 …
3
W
uj1 ‡ log uj c dx
2
ÿ1
W
u j`uj c dx
Recall the elementary inequality: Let 1 U a < 2 and b > 0.
b
a
2
u …1 ‡ jlog uj† U C…u ‡ 1†
1=2
:
Then
…u > 0†:
We obtain
1=6
jII j U Cku0 kL 1 …W†
1
U
4
…
…
2
W
u c dx ‡ 1
1
u j`uj c dx ‡
4
W
2
ÿ1
…
W
1=3 …
W
2
ÿ1
u j`uj c dx
1=2
u 2 c dx ‡ C
by (11).
Similarly, we have
…
jIII j U C uÿ1=2 j`ujc 1=2 u 1=2 j1 ‡ log ujc 1=3 v dx
W
…
UC
1
U
4
…
W
2
ÿ1
u j`uj c dx
1
u j`uj c dx ‡
4
W
2
ÿ1
and
…
jIV j U
…
U
Finally,
W
1
4
…
…
jI j U A
U
1
4
…
W
W
…
W
W
u
3=2
3
j1 ‡ log uj c dx
u 2 c dx ‡ C
juv log ujc 2=3 dx
W
U
by (16).
1=2 …
W
2=3
ju log uj 3=2 c dx
kvkL 3 …W†
u 2 c dx ‡ C:
jvt `vjc 5=6 dx
vt2 c dx ‡ A 2
…
W
j`vj 2 c 2=3 dx:
1=3
kvkL 6 …W†
Chemotactic collapse in a parabolic system
Here,
…
W
j`vj 2 c 2=3 dx ˆ ÿ
…
W
479
v` …c 2=3 `v†dx
… 2 ÿ1=3
v…ÿDv†c ÿ c
v`c `v dx
ˆ
3
W
…
…
2
v…ÿvt ‡ u ÿ v†c 2=3 dx ÿ
cÿ1=3 v`c `v dx
ˆ
3 W
W
…
…
1=6
1=2
1=2
1=6
vc uc dx ‡ vt c vc dx
U
W
2=3
W
…
2A
j`vjc 1=3 vc 1=6 dx
3 W
…
…
1
1
2
2
2
u
c
dx
‡
16A
kvk
‡
v 2 c dx
U
2
L …W†
8A 2 W
8A 2 W t
…
1
2A 2
kvkL2 2 …W† :
j`vj 2 c 2=3 dx ‡
‡ 32A 2 kvkL2 2 …W† ‡
2 W
9
‡
Therefore, it holds that
…
…
…
1
1
2
j`vj 2 c 2=3 dx U
v
c
dx
‡
u 2 c dx ‡ C;
4A 2 W t
4A 2 W
W
which implies
1
jI j U
2
…
W
vt2 c dx
1
‡
4
…
W
u 2 c dx ‡ C:
Inequality (22) has been proven.
r
We show a key fact for the proof of Theorems.
Proposition 4.2. Suppose Tmax < ‡y and let x0 A W and 0 < R f 1.
Then, if a solution …u; v† to (KS) satis®es
…
u log u dx < ‡y
…24†
sup
0Ut<Tmax B…x0 ; R†VW
it holds that
sup
0Ut<Tmax
for any r A …0; R†.
ku… ; t†kLy …B…x0 ; r†VW† < ‡y
…25†
480
Toshitaka Nagai et al.
Proof. We divide the argument in ®ve steps. Take R 0 A …0; R† and let
c ˆ …jx0 ; R 0 ; R † 6 .
Step 1 We show that (24) with Tmax < ‡y implies
… Tmax …
vt2 dxdt < ‡y:
…26†
0
B…x0 ; R 0 †VW
Therefore, taking R > 0 smaller, we can assume that
… Tmax …
vt2 dxdt < ‡y:
0
B…x0 ; R†VW
In fact, inequality (19) with
…
M ˆ sup
0Ut<Tmax B…x0 ; R†VW
…u log u ‡ eÿ1 †dx < ‡y
…27†
…28†
gives that
…
4K 2 M
u c dx U
log s
W
2
…
W
uÿ1 j`uj 2 c dx ‡ C ‡ 3s 2 jWj:
2
Therefore, taking s > 1 as 8K M=…log s† < 1=2, we have (26) by (22).
