Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity

Math. Nachr. 283, No. 11, 1664 – 1673 (2010) / DOI 10.1002/mana.200810838

Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity

Michael Winkler∗1

1 Fakultat fur Mathematik, Universitat Duisburg-Essen, 45117 Essen, Germany

Received 10 February 2008, revised 31 March 2009, accepted 11 July 2009Published online 23 September 2010

Key words Chemotaxis, global existence, boundednessMSC (2000) 35B35, 35B45, 35K55, 92C17

We consider the chemotaxis system{ut = Δu−∇ · (uχ(v)∇v), x ∈ Ω, t > 0,

vt = Δv − v + u, x ∈ Ω, t > 0,

under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ Rn. The chemotactic

sensitivity function is assumed to generalize the prototype

χ(v) =χ0

(1 + αv)2, v ≥ 0.

It is proved that no chemotactic collapse occurs in the sense that for any choice of nonnegative initial datau(·, 0) ∈ C0(Ω) and v(·, 0) ∈ W 1,r(Ω) (with some r > n), the corresponding initial-boundary value problempossesses a unique global solution that is uniformly bounded.

c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Chemotaxis is the directed movement of cells as a response to gradients of the concentration of a chemical signalsubstance. It is known to play an essential role in a large variety of fields within the life cycle of most multicellularorganisms, for instance in the governing of immune cell migration, in wound healing, in tumour growth or also inthe organization of embryonic cell positioning. For a comprehensive exposition of further examples illustratingthe outstanding biological relevance of chemotaxis, we refer the reader to the recent survey of Hillen and Painter([10]; cf. also the monograph [4]).

In the most frequently studied theoretical models for chemotaxis processes, one is concerned with two un-known functions u = u(x, t) and v = v(x, t), the former representing the density of the cell population andthe latter measuring the concentration of the chemoattractant. Around 1970, Keller and Segel ([14]) initiated afruitful and still continuing period of mathematical analysis of chemotaxis by introducing a system of PDEs, thegeneral form of which reads{

ut = ∇ · (A(u, v)u−B(u, v)∇v)

+ C(u, v),

vt = DΔv + E(u, v).(1.1)

Many particular cases and derivates of this Keller-Segel model have successfully been investigated up to now,where throughout one focus has been on the question whether the respective model allows for a chemotactic col-lapse, that is, if it possesses solutions that become unbounded in finite or infinite time (cf. [8,9,12,13,20–22,29],

∗ e-mail: [email protected]

c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Nachr. 283, No. 11 (2010) / www.mn-journal.com 1665

for instance). Such a blow-up is evidently connected to the phenomenon of cell aggregation. Therefore, its oc-currence is frequently employed to justify the biological relevance of a suggested model, although in the recentyears many results concerning pattern formation of bounded solutions have illustrated a rich dynamical structurealso of some of those versions of (1.1) for which no collapse happens (see [5,7,17,24–26,30], and the discussionin [11, Sect. 6.1.2]).

The present work is concerned with the question whether or not a chemotactic collapse occurs in the specialvariant of (1.1) addressed in the problem⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

ut = Δu−∇ · (uχ(v)∇v), x ∈ Ω, t > 0,

vt = Δv − v + u, x ∈ Ω, t > 0,∂u

∂ν=

∂v

∂ν= 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Ω,

(1.2)

where Ω ⊂ Rn is a bounded domain with smooth boundary, u0 and v0 are given nonnegative functions, and ∂

∂νdenotes differentiation with respect to the outward normal derivative on ∂Ω.

The function χ, often referred to as chemotactic sensitivity function, is supposed to generalize the standardchoice

χ(v) =χ0

(1 + αv)2, v ≥ 0, (1.3)

(cf. (1.4) below).As contrasted to the simplest choice χ ≡ 1 in the “minimal” Keller-Segel model thereby obtained from (1.2),

sensitivity laws of the form (1.3) are used to take into account the partial loss of the signal substance through itsbinding, for instance, to cell surface receptors (see [10,16], or [28] and the references in the latter).

