On a time-dependent bio-reactor model with chemotaxis

On a time-dependent bio-reactor model withchemotaxis

Le Dung

Division of Mathematics and Statistics, Department of Mathematics,

University of Texas at San Antonio, 6900 North Loop 1604 West, San Antonio, TX 78249, USA

Abstract

Long time dynamics of solutions to a strongly coupled system of parabolic equations

modeling the competition in bio-reactors with chemotaxis will be studied. If the pa-

rameters of the system are periodic, sufficient conditions for positive periodic solutions

will be derived in terms of parabolic eigenvalue problems.

� 2002 Elsevier Science Inc. All rights reserved.

Keywords: Cross-diffusion system; Global attractor; Periodic solutions; Index theory

1. Introduction

It is the purpose of this paper to study the existence of global attractors andthe existence of positive periodic solutions of a class of cross-diffusion systemswhich model the competition in bio-reactors with chemotactic effects. In par-ticular, on X ¼ ½0; 1�, let us consider the following time-dependent regular el-liptic operators

AiðtÞu ¼ �aiðx; tÞuxx þ biðx; tÞux þ ciðx; tÞu; i ¼ 0; . . . ;m;

and the parabolic system

Applied Mathematics and Computation 131 (2002) 531–558www.elsevier.com/locate/amc

E-mail address: [email protected] (L. Dung).

0096-3003/02/$ - see front matter � 2002 Elsevier Science Inc. All rights reserved.

PII: S0096 -3003 (01 )00168-0

mail to: [email protected]

ou0ot

þ A0ðtÞu0 ¼ f0ðx; t;~uuÞ; x 2 X;

ouiot

þ AiðtÞui þ ðuiUiðu0Þðu0ÞxÞx ¼ fiðx; t;~uuÞ; x 2 X ¼ ð0; 1Þ;

B0ðtÞu0ðx; tÞ ¼ Sðx; tÞ; BiðtÞuiðx; tÞ ¼ 0; x 2 oX ¼ f0; 1g;uiðx; 0Þ ¼ ui0ðxÞ; x 2 ð0; 1Þ;

ð1:1Þ

where ~uu ¼ ðu0; u1; . . . ; umÞ and the initial data ui0 are functions in H 1ð0; 1Þ. Theboundary conditions are also time-dependent and given by the operators

B0ðtÞu0 ¼ou0on

þ r0ðx; tÞu0; BiðtÞui ¼ouion

þ ri x; t;ou0on

� �ui;

which can be explicitly written as

� ðu0Þxð0; tÞ þ a0ðtÞu0ð0; tÞ ¼ S0ðtÞ; ðu0Þxð1; tÞ þ a1ðtÞu0ð1; tÞ ¼ S1ðtÞ;� ðuiÞxð0; tÞ þ bi0ðt; ðu0Þxð0; tÞÞuið0; tÞ ¼ 0;

ðuiÞxð1; tÞ þ bi1ðt; ðu0Þxð1; tÞÞuið1; tÞ ¼ 0: ð1:2ÞIn this form, (1.1) models the competition in a unstirred bio-reactor with u0representing the concentration of certain nutrient and ui the densities of thespecies of cells (or bacteria). In addition to the terms uxx; ux in the operator Aireflecting, respectively, the random diffusive flux and the convection effect in themodel, the term ðuiUiðu0Þðu0ÞxÞx reflects the chemotactic flux response of eachspecies to the presence of the nutrient u0. The function Ui, the so-called sensi-tivity rate, is included so that the sensitivity of cells to the nutrient may vary atdifferent level of nutrient concentration. In literature, Ui is usually considered tobe of constant sign so that the nutrient is assumed to be either attracting (ifUiðu0Þ > 0) or repelling (if Uiðu0Þ < 0). In this work, such restrictions will not beimposed so that results will cover a broader range of possibilities for chemo-tactical responses in applications. For example, it may happen that the nutrientis nutrious (thus, attracting) if its concentration is lower than certain thresholdlevel but becomes toxic (thus, repelling) if its concentration is too high. In fact,our assumption on the sensitivity rates Ui merely requires that they be contin-uously differentiable functions (see Condition (F.3)).

From a mathematical point of view, (1.1) is a strongly coupled system withthe couplings not only occurring in the reaction terms fi but also in the cross-diffusion terms ðuiUðu0Þðu0ÞxÞx, and even in the boundary condition (1.2). Thismakes the treatment completely different from (and more difficult than) thatfor reaction–diffusion systems without cross-diffusion terms. Since the pio-neering work of Keller and Segel [23], in contrast to a huge literature on re-action–diffusion problems, there have been only a few rigorous mathematicalstudies of (1.1). Here, we note the works of Schaff [28], and Lin et al. [26] (onsteady states), J€aager and Luckhaus [22], and Herrero and Vel�aasquez [20] (onfinite time blowup of solutions of a parabolic–elliptic system). Redlinger [27]

532 L. Dung / Appl. Math. Comput. 131 (2002) 531–558

proved the existence of the global attractor for a cross-diffusion system of twoequations with time-independent coefficients and no couplings in the boundaryconditions. Recently, Wang [30] made use of bifurcation techniques to studythe existence of positive steady states of a single species and time-independent(1.1) (m ¼ 1) and also obtained a result on global existence of time-dependentsolutions. In our earlier works [8,13], a homotopy technique for computing theindex of positive fixed points had been applied to (1.1) (mP 1 and the spatialdimension can be greater than 1) to derive sufficient conditions for the positivecoexistence steady states. The models in [8,13,30] follow the pioneering worksby Lauffenburger and coworkers [5–7] for the case of confined growth in atubular reactor supplied with a single diffusible growth-limiting nutrient en-tering at one boundary of the tube. The (scaled) model system for the nutrientconcentration S and bacterial populations ui is given by:

oSot

¼ Sxx � f1ðSÞu1 � f2ðSÞu2;

ouiot

¼ diðuiÞxx � dio

oxUiðSÞui

oSox

� �þ ½fiðSÞ � ki�ui

ð1:3Þ

with boundary conditions:

oSox

ð0; tÞ ¼ 0; Sð1; tÞ ¼ 1;

ouiox

ð0; tÞ ¼ 0 ¼ ouiox

ð1; tÞ � uið1; tÞUiðSð1; tÞÞoSox

ð1; tÞ;

and appropriate initial conditions. The authors assume that the chemotacticsensitivity U follows the receptor law:

UiðSÞ ¼ai

ðai þ SÞ2;

where a is a positive constant. Many different forms have been used in theliterature including constant U ¼ a and the log law, U ¼ a=ðaþ SÞ. Thefunctions fiðSÞ represent the functional response of the ith organism to nutrientconcentration S and typically are bounded functions satisfying fið0Þ ¼ 0,f 0i > 0. The constants ki are cell death rates.

It can be seen that (1.1) is the general form of the above system. The globalexistence result for (1.3) has been solved in [30] following a Moser–Alikakositeration technique to estimate the L1 norm of solutions and then apply thegeneral theory of strongly coupled parabolic systems by Amann [2–4]. In fact,the system (1.1) is normal parabolic and triangular in the terminology of [3];therefore [3, Theorem 5.2] asserts that one needs only to control the L1 normof the solution to obtain the global existence.

In Section 2, we will improve the global existence result of Wang in showingfurther that the solutions of the time-dependent (1.1) are in fact ultimatelyuniformly bounded and that the dynamical system of (autonomous) (1.1)

L. Dung / Appl. Math. Comput. 131 (2002) 531–558 533

possesses a global attractor in the space X ¼Qm

0 H1ð0; 1Þ. To this end, certain

compactness properties need to be established and we will show that highernorms, such asCm normwith m > 1, of the solution can be controlled.We are ableto obtain such estimates by using just standard techniques in the theory ofparabolic equations. Furthermore, the proof shows that one can get global ex-istence and ultimate boundedness of solutions if one can control weaker normsof solutions such as Lp norms for p > 1 but finite. The proof of these results arequite technical so that we will streamline only the main ideas and postpone thedetails to Section 4 for the convenience of the reader. An immediate consequenceof the obtained estimates is the existence of a global attractor for the dynamicalsystem associates (1.1) if the system is autonomous. We should remark that thestructure of this global attractor is still in general a widely open problem. Partialresults in this direction will be presented in a forthcoming paper where, underfurther assumptions on (1.1), we will show that the global attractor will consistof only one point, the steady-state solution (see also [14]).

On the other hand, in Section 3 where the parameters of (1.1) are assumed tobe periodic, the uniform estimates just derived allow us to employ the indextechnique of [8] to study the existence of periodic solutions. Here we will givesufficient conditions in terms of principal eigenvalues of certain parabolicequations for the existence of nontrivial periodic solutions of (1.1). Again, theuniqueness question is still generally unsolved.

2. Global existence and global attractor

We first consider the problem of global existence and ultimate boundednessof solutions. To achieve this, we will impose the following assumptions on thecoefficients and initial data of (1.1).

