Part

ial

Diff

ere

nti

al

Equati

ons

inM

ath

em

ati

cal

Bio

logy

On

the

Keller-

Segel

Syst

em

wit

h

Exte

rnal

Applica

tion

of

Chem

oatt

ract

ant

J.I

gnaci

oT

ello

1and

Mic

hael

Win

kle

r2

1-.

Un

ivers

idad

Poli

tecn

ica

de

Mad

rid

.S

pain

2-.

Ess

en

-Du

isb

urg

Un

ivers

ity.

Germ

any

Bedle

wo,

Septe

mb

er

12th

-17th

,2010

Conte

nts

Fir

stP

rob

lem

:A

Sys

tem

wit

hex

tern

alap

plic

atio

nof

chem

oatt

ract

ant

∂u ∂t

=∆u−χ∇·(u∇v

),x∈IR

2t>

0

−∆v

=u

+2πF

0δ(x

)x∈IR

2

Seco

nd

Pro

ble

m:

On

aC

hem

otax

issy

stem

wit

hlo

gist

icte

rm

∂u ∂t

=∆u−χ∇·(u∇v

)+λu

(1−u

),x∈

Ωt>

0

−∆v

+v

=u

x∈

Ω

Fir

stP

art

1-.

Intr

odu

ctio

n

2-.

Au

xilia

ryP

rob

lem

W(s,t

):=

1 π

∫ B√s(0

)ρu

(ρ,t

)dρ,

s>

0,t>

0,

3-.

Inst

anta

neo

us

Blo

wu

p

4-.

Cas

e∫ IR

2u

0>

8π−

4πF

0

5-.

Cas

e∫ IR

2u

0<

8π−

4πF

0

1In

troduct

ion

Ch

emot

axis

isth

eab

ilit

yof

mic

roor

gan

ism

sto

resp

ond

toch

emic

alsi

gnal

sby

mov

ing

alon

gth

egr

adie

ntof

the

chem

ical

sub

stan

ce,

eith

erto

war

dth

eh

igh

er

con

cent

rati

on(p

osit

ive

taxi

s)or

away

from

it(n

egat

ive

taxi

s).

-D

icty

oste

lium

dis

coid

eum

Bac

teri

a.S

eeK

elle

r-S

egel

[197

0],

[197

1]

-T

um

our-

ind

uce

dan

giog

enes

is.

See

An

der

son

-Ch

apla

in[1

997]

,[1

998]

A.

Ku

bo

inth

eco

nfe

ren

ce(T

ues

day

Aft

ern

oon

)

-A

stro

phy

sics

and

grav

itat

ion

alin

tera

ctio

nof

par

ticl

es.

See

Bile

r[1

995]

,

Bile

r-H

ilhor

st-N

adzi

eja

[199

4]

-M

orp

hog

enes

isfo

rmat

ion

ofth

eem

bry

o.S

eeM

erki

n-N

eed

ham

-Sle

eman

[200

5],

Bol

lenb

ach

-Kru

se-P

anta

zis-

Gon

zale

zG

aita

n-J

ulic

her

[200

7],

C.

Sti

nn

erin

the

Pos

ter

Ses

ion

ofth

eco

nfe

ren

ce.

Mat

hem

atic

alm

odel

sof

chem

otax

is

Kel

ler

and

Seg

el[1

970]

,[1

971]

(aft

erP

atla

k[1

953]

)

p=

bac

teri

a,w

=ch

emoa

ttra

ctan

t

∂u ∂t

=Q

(u,v

)+∇·D

(u,v

)∇u−uχ

(v)∇v

),

∂v ∂t

=d

∆v

+h

(u,v

)x∈

Ωt>

0

We

con

sid

erΩ

=R

2,

fast

diff

usi

on,

and

the

sim

plifi

edsy

stem

∂u ∂t

=∆u−χ∇·(u∇v

),x∈IR

2t>

0

−∆w

=u

+f

(x)

x∈IR

2

+in

itia

ld

atu

mfo

ru

wh

eref

isan

exte

rnal

app

licat

ion

ofch

emoa

ttra

ctan

t.

Jage

ran

dL

uck

hau

s[1

992]

,Ω⊂IR

2

∂u ∂t

=∆u−∇·(u∇v

),Ω×

(0,T

)

−∆v

=u−

1Ω×

(0,T

)

zero

flux

onth

eb

oun

dar

y

•if

∫ Ωu

0<

8π,

then

glob

also

luti

ons

exis

t

•if

∫ Ωu

0>

8π,

solu

tion

blo

ws

up

infin

ite

tim

e

•S

eeal

soH

erre

ro-V

elaz

quez

[199

6]

Cas

e−

∆v

=u

inIR

2.