It is not surprising that the decay at v = ∞ of χ as given by (1.3) has a dampening effect on the ability ofcells to move towards a higher signal concentration when this concentration is high; nevertheless, as pointed outin [10], the question whether this is sufficient to exclude blow-up appears to be open so far. A related work ofBiler ([3]) is concerned with positive functions χ decaying at v = ∞ which are such that vχ(v) is increasingwith v. Thus covering the case χ(v) = χ0

β+v for β > 0, in space dimension n = 2 his study excludes finite-timeblow-up. According to Nagai et al. ([23]), the same two-dimensional result is available for χ(v) = χ0

v if eitherχ0 < 1, or χ0 < 5

2 and the solution is radially symmetric. Apart from that, boundedness of solutions as wellas asymptotic properties have been asserted for rather general functions χ – also admitting singular behavior atv = 0 – when either n = 1 ([25]), or n = 2 and a production term C(u, v) ≡ C(u) as in (1.1) appears whichreflects a logistic growth dampening at large values of u ([1, 24]).

Our main result rules out any collapse in (1.2) for a class of functions χ that comprises the particular choicein (1.3). More precisely, we shall presuppose throughout this paper that χ is nonnegative and belongs toC1+δ([0,∞)) for some δ > 0, and that

χ(v) ≤ χ0

(1 + αv)kwith some χ0 > 0, α > 0 and k > 1 (1.4)

holds for all v ≥ 0. Under this assumption, we shall prove that

if n ≥ 1, u0 ∈ C0(Ω) and v0 ∈ W 1,r(Ω) for some r > n then (1.2) possesses a unique global classicalsolution that is bounded in Ω× (0,∞) (Theorem 3.2).

Let us emphasize that besides (1.4), we need to impose no further condition on either the space dimension, or onχ; in particular, neither any monotonicity assumption on χ(v) or vχ(v), nor any smallness requirement on χ0

will be necessary for our approach.In order to put our results in perspective within a more general modeling framework, we finally observe that

the first equation in (1.2) can be equivalently be rewritten as

ut = Δu−∇ · (u∇Φ(v)), x ∈ Ω, t > 0,

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1666 M. Winkler: Chemotaxis system

where Φ(v) = − ∫ ∞v

χ(s) ds is finite in view of (1.4). Then it can easily be checked that our boundedness resultapplies whenever Φ is smooth and nondecreasing on [0,∞) and satisfies

Φ(s) =sκ

1 + csκfor all s ≥ s0

with some κ > 0, c > 0 and s0 > 0; in particular, the biologically meaningful cases ([18, 19, 27])

Φ(s) =s

1 + cs, s ≥ 0, and Φ(s) =

s2

1 + cs2, s ≥ 0,

are covered as well, the former of course being nothing else than (1.3).

2 Preliminaries and local existence

Before starting our analysis, for later reference let us first briefly collect some well-known facts concerning theLaplacian in Ω supplemented with homogeneous Neumann boundary conditions (cf. [13] for more details and fora complete list of references). Firstly, the operator −Δ + 1 is sectorial in Lp(Ω) and therefore possesses closedfractional powers (−Δ + 1)θ, θ ∈ (0, 1), with dense domain D((−Δ + 1)θ). If m ∈ {0, 1}, p ∈ [1,∞] andq ∈ (1,∞) then with some constant c > 0, for all w ∈ D((−Δ + 1)θ) we have

‖w‖W m,p(Ω) ≤ c ‖(−Δ + 1)θw‖Lq(Ω), provided that m− n

p< 2θ − n

q. (2.1)

Moreover, for p < ∞ the associated heat semigroup (etΔ)t≥0 maps Lp(Ω) into D((−Δ + 1)θ) in any of thespaces Lq(Ω) for q ≥ p, and there exist c > 0 and ν > 0 such that the Lp-Lq estimates∥∥(−Δ + 1)θet(Δ−1)w

∥∥Lq(Ω)

≤ c t−θ−n2 ( 1

p− 1q )e−νt‖w‖Lp(Ω) for all w ∈ Lp(Ω) (2.2)

and ∥∥(−Δ + 1)θetΔw∥∥

Lq(Ω)≤ c t−θ−n

2 ( 1p− 1

q )e−νt‖w‖Lp(Ω) for all w ∈ Lp⊥(Ω) (2.3)

hold, where Lp⊥(Ω) := {w ∈ Lp(Ω) | ∫

Ωw = 0} denotes the orthogonal complement in Lp(Ω) of the null space

of the Neumann Laplacian.Finally, given p ∈ (1,∞), for any ε > 0 there exists cε > 0 such that

‖(−Δ + 1)θetΔ∇ · z‖Lp(Ω) ≤ cε t−θ− 12−εe−νt‖z‖Lp(Ω) (2.4)

is valid for all Rn-valued z ∈ Lp(Ω) ([13, Lemma 2.1]).