A.1. a0; b0; c0 2 C1ðð0; 1Þ Rþ;RÞ and ai; Si 2 C1ðRþ;RþÞ. Moreover, thenorms of these functions are uniformly bounded. That is,

ka0; b0; c0kC1 ; kðaiÞtk1; kðSiÞtk1 6C: ð2:1Þ

A.2. For i ¼ 1; . . . ;m, there exist constants Di; di such that

0 < di6 aiðx; tÞ6Di; ðx; tÞ 2 ½0; 1� Rþ: ð2:2ÞMoreover, ai is differentiable with respect to x and

kðaiÞx; bi; cik1 6C: ð2:3Þ

A.3. The functions Ui : R ! R are continuously differentiable and there existsa continuous function U : Rþ ! Rþ such that

jUiðxÞj; jU0iðxÞj6UðjxjÞ 8x 2 R; i ¼ 1; . . . ;m:

In fact, the assumption on the differentiability of the coefficients has beenimposed here because we want to include time-dependent boundary conditions


into our consideration. If this is not the case then one can see that certainH€oolder regularity would be sufficient for our argument.

Next, we allow the following growth conditions on the nonlinearities fi.

F.1. For ~uu ¼ ðu0; . . . ; umÞ, there exist some nonnegative continuous functionCð�Þ and r 2 ð1; 3Þ such that

jf0ðx; t;~uuÞj6Cðu0ÞXmi¼1

juij þ Cðu0Þ; ð2:4Þ

jfiðx; t;~uuÞj6Cðu0ÞXmi¼1

juijr þ Cðu0Þ; i ¼ 1; . . . ;m: ð2:5Þ

To deal with time-dependent boundary conditions, we will impose the fol-lowing regularity condition on (1.2).

BC.1. For i ¼ 1; . . . ;m, bi0ðt; pÞ;bi1ðt; pÞ are differentiable with respect to t; p,

and

obi

otðt; pÞ

�� and

obi

opðt; pÞ

��

are uniformly bounded in t and locally uniformly bounded in p.Assuming only the above conditions on (1.1) we will prove the main tech-

nical result of this paper (Theorem 2) which asserts that if the L1 norm of thecomponent u0 and L1 norm of other components are a priori ultimately uni-formly bounded (in the sense of the definition below) then, for some m > 1, theCm norm of the solution of the coupled system (1.1) also satisfies a similarestimate. We should remark that the assumption on the boundedness of u0 doesnot make the proof any easier since the coupling in the ith equation, i > 0,involves with the derivative of u0 which needs to be estimated appropriately.

To prove the ultimately uniform boundedness, it is convenient to introducethe following class of functions.

Definition 1. Consider the initial-boundary problem (1.1). Assume a prioritythat there exists a solution ~uu defined on a maximal existence interval I of Rþ.

A function xðtÞ defined on I is said to be in the class P~uu if for any compactsubset K � I there exists a positive constants C0, which may depend on theparameters of the system, the set K, and the H 1 norm of the initial value~uuð�; 0Þin general, such that

xðtÞ6C0 8t 2 K: ð2:6Þ

Moreover, if I ¼ ð0;1Þ then there exists a positive constant C1 dependingonly on the parameters of the system but not the initial value of ~uu such that

lim supt!1

xðtÞ6C1: ð2:7Þ


If x 2 P~uu and I ¼ ð0;1Þ, we also say that x is ultimately uniformly bounded.

If the solution~uu is already understood we will simply write P instead of P~uu.The main technical result of this paper is the following a priori estimates.

Theorem 2. Let~uu ¼ ðu0; . . . ; umÞ be a solution to (1.1) with its maximal existenceinterval I. Suppose that ku0ð�; tÞk1; kuið�; tÞk1 2 P for all i ¼ 1; . . . ;m. Thenthere exists m > 1 such that kuið�; tÞkCmð0;1Þ 2 P.That is, for any compact K � I , there exists a constant CðK; kuð�; 0ÞkH1ð0;1ÞÞ

such that

kuið�; tÞkCmð0;1Þ 6CðK; kuð�; 0ÞkH1ð0;1ÞÞ 8t 2 K: ð2:8Þ

This implies that the solution exists globally (I ¼ Rþ). Moreover, we can find aconstant C1 which does not depend on the initial data such that

lim supt!1

kuið�; tÞkCmð0;1Þ 6C1: ð2:9Þ

When blow-up results for chemotaxis systems of Keller–Segel [22,23] are re-called, it should be mentioned that Theorem 2 does not hold for these classicalevolution chemotaxis models since blow-up may occur with the L1-norm ofsolution conserved.

The proof of this theorem will be based on several lemmas which will bepresented in Section 4. We will outline only the main points of the proof here.Investigating the equation for u0 easily reveals that the derivative of u0 can becontrolled if one can estimate kuikp, the Lp norm of ui, for finite p > 1 andsufficiently large (Lemma 4.4). We then seek for Lp estimates for ui (Lemma4.5). In fact, this may be the most crucial step in our proof of Theorem 2.However, due to the coupling with u0 in the equation for ui and the estimate ofLemma 4.4, the standard bootstrapping technique in Lemma 4.5 leads us to anintegro-differential inequality for the kuikp. The form of inequality suggeststhat we study inequalities of the form

y0ðtÞ6 f ðt; yÞ; t 2 ð0;1Þ;

where f ðt; yÞ is a functional defined on Rþ CðRþ;RÞ. We then prove twoauxiliary results, which are interesting in themselves, Lemma A.1 and Propo-sition A.1, that assert the global boundedness and ultimately uniformboundedness for functions satisfying the above inequality.

We would like to emphasize that the assumption on the dimension of thedomain (n ¼ 1) is solely due to this technical difficulty in obtaining the Lp es-timate in Lemma 4.5. Almost all of our results and proofs hold true for (1.1)given on a bounded open subset of Rn for any dimension nP 1.

Having established the Lp estimates for ui, standard results for linear par-abolic equations will be used to give further estimates on u0 and its derivatives.


This regularity information is very important when we make use of the changeof variables in the equation for ui to reduce the Robin boundary condition(keeping in mind that it is coupled with ðu0Þx and time-dependent) to a ho-mogeneous Neumann condition. Once this is done, Theorem 2 follows fromstandard theory of linear parabolic equations.

Our proof in Section 4 is different from those in [27,30]. In [27], theboundary conditions are independent of time and not coupled. Furthermore, itwas assumed in [27] that the reaction term fi play a strong damping role, whileit is a source term in our problem. In [30], the setting is more related to oursand a rather complicated iteration technique following Moser–Alikakos isemployed to obtain global estimate for L1 norm of ui (in fact, by a similar useof this method, one can get stronger conclusion on the uniform estimates as in[11,12]. However, the bounds in [30] depend on the norms of the initial valuesso that only global existence results can be deduced from the general theory of[2–4]. We also mention here the work [10] where a Cm regularity theory forbounded solutions to a class of strongly coupled parabolic systems is studied.Here, we are able to obtain estimates for stronger norm (Cm with m > 1) andshow that they are ultimately uniformly bounded, and thus improve the globalexistence result of [30] in Corollary 4.

We next show that the boundedness assumption in Theorem 2 can be ver-ified if we assume further the following which are verified trivially by (1.3).

F.2. f0ðx; t;~uuÞ6 0 for any ðx; tÞ 2 ½0; 1� Rþ and ~uu 2 Rmþ. We assume thatfiðx; t;~uuÞ vanishes when ui ¼ 0. Moreover, f0ðx; t;~uuÞ ¼ 0 if ui ¼ 0 for all i 6¼ 0.

Also, there exist positive constants k; hi; h such that

Xmi¼0

hifiðx; t;~uuÞ6 kXmi¼1

hiui þ Cðu0Þ; ð2:10Þ

and

c0 � ðb0Þx þ ða0Þxx; ci � ðbiÞx þ ðaiÞxx � kP h > 0

8i ¼ 1; . . . ;m: ð2:11Þ

Regarding the boundary conditions we assume further that:

BC.2. For j ¼ 0; 1, set aðj; tÞ ¼ ajðtÞ,

bi j; t;ou0on

� �¼ bijðt; ðu0Þxðj; tÞÞ:

At the boundary x 2 oX ¼ f0; 1g, t > 0, we assume

b0ðx; tÞ � ða0Þxðx; tÞ þ a0ðx; tÞaðx; tÞP 0;

biðx; tÞ � ðaiÞxðx; tÞ þ aiðx; tÞbi x; t;

ou0on

� �þ Uiðu0Þ

ou0on

P 0; 16 i6m:


The proof of the following L1 estimates will be given in Section 4.

Lemma 2.1. Assume that the initial data ui0 are positive. Then uiðx; tÞ > 0 for allt 2 I and i ¼ 0; . . . ;m. Moreover, for some x 2 P, we have

ku0ð�; tÞk1 6xðtÞ 8t 2 I ; ð2:12Þkuið�; tÞk1 6xðtÞ 8t 2 I : ð2:13Þ

It is easy to check that all of our hypotheses hold trivially for system (1.3) con-sidered in [5–7,30]. We are now ready to state the main result of this section.

