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Partial differential equations

Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source

Existence globale et bornes pour les solutions classiques d’un système quasi linéaire, parabolique–elliptique, de chimiotaxie avec source logistique

Ali Khelghati, Khadijeh Baghaei

DepartmentofMathematics,PayameNoorUniversity(PNU),P.O.Box19395-3697,Tehran,Iran

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received29August2014 Acceptedafterrevision25July2015 Availableonline2October2015 PresentedbytheEditorialBoard

Weconsiderthequasilinearparabolic–ellipticchemotaxissystem u_t= ∇ ·(D(u)∇^u−

χ

u∇^v)+^g(u), x∈,t>0,

0=v−v+u, x∈,t>0,

under homogeneous Neumann boundaryconditions in asmooth bounded domain⊂ Rⁿ,n≥^{1.WeassumethatthefunctionsDandgaresmoothandsatisfy}

D(s) >0 fors≥0,D(s)≥CDs^m−1 fors>0, g(0)≥0,g(s)≤a−bs^γ, s>0

withsomeconstantsCD>0,m≥¹,a≥⁰,b>0 and

γ

>2.

Weprovethattheclassicalsolutionstotheabovesystemareuniformlyin-time-bounded withoutanyrestrictionsonmandb.ThisresultextendsoneoftherecentresultsbyWang etal.(2014)[16],whichasserttheboundednessofsolutionsfor

γ

>2 underthecondition b>b^∗withb^∗=^{0 form}≥²−²n andb^∗=⁽²(⁻2−^mm)n)⁻n²

χ

form<2−²n.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

r é s um é

Nousconsidéronslesystèmequasilinéaire,parabolique–elliptique,dechimiotaxie

ut= ∇ ·(D(u)∇u−

χ

u∇v)+g(u), x∈,t>0, 0=v−v+u, x∈,t>0,

avecdesconditionsaubordhomogènesdeNeumann,dansundomainelisse,borné⊂ Rⁿ,n≥^{1.NoussupposonsquelesfonctionsD et}

D(s) >0 pours≥0,D(s)≥CDs^m−1 pours>0, g(0)≥0,g(s)≤a−bs^γ, s>0

E-mailaddresses:[email protected](A. Khelghati),[email protected](K. Baghaei).

http://dx.doi.org/10.1016/j.crma.2015.08.006

1631-073X/©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

pourcertainesconstantesCD>0,m≥^1,a≥^0,b>0 et

γ

>2.

Nousdémontrons que lessolutions classiques dusystème ci-dessussont uniformément bornéesentemps,sansrestrictionsurmetb.Ceciétendunrésultatrécent deWanget al.(2014)[16],quibornelessolutionspour

γ

>2 souslaconditionb>b^∗,oùb^∗=^{0 si} m≥²−²netb^∗=⁽²(⁻2−^mm)n)⁻n²

χ

sim<2−n².

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

1. Introduction

InthisNote,westudythefollowinginitialboundaryvalueproblem:

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

u_t

= ∇ · (

D

(

u

)∇

^u

−

^S

(

u

)∇

^v

) +

^g

(

u

),

x

∈ ,

t

>

0

, τ

v_t

=

v

−

^v

+

^u

,

x

∈ ,

t

>

0

,

∂

u

∂ ν ⁼

∂

v

∂ ν ⁼

⁰

^,

^x

^{∈ ∂,}

^t

^>

⁰

^,

u

(

x

,

0

) =

^u0

,

v

(

x

,

0

) =

^v0

,

x

∈ ,

(1.1)

where

⊆

Rⁿ

,

n

≥

^{1,isaboundeddomainwithsmoothboundary,}

τ ∈ {

⁰

,

1

}

^and

ν

denotestheunitoutwardnormalvector to

∂

.WeassumethatS

(

u

) = χ

uwith

χ >

0 andthefunction DbelongstoC²

([

⁰

, ∞))

^{andsatisfies}

D

(

s

) >

0 fors

≥

⁰

,

D

(

s

) ≥

^CDs^m−1 fors

>

0 (1.2)

withsomeconstants C_D

>

0 andm

≥

^{1.Wealsoassumethat g isasmoothfunctionthatsatisfies}

g

(

0

) ≥

⁰

,

g

(

s

) ≤

^a

−

^{bsγ fors}

>

0 (1.3)

withconstantsa

≥

⁰

,

b

>

0 and

γ >

2.Moreover,u0

∈

^Cα

()

andv0

∈

^W1,r

()

forsome

α >

0 andr

>

n.