We first assert local-in-time existence of a classical solution. The arguments used here are quite standard (cf. also[31]), and so we may confine ourselves with an outline of the proof. A more detailed demonstration of a similarreasoning in a related situation can be found in [13, Theorem 3.1].

Lemma 2.1 Let the nonnegative functions u0 and v0 satisfy u0 ∈ C0(Ω) and v0 ∈ W 1,r(Ω) for some r > n.Then there exists Tmax ≤ ∞ and a uniquely determined pair (u, v) of nonnegative functions

u ∈ C0(Ω× [0, Tmax)) ∩ C2,1(Ω× (0, Tmax)),v ∈ C0(Ω× [0, Tmax)) ∩ L∞

loc([0, Tmax); W 1,q(Ω)) ∩ C2,1(Ω× (0, Tmax))

that solves (1.2) in the classical sense. If Tmax < ∞ then

‖u(·, t)‖L∞(Ω) + ‖v(·, t)‖L∞(Ω) →∞ as t ↗ Tmax. (2.5)

Moreover, the total masses of u and v evolve according to the identities∫Ω

u(x, t)dx =∫

Ω

u0(x)dx for all t ∈ (0, Tmax) (2.6)

c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com


and ∫Ω

v(x, t)dx =∫

Ω

u0(x)dx +(∫

Ω

v0(x)dx−∫

Ω

u0(x)dx

)· e−t for all t ∈ (0, Tmax), (2.7)

respectively.

P r o o f. Throughout we assume that χ has been extended to a C1+δ function (still named χ) on all of R, andadopt commonly used abbreviations like u(t) for u(·, t) without any danger of confusion. We consider the fixedpoint equation (u, v) = Ψ(u, v), where

Ψ(u, v) ≡(

Ψ1(u, v)Ψ2(u, v)

):=

(etΔu0 −

∫ t

0e(t−s)Δ∇ · (u(s)χ(v(s))∇v(s)

)ds

et(Δ−1)v0 +∫ t

0e(t−s)(Δ−1)u(s)ds

)

in the closed subset B := {(u, v) ∈ X | ‖(u, v)‖X ≤ R} of the space

X := C0([0, T ]; C0(Ω))× L∞((0, T ); W 1,r(Ω))

with norm ‖(u, v)‖X := ‖u‖L∞(Ω×(0,T )) + ‖v‖L∞((0,T );W 1,r(Ω)). From elementary properties of the heat semi-group it immediately follows that Ψ(u, v) ∈ X for all (u, v) ∈ X . Our goal is to show that Ψ becomes contractiveand maps B into itself if R is suitably large and T > 0 appropriately small.

To this end, we let a > 0 be such that ‖w‖L∞(Ω) ≤ a‖w‖W 1,r(Ω), which is possible since r > n, and letL(R) denote a Lipschitz constant for χ on the interval (−aR, aR). Fixing θ ∈ ( n

2r , 12 ) and then ε ∈ (0, 1

2 − θ),we thus obtain for all (u, v), (u, v) ∈ B by using (2.1) and (2.4) that

‖Ψ1(u, v)(t)−Ψ1(u, v)(t)‖L∞(Ω)

≤ c

∫ t

0

∥∥(−Δ + 1)θe(t−s)Δ∇ · [u(s)χ(v(s))∇v(s)− u(s)χ(v(s))∇v(s)]∥∥

Lr(Ω)ds

≤ c

∫ t

0

(t− s)−θ− 12−ε

∥∥u(s)χ(v(s))∇v(s)− u(s)χ(v(s))∇v(s)∥∥

Lr(Ω)ds

≤ c

∫ t

0

(t− s)−θ− 12−ε

{‖u(s)− u(s)‖L∞(Ω) · ‖χ(v(s))‖L∞(Ω) · ‖∇v(s)‖Lr(Ω)