Theorem 3. Assume that (A.1)–(A.3), (F.1), (F.2), (BC.1) and (BC.2) hold forsystem (1.1). Let X ¼ �mi¼0H

1ð0; 1Þ and Xþ be the positive cone of X. We assertthat

(i) For any given~uu0 ¼ ðu00; . . . ; um0 Þ 2 Xþ, there exists a unique maximal solution~uuðx; tÞ ¼ ðu0ðx; tÞ; . . . ; umðx; tÞÞ defined on ½0; T~uu0Þ for some T~uu0 > 0.

(ii) The above solution exists globally and stays nonnegative. That is, T~uu0 ¼ 1for all ~uu0 2 Xþ and uiðx; tÞP 0 for ðx; tÞ 2 ½0; 1� Rþ. Furthermore, forany small s > 0 and any 06 q6 r6 1, ~uu 2 Cqð½s;1Þ;�mi¼0C

2ð1�rÞ½0; 1�Þ.(iii) Let /ðt;~uu0Þ be the unique solution described above. Then /ðt; �Þ defines a

C0;1 semiflow on Xþ. For any t > 0, /ðt; �Þ is compact.(iv) There exists R > 0 such that, for any bounded W � Xþ, there is finite

T ðW Þ > 0 such that /ðt;W Þ is precompact and /ðt;W Þ � BXþðRÞ, the ballof radius R in Xþ, for all t > T ðW Þ.

Proof. It is easy to see that (1.1) is ‘‘normally parabolic’’ and a ‘‘triangularsystem’’ in the sense of [3,4]. Hence (i) follows from [4, Theorems 7.3 and 9.3].By Lemma 2.1 and Theorem 2, ~uuð�; tÞ stays positive if ~uu0 is given in Xþ andk~uuð�; tÞkX is bounded for all t 2 ½0; T~uu0Þ so that (ii) follows from [3, Theorem5.2]. (iii) is a consequence of the main Theorem of [4, p. 17]. Finally, recallingthe definition of functions of class P and the compactness of the imbeddingCm,!H 1 (m > 1), (iv) is already proved in Theorem 2. �

If (1.1) is autonomous, that is when its parameters are time-independent, wehave the following immediate consequence of (iv) of the above theorem (see[17]).

Corollary 4. Assume that (1.1) is autonomous and the conditions of Theorem 3are satisfied. Then the dynamical system associated with (1.1) possesses a globalattractor in Xþ.

We remark here that the above sharpens a result in [30] where only globalexistence result was proven.


3. Existence of positive periodic solutions

Having established a priori uniform estimates, we are now able to applyindex techniques to study the existence of positive periodic solutions if theparameters of (1.1) are periodic for some fixed period T. In addition to (F.1),(F.2), (BC.1) and (BC.2), the conditions that guarantee the H 1 estimates forsolutions, we will assume further that:

A.4. The coefficients of (1.1), aiðx; �Þ; biðx; �Þ; ciðx; �Þ; ai; bi; Si and fi, are periodicin t for some positive period T.

F.3. fi is continuous in its arguments. For i > 0, fiðx; t;~uuÞP 0 and fi has con-tinuous partial derivatives ofi=oui > 0.

Let

L0u ¼ouot

þ A0ðtÞu; Liðu0Þu ¼ouot

þ AiðtÞuþ ðuUiðu0Þðu0ÞxÞx;

i > 0: ð3:1Þ

Given w ¼ ðw0; . . . ;wmÞ in Xþ, we consider the Poincar�ee map or T-time mapSðwÞ ¼~uuð�; T Þ where ~uu ¼ ðu0; . . . ; umÞ is the solution of

L0u0 ¼ f0ðx; t;~uuÞ; Liðu0Þui ¼ fiðx; t;~uuÞ; 0 < x < 1; t > 0; i > 0;

B0ðtÞu0 ¼ S0ðtÞ;BiðtÞui ¼ 0; x 2 f0; 1g; t > 0; i > 0;

uiðx; 0Þ ¼ wiðxÞ; 0 < x < 1:

ð3:2Þ

We set up a fixed point equation in the positive cone Xþ of X :¼Qmþ1

0 H 1ð0; 1Þfor S and use the index techniques in [8] to derive sufficient conditions for theexistence of periodic solutions of (3.2). That is,

L0u0 ¼ f0ðx; t;~uuÞ; Liðu0Þui ¼ fiðx; t;~uuÞ; 0 < x < 1; t > 0; i > 0;

B0ðtÞu0 ¼ S0ðtÞ; BiðtÞui ¼ 0; x 2 f0; 1g; t > 0; i > 0;

uiðx; T Þ ¼ uiðx; 0Þ; 0 < x < 1:

ð3:3Þ

Thanks to Theorem 3 and (A.4), the following result is standard.

Lemma 3.1. In addition to the assumptions of Theorem 3, we assume further(A.4) and (F.3). Then S : Xþ ! Xþ is a well-defined completely continuous op-erator. Moreover, fixed points of S in Xþ give rise to positive periodic solutions of(3.3).

We also have the following result which is an immediate consequence of theultimate uniform boundedness (iv) of Theorem 3 and [17, Theorem 2.4.7].


Corollary 5. The discrete dynamical system generated by S possesses a globalattractor in Xþ.

Next, we will compute the fixed point index of the map S. The a priori es-timates derived in Theorem 2 now play a crucial role. To proceed, we considerthe Poincar�ee maps Sk, k 2 ½0; 1�, associate with the following family of para-bolic systems:

L0u0 ¼ f0ðx; t; k~uuÞ; LðkÞi ðu0Þui ¼ fiðx; t; k~uuÞ; x 2 ð0; 1Þ; t > 0; i > 0;

B0ðtÞu0 ¼ kS0; BiðtÞui ¼ 0 x 2 f0; 1g; t > 0; i > 0;

uiðx; 0Þ ¼ wiðxÞ; x 2 ð0; 1Þ;ð3:4Þ

where

LðkÞi ðu0Þu ¼ouot

þ AiðtÞuþ kðuUiðu0Þðu0ÞxÞx if i > 0: ð3:5Þ

Lemma 3.2. Let the assumptions of Lemma 3.1 hold. Then there is an R > 0independent of k such that

SkðUÞ ¼ U ; k 2 ½0; 1� ð3:6Þ

has no solution U 2 Xþ satisfying kUkX ¼ R.

Proof. Define fk and Uk for k 2 ½0; 1� by fk ¼ ðff0; . . . ; ffmÞ and UðkÞi ¼ kUi where

ffiðx; t;~uuÞ ¼ fiðx; t; k~uuÞ, 06 i6m. Then it is easy to check that if f satisfies (F.1)and (F.2) of Section 4, which we are assuming, then f and fk, with k 2 ½0; 1�,also satisfy these assumptions with a common set of constants hi; k; h in (F.2),which are independent of k 2 ½0; 1�. Similarly UðkÞ

i verifies (A.3). We see that(3.4) satisfies all the hypotheses, with the same set of constants, of Theorem 3.By (iv) of that theorem there exists a function x 2 P such that k~uuð�; tÞkX 6xðtÞfor any solution~uu of (3.4). If (3.6) holds and~uuðx; tÞ is the solution of (3.4) withw ¼ U then

kUkX ¼ kSkðUÞkX ¼ kSnkðUÞkX ¼ k~uuð�; nT ÞkX 6xðnT Þ 8n ¼ 1; 2; . . .

Since lim supt!1 xðtÞ6C1 for some constant C1 independent of the initialdata U and k, we can take R ¼ C1 þ 1 to complete the proof. �

The above result allows us to compute the index of the map S as follows.

Theorem 6. For r > 0, let Cr ¼ fu 2 Xþ : kuk < rg. With R given by Lemma3.2, we have

indðS;CRÞ ¼ þ1:

In particular, there is a fixed point of S in CR.


Proof.We consider the family of maps SkðUÞ defined by (3.4). fSkgkP 0 is a well-defined homotopy in CR thanks to Lemma 3.2. Hence,

indðS;CRÞ ¼ indðS0;CRÞ:

However, thanks to (F.3), S0 is just the Poincar�ee map of the linear system

ouiot

þ AiðtÞui ¼ 0; x 2 ð0; 1Þ; t > 0;

BiðtÞui ¼ 0; x 2 f0; 1g; t > 0;

uiðx; 0Þ ¼ wiðxÞ; x 2 ð0; 1Þ:

We then consider the homotopy I � kS0, k 2 ½0; 1�, which is well defined in CR.Indeed, if U ¼ ðu0; . . . ; umÞ satisfies kUkX ¼ R and U � kS0ðUÞ ¼ 0 for somek 2 ½0; 1� then ui > 0, for some i, and k > 0, k�1 P 1. But this implies that theprincipal eigenvalue of the Poincar�ee map corresponding to the ith equation isgreater than 1, a contradiction to [21, Lemma 14.2]. By the invariance of thedegree we conclude that indðS0;CRÞ ¼ 1 and complete our proof. �

The existence result in the above theorem is not interesting since the exis-tence of the global attractor in Corollary 5 and the asymptotic fixed pointtheorems (see [17, Section 2.6]) imply that there is at least one fixed point of Sin the global attractor. In fact, by similar argument to that of [29, Prop. 2.1](using the change of variables as in Lemma 4.4) we can show that there is aunique positive periodic solution S�ðx; tÞ of

L0S� ¼ 0; ðx; tÞ 2 ð0; 1Þ Rþ; B0S� ¼ S0ðtÞ; ðx; tÞ 2 f0; 1g Rþ:

ð3:7Þ

We observe that~uu� ¼ ðS�; 0; . . . ; 0Þ is a solution to (3.3) by virtue of (F.2). Werefer to it as ‘‘washout’’ periodic solution and w� ¼ ðS�ðx; 0Þ; 0; . . . ; 0Þ as thewashout fixed point of S. In the sequel we will study the existence of fixedpoints other than w�.