Problemsofthiskindareusedinmathematicalbiologytoillustratethemechanismofchemotaxis,thatis,themovement of cells towards the gradient of a substance called chemoattractant produced by the cellsthemselves. Here, u

=

^u

(

x

,

t

)

denotes thecell densityandv

=

^v

(

x

,

t

)

is theconcentration ofthechemicalsubstance. While thefunctions D and S are thediffusivityandchemotacticsensitivity,respectively,andg isthegrowthofu[7,4].

In the absence ofa logistic source, when D

≡

¹

,

S

(

u

) = χ

u with

χ >

0 and

τ >

0,for the one-dimensional case, it is shownthat blow-up phenomena cannot occur [11].Forthe two-dimensional case, itis proved that if

^u0

L¹()

<

⁴_χ^π, then allsolutionsare globalandbounded[10].Whilefor

^u0

L¹()

>

⁴_χ^π,solutions becomeunboundedeitherinfiniteor infinite time[5].Forhigher-dimensionalcase, itis shownthatforeach q

>

ⁿ₂ and p

>

n,there exists 0

>

0 such thatif

^u0

L^q()

<

and

^v0

L^p()

<

for

<

0,thenthecorrespondingsolutionsareglobalandbounded[18].Also,when

is aballfor

^u0

L¹()

>

0,itisprovedthatthereexistsolutionsblowupinfinitetime[22].ForthecaseD

≡

¹

,

S

(

s

) ≤

^c

(

s

+

¹

)

^q fors

≥

^{0 withc}

>

0 and

τ >

0,solutionsareglobalandboundedprovidedthatq

<

²_n andsolutionsblowupinfinitetime orinfinitetime ifS

(

s

) ≥

^c

(

s

+

¹

)

^q for s

≥

^{0 withc}

>

0 andq

>

²_n [6].When

isaboundedconvexdomaininRⁿ

,

n

≥

^2, and

τ >

0,thensolutionsareuniformly-in-timeboundedprovidedthat _D^S(₍^s_s⁾₎

≤

^c

(

s

+

¹

)

^α fors

≥

^{0 withc}

>

0 and

α <

²_n and other additionalconditionsarefulfilled[13],whereasforthecasewhere _D^S(₍^s_s⁾₎

≥

^c

(

s

+

¹

)

^α fors

≥

^{0 withc}

>

0 and

α >

²_n, thereexistsolutionsblowupeitherinfiniteorinfinitetime[19].Ifthesecondequationisreplacedwith0

=

v

−

^M

+

^u, where M denotes the meanvalue of initial data u₀,then under the conditions D

(

s

) ≥

^cDs^−p and S

(

s

) ≤

^cSs^q for s

≥

¹ with cD

,

cS

>

0

,

p

≥

^{0 andq}

∈

R^{, all solutions are globaland uniformlybounded provided that p}

+

^q

<

²_n, whereas for 0

<

D

(

s

) ≤

^cDs^−p andS

(

s

) ≥

^cSs

(

s

+

¹

)

^q−1 fors

≥

^{0 withc}D

,

cS

>

0

,

p

≥

⁰

,

q

>

0 and p

+

^q

>

²_n,thereexistradialsolutions thatbecomeunboundedinfinitetime[25].

