Does a ‘volume-filling effect’ always prevent chemotactic collapse?

Research Article

Received 11 October 2008 Published online 8 April 2009 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/mma.1146MOS subject classification: 92 C 17; 35 K 55; 35 B 35; 35 B 40

Does a ‘volume-filling effect’ always preventchemotactic collapse?

Michael Winkler∗†

Communicated by H. A. Levine

The parabolic–parabolic Keller–Segel system for chemotaxis phenomena,{ut =∇·(/(u)∇u)−∇ ·(w(u)∇v), x ∈X, t>0

vt =Dv−v+u, x ∈X, t>0

is considered under homogeneous Neumann boundary conditions in a smooth bounded domain X⊂Rn with n�2.It is proved that if w(u) //(u) grows faster than u2/n as u→∞ and some further technical conditions are fulfilled, then

there exist solutions that blow up in either finite or infinite time. Here, the total mass∫Xu(x, t) dx may attain arbitrarily

small positive values.In particular, in the framework of chemotaxis models incorporating a volume-filling effect in the sense of Painter

and Hillen (Can. Appl. Math. Q. 2002; 10(4):501–543), the results indicate how strongly the cellular movement must beinhibited at large cell densities in order to rule out chemotactic collapse. Copyright © 2009 John Wiley & Sons, Ltd.

Keywords: chemotaxis; global existence; boundedness; blow-up

1. Introduction

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

ut =∇ ·(�(u)∇u)−∇ ·(�(u)∇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 nonnegative initial data u0 ∈C0(�) and v0 ∈C1(�) in a bounded domain �⊂Rn with smooth boundary, where � / �� denotesdifferentiation with respect to the outward normal � on ��. The functions � and � are assumed to belong to C2([0,∞)) and tosatisfy �>0 on [0,∞),�(0)=0 and, for simplicity, that �>0 on (0,∞).

Partial differential equation systems of this type were introduced by Keller and Segel [1] as a first approach towards the modelingof the biological phenomenon of directed, or partially directed, movement of cells in response to a chemical signal. In typicalexamples, as a reaction to an internal or external stimulus, the cells within some population start to produce such a signal substanceand, at the same time, begin to move towards regions of higher concentration of this substance. As is known from experimentalobservations, such a movement may eventually lead to an aggregation of cells around one point, or few points, in space. Thisphenomenon, also referred to as chemotaxis, is believed to play an essential role in numerous processes of self-organization at themicroscopic level, not only in protozoal populations but also in higher developed organisms (cf. [2] for a recent survey).

In the framework of (1), one neglects all further physical and chemical circumstances other than the presence of the cellstheirselves and the signal, denoted in their respective densities by u(x, t) and v(x, t). One mathematical challenge in this context

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

12

Copyright © 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2010, 33 12–24

M. WINKLER

consists of finding out under which conditions and, if, in which sense cellular aggregation can indeed be described by the model (1),as simple as it stands. Here, one possible and frequently used mathematical notion of ‘aggregation’ is that (1) possesses solutionsfor which the cell density u becomes unbounded in space at some finite or infinite time, the so-called blow-up time. As, formally,(1) preserves the total cellular mass in the sense that

∫� u(x, t) dx =∫� u0(x) dx for all t>0, this concept appears to be meaningful,

because it implies that near the blow-up time the mass of an unbounded solution should essentially concentrate near those pointswhere u becomes large. However, the analysis of such blow-up solutions brings about several technical difficulties; even the mereproof of existence of unbounded solutions appears to be by far not trivial, unless the second equation in (1) is simplified usingfast-diffusive [3--5] or slow-diffusive limits [6, 7]. As to the full system (1), only few results are available that deal with existence[8--11] and asymptotic properties of blow-up solutions ([8, 12]; cf. also the survey [13]). Apart from that, from the point of view ofmodeling it is not completely clear how exploding cell densities are to be interpreted.

Accordingly, considerable effort has been made in developing models of Keller–Segel type that do not possess blowing up butexclusively bounded solutions [2]. One such approach was pursued by Hillen and Painter [14] and is based on a biased randomwalk analysis. Having as their main ingredient the assumption that the cells’ movement is inhibited near points where the cellsare densely packed, they derive a functional link between the self-diffusivity �(u) and the chemotactic sensitivity �(u) that, in anon-dimensionalized version, takes the form

�(u)=Q(u)−uQ′(u), �(u)=uQ(u), u�0 (2)

where Q(u) denotes the density-dependent probability for a cell to find space somewhere in its current neighborhood. Since thisprobability is basically unknown, different choices for Q are conceivable, each of these providing a certain version of (1) thatincorporates this so-called volume-filling effect. In [15], the authors propose the choice

Q(u)=A(u−u)+ (3)

with some A>0 and u>0, presuming that there exists some critical cell density u beyond which no further movement is possible. Infact, this model admits global bounded solutions only [15], and ‘describing aggregation’ amounts to studying dynamical propertiessuch as instability of constant steady states or the existence of attractors [16].