The abstract result in [8, Section 3.1] will be applied here to derive sufficientconditions for such solutions to exist. However, the fixed point problem for S isnot yet of the form considered in [8, Section 3.1] in order that the results can beapplied directly. Nevertheless, we will follow the same ideas to decouple thesystem.

We set M ¼ f0; . . . ;mg. For any subset b ofM we denote the set of positivefixed points of S on the ‘‘face’’ Xb ¼ �i2bH 1ð0; 1Þ by

Z0b ¼ f~ww ¼ ðw0; . . . ;wmÞ 2 X : Sð~wwÞ ¼ ~ww; wi > 0 if i 2 b; wi � 0 if i 2 bg:

Accordingly, the set of positive periodic solutions of (3.3) with b-componentpositive is denoted by


Zb ¼ f~uu 2 C1ðRþ;X Þ: ~uu solves ð3:3Þ; ui > 0 if i 2 b; ui � 0 if i 62 bg:

Due to the nonhomogeneity of the boundary condition for u0 we see that anysolution of (3.3) must satisfy u0 6¼ 0 so that Zb ¼ ; if either 0 62 b or b ¼ ;. IfZb 6¼ ; and b 6¼ M , we set a ¼ M n b and decompose X as X ¼ Xa � Xb and itselements as ~uu ¼ ðua; ubÞ. We also write, in an obvious manner, S in the formSð~uuÞ ¼ ðSaðua; ubÞ; Sbðua; ubÞÞ.

For each ~uu ¼ ðu0; . . . ; umÞ 2 Zb, let us consider the parabolic eigenvalueproblem

o/iot

þ AiðtÞ/i þ ð/iUiðu0Þðu0ÞxÞx ¼ /iofioui

ðx; t;~uuðx; tÞÞ þ l/i;

x 2 ð0; 1Þ; t > 0;

Bi/i ¼ 0; x 2 f0; 1g; t > 0;

/iðx; T Þ ¼ /iðx; 0Þ x 2 ð0; 1Þ:

ð3:8Þ

We assume the following:

BP. For b � M with 0 2 b, if Zb 6¼ ; then for any ~uu 2 Zb the principal eigen-value of (3.8) is nonzero for every i 2 a ¼ M nb.

As we remarked before, it is necessary that 0 2 b for Zb 6¼ ; so that 0 62 aand the 0th equation is not included in the above condition. We first show thatthe above condition implies some sort of isolatedness of Z0

b (compare with [8,Lemma 3.4]).

Lemma 3.3. If (BP) holds for some b � M then there exists r > 0 such that S hasno fixed point ðwa;wbÞ with kwakXa

6 r and wa > 0.

Proof. If the claim is not true then we can find a sequence fwðnÞg,wðnÞ ¼ ðwðnÞ

a ;wðnÞb Þ, of fixed points of S such that kwðnÞ

a kXa! 0 as n! 1. Let

~uuðnÞ ¼ ðuðnÞa ; uðnÞb Þ be the periodic solution of (3.3) corresponding to the initialdata wðnÞ. We then define wðnÞ

a ¼ uðnÞa =kwðnÞa kXa

and note that

L0uðnÞ0 ¼ f0ðx; t; ðuðnÞa ; uðnÞb ÞÞ; LiðuðnÞ0 ÞuðnÞi ¼ fiðx; t; ðuðnÞa ; uðnÞb ÞÞ; i 2 b;

ð3:9Þ

LiðuðnÞ0 ÞwðnÞi ¼ fiðx; t;~uu

ðnÞÞkwðnÞ

a kXa

¼Z 1

0

ofioua

ðx; t; ðsuðnÞa ; uðnÞb ÞÞwðnÞa ds; i 2 a ð3:10Þ

and BiwðnÞi ¼ 0. By a compactness and uniqueness argument we easily show that

uðnÞa converges to 0 in Xa. Similarly, uðnÞb converges to some ub with ð0; ubÞ 2 Zb,and wðnÞ

a converges to some T-periodic vector function wa 6¼ 0. From (3.10), weget


Liðu0Þwi ¼ wiofioui

ðx; t; ð0; ubÞÞ; Biwi ¼ 0; i 2 a:

Here we have used a consequence of (F.2) that ofi=oujðx; t;~uuÞ ¼ 0 if j 6¼ i and~uu ¼ ðu0; . . . ; umÞ with uj ¼ 0. Since wa 6¼ 0, wi > 0 for some i 2 a. By theuniqueness of positive eigenfunction for scalar parabolic problems [21], theabove equation contradicts (BP). �

Next, we employ a similar decoupling technique as in [8, Prop. 7] to computethe ‘‘face’’ index of S. For b � M as above, we consider the Poincar�ee map �SSassociated with the following parabolic system

L0u0 ¼ f0ðx; t; ð0; ubÞÞ; Liðu0Þui ¼ fiðx; t; ð0; ubÞÞ; i 2 b;

B0u0 ¼ S0ðtÞ; Biui ¼ 0; i 2 b;

Ljðu0Þuj ¼ ujofjouj

ðx; t; ð0; ubÞÞ; j 2 a; ð3:11Þ

Bjuj ¼ 0; j 2 a;

uaðx; 0Þ ¼ waðxÞ; ubðx; 0Þ ¼ wbðxÞ; x 2 ð0; 1Þ:

Lemma 3.4. Let r > 0 be as in Lemma 3.3 and Ub be an open neighborhood of Z0b

in Xb and CaðrÞ ¼ fua 2 Xa: ua P 0; kuakXa¼ rg. We assert that

indðS;CaðrÞ � UbÞ ¼ indð�SS;CaðrÞ � UbÞ: ð3:12Þ

Proof. For k 2 ½0; 1�, we consider the family of Poincar�ee maps Sk associatedwith the parabolic systems (see Remark 3.5 following the proof)

L0u0 ¼ f0ðx; t; ðkua; ubÞÞ; Liðu0Þui ¼ fiðx; t; ðkua; ubÞÞ; i 2 b;

Ljðu0Þuj ¼Z 1

0

ofjoua

ðx; t; ðksua; ubÞÞua ds; j 2 a;

uaðx; 0Þ ¼ waðxÞ; ubðx; 0Þ ¼ wbðxÞ; x 2 ð0; 1Þ

ð3:13Þ

with the same boundary conditions of (3.11). It is clear that S0 � �SS. Moreover,since

fjðx; t; ðkua; ubÞÞ ¼ kZ 1

0

ofjoua

ðx; t; ðksua; ubÞÞua ds; ð3:14Þ

we see that S1 is exactly S. Hence, to prove (3.12), we need only to show that Sk

gives rise to a well defined homotopy on E ¼ CaðrÞ � Ub.If there exists k 2 ð0; 1� such that Sk has a fixed point ðwa;wbÞ 2 oE, the

boundary of E relative to Xþ, then wa > 0 since otherwise wb 2 Z0b; but then

ð0;wbÞ 62 oE because wb 62 oUb. Thus, wa > 0. However, by using (3.14) andmultiplying the a-th equations in (3.13) by k, we can see that ðkwa;wbÞ is a fixedpoint of S with kkwak6 r, a contradiction to Lemma 3.3.


On the other hand, if S0 has a fixed point ðwa;wbÞ 2 oE then wb 2 Z0b and

wa > 0. However, by (3.13), we have that


ðx; t; ð0; ubÞÞ; ujðx; T Þ ¼ ujðx; 0Þ ¼ wjðxÞ; j 2 a

with uj > 0 for some j 2 a, a contradiction to (BP). Hence, the homotopy(3.13) is well defined on E. Our proof is complete. �

Remark 3.5. In the above proof, we have implicitly used the fact that (3.13) hasglobal solutions so that the Poicar�ee maps are well defined. By (3.14) one can seethat if ðua; ubÞ is a solution of (3.13) then ðkua; ubÞ is a solution of (3.2). As wealready showed (Theorem 3), the latter exists globally and thus (3.13) hasglobal solutions.