Inthepresenceoflogisticsource,when D

≡

¹

,

S

(

u

) = χ

uwith

χ >

0 and

τ =

^{0,itisprovedthat problem(1.1)admits} atleastoneglobalveryweaksolutionif

γ >

2

−

_n^{1 [17].Also,inthiscase,itisshownthatsolutionsareglobalandbounded} if

γ ₌

2 and b

>

ⁿ⁻_n²

χ

[14]. Thesame resultistrue for

τ >

0 and

γ ₌

2 providedthat n

≤

^{2 orn}

≥

^{3 andb}

>

b0 with b0 sufficientlylarge[12,20].Also,inthiscase,forn

≥

^{3,itisprovedthatthereexistsatleastoneglobalweaksolutionfor} arbitraryb

>

0[8].Moreover,inthiscasewhentheratio χ^b issufficientlylarge,itisshownthatforanychoiceofsuitably regular nonnegativeinitialdata,thereexistsa uniqueglobalclassicalsolution

(

u

,

v

)

suchthat

^u

(.,

t

) −

¹_b

L^∞()

→

^{0 and}

v

(.,

t

) −

¹_b

L^∞()

→

0 as t

→ ∞

[23].Ifthe firstequation iswritten asut

= −∇ · (

u

∇

v

) +

au

−

bu² and

τ =

0,then all solutions are globalin time forb

≥

^{1,andthere existsolutions blow up in finitetime ifb}

<

1. Theseresults havebeen obtained byWinklerforthe one-dimensional case[24] andLankeitforhigher-dimensional case[9].When

is aball in Rⁿ

,

n

≥

^{5,forthecaseD}

≡

^{1 andS}

(

u

) = χ

uwith

χ >

0,ifthesecondequationisreplacedwith0

=

v

+

^u

−

_||¹

u

(

x

,

t

)

dx and g satisfiesinthecondition g

(

s

) ≥ −

^{bsγ fors}

≥

^{0 withb}

≥

^{0 and}

γ >

1 andother additionalconditions,thenradially symmetricsolutionsblowupinfinitetimeprovidedthat1

< γ <

³₂

+

_2n¹₋₂ ^{[21].Also,forn}

≥

^{5 inthecasewhereD}

(

s

) ≤

^s−p

and S

(

s

) =

^{sq for s}

>

0 with p

,

q

∈

R ^{and g satisfies inthe condition g}

(

s

) ≥

^a

−

^{bsγ for s}

≥

^{0 with a}

,

b

≥

^{0 and}

γ >

1 andother additionalconditions, blow-up phenomena occur in finitetime if ²_n

−

¹

<

p

<

1

,

q

>

1 with2p

+

^3q

<

4 and 1

< γ <

⁽³⁻_2n^p₋⁾ⁿ₂⁻² [26]. Moreover, it is shown that under the conditions D

(

s

) ≥ (

s

+

¹

)

^−p and S

(

s

) ≤

^{sq for s}

≥

^{0 with} p

,

q

∈

R^{,allsolutions areglobalanduniformlyboundedprovidedthat p}

+

^q

<

_n² and

γ >

1 or p

+

^q

≥

_n²

,

b

>

^(p₍⁺_p+^q)_qⁿ₎⁻_n²

χ

and

γ ≥

^q

+

^{1 withq}

≥

^{1[26].Inthecasewhere}

τ >

0,undertheconditionsD

(

s

) ≥

^c1s^p

,

c2s^q

≤

^S

(

s

) ≤

^c3s^qfors

≥

^s0

>

1 withci

>

0

,

i

=

¹

,

2

,

3 and p

,

q

∈

R^{andg issmoothon}

[

⁰

, ∞ )

satisfying g

(

s

) ≤

^as

−

^{bs2 fors}

≥

^{0 with a}

≥

^{0 andb}

>

0,it isprovedthatthesolutionsareglobalandboundedprovidedthatq

<

1.Thisresultisindependentofthechoiceof p[2].

