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Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with

singular data

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2011 Nonlinearity 24 1433

(http://iopscience.iop.org/0951-7715/24/5/003)

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Nonlinearity 24 (2011) 1433–1449 doi:10.1088/0951-7715/24/5/003

Existence and asymptotic behaviour for theparabolic–parabolic Keller–Segel system withsingular data

Lucas C F Ferreira1 and Juliana C Precioso2

1 Universidade Estadual de Campinas, Departamento de Matematica, CEP 13083-859,Campinas-SP, Brazil2 Universidade Estadual Paulista, Departamento de Matematica, Sao Jose do Rio Preto-SP,CEP: 15054-000, Brazil

E-mail: lcff@ime.unicamp.br and precioso@ibilce.unesp.br

Received 23 June 2010, in final form 4 March 2011Published 31 March 2011Online at stacks.iop.org/Non/24/1433

Recommended by J A Sherratt

AbstractThis work considers the Keller–Segel system of parabolic–parabolic type in R

n

for n � 2. We prove existence results in a new framework and with initialdata in N −β

r,λ,∞ × B0∞,∞. This initial data class is larger than the previous

ones, e.g., Kozono–Sugiyama (2008 Indiana Univ. Math. J. 57 1467–500) andBiler (1998 Adv. Math. Sci. Appl. 8 715–43), and covers physical cases ofinitial aggregation at points (Diracs) and on filaments. Self-similar solutionsare obtained for initial data with the correct homogeneity and a certain value ofparameter γ . We also show an asymptotic behaviour result, which provides abasin of attraction around each self-similar solution.

Mathematics Subject Classification: 35K45, 35Q92, 92C15, 35A01, 35B40,35C06, 42B35

1. Introduction

We consider a chemotaxis model introduced by Keller–Segel which models aggregation ofbiological species. It describes a situation where organisms (e.g. amoebae and bacteria) movetowards high concentration of food molecules (e.g. glucose) or of a chemical secreted bythemselves. The mathematical structure of the Keller–Segel model consists in a coupledsystem of parabolic–parabolic type which reads as follows:

∂tu = ∇ · (∇u − u∇v) , in x ∈ Rn, t ∈ (0, ∞),

∂tv = �v − γ v + κu, in x ∈ Rn, t ∈ (0, ∞),

u|t=0 = u0, v|t=0 = v0, in x ∈ Rn,

(1.1)

0951-7715/11/051433+17$33.00 © 2011 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA 1433

1434 L C F Ferreira and J C Precioso

where n � 2, u(x, t) represents the density of cells and v(x, t) is the concentration of thechemoattractant. The parameters γ � 0 and κ � 0 denote the decay and production rate ofthe attractant, respectively. Without loss of generality we assume κ = 1.

Assuming that the chemoattractant concentration is time independent, one obtains asimplified version of (1.1) which is the parabolic–elliptic variant of (1.1) and reads as

∂tu = ∇ · (∇u − u∇v) , in x ∈ Rn, t ∈ (0, ∞),

−�v + γ v = u, in x ∈ Rn, t ∈ (0, ∞),

u|t=0 = u0, in x ∈ Rn.

(1.2)

For this particular situation, we would like to mention the works [3, 4, 5, 11] (see also theirreferences). Existence of global solutions for (1.2) with fractional dissipation (−�)

d2 was

addressed by [5, 11] with u0 ∈ L1(R) ∩ H 1(R) ⊂ L∞(R) and small data u0 ∈ B1−d2,r (R2),

respectively. For the value γ = 0, (1.2) boils down to a model that describes gravitationalinteraction of many particles with themselves. This case was studied in [3, 4] where the authorsobtained existence of forward self-similar solutions by employing Morrey and PMa spaces,respectively.

The Keller–Segel model has attracted the attention of many authors, see, e.g. [2, 6, 9, 14–16, 19–21, 30] and references therein. Existence of global bounded mild solutions withsmall data u0 ∈ L1 ∩ L∞ and ∇v0 ∈ L1 ∩ L∞ was proved in [26]. In the frameworkof Lebesgue spaces, the authors of [7] constructed global positive solutions in R

2 for(1.1) by means of entropy methods; among other conditions, they assume positive data[u0, v0] ∈ L1(R2) × H 1(R2) with small mass. For n � 3, paper [9] proved existence of weakpositive solutions for (1.1) by taking small data u0 ∈ Lr and ∇v0 ∈ Ln with n/2 < r � n.

Note that the latter class is scaling invariant (see (3.2)) only when r = n/2. Assuming furtherthat u0 ∈ L1, this endpoint case was treated for the model (1.2) in [10]. Weak solutions inL∞((0, ∞); L∞(Rn)) were obtained in [30] for a degenerated quasi-linear version of (1.1)with non-negative data [u0, v0] ∈ L1 ∩ L∞ × L1 ∩ H 1 ∩ W 1,∞. Taking θ1 = 0 andθ = 2 in [33, theorem 1.1], one obtains existence of small global solution for (1.1) with

data B−(2− n

p)

p,∞ × Bnp

p,∞. The results of [2] yield global mild solutions for n �= 3 and initial datain a certain class of Morrey spaces invariant by (3.2), namely:[u0, ∇v0] ∈ M × L2, for n = 2,

[u0, ∇v0] ∈ Mp,n−2p × M2p,n−2p, for n � 4 and3

2< q � n

2,

(1.3)

where M is the space of finite measures and Mp,λ is defined in (2.1) (see also (1.5)). Indeedthe author of [2] works with local-in-time solutions; however, since his estimates are timeindependent, solutions can be made global-in-time under smallness assumption on data.

Recently in [19], for n � 3 and max{1, n/4} < r < n/2, existence of global mildsolutions for (1.1) was proved with small data [u0, v0] belonging to class Hn/r−2,r × Hn/r,r ,which is also invariant by (3.2). In [21] the results of [19] were improved by considering datain L(n/2,∞) × BMO and n � 3, where BMO stand for the space of functions with boundedmean oscillation and L(p,∞) (also named weak-Lp) is the space of all functions f such that‖f ‖∗

(p,∞) = supt>0 tλ1/p

f (t) < ∞; here λf (t) = |{x ∈ Rn : |f (x)| > t}| denotes the

distribution function of f. Note that the continuous inclusions Hn/r−2,r ⊂ Ln/2 ⊂ L(n/2,∞)

and Hn/r,r ⊂ BMO hold true for n/4 < r < n/2. Reference [21] also proved the existence ofself-similar solutions for (1.1) when γ = 0. In the same case a non-existence result of finitetime self-similar solution of backward type can be found in [20]. We refer the reader to [27, 28]for methods of constructing forward self-similar solutions different from [21]. In particular,the author of [28] showed that if the profile of a classical self-similar solution is bounded,

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data 1435

vanishes at infinity and possesses finite mass in R2, then that profile is radially symmetric

about the origin.It is an open problem to know whether smooth solutions for (1.1) blow up (or not) at a

finite time T . In connection with this problem, it arises a motivation for studying system (1.1)in spaces that contain singular functions and self-similar solutions. In this direction, to thebest of our knowledge, the larger existence classes for (1.1) are those of [2, 21, 33]. We referto [2, 12–16, 24] for some blow up results, with (or not) radial and self-similarity symmetry,for (1.1) or its simplified versions.

