Global solutions in a fully parabolic chemotaxis system with singular sensitivity

Research Article

Received 15 April 2010 Published online 24 August 2010 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1346MOS subject classification: 92 C 17; 35 K 55; 35 B 35; 35 B 40

Global solutions in a fully parabolicchemotaxis system with singular sensitivity

Michael Winkler∗†

Communicated by Howard A. Levine

The Neumann boundary value problem for the chemotaxis system⎧⎨⎩

ut = Du−v∇·(u

v∇v)

, x ∈X, t>0,

vt = Dv−v+u, x ∈X, t>0,

is considered in a smooth bounded domain X⊂Rn, n�2, with initial data u0 ∈C0(X̄) and v0 ∈W1,∞(X) satisfying u0�0and v0>0 in X̄. It is shown that if 0<v<

√2 / n then for any such data there exists a global-in-time classical solution,

generalizing a previous result which asserts the same for n=2 only. Furthermore, it is seen that the range of admissiblev can be enlarged upon relaxing the solution concept. More precisely, global existence of weak solutions is establishedwhenever 0<v<

√(n+2) / (3n−4). Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: chemotaxis; global existence; weak solutions; singularity formation

1. Introduction

We consider the Neumann initial boundary value problem for two coupled parabolic equations,⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

ut = �u−�∇ ·(u

v∇v)

, x ∈�, t>0,

vt = �v−v+u, x ∈�, t>0,

�u

��= �v

��=0, x ∈��, t>0,

u(x, 0) = u0(x), v(x, 0)=v0(x), x ∈�,

(1)

with parameter �>0, in a bounded domain �⊂Rn with smooth outward normal vector field � on ��. The initial functions u0 ∈C0(�̄)and v0 ∈W1,∞(�) are assumed to satisfy u0�0 and v0>0 in �̄.

This problem is used in the theoretical description of chemotaxis, the biological phenomenon of oriented, or partially oriented,movement of cells in response to a chemical signal that is produced by themselves. Being a special case of a more general PDEsystem originally introduced by Keller and Segel [1], (1) especially focuses on the modeling of those chemotaxis processes wheremovement towards higher signal concentrations is inhibited at points where these concentrations are high; in [2, 3] and [4], thereader can find both a derivation of the crucial first equation in (1) and a number of examples where incorporating the above formof signal dependence of chemotactic sensitivity is used as an essential ingredient in the respective model. For various relatives of(1) with the second equation replaced by an ODE, quite a rich understanding has been gained both at a numerical [3, 5] and atan analytical level [2, 5]; however, the knowledge appears to be much less complete as soon as diffusion is accounted for in theequation for the signal.

Mathematically, the system (1) of two diffusion equations may be viewed as a borderline case among basically well-understoodmodels. To demonstrate this, let us replace the sensitivity function �(v), chosen as �(v)=� / v in (1), by ��,�(v)=� / (v+�)1+� for ��0

and ��0. Then known results state that if n=2, �>0, ��0 and ��(1+�)��, then all reasonable solutions of (1) are global in time

Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany∗Correspondence to: Michael Winkler, Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany.†E-mail: [email protected]

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Copyright © 2010 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2011, 34 176–190

M. WINKLER

[6, Theorem 3 and Remark 4]. Moreover, in the case when both � and � are positive, the result in [7] goes beyond this and statesthat for arbitrary n�1 and �>0, all solutions will be global and even bounded. Hence, either of the sets of conditions n�1, �>0,�>0, �>0, and n=2, �>0, �=0, ��1 is sufficient to prevent a chemotactic collapse in the sense of finite-time blow-up such as itmay occur in the so-called minimal Keller–Segel model formally obtained when �=−1 [8].

However, in the limit case �=0 and �=0 to be considered here, it seems that the picture is more involved: whereas all solutionsof (1) are global in time when either n=1 [9], or when n=2 and ��1 [6, 10], the results in [11] give rise to the conjecture thatunbounded solutions might exist for large values of �. In that work, namely, the authors consider the parabolic–elliptic analogue of(1) obtained upon replacing the second PDE with the elliptic equation 0=�v−v+u, and it is shown that if n�3 and �>2n / (n−2)then finite-time blow-up does occur for some solutions. In addition, it is proved there that in the same simplified problem, all radiallysymmetric solutions are global in time if either n�3 and �<2 / (n−2), or n=2 and �>0 is arbitrary [11]. In the two-dimensionalsetting, the latter may be regarded as an indication for absence of blow-up also in the full parabolic–parabolic problem (1) for any�>0.

It is the purpose of the present work to investigate the question of global existence of solutions to (1), which has been postedas an open problem in [4]. The mathematical challenging issue behind this is to estimate the effectiveness of the growth inhibitionthat is to be expected when passing from the minimal Keller–Segel model to (1) by replacing �(v)≡const. with �(v)=� / v.

Our first result in this direction states that

• if �<√

2 / n then (1) has a global classical solution (Theorem 3.5).

In respect of classical solvability, this essentially extends the mentioned two-dimensional result to arbitrary space dimensions. Inparticular, as to the biologically relevant case n=3, this means that global smooth solutions exist for small � below an explicitthreshold.

Next, to gain further insight into the global dynamics of (1) for a wider range of �, we introduce a generalized solution concept.This will allow us to also include solutions that might become unbounded in space at some time, but continue to exist as a solutionin a meaningful sense afterwards. More precisely, we consider a notion of weak solutions that essentially is the natural one obtainedupon carrying over as many derivatives as possible to sufficiently smooth test functions (see Definition 4.1). As to solvability withinthis framework, we shall derive that

• if �<√

(n+2) / (3n−4) then there exists a global weak solution of (1) (Theorem 4.6).

As we shall see below, this solution has additional regularity properties; for instance, there exists �=�(n,�)>1 such thatu∈L�

loc([0,∞); L�(�)). However, we believe that our explicit lower estimates for such � are not optimal. Therefore, we prefer notto present them here, and rather refer the reader to Corollary 4.3 and the proofs of Lemmas 4.4 and 4.5, where supplementaryinformation about integrability of our weak solutions and their gradients can be found.