It is the purpose of the present paper to relax the above ‘decay’ assumption that Q(u) be identically zero for u large enough, andinvestigate the question to which extent (1) is then still able to prevent a chemotactic collapse in the sense of blow-up. Specifically,we think of Q(u) to be positive for all u∈ [0,∞) and to satisfy Q(u)→0 as u→∞, including algebraic or exponential decay, forinstance. Such generalizations were previously suggested in [17] and seem to be adequate in view of the lack of any experimentalhint about a reasonable value for u in (3), or about the particular behavior of Q(u) near u= u in this particular model.

Our main results, actually not requiring that � and � be connected via (2), indicate that the asymptotic behavior of the quotient�(u) / �(u) for large u is crucial. The main purpose is to show that

• If � is a ball in Rn for some n�2 and

�(u)

�(u)grows faster than u2/n as u→∞ (4)

in a certain sense then for any m>0, (1) possesses unbounded solutions having mass∫� u(x, t) dx ≡m

by proving the claimed conclusion under various, technically inspired, specifications of (4); as will be proved in Corollary 5.2, preciseconditions that are sufficient for the occurrence of blow-up are

• n=2 and �(u) / �(u)�c0u ln u for some c0>0 and sufficiently large u;• n�3 and u−��(u) / �(u)→c0>0 as u→∞ with some �>2 / n;• n�3 and

lim infu→∞

u

(�

�

)′(u)(

�

�

)(u)

>2

n

To illustrate this, let us suppose that � and � are given by (2) with Q(u)u−� for large u for some �>0. Then �(u) / �(u)1 / (1+�)uand hence the above results imply that blow-up occurs for some data when n�3. On the other hand, if Q decays exponentially,Q(u)e−�u for some �>0, then �(u) / �(u)1 / �, so that none of the above criteria is fulfilled, and accordingly no collapse isasserted. However, if Q(u)e−�u�

with positive � and � then �(u) / �(u) (1 / ��)u1−�; accordingly, we conclude that if n�3 and�<(n−2) / n here then blow-up solutions exist.

The number 2 / n in (4) cannot be diminished; this is implied by the outcome in [18], where it was proved that if �≡1 andlim supu→∞ u−��(u)<∞ for some �<2 / n then all solutions remain bounded. It is an interesting open question whether 2 / n isindeed critical with respect to collapse prevention also when �(u) decays to zero as u→∞. Proving this conjecture seems to beconnected to overcoming some technical complications stemming from the degeneracy in the diffusion part (cf. [17]). To the bestof our knowledge, affirmative results are available only under stronger growth restrictions of � / � [19], or for elliptic–parabolicsimplifications of (1), but then only under the additional hypothesis that � decays at most at an algebraic rate [20, 21].


13

M. WINKLER

In space dimension n=2, our requirement �(u) / �(u)�c0u ln u at first glance seems far away from being optimal, for even in thestandard Keller–Segel model with �≡1 and �(u)=u collapse is known to occur. However, in this case the appearance of blow-upis coupled to a large total mass of cells [9, 22], whereas we assert blow-up for arbitrarily small mass.

Let us mention that also in the spatially one-dimensional setting not considered here, it seems that the number 2 / n=2 charac-terizes a critical growth rate of the ratio �(u) / �(u). This is strongly indicated by the very recent result in [11]: There, the authorsconsider �(u)≡u and any smooth � satisfying �(u)�c(1+u)−p for all u�0 with some constants c and p, and prove that someblow-up solutions exist if p>1, whereas for p<1 only global solutions are likely to be expected [11, Remark 2].

Our strategy of proof is closely related to the idea of showing blow-up for low energy solutions, which has turned out to be apowerful master concept in a number of semilinear and quasilinear parabolic problems, preferably with superlinear nonlinearities (cf.[23] for a survey). Within the present framework, our approach is inspired by the arguments in [9, 24]: We first identify an appropriateLyapunov functional F for (1) in Section 2 and thereby prove that bounded solutions of (1) approach certain steady-state solutionsin an appropriate sense. Restricting to the radially symmetric setting henceforth, we next show in Section 3 that if � / � satisfiessome growth hypotheses then F, when evaluated at such equilibria, is uniformly bounded from below. In Section 4, however, weprove that suitably fast growth of � / � implies the existence of smooth initial data at which F attains arbitrarily large negativevalues. For such initial data, using the Lyapunov property of F we easily conclude in Section 5 that the corresponding solutionscannot remain bounded.

2. Linkingx-limit sets and steady states via a Lyapunov functional

For our purpose, a highly favorable structural property of the considered form of the Keller–Segel system will be that

F(u, v) :=∫

�

(1

2|∇v|2 + 1

2v2 −uv+G(u)

)

acts as a Lyapunov functional for (1) in that F(u(·, t), v(·, t)) is nonincreasing along trajectories and thus plays a role similar to that ofenergy in physics. Here, for any s0>0 one may define the positive function G=Gs0 on (0,∞) by

G(s) :=∫ s

s0

∫ �

s0

�()

�()dd�, s>0

The use of a Lyapunov functional has proved to be a strong tool in a large number of models of Keller–Segel type, also in simplifiedversions where the second equation in (1) is stationary [17]; for an overview, we refer to the surveys [2, 13].