Step 2 Multiplying uc by the ®rst equation of (KS), we have
…
…
…
1 d
u 2 c dx ‡ j`uj 2 c dx ‡ u`u `c dx
2 dt W
W
W
…
…
uc`v `u dx ‡ u 2 `v `c dx:
ˆ
W
W
From the second equation of (KS) follows that
…
…
1
uc`v `u dx ˆ
c`v `u 2 dx
2 W
W
…
…
1
1
2
u chv dx ÿ
u 2 `v `c dx
ˆÿ
2 W
2 W
…
…
…
1
1
1
3
2
u c dx ÿ
u …vt ‡ v†c dx ÿ
u 2 `v `c dx
ˆ
2 W
2 W
2 W
…
…
…
1
1
1
3
2
u c dx ÿ
u vt c dx ÿ
u 2 `v `c dx:
U
2 W
2 W
2 W
Therefore, in use of
…
…
…
2
2
u `v `c dx ˆ ÿ v`u `c dx ÿ u 2 vhc dx
W
W
W
…29†
Chemotactic collapse in a parabolic system
and
…
1
u`u `c dx ˆ ÿ
2
W
…
W
481
u 2 hc dx;
we obtain
…
…
…
…
1d
1
1
u 2 c dx ‡ j`uj 2 c dx U
u 3 c dx ÿ
u 2 vt c dx
2 dt W
2
2
W
W
W
…
…
1
1
v`u 2 `c dx ‡
u 2 …v ‡ 1†hc dx: …30†
‡
2 W
2 W
Here, last three terms of the right-hand side are dominated as follows.
First, inequality (16) gives that
…
…
1 B
2
u …v ‡ 1†Dc dx U
…v ‡ 1† u 2 c 2=3 dx
2 W
2 W
…
2=3
B
1=3
3
u c dx
U …kvkL 3 …W† ‡ jWj †
2
W
…
1
U
u 3 c dx ‡ C:
3 W
Similarly,
…
…
1 2
U 3A v uc 1=3 j`ujc 1=2 dx
v`u
`c
dx
2 W
W
…
1=3 …
1=2
u 3 c dx
j`uj 2 c
U 3AkvkL 6 …W†
U
1
8
…
W
W
j`uj 2 c dx ‡
1
3
…
W
W
u 3 c dx ‡ C:
Finally, Gagliardo-Nirenberg's inequality
1=2
1=2
kwkL 4 …W† U K…k`wkL 2 …W† kwkL 2 …W† ‡ kwkL 2 …W† †
to w ˆ uc 1=2 implies that
(…
)1=2 …
…
1=2
1
1 2
2
4 2
u vt c dx U
v dx
u c dx
2 W
2 B…x0 ; R†VW t
W
(…
UC
B…x0 ; R†VW
)1=2
vt2
dx
…k`…uc 1=2 †kL 2 …W† kuc 1=2 kL 2 …W† ‡ kuc 1=2 kL2 2 …W† †
482
Toshitaka Nagai et al.
U
1
16
…
W
j`…uc 1=2 †j 2 dx ‡ C
(…
‡C
1
U
16
B…x0 ; R†VW
vt2 dx kuc 1=2 kL2 2 …W†
)1=2
B…x0 ; R†VW
…
W
…
j`…uc
1=2
kuc 1=2 kL2 2 …W†
vt2 dx
!
…
2
†j dx ‡ C
B…x0 ; R†VW
vt2
dx ‡ 1 kuc 1=2 kL2 2 …W†
Here, we have
…
j`…uc
W
so that
1=2
…
A2
†j dx U 2 j`uj c dx ‡
u 2 c 2=3 dx
2 W
W
…
…
16
2
u 3 c dx ‡ C;
U 2 j`uj c dx ‡
3 W
W
…
2
2
…
…
…
1 2
U 1 j`uj 2 c dx ‡ 1 u 3 c dx
u
v
c
dx
t
2 W
8 W
3 W
! …
…
‡C
B…x0 ; R†VW
vt2 dx ‡ 1
W
u 2 c dx ‡ C:
In this way, inequality (30) has been reduced to
1 d
2 dt
…
W
u 2 c dx ‡
…
U2
W
…
U3
W
3
4
…
W
j`uj 2 c dx
3
…
u c dx ‡ C
3
B…x0 ; R†VW
…
u c dx ‡ C
B…x0 ; R†VW
vt2
vt2
!…
dx ‡ 1
…
dx W
W
u 2 c dx ‡ C
!
2
u c dx ‡ 1 :
We can make use of (20) for the ®rst term of the right-hand side.
…31†
It holds
that
…
72K 2 M
u c dx U
log s
W
3
…
W
j`uj 2 c dx ‡ C ‡ 10jWjs 3 ;
where M > 0 is the constant de®ned in (28). Making s > 1 large, this term is
absorbed into the left-hand side of (31). We obtain
Chemotactic collapse in a parabolic system
d
dt
…
W
…
2
u c dx ‡
In particular, g…t† ˆ
…
2
W
„
W
j`uj c dx U C
B…x0 ; R†VW
vt2
483
!