We have to leave open here whether blow-up with respect to the norm in L∞(�) may or may not occur for large �, includingthe question of criticality of the particular value �=√

2 / n arising here. If such blow-up solutions indeed exist, our results indicatethat the decay of �(v) for large v should influence the explosion mechanism to some extent, and it would of course be interestingto see quantitative details of these effects. Moreover, our approach does not rule out the possibility that solutions, though existingglobally and being smooth throughout, may become unbounded as t →∞. We believe, but cannot prove, that such a blow-up ininfinite time does not occur, at least not for �<

√2 / n.

The plan of the paper is as follows: in Section 2 we will provide some preparatory material, including a useful integral identity(Lemma 2.3) that will be fundamental for our subsequent analysis. We then consider the case of small � in Section 3 and demonstrateglobal existence of classical solutions. Our approach here differs from well-established regularity-providing procedures in the caseof signal-independent sensitivity functions as can be found in [12, 13] or [14], for instance: namely, our initial step will rely on acertain weighted integral estimate (Corollary 3.2) which is turned into a bound for u(·, t) in L∞(�) for 0<t<T<∞ by a recursiveargument. Next, Section 4 is devoted to the study of weak solutions, where the basic difficulty consists of proving a bound for(u / v)∇v in L�(�×(0, T)) for some �>1 (Lemma 4.4), which itself is prepared by an estimate for u in L�(�×(0, T)) for some �>2−(1 / n)(Lemma 4.5).

Let us finally mention that all of our methods can be extended so as to cover the case when the first equation in (1) is replacedwith the more general equation ut =�u−�∇ ·((u / (v+�))∇v) for any ��0, as proposed in [3] and [4], but for simplicity in notationwe refrain from giving the technical details here.

2. Preliminaries

We intend to construct a solution of (1) as the limit of a sequence of solutions to the regularized problems⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

u�t = �u�−�∇ ·(

u�

��(v�)∇v�

), x ∈�, t>0,

v�t = �v�−v�+u�, x ∈�, t>0,

�u�

��= �v�

��=0, x ∈��, t>0,

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

(2)


17

7

M. WINKLER

where �∈ (0, 1) and

��(s) :=s+�(1+s2), vs�0. (3)

Then �(s) :=� / ��(s) is smooth and nonnegative in [0,∞) and satisfies lim sups→∞ s2�(s)<∞. Therefore, we may invoke Theorem 2.2in [7] to obtain global existence of classical solutions to (2):

Lemma 2.1For all �∈ (0, 1), (2) possesses a global bounded solution (u�, v�) satisfying u��0 and v��0 in �×(0,∞).

For later reference, let us note that for each �∈ (0, 1), u� and v� enjoy the properties∫�

u�(x, t)dx =∫

�u0 for all t>0 (4)

and ∫�

v�(x, t)dx =∫

�u0 +

(∫�

v0 −∫

�u0

)·e−t for all t>0 (5)

hold; in fact, these mass identities immediately result from integrating the first and the second PDE in (2).Moreover, since we assume v0 to be strictly positive, we can strengthen the above nonnegativity statement as follows.

Lemma 2.2For any �∈ (0, 1), we have

v�(x, t)�(

infy∈�

v0(y)

)·e−t for all t>0 and x ∈�. (6)

ProofBy nonnegativity of u�, the right-hand side of (6) is a spatially homogeneous subsolution of the second PDE in (2). Therefore, theclaim results from the comparison principle. �

The following lemma will serve as a source for several �-independent integral estimates in the sequel. It will be applied withcertain parameters p>1 to provide global smooth solutions in Section 3 (cf. Lemma 3.1), and for p∈ (0, 1) in establishing globalweak solvability in Section 4 (Lemma 4.2 and Corollary 4.3).

Lemma 2.3Let p∈R and q∈R. Then for all �∈ (0, 1), the identity

d

dt

∫�

up� vq

� +q

∫�

up� vq

� −q

∫�

up+1� vq−1

� = −p(p−1)

∫�

up−2� vq

� |∇u�|2 +∫

�up

� vq−2� ·

[−q(q−1)+pq�· v�

��(v�)

]·|∇v�|2

+∫

�up−1

� vq−1� ·

[−2pq+p(p−1)� · v�

��(v�)

]∇u� ·∇v� (7)

holds for all t>0.

ProofBy direct computation using (2), we find

d

dt

∫�

up� vq

� = p

∫�

up−1� vq

� u�t +q

∫�

up� vq−1

� v�t

= p

∫�

up−1� vq

� �u�−p�∫

�up−1

� vq� ∇ ·

(u�

��(v�)∇v�

)+q

∫�

up� vq−1

� �v�−q

∫�

up� vq

� +q

∫�

up+1� vq−1

� . (8)

Integrating by parts, we see that

p

∫�

up−1� vq

� �u� =−p(p−1)

∫�

up−2� vq

� |∇u�|2 −pq

∫�

up−1� vq−1

� ∇u� ·∇v� (9)

and

−p�∫

�up−1

� vq� ∇ ·

(u�

��(v�)∇v�

)=p(p−1)�

∫�

up−1�

vq�

��(v�)∇u� ·∇v�+pq�

∫�

up�

vq−1�

��(v�)|∇v�|2 (10)

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8


M. WINKLER

as well as

q

∫�

up� vq−1

� �v� =−pq

∫�

up−1� vq−1

� ∇u� ·∇v�−q(q−1)

∫�

up� vq−2

� |∇v�|2, (11)

because �u� / ��=�v� / ��=0 on ��. Inserting (9)–(11) into (8), after obvious rearrangements we end up with (7). �

The following regularity properties of v� in dependence on boundedness features of u� can be derived by a straightforwardapplication of well-known smoothing estimates for the heat semigroup under homogeneous Neumann boundary conditions(cf. [15, Lemma 1.3] for sharp versions thereof, for instance).

Lemma 2.4Let T>0 and 1�,�∞.

(i) If (n / 2)((1 / )−(1 / ))<1 then there exists C>0 such that

‖v�(·, t)‖L(�)�C

(1+ sup

s∈(0,t)‖u�(·, s)‖L(�)

)for all t ∈ (0, T) and �∈ (0, 1). (12)

(ii) If (1 / 2)+(n / 2)((1 / )−(1 / ))<1 then

‖∇v�(·, t)‖L(�)�C

(1+ sup

s∈(0,t)‖u�(·, s)‖L(�)

)for all t ∈ (0, T) and �∈ (0, 1) (13)

is valid with some C>0.