Lemma 2.1Let T ∈ (0,∞] and suppose that (u, v) is a classical solution of (1) in �×(0, T) with initial data (u0, v0) satisfying infx∈� u0(x)>0. Then

∫ t

0

∫�

v2t +

∫ t

0

∫�

�(u) ·∣∣∣∣�(u)

�(u)∇u−∇v

∣∣∣∣2 +F(u(·, t), v(·, t))=F(u0, v0) for all t ∈ (0, T) (5)

ProofWe perform a straightforward extension of the respective computations for the classical Keller–Segel model where �≡1 and �(u)=u[25]. By the strong maximum principle applied to the first equation in (1), u inherits strict positivity from its initial data. Therefore,�(u) / �(u) and hence G(u) and G′(u)≡ (d / du)G(u) are continuous in �×[0, T), so that the first equation in (1) yields

∫�

G(u)

∣∣∣∣t0

=∫ t

0

∫�

G′(u)∇ ·(�(u)∇u−�(u)∇v)

= −∫ t

0

∫�

G′′(u)∇u ·(�(u)∇u−�(u)∇v)

= −∫ t

0

∫�

�2(u)

�(u)|∇u|2 +

∫ t

0

∫�

�(u)∇u ·∇v

As

�2(u)

�(u)|∇u|2 =�(u)

∣∣∣∣�(u)

�(u)∇u−∇v

∣∣∣∣2 −�(u)|∇v|2 +2�(u)∇u ·∇v

we obtain

∫�

G(u)

∣∣∣∣t0=−

∫ t

0

∫�

�(u)

∣∣∣∣�(u)

�(u)∇u−∇v

∣∣∣∣2 +∫ t

0

∫�

�(u)|∇v|2 −∫ t

0

∫�

�(u)∇u ·∇v (6)14


M. WINKLER

Here, the last term can be rewritten using the first equation in (1) according to

−∫ t

0

∫�

�(u)∇u ·∇v =∫ t

0

∫�

∇ ·(�(u)∇u) ·v

=∫ t

0

∫�

utv+∫ t

0

∫�

∇ ·(�(u)∇v) ·v

=∫ t

0

∫�

utv−∫ t

0

∫�

�(u)|∇v|2 (7)

where

∫ t

0

∫�

utv =∫

�uv

∣∣∣∣t0−∫ t

0

∫�

uvt

=∫

�uv

∣∣∣∣t0−∫ t

0

∫�

(vt −�v+v) ·vt

=∫

�uv

∣∣∣∣t0−∫ t

0

∫�

v2t − 1

2

∫�

|∇v|2∣∣∣∣t0− 1

2=∫

�v2∣∣∣∣t0

(8)

Combining (6) and (7) with (8), after an obvious reorganization we end up with (5). �

In what follows, an important role is played by the properties of solutions (u∞, v∞) of

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−�v∞+v∞ =u∞, x ∈�

�(u∞)∇u∞ =�(u∞)∇v∞, x ∈�

�v∞��

=0, x ∈��

(9)

Evidently, such function pairs are stationary solutions of (1) in the classical sense.The identity (5) will allow us to establish a connection between the set Em of such equilibria that have mass m>0,

Em :={

(u∞, v∞)∈ (C2(�))2∣∣∣∣(u∞, v∞) solves (9) and

∫�

u∞ =m

}(10)

and the -limit set of bounded solutions of (1) given by

(u, v) :={(u∞, v∞)∈ (C2(�))2|∃tk →∞ such that as k →∞, (u(·, tk), v(·, tk))→ (u∞, v∞) in (C2(�))2} (11)

Lemma 2.2Suppose (u, v) is a global bounded solution of (1) with u0>0 in � and

∫� u0 =m. Then (u, v)∩Em =∅.

ProofSince u is bounded, scalar parabolic regularity theory applied to the second, the first, the second and again the first equation in

(1) [26] implies boundedness of v in C1+�,(1+�)/2(�×[1,∞)), then of u in C�,�/2(�×[1,∞)), then of v in C2+� ,1+�/2(�×[1,∞)) andfinally of u in C2+�,1+�/2(�×[1,∞)) for some �>0. Moreover, the boundedness of (u, v) entails that F(u(·, t), v(·, t)) is bounded frombelow for all times, so that Lemma 2.1 guarantees

∫ ∞

0

∫�

v2t +

∫ ∞

0�(u)

∣∣∣∣ �(u)

�(u)∇u−∇v

∣∣∣∣2 <∞

Using this together with the Arzelà–Ascoli theorem, we can extract a sequence of times tk →∞ such that

vt(·, tk)→0 in L2(�) (12)

and

�(u(·, tk))

∣∣∣∣�(u(·, tk))

�(u(·, tk))∇u(·, tk)−∇v(·, tk)

∣∣∣∣2 →0 a.e. in � (13)

as well as

u(·, tk)→u∞ in C2(�) (14)


15

M. WINKLER

and

v(·, tk)→v∞ in C2(�) (15)

hold with some nonnegative u∞ and v∞ belonging to C2(�).Clearly, from (15) we know that �v∞ / ��=0 on ��, whereas (12), (14) and (15) imply upon letting t = tk →∞ in vt =�v−v+u

that 0=�v∞−v∞+u∞ in �. The mass requirement in the definition of Em is an immediate consequence of the obvious massconservation property

∫� u(x, t) dx ≡∫� u0.

