R E S E A R C H
Open Access
Large time behavior for a
multidimensional chemotaxis model
Lan Luo
**Correspondence:
[email protected] School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P.R. China
Abstract
We investigate the global existence and large time behavior of the solutions to the Cauchy problem of a multidimensional chemotaxis model with initial data close to a constant state. We also obtain the time decay rate ofkth-order derivatives of these solutions.
MSC: 35G25; 35M10
Keywords: chemotaxis model; global existence; large time behavior; time decay rate
1 Introduction
In this paper, we consider the Cauchy problem of the multidimensional chemotaxis model
⎧ ⎨ ⎩
∂tu=u+∇ ·(u∇lnv),
∂tv=uv–μv,
(.)
where (t,x)∈R+×Rn,u(t,x) andv(t,x) denote the cell density and the chemical concen-tration, respectively, andμis a constant.
The system (.) was proposed by Othmer and Stevens [] to describe the chemotac-tic movement of parchemotac-ticles where the chemicals are non-diffusible and can modify the lo-cal environment for succeeding passages. For example, myxobacteria produce lime over which their cohorts can move more readily and ants can follow trails left by predecessors []. One direct application of (.) is to model haptotaxis where cells move toward an in-creasing concentration of immobilized signals such as surface or matrix-bound adhesive molecules.
For the sake of simplicity, we setw=μt+lnv. Therefore we get from (.)
⎧ ⎨ ⎩
∂tu=u+∇ ·(u∇w),
∂tw=u.
(.)
Lettingp¯> be a constant, we substitute
u=p+p¯, q=∇w, (.)
into the system (.) to get
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂tp–p¯divq–p=div(pq),
∂tq–∇p= ,
∇ ×q= .
(.)
To represent the results in this paper, the following notations are needed. We use (·,·) to denote the standard inner products inL(Rn) and·is the corresponding norm inL(Rn). For anyr∈[,∞],Lr(Rn) is the usual Lebesgue space on Rn. We define the operators withs∈Rby
sf(x) :=
Rn|
ξ|sfˆ(ξ)eπix·ξdξ,
wherefˆ(ξ) =Ff(ξ) is the Fourier transform off(x). We shall useDkwith an integerk≥ to stand for the usual spatial derivatives of order k. Whenk< ork is not a positive integer,Dkstands fork. We then recall the homogeneous Besov spaces. Letφ∈Cc∞(Rnξ)
be such thatφ(ξ) = when|ξ| ≤ andφ(ξ) = when|ξ| ≥. Letϕ(ξ) =φ(ξ) –φ(ξ) and
ϕj(ξ) =ϕ(ξj) for anyj∈Z. Then
j∈Zϕj(ξ) = ifξ= . We definej:=F–(ϕj)∗f, then, for ∈Rand ≤r,s≤ ∞, we define the homogeneous Besov spacesB˙r,s(Rn) with the norm · B˙r,sdefined by
fB˙r,s:= j∈Z
sj jfsLr
s
, fB˙r,∞:=sup
j∈Z
j
jfLr.
We also usek(p, q)(t):=kp(t)+kq(t) and denote byCa generic positive
constant which varies from line to line. For the energy estimates, the instant energy func-tional for a solution (p, q)(t,x) to the system (.), denoted by EN(t) has the equivalent relation with some integerN≥,
EN(t)∼ N
k=
Dk(p, q)(t). (.)
In addition, the corresponding dissipation functional denoted byDN(t) satisfies
DN(t)∼ N
k=
Dk(p, q)(t)+DN+p(t). (.)
Here the notationA∼Bis used to denote that there exist two generic positive constants
C<Csuch thatCB≤A≤CB.
With the above preparation, the main results in this paper can be stated as follows.
Theorem . There exists> such that ifEN()≤with N>n+ and n≥,there is a
unique global solution(p, q) = (p, q)(t,x)to the system(.)with(p, q)(,x) = (p(x), q(x))
that satisfies
d
This global solution has the following time decay:
(p, q)(t)≤C( +t)– for any∈[,N]. (.)
If we further assume(p, q)B˙–,∞<∞with ∈(,
n
],we have
(p, q)(t)≤C( +t)–(+ ) for any∈[– ,N– ]. (.)
Remark . Similar time decay rates to (.) and (.) were obtained when the initial data in the additional negative homogeneous Sobolev spaceH˙– (Rn) with ∈(,n) for equa-tion (.) with some perturbaequa-tion term in [] the referee told us. Compared to the results in [], our results can include the endpoint cases while the equation considered in [] is a little more complicated. In some sense our results generalize their partial time decay rates results and we believe that our proof methods can also apply to the equation considered in [] to obtain the time decay rates of the endpoints. In addition, the other small assump-tions of the initial data are imposed in [, ], some time decay rates results can be obtained and our results also improve the time decay rate results in [, ]. As to the chemotaxis sig-nificance of the time decay rates of equation (.), we refer the reader to [–], where the authors give more discussions.