The aim of this paper is to study global existence and long time behaviour of solutionsfor the IVP (1.1) with small initial data belonging to a new scaling-invariant class, namelyN −β

r,λ,∞ × B0∞,∞ with n � 2, 0 � λ < n, and β = 2 − n−λ

r(see section 2 for more details

on these spaces). Here N sp,λ,q is the Banach space of all tempered distributions modulo

polynomials with norm given by

‖f ‖N sp,λ,q

=

(∑

k∈Z

(2ks ‖ϕk ∗ f ‖p,λ)q

) 1q

, 1 � p � ∞, 1 � q < ∞, s ∈ R,

supk∈Z

(2ks ‖ϕk ∗ f ‖p,λ), 1 � p � ∞, q = ∞, s ∈ R,

(1.4)

where ∗ is the convolution product and ‖ · ‖p,λ denotes the norm of the Morrey space Mp,λ:

‖f ‖p,λ = supx0∈Rn,R>0

R− λ

p ‖f ‖Lp(BR(x0)), with BR(x0) = {x ∈ R

n : |x − x0| < R}. (1.5)

Observe that (1.5) with λ = 0 gives the Lp-norm. Above {ϕk}k∈Z is given by ϕk(ξ) = ϕ(2−kξ)

where ϕ ∈ C∞0 , supp(ϕ) ⊂ {ξ ∈ R

n : 2−1 < |ξ | < 2}, and

∞∑k=−∞

ϕ(2−kξ) =∞∑

k=−∞ϕk(ξ) = 1, for all ξ �= 0.

The symbol ϕ stands for the Fourier transform of ϕ and note that {ϕk}k∈Z is a Littlewood–Paley partition of unity subordinated to a dyadic decomposition of R

n. The homogeneousBesov space Bs

p,q corresponds to the particular case λ = 0, that is, Bsp,q = N s

p,0,q and soB0

∞,∞ = N 0∞,0,∞. The space N s

p,λ,q is called homogeneous Besov–Morrey space and wasinitially introduced in the interesting work [22] to study Navier–Stokes equations by means ofKato’s approach involving two norms. These spaces contain strongly singular functions and,since we have the continuous inclusions

L(n/2,∞) ⊂ Mp,n−2p ⊂ N −β

r,λ,∞; BMO ⊂ B0∞,∞; B

−(2− np)

p,∞ ⊂ N −β

r,λ,∞

and Bnp

p,∞ ⊂ B0∞,∞, (1.6)

our initial data class is larger than those of [2, 21, 33] and the other ones mentioned above.Indeed we can extend the class B0

∞,∞ of v0 to a little larger one (see remark 3.2).Despite the fact that system (1.1) does not have a scaling property when γ �= 0, one can

consider an ‘intrinsic scaling’ that is inherited from the case γ = 0 (see (3.1)). We constructmild solutions by means of a fixed point argument in a time-dependent space X1 × X2, whosenorm is invariant by that intrinsic scaling. In particular, taking u0 homogeneous of degree −2and v0 of degree 0, solutions are forward self-similar when γ = 0, i.e. they are invariant by(3.1). Precisely, for q �= r, α = 2 − n−λ

qand β = 2 − n−λ

r, the Banach spaces X1 and X2 are

1436 L C F Ferreira and J C Precioso

defined by

X1 ={u ∈ BC

((0, ∞); N −β

r,λ,∞)

: tβ

2 u ∈ BC((0, ∞); Mr,λ

)and t

α2 u ∈ BC

((0, ∞); Mq,λ

) }, (1.7)

X2 ={v ∈ BC

((0, ∞); B0

∞,∞)

: t12 ∇xv ∈ BC

((0, ∞); L∞)}

, (1.8)

with norms

‖u‖X1= sup

t>0‖u(·, t)‖N −β

r,λ,∞+ sup

t>0t

β

2 ‖u(·, t)‖r,λ + supt>0

tα2 ‖u(·, t)‖q,λ , (1.9)

‖v‖X2= sup

t>0‖v(·, t)‖B0∞,∞ + sup

t>0t

12 ‖∇v(·, t)‖L∞ . (1.10)

Furthermore, in the spirit of [4] we prove an asymptotic stability result that provides a criteriafor dissipation of small perturbations of initial data as t → ∞ (see theorem 3.5). Appliedto the case γ = 0, this result implies existence of an attractor-basin around each self-similarsolution according to the convergence (3.14). It is worth to note that the kind of stabilitystudied here is different from the Lyapunov one studied in [21].

Let us recall the following phenomenological fact, known as Keller–Segel law, concerningsystem (1.1): for n = 3 the collapse of cells to density infinite lines and points is possible;however, concentration on a sheet cannot occur. Nevertheless, cells moving within a plancan aggregate at points but not on lines (see [6]). So far it is an open problem to describemathematically the connection between dimension and biological pattern aggregation for(1.1), see [11] for more details. For the model (1.2) this picture is stated mathematicallyin the particular case n = 2 and γ = 0; precisely, there exists a threshold value for the massof u0 that decides if solutions blow up or not in a finite time (see e.g. [7, 16]).

It is natural to consider the inverse problem associated with Keller–Segel law, that is: howdoes system (1.1) behave when cells are initially concentrated on lines or at points? Dependingon dimension n, our existence theorem allows us to take as initial data measure concentrated atpoints (sum of Diracs), filaments or rings (see remark 3.3). From a physical point of view, thesecases can be regarded as an initial aggregation of cells around points or on lines. The decaysgiven by norm of X1 ×X2 show that system (1.1) tends to dissipate initial small concentrationsin our framework.

Finally, we comment on some technical difficulties that we overcame to obtain our results.Kato’s approach consists in constructing a fixed point argument using a suitable time-dependentspace, whose norm is composed of two parts. Here, it was necessary to perform an adaptationof that approach in order to take [u0, v0] ∈ N −β

r,λ,∞ × B0∞,∞. More precisely, in order to

avoid singular factors by handling some norms of the nonlinear term B(u, v) (see lemma 4.4and (4.17)), we work in the time-dependent space X1 � u containing three norms, namelythe natural one supt>0 ‖u(·, t)‖N −β

r,λ,∞and two others (see (1.9)). We need to split the integral

(4.16) into two parts and use a different combination of two norms to estimate each one.A similar argument was also needed in deriving the estimate (4.13) for the linear term T (u)

(see lemma 4.3 and (4.10)). These difficulties were naturally propagated to the proof ofthe asymptotic stability result, because (in some parts) it depends intrinsically on argumentsemployed to prove lemmas 4.4 and 4.3, see (4.26) and (4.29)–(4.30). Thus, beyond workingin a distinct framework, the adaptation of the stability arguments from [4] made here is not astraightforward matter and involves certain care.