In order to prove the second identity in (9), we first note that the nonnegativity of u∞ implies that ∇u∞ ≡0 holds in the set{u∞ =0} of zeros of u∞. Thus, �(u∞)∇u∞ =�(u∞)∇v∞ =0 in {u∞ =0}, because �(0)=0. For fixed x ∈{u∞>0}, however, we havelim infk→∞ �(u(x, tk))>0, whence from (13) we infer that

�(u(·, tk))

�(u(·, tk))∇u(·, tk)−∇v(·, tk)→0 a.e. in {u∞>0}

and therefore

�(u∞)∇u∞ =�(u∞)∇v∞ a.e. in {u∞>0}As both sides of this equation are continuous in �, this proves that �(u∞)∇u∞ ≡�(u∞)∇v∞ also holds in the whole set {u∞>0},as desired. �

RemarkIt can be shown that actually each u∞ ∈Em must be positive throughout �. In fact, suppose that some component C of {u∞>0}does not coincide with �. Then there exist x0 ∈�C and a sequence of points xj ∈C such that xj →x0. Writing g(s) :=∫ s

s0(�(�) / �(�)) d�

for s>0, we know from (9) that ∇(g(u∞)−v∞)≡0 and hence g(u∞)−v∞ ≡� in C with some �∈R. Since �(0)>0=�(0) and�∈C1([0,∞)), it is clear that g(0)=−∞, so that taking j→∞ in g(u∞(xj))−v∞(xj)=� yields a contradiction. Accordingly, C= �,

which means that u∞>0 in �.

3. Lower bounds for steady-state energies

The goal of this section is to assert a lower bound for the values of F(u, v) for all possible members (u, v) of Em. According to slightlydifferent technical approaches, we distinguish between the cases n=2 and n�3.

As a first step, let us make a simple but useful observation that allows us to rewrite the energy F(u, v) of a solution (u, v) of (9)without the ‘mixed’ term

∫� uv.

Lemma 3.1If (u, v) is a solution of (9) and s0>0 then

F(u, v)=−1

2

∫�

|∇v|2 − 1

2

∫�

v2 +∫

�Gs0 (u) (16)

ProofWe multiply the first equation in (9) by v to obtain

∫� |∇v|2 +∫� v2 =∫� uv. Inserting this into the definition of F immediately results

in (16). �

The following preliminary estimate for radial steady states will be a common ingredient for both cases n=2 and n�3. It is inspiredby the classical proof of Pohozaev’s identity.

Lemma 3.2Let �=BR(0), s0>0 and

H(s) :=∫ s

s0

��(�)

�(�)d� for s>0 (17)

Then for all nonnegative and nonincreasing �∈C∞([0, R]) satisfying �′(0)=0 and �(R)=0, the inequality

n−2

2

∫�

�(|x|)|∇v|2 − 1

2

∫�

|x|�′(|x|)|∇v|2�∫

�|x|�(|x|) ·(v+s0) ·|∇v|+n

∫{u>s0}

�(|x|)H(u) (18)

is valid for each radially symmetric solution (u, v) of (9).

ProofWe multiply �v =v−u by �(|x|)(x ·∇v) and integrate over � to see that∫

��(|x|)(x ·∇v)�v =

∫�

�(|x|)v(x ·∇v)−∫

��(|x|)u(x ·∇v) (19)1

6


M. WINKLER

Since �(R)=0, two integrations by parts on the left yield∫�

�(|x|)(x ·∇v)�v =−∫

��(|x|)|∇v|2 −

∫�

�′(|x|)|x| (x ·∇v)2 − 1

2

∫�

�(|x|)(x ·∇|∇v|2)

and

−1

2

∫�

�(|x|)(x ·∇|∇v|2)= n

2

∫�

�(|x|)|∇v|2 + 1

2

∫�

|x|�′(|x|)|∇v|2

Thereupon, (19) turns into the identity

n−2

2

∫�

�(|x|)|∇v|2 −∫

�

�′(|x|)|x| (x ·∇v)2 + 1

2

∫�

|x|�′(|x|)|∇v|2 =∫

��(|x|)v(x ·∇v)−

∫�

�(|x|)u(x ·∇v)