From Theorem ., (.) and (.), we have the following results on the asymptotic be-havior of solutions to the system (.).
Corollary . There exists> such that ifNk=Dk(u
–p¯,∇lnv) ≤ with N > n
+ and n≥,then the Cauchy problem(.)has a global solution(u,v) = (u,v)(t,x)with
(u,v)(,x) = (u(x),v(x))satisfying for all t>
N
k=
Dk(u–p¯,∇lnv)(t)≤C.
This global solution has the following time decay:
(u–p¯,∇lnv)(t)≤C( +t)– for any∈[,N].
If we further assume(u–p¯,∇lnv)B˙–,∞<∞with ∈(,
n
],we have
(u–p¯,∇lnv)(t)≤C( +t)–(+ ) for any∈[– ,N– ].
was in the Chemin-Lerner space framework (seee.g.[, ]). In [], the authors obtained the global weak existence and partial time decay of the solutions to the system (.) near some constant state in the slightly low regular Sobolev space. It was studied in [] and a comprehensive qualitative and numerical analysis were provided. We will not attempt to exhaust references in this paper. We refer reader to [, –, –, ] and the references therein for more discussions in this direction.
In this paper we consider precise large time behavior of the global solutions to the sys-tem (.) near some constant state, motivated by [, , ]. We first construct the uniform estimates, which are used to obtain global existence and time decay rate under the as-sumption of initial energy being small enough. Once a global solution are constructed, by using a time-weighted methods, we can prove the time decay rate of thekth-order deriva-tives of these solutions approaching a constant state at a time decay rateO(t–k) without
additional assumption. We remark that the time-weighted methods was also applied to prove the time decay rate for the fluid system [, ]. If we further assume that the initial data is in the negative Besov space (not necessary small) by a similar interpolation to [, ], we can obtain the fast time decay rate.
2 Some basic estimates
In this section we first represent some elementary inequalities and then we use these re-sults to handle the nonlinear term.
Lemma .([]) For any f,g∈L∞(Rn)∩Hk(Rn),we have
Dk(fg)≤CfL∞Dkg+gL∞Dkf. (.)
Lemma .([]) Suppose that m= .We have the interpolation estimate kgLr≤Cmg
–θ
L g
θ
˙
B,∞≤C mg–θ
L g
θ
L, (.)
where <θ< and≤r≤ ∞and
k+n
–
n
r =m( –θ) + θ. (.)
Lemma . If≤k≤N– ,it follows that Dkdiv(pq),divDkq≤CEN(t)Dk+(p, q)
(.)
and
Dkdiv(pq),Dkp≤CE N(t)
N
m=k+
Dm(p, q). (.)
We have
Proof By (.) and Lemma ., we have, forθ= (n – )/[n],
pL∞≤CDp(–θ)D[
n
]+pθ≤CEN(t). (.)
It follows from this and (.) that
Dkdiv(pq),divDkq≤C(p, q)L∞Dk+(p, q)≤C
EN(t)Dk+(p, q)
.
It follows from (.) that
Dkdiv(pq),Dkp=Dk(pq),Dk+p
≤C(p, q)L∞Dk(p, q)Dk+p. (.)
Then we see from (.) that (.) holds. Next we prove (.). Ifk= , (.) follows from (.) and (.). If ≤k≤n, it follows from (.) that
pL∞kp≤Cαp k
k+[n]+p
k+k+pkk+pk+, (.)
whereα=n
– ([ n ] + –
n )
k∈[,
n ).
Ifα∈Z, by (.) and (.), we get (.). Ifα∈/Z, it follows from (.) that
αp≤[α]p[α]+–α[α]+pα–[α]. (.)
By this, (.) and (.), we obtain (.). If n
<k≤N– , we have from (.)
pL∞kp≤Cp–θk+p
θ
k+p–θαpθ, (.)
whereθ=(kn+) andα= ( –n)(k+ )∈(,k+ ). Ifα∈Z, by (.) and (.), we get (.).
Ifα∈/Z, we use (.), (.) and (.) to obtain (.).
3 Energy analysis
In this section we first deduce the uniform estimates of the system (.) and make use of the time-weighted methods to the time decay rate (.). Then we use the properties of the Besov space and the interpolation to get the fast time decay rate.
Theorem . Letting(p, q) = (p, q)(t,x)be the solution to the system(.)satisfyingEN(t) <
,we have the estimate
d
dtEN(t) +CDN(t)≤. (.)
HereEN(t)andDN(t)are defined as(.)and(.).