The outline of this manuscript is as follows. In the next section we review some basicproperties about Besov–Morrey spaces. We state our results in section 3 and prove them insection 4.

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data 1437

2. Preliminaries

In this section we summarize some basic properties of Morrey, Besov and Besov–Morreyspaces that we will need in this work. For a deeper discussion on these spaces, see[1, 17, 22, 23, 29, 32]. In order to facilitate the exposition, some definitions given inintroduction will be repeated below.

For 1 � p < ∞ and 0 � λ < n, the Morrey space Mp,λ = Mp,λ(Rn) is defined as

Mp,λ = {f ∈ L

p

loc(Rn) : ‖f ‖p,λ < ∞}

, (2.1)

with norm given by

‖f ‖p,λ = sup{R

− λp ‖f ‖p;x,R ; x ∈ R

n, R > 0}

, (2.2)

where ‖f ‖p;x,R denotes the Lp-norm of f on the ball BR(x) ⊂ Rn with centre x and radius R.

The space Mp,λ endowed with ‖f ‖p,λ is a Banach space. When p = 1, M1,λ is a spaceof signed measures with ‖ · ‖1;x,R standing for total variation of f on BR(x). For p > 1Mp,0 = Lp, and M1,0 = M is the space of the finite measures. Whereas p = ∞ we caninclude L∞ among the Morrey spaces and then M∞,λ simply means L∞. The norm ‖ · ‖p,λ

presents the following scaling relation:

‖f (σx)‖p,λ = σ−τp,λ ‖f (x)‖p,λ , with τp,λ = n − λ

p. (2.3)

In the following we recall Holder inequality in Morrey spaces:

Lemma 2.1. Let 1 � pi � ∞ and 0 � λi < n, i = 1, 2, 3. If 1p3

= 1p1

+ 1p2

and λ3p3

= λ1p1

+ λ2p2

then

‖fg‖p3,λ3� ‖f ‖p1,λ1

‖g‖p2,λ2. (2.4)

Let ϕ ∈ C∞0 with supp(ϕ) ⊂ D0 = {ξ ∈ R

n : 2−1 < |ξ | < 2} such that∞∑

k=−∞ϕ(2−kξ) =

∞∑k=−∞

ϕk(ξ) = 1, for all ξ �= 0,

where ϕk is defined by means of Fourier transform as ϕk(ξ) = ϕ(2−kξ) for all integer k. Forf ∈ S ′

we define the quantity

‖f ‖N sp,λ,q

=

(∑

k∈Z

(2ks ‖ϕk ∗ f ‖p,λ)q

) 1q

, 1 � p � ∞, 1 � q < ∞, s ∈ R,

supk∈Z(2ks ‖ϕk ∗ f ‖p,λ), 1 � p � ∞, q = ∞, s ∈ R.

Let P denote polynomials with n variables. The homogeneous Besov–Morrey space N sp,λ,q is

the set of equivalence classes f ∈ S ′/P such that ‖f ‖N s

p,λ,q< ∞. The pair (N s

p,λ,q , ‖ · ‖N sp,λ,q

)

is a Banach space and can be described as a real interpolation space, namely

(Ms1p,λ, Ms2

p,λ)θ,q = N sp,λ,q (2.5)

where θ ∈ (0, 1), s = (1−θ)s1 +θs2 with s1 �= s2, and Msp,λ = (−�)−s/2Mp,λ. In particular,

Msp,0 ≡ H s

p and N sp,0,q ≡ Bs

p,q are homogeneous Sobolev and Besov spaces, respectively.Thus it follows from (2.5) with λ = 0 that

(H s1p , H s2

p )θ,q = Bsp,q . (2.6)

Let 1 � q, q1, q2 � ∞ and θ ∈ (0, 1) such that s = (1 − θ)s1 + θs2. The following propertyholds true:

(N s1p,λ,q1

, N s2p,λ,q2

)θ,q = N sp,λ,q . (2.7)

1438 L C F Ferreira and J C Precioso

The inclusion N sp,λ,q2

⊂ N sp,λ,q1

is continuous for 1 � q2 � q1 � ∞, and we have the Sobolevtype embedding

N s2p2,λ,q2

⊂ N s1p1,λ,q1

, for p1 > p2 and s1 − n − λ

p1= s2 − n − λ

p2. (2.8)

Moreover,

(−�)l/2N sp,λ,q = N s−l

p,λ,q for all l ∈ R, 1 � p, q � ∞. (2.9)

Finally, we recall the inclusions

B0∞,2 ⊂ BMO ⊂ B0

∞,∞, (2.10)

where BMO stand for the set of functions of bounded mean oscillation.The heat semigroup {G(t)}t�0 is the family of convolution operators with respective

kernels gt (x) = (4πt)−n2 e− |x|2

4t , i.e. G(t)f = gt ∗ f. For 1 � p < ∞ and s > 0 we have thefollowing equivalence (see [23]):

‖f ‖N −sp,λ,∞

∼= supt>0

ts2 ‖G(t)f ‖p,λ . (2.11)

According to our notation, we have N s∞,λ,∞ ≡ Bs

∞,∞ (since M∞,λ ≡ L∞), and (2.11) is alsovalid to the case p = ∞ (see [8]), that is

‖f ‖B−s∞,∞∼= sup

t>0t

s2 ‖G(t)f ‖L∞ for s > 0. (2.12)

In the following we compile some estimates for {G(t)}t�0 on the spaces defined through thissection. Particularly, estimate (2.15) seems to be stated in the literature only for p1 = p2 and|k| = 1 (see [22]). Using Mp,λ-estimates for {G(t)}t�0 [17, 31], we extend it to other casesand give a simpler proof than [22] for reader’s convenience.

Lemma 2.2. Let k be a multiindex, s1 � s2, si ∈ R, 1 � q � ∞, 1 � p1 � p2 � ∞,0 � λ < n and τpi ,λ be as in (2.3). There exists a constant C > 0 such that∥∥∇k

xG(t)f∥∥

p2,λ� Ct−

|k|2 − 1

2 (τp1 ,λ−τp2 ,λ) ‖f ‖p1,λ(2.13)

‖G(t)f ‖Bs2p2 ,q

� Ct− s2−s1

2 − n2 ( 1

p1− 1

p2) ‖f ‖B

s1p1 ,q

, (2.14)∥∥∇kxG(t)f

∥∥N s2

p2 ,λ,q

� Ct−|k|+(s2−s1)

2 − 12 (τp1 ,λ−τp2 ,λ) ‖f ‖N s1

p1 ,λ,q, (2.15)

‖∇xG(t)f ‖L∞ � Ct−12 ‖f ‖B0∞,∞ , (2.16)

for all f ∈ S ′.

