Blow-Up of Solutions to the Patlak-Keller-Segel Equation in Dimension $\nu\geq2$

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by  Li Chen

BLOW-UP OF SOLUTIONS TO THE PATLAK-KELLER-SEGEL
EQUATION IN DIMENSION ν ≥2
LICHENANDHEINZSIEDENTOP
Abstract. Weproveablow-upcriterionforthesolutionstotheν-dimensional
7 Patlak-Keller-Segel equation in the whole space. The condition is new in
1 dimension three and higher. In dimension two it is exactly Dolbeault’s and
0 Perthame’s blow-up condition, i.e., blow-up occurs, if the total mass exceeds
2 8π.

r
a
M
1. Introduction

1
It is well-known that Patlak-Keller-Segel type equations describing chemotaxis

] (Patlak [11] and Keller and Segel [7, 8]) allow for both diffusion and aggregation
P
phenomena: depending on the initial data, the solution might exist globally in
A
time or blowupin finite time. Since J¨ager’sandLuckhaus’[6]pioneeringworkthe
.
h analysisofthissystemproliferated. Itistooambitioustomentionalltheimportant
t results in this short article, instead we refer to the review of Wang et al [14].
a
m We will focus on the blow-up of solutions of the following system
[ ∂ ρ−∇·(∇ρ−ρ∇c)=0,
t
2 (1) −∆c=ρ−Zδ, x∈Rν,t≥0,
v
ρ(x,0)=ρ (x)≥0,
0
1
3 where δ is the Dirac delta function at the origin. The parameter Z ∈R represents
6 the strength of the external source. The source is repulsive, if Z is positive, and
4
attractive, if Z is negative.
0
For sake of simplicity, we write
.
1
0 M (t):= |x|µρ(x,t)dx, µ=0,1,···
µ
7 ZRν
1 forthe momentsofasolutionρattime t. The system(1)conservesthetotalmass,
:
v i.e., for all times t for which the solution exists
Xi M (t)=M (0)=:M .
0 0 0
r For ν =2 andZ =0,Dolbeault and Perthame[4] and– later in greaterdetail –
a
Blanchetetal[1]showedthat,ifthemassM exceeds8π,thenanyclassicalsolution
0
blows up in finite time, while, if M <8π, then a classical solution exists globally,
0
since the diffusion dominates the aggregation which follows from the logarithmic
Hardy-Littlewood-Sobolev inequality.
Also for ν = 2, Wolansky and Espejo [15], show that adding a repulsive point
source will slow down the blow-up compared to Z = 0, where the blow-up occurs
for masses that exceed 8π, while adding an attractive point source will enhance
blow-up. This generalizes [4].
Inhigherdimensions,i.e.,ν >2,itisknown(Perthame[12,Chapter6],Chapter
6)thatthereexists aconstantc suchthat, ifthe initialdatumfulfills kρ0kLν/2 ≤c,
Date:February28,2016.
1991 Mathematics Subject Classification. 35Q92.
Keywords and phrases. Patlak-Keller-Segelequation, blow-up,higherdimension.
1
2 L.CHENANDH.SIEDENTOP
then a solution exists globally. On the other hand there exists a (small) constant c
such that, if the initial datum fulfills
ν
(2) M <cMν−2,
2 0
then all classical solutions blow up in finite time. However, the results are much
weakerthanthe correspondingversionin dimension two. This paper is a contribu-
tion to fill this gap.
A tool often used to show blow-up is the time evolution of the second moment
M (t): if one can show that the time derivative is strictly negative, then the finite
2
time blow-up is proved. However, the second moment is – in some sense – only
natural in dimension two.
The blow-up of solutions has also been studied for other variants of the Patlak-
Keller-Segelsystem. Anexampleis∂ ρ−∇·(∇ρm−ρ∇c)=0,whereaporousmedia
t
typedegeneratediffusionisincluded. Sugiyama[13]andBlanchetetal[1]obtained
both existence and blow-up results for m=2− 2, and Chen et al [2] for m= 2ν .
ν ν+2
Furthermore, Chen and Wang [3] obtained sharp results on the occurrence of the
blow-up and existence for m in between 2− 2 and 2ν . In particular they were
ν ν+2