Proof(i) We represent v� according to

v�(·, t)=et(�−1)v0 +∫ t

0e(t−s)(�−1)u�(·, s)ds, t>0, (14)

where (et�)t�0 denotes the Neumann heat semigroup. By standard smoothing estimates, we see that if � then

‖v�(·, t)‖L(�)�c1

(‖v0‖L∞(�) +

∫ t

0(t−s)−

n2 ( 1

− 1 )‖u�(·, s)‖L(�) ds

)

for some c1>0. This proves (12) in the case �, whereas for < the assertion results from this and Hölder’s inequality.(ii) Applying ∇ to both sides in (14) and invoking corresponding smoothing properties involving gradients [15, Lemma 1.3], we

similarly find that

‖∇v�(·, t)‖L(�)�c2

(‖∇v0‖L∞(�) +

∫ t

0(t−s)−

12 − n

2 ( 1 − 1

)‖u�(·, s)‖L(�) ds

)

with a certain c2>0 and conclude as before. �

3. Global smooth solutions for v<√

2 / n

The technical advantage of small values of � is that these will allow us to pick some p>1 in Lemma 2.3 in such a way that theright-hand side in (7) becomes nonpositive. Relabeling q=−r, we can thereupon derive the following.

Lemma 3.1Assume that �<1. Then for all p∈ (1, 1 / �2), each r ∈ (r−(p), r+(p)) and any T>0 one can find C>0 such that∫

�up

� (x, t)v−r� (x, t)dx�C for all t ∈ (0, T) and �∈ (0, 1) (15)

and ∫ T

0

∫�

up+1� v−r−1

� �C for all �∈ (0, 1), (16)

where

r±(p) := p−1

2

(1±

√1−p�2

).


17

9

M. WINKLER

ProofChoosing q :=−r in (7), we obtain

I := d

dt

∫�

up� v−r

� −r

∫�

up� v−r

� +r

∫�

up+1� v−r−1

�

= −p(p−1)

∫�

up−2� v−r

� |∇u�|2 −∫

�up

� v−r−2�

[r(r+1)+pr�· v�

��(v�)

]·|∇v�|2

+∫

�up−1

� v−r−1�

[2pr+p(p−1)�

v�

��(v�)

]∇u� ·∇v� (17)

for t>0. Here, Young’s inequality says that we can estimate the last term according to∣∣∣∣∫

�up−1

� v−r−1�

[2pr+p(p−1)�

v�

��(v�)

]∇u� ·∇v�

∣∣∣∣�p(p−1)

∫�

up−2� v−r

� |∇u�|2 + 1

4p(p−1)

∫�

up� v−r−2

�

[2pr+p(p−1)�· v�

��(v�)

]2·|∇v�|2.

Therefore (17) yields

I�−∫

�up

� v−r−2� h�(v�)|∇v�|2, (18)

where we have set

h�(s) := r(r+1)+pr�· s

��(s)−

[2pr+p(p−1)�· s

��(s)

]2

4p(p−1)for s�0.

By direct computation, for s�0 and �∈ (0, 1) we find

4(p−1)h�(s) = 4(p−1)r(r+1)+4p(p−1)r�· s

��(s)−4pr2 −p(p−1)2�2 s2

�2� (s)

−4p(p−1)r�· s

��(s)

= −4r2 +4(p−1)r−p(p−1)2�2 · s2

�2� (s)

.

As ��(s)�s for all s�0, we thus obtain

4(p−1)h�(s) � −4r2 +4(p−1)r−p(p−1)2�2

= 4(r+(p)−r)(r−r−(p))

> 0

for all s�0 and �∈ (0, 1). In view of (18), this shows that I�0, so that (17) implies

d

dt

∫�

up� v−r

� +r

∫�

up+1� v−r−1

� �r

∫�

up� v−r

� (19)

for all t>0 and �∈ (0, 1). Observing that r is positive, upon integrating this differential inequality we first infer that∫�

up� v−r

� �(∫

�up

0v−r0

)·erT for all t ∈ (0, T) and �∈ (0, 1)

and then conclude, again using (19), that both (15) and (16) hold for a suitable C>0. �

Let us state an immediate consequence for the available p and r in the case when �<√

2 / n.

Corollary 3.2Assume that �<

√2 / n. Then there exist p>n / 2 and r ∈ (0, n / 2) with the property that for all T>0 one can pick C>0 such that∫

�up

� (x, t)v−r� (x, t)dx�C for all t ∈ (0, T) and �∈ (0, 1).

Using that boundedness properties of u� imply (even better) boundedness properties of v� by Lemma 2.4, in the latter estimatewe actually can get rid of the factor involving v�:

Corollary 3.3Suppose that �<

√2 / n. Then there exists p>n / 2 such that for all T>0 one can find C>0 satisfying∫

�up

� (x, t)dx�C for all t ∈ (0, T) and �∈ (0, 1). (20)

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0


M. WINKLER

ProofAccording to Corollary 3.2, we pick p0>n / 2 and r ∈ (0, n / 2) such that for all T>0,∫

�up0

� (x, t)v−r� (x, t)dx�c1 for all t ∈ (0, T) and �∈ (0, 1) (21)

holds with some c1 =c1(T)>0. Since r<n / 2 and p0>n / 2, it is possible to fix p∈ (n / 2, p0) such that p<n(p0 −r) / (n−2r). ApplyingHölder’s inequality, we find that

∫�

up� �

(∫�

up0� v−r

�

) pp0 ·(∫

�v

prp0−p�

) p0−pp0

,

so that (21) implies

‖u�(·, t)‖Lp(�)�c1

p01 ‖v�(·, t)‖

rp0

Lpr

p0−p (�)for all t ∈ (0, T) and �∈ (0, 1). (22)

Now since (n / 2)((1 / p)−((p0 −p) / pr))<1 in view of our restriction p<n(p0 −r) / (n−2r), it follows from Lemma 2.4 that

supt∈(0,T)

‖v�(·, t)‖L

prp0−p (�)

�c2

(1+ sup

t∈(0,T)‖u�(·, t)‖Lp(�)

)

and accordingly, by (22),

supt∈(0,T)

‖u�(·, t)‖Lp(�)�c3

⎛⎝1+

(sup

t∈(0,T)‖u�(·, t)‖Lp(�)

) rp0

⎞⎠

with appropriate constants c2 and c3. Upon the observation that (r / p0)<1 due to p0>(n / 2)>r, we can conclude the proof. �

We proceed to turn this into an estimate in L∞(�×(0, T)) by a recursive argument. Note that the following lemma applies to any�>0—in fact, it would hold for negative � as well.