…
dx 2
W
u c dx ‡ 1 :
u 2 c dx solves
dg
U hg ‡ C
dt
…0 U t < Tmax †
„T
with a continuous function h…t† V 0 satisfying 0 max h…t†dt < ‡y.
…
u 2 c dx < ‡y:
sup g…t† ˆ sup
0Ut<Tmax
This implies
0Ut<Tmax W
…32†
Setp 3 We take R 00 A …0; R 0 † and set c1 ˆ …jx0 ; R 00 ; R 0 † 6 . Multiplying u 2 c1
by the ®rst equation of (KS), we have
…
…
…
1 d
2
3
u c1 dx ‡ 2 uj`uj c1 dx ‡ u 2 `u `c1 dx
3 dt W
W
W
…
…
ˆ 2 u 2 c1 `v `u dx ‡ u 3 `v `c1 dx:
W
This means that
1 d
3 dt
…
W
…
…
8
2
2
w c1 dx ‡
j`wj c1 dx ‡
w`w `c1 dx
9 W
3 W
W
…
…
4
wc1 `v `w dx ‡ w 2 `v `c1 dx
ˆ
3 W
W
2
…33†
for w ˆ u 3=2 . From (32) and
w log w U 3w 4=3 ˆ 3u 2
we have
…
sup
0Ut<Tmax B…x0 ; R 0 †VW
…w log w†dx < ‡y:
Relation (33) is similar to (29). Inequality (20) holds with u replaced by
w, and kwkL 1 …B…x0 ; R 0 †VW† U C follows from (32). Finally, if we make use of the
second equation of (KS), we obtain
…
W
wc1 `v `w dx U
U
1
2
…
…
W
W
w 8=3 c1 dx ÿ
w 3 c1 dx ÿ
1
2
…
1
2
W
…
W
w 2 vt c1 dx ÿ
w 2 vt c1 dx ÿ
1
2
…
1
2
W
…
W
w 2 `v `c1 dx
w 2 `v `c1 dx ‡ C
484
Toshitaka Nagai et al.
similarly to (30).
Step 2 and get
Under those circumstances we can repeat the arguments in
…
…
3
sup
0Ut<Tmax B…x0 ; R 00 †VW
u dx ˆ
sup
0Ut<Tmax B…x0 ; R 00 †VW
w 2 dx < ‡y:
…34†
If we repeat the arguments once more, we get
sup
0Ut<Tmax
ku… ; t†kL 4 …B…x0 ; r†VW† < ‡y:
…35†
for any r A …0; R†.
Step 4 Put u1 ˆ uwB…x0 ; r† , u2 ˆ u ÿ u1 , and let v1 , v2 be the solutions for
8
in W …0; Tmax †;
vt ˆ hv ÿ v ‡ f
>
>
>
<
qv
ˆ0
on qW …0; Tmax †;
> qn
>
>
:
v… ; 0† ˆ 0
in W:
with f ˆ u1 , u2 , respectively.
…t …
v2 ˆ
It holds that
0 WnB…x0 ; r†
G…x; y; t ÿ s†u… y; s†dyds;
so that
sup
0Ut<Tmax
kv2 … ; t†kW 2; y …B…x0 ; r 0 †VW† < ‡y
for r 0 A …0; r† by (10) and (11).
To handle with v1 …x; t†, we recall the operator Ap in Section 3.
5=6 < b < 1 and p ˆ 3. Then we have
sup
0Ut<Tmax
kv1 … ; t†kX b ˆ
p
U
sup
0Ut<Tmax
by (34).
kApb v1 … ; t†kL p …W†
sup
…t
0Ut<Tmax 0
UC
Let
sup
kApb Tp …t ÿ s†u1 … ; s†kL p …W† ds
…t
0Ut<Tmax 0
…t ÿ s†ÿb ku1 … ; s†kL p …W† ds < ‡y
Inclusion Xpb H C 1 …W† holds and hence
sup
0Ut<Tmax
kv… ; t†kC 1 …B…x0 ; r†VW† < ‡y
for any r A …0; R†.