+ ‖u(s)‖L∞(Ω) · ‖χ(v(s))− χ(v(s))‖L∞(Ω) · ‖∇v(s)‖Lr(Ω)

+ ‖u(s)‖L∞(Ω) · ‖χ(v(s))‖L∞(Ω) · ‖∇v(s)−∇v(s)‖Lr(Ω)

}ds

≤ c(2‖χ‖L∞(R) ·R + R2 · L(R)

) · T 12−θ−ε · ‖(u, v)− (u, v)‖X

(2.8)

for all t ∈ [0, T ]. Similarly, picking ρ ∈ ( 12 , 1) we find

‖Ψ2(u, v)(t)−Ψ2(u, v)(t)‖W 1,r(Ω)

≤ c

∫ t

0

∥∥(−Δ + 1)ρe(t−s)(Δ−1)(u(s)− u(s))∥∥

Lr(Ω)ds

≤ c

∫ t

0

(t− s)−ρ‖u(s)− u(s)‖Lr(Ω) ds

≤ c T 1−ρ‖(u, v)− (u, v)‖X for all t ∈ [0, T ].

(2.9)

Inserting (u, v) ≡ (0, 0) into (2.8) and (2.9), we in particular obtain that Ψ maps B into itself if we chooseR sufficiently large and then T small; apart from that, (2.8) and (2.9) show that after possibly diminishing T ,Ψ becomes a contraction. Hence, Ψ possesses a fixed point which evidently is a weak solution of (1.2) in thesense of ([15, p. 136]). The above choice of T (depending on u0 and v0 only through their norms in L∞(Ω) andW 1,r(Ω), respectively) moreover ensures that (u, v) can be extended up to some maximal Tmax ≤ ∞, wherenecessarily (2.5) holds if Tmax < ∞.



From standard parabolic regularity arguments ([15]) it follows that (u, v) satisfies the regularity propertieslisted in the formulation of the lemma, and solves (1.2) classically. Accordingly, the maximum principle appliesto yield u ≥ 0 and v ≥ 0 in Ω× (0, Tmax), whereas (2.6) and (2.7) immediately result upon integration.

Finally, to see that two solutions (u, v) and (u, v) of (1.2) in Ω× [0, T ] from the indicated class must coincide,let w := u− u and z := v − v. Then the easily obtained identities∫

Ω

z2t +

d

dt

∫Ω

[12

∫Ω

|∇z|2 +12

∫Ω

z2

]= −

∫Ω

∇w · ∇z −∫

Ω

wz +∫

Ω

w2

and12

d

dt

∫Ω

w2 +∫

Ω

|∇w|2 =∫

Ω

[uχ(v)∇v − uχ(v∇v)] · ∇w

along with straightforward estimating techniques relying on the local Lipschitz continuity of χ lead to the in-equality

d

dt

[∫Ω

|∇z|2 +∫

Ω

z2 +∫

Ω

w2

]≤ c(T )

[∫Ω

|∇z|2 +∫

Ω

z2 +∫

Ω

w2

]

for all t ∈ (0, T ) with some c(T ) > 0. Consequently, the Gronwall lemma ensures z ≡ 0 and w ≡ 0, asdesired.

3 Boundedness

The main step towards global boundedness of solutions is carried out in the next lemma. It demonstrates uniformboundedness of the population density u(·, t) in any of the spaces Lp(Ω) with p < ∞. This is accomplished byproviding some associated weighted bounds involving weight functions which depend on the signal density v,but which are uniformly bounded both from above and below by positive constants.

Lemma 3.1 For all p > 1 there exists a constant C(p) > 0 such that the first component of the solution of(1.2) satisfies

‖u(·, t)‖Lp(Ω) ≤ C(p) for all t ∈ (0, Tmax). (3.1)

P r o o f. Given p > 1, we fix κ > 0 small such that

κ <p− 18p

and κ < 2k − 2, (3.2)

where k > 1 is the constant from (1.4). Then we pick β > α large enough fulfilling

β >

√2p(p− 1)κ(κ + 1)

χ0 (3.3)

and define

ϕ(s) := e(1+βs)−κ

for s ≥ 0.