Lemma 3.4Let �>0, and suppose that there exist p0>(n / 2), T>0 and c1>0 such that∫

�up0

� (x, t)dx�c1 for all t ∈ (0, T) and all �∈ (0, 1). (23)

Then there exists c2>0 with the property

‖u�(·, t)‖L∞(�)�c2 for all t ∈ (0, T) and all �∈ (0, 1). (24)

ProofWe first claim that

‖u�(·, t)‖Lp(�)�c3 for all t ∈ (0, T) and all �∈ (0, 1) (25)

for some p>1 and c3>0 implies that ⎧⎨⎩

for all p̄<np

2(n−p)if p�n,

for p̄=∞ if p>n,

we can find c4>0 such that

‖u�(·, t)‖Lp̄(�)�c4 for all t ∈ (0, T) and all �∈ (0, 1). (26)

To show this, we first consider the case p�n. Then, if (1 / 2)+(n / 2)((1 / p)−(1 / ))<1, that is, if <np / (n−p), in view of Lemma 2.4and (25), we can find c5()>0 fulfilling

‖∇v�(·, t)‖L(�)�c5() for all t ∈ (0, T) and all �∈ (0, 1). (27)

Combining (25) with (27) and invoking Lemma 2.2, from Hölder’s inequality we gain some c6>0 such that∥∥∥∥ u�(·, t)

��(v�(·, t))∇v�(·, t)

∥∥∥∥L(�)

� c6‖u�(·, t)‖Lp(�)‖∇v�(·, t)‖L

pp− (�)

� c6c3c5

(p

p−

)for all t ∈ (0, T) and all �∈ (0, 1) (28)


18

1

M. WINKLER

whenever is such that p / (p−)<np / (n−p), that is, when <np / (2n−p). We now use the representation formula

u�(·, t)=et�u0 −�∫ t

0e(t−s)�∇ ·

(u�(·, s)

��(v�(·, s))∇v�(·, s)

)ds, t>0 (29)

and the smoothing estimate [15, Lemma 1.3]

‖e��∇ ·w‖Lp̄(�)�c7�− 1

2 − n2

(1 − 1

p̄

)‖w‖L(�), (30)

valid for all w ∈C1(�̄), �∈ (0, T) and p̄�>1 with some c7>0. We thus find that if <np / (2n−p) then

‖u�(·, t)‖Lp̄(�)�c8

(‖u0‖L∞(�) +

∫ t

0(t−s)

− 12 − n

2

(1 − 1

p̄

)ds

)

and, consequently,

‖u�(·, t)‖Lp̄(�)�c10 for all t ∈ (0, T) and all �∈ (0, 1)

with some c8 and c9, provided that (1 / 2)+(n / 2)((1 / )−(1 / p̄))<1 or, equivalently, p̄<n / (n−). Choosing close to np / (2n−p),we thereby see that (26) will hold for any p̄<np / (2(n−p)) if p�n.

In the case p>n the procedure is quite similar. Then (27) holds for =∞ and, consequently, (28) is true for =p>n. Therefore,if we pick m>n large and then �>0 small such that 0<(�−(n / 2m))<((1 / 2)−(n / 2p)) then applying the fractional power (−�+1)�

to the integral in (29) shows that

‖u�(·, t)‖L∞(�) � ‖et�u0‖L∞(�) +c10

∫ t

0

∥∥∥∥(−�+1)�e(t−s)�∇ ·(

u�(·, s)

��(v�(·, s))∇v�(·, s)

)∥∥∥∥Lm(�)

ds

� ‖u0‖L∞(�) +c11

∫ t

0(t−s)−�

∥∥∥∥e12 (t−s)�∇ ·

(u�(·, s)

��(v�(·, s))∇v�(·, s)

)∥∥∥∥Lm(�)

ds

� ‖u0‖L∞(�) +c12

∫ t

0(t−s)

−�− 12 − n

2

(1p − 1

m

)ds (31)

with constants c10, c11, and c12, where we have used the maximum principle and the well-known fact that the domain of definitionof (−�+1)� in Lm(�) is embedded into L∞(�) when �−(n / 2m)>0. As �+(1 / 2)+(n / 2)((1 / p)−(1 / m))<1 by our choice of �, weconclude that (26) is valid with p̄=∞ if p>n, as claimed.

Now the rest of the proof is a straightforward iteration of the implication (25) ⇒ (26). Namely, if we fix �>0 small such thatp0>(n / 2)+� and define

pk+1 :=

⎧⎪⎨⎪⎩

npk

2(n−pk +�)if pk�n

∞ if pk>n

⎫⎪⎬⎪⎭ for k =0, 1, 2,. . . ,

then we easily see than pk+1�pk for all k�0, and that pk =∞ holds for all sufficiently large k. Moreover, successive applications ofthe above result to p=pk and p̄ :=pk+1 shows that (u�)�∈(0,1) is bounded in L∞((0, T); Lpk (�)) for all k and hence, in particular, inL∞((0, T); L∞(�)). �

Now standard parabolic theory allows us to turn the above L∞ estimate into global solvability in the classical pointwise sense.

Theorem 3.5Suppose that �<

√2 / n. Then for all u0 ∈C0(�̄) and v0 ∈W1,∞(�) satisfying u0�0 and v0>0 in �̄, (1) has a global classical solution.