See

Bile

r[1

995a

][1

995b

],B

lan

chet

-Dol

bea

ult

-

Per

tham

e[2

006]

Bla

nch

et-C

arri

llo-M

asm

oun

di

[200

8],

Nai

to-S

uzy

ki[2

004]

,

Vel

azqu

ez[2

002]

,[2

004]

.

Un

der

assu

mp

tion

s∫ IR

2(1

+|x|2 )u

0<∞,

∫ IR2u

0lo

gu

0<∞

-T

he

sub

crit

ical

case

∫ IR2u

0<

8π−→

glob

alex

iste

nce

.

See

Bla

nch

et-D

olb

eau

lt-P

erth

ame

[200

6].

-T

he

sup

ercr

itic

alca

se∫ IR

2u

0>

8πso

luti

ons

blo

ws

up

asa

dir

acfu

nct

ion

s.

Vel

azqu

ez[2

002]

-T

he

crit

ical

case

∫ IR2u

0=

8π.

Th

eb

low

up

att

=∞

wit

hsi

mila

rp

rofil

e

than

inth

esu

per

crit

ical

case

.B

lan

chet

-Car

rillo

-Mas

mou

nd

i[2

008]

.

Th

em

ath

em

ati

cal

mod

el

-u

con

cent

rati

onof

livin

gor

gan

ism

s

-v

con

cent

rati

onof

chem

oatt

ract

ant

sub

stan

ce

-w

eas

sum

eχ

=1

∂u ∂t

=∆u−∇·(u∇v

)x∈IR

2,t>

0

−∆v

=u

+f

(x),

x∈IR

2,

wh

ere

f(x

)=

lim ε→0

f 0 |ωε|I

ωε

=f 0δ(x

)

wh

eref 0≥

0.

We

intr

odu

ceh

igh

ergr

adie

ntofv

.W

ed

ont

incr

ese

the

mas

sofu

.

2A

uxilia

ryP

roble

m

Rad

ially

sym

met

ric

solu

tionu

=u

(r,t

)

W(s,t

):=

1 π

∫ B√s(0

)ρu

(ρ,t

)dρ,

s>

0,t>

0,

wh

ich

sati

sfies

Wt

=4sW

ss+WW

s+

2F0W

s,s>

0,t>

0,

W(0,t

)=

0,lim s→∞W

(s,t

)=

µ π,

t>

0,

W(s,0

)=W

0(s

),s>

0,

wit

hF

0:=

f 0 2πan

d

W0(s

):=

2∫√

s0

ρu

0(ρ

)dρ,

s>

0.

We

may

con

sid

erth

ecu

t-off

fun

ctio

n

χ∈C∞

([0,∞

)),

χ≡

0on

0,1 2

,χ≡

1on

[1,∞

),χ′≥

0on

[0,∞

)

ε∈

(0,1

)χ

(ε) (s

):=χ

s ε

,s≥

0.

Th

en W

(ε)

t=

4sW

(ε)

ss+χ

(ε) (s

)W(ε

) W(ε

)s

+2F

0χ

(ε) (s

)W(ε

)s,

s>

0,t>

0,

W(ε

) (0,t

)=

0,lim s→∞W

(ε) (s,t

)=

µ π,

t>

0,

W(ε

) (s,0

)=W

0(s

),s>

0,

-T

her

eex

ists

au

niq

ue

solu

tionW

(ε)

toth

eε-

pro

ble

m

-W

(ε)

isn

on-i

ncr

easi

ng

wit

hre

spec

ttoε

-W

(ε)→

Wp

oint

wis

ean

dC

2in

Com

pac

tse

ts.

-W∈L∞

((0,∞

)×

(0,∞

))is

aw

eak

solu

tion

ofth

ep

rob

lem

inth

e

follo

win

gse

nse

W(s,t

)→

µ πas

s→∞

for

allt>

0

−∫ ∞ 0

∫ ∞ 0ζ tW−

∫ ∞ 0ζ

(·,0)W

0=

4∫ ∞ 0

∫ ∞ 0(sζ

) ssW−

1 2

∫ ∞ 0

∫ ∞ 0ζ sW

2−f 0 2π

∫ ∞ 0

∫ ∞ 0ζ sW

forζ∈C∞ 0

([0,∞

)×

[0,∞

)),

wh

ereW

0(s

):=

1 π∫ B√s(0

)u

0(x

)dx

fors≥

0.

