ERRATUM TO: SINGULARITY FORMATION IN CHEMOTAXIS - A CONJECTURE
OF NAGAI
MATTHEW A. HALVERSON & HOWARD A. LEVINE & JOANNA RENCLAWOWICZ
Abstract. In [1] we considered the problem ut = uxx − (uvx )x , vt = u − av on the interval I = [0, 1] where
ux , vx = 0 at the end points, u(x, 0), v(x, 0) are prescribed and a > 0 (It was claimed that there were solutions
that blow up in finite time [1] in every neighborhood of the spatially homogeneous steady state (u, v) =
(µ, µ/a) if µ > a.) Here we correct an estimate and reduce Nagai’s conjecture to the following statement.
j n−j
n−1
Let σ = a/(µ − a), ρ1 = 1. If limn→+∞ ρn exists, where for n ≥ 2, ρn
n ≡ 1/(n − 1)
j=1 (1 + σ/j)ρj ρn−j ,
then the blow up assertion holds.
P
1. Introduction
In [1] we studied the system ut = uxx − (uvx )x , vt = u − av on the interval I = [0, 1] where ux , vx = 0
at the end points, u(x, 0), v(x, 0) are prescribed and a > 0. Nagai and Nakaki [2] showed that there are
solutions that are unbounded in finite or in infinite time.1 We claimed that there were initial conditions for
which solutions failed to exist for all time. In our proof we used a differential inequality, the derivation of
which was unfortunately flawed. We correct this, and make more precise the statement proved in [1].
2. Approximate Solution
The notation of [1] is in force here. Because system ut = uxx − (uvx )x , vt = u − av is autonomous, we
can assume the initial values are prescribed at t = 0 and that the blow up time, when it exists, is positive.
a
0
2
0
As in [1], define, for any sequence z(t) = {zn (t)}∞
n=1 , Gn (z, z ) = (1/2)C n{(Mz ∗ z )n + n 2 (z ∗ z)n } and
0
2
0
0
∞
Hn (z, z ) = (1/2)C n{[(Tn Mz, z ) − (Mz, Tn z )] + an(z, Tn z)} where Mz(t) = {nzn (t)}n=1 and Tk z(t) =
Pn−1
∞
{zn+k (t)}∞
n=1 . Here |z| = {|zn |}n=1 and (z ∗ w)n =
k=1 zk wn−k . (The sum is zero if n = 1.)
2
The infinite system of ordinary differential equations for the cosine coefficients h(t) = {hn (t)}∞
n=1 is :
Ln hn ≡ h00n + (C 2 n2 + a)h0n − (µ − a)C 2 n2 hn = Gn (h, h0 ) + Hn (h, h0 ).
The infinite system of ordinary differential equations satisfied by the cosine coefficients for the approximate
0
nλt ∞
problem, g(t) = {gn (t)}∞
}n=1
n=1 , satisfies Ln gn = Gn (g, g ). The particular sequence g(t) ≡ {gn (t) = an e
satisfies this system for a1 > 0, and for n ≥ 2 and any integer M > 0 with C = 2πM , µ > a if
(2.1)
2λ[n − a/(4π 2 M 2 )]an =
n−1
1 X
[λ(n − k)k + ak]ak an−k
n−1
k=1
2
2
2
where λ is the positive root of λ + (4π M + a)λ − (µ − a)4π 2 M 2 = 0. There are positive constants, a, b, , δ
with an ≤ nan ≤ bδ n for all positive integers [1]. From this, it follows that lim inf n→+∞ [(− ln nan )/(nλ)] ≡
Tb and lim supn→+∞ [(− ln nan )/(nλ)] ≡ Tb are finite. Hence there is a subsequence {ank }∞
k=1 such that
Date: June 8, 2005.
Key words and phrases. chemotaxis, finite time singularity formation, Keller-Segel model.
1The Nagai conjecture states that if µ > a, there are spatially nonhomogeneous solutions beginning in every small neighborhood of (µ, µ/a) which cannot exist for all time.