Wangetal.[16] studied problem(1.1) undertheconditions(1.2) and(1.3) with S

(

u

) = χ

u and

τ =

^{0, where}

χ

issome positive constant. Theirresults improvethe recent resultin [3], whichasserts the boundednessof solutions with

γ =

² undertheconditionb

> χ (

1

−

_n(1−²_m)+

)

,or,equivalently,m

>

1

−

_n(χ²₋^χ_b)+^{.Infact,Wangetal.provedthatfor}

γ ≥

^{2 under} theconditionb

>

b^∗ withb^∗

=

^{0 form}

≥

²

−

²_n ^andb∗

=

⁽²₍⁻₂₋^m_m⁾ⁿ₎⁻_n²

χ

form

<

2

−

²_n^{,problem(1.1)hasaunique}nonnegative classicalsolution,whichisglobalandbounded.Theyalsoprovedthatthesameresultistrueprovidedthat 1

< γ <

2 and m

>

2

−

²_n^{.Besides,forthecase}

γ ₌

2,undertheconditions0

<

b

≤

⁽²₍⁻₂₋^m_m⁾ⁿ₎⁻_n²

χ

andm

<

2

−

²_n^{,they provedthatproblem} (1.1)hasatleastonenonnegativeglobalsolutionintheweaksense.

Inthepresentpaper,wewillstudyproblem(1.1)with

τ =

^{0 undertheconditions(1.2)and(1.3).Wewillprovethatfor}

γ >

2,problem(1.1) hasauniquesolution,whichisuniformlyin-time-bounded.Wedonotknow,forthelimitcase

γ =

² undertheconditions0

<

b

≤

⁽²₍⁻₂₋^m_m⁾ⁿ₎⁻_n²

χ

andm

<

2

−

²_n^{,whethersolutionsexistglobally orblowupinfinitetime.So,the} globalexistence ortheblowingupofsolutionswillremainanopenquestioninthecasewherem

<

2

−

²_n ^and1

< γ _≤

2.

Inthenextsection,wewillprovetheaboveresult.

2. Proofofmainresult

Here,westatethestandardwell-posednessandclassicalsolvabilityresult.

Lemma2.1.LetfunctionsD andg satisfy(1.2)and(1.3).Moreover,weassumethatu0

∈

^Cα

()

andv0

∈

^W1,r

()

arenonnegative functionsforsome

α >

0andr

>

n.Thenproblem(1.1)hasauniquenon-negativeclassicalsolutionthatcanbeextendeduptoits maximalexistencetimeT_max

∈ (

0

, ∞]

^{.Inaddition,ifT}max

< +∞

^,then

t→lim^Tmax

^u

(.,

t

)

L^∞()

= ∞ .

Fordetailsoftheproof,wereferthereaderto[25,16].

Inordertoprovetheboundednessofthesolution,weneedthefollowinglemma,whichisgivenin[15].

Lemma2.2.Lety beapositiveabsolutelycontinuousfunctionon

(

0

, ∞ )

thatsatisfies

_dy

dt

+

^Ayp

≤

^B

,

y

(

0

) =

y0

withsomeconstantsA

>

0

,

B

≥

^0andp

>

1.Thenfort

>

0,wehave y

(

t

) ≤

^max

y₀

,

^B

A

¹_p

.

Althoughtheproofofthefollowinglemmaisgivenin[16],wepresentithereforcompleteness.

Lemma2.3.Assumethatthefunctiong satisfies(1.3).Thenforallt

∈ (

0

,

T_max

)

,thereexistsaconstantC

>

0suchthat

^u

(.,

t

)

L¹()

≤

^C

.

(2.1)

Proof. Integratingthefirstequationin(1.1)andusing(1.3),weget d

dt

udx

=

g

(

u

)

dx

≤

(

a

−

^buγ

)

dx

.

Hence, d dt

udx

+

^b

u^γdx

≤

^a

| | .

(2.2)

MakinguseofHölder’sinequality,weobtain

u^γdx

≥

udx

γ

| |

^1−γ

.

Combiningthisinequalitywith(2.2)gives d

dt

udx

+

b

||

^1−γ

udx

γ

≤

a

||.