We observe that (3.11), the system that defines �SS, is already decoupled. Thatis, for any given ðwa;wbÞ 2 Xþ, we can solve the b-components of �SS ¼ ð�SSa; �SSbÞfrom the subsystem

L0u0 ¼ f0ðx; t; ð0; ubÞÞ; Liðu0Þui ¼ fiðx; t; ð0; ubÞÞ; i 2 b;

B0u0 ¼ S0; Biui ¼ 0; i 2 b;

ubðx; 0Þ ¼ wbðxÞ; x 2 ð0; 1Þ;ð3:15Þ

and then substitute the result ubðx; tÞ, which depends solely on wb (and existsglobally, see Remark 3.5), into the rest of system (3.11)


ðx; t; ð0; ubðx; tÞÞÞ; j 2 a;

Bjuj ¼ 0; j 2 a;

uaðx; 0Þ ¼ waðxÞ; x 2 ð0; 1Þ:

ð3:16Þ

We then solve this linear parabolic system to obtain the a-components of �SS. Infact, for the decompositions �SS ¼ ð�SSa; �SSbÞ and S ¼ ðSa; SbÞ, (3.15) is obtained bysetting ui;wi, i 2 a, to be zero in (3.3), the system that defines S. Hence, it isclear that we can write �SSbðwÞ ¼ Sbð0;wbÞ.

On the other hand, since (3.16) is linear in uj, j 2 a, the Poincar�ee map BaðwbÞof (3.16) is also linear (although it is nonlinear in wb). Moreover, because Ljðu0Þand the right-hand side of (3.16) depend on wb (via ub) we can write �SSaðwa;wbÞ ¼ BaðwbÞwa. Thus,

�SSðwÞ ¼ �SSðwa;wbÞ ¼ ðBaðwbÞwa; Sð0;wbÞÞ: ð3:17Þ

By (iv) of Theorem 3, the above maps are completely continuous. Furthermore,concerning the eigenvalue of Ba, we have:


Lemma 3.6. If K is an eigenvalue of BaðwbÞ corresponding to a positive eigen-vector / 2 Xa then at least one of the components /j of / is positive and satisfies

Ljðu0Þ/j ¼ /jofjouj

ðx; t; ð0; ubðx; tÞÞÞ þ l/j; x 2 ð0; 1Þ; t > 0;

Bj/j ¼ 0; j 2 a;

/jðx; 0Þ ¼ /jðx; T Þ; x 2 ð0; 1Þ;

ð3:18Þ

where l ¼ �1T logK.

The first assertion is obvious and the relation between K and l is well known(see, e.g., [21, Prop. 14.4]).

If ub is periodic and solves (3.15) then it is clear that ð0; ubÞ 2 Zb. Thecondition (BP) then states that l 6¼ 0 so that K 6¼ 1. The form of �SS as in (3.17)and the above discussion allow us to apply [8, Prop. 8] to compute the index of�SS as follows.

Proposition 7. Assume (BP). Let Ub and CaðrÞ be as in Lemma 3.4. Then theory

(i) indð�SS;CaðrÞ � UbÞ ¼ indðSjUb;UbÞ if for any ð0;wbÞ 2 Z0

b, BaðwbÞ has no pos-itive eigenvector corresponding to an eigenvalue greater than one.

(ii) indð�SS;CaðrÞ � UbÞ ¼ 0 if there exists an element pa 2 Xa such that for anyð0;wbÞ 2 Z0

b, we have

v 6¼ BaðwbÞvþ tpa 8v 2 oCaðrÞ 8t > 0: ð3:19Þ

Remark 3.7. As we remarked in [8, Prop. 9, Section 3], the assumptions in (i)and (ii) of the above proposition are closely related to the spectral radius of thelinear map BaðwbÞ. In fact, as a consequence of maximum principles, one caneasily show that BaðwbÞ is a positive map on Xa (see [21, Section II.13]). How-ever, BaðwbÞ is not strongly positive unless (3.16) consists of only one equation,that is when a is a singleton. If this is the case then the condition in (i) (resp. (ii))simply requires that the spectral radius sprðBaðwbÞÞ < 1 (resp. > 1) (see [2]).

Remark 3.8. If BaðwbÞ is not strongly positive but Z0b consists of only one el-

ement wb and BaðwbÞ possesses a positive eigenvector pa corresponding to aneigenvalue k > 1 then it is easy to show that pa verifies (ii) [8, Remark 3.5].

We are now ready to discuss the existence of periodic solutions of (3.2). Form arbitrary we give sufficient conditions for semitrivial (or single population)solutions to exist. If m ¼ 2, (3.2) consists of three equations and we will derivesufficient conditions for positive periodic solutions.


3.1. Single population periodic solutions

We first apply the above index results to the case b ¼ f0g. In this case, Z0f0g is

the set of fixed points with w0 > 0 and wi ¼ 0 for all i 6¼ 0. As we remarkedafter Theorem 6, the system reduces to equation (3.7) so that Z0f0g consistsexactly of the washout fixed point w� ¼ ðS�ðx; 0Þ; 0; . . . ; 0Þ. The correspondingwashout periodic solution is ~uu� ¼ ðS�ðx; tÞ; 0; . . . ; 0Þ.

Because Z0f0g is now a singleton, Remark 3.8 and the eigenvalue relationstated in Lemma 3.6 suggest that we consider the following parabolic eigen-value problem:

oUiot

� AiðtÞUi þ ðUiUiðS�x ÞS�x Þx ¼ Uio

ouifiðx; t;~uu�Þ þ lUi; x 2 X;

oUion

þ ri x;oS�

on

� �Ui ¼ 0; x 2 oX;

Uiðx; T Þ ¼ Uiðx; 0Þ; x 2 ð0; 1Þ

ð3:20Þ

for i ¼ 1; . . . ;m. Our principal assumption concerns these eigenvalue problems.We consider the following conditions.

ðEiÞ The principal eigenvalue of (3.20) is negative. We say that ðEÞ holds if (Ei)holds for 16 i6m.

By Lemma 3.6, (Ei) asserts that Bf0gðw�Þ has a positive eigenvector corre-sponding to an eigenvalue greater than one. In the same way, the condition(BP), for b ¼ f0g, is now of the form BP0: The principal eigenvalue of (3.20) isnonzero for 16 i6m.

By similar arguments as in the elliptic case in [8, Corollary 10], using Lemma3.6, Remark 3.8, (ii) of Proposition 7, and the index result of Theorem 6, weeasily show that:

Corollary 8. Assume (BP0). If for some i, 16 i6m, (Ei) holds then theory

(i) indðS; Z0f0gÞ ¼ 0.(ii) for Z0

þ ¼ fw ¼ ðw0; . . . ;wmÞ : w ¼ SðwÞ; wi > 0 for some i > 0g, ind ðS;Z0þÞ ¼ 1.

(iii) there exists a semi-trivial (single-population) periodic solution of (3.3).

3.2. The case of two species

We now turn to the two species case, that is m ¼ 2. It is assumed that fori ¼ 1; 2, the principal eigenvalue of the eigenvalue problem (3.20) is negative.Corollary 8 then implies the existence of at least one single-population periodic


solution for each of the two populations. Thus, Zf0;1g and Zf0;2g, the sets ofsingle-population periodic solutions for which u1 > 0 or u2 > 0, are nonempty.

Let b ¼ f0; 1g and a ¼ f0; 1; 2g n b ¼ f2g. Let ðu0; u1; 0Þ 2 Zb be a periodicsolution of (3.3). The system (3.16) now reduces to one scalar parabolicequation

ouot

þ A2ðtÞuþ ðuU2ðu0Þðu0ÞxÞx ¼ uof2ou2

ðx; t; u0; u1; 0Þ; x 2 ð0; 1Þ; t > 0;

B2u ¼ 0; x 2 f0; 1g; t > 0;

uðx; 0Þ ¼ wðxÞ; x 2 ð0; 1Þ:

So, BaðwbÞ is now a strongly positive map on H 1ð0; 1Þ (see [21]). Therefore,from Remark 3.7 and Lemma 3.6, in order to apply Proposition 7 we need toinvestigate the principal eigenvalue of the periodic parabolic problem

o/ot

þ A2ðtÞ/ þ ð/U2ðu0Þðu0ÞxÞx ¼ /of2ou2

ðx; t; u0; u1; 0Þ þ l/;

x 2 ð0; 1Þ; t > 0;

B2/ ¼ 0; x 2 f0; 1g; t > 0;

/ðx; 0Þ ¼ /ðx; T Þ; x 2 ð0; 1Þ

for u ¼ ðu0; u1; 0Þ 2 Zb. The spectral radius of BaðwbÞ (wbðxÞ ¼ ðu0ðx; 0Þ;u1ðx; 0ÞÞ) is given by e�lT. Similar analysis applies to the case b ¼ f0; 2g. ByProposition 7, we are led to the following conditions.

Let Zi ¼ Zf0;ig, and denote the elements of Zi by UUi, that is, UU1 ¼ ðuu01; uu1; 0Þand UU2 ¼ ðuu02; 0; uu2Þ. We consider:

ðDþÞ For each i ¼ 1; 2, and for any UUi 2 Zi, i; j 2 f1; 2g with j 6¼ i, the principaleigenvalue of

o/ot

þ AiðtÞ/ þ ð/Uiðuu0jÞðuu0jÞxÞx ¼ /o

ouifiðx; t; UUjðx; tÞÞ þ l/;

o/on

þ ri�x; t;

oðuu0jÞon

�/ ¼ 0;

/ðx; 0Þ ¼ /ðx; T Þ;

ð3:21Þ

is positive.