Using both PDEs in (1.2) along with the resprecitve boundary conditions, we obtain upon differentiation andintegration by parts that

1p

d

dt

∫Ω

upϕ(v) =∫

Ω

up−1ϕ(v)ut +1p

∫Ω

upϕ′(v)vt

=∫

Ω

up−1ϕ(v)Δu−∫

Ω

up−1ϕ(v)∇ · (uχ(v)∇v)

+1p

∫Ω

upϕ′(v)Δv − 1p

∫Ω

upvϕ′(v) +1p

∫Ω

up+1ϕ′(v) =



= −(p− 1)∫

Ω

up−2ϕ(v)|∇u|2 −∫

Ω

up−1ϕ′(v)∇u · ∇v

+ (p− 1)∫

Ω

up−1ϕ(v)χ(v)∇u · ∇v +∫

Ω

upϕ′(v)χ(v)|∇v|2

−∫

Ω

up−1ϕ′(v)∇u · ∇v − 1p

∫Ω

upϕ′′(v)|∇v|2

− 1p

∫Ω

upvϕ′(v) +1p

∫Ω

up+1ϕ′(v).

Since v ≥ 0 and χ(s) ≥ 0 as well as ϕ′(s) ≤ 0 for all s ≥ 0, we thus have

1p

d

dt

∫Ω

upϕ(v) + (p− 1)∫

Ω

up−2ϕ(v)|∇u|2 +1p

∫Ω

upϕ′′(v)|∇v|2

≤ −2∫

Ω

up−1ϕ′(v)∇u · ∇v + (p− 1)∫

Ω

up−1ϕ(v)χ(v)∇u · ∇v − 1p

∫Ω

upvϕ′(v).(3.4)

Here, since

−sϕ′(s) = κβs(1 + βs)−κ−1 e(1+βs)−κ ≤ κ(1 + βs)−κe(1+βs)−κ

= κϕ(s)

for all s ≥ 0, we have

−1p

∫Ω

upvϕ′(v) ≤ κ

p

∫Ω

upϕ(v).

Moreover, by Young’s inequality,

−2∫

Ω

up−1ϕ′(v)∇u · ∇v ≤ p− 14

∫Ω

up−2ϕ(v)|∇u|2 +4

p− 1

∫Ω

up ϕ′2(v)ϕ(v)

|∇v|2

and

(p− 1)∫

Ω

up−1ϕ(v)χ(v)∇u · ∇v ≤ p− 14

∫Ω

up−2ϕ(v)|∇u|2

+ (p− 1)∫

Ω

upϕ(v)χ2(v)|∇v|2

≤ p− 14

∫Ω

up−2ϕ(v)|∇u|2

+ (p− 1)χ20

∫Ω

up(1 + αv)−2kϕ(v)|∇v|2,

where we also have used (1.4). Thus, (3.4) yields

1p

d

dt

∫Ω

upϕ(v) +p− 1

2

∫Ω

up−2ϕ(v)|∇u|2 +1p

∫Ω

upϕ′′(v)|∇v|2

≤ 4p− 1

∫Ω

up ϕ′2(v)ϕ(v)

|∇v|2 + (p− 1)χ20

∫Ω

up(1 + αv)−2kϕ(v)|∇v|2

+κ

p

∫Ω

upϕ(v).