ProofAs (u�)�∈(0,1) is bounded in L∞

loc(�̄×[0,∞)), standard parabolic Schauder estimates [16] imply that both (u�)�∈(0,1) and (v�)�∈(0,1)

are bounded in C2+,1+

2loc (�̄×(0,∞)) for some >0, whence along a suitable sequence of numbers �=�k ↘0 we have u� →u and

v� →v in C2,1loc (�̄×(0,∞)) for some limit pair (u, v) that solves the PDE system and the boundary conditions in (1). Also by parabolic

regularity theory (or by going back to (27) for =∞), we obtain that ((u� / ��(v�))∇v�)�∈(0,1) is bounded in L∞loc(�̄×[0,∞)), whence

arguing as in (31), for each �>0 we can find �>0 such that

‖u�(·, t)−et�u0‖L∞(�)<� and ‖v�(·, t)−e(t−1)�v0‖L∞(�)<�

for all t ∈ (0,�). We, therefore, easily conclude that for any T>0, both u� and v� converge to their respective limits uniformly in�×(0, T) as �=�k ↘0, and that (u, v) also satisfies the initial conditions in (1). �

18

2


M. WINKLER

4. Weak solutions for v<√

(n+2) / (3n−4)

By straightforward testing procedures, we obtain a natural concept of weak solutions.

Definition 4.1Assume that �>0, u0 ∈L1(�), v0 ∈L1(�), and T>0. By a weak solution of (1) in �×(0, T) we mean a pair (u, v) of nonnegative functions

u∈L1loc(�̄×[0, T)), v ∈L1

loc(�̄×[0, T))

with the additional property

u

v∇v ∈L1

loc(�̄×[0, T)) (32)

that satisfy

−∫ T

0

∫�

u t −∫ T

0

∫�

u� −�∫ T

0

∫�

u

v∇v ·∇ =

∫�

u0 (·, 0)

as well as

−∫ T

0

∫�

v t −∫ T

0

∫�

v� +∫ T

0

∫�

v −∫ T

0

∫�

u =∫

�v0 (·, 0)

for all ∈C∞0 (�̄×[0,∞)) with � / ��=0 on ��×(0, T).

If u and v are defined in �×(0,∞) then (u, v) will be said to be a global weak solution of (1) if (u, v) is a weak solution of (1) in�×(0, T) for all T>0.

The main obstacles in proving global existence of weak solutions will be connected to the above regularity requirements, inparticular to (32). In order to achieve appropriate estimates also in the case �>1, we need to develop a strategy other than thatleading to Lemma 3.1. Our approach will still rely on Lemma 2.3, but we shall now choose p to be smaller than one and prove thatthe right-hand side of (7) will be positive for appropriate q.

Lemma 4.2Given �>0 and p∈ (0, 1) satisfying p<1 / �2, let q+(p) and q−(p) be defined by

q±(p) := 1−p

2

(1±

√1−p�2

).

Then for all q∈ (q−(p), q+(p)) there exists C>0 such that for any �∈ (0, 1) we have∫ t

0

∫�

up−2� vq

� |∇u�|2 +∫ t

0

∫�

up� vq−2

� |∇v�|2 +∫ t

0

∫�

up+1� vq−1

� �C

∫�

up� (·, t)vq

� (·, t) for all t>0. (33)

ProofAs in the proof of Lemma 3.1, the basic idea is to use Lemma 2.3, but now with reversed sign of the factor in front of those integralsin (7) that involve gradients: Indeed, from (7) we obtain

I := d

dt

∫�

up� vq

� +q

∫�

up� vq

� −q

∫�

up+1� vq−1

�

= p(1−p)

∫�

up−2� vq

� |∇u�|2 +∫

�up

� vq−2�

[q(1−q)+pq�

v�

��(v�)

]·|∇v�|2 −

∫�

up−1� vq−1

�

[2pq+p(1−p)�

v�

��(v�)

]∇u� ·∇v� (34)

for t>0, and our goal is to estimate the right-hand side from below, up to lower order terms, by positive multiples of the integralsinvolving squares of gradients. To this end, we note that Young’s inequality ensures that for all �∈ (0, 1) and >0,∣∣∣∣−∫

�up−1

� vq−1�

[2pq+p(1−p)�

v�

��(v�)

]∇u� ·∇v�

∣∣∣∣��p(1−p)

∫�

up−2� vq

� |∇u�|2 + 1

4�p(1−p)

∫�

up� vq−2

�

[2pq+p(1−p)�

v�

��(v�)

]2|∇v�|2.

Thus,

I�(1−�−)p(1−p)

∫�

up−2� vq

� |∇u�|2 +∫

�up

� vq−2� h�,�,(v�)|∇v�|2, (35)

where

h�,�,(s)=q(1−q)+pq�s

��(s)−

[2pq+p(1−p)�

s

��(s)

]2

4�p(1−p)−.


18

3

M. WINKLER

Therefore, (33) will result upon an integration as soon as we have shown that for all q∈ (q−(p), q+(p)) we can pick �∈ (0, 1), >0 andc>0 such that

h�,�,(s)�c for all s�0 and �∈ (0, 1). (36)

For this purpose, we compute

4�(1−p)(h�,�,(s)+) = 4�(1−p)q(1−q)+4�(1−p) ·pq�· s

��(s)−4pq2 −p(1−p)2�2 · s2

�2� (s)

−4p(1−p)q�· s

��(s)

= 4�(1−p)q(1−q)−4pq2 −4(1−�)p(1−p)q�· s

��(s)−p(1−p)2�2 · s2

�2� (s)

.

We can now use that ��(s)�s for all s�0 to estimate

4�(1−p)(

h�,�,(s)+)�4�(1−p)q(1−q)−4pq2 −4(1−�)p(1−p)q�−p(1−p)2�2 =: J(�) (37)

for all s�0, �∈ (0, 1), �∈ (0, 1), and >0. Here, as �↗1 we have

J(�) → 4(1−p)q(1−q)−4pq2 −p(1−p)2�2

= −4q2 +4(1−p)q−p(1−p)2�2

= 4(q+(p)−q)(q−q−(p))

> 0,

whence it follows from (37) that (36) holds if we first pick �∈ (0, 1) sufficiently close to 1 and then >0 appropriately small. �

We shall need the following consequence.