ðD�Þ For each i, the eigenvalues of (3.21) are all negative.In biological terms, (Dþ) says that every u1-single population periodic so-

lutions is stable to invasion by u2 and conversely. (D�) says that every u1-singlepopulation periodic solutions is unstable to invasion by u2 and conversely. Ineither case, we might expect the existence of a positive periodic solutionðu0; u1; u2Þ with ui > 0 for i ¼ 1; 2. The main result of this section asserts thatthis is the case.


Theorem 9. Let m ¼ 2 and assume (Ei) holds for i ¼ 1; 2 and either (Dþ) or (D�)holds. Then the system (3.3) has at least one positive periodic solution.

Proof. First of all, since u0 must be nonzero, we have Zf1;2g ¼ ;. By Corollary 8,indðS; Z0f0gÞ ¼ 0. Let b ¼ f0; 1g or f0; 2g. If (Dþ) (resp. (D�)) holds then thecondition of (i) (resp. (ii)) of Proposition 7 holds so that indðS; Z0

bÞ ¼indðSjXb

; Z0bÞ (resp. 0).

Since (Ei) holds for i ¼ 1; 2, by (ii) of Corollary 8 (with m ¼ 1), we see thatfor the face index, indðSjXb

; Z0bÞ ¼ 1. In both cases, we observe that the sum of

the face indices of S is either 0 or 2 and not equal to indðS;CðRÞÞ ¼ 1. From theaddition property of index we conclude that the set of positive fixed points of Smust be nonempty. Our proof is complete. �

4. Proofs of technical results

4.1. Proof of Theorem 2

In this section we will prove the main technical result, Theorem 2. In theproof we will use the notations xðtÞ;x1ðtÞ; . . . to denote various continuousfunctions in the class P. The proof of this theorem will base on several lemmas.Because the boundary conditions of (1.1) are time-dependent and nonlinear ingeneral, some standard results on parabolic equation theory cannot apply di-rectly. We will use a simple change of variables (see [18, p. 174]) to reduce thetime-dependent Robin boundary condition to a homogeneous Neumann one.In particular, we shall have to deal with

ouot

¼ aðx; tÞuxx þ bðx; tÞux þ cðx; tÞuþ f ðx; tÞ; x 2 ½0; 1�; t > 0;

� uxð0; tÞ þ a0ðtÞuð0; tÞ ¼ 0; uxð1; tÞ þ a1ðtÞuð1; tÞ ¼ 0; t > 0;

uðx; 0Þ ¼ u0ðxÞ; x 2 ½0; 1�:

ð4:1Þ

We easily check that:

Lemma 4.1. Let u be a solution to (4.1). Suppose that there exist differentiablefunctions r0ðx; tÞ; r1ðx; tÞ such that r0ð0; tÞ ¼ a0ðtÞ, r1ð1; tÞ ¼ a1ðtÞ. We set

Rðx; tÞ ¼Z x

0

fsr1ðs; tÞ þ ðs� 1Þr0ðs; tÞgds; and

wðx; tÞ ¼ eRðx;tÞuðx; tÞ: ð4:2Þ

Then w satisfies the homogeneous Neumann boundary condition problem


owot

¼ awxx þ �bbwx þ �ccwþ eRf ðx; tÞ; x 2 ½0; 1�; t > 0;

owon

ðx; tÞ ¼ 0; x 2 f0; 1g; t > 0;

wðx; 0Þ ¼ eRðx;0Þu0ðxÞ; x 2 ½0; 1�;

ð4:3Þ

where �bb ¼ b� 2aRx and �cc ¼ c� bRx þ aðR2x � RxxÞ þ Rt.

Depending on the smoothness of ri, various norms of u and w are equivalent.For example, if riðx; tÞ is continuous and bounded in x; t then Rð�; tÞ is of classC1. So, the H 1 norm of u can be majorized by that of w and many properties of(4.1) can be carried over to (4.3). Therefore, we will study (4.3).

First, for any t > sP 0, we let Qt ¼ ð0; 1Þ ½0; t� and Qs;t ¼ ð0; 1Þ ½s; t�.For r 2 ð1;1Þ, let W 2;1

r ðQÞ, with Q being one of the cylinders Qt;Qs;t, be theBanach space of functions u 2 LrðQÞ having generalized derivatives ut; ux; uxxwith finite LrðQÞ norms (see [25, p. 5]). For sP 0, r 2 ð1;1Þ we also make useof the fractional order Sobolev spaces W s

r ¼ W sr ð0; 1Þ (see, e.g., [1,25] for a

definition).Let us consider the parabolic equation

ovot

¼ AðtÞvþ f ðx; tÞ; x 2 ð0; 1Þ; t > 0;

vxð0; tÞ ¼ vxð1; tÞ ¼ 0 t > 0;

vðx; 0Þ ¼ v0ðxÞ

ð4:4Þ

where AðtÞ is a uniformly regular elliptic operator, with domain of definitionW 2r ð0; 1Þ. If the coefficients of the operator AðtÞ are uniformly H€oolder contin-

uous in a cylinder Q then it is well known that (see, e.g., [16, Sections II.16–17]), for each t > 0, r > 1 and any bP 0, the fractional power AbðtÞ, with itsdomain of definition DðAb

r ðtÞÞ in Lrð0; 1Þ, of AðtÞ is well defined [16]. We recallthe following imbedding (see [19]).

DðAbr ðtÞÞ � Clð0; 1Þ for 2b > l þ 1=r: ð4:5Þ

Next, we collect some well-known facts about (4.4) (see [16,25]).

Lemma 4.2. Let r 2 ð1;1Þ. Assume that the coefficients of AðtÞ are boundedand H€oolder continuous. Let u be any solution u of (4.4). theory

(i) For t > sP 0. If f 2 LrðQs;tÞ for some r > 3 then we have

kukW 2;1r ðQs;tÞ 6C kf kLrðQs;tÞ

þ kuð�; sÞk

W 2�1=rr ð0;1Þ

; ð4:6Þ

where the constant C depends on the cylinder Qs;t but remains bounded if thelength t � s of the cylinder Qs;t is bounded and the coefficients of AðtÞ are uni-formly bounded in Qs;t.


(ii) Let r > 1 and f ð�; tÞ 2 Lrð0; 1Þ. For some fixed t0 > 0 and any b 2 ½0; 1�, wehave

kAbðt0ÞvðtÞkr6Cbt�be�dtkv0kr þ Cb

Z t

0

ðt � sÞ�be�dðt�sÞkf ð�; sÞkr ds ð4:7Þ

for some constants d;Cb > 0.

Remark 4.3. The proof of (i) can be found in [25, Theorem 9.1] where Dirichletboundary condition was considered, but the result holds as well for Neumannboundary condition (see [25, p. 351]).

Going back to the solution ~uu of Theorem 2, we first have the followingestimate for the spatial derivative of u0.

Lemma 4.4. For some x 2 P and d > 0, r > 1, b 2 ð0; 1Þ and lP 0 such that2b > l þ 1=r, we have

ku0ð�; tÞkClð0;1Þ 6xðtÞ þZ t

0

ðt � sÞ�be�dðt�sÞxðsÞ

Xmi¼1

kuið�; sÞkr ds: ð4:8Þ

Proof. The proof is elementary, using the change of variables as in Lemma 4.1so that (ii) of Lemma 4.2 can be applied here. Recalling the growth condition(F.1) and the imbedding (4.5) and (4.8) then follows. �

Next, we will make use of Lemma A.1 and Proposition A.1 in Appendix A toshow that the Lp norms of the solution are of class P for any pP 1. In fact, thisis the crucial step in proving Theorem 2.

Lemma 4.5. For any finite pP 1, there exists a function x 2 P such that

kuið�; tÞkp 6xðtÞ; i ¼ 1; . . . ;m: ð4:9Þ

Proof.We first assume that m ¼ 1. The proof is by induction on p. We supposethat (4.9) holds for some pP 1. Let us denote v ¼ u0 and u ¼ upi . We multiplythe equation for ui by u

2p�1i and integrate over ½0; 1�. Using integration by parts

and dropping negative boundary integrals, we easily derive (using the growthcondition of (F.1))

d

dt

ZXu2 dxþ d

ZXu2xdx6Cp

ZXðjuuxUðvÞvxj þ jðb� axÞuxuj þ u2 þ urp þ 1Þdx;