3In

stanta

neous

Blo

wup

Lem

ma

1

LetF

0>

0,δ>

2−F

02

such

thatδ∈

(0,1

).T

hen

ther

eex

ist

pos

itiv

e

con

stan

tsa,b,ξ,k

0,K

0su

chth

atfo

ran

yγ>

0,

ϕ(s

):=

a γδs−

δ−b

if0<s<

ξ γ,

e−γs

ifs≥

ξ γ,

ϕ(s

)∈W

2,∞

loc

((0,∞

))

4sϕss

+(8−

2F0)ϕ

s≥k

0γϕ

a.e.

in(0,∞

)

∫ ∞ 0ϕ

2(s

)|ϕs(s)|−

1ds≤K

0

γ2.

Th

eore

m1.

LetF

0>

0,an

dth

atW

0

-.W

0∈W

1,∞

((0,∞

)),

-.W

0s≥

0in

(0,∞

)

-.W

0(s

)→

µ πass→∞

for

som

eµ>

0.

Th

enfo

ran

yp

osit

iveα>

2−F

02

and

anyt 0≥

0,

sup

s>0,t∈

(t0,t

0+τ)

W(s,t

)

sα=∞

for

allτ>

0.

Inp

arti

cula

r

‖Ws‖L∞

((0,∞

)×(t

0,t

0+τ))

=∞

for

allτ>

0.

wh

ich

imp

lies

‖u‖ L∞

(IR

2×

(t0,t

0+τ))

=∞

for

allτ>

0.

Idea

of

the

pro

of.

We

con

sid

er

y(t

):=

∫ ∞ 0ϕ

(s)W

(s,t

)ds,

t>

0,

and

y(ε

) (t)

:=∫ ∞ 0

ϕ(s

)χεW

(s,t

)ds,

t>

0,

mu

ltip

lyth

eε−

pro

ble

mbyχεϕ

and

inte

grat

eby

par

tsan

dta

kelim

its

as

ε−→

0af

ter

tech

nic

ales

tim

ates

we

arri

veto

y(t

)≥y

(t1)

+k

0γ∫ t t 1

∫ ∞ 0ϕW−

1 2

∫ t t 1

∫ ∞ 0ϕsW

2fo

ral

lt∈

(t1,t

0+τ

).

Not

ice

that

y2(t

)=

(∫ ∞ 0

ϕW

)2≤

( ∫ ∞ 0ϕ

2|ϕ

s|−1) ·( ∫ ∞ 0

|ϕs|W

2) ≤

K0

γ2·∫ ∞ 0|ϕ

s|W2

fort>

0,

ther

efor

ew

eh

ave

y(t

)≥y

(t1)

+∫ t t 1Ay

(t)

+By

2(t

)dt

for

allt∈

(t1,t

0+τ

).

for

A:=k

0γ

and

B:=

γ2

K0.

By

com

par

ison

wit

hth

efo

llow

ing

equ

atio

n

z′=Az

+Bz2

z(t 1

)=y

(t1)

we

hav

efin

ite

tim

eb

low

up

forT≤

C γ.

4Form

ati

on

of

Dir

ac-

typ

esi

ngula

riti

es

forµ>

8π−

4πF

0

Th

eore

m2.

LetF

0≥

0 µ:=

∫ IR2u

0(x

)dx<∞,

such

thatµ>

0.T

hen

,fo

rµ>

8π−

4πF

0,u

sati

sfies

u(x,t

)→

µδ(x

)ast→∞.

Th

eP

roof

of

the

Th

eore

mis

giv

en

in3

step

s.

Lem

ma

3

LetF

0≥

0,an

dW

0su

chth

at (H

1)W

0∈W

1,∞

((0,∞

)),

(H2)

W0s≥

0in

(0,∞

)as

wel

las

(H3)

W0(s

)→

µ πass→∞.

Th

enfo

ral

lµ∈

(8π−

4πF

0,µ

)th

ere

exis

ts 0>

0an

dW

0

W0(s

):=

0ifs∈

[0,s

0],

a−

1b+csβ

ifs>s 0,

such

that

W0∈W

1,∞

((0,∞

))∩C

2([

0,∞

)\s

0)

limin

fs

s 0W

0s(s

)>

0,W

0(s

)→

µ πas

s→∞,

4sW

0ss

+W

0W

0s+

2F0W

0s=

0in

(s0,∞

)

W0≤W

0in

(0,∞

)

For

a:=

µ πb

:=1

2(a

+2F

0−

4)β

:=a

+2F

0−

4

4s 0

:=

1 a−b

c

1 β

≥s 1.

for

som

ela

rges 1>

0su

chth

at

W0(s

)≥a

for

alls≥s 1,

andc>

0.