Proof. For (2.13) we refer the reader to [17, 31]. Estimate (2.13) still holds true for fractionalderivatives, that is, with d � 0 and (−�)d/2 in place of |k| and ∇k

x . From the latter commentand using (2.13) with λ = 0 and λ �= 0, one can obtain versions of (2.14) in the frameworks H s

p

(see [18]) and Msp,λ, respectively; because (−�)d/2 commutes withG(t). Now, in view of (2.6)

and (2.5), (2.14)–(2.15) follow from interpolation arguments. Finally, taking�(x) = ∇xg1(x),

it follows from the proof of theorem 1 of [29, p 156] (see also [8, proposition 1]) that

supt>0

t1/2 ‖∇xG(t)f ‖L∞ = supt>0

∥∥∥[t−

n2 �(xt−1/2)

]∗ f

∥∥∥L∞

� C ‖f ‖B0∞,∞ ,

which is equivalent to (2.16).

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data 1439

3. Results

The purpose of this section is to present our existence and long time behaviour results for theCauchy problem (1.1). Before stating them, we perform a scaling analysis for finding suitablespaces to study (1.1).

Note that for γ �= 0, system (1.1) has not a scaling property. Despite this, there is an‘intrinsic scaling’ that is inherited from the case γ = 0. Let us take temporarily γ = 0and assume that [u, v] is a classical solution for (1.1). It is not difficult to check thatuσ := σ 2u(σx, σ 2t) and vσ := v(σx, σ 2t) also satisfies (1.1). The map

[u, v] → [uσ , vσ ] (3.1)

is called the scaling of (1.1) (when γ = 0) and solutions invariant by it are named self-similarsolutions. Formally, making t → 0+ in (3.1) one obtains

u0 → σ 2u0(σx) and v0 → v0(σx). (3.2)

Thus, for [u, v] to be self-similar, the data u0 and v0 must be homogeneous functions of degrees−2 and 0, respectively. The last observations motivate us to choose the following space ofinitial data, which is invariant by (3.2):

u0 ∈ N −β

r,λ,∞ and v0 ∈ B0∞,∞, where β = 2 − τr,λ and τr,λ is given by (2.3). (3.3)

Let α = 2 − τq,λ with q �= r . As said in introduction, we will look for global-in-timesolutions [u(x, t), v(x, t)] in the scaling-invariant Kato class:

X1 ={u ∈ BC

((0, ∞); N −β

r,λ,∞)

: tβ

2 u ∈ BC((0, ∞); Mr,λ

)and

tα2 u ∈ BC

((0, ∞); Mq,λ

) }, (3.4)

X2 ={v ∈ BC

((0, ∞); B0

∞,∞)

: t12 ∇xv ∈ BC

((0, ∞); L∞)}

, (3.5)

which are Banach spaces with norms

‖u‖X1= sup

t>0‖u(·, t)‖N −β

r,λ,∞+ sup

t>0t

β

2 ‖u(·, t)‖r,λ + supt>0

tα2 ‖u(·, t)‖q,λ , (3.6)

‖v‖X2= sup

t>0‖v(·, t)‖B0∞,∞ + sup

t>0t

12 ‖∇v(·, t)‖L∞ . (3.7)

Note that a pair [u, v] ∈X1×X2 is continuous at t > 0, but not necessarily at t = 0+. However,our existence theorem will provide continuous solutions at t = 0+ in the sense of S ′(Rn). Herewe prefer not to assume previously this property for functions in the class X1 × X2.

In according to Duhamel’s principle, we will work with the following integral formulationfor (1.1):

u(t) = G(t)u0 −∫ t

0∇ · G(t − s)(u∇v)(s) ds, (3.8)

v(t) = G(t)v0 +∫ t

0G(t − s)u(s) ds, with G(t)v0 = e−tγ G(t)v0. (3.9)

A pair [u, v] satisfying (3.8)–(3.9) is called a mild solution for the Cauchy problem (1.1). Inthe rest of this work, we will use the following notation for the bilinear and linear operatorsappearing in (3.8)–(3.9):

B(u, v) = −∫ t

0∇ · G(t − s)(u∇v)(s) ds and T (u) =

∫ t

0G(t − s)u(s) ds. (3.10)

1440 L C F Ferreira and J C Precioso

We are ready to state our well-posedness result:

Theorem 3.1. Assume that n � 2, 0 � λ � n − 2, p0 = n−λ2 and p0 < r < 2p0 < q < ∞,

and suppose that the initial data belong to class (3.3). Let {K1, K2} and {K3, K4, K5} be as inlemmas 4.3 and 4.4, respectively, and τ = K1 +K2, K = K3 +K4 +K5, and 0 < ε < 1

2K(1+2τ).

There exist δ1 = δ1(ε) and δ2 = δ2(ε) such that system (3.8)–(3.9) has a global mild solution[u, v] ∈ X1 × X2 provided that ‖u0‖N −β

r,λ,∞� δ1 and ‖v0‖B0∞,∞ � δ2. The solution is the unique

one satisfying ‖u‖X1 � 2ε and ‖v‖X2 � (1 + 2τ)ε. Moreover, [u(t), v(t)] ⇀ [u0, v0] inS ′(Rn) as t → 0+.

In the next remark we comment as a slight modification of our results extends the initialdata class for v0:

Remark 3.2. In fact we can take the initial chemoattractant concentration v0 in a more singularclass, namely v0 ∈ I0 ≡ {v0 ∈ S ′

/P : ∇v0 ∈ B−1∞,∞} with norm ‖v0‖I0 = ‖∇v0‖B−1∞,∞ . In view

of (2.12) and (2.16), note that

‖∇xv0‖B−1∞,∞ � supt>0

t12 ‖G(t)∇xv0‖L∞ = sup

t>0t

12 ‖∇xG(t)v0‖L∞ � C ‖v0‖B0∞,∞ , (3.11)

and then B0∞,∞ ⊂ I0. Moreover, since M2p,n−2p ⊂ B−1

∞,∞, we have that

{v0 ∈ S ′/P : ∇v0 ∈ M2p,n−2p} ⊂ I0. (3.12)

Remark 3.3 (Concentrated measures as initial data). The Besov–Morrey space N −β

r,λ,∞contains measures concentrated on a smooth λ-dimensional manifold in R

n. Thus, we canconsider the initial data u0 as such measures. In the case n = 2 one can take λ = 0 and u0 asa measure concentrated on a countable set of points, that is, a countable sum of Diracs. Whenn = 3, measures concentrated on filaments, rings and more generally smooth curves can beconsidered as initial data u0.