ð4:10Þ

where rp ¼ ðr þ 2p � 1Þ=p. Applying the Young inequality to the second termin the integral on the right-hand side, we obtain


y0ðtÞ þ d2

ZXu2x dx6C

ZXðjuuxUðvÞvxj þ u2 þ urp þ 1Þdx; ð4:11Þ

where yðtÞ ¼R

X u2ðx; tÞdx. By [9, Lemma 2.4], for any k 2 ð1; 4Þ and e > 0, we

have thatZXuk dx6 e

ZXu2x dx

�þ kuk21

þ CðeÞkuklðkÞ1 ð4:12Þ

for some positive constants CðeÞ; lðkÞ. Since rp6 r þ 1 for any pP 1 and r < 3,we can use the above inequality with e ¼ d=8 in the integrals of urp and u2 onthe right-hand side of (4.11). Recalling the induction assumption kuð�; tÞk1 2P, we obtain

y0ðtÞ þ d2

ZXu2x dx6Cp

ZXjuuxUðvÞvxjdxþ x0ðtÞ ð4:13Þ

for some x0 2 P. We next estimate the integral of uuxUðvÞvx. By (2.13),UðvÞ6x1ðtÞ for some x1 2 P. Moreover, we have the estimate (see [25,pp. 62–64])

kuk1 6C kuk1

þ kuk1=31 kuxk2=32

: ð4:14Þ

Using the Young inequality, the above estimate and the induction hypothesis,we haveR

X juuxUðvÞvxjdx 6 e=2R

X u2x dxþ CðeÞx1ðtÞ

RX u

2v2x dx6 e=2

RX u

2x dxþ CðeÞx1ðtÞkuk21

RX v

2x dx

ð4:14Þ 6 e=2R

X u2x dxþ CðeÞx2ðtÞ 1þ

RX u

2x dx

� �2=3n oRX v

2x dx

ðYoung inequalityÞ 6 eR

X u2x dxþ CðeÞx3ðtÞ kvxk22 þ kvxk62

n o:

ð4:15Þ

Using (4.8), with b 2 ð0; 1Þ and r 2 ðp; 2pÞ chosen such that 2b > 1þ 1=r, we get

kvxð�; tÞk1 6x4ðtÞ þ CZ t

0

ðt � sÞ�be�dðt�sÞx4ðsÞkuið�; sÞkr ds ð4:16Þ

for some x4 2 P. By H€oolder inequality,

kuikr ¼ kuk1=pr=p 6 kuk1=p�h1 kukh

2; h ¼ 1=p � 1=r1� 1=2

:

Note that h can be arbitrarily small if r is close to p. From now on, we willchoose r > p such that 6h < 1. Using the above in (4.16) we obtain

kvxð�; tÞk1 6x4ðtÞ þZ t

0

ðt � sÞ�be�dðt�sÞx4ðsÞyhðsÞds:


Using this and (4.15) in (4.13), we have that

y0ðtÞ þ d2

ZXu2x dx6x6ðtÞ þ x6ðtÞfK2ðtÞ þ K6ðtÞg; ð4:17Þ

where

KðtÞ ¼Z t

0

ðt � sÞ�be�dðt�sÞx4ðsÞyhðsÞds and x6 2 P:

By (4.12) and the induction assumption,R

X u2 dx6 d

2

RX u

2x dxþ x7ðtÞ for some

function x7 2 P. We thus deduce the following integro-differential inequality:

y0ðtÞ6 � yðtÞ þ x8ðtÞ þ x8ðtÞfK2ðtÞ þ K6ðtÞg: ð4:18Þ

We will show that Lemma A.1 and Proposition A.1 in Appendix A can be usedhere to show that y is globally bounded and, more importantly, ultimatelyuniformly bounded. This implies that kuik2p 2 P and completes the proof byinduction. We define the functional

f ðt; yÞ ¼ �yðtÞ þ x8ðtÞ þ x8ðtÞfK2ðtÞ þ K6ðtÞg: ð4:19Þ

Since x8 2 P, we can find positive constants Cx, which may still depend on theinitial data, such that x8ðtÞ6Cx.

Let

C1 :¼ supt>0

Z t

0

ðt � sÞ�be�dðt�sÞ ds6

Z 1

0

s�be�ds ds < 1;

because b 2 ð0; 1Þ and d > 0.We then set

F ðy; Y Þ ¼ �y þ Cx þ CxfðC1Y hÞ2 þ ðC1Y hÞ6g:

Obviously f ; F satisfy the conditions (F.1),(F.2) if 6h < 1. Hence, Lemma A.1applies and gives

yðtÞ6Cð~uuð�; 0ÞÞ 8t > 0 ð4:20Þ

for some constant Cð~uuð�; 0ÞÞ which may still depend on the initial data since Fdoes. We have shown that yðtÞ is globally bounded. We now seek uniformestimates. By Definition 1, we can find s1 > 0 such that xðsÞ6 �CC1 ¼ C1 þ 1 ifs > s1. Let t > s P s1 and assume that yðsÞ6 Y for all s 2 ½s; t�. Let us write

KðtÞ ¼Z s

0

ðt � sÞ�be�dðt�sÞx4ðsÞyhðsÞdsþ

Z t

sðt � sÞ�b

e�dðt�sÞx4ðsÞyhðsÞds

¼ J1 þ J2:

By (4.20), x4ðsÞyhðsÞ6Cð~uuð�; 0ÞÞ for every s so that we can find s0 > s such thatJ1 6 1 if t > s0. We then have


KðtÞ6 1þ �CC1C0Y h; where C0 ¼ supt>T

Z t

Tðt � sÞ�b

e�dðt�sÞ ds < 1:

Thus, for t > s0 we have f ðt; yÞ6GðyðtÞ; Y Þ, with

GðyðtÞ; Y Þ ¼ �yðtÞ þ �CC1

þ �CC1fð1þ �CC1C0Y hÞ2 þ ð1þ �CC1C0Y hÞ6g: ð4:21Þ

We see that G is independent of the initial data and satisfies (G.1)–(G.3) if6h < 1. Therefore, Proposition A.1 can be applied here to give (4.9). For thecase of more than two equations, that is m > 1, we argue similarly by definingyðtÞ ¼

Pmi¼1

RX u

pi dx and adding the inequalities (4.18). �

With all these preparations in hand, we now can show that the Cm normof ui, for some m > 1, is ultimately uniformly bounded.

Proof of Theorem 2. We first apply (i) of Lemma 4.2 to the equation for u0.Since kuið�; tÞkp 2 P for any p large, we see that f0 2 LqðQs;tÞ for any q > 1. Infact, with s ¼ t � 1, kf0kLqðQs;tÞ, as a function in t, is in the class P. Hence,

ku0kW 2;1q ðQsÞ 6C kf0kLqðQsÞ

þ ku0ð�; sÞkW 2�1=q

q ð0;1Þ

: ð4:22Þ

Choosing b 2 ð0; 1Þ (close to 1) and r sufficiently large such that 2b > 2�1=qþ 1=r, Lemma 4.4 states that the norm of u0ð�; tÞ in C2�1=qð0; 1Þ, andtherefore W 2�1=q

q ð0; 1Þ, is in the class P for any q > 1. We then conclude thatku0kW 2;1

q ðQs;tÞ 2 P for any q > 1. So,Z t

t�1

ZX

ou0ot

ðx; sÞ��

��q�þ jðu0Þxxðx; sÞj

q�dxds6xðtÞ 8t 2 I ð4:23Þ

for some x 2 P. We now turn to the equation for ui. At x ¼ 0; 1, ðu0Þx can besolved in terms of u0 from the boundary condition for u0. Thus, the boundarycondition for u ¼ ui can be rewritten as

�uxð0; tÞ þ �rr0ðt; u0ð0; tÞÞuð0Þ ¼ 0; uxð1; tÞ þ �rr1ðt; u0ð1; tÞÞuð1; tÞ ¼ 0

for some functions �rri enjoying the same properties of bi in (BC.1). We now usea change of variables w ¼ eRui as in Lemma 4.1 with

Rðx; tÞ ¼Z x

0

ðs�rr1ðt; u0ðs; tÞÞ þ ðs� 1Þ�rr0ðt; u0ðs; tÞÞÞds

to see that w satisfies

owot

¼ awxx � bbwx þ ff ðx; tÞ

with Neumann boundary condition. Here, for �bb ¼ ðbþ Uðu0Þðu0ÞxÞ,


bb ¼ �bb� 2aRx ¼ bþ Uðu0Þðu0Þx � 2aðxr1ðx; tÞ þ ðx� 1Þr0ðx; tÞÞ;

ff ¼ eR½f ðx; t;~uuÞ � uðcþ U0ðu0Þðu0Þ2x þ Uðu0Þðu0ÞxxÞ�þ uð��bbRx þ aðR2x � RxxÞ þ RtÞ:

Note that the coefficient bb depends only on u0; ðu0Þx whose H€oolder norms, asfunctions in t, are in the class P. Therefore, bb is H€oolder continuous with ul-timately uniformly bounded norm. Let AAðtÞ be the operator correspondingto the equation above. By Lemma 4.2, we have for any fixed t0 > 0 and s ¼t � 1 > 0

k AAbðt0ÞwðtÞkr6CkwðsÞkr þ Cb

Z t

sðt � sÞ�b

e�dðt�sÞkff ð�; sÞkr ds ð4:24Þ

for some fixed constants C; d;Cb > 0. By H€oolder inequality we can estimate thesecond term as follows:

Z t

sðt � sÞ�b

e�dðt�sÞkff ð�; sÞkr ds6Z t

sðt

�� sÞ�qbe�qdðt�sÞ ds

�1=q

kff kLrðQs;tÞ;