Step 5 Take r 0 A …0; r† and put c1 ˆ …jx0 ; r 0 ; r † 6 .
the ®rst equation of (KS) and get
…36†
We multiply u p c1p‡1 by
Chemotactic collapse in a parabolic system
d 1
dt p ‡ 1
…
W
…uc1 † p‡1 dx ˆ ÿ
…
W
`…u p c1p‡1 † `u dx ‡
…
W
485
u`…u p c1p‡1 † `v dx
ˆ ÿI ‡ II :
Here,
Iˆ
ˆ
ˆ
…
W
… pu pÿ1 c1p‡1 `u ‡ u p `c1p‡1 † `u dx
…
4p
…p ‡ 1† 2
4p
…p ‡ 1†
‡
(
4
p‡1
W
…
2
…
W
j`u… p‡1†=2 j 2 c1p‡1 dx ‡
…
W
`c1p‡1 `u p‡1 dx
j`u… p‡1†=2 j 2 c1p‡1 dx
… p‡1†=2
W
1
p‡1
c1
… p‡1†=2
`u… p‡1†=2 u… p‡1†=2 `c1
dx
)…
2
ÿ
j`u… p‡1†=2 j 2 c1p‡1 dx
ˆ
2
p
‡
1
… p ‡ 1†
W
…
…
2
2
… p‡1†=2 2
j`…uc1 †… p‡1†=2 j 2 dx ÿ
u p‡1 j`c1
j dx
‡
p‡1 W
p‡1 W
…
…
2
p‡1
j`…uc1 †… p‡1†=2 j 2 dx ÿ
u p‡1 c1pÿ1 j`c1 j 2 dx
V
p‡1 W
2
W
…
…
2
2
A … p ‡ 1†
… p‡1†=2 2
j`…uc1 †
j dx ÿ
…uc1 † p‡…2=3† u 1=3 dx
V
p‡1 W
2
W
…
2=3
…
2
A 2 … p ‡ 1†
1=3
ku0 kL 1 …W†
j`…uc1 †… p‡1†=2 j 2 dx ÿ
…uc1 † 1‡…3=2†p dx
:
V
p‡1 W
2
W
4p
Furthermore, (36) implies
…
II U C ju`…u p c1p‡1 †jdx
W
…
…
p
p‡1
p‡1 p
j`…uc1 † jdx ‡ … p ‡ 1† u c1 j`c1 jdx
UC
p‡1 W
W
…
…
2p
…uc1 †… p‡1†=2 j`…uc1 †… p‡1†=2 jdx ‡ CA…p ‡ 1† …uc1 † p‡…5=6† u 1=6 dx
UC
p‡1 W
W
…
…
1
j`…uc1 †… p‡1†=2 j 2 dx ‡ C… p ‡ 1† …uc1 † p‡1 dx
U
p‡1 W
W
…
5=6
1=6
…uc1 † 1‡…6=5†p
:
‡ CA… p ‡ 1†ku0 kL 1 …W†
W
486
Toshitaka Nagai et al.
It holds that
d
dt
…
W
u1p‡1 dx U ÿ
…
… p‡1†=2 2
W
j dx ‡ C…p ‡ 1† 2
j`u1
‡ C… p ‡ 1†
2
…
W
1‡…3=2†p
u1
…
W
u1p‡1 dx
2=3 …
5=6 !
1‡…6=5†p
dx
‡
u1
dx
;
…37†
W
where u1 ˆ uc1 . Here, C > 0 is independent of p V 1 and we can apply an
iteration scheme of Moser's type (see Alikakos [2]). To this end we make use
of Gagliardo-Nirenberg's inequality in the form of
1=q
kwkL q …W† U K…k`wkL2 2 …W† ‡ kwkL2 2 …W† †…1ÿ…1=q††=2 kwkL 1 …W† ;
where K > 0 independent of q A ‰1; q0 Š for given q0 > 1.
3p ‡ 2
5
… p‡1†=2
A ;3 .
and q ˆ
First, apply (38) for w ˆ u1
p‡1
2
C… p ‡ 1†
2
…
W
1‡…3=2†p
u1
U C… p ‡ 1†
Because
…
W
We have
2=3
… p‡1†=2 2
j`u1
j dx
…
‡
W
u1p‡1
dx
…2p‡1†=…3p‡3† …
W
… p‡1†=2
u1
dx
2=3
:
2p ‡ 1 2
< , the right-hand side is dominated by
3p ‡ 3 3
C…p ‡ 1† 2
U
2
dx
…38†
1
6
…
… p‡1†=2 2
W
j ‡ u1p‡1 †dx ‡ 1
…j`u1
…
j ‡ u1p‡1 †dx ‡ 1 ‡ C…p ‡ 1† 6
…j`u1
… p‡1†=2
and q ˆ
Second, apply (38) for w ˆ u1
C… p ‡ 1† 2
…
W
1‡…6=5†p
u1
U C… p ‡ 1†
2
U C… p ‡ 1†
…
W
2
…
W
W
… p‡1†=2 2
W
2=3 …
2=3
… p‡1†=2
u1
dx
dx
…
… p‡1†=2
W
u1
12p ‡ 10
22 12
A
;
.