In the case γ = 0, the existence of self-similar solutions for (1.1) follows from theorem 3.1and the fact that the classes (3.3)–(3.5) contain functions with the correct homogeneity:

Corollary 3.4. Under the hypotheses of theorem 3.1. Assume that u0 ∈ N −β

r,λ,∞ and v0 ∈ B0∞,∞

are homogeneous functions with degree −2 and 0, respectively. Then the solution [u, v]obtained through theorem 3.1 is self-similar, that is,

u(x, t) = σ 2u(σx, σ 2t) and v(x, t) = v(σx, σ 2t)

for all σ > 0, t > 0 and x ∈ Rn.

We also provide an asymptotic stability result for solutions of system (1.1). In addition torefining the time decays obtained in theorem 3.1, next theorem shows that certain perturbationsof the initial data vanish as t → ∞.

Theorem 3.5. Under the hypotheses of theorem 3.1. Assume that [u, v] and [u, v] aretwo solutions given by theorem 3.1 corresponding to the initial data [u0, v0] and [u0, v0],respectively. We have that

limt→+∞

(t

β

2 ‖G(t)(u0 − u0)‖r,λ + tα2 ‖G(t)(u0 − u0)‖q,λ + t

12∥∥∇G(t)(v0 − v0)

∥∥L∞

)= 0

(3.13)

if only if

limt→+∞ t

β

2 ‖u(·, t) − u(·, t)‖r,λ = limt→+∞ t

α2 ‖u(·, t) − u(·, t)‖q,λ

= limt→+∞ t

12 ‖∇(v − v)(·, t)‖L∞ = 0. (3.14)

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data 1441

As a consequence, in the case γ = 0 theorem 3.5 furnishes an attractor-basin around eachself-similar solution. Precisely, we have:

Corollary 3.6 (Basin of attraction). Under the hypotheses of theorem 3.1. Assume that u0

is a homogeneous function of degree −2, v0 is one of degree 0 and [φ, ψ] ∈ C∞0 × C∞

0 . Let[u, v] and [θ, ϕ] be mild solutions with data [u0, v0] and [u0 + φ, v0 + ψ], respectively. Notethat [u0 − u0, v0 − v0] = [φ, ψ] verify (3.13). Then, in the case γ = 0 the perturbed solution[θ, ϕ] is attracted to the self-similar solution [u, v] in the sense of (3.14).

4. Proofs

In this section, we present the proofs of the results stated in section 3. Let us start with anabstract fixed point lemma that will be useful for our purposes.

Lemma 4.1. Let X be a Banach space with norm ‖ · ‖X and T : X → Y a continuous linearmap with norm τ. Assume that B : X × Y → X is a continuous bilinear map, that is, there isK > 0 such that

‖B(x1, y1)‖X � K ‖x1‖X ‖y1‖Y , for all (x1, y1) ∈ X × Y. (4.1)

Let 0 < ε < 12K(1+2τ)

and B2ε = {x ∈ X : ‖x‖X � 2ε}. If ‖y‖X � ε and ‖y‖Y � ε then thereexists an unique solution x ∈ B2ε for equation x = y + B(x, H(x)), where H(x) = y + T (x).

Proof. Consider the mapping F : X → X given by F(x) = y + B(x, H(x)). For x ∈ B2ε,we have that

‖F(x)‖X � ‖y‖X + ‖B(x, H(x))‖X

� ‖y‖X + K ‖x‖X ‖H(x)‖Y

� ‖y‖X + K ‖x‖X

(‖y‖Y + ‖T (x)‖Y

)� ε + K ‖x‖X (ε + τ ‖x‖X) � ε + 2Kε(ε + 2τε) < 2ε,

by the choice of ε, and then F(B2ε) ⊂ B2ε. Next, we take x, z ∈ B2ε and estimate

‖F(x) − F(z)‖X = ‖B(x, H(x)) − B(z, H(z))‖X

= ‖B((x − z), H(x)) + B(z, H(x) − H(z))‖X

� K||x − z||X ‖H(x)‖Y + K ‖z‖X ‖H(x) − H(z)‖Y

� K ‖x − z‖X

(‖y‖Y + τ ‖x‖X

)+ K ‖z‖X (τ ‖x − z‖X)

= K(‖y‖Y + τ ‖x‖X + τ ‖z‖X

) ‖x − z‖X

� K(ε + 4τε) ‖x − z‖X .

Since K(ε + 4τε) < 2K(1 + 2τ)ε < 1, F is a contraction in B2ε, and then Banach’s fixed pointtheorem concludes the proof.

Remark 4.2. It will be useful in the proof of corollary 3.4 to note that the solution obtainedthrough lemma 4.1 is the limit in X of the sequence of iterates x1 = y and xn+1 = F(xn), n � 1.

4.1. Estimates for T (·)Lemma 4.3. Under the hypotheses of theorem 3.1. There exist constants K1, K2 > 0 suchthat

supt>0

‖T (u)‖B0∞,∞ � K1 supt>0

tα2 ‖u(·, t)‖q,λ , (4.2)

supt>0

t12 ‖∇xT (u)‖L∞ � K2

(supt>0

tβ

2 ‖u(·, t)‖r,λ + supt>0

tα2 ‖u(·, t)‖q,λ

), (4.3)

for all function u ∈ X1.

1442 L C F Ferreira and J C Precioso

Proof. Note that the hypotheses of theorem 3.1 imply

0 � β < 1 < α < 2, τq,λ = 2 − α < 1 and τr,λ = 2 − β � 2. (4.4)

Using the continuous inclusion L∞ ⊂ B0∞,∞ it follows that

‖T (u)(t)‖B0∞,∞ �∫ t

0

∥∥G(t − s)u(s)∥∥

B0∞,∞ds

�∫ t

0e−(t−s)γ ‖G(t − s)u(s)‖L∞ ds. (4.5)

We recall L∞ = M∞,λ and employ (2.13) in (4.5) with k = 0, p2 = ∞, p1 = q to obtain

||T (u)(t)||B0∞,∞ � C

∫ t

0e−(t−s)γ (t − s)−

τq,λ

2 ||u(s)||q,λds

� C

∫ t

0e−(t−s)γ (t − s)−

τq,λ

2 s− α2 ds sup

t>0t

α2 ||u(t)||q,λ (4.6)

= Ct1− τq,λ

2 − α2

∫ 1

0(1 − z)−

τq,λ

2 z− α2 dz sup

t>0t

α2 ||u(t)||q,λ

= K1 supt>0

tα2 ||u(t)||q,λ, (4.7)

where we have used the change of variable t = sz in (4.6), conditions (4.4) and that1 − τq,λ/2 − α/2 = 0. Estimate (4.2) follows by taking supt>0 in left-hand side of (4.7).