We now use the radial symmetry of v which guarantees that (x ·∇v)2 =|x|2|∇v|2, so that

n−2

2

∫�

�(|x|)|∇v|2 − 1

2

∫�

|x|�′(|x|)|∇v|2 =∫

��(|x|)v(x ·∇v)−

∫�

�(|x|)u(x ·∇v) (20)

In order to find an upper bound for the second term on the right, we recall the definition of H and split the integral in questionaccording to

−∫

��(|x|)u(x ·∇v)=−

∫{u�s0}

�(|x|)u(x ·∇v)−∫{u>s0}

�(|x|)(x ·∇H(u)) (21)

Here, integrating by parts we obtain

−∫{u>s0}

�(|x|)(x ·∇H(u)) = n

∫{u>s0}

�(|x|)H(u)+∫{u>s0}

|x|�′(|x|)H(u)

� n

∫{u>s0}

�(|x|)H(u) (22)

because H(x)≡0 on �{u>s0}∩�, �(R)=0 and �′�0. Using the obvious estimates

−∫{u�s0}

�(|x|)u(x ·∇v)�∫

�|x|�(|x|)s0|∇v|

and ∫�

�(|x|)v(x ·∇v)�∫

�|x|�(|x|)v|∇v|

we thus infer from (20)–(22) that (18) holds. �

3.1. The case n=2

We now concentrate on the two-dimensional case first. In this, the first term on the left of (18) vanishes, so that taking advantagefrom this estimate means using the second term on its left appropriately. This is the main technical goal in the proof of the followingstatement.

Lemma 3.3Assume that �=BR(0)⊂R2 for some R>0. If ∫ s

s0

��(�)

�(�)d��K

s

ln sfor all s�s0 (23)

holds with some K>0 and s0>1, then for all m>0 there exists C =C(R, n, K, s0)>0 such that

F(u, v)�−C (24)

is valid for all radial solutions (u, v) of (9).

ProofFor �∈ (0, 1), we let

�(r) := lnR2 +�

r2 +�, r ∈ [0, R]


17

M. WINKLER

Then � is smooth and nonnegative in [0, R] with �(R)=0 and �′(r)=−2r / (r2 +�), so that �′�0 on [0, R] and �′(0)=0. Thus, from (18)we obtain, again abbreviating H(s)=∫ s

s0(��(�) / �(�)) d� and using Young’s inequality,

∫�

|x|2|x|2 +�

·|∇v|2 �∫

�|x| ln

(R2 +�

|x|2 +�

)·(v+s0) ·|∇v|+2

∫{u>s0}

ln

(R2 +�

|x|2 +�

)·H(u)

� 1

2

∫�

|x|2|x|2 +�

·|∇v|2 +∫

�(|x|2 +�) · ln2

(R2 +�

|x|2 +�

)·v2

+s20

∫�

(|x|2 +�) · ln2

(R2 +�

|x|2 +�

)+2

∫{u>s0}

ln

(R2 +�

|x|2 +�

)·H(u)

As it can easily be checked that (r2 +�) · ln2(R2 +�) / (r2 +�)�4(R2 +�)e−2 for all r ∈ [0, R], we thus find

1

2

∫�

|x|2|x|2 +�

·|∇v|2�4(R2 +1)e−2∫

�v2 +4(R2 +1)e−2s2

0|�|+2

∫{u>s0}

ln

(R2 +�

|x|2 +�

)·H(u) (25)

Now in Young’s inequality in the form

ab� 1

ee a + 1

b ln b

valid for all positive a, b and , we pick any ∈ (0, 1) and thus estimate

2

∫{u>s0}

ln

(R2 +�

|x|2 +�

)H(u) � 2

e

∫�

(R2 +�

|x|2 +�

)

+ 2

∫{u>s0}

H(u) ln H(u)

� 2(R2 +1)

e

∫�

|x|−2 + 2

∫{u>s0}

H(u) ln H(u)

for all �∈ (0, 1), where the first integral on the right is finite since <1. According to (23), we have

H(s) ln H(s) � Ks

ln s· ln

Ks

ln s�K

s

ln s

(ln s+ ln

K

ln s0

)

� K(1+c1)s for all s>s0

with c1 :=max{0, (ln K / ln s0) / ln s0}. As∫{u>s0} u�

∫� u=m, (25) therefore implies

1

2

∫�

|x|2|x|2 +�

|∇v|2�c2

∫�

v2 +c3

where

c2 =4(R2 +1)e−2

and

c3 =4(R2 +1)e−2s20|�|+ 2(R2 +1)

e·∫

�|x|−2 + 2K(1+c1)m

In the limit �→0, Fatou’s lemma thus yields

1

2

∫�

|∇v|2�c2

∫�

v2 +c3

Hence, by (16) and the nonnegativity of G,

F(u, v) � 1

2

∫�

|∇v|2 −∫

�|∇v|2 − 1

2

∫�

v2

� 1

2

∫�

|∇v|2 −(

2c2 + 1

2

)∫�

v2 −2c3

Finally, from Ehrling’s lemma we gain some c4>0 such that(2c2 + 1

2

)∫�

w2�1

2

∫�

|∇w|2 +c4

(∫�

w

)2for all w ∈W1,2(�)1

8


M. WINKLER

and thereby conclude, recalling∫� v =m, that

F(u, v)�−c4m2 −2c3

and finish the proof. �

3.2. The case n�3

In the three-dimensional situation, the first term on the left of (18) will be essentially responsible for the fact that a correspondinglower bound for all steady-state energies can be proved under a less restrictive growth restriction on � / �.