2The spatially homogeneous solution is given by V (t) = µ/a + (v − µ/a) exp(−at), U (t) = µ. One sets ψ(x, t) = v(x, t) −
0
V (t), u(x, t) = µ + ψt + aψ. Then h(t) is the sequence of cosine coefficients for ψ(x, t).
1
2
MATTHEW A. HALVERSON & HOWARD A. LEVINE & JOANNA RENCLAWOWICZ
limk→+∞ [(− ln nk ank )/(nk λ)] ≡ Tb . For this sequence, limk→+∞ nk ank exp (nk λTb ) = 1. Set an = (An /n) exp (−nλTb ).
On the subsequence, Ank → 1 and
(2.2)
lim
∞
X
t↑Tb
Ank e−nk λ(Tb −t) = +∞
∞
X
Ank e−nk λ(Tb −t)
< +∞ (for any δ > 0).
t↑Tb
nk 1+δ
and
lim
k=1
k=1
Now Tb must be the blow up time for the approximate solution g(t) in the space `11 (0, Tb ) × `1 (0, Tb ). (A
sequence {an } is in `11 if {nan } is in `1 .) To see this, note that as long as t is in the existence interval,
∞
∞
∞
X
X
X
kMg(t)k`1 + kg 0 (t)k`1 =
nan (1 + λ)enλt ≥ (1 + λ)
nk ank enk λt = (1 + λ)
Ak e−nk λ(Tb −t) .
n=1
k=1
k=1
Consequently, from the first equation in (2.2), g(·) must blow up at some time possibly earlier than Tb . If
t < Tb then lim inf n→+∞ [(− ln nan )/(nλ)] ≡ Tb > Tb − δ > t for some positive δ. Therefore, for sufficiently
P∞
P∞
large N , n=N nan enλt ≤ n=N ne−nλ(Tb −δ−t) < ∞.
∞
∞
Set σ = a/λ. Let {ln[nan /(2an1 )]/n}∞
n=1 = {ln An /n}n=1 ≡ {pn /n}n=1 . The pn satisfy p1 = − ln 2, and
P
n−1
1
2
2
pn
(pj +pn−j )
for n ≥ 2 [1 − a/(4π M n)]e = n−1 j=1 (1 + σ/j)e
. Then we have
Theorem 1. Nagai’s Conjecture in `11 × `1 . Let limn→+∞ pnn exist. The corresponding solution of the Nagai
problem for which hn (0) = gn (0) and h0n (0) = gn0 (0) for all n, cannot both exist and be `1 regular on [0, ∞).
(A solution of the Nagai-Nakaki problem is `1 regular on an interval I = [0, Tb ) if it exists there and if
(kMh(s)k`1 + kh0 (s)k`1 ) is uniformly bounded on compact subsets I.)