(3.5)

Our goal is to demonstrate that our assumptions on κ and β guarantee that the terms on the right-hand sidecontaining |∇v|2 are dominated by 1

p

∫Ω

upϕ′′(v)|∇v|2. To this end, we compute

I1 :=4

p− 1ϕ′2(s)ϕ(s)

=4

p− 1· β2κ2(1 + βs)−2κ−2e(1+βs)−κ

,

I2 := (p− 1)χ20(1 + αs)−2kϕ(s) = (p− 1)χ2

0(1 + αs)−2ke(1+βs)−κ



and

I3 :=1p

ϕ′′(s) =1p· β2κ(κ + 1)(1 + βs)−κ−2e(1+βs)−κ

+1p· β2κ2(1 + βs)−2κ−2e(1+βs)−κ

for s ≥ 0. Hence,

I112I3

≤4

p−1 · β2κ2(1 + βs)−2κ−2e(1+βs)−κ

12 · 1

p · β2κ(κ + 1)(1 + βs)−κ−2e(1+βs)−κ

=8pκ

(p− 1)(κ + 1)· (1 + βs)−κ

≤ 8pκ

p− 1

(3.6)

holds for s ≥ 0 due to the first restriction in (3.2) and the fact that κ > 0. Next,

I212I3

≤ (p− 1)χ20(1 + αs)−2ke(1+βs)−κ

12 · 1

p · β2κ(κ + 1)(1 + βs)−κ−2e(1+βs)−κ

=2p(p− 1)χ2

0

β2κ(κ + 1)· (1 + αs)−2k · (1 + βs)κ+2.

As κ + 2 < 2k by (3.2) and since β > α, the function ψ defined for s ≥ 0 by ψ(s) := (1 + αs)−2k(1 + βs)κ+2

satisfies

ψ′(s) = (1 + αs)−2k−1 · (1 + βs)κ+1 ·{− 2k(1 + βs) + (κ + 2)(1 + αs)

}≤ (1 + αs)−2k−1 · (1 + βs)κ+1 ·

{− (2k − κ− 2)− [2kβ − (κ + 2)α]s

}≤ 0 for all s > 0,

so that ψ(s) ≤ ψ(0) = 1 for all s ≥ 0. Therefore, in view of (3.3),

I212I3

≤ 2p(p− 1)χ20

β2κ(κ + 1)≤ 1 (3.7)

holds whenever s ≥ 0. From (3.6) and (3.7) we find that

4p− 1

∫Ω

up ϕ′2(v)ϕ(v)

|∇v|2 + (p− 1)χ20

∫Ω

up(1 + αv)−2kϕ(v)|∇v|2 ≤ 1p

∫Ω

upϕ′′(v)|∇v|2

and hence, by (3.5),

1p

d

dt

∫Ω

upϕ(v) +p− 1

2

∫Ω

up−2ϕ(v)|∇u|2 ≤ κ

p

∫Ω

upϕ(v) (3.8)

for all t ∈ (0, Tmax). Now since for any q > 0 there exists c(q) > 0 such that ‖w‖W 1,2(Ω) ≤ c(q)(‖∇w‖L2(Ω) +‖w‖Lq(Ω)) holds for all w ∈ W 1,2(Ω), we obtain from the Gagliardo-Nirenberg inequality ([6]) and the fact thatϕ(s) ≤ e for all s ≥ 0 that∫

Ω

upϕ(v) ≤ e

∫Ω

up

= e‖u p2 ‖2L2(Ω)

≤ e · cGN‖up2 ‖2a

W 1,2(Ω) · ‖up2 ‖2(1−a)

L2p (Ω)

≤ e · cGN · (c( 2p ))2a

(‖∇u

p2 ‖L2(Ω) + ‖u p

2 ‖L

2p (Ω)

)2a

· ‖u p2 ‖2(1−a)

L2p (Ω)

(3.9)



holds with some positive constant cGN and

a =2np − n

22np + 1− n

2

∈ (0, 1). (3.10)

According to the mass conservation property (2.6), we have

‖u p2 (·, t)‖

2p

L2p (Ω)

=∫

Ω

u(x, t)dx ≡∫

Ω

u0(x)dx for all t ∈ (0, Tmax),

so that from (3.9) we infer the existence of some c1 > 0 such that∫Ω

upϕ(v) ≤ c1

(‖∇u

p2 ‖2L2(Ω) + 1

)a

.