5p ‡ 5
10 5
dx ‡ 1
2
:
We have
5=6
… p‡1†=2 2
j`u1
j dx ‡
… p‡1†=2 2
…j`u1
j
‡
…
W
u1p‡1
u1p‡1 †dx
dx
…7p‡5†=…12p‡12† …
‡1
W
7=12 …
W
… p‡1†=2
u1
… p‡1†=2
u1
5=6
dx
5=6
dx
Chemotactic collapse in a parabolic system
1
U
6
1
U
6
…
W
…
W
… p‡1†=2 2
…j`u1
j
… p‡1†=2 2
…j`u1
j
‡
u1p‡1 †dx
‡
u1p‡1 †dx
‡ 1 ‡ C… p ‡ 1†
‡ 1 ‡ C… p ‡ 1†
… p‡1†=2
Finally, apply (38) for w ˆ u1
…
C…p ‡ 1† 2 u1p‡1 dx
W
U C…p ‡ 1†
…
1
6
U
W
2
…
W
… p‡1†=2 2
…j`u1
j
and q ˆ 2.
‡
u1p‡1 †dx
24=5
6
487
…
…
W
W
… p‡1†=2
u1
… p‡1†=2
u1
dx
2
dx ‡ 1
2
:
We have
1=2 …
W
… p‡1†=2
u1
dx
…
2
… p‡1†=2
j ‡ u1p‡1 †dx ‡ 1 ‡ C…p ‡ 1† 4
u1
dx :
… p‡1†=2 2
…j`u1
W
Inequality (37) has been reduced to
…
…
d
1
… p‡1†=2 2
p‡1
u
dx ‡
j`u1
j dx
dt W 1
2 W
…
2
…
1
… p‡1†=2
u1p‡1 dx ‡ C…p ‡ 1† 6
u1
dx ‡ 1 :
U
2 W
W
However, again (38) for q ˆ 2 implies
kwkL2 2 …W† U K 2 …k`wkL2 2 …W† ‡ kwkL2 2 …W† † 1=2 kwkL 1 …W†
1
…k`wkL2 2 …W† ‡ kwkL2 2 …W† † ‡ CkwkL2 1 …W†
2
U
and hence
… p‡1†=2 2
kL 2 …W†
ku1
… p‡1†=2 2
kL 2 …W†
U 12k`u1
… p‡1†=2 2
kL 1 …W† :
‡ Cku1
We obtain
d
dt
and hence
sup
0Ut<Tmax
…
W
u1p‡1
…
W
1
dx ‡
4
u1p‡1
…
W
u1p‡1
dx U C…p ‡ 1†
6
…
W
… p‡1†=2
u1
dx ‡ 1
2
dx ‡ 1
(
U C max … p ‡ 1†
6
…
sup
0Ut<Tmax
W
… p‡1†=2
u1
dx ‡ 1
2
)
; ku0 kLp‡1
y …W† jWj
‡1 :
488
Toshitaka Nagai et al.
Therefore,
…
Fk ˆ
sup
k
0Ut<Tmax W
u12 dx ‡ 1
satis®es
Fk‡1 U C maxf2 6…k‡1† Fk2 ; …jWj ‡ 1†…ku0 kLy …W† ‡ 1† 2
U C2 6…k‡1† maxfFk2 ; …ku0 kLy …W† ‡ 1† 2
k‡1
k‡1
g
g
…39†
for k ˆ 1; 2; . . . :
Let d ˆ ku0 kLy …W† ‡ 1.
Then, (39) is reduced to
Pk
kÿl
kÿ1
k‡1
Fk‡1 U C 2kÿ3 2 lˆ2 6…l‡1†2 maxfF22 ; d 2 g
for k ˆ 2; 3; . . . : We have
…
sup
0Ut<Tmax
W
k‡1
u12
dx
1=2 k‡1
1=2 k‡1
U Fk‡1
U C …2kÿ3†=2
and letting k ! ‡y,
sup
0Ut<Tmax
follows.
ku1 … ; t†kLy …W† U C max
8
<
:
k‡1
2
6
Py
jˆ1
j2ÿj
1=4
maxfF2 ; dg;
!1=4
sup
0Ut<Tmax
ku1 … ; t†kL4 4 …W† ‡ 1
;d
9
=
;
In use of (35), we obtain
sup
0Ut<Tmax
ku1 … ; t†kLy …W† ˆ
sup
0Ut<Tmax
ku… ; t†c1 kLy …W† < ‡y:
Since r 0 A …0; r† and r A …0; R† are arbitrary, we have (25).