Lemma 3.4Let n�3 and �=BR(0) with some R>0. Suppose that∫ s

s0

��(�)

�(�)d�� n−2−�

n

∫ s

s0

∫ �

s0

�()

�()dd�+Ks for all s�s0 (26)

holds for some s0>1 and �∈ (0, 1). Then for each m>0 one can find C =C(R, m, n,�, s0)>0 with the property that

F(u, v)�−C (27)

is satisfied for all radial solutions (u, v) of (9).

ProofWe fix a nondecreasing �0 ∈C∞(R) such that �0 ≡0 in (−∞, 1) and �0 ≡1 on (2,∞), and let �(r)≡�k(r) :=�0(k(R−r)) for r ∈ [0, R] andk ∈N large satisfying k>2 / R. Then Lemma 3.2 ensures that with H(s)=∫ s

s0��(�) / �(�) d�, we have

n−2

2

∫�

�k(|x|)|∇v|2 �∫

�|x|�k(|x|) ·(v+s0) ·|∇v|+n

∫{u>s0}

�k(|x|)H(u)

�∫

�|x|(v+s0)|∇v|+n

∫{u>s0}

H(u)

for all such k, whence by Fatou’s lemma we gain

n−2

2

∫�

|∇v|2�∫

�|x|(v+s0)|∇v|+n

∫{u>s0}

H(u)

With � taken from (26), we use Young’s inequality to estimate∫�

|x|(v+s0)|∇v| � �

4

∫�

|∇v|2 + 1

�

∫�

|x|2(v+s0)2

� �

4

∫�

|∇v|2 + 2R2

�

∫�

v2 + 2s20R2|�|

�

so that

n−2−�

2

∫�

|∇v|2�− �

4

∫�

|∇v|2 + 2R2

�

∫�

v2 + 2s20R2|�|

�+n

∫{u>s0}

H(u)

Rearranging this, from (16) we infer that

F(u, v)� �

4(n−2−�)

∫�

|∇v|2 −c1

∫�

v2 −c2 − n

n−2−�

∫{u>s0}

H(u)+∫

�G(u)

with

c1 = 2R2

�(n−2−�)+ 1

2and c2 = 2s2

0R2|�|�(n−2−�)

Now by Ehrling’s lemma, we have

c1

∫�

v2� �

4(n−2−�)

∫�

|∇v|2 +c3

(∫�

v

)2

for some c3>0. In view of the mass identity∫� v =m, we thus find

F(u, v)�−c3m2 −c2 − n

n−2−�

∫{u>s0}

H(u)+∫

�G(u)


19

M. WINKLER

As G(s)�0 for s�s0 and

G(s)− n

n−2−�H(s)�− nK

n−2−�s for all s�s0

by (26), we conclude that

F(u, v)�−c3m2 −c2 − nKm

n−2−�

because∫� u=m. �

4. Initial data with large negative energy

We next assert that smooth initial data with arbitrarily large negative energies exist. The construction partly parallels that in [18,Lemma 5.2]. However, the growth assumptions on � / � are weaker here, especially in the case n=2; moreover, we provide initialdata in C∞ here.

Lemma 4.1Let n�2, R>0 and �=BR(0), and suppose that there exist k>0 and s0>1 such that

∫ s

s0

∫ �

s0

�()

�()dd��

⎧⎪⎨⎪⎩

ks(ln s)� if n=2 with some �∈ (0, 1)

ks2−� if n�3 with some �>2

n

(28)

holds for all s�s0. Then for each m>0 and C>0 one can find positive (u0, v0)∈ (C∞(�))2 satisfying∫� u0 =m and

F(u0, v0)<−C (29)

ProofFirst, in the case n�3 we evidently may assume that (28) is valid with some �>2 / n satisfying �<1. We then pick positive numbers�, � and such that

�>n, �∈ ((1−�)n, n−2) and >n

2(30)

which is possible because �>2 / n. For small �>0, we define the smooth functions u� and v� by

u�(x) :=a� ·��−n ·(|x|2 +�2)−�/2

and

v�(x) :=� −� ·(|x|2 +�2)− /2

for x ∈ �, where

a� := �n−�m∫�(|x|2 +�2)−�/2 dx

As upon the substitution r =�s we see that

��−N∫ R

0rN−1(r2 +�2)−�/2 dr →A(N,�) :=

∫ ∞

0sN−1(s2 +1)−�/2 ds as �→0 (31)

whenever N>0 and �>N, it can easily be checked that a� →a0 :=m / n ·A(n,�) as �→0, where n denotes the (n−1)-dimensionalsurface area of the unit sphere in Rn. In particular, a� is bounded from above and below by a positive constant, uniformly withrespect to �∈ (0, 1).