Since ϕ(s) ≥ 1 for s ≥ 0, this entails

p− 12

∫Ω

up−2ϕ(v)|∇u|2 ≥ p− 12

∫Ω

up−2|∇u|2

=2(p− 1)

p2

∫Ω

|∇up2 |2

≥ 2(p− 1)

p2c1a1

(∫Ω

upϕ(v)) 1

a

− 2(p− 1)p2

,

so that (3.8) gives

1p

d

dt

∫Ω

upϕ(v) ≤ −2(p− 1)

p2c1a1

(∫Ω

upϕ(v)) 1

a

+2(p− 1)

p2+

κ

p

∫Ω

upϕ(v)

for all t ∈ (0, Tmax). Consequently, y(t) :=∫Ω

up(x, t)ϕ(v(x, t)) dx satisfies

y′(t) ≤ −Ay1a (t) + By(t) + C

with certain positive constants A, B and C. In view of (3.10), we have 1a > 1 and thus a standard ODE com-

parison argument implies boundedness of y on (0, Tmax). In particular, again using that ϕ(s) ≥ 1 for s ≥ 0, weconclude that ‖u(·, t)‖Lp(Ω) ≤ C(p) holds for all t ∈ (0, Tmax) and some C(p) > 0.

We now derive from the above lemma that the local solution (u, v) constructed in Lemma 2.1 is uniformlybounded and hence, in particular, actually global in time. There are two well-established methods to achievethis in the literature: One way to derive such L∞ bounds is based on an iteration of appropriate Lp norms ([2]);another one uses a series of standard semigroup arguments. Although both techniques would equally fit to thepresent situation, we apply the latter one in the proof of our final result. In order to keep the exposition essentiallyself-contained, let us include a short proof here.

Theorem 3.2 The solution (u, v) of (1.2) is global and bounded.

P r o o f. In view of Lemma 2.1 it is sufficient to make sure that for any τ ∈ (0, Tmax),

‖u(·, t)‖L∞(Ω) + ‖v(·, t)‖L∞(Ω) ≤ c(τ) for all t ∈ (τ, Tmax) (3.11)

holds with some c(τ) > 0. To this end, we let τ ∈ (0, Tmax) be given such that τ < 1, and fix positive numbersθ ∈ (0, 1

2 ) and ε ∈ (0, 12 − θ) and then p > n

2θ . Moreover, we let ρ ∈ ( 12 , 1) and pick q > 1 large such that

2ρ− nq > 1− n

2p , that is, q > 2np(2ρ−1)p+n . Then from the representation formula

v(·, t) = et(Δ−1)v0 +∫ t

0

e(t−s)Δ)u(·, s) ds, t ∈ (0, Tmax),

we obtain in view of (2.1), (2.2) and Lemma 3.1



‖v(·, t)‖W 1,2p(Ω) ≤ c ‖(−Δ + 1)ρv(·, t)‖Lq(Ω)

≤ c t−ρe−νt‖v0‖Lq(Ω)

+ c

∫ t

0

(t− s)−ρe−ν(t−s)‖u(·, s)‖Lq(Ω) ds (3.12)

≤ c t−ρ + c

∫ t

0

(t− s)−ρe−ν(t−s)ds

≤ c t−ρ + c

∫ ∞

0

σ−ρe−νσdσ

≤ c (t−ρ + 1) for all t ∈ (0, Tmax), (3.13)

where here and in the sequel c denotes a generic constant that may vary from line to line. As a consequence ofthis, (1.4) and again Lemma 3.1, using the Holder inequality we find

‖u(·, s)χ(v(·, s))∇v(·, s)‖Lp(Ω) ≤ χ0 · ‖u(·, s)‖L2p(Ω) · ‖∇v(·, s)‖L2pΩ)

≤ c τ−ρ for all s ∈ ( τ2 , Tmax).

Therefore, writing m :=∫Ω

u0(x) dx and employing the variation of constants formula for u,

u(·, t)−m = e(t− τ2 )Δ(u(·, τ

2 )−m)

−∫ t

τ2

e(t−s)Δ∇ · (u(·, s)χ(v(·, s))∇v(·, s)ds, t ∈ (τ2 , Tmax

),

and recalling (2.1), (2.3), (2.6) and (2.4), we infer that for all t ∈ (τ, Tmax) we have

‖u(·, t)−m‖L∞(Ω) ≤ c ‖(−Δ + 1)θ(u(·, t)−m)‖Lp(Ω)

≤ c (t− τ2 )−θe−ν(t− τ

2 )‖u(·, τ2 )−m‖Lp(Ω)

+c

∫ t

0

(t− s)−θ− 12−εe−ν(t−s)‖u(·, s)χ(v(·, s))∇v(·, s)‖Lp(Ω) ds

≤ c τ−θ + c τ−ρ ·∫ ∞

0

σ−θ− 12−εe−νσdσ.