Lem

ma

4

LetF

0≥

0,an

das

sum

eth

atψ∈C

2((

0,∞

))is

an

onn

egat

ive

solu

tion

of

0=

4sψss

+ψψs

+2F

0ψs,

s>

0,

wit

hth

ead

dit

ion

alp

rop

erti

esψs≥

0on

(0,∞

)an

d

ψ(s

)

µ πass→∞

wit

hso

meµ≥

0.In

that

case

,if

µ>

8π−

4πF

0,

then

ψ≡µ π

in(0,∞

).

Lem

ma

5

LetF

0≥

0,an

dfo

rso

me

µ>

8π−

4πF

0

we

hav

e (H

1)W

0∈W

1,∞

((0,∞

)),

(H2)

W0s≥

0in

(0,∞

)as

wel

las

(H3)

W0(s

)→

µ πass→∞.

Th

en,W

sati

sfies

W(s,t

)→

µ πast→∞,

the

conv

erge

nce

bei

ng

un

ifor

mon

com

pac

tsu

bse

tsof

(0,∞

).

Idea

of

the

pro

of:

–L

etW

be

the

solu

tion

for

init

ial

dat

aW

0

–S

inceW

0<W

0=⇒

W<W

–W

t≥

0in

(0,∞

)×

(0,∞

).

–W

(s,t

)

ψ(s

)ast→∞

–L

emm

a4

end

sth

ep

roof

.

5E

merg

ence

of

mild

singula

riti

es

forµ<

8π−

4πF

0

Th

eore

m3:

LetF

0≥

0,an

dW

0su

chth

at (H

1)W

0∈W

1,∞

((0,∞

)),

(H2)

W0s≥

0in

(0,∞

)as

wel

las

(H3)

W0(s

)→

µ πass→∞.

forµ<

8π−

4πF

0.

Th

en,

for

allτ>

0th

ere

exis

tsC>

0su

chth

at

Ws(s,t)≤C

(1+s−

F0 2)

for

alls>

0an

dan

yt>τ.

ther

efor

e

u(x,t

)≤C|x|−F

0fo

ral

lx∈IR

2an

dt≥τ.

Mor

eove

r,fo

rp∈

[1,

2 F0)

andτ>

0

‖u(·,t)‖ L

p(B

1(0

))≤C

for

allt≥τ.

Seco

nd

Part

Conte

nts

1.-

Glo

bal

bou

nd

edso

luti

ons

2.-

Wea

kgl

obal

solu

tion

s

3.-

Ste

ady

stat

es

4.-

Asy

mp

toti

cb

ehav

iou

r

Th

em

ath

em

ati

cal

mod

el

∂u ∂t

=∆u−∇·(uχ∇v

)+λu

(1−u

),x∈

Ω,t>

0

−∆v

+v

=u,

x∈

Ω,

∂u

∂n

=∂v

∂n

=0,

x∈∂

Ω,

u(x,0

)=u

0(x

)≥

0,x∈

Ω,

Glo

bal

bounded

solu

tions

–A

ssu

mp

tion

s:

-0≤u

0≤c<∞

;G

lob

alC

lass

ical

-λ>

0fo

rn

=1,

2;=⇒

Sol

uti

onex

ists

-λ>

n−

2nχ

forn≥

3.

λ=

0b

low

su

pfo

rn≥

3an

dfo

rn

=2

if∫ Ωu

0>c(

Ω)

(Her

rero

-

Vel

azqu

ez96

)

Th

ere

sult

isva

lidif

we

rep

laceu

(1−u

)by

h(u

)sa

tisf

yin

gh

(u)≤

s 0−s 1u

2.

2.-

Glo

bal

weak

solu

tions

for

arb

itra

ryλ>

0

(u,v

)is

aweaksolution

toth

ep

rob

lem

in(0,T

)if

u∈L

1((

0,T

);W

1,1(Ω

)),

v∈L

1((

0,T

);W

1,1(Ω

));

such

that

u∇v∈L

1((

0,T

);L

1(Ω

)),

λu

(1−u

)∈L

1((

0,T

);L

1(Ω

))

and

−∫ T 0

∫ Ωuϕt+

∫ T 0

∫ Ω∇u·∇ϕ−χ

∫ T 0

∫ Ωu∇v·∇ϕ

=∫ Ωu

0ϕ

(0)+

∫ T 0

∫ Ωλu

(1−u

)ϕ;

T 0

∫ Ω∇v·∇

ψ+

∫ T 0

∫ Ωvψ

=∫ T 0

∫ Ωuψ

(1)

forϕ,ψ∈C∞ 0

(Ω×

[0,T

)).