The choice of a� immediately implies that∫� u� =m. Moreover, by straightforward computations using (31) we obtain

�−n+2�+2∫

�|∇v�|2 = n 2�−n+2 +2

∫ R

0rn+1(r2 +�2)− −2 dr

→ n 2 ·A(n+2, 2 +4) as �→0

and

�−n+2�∫

�v2� = n�−n+2

∫ R

0rn−1(r2 +�2)− dr

→ n ·A(n, 2 ) as �→0

20


M. WINKLER

Similarly,

��∫

�u�v� = na� ·�−n+�+

∫ R

0rn−1(r2 +�2)−(�+ )/2 dr

→ na0 ·A(n,�+ ) as �→0

and

�(1−�)n∫

�G(u�) � nka2−�

� ·�(1−�)n+(2−�)(�−n)∫ R

0rn−1(r2 +�2)−(2−�)�/2 dr

→ nka2−�0 ·A(n, (2−�)�) as �→0

Since �>0 and

�>−n+2�+2, �>−n+2� and �>(1−�)n

according to (30), it follows that F(u�, v�)→−∞ as �→0, whence (29) is true for all sufficiently small �>0.In the case n=2 we define u� as above, but let

v�(x) :=(

lnR

�

)−�

· lnR2

|x|2 +�2

for �∈ (0, R / 2) this time, where �∈ (0, 1) is small enough fulfilling �<1−�. Then, again substituting r =�s, we find∫�

|∇v�|2 = 8�·(

lnR

�

)−2�

·∫ R

0r3(r2 +�2)−2 dr

= 8�·(

lnR

�

)−2�

·∫ R/�

0s3(s2 +1)−2 ds

� 8�·(

lnR

�

)−2�

·(

1+ lnR

�

)

whereas clearly

∫�

v2� = 2�·

(ln

R

�

)−2�

·∫ R

0r

(ln

R2

r2 +�2

)2

dr

� 8�·(ln 2)−2� ·∫ R

0r

(ln

R

r

)2dr

for all �∈ (0, R / 2). Moreover,(ln

R

�

)�−1·∫

�u�v� = 2�a� ·��−2 ·

(ln

R

�

)−1·∫ R

0r(r2 +�2)−�/2 · ln

R2

r2 +�2dr

= 2�a� ·(

lnR

�

)−1·∫ R/�

0s(s2 +1)−�/2 ·

(2 ln

R

�− ln(s2 +1)

)ds

= 4�a� ·∫ R/�

0s(s2 +1)−�/2 ds−2�a� ·

(ln

R

�

)−1·∫ R/�

0s(s2 +1)−�/2 ln(s2 +1) ds

→ 4�a0 ·∫ ∞

0s(s2 +1)−�/2 ds as �→0

and, by (28), ∫�

G(u�) � 2�ka� ·��−2 ·∫ R

0r(r2 +�2)−�/2 ·(ln(a��

�−2(r2 +�2)−�/2))� dr

= 2�ka� ·∫ R/�

0s(s2 +1)−�/2 ·(ln(a��

−2(s2 +1)−�/2))� ds

� 2�ka� ·(

2 ln

√a�

�

)�

·∫ ∞

0s(s2 +1)−�/2 ds


21

M. WINKLER

It therefore follows that

F(u�, v�)�−c2

(ln

R

�

)1−�

+c3

(1+

(ln

R

�

)1−2�

+(

ln

√a�

�

)�)

for all �∈ (0, R / 2) with positive constants c2 and c3. Since

1−�>0, 1−�>1−2� and 1−�>�

due to our choice of �, we again infer that F(u�, v�)→−∞ as �→0 and conclude as before. �

5. Blow-up

It is not the purpose of the present paper to develop a refined existence and uniqueness theory under optimal regularity assumptionson �, �, �� and the initial data. As we intend to use (u0, v0) as provided by Lemma 4.1 as initial data, the only element fromexistence theory that we need here is the fact that if � is a ball in Rn then for any positive (u0, v0)∈ (C∞(�))2, there exists a maximalexistence time Tmax�∞ such that (1) possesses at least one classical solution (u, v) in �×(0, Tmax), and that the alternative

either Tmax =∞ or lim supt↗Tmax

‖u(·, t)‖L∞(�) =∞ (32)

holds. The existence of at least one such solution can be demonstrated by means of either Schauder’s fixed point theorem orgeneral theory of quasilinear parabolic systems ([27]; cf. also [16, 19, 28] for corresponding procedures in closely related problems);the existence of a maximal existence time along with its property (32) can be deduced from standard extendibility arguments.