able to identify very large classes of initial conditions for which they can predict
blow-up in finite time respectively existence of solutions. Another variant is the
modelwithnonlinearchemotaxissensitivity∂ ρ−∇·(∇ρ−ρα∇c)=0. Horstmann
t
and Winkler [5] – and later many others – studied both blow-up and existence of
the solutions for different α.
Inthis paper,we treatthe ν-dimensionalsystemwitha pointsourceofstrength
Z. We have the following sufficient condition for blow-up:
Theorem 1. For ν ≥2, assume that the initial datum ρ satisfies
0
(3) Mν−ν2(0)< 1 M2−ν2 − Z M1−ν2,
ν (ν−1)2ν|Sν−1| 0 2(ν−1)|Sν−1| 0
where |Sν−1| is the volume of ν − 1-dimensional sphere. Then there is no clas-
sical solution that exists for all times, i.e., there exists a finite T > 0 such that
limt→T−kρ(·,t)k∞ =+∞.
This has two immediate consequences:
No point source: Without an external point source, i.e., Z = 0, the condi-
tion (3) is a blow-up criterion for the multi-dimensional parabolic-elliptic
Patlak-Keller-Segelsystem, i.e., if, initially
Mν−ν2(0)< 1 M2−ν2,
ν (ν−1)2ν|Sν−1| 0
then the solution blows up in finite time. This is actually the condition we
are looking for, since for ν = 2, it is exactly Dolbeault’s and Perthame’s
condition [4]
1
1< M .
0
8π
With point source and ν =2: In dimension two, the condition becomes
Z 1
1+ < M ,
0
4π 8π
which is exactly Wolansky’s and Espejo’s blow-up condition [15].
Remark 1. It is a interesting question whether the reverse inequality, i.e.,
Mν−ν2(0)> 1 M2−ν2 − Z M1−ν2,
ν (ν−1)2ν|Sν−1| 0 2(ν−1)|Sν−1| 0
KELLER-SEGEL EQUATION 3
would already imply existence of a classical solution. It has an affirmative answer
in dimension two but is open in higher dimensions.
2. Proof of the main result
Proof. By using the fundamental solution Φ of the Poisson equation, the system
(1) can be rewritten into the following form, for ν ≥2,
(4) ∂ ρ=∆ρ−∇[ρK∗(ρ−Zδ)], x∈Rν,
t
where
1 x
K=∇Φ=− .
|Sν−1||x|ν
Multiplication of (4) by |x|ν and integration gives
d
|x|νρ(x)dx
dtZRν
νZ
=2ν(ν−1) |x|ν−2ρ(x)dx+ ρ(x)dx
ZRν |Sν−1|ZRν
ν x x−y
− |x|ν−1 ρ(x)ρ(y)dxdy
(5) |Sν−1|ZRνZRν |x||x−y|ν
νZ
=2ν(ν−1) |x|ν−2ρ(x)dx+ ρ(x)dx
Z |Sν−1|ZRν
ν x y x−y
− (|x|ν−1 −|y|ν−1 ) ρ(x)ρ(y)dxdy
2|Sν−1|ZRν ZRν |x| |y| |x−y|ν
=:V
| {z }
We estimate V from below. To this end we set r := |x|, s := |y|, and u :=
x·y/(|x||y|). Thus
rν −(rν−1s+rsν−1)u+sν
V =
(r2+s2−2rsu)ν/2
(6)
(r/s)ν/2+(s/r)ν/2−[(r/s)ν/2−1+(s/r)ν/2−1]u
= .
{r/s+s/r−2u}ν/2
The right hand side of (6) is monotone increasing in u. To see this we first write
τ :=r/s andremarkthatthederivativeofV withrespectuis–uptoanirrelevant
non-negative factor –
1
− τν2−1+τ−(ν2−1) (τ + −2u)+ν τν2 +τ−ν2 −(τν2−1+τ−(ν2−1))u
τ
(cid:16) (cid:17) (cid:16) (cid:17)
1
≥(2−ν) τν2−1+τ−(ν2−1) +ν τν2 +τ−ν2 − τ + τν2−1+τ−(ν2−1)
(7) (cid:16) (cid:17) (cid:0) (cid:1) (cid:18) τ(cid:19)(cid:16) (cid:17)
=(2−ν) τν2−1+τ−(ν2−1) +(ν−1) τν2 +τ−ν2 − τν2−2+τ−(ν2−2)
(cid:16) (cid:17) (cid:0) (cid:1) (cid:16) (cid:17)
≥(2−ν) τν2−1+τ−(ν2−1) +(ν−2) τν2 +τ−ν2 ≥0
(cid:16) (cid:17) (cid:0) (cid:1)
where we have used that τα +τ−α is monotone increasing in α for positive α
and τ. Thus
(r/s)ν/2+(s/r)ν/2+(r/s)ν/2−1+(s/r)ν/2−1
V ≥
{r/s+s/r+2}ν/2
(r/s)ν2 +(s/r)ν2 +(r/s)ν2−1+(s/r)ν2−1
= ≥ min f(τ)
( r/s+ s/r)ν τ∈[0,1]
p p
4 L.CHENANDH.SIEDENTOP
with
1+τν +τ +τν−1 1+τν−1
f(τ)= = .
(1+τ)ν (1+τ)ν−1
Next we prove that f is decreasing and the minimum achieved at τ =1. In fact,
(ν−1)τν−2(1+τ)ν−1−(ν −1)(1+τ)ν−2(1+τν−1)
′
f (τ)=
(1+τ)2(ν−1)
ν−1
= [τν−2−1]≤0.
(1+τ)ν
Thus
V ≥f(1)=22−ν
and therefore
d ν21−ν νZ
(8) dtMν ≤2ν(ν−1)Mν−2− |Sν−1|M20+ |Sν−1|M0.
Estimating by Ho¨lder’s inequality yields
(9) M′ ≤2ν(ν−1)Mν−ν2Mν2 − ν21−νM2+ νZ M .
ν ν 0 |Sν−1| 0 |Sν−1| 0