In order to derive (4.3) we split the integral (3.10) as follows:

‖∇xT (u)‖L∞ �∫ t

0

∥∥∇xG(t − s)u(s)∥∥

L∞ ds (4.8)

�∫ t/2

0e−(t−s)γ ‖∇xG(t − s)u(s)‖L∞ ds

+∫ t

t/2e−(t−s)γ ‖∇xG(t − s)u(s)‖L∞ ds. (4.9)

Now we estimate the first and second integrals in (4.9) using (2.13) with |k| = 1, p2 = ∞,

p1 = r and |k| = 1, p2 = ∞, p1 = q, respectively. Recalling (4.4) we obtain

rhs of (4.9) � C

(∫ t/2

0(t − s)−

12 − τr,λ

2 ‖u(s)‖r,λ ds +∫ t

t/2(t − s)−

12 − τq,λ

2 ‖u(s)‖q,λ ds

)� C

∫ t/2

0(t − s)−

12 − τr,λ

2 s− β

2 ds supt>0

tβ

2 ‖u(t)‖r,λ

+ C

∫ t

t/2(t − s)−

12 − τq,λ

2 s− α2 ds sup

t>0t

α2 ‖u(t)‖q,λ

� C(t/2)−12 − τr,λ

2

∫ t/2

0s− β

2 ds supt>0

tβ

2 ‖u(t)‖r,λ

+ C(t/2)−α2

∫ t

t/2(t − s)−

12 − τq,λ

2 ds supt>0

tα2 ‖u(t)‖q,λ

= K2 t−12

(supt>0

tβ

2 ‖u(t)‖r,λ + supt>0

tα2 ||u(t)||q,λ

), (4.10)

which yields (4.3).

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data 1443

4.2. Estimates for B(·, ·)Lemma 4.4. Under the hypotheses of theorem 3.1. There exist constants K3, K4, K5 > 0 suchthat

supt>0

‖B(u, v)‖N −β

r,λ,∞� K3 sup

t>0‖u(·, t)‖N −β

r,λ,∞supt>0

t12 ‖∇v(·, t)‖L∞ , (4.11)

supt>0

tβ

2 ‖B(u, v)‖r,λ � K4 supt>0

tβ

2 ‖u(·, t)‖r,λ supt>0

t12 ‖∇v(·, t)‖L∞ , (4.12)

supt>0

tα2 ‖B(u, v)‖q,λ � K5(sup

t>0t

β

2 ‖u(·, t)‖r,λ + supt>0

tα2 ‖u(·, t)‖q,λ) sup

t>0t

12 ‖∇v(·, t)‖L∞ ,

(4.13)

for all pair [u, v] ∈ X1 × X2.

Proof. We obtain (4.11) by proceeding in the following way:

‖B(u, v)‖N −β

r,λ,∞�

∫ t

0‖∇ · G(t − s)(u∇v)(s)‖N −β

r,λ,∞ds

� C

∫ t

0(t − s)−

12 ‖(u∇v)(s)‖N −β

r,λ,∞ds (by (2.15))

� C

∫ t

0(t − s)−

12 ‖u(s)‖N −β

r,λ,∞‖∇v(s)‖L∞ ds

� C

∫ t

0(t − s)−

12 s− 1

2 ds supt>0

‖u(·, t)‖N −β

r,λ,∞supt>0

t12 ‖∇v(·, t)‖L∞ .

= K3 supt>0

‖u(·, t)‖N −β

r,λ,∞supt>0

t12 ‖∇v(·, t)‖L∞ .

For (4.12), we apply (2.13) with |k| = 1 and p1 = p2 = p to obtain

‖B(u, v)‖r,λ �∫ t

0‖∇ · G(t − s)(u∇v)(s)‖r,λ ds (4.14)

� C

∫ t

0(t − s)−

12 ‖(u∇v)(s)‖r,λ ds

� C

∫ t

0(t − s)−

12 ‖u(s)‖r,λ ‖∇v(s)‖L∞ ds

� C

∫ t

0(t − s)−

12 s− 1

2 − β

2 ds supt>0

tβ

2 ‖u(·, t)‖r,λ supt>0

t12 ‖∇v(·, t)‖L∞ . (4.15)

Recalling that β < 1, the change of variable s = tz in (4.15) yields

‖B(u, v)‖r,λ � Ct1− 12 − 1

2 − β

2

∫ 1

0(1 − z)−

12 z− 1

2 − β

2 dz supt>0

tβ

2 ‖u(·, t)‖r,λ supt>0

t12 ‖∇v(·, t)‖L∞

= K4t− β

2 supt>0

tβ

2 ‖u(·, t)‖r,λ supt>0

t12 ‖∇v(·, t)‖L∞ ,

which implies (4.12). Next we prove (4.13). To do so, we first split B(u, v) into two parts

‖B(u, v)‖q,λ �∫ t

0‖∇ · G(t − s)(u∇v)(s)‖q,λ ds (4.16)

=∫ t/2

0‖∇ · G(t − s)(u∇v)(s)‖q,λ ds +

∫ t

t/2‖∇ · G(t − s)(u∇v)(s)‖q,λ ds.

1444 L C F Ferreira and J C Precioso

Proceeding as in the proof of (4.3) and using (4.4) we estimate the rhs of (4.16) by

� C

(∫ t/2

0(t − s)−

12 − 1

2 (τr,λ−τq,λ) ‖(u∇v)(s)‖r,λ ds +∫ t

t/2(t − s)−

12 ‖(u∇v)(s)‖q,λ ds

)� C

∫ t/2

0(t − s)−

12 − 1

2 (τr,λ−τq,λ) ‖u(s)‖r,λ ‖∇v(s)‖L∞ ds

+ C

∫ t

t/2(t − s)−

12 ‖u(s)‖q,λ ‖∇v(s)‖L∞ ds

� C

∫ t/2

0(t − s)−

12 − 1

2 (τr,λ−τq,λ)s− 12 − β

2 ds supt>0

tβ

2 ‖u(·, t)‖r,λ supt>0

t12 ‖∇v(·, t)‖L∞

+ C

∫ t

t/2(t − s)−

12 s− 1

2 − α2 ds sup

t>0t

α2 ‖u(·, t)‖q,λ sup

t>0t

12 ‖∇v(·, t)‖L∞

� K5t− α

2

(supt>0

tβ

2 ‖u(·, t)‖r,λ + supt>0

tα2 ‖u(·, t)‖q,λ

)supt>0

t12 ‖∇v(·, t)‖L∞ , (4.17)

which is equivalent to (4.13).

4.3. Proof of theorem 3.1

Take X = X1 and Y = X2 with X1 and X2 defined by (3.4) and (3.5). Let y = G(t)v0 andy = G(t)u0. For [u, v] ∈ X1 × X2 we denote τ = K1 + K2, K = K3 + K4 + K5,

v = H(u) = y + T (u) (4.18)

and

B(u, v) =∫ t

0∇ · G(t − s)(u · ∇v)(s) ds.