Corollary 4.3Let �>0 and p∈ (0, 1) be such that p<1 / �2, and define q±(p) as in Lemma 4.2. Then for all q∈ (q−(p), q+(p)) and each T>0 thereexists C1>0 such that

∫ T

0

∫�

up−2� vq

� |∇u�|2 +∫ T

0

∫�

up� vq−2

� |∇v�|2�C1 (38)

for all �∈ (0, 1). In particular, for all such p and each T>0 one can find C2>0 with the property

∫ T

0

∫�

up−2� |∇u�|2 +

∫ T

0

∫�

up� v−2

� |∇v�|2�C2 (39)

for all �∈ (0, 1).

ProofWe apply Lemma 4.2 and use Hölder’s inequality and the fact that p+q<p+q+(p)<1 in estimating

∫�

up� (·, t)vq

� (·, t)�(∫

�u�(·, t)

)p(∫�

v�(·, t)

)q|�|

11−p−q

for all t>0 and �∈ (0, 1). Thereupon, (33) implies (38), which in turn entails (39) in view of Lemma 2.2. �

Now if � lies below another threshold (which is larger than that in the previous section), using the above along with results onmaximal Sobolev regularity in scalar parabolic equations [17], we can find a space–time integral estimate for some power of u� thatwill be large enough to yield the desired regularity of (u / v)∇v.

Lemma 4.4Suppose that �<

√(n+2) / (3n−4). Then there exists �>2−(1 / n) with the property that for all T>0 one can find C>0 such that

∫ T

0

∫�

u��C for all �∈ (0, 1). (40)

ProofLet us first consider the case n=2, in which the condition �<

√(n+2) / (3n−4)=√

2 allows us to pick some p<1 such that p<1 / �2

and p>1 / 2. The latter requirement ensures that we can now fix a number �<p+1 satisfying �>2−(1 / n)>1. Then in view of the

18

4


M. WINKLER

mass conservation property (4), Corollary 4.3 yields uniform boundedness of up2� in L2((0, T); W1,2(�)) and hence in L2((0, T); L�(�)) for

all �∈ (0,∞) by the Sobolev embedding theorem. This means that for such �,

∫ T

0‖u�(·, t)‖p

Lp�2 (�)

dt�c1 for all �∈ (0, 1) (41)

holds with some c1>0. Specifying � :=2 / (p+1−�) here, we have �∈ (0,∞) and p� / 2>� because of �<p+1 and p<1<�, and thuswe may interpolate between (4) and (41) to obtain

∫ T

0‖u�(·, t)‖�

L�(�) dt�∫ T

0‖u�(·, t)‖p

Lp�2 (�)

‖u�(·, t)‖�−pL1(�)

�c1

(∫�

u0

)�−p

for all �∈ (0, 1), which proves (40) in the two-dimensional setting.If n�4, the hypothesis �<

√(n+2) / (3n−4) enables us to choose some p>((3n−4) / (n+2))�1 such that p<1 / �2. As p>(3n−4) /

(n+2), it is then possible to pick some �>2−(1 / n) fulfilling �<((n+2)(p+1)) / 2n, where we evidently can also achieve that�>(p+1) / 2 and �<n / 2, the latter because of the fact that n�4.

We now apply Lemma 3.1 with r := (p−1) / 2 to obtain

∫ T

0

∫�

up+1� v

− p+12

� �c2 for all �∈ (0, 1) (42)

with some c2>0. By parabolic Sobolev regularity theory [17, Theorem 3.1] applied to the second equation in (2), we furthermorefind c3>0 such that

‖v�−et(�−1)v0‖L�((0,T);W2,�(�))�c3‖u�‖L�(�×(0,T)).

Hence, invoking the embedding W2,�(�) ↪→Ln�

n−2� (�), valid since �<n / 2, and the fact that et(�−1)v0 is bounded since v0 ∈L∞(�),

‖v�‖L�((0,T);L

n�n−2� (�))

�c4(1+‖u�‖L�(�×(0,T))) (43)

for all �∈ (0, 1) with suitable c4>0. Also, Lemma 2.4 implies that for any ∈ (1, n / (n−2)) there exists c5>0 such that

‖v�‖L∞((0,T);L(�))�c5 for all �∈ (0, 1).

In conjunction with (42), for all such this gives

∫ T

0

∫�

u��

(∫ T

0

∫�

up+1� v

− p+12

�

) �p+1

·(∫ T

0‖v�(·, t)‖

(p+1)�2(p+1−�)

L(p+1)�

2(p+1−�) (�)

) p+1−�p+1

� c�

p+12

(∫ T

0‖v�(·, t)‖

(p+1)�2(p+1−�) ·aL

n�n−2�

(�)·c

(p+1)�2(p+1−�) ·(1−a)

5

) p+1−�p+1

(44)

by Hölder’s inequality, where

a=1

− 2(p+1−�)

(p+1)�1

− n−2�

n�

. (45)

Choosing :=n(2�−p−1) / 4(p+1−�) here, we have >0 since �>(p+1) / 2, and moreover <n / (n−2) because of the fact that�<((n+2)(p+1)) / 2n implies

n−2

n<

(n−2)

((n+2)(p+1)

n−p−1

)

4

(p+1− (n+2)(p+1)

2n

) =(n−2) · 2

n

4

(1− n+2

2n

) =1.

Moreover, by (45) we then have (((p+1)�) / (2(p+1−�))) ·a=� and therefore (44) combined with (43) shows that

∫ T

0

∫�

u��c6

⎛⎜⎝1+

(∫ T

0

∫�

u��

) p+1−�p+1

⎞⎟⎠ for all �∈ (0, 1)

with some c6>0. This proves (40) if the spatial dimension is at least four.


18

5

M. WINKLER

In the remaining case n=3 we apply a mixture of both above procedures. Let us first pick �>2−(1 / n)= (5 / 3) close enough to5 / 3 such that �<2 and

�<

10

(1+ 1

�2

)

11+ 1

�2

, (46)

which is possible since now our assumption on � precisely means that 1 / �2>1. Next, we can fix a number p>1 satisfying p<1 / �2,

p>11�−10

10−�(47)

and

p<7�−6

6−�, (48)

since (46) ensures that (11�−10) / (10−�)<1 / �2, and since (7�−6) / (6−�)>(11�−10) / (10−�) thanks to the positivity of �. Wefinally choose some small �∈ (0, 2 / 3) fulfilling

�<5

3− (p+1)�

6(p+1−�)(49)

and

�<1

3+ 2(p+1−�)

(p+1)�, (50)

where we note that �<p+1 because �<2 and p>1, and that the expression on the right of (49) is positive due to our restriction(47). We now apply Corollary 4.3 to obtain c7>0 and c8>0 such that∫ T

0‖u

1−�2

� (·, t)‖2W1,2(�) dt�c7

and, by an embedding estimate, ∫ T

0‖u�(·, t)‖1−�

L3(1−�)(�)dt =

∫ T

0‖u

1−�2

� (·, t)‖2L6(�) dt�c8.