Becauseof

γ >

2,wecanapplyLemma 2.2andobtainthedesiredresult. 2

Lemma2.4.Assumethatthefunctiong satisfies(1.3).Thenforallk

>

1andt

∈ (

0

,

Tmax

)

,thereexistsaconstantC

>

0suchthat

^u

(.,

t

)

L^k()

≤

^C

.

Proof. Atfirst,wecompute d

dt

u^kdx

=

k

u^k−1utdx

=

^k

u^k−1

∇ ·

^D

(

u

) ∇

^u

− χ

u

∇

^v

dx

+

^k

u^k−1g

(

u

)

dx

≤ −

^k

(

k

−

¹

)

u^k−2D

(

u

)|∇

^u

|

^2dx

+

^k

(

k

−

¹

) χ

u^k−1

∇

^u

· ∇

^vdx

+

^ak

u^k−1dx

−

^bk

u^k+γ−1dx

.

(2.3)

Wemakeuseofintegratingbypartsandusethesecondequationin(1.1)toobtain k

(

k

−

¹

) χ

u^k−1

∇

^u

· ∇

^vdx

= (

k

−

¹

) χ

∇

^uk

· ∇

^vdx

= −(

^k

−

¹

) χ

u^k

vdx

= −(

k

−

1

) χ

u^k

(

v

−

u

)

dx

≤ (

k

−

¹

) χ

u^k+1dx

.

Substitutingthisinequalityinto(2.3),weget:

d dt

u^kdx

+

^k

(

k

−

¹

)

u^k−2D

(

u

) |∇

^u

|

^2dx

≤ (

k

−

¹

) χ

u^k+1dx

+

^ak

u^k−1dx

−

^bk

u^k+γ−1dx

.

(2.4)

Becauseof

γ >

2,wecanusetheYounginequalitywithexponentss

=

^k+_k^γ₊⁻₁^{1 ands}

=

^k+_γ^γ₋⁻₂^{1.Thus,weobtain}

u^k+1dx

≤

^b

χ

u^k+γ−1dx

+

C1

,

whereC1

= (

_χ^bs

)

^−s

s

(

s

)

⁻¹

||

isapositiveconstant.Hence,wecanwrite

−

^bk

u^k+γ−1dx

≤ − χ

k

u^k+1dx

+

^C2

withC2

= χ

kC1.Combiningthelastinequalitywith(2.4)gives d

dt

u^kdx

+

^k

(

k

−

¹

)

u^k−2D

(

u

)|∇

^u

|

^2dx

≤ − χ

u^k+1dx

+

^ak

u^k−1dx

+

^C2

.

(2.5)

WenowmakeuseofYoung’sinequalitytothesecondtermontheright-handsideof(2.5)toget ak

u^k−1dx

≤ χ

2

u^k+1dx

+

C3

,

whereC₃

=

_k+²₁

(

ak

)

^k+12 _χ²⁽₍^k_k⁻₊¹₁⁾₎ ^k−1₂

||

^{isapositiveconstant.}Substitutingthisinequalityinto(2.5) yields d

dt

u^kdx

+ χ

2

u^k+1dx

≤

^C4 (2.6)

withC₄

=

^C2

+

^C3.WenowmakeuseofHölder’sinequalitytoobtain

u^k+1dx

≥

u^kdx

^k+1

k

| |

^−1k

.

Thisinequalityalongwith(2.6)yields

d dt

u^kdx

+ χ

2

| |

^−1k

u^kdx

^k+1

k

≤

^C4

.

By applying Lemma 2.2, the last inequality yields

^u

L^k(), which is bounded for all t

∈ (

0

,

T_max

)

. This completes the proof. 2

ByusingLemma 2.4andAlikakos’iterativetechnique[1](seealso[13,LemmaA.1]),wecaninferourmainresult.

Theorem2.5.Letu0

∈

^Cα

()

andv0

∈

^W1,r

()

benonnegativefunctionsforsome

α >

0andr

>

n.Moreover,weassumethatfunc- tionsD andg satisfy(1.2)and(1.3).Thenproblem(1.1)admitsauniqueglobalclassicalsolutionwhichisuniformlyin-time-bounded.

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