Boundedness and exponential stabilization in a signal transduction model with diffusion

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2008; 31:1905–1921Published online 16 May 2008 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/mma.1010MOS subject classification: 35K 57; 35K 50; 35B 40

Boundedness and exponential stabilization in a signaltransduction model with diffusion

Michael Winkler∗,†

Department of Applied Mathematics and Statistics, Comenius University, 84248 Bratislava, Slovakia

Communicated by M. Fila

SUMMARY

The influence of diffusion in a model arising in the description of signal transduction pathways in livingcells is investigated. It is proved that all solutions of the corresponding semilinear parabolic system,consisting of four equations, are global in time and bounded. Under the additional assumption that certaintwo of the diffusion coefficients are equal, it is furthermore demonstrated that all solutions approach aspatially homogeneous steady state as t→∞. This equilibrium is uniquely determined by the initial data,and the rate of convergence is shown to be at least exponential. Copyright q 2008 John Wiley & Sons,Ltd.

KEY WORDS: parabolic system; asymptotic behavior; stabilization

1. INTRODUCTION

Processing information and finding appropriate responses to environmental signals belong to themost vital tasks that any living organism has to cope with. At the level of single cells, this complexoperation is carried out within the so-called signalling networks, the function of which forms theprecondition for any kind of reacting behavior of the living being, such as directed movement (see[1] for a discussion of associated multiscale aspects). Understanding the underlying mechanismshas been the objective of a great number of studies in theoretical and mathematical biology (see[2, 3] and the references therein, for instance). One of its basic requirements consists of detectingthose among the numerous components creating the respective physical and chemical frameworksthat are essentially responsible for the function of such signal transduction pathways.

Typically, the immediate process of transduction takes place along time scales that are rathershort compared with those arising in, e.g. metabolic systems; influences of slowly varying quantitiessuch as changes in temperature are thus usually neglected at first glance. Accordingly, in order

∗Correspondence to: Michael Winkler, Department of Applied Mathematics and Statistics, Comenius University,84248 Bratislava, Slovakia.

†E-mail: micha [email protected]

Copyright q 2008 John Wiley & Sons, Ltd. Received 21 December 2007

1906 M. WINKLER

to establish simple but still reasonable models, signal transduction pathways are often treated asnetworks of coupled modules, each of which basically consists of an activation/deactivation cyclemade up by an enzymatically regulated chemical reaction of the general type

X+Ek1�k−1

Ck2→E+X� (1)

Here, X and X� denote a substrate protein and its product, respectively, E a catalyzing enzymeand C an intermediate complex formed by X and E .

In [3], a mathematical model for (1) consisting of the four ordinary differential equations (ODEs)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dX

dt=−k1EX+k−1C, t>0

dE

dt=−k1EX+(k−1+k2)C, t>0

dC

dt=k1EX−(k−1+k2)C, t>0

dX�

dt=k2C, t>0

(2)

is considered, describing the evolution of the total masses of the participants. Mainly focusingon the problem of parameter identification, the authors in [3] discuss the relationship betweensolutions to (2) and some further simplifications thereof, the latter being frequently employed asthey allow for detection of the a priori unknown constants k1,k−1 and k2 from experimental data.

Within the above model (2) it is tacitly assumed that all concentrations are spatially homogeneousand that the reactions take place simultaneously throughout the available region. It is the purpose ofthis study to include the previously neglected role of diffusion in the model. Accordingly, insteadof (2), we shall consider the parabolic system⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

Xt =DX�X−k1EX+k−1C, x ∈�, t>0

Et =DE�E−k1EX+(k−1+k2)C, x ∈�, t>0

Ct =DC�C+k1EX−(k−1+k2)C, x ∈�, t>0

X�t =DX��X�+k2C, x ∈�, t>0

(3)

in a bounded reaction volume � with smooth boundary, the diffusion coefficients DX , DE , DCand DX� immediately reflecting the strength of (linear) diffusion of the respective substances.

To turn (3) into a proper parabolic problem, we additionally impose no-flux conditions for allchemicals on the boundary of � as well as initial conditions according to⎧⎪⎨

⎪⎩�X��

= �E��

= �C��

= �X�

��=0, x ∈��, t>0

X (x,0)= X0(x),E(x,0)=E0(x),C(x,0)=C0(x), X�(x,0)= X�

0(x), x ∈�

(4)

where �/�� denotes differentiation with respect to the outward normal vector field at ��, and theinitial data X0,E0,C0 and X�

0 are supposed to be given nonnegative and continuous functionsin �.