Using the mass conservation property (4) and the Hölder inequality, from this we infer that∫ T

0‖u�(·, t)‖�

L�(�)dt�c9 (51)

holds for

� := 12(p+1−�)

(p+1)�+ 2

3

and � :=2

3−�

1− 1

�

=2

3−�

1

3− 2(p+1−�)

(p+1)�

(52)

and some c9>0. Here it can easily be checked that �>1 by (48) and �>1 according to (50). By parabolic regularity theory, (51) entails

a bound for v� in L�((0, T); W2,�(�)) and thus also in L�((0, T); L3�

3−2� (�)), because we have �< 32 by (52) and hence W2,�(�) ↪→L

3�3−2� (�).

Computing

3�

3−2�= 1

1

�− 2

3

= (p+1)�

2(p+1−�)

and using (49) in estimating

2(p+1−�)

(p+1)�·�>2(p+1−�)

(p+1)�·

2

3−(

5

3− (p+1)�

6(p+1−�)

)1

3− 2(p+1−�)

(p+1)�

=1,

we thereby obtain ∫ T

0‖v�(·, t)‖

(p+1)�2(p+1−�)

L(p+1)�

2(p+1−�) (�)

dt�c10 (53)

with some c10>0. We now only need to observe that the first line in (44) is still valid to derive (40) from (53). �

18

6


M. WINKLER

We can next establish sufficient regularity of the cross-diffusion term in (2).

Lemma 4.5If �<

√(n+2) / (3n−4) then there exists �>1 such that for all T>0,

∫ T

0

∫�

∣∣∣∣ u�

��(v�)∇v�

∣∣∣∣� �C for all �∈ (0, 1) (54)

holds for some C>0 independent of �.

ProofAccording to Lemma 4.4, we can pick �∈ (2−(1 / n), 2) such that

(u�)�∈(0,1) is bounded in L�loc(�̄×[0,∞)). (55)

We then can find ∈ (1,�) satisfying >n� / (n�−(n−�)), which is possible since �>2−(1 / n)>2n / (n+1). By a continuity argument,we can now fix �0 ∈ (1,�) such that

>n��0

n�−�0(n−�)for all �∈ [1,�0], (56)

where we can also achieve

�0<�−1

1− 1

, (57)

because �>2−(1 / n) and <�<2�n imply that (�−1) / (1−(1 / ))>1. Next, since (56) and �0<� guarantee that

>n�0

n−�0

(n

�−1)

= �0

�− n−1

n�0 +(�−1)

(�0

�−1

)

>�0

�− n−1

n�0

,

we see upon rearranging that (n−1) / n>((�0 / )−(� / n)) / (�−�0). Hence, for all ∈ (0, n / (n−1)] we have 1 / >((�0 / )−(� / n)) /(�−�0) or, equivalently,

�

(1

+ 1

n

)>�0

(1

+ 1

).

Therefore, for all �∈ [1,�0] and ∈ (0, n / (n−1)] the expression

I(�,) := �−1

�

(1− 1

)−�

(1

+ 1

n

)−1

�

(1

+ 1

n

)−�

(1

+ 1

)

is well defined. Since

I

(1,

n

n−1

)= �−1

1− 1

− �−1

�− n−1

n− 1

=(�−1)(�−2+ 1

n)(

1− 1

)(�− n−1

n− 1

)>0

in view of (56) and the fact that �>2−(1 / n), again concluding by continuity we can fix some �∈ (1,�0) and ∈ (0, n / (n−1)) suchthat I(�,)>0. Then s := (�−1) / �(1−(1 / )) satisfies s>1 by (57), and by positivity of I(�,), s′ :=1 / (1−(1 / s)) fulfills

1

s′ =1− 1

s>1−

�

(1

+ 1

n

)−�

(1

+ 1

)

�

(1

+ 1

n

)−1

=�

(1

+ 1

)−1

�

(1

+ 1

n

)−1

. (58)


18

7

M. WINKLER

We now let T>0 be given and apply Hölder’s inequality to estimate, using that ��(v�)�v� and Lemma 2.2,

∫ T

0

∫�

∣∣∣∣ u�

��(v�)∇v�

∣∣∣∣� � c1

∫ T

0

∫�

|u�∇v�|�

� c1

∫ T

0‖u�(·, t)‖�

L(�)‖∇v�(·, t)‖�

L�

−� (�)dt

� c1

(∫ T

0‖u�(·, t)‖�s

L(�)dt

) 1s

·(∫ T

0‖∇v�(·, t)‖�s′

L�

−� (�)dt

) 1s′

(59)

with some c1>0 independent of �∈ (0, 1). Here, recalling 1<<� we can interpolate

∫ T

0‖u�(·, t)‖�s

L(�)dt�

∫ T

0‖u�(·, t)‖�sa

L�(�)‖u�(·, t)‖�s(1−a)L1(�)

dt

with a= (1−(1 / )) / (1−(1 / �)). Since

�sa=� · �−1

�

(1− 1

) ·1− 1

1− 1

�

=�,

we infer from (55) and the mass identity (4) that for some c2>0,

∫ T

0‖u�(·, t)‖�s

L(�)dt�c2 for all �∈ (0, 1). (60)

Next, from Lemma 2.4 and (4) we know that

(∇v�)�∈(0,1) is bounded in L∞loc([0,∞); L(�)), (61)

because <n / (n−1). Also, in view of (55) and [17, Theorem 3.1], (v�−et(�−1)v0)�∈(0,1) is bounded in L�((0, T); W2,�(�)) and hence

in L�((0, T); W1, n�n−� (�)). Since by semigroup estimates [15, Lemma 1.3] we have ‖∇et(�−1)v0‖L∞(�)�c3t−