Recalling (3.6)–(3.7), lemmas 4.3 and 4.4 yields, respectively,

‖T (u)‖Y � (K1 + K2)

(supt>0

tβ

2 ‖u(·, t)‖r,λ + supt>0

tα2 ‖u(·, t)‖q,λ

)= τ ‖u‖X

and

‖B(u, v)‖X � (K3 + K4 + K5)

(supt>0

‖u(·, t)‖N −β

r,λ,∞+ sup

t>0t

β

2 ‖u(·, t)‖r,λ + supt>0

tα2 ‖u(·, t)‖q,λ

)× sup

t>0t

12 ‖∇v(·, t)‖L∞

= K ‖u‖X supt>0

t12 ‖∇v(·, t)‖L∞ � K ‖u‖X ‖v‖Y .

On the other hand, estimate (2.15) in lemma 2.2, the equivalence (2.11) and inclusionN −β

r,λ,∞ ⊂ N −αq,λ,∞ (when r < q) yield

‖y‖X1= sup

t>0‖G(t)u0‖N −β

r,λ,∞+ sup

t>0t

β

2 ‖G(t)u0‖r,λ + supt>0

tα2 ‖G(t)u0‖q,λ

� supt>0

‖G(t)u0‖N −β

r,λ,∞+ C ‖u0‖N −β

r,λ,∞+ C ‖u0‖N −α

q,λ,∞

� C1 ‖u0‖N −β

r,λ,∞� ε

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data 1445

and

‖y‖X2= sup

t>0

∥∥G(t)v0

∥∥B0∞,∞

+ supt>0

t12∥∥∇G(t)v0

∥∥L∞

� supt>0

‖G(t)v0‖B0∞,∞ + supt>0

t12 ‖∇G(t)v0‖L∞

� C2 ‖v0‖B0∞,∞ � ε,

provided that ‖u0‖N −β

r,λ,∞� δ1 = ε

C1and ‖v0‖B0∞,∞ � δ2 = ε

C2. If 0 < ε < 1

2K(1+2τ)then

lemma 4.1 implies that there exist a unique solution [u, v] ∈ X1 × X2 of (3.8)–(3.9) such that‖u‖X1 � 2ε and

‖v‖X2= ‖H(u)‖X2

� ‖y‖X2+ τ ‖u‖X1

� (1 + 2τ)ε. (by (4.18))

The convergence in S ′(Rn) as t → 0+ follows from standard arguments, see e.g. [22].

4.4. Proof of corollary 3.4

Recall that lemma 4.1 was used to prove theorem 3.1. Thus, in view of remark 4.2, the solution[u, v] given by theorem 3.1 is the limit of the following Picard sequence:

{u1 = G(t)u0

v1 = G(t)v0and

um+1 = G(t)u0 −

∫ t

0∇ · G(t − s)(um∇vm)(s) ds,

vm+1 = G(t)u0 +∫ t

0G(t − s)um(s) ds,

(4.19)

where m ∈ N. Let u0 be homogeneous of degree −2 and v0 of degree 0. It is a simple matterto check that the pair [u1, v1] is invariant by (3.1), that is

u1(x, t) = σ 2u1(σx, σ 2t) and v1(x, t) = v1(σx, σ 2t). (4.20)

By employing an induction argument, one can verify that [um, vm] also satisfies the scalingproperty (4.20) for all m. Since the solution [u, v] is the limit in X1 × X2 of thesequence {[um, vm]}m∈N and the norm ‖ · ‖X1×X2 is scaling invariant, it follows that [u, v] isself-similar, i.e.

u(x, t) = σ 2u(σx, σ t) and v(x, t) = v(σx, σ 2t),

for all σ > 0, x ∈ Rn and t > 0.

4.5. Proof of theorem 3.5

Firstly we will obtain (3.14) from (3.13). Let [u, v] and [u, v] be two mild solutions givenby theorem 3.1. Subtracting the equations verified by u and u, and afterwards computing thenorm ‖ · ‖p,λ, we obtain

tβ

2 ‖(u − u)(t)‖r,λ � tβ

2 ‖G(t)(u0 − u0)‖r,λ + tβ

2

∫ t

0‖∇ · G(t − s)(u∇v − u∇v)‖r,λ ds

:= tβ

2 ‖G(t)(u0 − u0)‖r,λ + I (t). (4.21)

The integral I is estimated as follows:

I (t) = tβ

2

∫ t

0‖∇ · G(t − s)(u∇v − u∇v)‖r,λ ds

� Ctβ

2

∫ t

0(t − s)−

12(‖(u − u)(s)‖r,λ ‖∇v(s)‖L∞ + ‖∇(v − v)(s)‖L∞ ‖u(s)‖r,λ

)ds

1446 L C F Ferreira and J C Precioso

� CI tβ

2

∫ t

0(t − s)−

12 s− 1

2 − β

2 sβ

2 ‖(u − u)(s)‖r,λ ds supt>0

t12 ‖∇v(·, t)‖L∞

+ CI tβ

2

∫ t

0(t − s)−

12 s− 1

2 − β

2 s12 ‖∇(v − v)(s)‖L∞ ds sup

t>0t

β

2 ‖u(·, t)‖r,λ . (4.22)

We recall

‖u‖X1, ‖u‖X1

� 2ε and ‖v‖X2, ‖v‖X2

� ε(1 + 2τ) (4.23)

and use the substitution s = zt in the last integrals to obtain

I (t) � ε(1 + 2τ)CI

∫ 1

0(1 − z)−

12 z− 1

2 − β

2 (tz)β

2 ‖(u − u)(tz)‖r,λ dz (4.24)

+ 2εCI

∫ 1

0(1 − z)−

12 z− 1

2 − β

2 (tz)12 ||∇(v − v)(tz)||L∞ dz.

In the following we deal with the difference (v−v). For this purpose, subtract the equationfor v from the one for v, and take the norm t

12 ||∇(·)||L∞ of both sides, to obtain

t12 ‖∇(v − v)(t)‖L∞ � t

12∥∥∇G(t)(v0 − v0)

∥∥L∞ + J (t), (4.25)

where J is handled in the following way:

J (t) � CJ t12

∫ t/2

0(t − s)−

12 − τr,λ

2 s− β

2 sβ

2 ‖(u − u)(s)‖r,λ ds

+ CJ t12

∫ t

t/2(t − s)−

12 − τq,λ

2 s− α2 s

α2 ‖(u − u)‖q,λ ds (4.26)

Using again the change of variable s = zt, the inequality (4.26) becomes

J (t) � CJ

∫ 1/2

0(1 − z)−

12 − τr,λ

2 z− β

2 (tz)β

2 ‖(u − u)(tz)‖r,λ dz (4.27)

+ CJ

∫ 1

1/2(1 − z)−

12 − τq,λ

2 z− α2 (tz)

α2 ‖(u − u)(tz)‖q,λ dz.