In particular, we have a shrinking ν-th moment, if the initial moments fulfill
Mν−ν2(0)< 1 M2−ν2 − Z M1−ν2,
ν (ν−1)2ν|Sν−1| 0 2(ν−1)|Sν−1| 0
However, a shrinking ν-the moment implies blow-up. (cid:3)
At this point we would like to remark that the strategy of the proof, namely
multiplication with an appropriate power based on a dimensional analysis, has
been previously used in the context of effective quantum models and dates back –
at least – to an unpublished observation of Benguria to bound the excess charge
of atoms. Later, Lieb [10] extended the argument to the quantum case; Lenzmann
and Lewin [9] used it in the time dependent setting.
Inconclusion,we wouldlike to point thatour blow-upcondition(3) implies (2).
ν−2 2
To see this, we first note that M ≤ M ν Mν by interpolation. Estimating the
2 0 ν
right hand side by using (3) gives (2) with the extra bonus of a definite constant
insteadofanuncontrolledone. Finally,notethatthisisonlynecessarywhenν >2,
sinceInequality(2)isanempty statementindimensiontwowhereasInequality(3)
remains meaningful.
Acknowledgment: WethankGeorgiosPsaradakisforcriticalreadingofthemanuscript.
WeacknowledgesupportbytheDeutscheForschungsgemeinschaftthroughthegrantsCH
955/4-1 and SI 348/15-1.
References
[1] AdrienBlanchet,JeanDolbeault,andBenoˆıtPerthame.Two-dimensionalKeller-Segelmodel:
optimal critical mass and qualitative properties of the solutions. Electron. J. Differential
Equations,pages No.44,32,2006.
[2] LiChen,Jian-GuoLiu,andJinhuanWang.MultidimensionaldegenerateKeller-Segelsystem
withcriticaldiffusionexponent2n/(n+2).SIAM J. Math. Anal.,44(2):1077–1102, 2012.
[3] LiChenandJinhuanWang.Exactcriterionforglobalexistenceandblowuptoadegenerate
Keller-Segelsystem.Doc. Math.,19:103–120, 2014.
[4] Jean Dolbeault and Benoˆıt Perthame. Optimal critical mass in the two-dimensional Keller-
SegelmodelinR2.C. R. Math. Acad. Sci. Paris,339(9):611–616, 2004.
[5] DirkHorstmannandMichael Winkler.Boundedness vs.blow-upinachemotaxis system.J.
DifferentialEquations,215(1):52–107, 2005.
[6] W. Ja¨ger and S. Luckhaus. On explosions of solutions to a system of partial differential
equations modellingchemotaxis. Trans. Amer. Math. Soc.,329(2):819–824, 1992.
KELLER-SEGEL EQUATION 5
[7] Evelyn F. Keller and Lee A. Segel. Initiation of slimemoldaggregation viewed as an insta-
bility.Journal of Theoretical Biology,26(3):399–415, 1970.
[8] Evelyn F. Keller and Lee A. Segel. Model for chemotaxis. Journal of Theoretical Biology,
30(2):225–234, 1971.
[9] Enno Lenzmann and Mathieu Lewin. Dynamical ionization bounds for atoms. Anal. PDE,
6(5):1183–1211, 2013.
[10] Elliott H. Lieb. Bound on the maximum negative ionization of atoms and molecules. Phys.
Rev. A,29(6):3018–3028, June1984.
[11] CliffordSPatlak. Random walkwithpersistence andexternal bias.The Bulletin of Mathe-
matical Biophysics,15(3):311–338, 1953.
[12] BenoˆıtPerthame.Transport equations inbiology.FrontiersinMathematics.Birkh¨auserVer-
lag,Basel,2007.
[13] Yoshie Sugiyama. Global existence in sub-critical cases and finite time blow-up in super-
critical cases todegenerate Keller-Segelsystems. Differential Integral Equations, 19(8):841–
876,2006.
[14] JinhuanWang,LiChen,andLiangHong.ParabolicelliptictypeKeller-Segelsystemonthe
wholespacecase.Discrete and Continuous Dynamical Systems,36(2):1061–1084, 2016.
[15] GershonWolanskyandElioEspejo.ThePatlak-Keller-SegelmodelofchemotaxisonR2with
singulardriftandmortalityrate.Nonlinearity,26(8):2315, 2013.
MathematischesInstitut, Universita¨tMannheim,A5 6,68131Mannheim,Germany
E-mail address: [email protected]
MathematischesInstitut,Ludwig-Maximilians-Universita¨tMu¨nchen,Theresienstraße
39,80333Mu¨nchen,Germany
E-mail address: [email protected]