Ifu

0∈Lγ(Ω

)fo

rγ∈

(1,

χ(χ−λ

) +)

then

ther

eex

ists

agl

obal

wea

kso

luti

on

sati

sfyi

ng

u∈L∞

((0,∞

);Lγ(Ω

))∩Lγ

+1

loc

([0,∞

);Lγ

+1(Ω

))∩Lp loc(

[0,∞

);W

1,p(Ω

)),

∇uγ 2∈L

2 loc(

[0,∞

);L

2(Ω

))

v∈L∞

((0,∞

);W

2,γ(Ω

))∩Lγ

+1

loc

([0,∞

);W

2,γ

+1(Ω

))

for

p∈

(1,2 3

(1+

min2,

χ(χ−λ

) +)

).

3.-

Ste

ady

state

s

0=

∆u−χ∇·(u∇v

)+λu

(1−u

)in

Ω,

0=

∆v−v

+u

inΩ,

∂u∂n

=∂v∂n

=0,

in∂

Ω

Defi

nit

ion

.(u,v

)is

ast

ead

yst

ate

ofth

ep

rob

lem

ifu,v≥

0;

u,v∈W

1,1(Ω

),u∇v∈L

1(Ω

),u

(1−u

)∈L

1(Ω

),

and

sati

sfy

the

iden

titi

es∫ Ω∇u·∇

ϕ−χ

∫ Ωu∇v·∇

ϕ=

∫ Ωλu

(1−u

)ϕan

d∫ Ω∇v·∇

ψ+

∫ Ωvψ

=∫ Ωuψ

for

allϕ∈C∞

(Ω)

andψ∈C∞

(Ω).

Reg

ula

rity

ofst

ead

yst

ates

Lem

ma

i)U

nd

eras

sum

pti

on

λ>

0an

dn≤

4or

λ>n−

4

n−

2χ

andn>

4;

the

solu

tion

isb

oun

ded

andu,v∈C

1,α( Ω

)fo

rα∈

(0,1

).

ii)F

oran

yn

onco

nst

ant

solu

tion

(u,v

),w

eh

ave

expχ

(min

x∈Ωv

(x)−

max

x∈Ω

v(x

))≤u

(x)≤expχ

(max

x∈Ω

v(x

)−m

inx∈Ωv

(x))

inΩ.

Inp

arti

cula

r,ifv

isb

oun

dedu,v∈C

1,α(Ω

)fo

ral

lα∈

(0,1

).

4.-

Asy

mpto

tic

behavio

r

Th

eore

m.

Un

der

assu

mp

tion

s

λ>

2χ,

u0∈C

0(Ω

),0<u<c

the

un

iqu

eso

luti

on(u,v

)sa

tisfi

es

|u−

1|L∞

(Ω)

+|v−

1|L∞

(Ω)−→

0ast−→∞.

Idea

of

the

pro

of:

ut−

∆u

=−χ∇u·∇

v+χu

(u−v

)+λu

(1−u

)

−∆v

+v

=u

We

con

sid

erth

efo

llow

ing

syst

emof

equ

atio

ns

ut

=χu

(u−u

+λ χ

(1−u

)),

ut

=χu

(u−u

+λ χ

(1−u

)).

Ste

p1.

-u

andu

exis

tfo

rt∈

(0,∞

);

Ste

p2.

-u≤u

;

Ste

p3.

-0<u≤

1≤u

;

Ste

p4.

-lim

t→∞|u−u|=

0.

ut u

=χ

(u−u

+λ χ

(1−u

)),

ut u

=χ

(u−u

+λ χ

(1−u

)).

Su

btr

acti

ng

d dt(L

nu u

)=χ

2(u−u

)+λ χ

(u−u

) =

(2χ−λ

)(u−u

).

Inte

grat

ing

Lnu u≤e−

αα

0t Lnu

0

u0

and

taki

ng

limit

s,w

eob

tain lim t→∞Lnu u

=0.

Ste

p5.

-(3

)+

(4)

imp

lies |u−

1|+|u−

1|→

0ast→∞.

By

com

par

ison

,ifu

0<u

0<u

0,u

isa

sub

solu

tion

andu

isa

sup

erso

-

luti

on.

DZ

IEK

UJE

ZA

UW

AG

E