The proof is complete.
r
Theorems 2 and 3 are immediate consequences of the following.
Proposition 4.3. Let …u; v† be a solution to (KS) and Tmax < ‡y. Then,
any x0 A B and 0 < R f 1 admit
…
1
u…x; t†dx V
:
…40†
lim sup
2
16K
t"Tmax
B…x0 ; R†VW
Proof. Take r A …0; R† and c ˆ …fx0 ; r; R † 6 .
(18) implies
If (40) does not hold, then
Chemotactic collapse in a parabolic system
…
W
…
u 2 c dx U 2K 2
1
U
8
for 0 < Tmax ÿ t f 1.
…
B…x0 ; R†VW
…
W
u dx W
489
uÿ1 j`uj 2 c dx ‡ CkukL2 1 …W†
uÿ1 j`uj 2 c dx ‡ C
Then (22) gives
…
lim sup
t"Tmax
W
…u log u†c dx < ‡y;
and hence
lim sup ku… ; t†kLy …B…x0 ; r 0 †VW† < ‡y
t"Tmax
follows from Lemma 4.2, where r 0 A …0; r†.
complete.
Proof of Theorem 3.
We get x0 B B and the proof is
r
In this case it holds that
u ˆ u…jxj; t†:
…41†
If x0 A Bnf0g, we have S 1 fxj jxj ˆ jx0 jg H B.
Given a positive integer m, we take 0 < R f 1 and x1 ; . . . ; xm A S satisfying B…xi ; R† V B…xj ; R† ˆ q for i 0 j. Relation (40) admits a sequence
tk " Tmax satisfying
…
B…xj ; R†VW
u…x; tk †dx >
1
18K 2
for j ˆ 1 and hence for j ˆ 2; . . . ; m by (41). Therefore,
m …
X
m
ku… ; tk †kL 1 …W† V
u…x; tk †dx >
2
18K
jˆ1 B…xj ; R†VW
follows, which contradicts (11) if m V 18K 2 ku0 kL 1 …W† .
Proof of Theorem 2.
… Tmax …
0
follows.
W
The proof is complete.
r
If the solution satis®es (1), then
…vt2 ‡ uj`…log u ÿ v†j 2 †dxdt < ‡y
490
Toshitaka Nagai et al.
Take x0 A B, 0 < R f 1 and let j ˆ jx0 ; R=2; R . We have
…
…
…
d
ˆ ut j dx ˆ …`u ÿ u`v† `j dx
uj
dx
dt
W
W
W
…
uj`…log u ÿ v†jdx
UC
A…x0 ; R=2; R†
U C kukL 1 …W† ‡
…
W
2
uj`…log u ÿ v†j dx
…42†
and hence
… Tmax …
d
dt uj dxdt < ‡y:
0
W
„
This assures the existence of limt"Tmax W uj dx. In use of (40) we have
…
…
u V lim
uj dx
lim inf
t"Tmax
t"Tmax W
B…x0 ; R†VW
…
V lim sup
t"Tmax
V
B…x0 ; R=2†VW
u dx
1
:
16K 2
Since x0 A B and 0 < R f 1 is arbitrary, this implies that
]B U 16K 2 ku0 kL 1 …W† < ‡y
by (11), and in particular, any blow-up point is isolated.
5.
r
Isolated blowup points
In this section we study the behavior of u around the isolated blowup
points more precisely and prove Theorem 1.
We ®rst note the following.
Lemma 5.1. Let …u; v† be a solution to (KS) and x0 A BI . Then there exist
0 < R f 1 and y A …0; 1=2† such that
kukC 2‡2y; 1‡y ……A…x0 ; r; R†VW†‰0; Tmax †† ‡ kvkC 2‡2y; 1‡y ……A…x0 ; r; R†VW†‰0; Tmax †† < ‡y
for any r A …0; R†.
Proof. Because x0 A BI , there exists R0 > 0 such that
sup …ku… ; t†kLy …A…x0 ; r0 ; R0 †VW† ‡ kv… ; t†kLy …A…x0 ; r0 ; R0 †VW† † < ‡y
0Ut<Tmax
…43†
Chemotactic collapse in a parabolic system
491
for any r0 A …0; R0 †. Then the parabolic estimate for the second equation of
(KS) (see [26]) gives that
sup
0Ut<Tmax
k`v… ; t†kLy …A…x0 ; r; R†VW† < ‡y
for R and r in r0 < r < R < R0 and the standard theory for the ®rst equation
(see Theorem 10.1 of Chapter IV of [17]) applies; R 0 and r 0 in r < r 0 < R 0 < R
admit y A …0; 1=2† such that
kukC 2y; y ……A…x0 ; r 0 ; R 0 †VW†‰0; Tmax †† < ‡y:
Now Theorem 10.1 in Section IV of [17] is available for the second and the
®rst equation in turn, and, given R 00 and r 00 in r 0 < r 00 < R 00 < R 0 we have
y 0 A …0; 1=2† such that
kvkC 2‡2y 0 ; 1‡y 0 ……A…x0 ; r 00 ; R 00 †VW†‰0; Tmax †† < ‡y
and
kukC 2‡2y 0 ; 1‡y 0 ……A…x0 ; r 00 ; R 00 †VW†‰0; Tmax †† < ‡y:
Since r 00 is arbitrary, proof is complete.