Proof Taking the derivativeDkof (.) with ≤k≤N, we can obtain
d dtD
For the system (.), we have for ≤k≤N–
d dtdivD
kq– d
dt
Dkp,divDkq+p¯divDkq–∇Dkp
= –Dkdiv(pq),divDkq. (.)
For anyκ> small enough, we have from (.) and (.)
d dt
N
k=
Dkp+p¯Dkq+κ N–
k=
divDkq– Dkp,divDkq
+ N–
k=
( –κ)∇Dkp+ κp¯divDkq+ DN+p
= N
k=
Dkdiv(pq),Dkp– κ
N–
k=
Dkdiv(pq),divDkq. (.)
Notice that the equality∇xq=divq+∇ ×qholds on p. of []. By the fact that∇ ×q= , we havedivDkq=Dk+q. By this, for anyκ> small enough, one
has
EN(t) = N
k=
Dkp+p¯Dkq+κ N–
k=
divDkq– Dkp,divDkq
∼
N
k=
Dk(p, q),
DN(t) = N–
k=
( –κ)∇Dkp+ κp¯divDkq+ DN+p
∼
N
k=
Dk(p, q)+DN+p.
By Lemma . we have
N
k=
Dkdiv(pq),Dkp +
N–
k=
Dkdiv(pq),divDkq ≤C
EN(t)DN(t).
SinceEN(t) <, by these, we see from (.) that (.) holds.
Remark . By Theorem . we have deriveda prioriestimates. Based on this uniform estimate the global existence is proved by the standard continuation argument together with the local in time existence theory; for example, see [].
Lemma . Assume that (p, q) = (p, q)(t,x)be the solution to the system(.)satisfying EN(t) <.For any fixed m∈ {, , . . . ,N– },we have the following estimate:
d dtE
m(t) +CDm(t)≤. (.)
HereEm(t)andDm(t)are defined by
Em(t) = N
k=m
Dkp+p¯Dkq+κ
N–
k=m
divDkq– Dkp,divDkq, (.)
Dm(t) = N–
k=m
( –κ)∇Dkp+ κp¯divDkq+DN+p, (.)
whereκ> is small enough.
Proof For anyκ> small enough, we have from (.) and (.)
d dt
N
k=m
Dkp+p¯Dkq+κ N–
k=m
divDkq– Dkp,divDkq
+ N–
k=m
( –κ)∇Dkp+ κp¯divDkq+ DN+p
= N
k=m
Dkdiv(pq),Dkp– κ
N–
k=m
Dkdiv(pq),divDkq. (.)
SincedivDkq=Dk+qandκ> is small enough, one has from (.) and (.) that
Em(t)∼ N
k=m
Dk(p, q), and Dm(t)∼ N
k=m+
Dk(p, q)+DN+p. (.)
By Lemma . we have
N
k=m
Dkdiv(pq),Dkp +
N–
k=m
Dkdiv(pq),divDkq ≤C
EN(t)Dm(t).
SinceEN(t) <, by these, we see from (.) that (.) holds.
In the following we will use the time-weighted methods to show the time decay of the solutions to the system (.).
Theorem . Assume that(p, q) = (p, q)(t,x)be the solution to the system(.)satisfying EN(t) <.We have the time decay estimate,for any m∈[,N],
Proof For anym∈ {, . . . ,N– }, we see from (.) that
d dt
( +t)mEm(t)–m( +t)m–Em(t) + ( +t)mDm(t)≤. (.) Letting the constantsCm> andCmCm+be large enough, we define the functional
E(t) = N–
m=
Cm( +t)mEm(t). (.)
By (.), (.) and (.) withm= , we see that
E(t) +C
N–
m=
t
( +s)mDm(s)ds≤E(). (.)
SinceEN(t) <, for anym∈ {, , . . . ,N– }, we have from (.) and (.)
Dm(p, q)≤CEm(t)≤C( +t)–m. (.)
By (.) and (.) one has
d dtD
Np
+p¯DNq+C∇DNp
≤Cε(p, q)
L∞D
N(p, q)+ε∇DNp.
Takingε> small enough, we have
d dtD
Np
+p¯DNq+C∇DNp≤Cε(p, q)
L∞D
N(p, q)
. (.)
It follows from (.) that
d dt
( +t)NDNp+p¯DNq–CN( +t)N–DN(p, q)
≤C( +t)N(p, q)L∞DN(p, q). (.)
By (.) and (.), one has(p, q)
L∞≤C( +t)–. It follows from (.) that ( +t)NDN(p, q)≤C
t
( +s)N–DN(p, q)(s)ds+CDN(p, q).
SinceEN(t) <, we have from (.) and the definition ofDN–(s) that
C t
( +s)N–DN(p, q)(s)ds+CDN(p, q)
≤C.