Theorem 5.1Let n�2 and �⊂Rn be a ball, and suppose that there exist s0>1, �∈ (0, 1), K>0 and k>0 such that

∫ s

s0

��(�)

�(�)d��

⎧⎪⎪⎨⎪⎪⎩

Ks

ln sif n=2

n−2−�

n

∫ s

s0

∫ �

s0

�()

�()dd�+Ks if n�3

(33)

as well as

∫ s

s0

∫ �

s0

�()

�()dd��

⎧⎪⎨⎪⎩

ks(ln s)� with some �∈ (0, 1) if n=2

ks2−� with some �>2

nif n�3

(34)

hold for all s�s0. Then for each m>0 there exist initial data (u0, v0)∈ (C∞(�))2 with∫� u0 =m such that the corresponding solution

(u, v) blows up in either finite or infinite time.

ProofLet m>0 be given. From Lemma 2.2 we know that each global bounded solution (u, v) of (1) with positive initial data gives rise toa steady-state solution (u∞, v∞) of (9) satisfying F(u∞, v∞)�F(u(·, 0), v(·, 0)). By Lemmas 3.4 and 3.3, this entails that for some C>0we have F(u(·, 0), v(·, 0))�−C whenever (u, v) is global and bounded. But Lemma 4.1 says that there exist smooth positive initial data(u0, v0) with

∫� u0 =m but F(u0, v0)<−C. In view of (32), the corresponding solution of (u, v) evidently must blow up. �

Let us finally specify some conditions on � / � that are sufficient to guarantee (33) and (34) but easier to verify.

Corollary 5.2In each ball �⊂Rn, n�2, for any m>0 there exist unbounded solutions of (1) having mass

∫� u(x, t)≡m, provided that one of the

following hypotheses is satisfied.

(i) n=2 and there exist c0>0 and s0>1 such that

�(s)

�(s)�c0s ln s for all s�s0 (35)

holds.(ii) n�3 and for some c0>0 and s0>1, the lower estimate

�(s)

�(s)�c0s for all s�s0 (36)

is valid.

22


M. WINKLER

(iii) n�3 and for some c0>0 and �>2 / n we have

s−� �(s)

�(s)→c0 as s→∞ (37)

(iv) n�3 and

lim infs→∞

s

(�

�

)′(s)(

�

�

)(s)

>2

n(38)

Proof

(i) We may assume that s0�e2. From (35) we obtain∫ s

s0(��(�) / �(�)) d��1 / c0

∫ ss0

d� / ln�, and since

d

ds

(∫ s

s0

d�

ln�− 2s

ln s

)= 2− ln s

(ln s)2�0 for all s�e2

(33) follows with K :=2 / c0 upon integrating this.In order to show that (34) actually holds for all �∈ (0, 1), we observe that because of s0�e2, we have

d2

ds2(s(ln s)�) = �(ln s)�

s ln s·(

1− 1−�

ln s

)��(ln s)�

s

(1− 1−�

2

)

� � ·2�

s ln s· 1+�

2for all s�s0

Integrating this twice, we easily derive (34) upon the choice k :=21−� / �(1+�)c0.(ii) As, by (36),

∫ s

s0

��(�)

�(�)d�� 1

c0(s−s0)� 1

c0s for all s�s0

and since the first term on the right of (33) is nonnegative, we may let K :=1 / c0 and see that (33) is satisfied, whereas (34)is trivial here.

(iii) In view of (ii) we only need to consider the case when �<1, in which we fix

�∈

⎛⎜⎝1,

1− 2

n1−�

⎞⎟⎠

and let c1>0 be such that c1<c0<�c1. Then (37) guarantees that for some s0>1, we have

c1s��(s)

�(s)��c1s� for all s�s0 (39)

We now pick �∈ (0, 1) such that �<n−2−n(1−�)� and use (39) to estimate

∫ s

s0

∫ �

s0

�()

�()dd�− n

n−2−�

∫ s

s0

��(�)

�(�)d� � 1

�c1(1−�)(2−�)(s2−�−s2−�

0 )− s1−�0

�c1(1−�)(s−s0)− n

(n−2−�)c1(2−�)(s2−�−s2−�

0 )

� 1

c1(2−�)

(1

�(1−�)− n

n−2−�

)s2−�− s1−�

0�c1(1−�)

s− s2−�0

�c1(1−�)(2−�)

for all s�s0, where according to the choice of �, the first term on the right is nonnegative. Hence, it follows that (33) is trueif we set

K := (3−�)(n−2−�)s1−�0

�c1(1−�)(2−�)n

for instance, whereas (34) is obvious.(iv) We note that (38) implies the existence of �∈ (0, 1) and s0>1 such that

d

ds

s�(s)

�(s)� n−2−�

n· �(s)

�(s)for all s�s0 (40)


23

M. WINKLER

From this, we immediately derive (33) with K :=0 upon two integrations. Rewriting (40) in the equivalent form

d

ds

�(s)

�(s)�(s)

�(s)

�2+�

nsfor all s�s0

again by integration we also obtain (34) with � := (2+�) / n∈ (2 / n, 1) and

k := s�0�(s0)

(1−�)(2−�)�(s0)�

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