r
An immediate consequence is the following.
Lemma 5.2.
have
Let x0 A BI and j ˆ jx0 ; R 0 ; R for 0 < R 0 < R f 1.
sup
0Ut<Tmax
and
…
lim sup
t"Tmax
W
Wj …t† < ‡y
Then we
…44†
j`vj 2 j dx ˆ ‡y:
…45†
Proof. Recall (6) and put
F …t† ˆ Wj …t† ÿ
…t
0
…
R1 …u; v; j†ds ÿ
Relations (11) and (43) imply
…
uj dx U ku0 k 1
and
L …W†
W
sup
0Ut<Tmax
W
uj dx:
jR1 …u; v; j†j < ‡y;
respectively. By Lemma 2.1, F is monotone decreasing in ‰0; Tmax † and (44)
follows. Then we have
492
Toshitaka Nagai et al.
…
W
…
…u log u†j dx U C ‡
and
W
uvj dx;
…
lim sup
t"Tmax
W
uvj dx ˆ ‡y
follows from Proposition 4.2. In use of Young's inequality we have
…
…
…
1
e av j dx
a uvj dx U …u log u†j dx ‡
e W
W
W
…
…
1
e av j dx
U Wj ‡ uvj dx ‡
e W
W
…
…
1
e av j dx;
U C ‡ uvj dx ‡
e W
W
and hence
…
1
…a ÿ 1† uvj dx U
e
W
If a > 1, we have
…
lim sup
t"Tmax
W
…
W
e av j dx ‡ C:
e av j dx ˆ ‡y;
which implies (45) by the following Lemma.
r
Lemma 5.3. Let a > 0, x0 A BI , and j ˆ jx0 ; R 0 ; R for 0 < R 0 < R f 1.
Then, the inequality
2…
…
a
e av j dx U C exp
j`vj 2 j dx
…46†
8p W
W
holds on ‰0; Tmax †.
If x0 A W, then
2 …
…
a
2
av
e j dx U C exp
j`vj j dx :
16p W
W
…47†
Proof. We recall the following inequalities due to Moser [18] and Chang
and Yang [4]: There exists a constant K determined by W such that
…
…
1
1
e w dx U k`wkL2 2 …W† ‡
w dx ‡ K
log
2p
jWj W
W
for w A X , where
Chemotactic collapse in a parabolic system
p ˆ
4p
8p
493
if X ˆ H 1 …W†
if X ˆ H01 …W†:
Because x0 A BI , we have
sup
0Ut<Tmax
Therefore, we get
…
…
e av j dx U
W
kv… ; t†kLy …A…x0 ; R 0 ; R†VW† < ‡y:
B…x0 ; R 0 †VW
e av dx ‡
…
A…x0 ; R 0 ; R†VW
e av j dx
a2
k`vkL2 2 …B…x0 ; R 0 †VW† ‡ C
U C exp
8p
2…
a
j`vj 2 j dx
U C exp
8p W
by (43). This shows (46).
proof is complete.
‡C
A similar calculation gives (47) if x0 A W. The
r
The following lemma is a modi®cation of [21].
Lemma 5.4. We have
…
…
…
v
uvj dx U …u log u†j dx ‡ Mj log
e j dx ÿ Mj log Mj ;
W
where Mj ˆ
W
„
W
W
…48†
uj dx.
Proof. Since ÿlog s is convex, Jensen's inequality applies as
… v
…
1
e u
e v j dx ˆ ÿlog
j dx
ÿlog
Mj W
W u Mj
… 1 v u
ÿlog e
j dx
U
u
Mj
W
v … 1
e
u log
j dx:
ˆÿ
u
Mj W
This means (48).
r
This implies the following.
Lemma 5.5. Suppose Tmax < ‡y and take x0 A BI , and 0 < R 0 < R f 1.
Then the relation
494
Toshitaka Nagai et al.
…
lim
t"Tmax W
uj dx V m
follows, where m is the constant in Theorem 1 and j ˆ jx0 ; R 0 ; R .