3. Estimate
Inequality (7.5) of [1] is incorrect. The correct form of the upper bound for the norm of g − h ≡ w,
kMw(t)k`1 + kw0 (t)k`1 , is based on the following (infinite) system of ordinary differential equations:
(3.1)
Ln wn = Gn (h − g, h0 ) + Gn (g, h0 − g 0 ) + Hn (h, h0 ) = Gn (w, h0 ) + Gn (g, w0 ) + Hn (h, h0 )
and, for some B > 0 depending perhaps on τ but not on w, w0 , h, h0 , g, g 0 , is given by
Z t
(kMh(s)k`1 + kh0 (s)k`1 )2
0
√
kMw(t)k`1 + kw (t)k`1 ≤ I(t) + J(t) + B
ds
t−s
0
(3.2)
Z t
(kMw(s)k`1 + kw0 (s)k`1 )(kMh(s)k`1 + kh0 (s)k`1 )
√
ds,
+B
t−s
0
where
Z tX
Z tX
∞
∞
2
0
−dn2 (t−s)
I(t) + J(t) ≡
M(|g | ∗ M|w|)n e
ds +
M2 (|g| ∗ |w|)n e−dn (t−s) ds,
0 n=1
0 n=1
and where d > 0 is the positive constant in [1], Lemma 1. We have,
Z tX
Z tX
∞ n−1
∞ X
∞
X
2
0
−dn2 (t−s)
I(t) =
n(n − k)|gk ||wn−k |e
ds =
n(n + k)|gk0 ||wn |e−d(n+k) (t−s) ds,
0 n=1 k=1
≤
Z tX
∞
0 k=1 n=1
2
|gk0 |e−(d/2)k (t−s)
0 k=1
Z
≤c
t
P∞
k=1
Z t X
∞
0
−(d/2)(n+k)2 (t−s)
(n + k)e
n|wn | ds,
n=1
0
≤c
X
∞
k=1
|gk0 |e−(d/2)k
√
t−s
2
(t−s)
kMw(s)k`1 ds ≤ c
−[(d/2)k2 +kλ](t−s)
Ak e
Z tX
∞
Ak e−(d/2)k
0 k=1
Z t
kMw(s)k`1
√
ds ≡ c
t−s
W(t − s)
0
2
(t−s)−λk(Tb −s) kMw(s)k`1
kMw(s)k`1
√
ds.
t−s
√
t−s
ds,
SINGULARITY FORMATION IN CHEMOTAXIS
3
It follows that for any p ≥ 1, W ∈ Lp (0, 1). In the same manner,
Z tX
Z tX
∞ n−1
∞ X
∞
X
2
2
J(t) =
n2 |gk ||wn−k |e−dn (t−s) ds ≤
(n + k)2 |gk ||wn |e−d(n+k) (t−s) ds
0 n=1 k=1
≤
Z tX
∞
0 k=1
0 k=1 n=1
−(d/2)k2 (t−s)
|kgk |e
X
∞
2
(n + k) −(d/2)(n+k)2 (t−s)
e
n|wn | ds.
kn
n=1
From the inequality (k + l)/kl ≤ 2:
Z t X
Z t
∞
2
kMw(s)k`1
kMw(s)k`1
√
J(t) ≤ c0
ds ≡ c0
W(t − s) √
ds.
Ak e−[(d/2)k +kλ](t−s)
t
−
s
t−s
0
0
k=1
P∞
In view of (2.2), limt↑Tb k=1 Ak k −2 e−kλ(Tb −t) < +∞. Thus W(t) is in every Lp [0, Tb ] space for 1 ≤
Rt
√
p < ∞. With f (t) = kMw(t)k`1 + kw0 (t)k`1 , we see f (t) ≤ 0 W(t − s)f (s)/ t − s ds + Φ(h(t)). From
Hölder’s inequality with 1/p + 1/r + 1/q = 1 and 1 < r < 2, there is a constant K > 0 such that f (t) ≤
Rt
K[ 0 f (s)q ds]1/q + Φ(h(t)) on [0, Tb ). From Gronwall, if h is global, f (t) is bounded on [0, Tb ). From the
first sum in (2.2) and the triangle inequality, this is impossible.
References
[1] H. A. Levine and J. Renclawowicz, Singularity formation in chemotaxis - a conjecture of nagai, SIAM J. Appl. Math.,
(2004), pp. 336–330.
[2] T. Nagai and T. Nakaki, Stability of constant steady states and existence of unbounded solutions in time to a reactiondiffusion equation modelling chemotaxis, Nonlinear Analysis, 58 (2004), pp. 657–681.
Author addresses
Matthew A. Halverson
mhalver@iastate.edu
and
Howard A. Levine
Department of Mathematics
Iowa State University
Ames, Iowa, 50011
United States of America
halevine@iastate.edu
and
Joanna Renclawowicz
Institute of Mathematics
Polish Academy of Sciences
Śniadeckich 8, 00-956
Warsaw, Poland
jr@impan.gov.pl