By the above estimates one has
DN(p, q)≤C( +t)–N.
To show the fast time decay under the additional assumption, for this, we need to show the propagation of the solutions inB˙–,∞.
Lemma . For any ∈(,n],for the system(.),we have d
dt
p˙
B–,∞+p¯q
˙
B–,∞
+C∇p˙
B–,∞≤Cp
L n
x
q. (.)
Proof Taking the operationjon the sides of (.), we have
d dtjp
–p¯(
jdivq,jp) +∇jp=
divj(pq),jp
and
d dtjq
– (∇
jp,jq) = .
By these we have
d dt
jp+p¯jq
+∇jp=
divj(pq),jp
. (.)
We have
divj(pq),jp=j(pq),∇jp≤ jCpqLr∇jp
≤ jCεp
L n
x
q+ε∇jp.
Here we have used the Bernstein type inequality in [] for =n r–
n ∈(,
n
] andr∈[, ).
By using the definition ofB˙–,∞and takingε> small enough, it follows from (.) that
(.) holds.
To finish the proof of the fast time decay we will make use of an interpolation method similar to [, ].
Proof of(.)in Theorem. By (.) we have, for any ∈(,n],
(p, q)(t)B˙–
,∞≤C(p, q)
˙
B–,∞+C
t
p
L n
x
qds. (.)
For ∈(,n–
], we have ≤[ n – ] <
n and
p
L n
x
q≤CD[n– ]p(–θ)D[n– ]+pθq≤CEN(s)DN(s). (.)
Hereθ=n– – [n – ]∈(, ) for n– ∈/Zby Lemma . andθ = for n – ∈Z. If
EN()≤, it follows from (.) that for anyt>
EN(t) +C
t
The above three inequalities imply that if(p, q)B˙–,∞≤C
(p, q)(t)B˙–
,∞≤C(p, q)
˙
B–,∞+
≤C.
For fixedm∈ {, , , . . . ,N– }we have from (.)
Dm(p, q)≤CDm+(p, q)
+m
+m+(p, q) +m+
˙
B–,∞ ≤CD
m+(p, q) ++mm+.
It follows from this, (.), (.) and (.) that
Em(t)+ +m
≤C N
k=m
Dk(p, q)
+ +m
≤C N–
k=m
Dk(p, q)
+ +m
+CDN(p, q)(+
+m)
≤C
N
k=m
Dk(p, q)+CDN(p, q)≤CDm(t).
We have from (.) and the above inequality, form∈ {, , , . . . ,N– },
d dtE
m(t) +CEm(t)+ +m≤. (.)
By using (.), we solve this inequality to obtain
N
k=m
Dk(p, q)≤C( +t)–(m+ ). (.)
By using Lemma ., we can obtain (.) for ∈(,n– ].
Next we consider the case ∈(n– ,n) and(p, q)B˙–,∞ <∞. In this case, we have
(p, q)˙
B–,∞ <∞for any
∈(, ) by interpolation:
(p, q)˙
B–,∞ ≤C(p, q)
/
˙
B–,∞(p, q) ( – )/
˙
B,∞
≤C(p, q)
/
˙
B–,∞(p, q) ( – )/
<∞. (.)
Then (.) holds for any ∈(,n– ] and anym∈ {, , , . . . ,N– }. We set =n– with
n≥. By (.) and (.) we have
(p, q)(s) L n
x
(p, q)(s)
≤CD[n– ](p, q)(–θ)D[n– ]+(p, q)θ(p, q)(s)
≤C( +s)–(–θ)([n– ]+ )( +s)–θ([n– ]++ )( +s)–
Since ∈(n– ,n) and n≥, one has n – +n– > . It follows from (.) that
(p, q)(t)B˙–,∞<∞for anyt≥. By this we may repeat the arguments leading to (.)
and prove that (.) holds for the case ∈(n– ,n).
Finally if =n and(p, q)B˙,–∞<∞, it follows from (.) that(p, q)B˙–
,∞<∞for any ∈(,n). Assume that =n–δ
withn≥ andδ> small enough such that
(p, q)(s) L n
x
(p, q)(s)=(p, q)(s)≤C( +s)–(n–δ)≤C( +s)–/.
By this we see from (.) that(p, q)(t)B˙–,∞<∞. By similar arguments to (.), we can
obtain (.) for the case =n. By using Lemma ., we can obtain (.) for any ∈(,n].
This completes the proof of Theorem ..
Competing interests
The author declares that she has no competing interests.
Acknowledgements
We would like to thank the referees for their helpful comments and pointing us out [3, 15]. The research of the author was supported by the NNSFC Grant 11371151.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 16 January 2017 Accepted: 14 March 2017
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