Proof. In use of (43), we have
…
d
dt uj dx U C
W
similarly to (42), so that limt"Tmax kujkL 1 …W† exists.
Suppose
lim Mj …t† ˆ lim kujkL 1 …W† < m :
t"Tmax
…49†
t"Tmax
In the case that x0 A W we have (47). Inequality (48) implies
…
…
1
2
2
…j`vj ‡ v †j dx ˆ Wj ÿ …u log u ÿ uv†j dx
2 W
W
…
v
e j dx ÿ Mj log Mj
U Wj ‡ Mj log
UC ‡
by (44).
Mj
16p
W
…
W
j`vj 2 j dx ‡ Mj log
C
Mj
It follows that
…
Mj
1
C
1ÿ
j`vj 2 j dx U C ‡ Mj log
:
8p
2
Mj
W
Therefore, (49) with m ˆ 8p gives
…
…
j`vj 2 dx U lim sup j`vj 2 j dx < ‡y:
lim sup
t"Tmax
t"Tmax
B…x0 ; R=2†VW
W
This contradicts (45) with R replaced by R=2.
The case x0 A qW can be treated similarly and the proof is complete. r
We are able to give the following.
Proof of Theorem 1.
lemma, the value
Let x0 A BI and j ˆ jx0 ; R=2; R . From above
…
m…x0 ; R† ˆ lim
t"Tmax W
u…x; t†jR …x†dx V m
exists for any x0 A BI and 0 < R f 1. Moreover,
…
u…x; t†…jR …x† ÿ jR=2 …x††dx V 0
m…x0 ; R† ÿ m…x0 ; R=2† ˆ lim
t"Tmax W
Chemotactic collapse in a parabolic system
495
and there exists
m…x0 † ˆ lim m…x0 ; R=2 k † V m :
k!y
Inequality (43) implies
sup
A…x0 ; r; R†VW‰0; Tmax †
for 0 < r < R.
Therefore,
f …x† ˆ u…x; 0† ‡
jut j < ‡y
… Tmax
0
ut …x; t†dt
ˆ lim u…x; t† V 0
…50†
t"Tmax
exists for x A B…x0 ; R† V Wnfx0 g. In use of (43) again, convergence (50) holds
in the sense C…A…x0 ; r; R† V W†, where r 2 …0; R†. Also f A L 1 …W† follows from
(11).
For simplicity we set E ˆ B…x0 ; R† V W. Given x A C…E†, we have
…
…
ux dx ÿ m…x0 †x…x0 † ÿ f x dx
E
E
…
ˆ x…x0 †
…
ÿ
E
E
ujR=2 k dx ÿ m…x0 † ‡
…
x f jR=2 k dx ‡
E
…
E
…x ÿ x…x0 ††ujR=2 k dx
x…u ÿ f †…1 ÿ jR=2 k †dx
for k ˆ 1; 2; 3; . . . : It follows that
…
…
ux dx ÿ m…x0 †x…x0 † ÿ f x dx
E
E
…
U kxkLy …E† ujR=2 k dx ÿ m…x0 † ‡ ku0 kL 1 …W† kx ÿ x…x0 †kLy …B…x0 ; R=2 k †VW†
E
…
‡ kxkLy …E†
E
f jR=2 k dx ‡ ku ÿ f kLy …A…x0 ; R=2 k‡1 ; R=2 k †VW†
and hence
…
…
lim sup ux dx ÿ m…x0 †x…x0 † ÿ f x dx
t"Tmax
E
E
U kxkLy …E† jm…x0 ; R=2 k † ÿ m…x0 †j
‡ ku0 kL 1 …W† kx ÿ x…x0 †kLy …B…x0 ; R=2 k † V W† ‡ kxkLy …E†
…
E
f jR=2 k dx:
496
Toshitaka Nagai et al.
Making k ! ‡y, we get
…
…
ux dx ˆ m…x0 †x…x0 † ‡ f x dx
lim
t"Tmax E
E
by f A L 1 …E†. This means (2) and proof is complete.
r
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system of chemotaxis, to appear in J. Korean Math. Soc.
[26] H. Tanabe, Functional Analytic Methods for Partial Di¨erential Equations, Dekker, New
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Japonica 45 (1997), 241±265.
T. Nagai
Department of Mathematics
Faculty of Science
Hiroshima University
Higashi-Hiroshima 739-8526, Japan
T. Senba
Department of Applied Mathematics
Faculty of Technology
Miyazaki University
Miyazaki 889-2192, Japan
T. Suzuki
Department of Mathematics
Graduate School of Science
Osaka University
Toyonaka 560-0043, Japan