Now we estimate the norm ‖ · ‖q,λ of tα2 (u − u). We have that

tα2 ‖(u − u)(tz)‖q,λ � t

α2 ‖∇G(t)(u0 − u0)‖q,λ + �1(t) + �2(t), (4.28)

where �1(t) and �2(t) are bounded as follows:

�1(t) = C�1 tα2

∫ t/2

0(t − s)−

12 − 1

2 (τr,λ−τq,λ) ‖(u − u)(tz)‖r,λ ‖∇v(s)‖L∞ ds

+ C�1 tα2

∫ t/2

0(t − s)−

12 − 1

2 (τr,λ−τq,λ) ‖∇(v − v)(s)‖L∞ ‖u(s)‖r,λ ds

� C�1 tα2

∫ t/2

0(t − s)−

12 − 1

2 (τr,λ−τq,λ)s− 12 − β

2 sβ

2 ‖(u − u)(s)‖r,λ ds supt>0

t12 ‖∇v(t)‖L∞

+ C�1 tα2

∫ t/2

0(t − s)−

12 − 1

2 (τr,λ−τq,λ)s− 12 − β

2 s12 ‖∇(v − v)(s)‖L∞ ds sup

t>0t

β

2 ‖u(t)‖r,λ

� C�1ε(1 + 2τ)

∫ 1/2

0(1 − z)−

12 − 1

2 (τr,λ−τq,λ)z− 12 − β

2

((tz)

β

2 ‖(u − u)(tz)‖r,λ

)dz

+ C�1 2ε

∫ 1/2

0(1 − z)−

12 − 1

2 (τr,λ−τq,λ)z− 12 − β

2

((tz)

12 ‖∇(v − v)(tz)‖L∞

)dz, (4.29)

Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data 1447

because of (4.23) and the change s = tz. For the term �2(t) we have

�2(t) = C�2 tα2

∫ t

t/2(t − s)−

12(‖(u − u)(s)‖q,λ ‖∇v(s)‖L∞ + ‖∇(v − v)(s)‖L∞ ‖u(s)‖q,λ

)ds

� C�2 tα2

∫ t

t/2(t − s)−

12 s− 1

2 − α2(s

α2 ‖(u − u)(s)‖q,λ

)ds sup

t>0t

12 ‖∇v(t)‖L∞

+ C�2 tα2

∫ t

t/2(t − s)−

12 s− 1

2 − α2

(s

12 ‖∇(v − v)(s)‖L∞

)ds sup

t>0t

α2 ‖u(t)‖q,λ

� C�2ε(1 + 2τ)

∫ 1

1/2(1 − z)−

12 z− 1

2 − α2((tz)

α2 ‖(u − u)(tz)‖q,λ

)dz

+ C�2 2ε

∫ 1

1/2(1 − z)−

12 z− 1

2 − α2

((tz)

12 ‖∇(v − v)(tz)‖L∞

)dz. (4.30)

Now we define

A1 = lim supt→∞

tβ

2 ‖u(t) − u(t)‖r,λ , A2 = lim supt→∞

t12 ‖∇(v − v)(t)‖L∞

and

A3 = lim supt→∞

tα2 ‖u(t) − u(t)‖q,λ .

Note that A1, A2, A3 < ∞ from (4.23). We calculate the superior limit in (4.21), (4.25) and(4.28), and then use (4.24), (4.27) and (4.29)–(4.30), respectively, in order to obtain

A1 � 0 + ε(1 + 2τ) CI

∫ 1

0(1 − z)−

12 z− 1

2 − β

2 dz A1 + 2ε CI

∫ 1

0(1 − z)−

12 z− 1

2 − β

2 dzA2

� ε(1 + 2τ) K4 A1 + 2ε K4 A2, (4.31)

A2 � 0 + CJ

(∫ 1/2

0(1 − z)−

12 − τr,λ

2 z− β

2 dz

)A1 + CJ

(∫ 1

1/2(1 − z)−

12 − τq,λ

2 z− α2 dz

)A3

� CJ

(∫ 1/2

0(1 − z)−

12 − τr,λ

2 z− β

2 dz +∫ 1

1/2(1 − z)−

12 − τq,λ

2 z− α2 dz

)(A1 + A3)

= K2(A1 + A3), (4.32)

and

A3 � 0 + ε(1 + 2τ) C�1

∫ 1/2

0(1 − z)−

12 − 1

2 (τr,λ−τq,λ)z− 12 − β

2 dz A1

+ 2ε C�1

∫ 1/2

0(1 − z)−

12 − 1

2 (τr,λ−τq,λ)z− 12 − β

2 dz A2

+ ε(1 + 2τ) C�2

∫ 1

1/2(1 − z)−

12 z− 1

2 − α2 dz A3 + 2ε C�2

∫ 1

1/2(1 − z)−

12 z− 1

2 − α2 dz A2

= [ε(1 + 2τ) A1 + 2εA2] C�1

∫ 1/2

0(1 − z)−

12 − 1

2 (τr,λ−τq,λ)z− 12 − β

2 dz

+ [ε(1 + 2τ) A3 + 2εA2] C�2

∫ 1

1/2(1 − z)−

12 z− 1

2 − α2 dz

� K5 [ε(1 + 2τ) (A1 + A3) + 2εA2] . (4.33)

1448 L C F Ferreira and J C Precioso

From hypotheses of theorem 3.1, τ = K1 + K2 and K = K3 + K4 + K5. Adding (4.31) and(4.33), and inserting (4.32) within result we obtain

A1 + A3 � ε(1 + 2τ) K4 A1 + 2ε K4 A2 + K5 [ε(1 + 2τ) (A1 + A3) + 2εA2]

� Kε(1 + 2τ)(A1 + A3) + 2εKA2

� Kε(1 + 2τ)(A1 + A3) + 2εKτ(A1 + A3)

� Kε(1 + 4τ)(A1 + A3).

Since Kε(1 + 4τ) < 1, it follows that A1 = A3 = 0 and 0 � A2 � τ(A1 + A3) = 0.

It remains to prove the reciprocal statement, that is, (3.14) implies (3.13). Proceeding asin the proof of (4.21), (4.25) and (4.28), and using the hypothesis A1 = A2 = A3 = 0, oneobtains

limt→+∞ sup t

β

2 ‖G(t)(u0 − u0)‖r,λ � A1 + limt→+∞ sup I (t)

� A1 + ε(1 + 2τ) K4 A1 + 2ε K4 A2

= 0 + 0 + 0 = 0,

limt→+∞ sup t

α2 ‖G(t)(u0 − u0)‖q,λ � A3 + lim

t→+∞ sup (�1(t) + �2(t))

� A3 + K5 [ε(1 + 2τ) (A1 + A3) + 2εA2]

= 0 + 0 + 0 = 0,

and finally,

limt→+∞ sup t

12∥∥∇G(t)(v0 − v0)

∥∥L∞ � A2 + lim

t→+∞ sup J (t)

� A2 + K2(A1 + A3)

= 0 + 0 + 0 = 0.

Acknowledgments

L C F Ferreira was supported by FAPESP and CNPQ, Brazil. J C Precioso was supported byProject FAPESP 2007/51490-7, Brazil.

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