Since θ + 12 + ε < 1 by assumption, this proves the desired L∞(Ω) bound for u, whereas the inequality (3.12)

yields ‖v(·, t)‖L∞(Ω) ≤ c τ−ρ for t ∈ (τ, Tmax) due to the fact that p > n2θ > n

2 implies W 1,2p(Ω) ↪→L∞(Ω).

Acknowledgements The author would like to thank T. Hillen for drawing his attention to the question pursued here.

References[1] M. Aida, K. Osaki, T. Tsujikawa, A.Yagi, and M. Mimura, Chemotaxis and growth system with singular sensitivity

function, Nonlinear Anal. Real World Appl. 6, No. 2, 323–336 (2005).[2] N. D. Alikakos, Lp bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4, 827–

868 (1979).[3] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl. 9 (1), 347–359 (1999).[4] M. Eisenbach, Chemotaxis (Imperial College Press, London, 2004)[5] E. Feireisl, Ph. Laurencot, and H. Petzeltova, On convergence to equilibria for the Keller-Segel chemotaxis model, to

appear in J. Differential Equations.[6] A. Friedman, Partial Differential Equations (Holt, Rinehart & Winston, New York, 1969).[7] M. Funaki, M. Mimura, and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces

and Free Boundaries 8, 223–245 (2006).[8] H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr. 195,

77–114 (1998).



[9] M. A. Herrero and J. J. L. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl.Sci. (5) 24, 663–683 (1997).

[10] T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, to appear in J. Math. Biol.[11] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresb. Deutsch.

Math.-Verein. 105 (3), 103–165 (2003).[12] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math.

12, 159–177 (2001).[13] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (1),

52–107 (2005).[14] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26, 399–415

(1970).[15] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type (Amer.

Math. Soc., Providence, RI, 1968).[16] H. A. Levine and B. D. Sleeman, A system of reaction-diffusion equations arising in the theory of reinforced random

walks, SIAM J. Appl. Math. 57, 683–730 (1997).[17] M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Phys. A 230

(3–4), 499–543 (1996).[18] J. D. Murray, Mathematical Biology I: An Introduction, 3rd ed., Series: Interdisciplinary Applied Mathematics Vol. 17

(Springer, Heidelberg – New York, 2002).[19] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd ed., Series: Interdisciplinary

Applied Mathematics Vol. 18 (Springer, Heidelberg – New York, 2003).[20] T. Nagai, Global Existence and Blowup of Solutions to a Chemotaxis System, Nonlinear Anal. 47, 777–787 (2001).[21] T. Nagai, T. Senba, and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima

Math. J. 30, 463–497 (2000).[22] T. Nagai, T. Senba, and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,

Funkcial. Ekvac. 40, 411–433 (1997).[23] T. Nagai, T. Senba, and K. Yoshida, Global existence of solutions to the parabolic system of chemotaxis, RIMS

Kokyuroku 1009, 22–28 (1997).[24] K. Osaki, T. Tsujikawa, A. Yagi, and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,

Nonlinear Anal., Theory Methods Appl. 51, 119–144 (2002).[25] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac. 44,

441–469 (2001).[26] A. B. Potapov and T. Hillen, Metatsability in Chemotaxis Models, J. Dyn. Differ. Equations 17 (2), 293–330 (2005).[27] R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc. 292, 531–556 (1985).[28] L. A. Segel, Incorporation of receptor kinetics into a model for bacterial chemotaxis, J. Theor. Biol. 57 (1), 23–42

(1976).[29] T. Senba and T. Suzuki, Parabolic system of chemotaxis: blowup in a finite and the infinite time, Methods Appl. Anal.

8, 349–367 (2001).[30] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial Differ. Equations 32 (6), 849–877

(2007).[31] A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Jap. 45, 241–265 (1997).


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