ð4:25Þ

where ð1=qÞ þ ð1=rÞ ¼ 1. From the definition of R; ff , and the facts thatku0ð�; tÞk1, kðu0Þxð�; tÞk1, kuið�; tÞkp are functions in the class P for any p, and(4.23) holds for any q, it is not difficult to see that ff 2 LrðQs;tÞ andkff kLrðQs;tÞ 2 P for any large r. Therefore, given any b 2 ð0; 1Þ, if we choose rlarge enough such that q ¼ r=ðr � 1Þ sufficiently close to 1 then it is easy to seethat the integral on the right-hand side of (4.25) is finite. Moreover, thequantity on the right-hand side is in the class P. Using this in (4.24), we haveshown that for Y ¼ Dð AAb

r ðt0ÞÞ, kwðtÞkY 2 P for any b 2 ð0; 1Þ and r > 1. Wehave the same estimate for kuðtÞkY since we have shown that u0ð�; tÞ 2 C1;cð0; 1Þfor any c 2 ð0; 1Þ. Using the imbedding (4.5) with m ¼ 2b � 1=r > 0 and b; rchosen such that m > 1, we prove the theorem. �

4.2. Proof of Lemma 2.1

By (F.2), since fiðx; t;~uuÞ � 0 when ui ¼ 0 and f0ðx; t;~uuÞ ¼ 0 if ui ¼ 0 for alli 6¼ 0, the positiveness of ui is just a simple application of invariant principles(see, e.g., [24] and [27, Prop. 3.1]). We shall prove only (2.12) and (2.13). Asin the proof of Lemma 4.4, for any t > 0, let u�ðx; tÞ satisfy

� A0ðtÞv ¼ 0; x 2 ð0; 1Þ; �vxð0Þ þ a0ðtÞvð0Þ ¼ S0ðtÞ;vxð1Þ þ a1ðtÞvð1Þ ¼ S1ðtÞ:


Let w be the solution to the linear parabolic equation

owot

¼ A0ðtÞwþ j ou�

otj; x 2 ð0; 1Þ; t > 0;

� wxð0; tÞ þ a0ðtÞwð0; tÞ ¼ 0; wxð1; tÞ þ a1ðtÞwð1; tÞ ¼ 0; t > 0;

wðx; 0Þ ¼ maxfu0ðx; 0Þ � u�ðx; 0Þ; 0g:ð4:26Þ

By comparison principles, wðx; tÞP 0. We first give a bound for the L1 normof w. Integrating the equation of w over X ¼ ½0; 1� and using integration byparts and the boundary condition, we easily get

d

dt

ZXwdx ¼ �

ZoXðb� ax þ aaÞwdr �

ZXðc� bx þ axxÞwþ ou�

ot

��dx:

Here we have used the fact that bwx ¼ ðbwÞx � bxw. Since ðb� ax þ aaÞwP 0on oX ¼ f0; 1g, ðc� bx þ axxÞP h > 0, (by (BC.2) and (2.11) of (F.1)), wededuce the following inequality:

d

dt

ZXwdxþ h

ZXwdx6

ZX

ou�

ot

��dx:

From uniform regularity assumption (A.1), it follows easily that kou�=otðx;tÞk1 is also bounded. Hence, by integrating the above inequality, we obtainkwð�; tÞk1 6x1ðtÞ for some x1 2 P. A similar argument as in [15, Lemmas 3.2and 3.3] shows that kwð�; tÞkH1 6x2ðtÞ for some x2 2 P. By Sobolev imbed-ding theorems, we have kwk1 6x3ðtÞ. Since �wwðx; tÞ ¼ u0ðx; tÞ � u�ðx; tÞ6wðx; tÞthanks to the fact that f0 6 0 and comparison principles, we obtain (2.12).

For (2.13), we multiply the equations of ui by hi, integrate over ½0; 1� and addthe results. By a similar argument as above, using (2.11) of (F.2) and (BC.2), weget

y0ðtÞ þ hyðtÞ6Kðu0Þ;

where yðtÞ ¼Pmi¼0

R 1

0hiuiðx; tÞdx and Kðu0Þ can be majorized by ku0ð�; tÞk1.

We now integrate the above inequality and use (2.12) to obtain (2.13). �

Appendix A

The following auxiliary results are useful in getting estimates for differentialinequalities which involve integrals such as those encountered in the proof ofLemma 4.5 of Section 4.

For a function y : Rþ ! R, let us consider the following inequalities:

y0ðtÞ6 f ðt; yÞ; yð0Þ ¼ y0; t 2 ð0;1Þ; ðA:1Þ


where f is a functional from Rþ CðRþ;RÞ into R. The following lemma isstandard and gives a global estimate for y which is still dependent on the initialdata.

Lemma A.1. Assume (A.1) and

F.1. Suppose that there is a function F ðy; Y Þ : R2 ! R such that f ðt; yÞ6F ðyðtÞ; Y Þ if yðsÞ6 Y for all s 2 ½0; t�.

F.2. There exists a real M such that F ðy; Y Þ < 0 if y ¼ Y PM .Then there exists finite M0 such that yðtÞ6M0 for all tP 0.

Proof. The proof is standard. We set M0 ¼ maxfy0;Mg þ 1 and show thatyðtÞ6M0 for all t. If this is not true then there exist t > 0 such that yðsÞ < M0

for all s 2 ½0; tÞ and yðtÞ ¼ M0 with y0ðtÞP 0. But, by (F.1) and then (F.2), wehave

y0ðtÞ6 f ðt; yÞ6 F ðM0;M0Þ < 0:

This contradiction completes the proof. �

Remark A.2. In (F.1), the inequality f ðt; yÞ6 F ðy; Y Þ is not pointwise. It re-quires that yðtÞ6 Y on the interval ½0; t�. Such a situation usually occurs whenf ðt; yÞ contains integrals of yðtÞ over ½0; t�.

The above constant M0 still depends on the initial data y0. Moreover, thefunction F may also depend on y0. Next, we consider conditions which guar-antee uniform estimates for yðtÞ.

Proposition A.1. Assume (A.1) and

G.1. There exists a Lipschitz function Gðy; Y Þ : R2 ! R such that for s suffi-ciently large, if t > s and yðsÞ6 Y for every s 2 ½s; t� then there exists s0 P s suchthat

f ðt; yÞ6GðyðtÞ; Y Þ if tP s0 P s; ðA:2Þ

G.2. The set fz : Gðz; zÞ ¼ 0g is not empty and z� ¼ supfz : Gðz; zÞ ¼ 0g < 1.Moreover, GðM ;MÞ < 0 for all M > z�,

G.3. For y; Y P z�, Gðy; Y Þ is increasing in Y and decreasing in y. If M0 ¼lim supt!1 yðtÞ < 1 then

lim supt!1

yðtÞ6 z�: ðA:3Þ

Proof. If M0 6 z� then there is nothing to prove. So, assume that M0 > z�.


First, let M > z�. Since Gðz�; z�Þ ¼ 0, we have Gðz�;MÞ > 0 (because Gðz�; �Þis increasing, by (G.3)). This and the fact that GðM ;MÞ < 0 imply the existenceof a number z 2 ðz�;MÞ such that Gðz;MÞ ¼ 0. Let zðMÞ be the largest of such zin ðz�;MÞ. By (G.3), we have

GðzðMÞ;MÞ ¼ 0 and Gðy;MÞ < 0 8y 2 ðzðMÞ;M �: ðA:4Þ

Now, for t large, says tP T , we have that yðtÞ6M for some M > z�. By (G.1),we can find T0 P T such that

y0ðtÞ6GðyðtÞ;MÞ; tP T0; yðT0Þ6M :

Comparing yðtÞ with the solution of Y 0ðtÞ ¼ GðY ðtÞ;MÞ, t > T0 and Y ðT0Þ ¼ M ,we conclude that yðtÞ6 Y ðtÞ for all tP T0. From (A.4) we see that Y ðtÞ ! zðMÞas t! 1. Thus, for any given e > 0, there exist T1 > T0 and e1 2 ð0; eÞ such thatzðMÞ þ e1 < M and yðtÞ6 Y ðtÞ6 zðMÞ þ e1 for all t > T1.

Since zðMÞ > z�, the above argument can be repeated with zðMÞ þ e1 in placeofM to show that there exist sequences of positive numbers fTjg; fejg and fkjgsuch that k0 ¼ M , limj!1 ej ¼ 0, limj!1 Tj ¼ 1 and

kjþ1 ¼ zðkjÞ þ ej < kj; yðtÞ6 kj 8tP Tj:

Since kj is decreasing and bounded from below by z�, kj converges to somezP z� satisfying Gðz; zÞ ¼ 0 (because Gðkjþ1 � ej; kjÞ ¼ 0 for all j and ej ! 0).Since z� is the largest of such solutions, we must have z ¼ z�. Thus,lim supt!1 yðtÞ6 z�. �

Remark A.3. Condition (G.3) is only used to guarantee the existence of zðMÞthat has the property (A.4). One can see that the proof works well withfunctions satisfying (A.4) for any given M > z�.

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On a time-dependent bio-reactor model with chemotaxis