12 ‖v0‖L∞(�) for some c3>0,

recalling our restriction �<2 we conclude that (v�)�∈(0,1) is bounded in L�((0, T); W1, n�n−� (�)). We can thus once more employ Hölder’s

inequality to interpolate between the latter estimate and (61). As a result, writing

b :=1

− −�

�1

− n−�

n�

,

for some c3>0 we have

∫ T

0‖∇v�(·, t)‖�s′

L�

−� (�)dt �

∫ T

0‖∇v�(·, t)‖�s′b

Ln�

n−� (�)‖∇v�(·, t)‖�s′(1−b)

L(�) dt

� c4 for all �∈ (0, 1)

with an appropriate c4>0, because according to (58),

�s′b<�·�

(1

+ 1

n

)−1

�

(1

+ 1

)−1

·1

− −�

�1

− n−�

n�

=�.

Together with (60) and (59), this establishes (54). �

On the basis of the two preceding lemmata, performing standard compactness arguments and extraction procedures we cannow find a sequence of solutions to (2) that converges to a weak solution of (1) in an appropriate sense.

Theorem 4.6Assume that n�2 and �<

√(n+2) / (3n−4). Then for all u0 ∈C0(�̄) and v0 ∈W1,∞(�) with u0�0 and v0>0 in �̄, there exists a global

weak solution (u, v) of (1).

18

8


M. WINKLER

ProofAccording to Lemma 4.4 and Lemma 4.5, for any T>0,

(u�)�∈(0,1) is weakly precompact in L�(�×(0, T)) (62)

and (u�

��(v�)

)�∈(0,1)

is weakly precompact in L�(�×(0, T)) (63)

for some �>1. Moreover, from parabolic regularity theory [17, Theorem 3.1] and the inclusion v0 ∈W1,∞(�) we gain (cf. also theproof of Lemma 4.5)

‖v�‖L�((0,T);W2,�(�)) +‖v�t‖L�(�×(0,T))�c1 for all �∈ (0, 1)

with some c1>0, so that the Aubin–Lions lemma [18] ensures that

(v�)�∈(0,1) is strongly precompact in L�((0, T); W1,�(�)). (64)

To derive some type of strong compactness property for (u�)�∈(0,1), we apply Corollary 4.3 to obtain that for some p∈ (0, 1) andc2>0, ∫ T

0

∫�

up−2� |∇u�|2 +

∫ T

0

∫�

up� v−2

� |∇v�|2�c2 for all �∈ (0, 1). (65)

We now fix m∈N large such that Wm,2(�) ↪→W1,∞(�) and let �∈Wm,2(�). Then multiplying the first equation in (2) by (u�+1)p2 −1�

and integrating over � we see that

2

p

∫�

�t(u�+1)p2 � =

(1− p

2

)∫�

(u�+1)p2 −2|∇u�|2�−

∫�

(u�+1)p2 −1∇u� ·∇�

+(

1− p

2

)�∫

�(u�+1)

p2 −2 u�

��(v�)∇u� ·∇v��−�

∫�

(u�+1)p2 −1 u�

��(v�)∇v� ·∇�

and hence, using Hölder’s and Young’s inequalities,

2

p

∣∣∣∣∫

��t(u�+1)

p2 �

∣∣∣∣�c3‖�‖W1,∞(�) ·(∫

�(u�+1)

p2 −2|∇u�|2 +

∫�

(u�+1)p2 +

∫�

(u�+1)p2 −2 u2

�

��(v�)|∇v�|2

)

with some c3>0. As ��(v�)�v�, (u�+1)p2 �(u�+1)p�u�+1 and (u�+1)p−2u2

� �up� , we therefore obtain

2

p‖�t(u�+1)

p2 ‖L1((0,T);(Wm,2(�))�) = 2

p

∫ T

0sup

‖�‖Wm,2(�)�1

∣∣∣∣∫

��t(u�+1)

p2 (·, t)�

∣∣∣∣dt

� c4

(∫ T

0

∫�

(u�+1)p−2|∇u�|2 +∫ T

0

∫�

(u�+1)+∫ T

0

∫�

up� v−2

� |∇v�|2)

for an appropriate c4>0. Due to (65) and (4), this entails that (�t(u�+1)p2 )�∈(0,1) is bounded in L1((0, T); (W2,m(�))�), whereas also by

(65) and (4), ((u�+1)p2 )�∈(0,1) is bounded in L2((0, T); W1,2(�)). By a variant of the Aubin–Lions lemma [18], this guarantees that

((u�+1)p2 )�∈(0,1) is strongly precompact in L1(�×(0, T)). (66)

Now according to (66) and (64), we can extract a sequence of numbers �=�k ↘0 along which

u� →u, v� →v, and ∇v� →∇v a.e. in �×(0,∞), (67)

u� ⇀u in L�(�×(0, T)) for all T>0, (68)

v� →v in L�((0, T); W1,�(�)) for all T>0 (69)

and

u�

��(v�)∇v� ⇀w in L�(�×(0, T)) for all T>0 (70)

hold for some limit functions u�0, v�0 and w defined in �×(0,∞). Here, (67) and Egorov’s theorem guarantee that

w = u

v∇v. (71)


18

9

M. WINKLER

Now given T>0 and ∈C∞0 (�̄×[0, T)) with � / ��=0 on ��×(0, T), from (2) we obtain upon obvious testing procedures the

identities

−∫ T

0

∫�

u� t −∫ T

0

∫�

u�� −�∫ T

0

∫�

u�

��(v�)∇v� ·∇ =

∫�

u0 (·, 0) (72)

and

−∫ T

0

∫�

v� t −∫ T

0

∫�

v�� +∫ T

0

∫�

v� −∫ T

0

∫�

u� =∫

�v0 (·, 0) (73)

for all �∈ (0, 1). Thanks to (68)–(71), we can take �=�k ↘0 here to find that each of the terms in (72) and (73) converges toits expected limit, whereby it follows that (u, v) is a weak solution of (1) in �×(0, T) in the sense of Definition 4.1 for arbitraryT>0. �

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Global solutions in a fully parabolic chemotaxis system with singular sensitivity