Copyright q 2008 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2008; 31:1905–1921DOI: 10.1002/mma

SIGNAL TRANSDUCTION MODEL WITH DIFFUSION 1907

Throughout we assume that the parameters k1, k−1 and k2 have already been determined fromwhatever source and, thus, are given positive constants. Starting at this point, we wish to investigatethe role played by diffusion in the above refined model for (1), especially in respect of possibleformation of singularities and of the asymptotic behavior of solutions. As the nonlinearities in both(2) and (3) formally exhibit superlinear growth in the variable (X,E,C, X�), the question of globalexistence of solutions already appears to be nontrivial; moreover, even from global solvabilityof (2) one cannot directly infer that all solutions to (3) and (4) are global in time: Numerousexamples illustrating the possibility of ‘diffusion-induced blow-up’ in various superlinear parabolicsystems similar to (3) and (4) have been found in the past two decades (cf. [4, 5], for instance,and Section 33 in [6] for a recent survey).

Our first main result, however, asserts that no singularity formation can occur in finite time;more precisely, Corollary 3.2 will reveal that

• All solutions to (3) and (4) are global in time and uniformly bounded.

The next natural task is to examine the large time asymptotics of solutions. In Corollary 3.6, we canrule out any type of oscillatory behavior in (3) and (4) and prove exponentially fast stabilizationtowards a spatially homogeneous steady state under additional hypotheses, essentially requiringthat the enzyme E and the complex C diffuse at the same rate:

• If DE =DC and E0+C0 /≡0 then the solutions to (3) and (4) satisfy (X,E,C, X�)→(0,E∞,0, X�∞) in L∞(�) as t→∞ at an exponential rate, where the positive constants E∞and X�∞ are determined by

E∞ = k1k−1|�|

∫�(E0(x)+C0(x))dx (5)

X�∞ = k−1

k1|�|∫

�X�0+ k1k2

k−1|�|∫ ∞

0

∫�C(x, t)dx dt (6)

This indicates that—under the extra assumption DE =DC—the behavior of solutions to (3) and(4) does not differ significantly from that of solutions to (2), at least not beyond some initialstage, spatial inhomogeneities being dampened exponentially fast (see also the remark followingCorollary 3.6). This can be regarded as a part of a rigorous justification of the ‘no diffusion’assumption made in [3].

Let us mention that, as it turns out, the stabilization rates of both X and C can be boundedfrom below independent of the size of �. This goes beyond known results on general semilinearparabolic systems with no-flux boundary conditions, according to which exponentially fast decay ofbounded solutions to spatially homogeneous functions can be proved for any smooth nonlinearity,provided � is small enough compared with the uniform bound of the solution in question [7].

As an open problem, we have to leave unanswered the question whether the restriction DE =DCimposed here is of purely technical nature, or whether a more complicated dynamical picture hasto be drawn when DE and DC significantly differ from each other. For instance, examples knownfor a long time [8] show that unequal diffusion constants in fact can destabilize an otherwise stable(constant) equilibrium of parabolic systems in mathematical biology.

In order to simplify the exposition, we find it convenient to reduce the number of parameters in(3) and (4) using the substitutions u(x, t) :=aX (x,bt), v(x, t) :=aE(x,bt), w(x, t) :=aC(x,bt),


1908 M. WINKLER

z(x, t) :=aX�(x,bt), where a :=k1/k−1 and b :=1/k−1. In fact, this rescaling transforms (3), (4)into the problem

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ut =Du�u−uv+w, x ∈�, t>0

vt =Dv�v−uv+lw, x ∈�, t>0

wt =Dw�w+uv−lw, x ∈�, t>0

�u��

= �v

��= �w

��=0, x ∈��, t>0

u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x), x ∈�

(7)

supplemented by the scalar problem

⎧⎪⎪⎪⎨⎪⎪⎪⎩

zt =Dz�z+l ′w, x ∈�, t>0

�z��

=0, x ∈��, t>0

z(x,0)= z0(x), x ∈�

(8)

with the new diffusion coefficients Du =DX/k−1, Dv =DE/k−1, Dw =DE/k−1 and Dz =DX�/k−1as well as the nonnegative continuous initial data u0=k1/k−1X0, v0=k1/k−1E0, w0=k1/k−1C0and z0=k1/k−1X�

0. It will be important for our analysis to note that the parameter l=(k−1+k2)/k−1satisfies l>1. Moreover, l ′ =k2/k−1 is positive.

All our results will be derived for (7) and (8) first and then translated back to the originalvariables.

2. BOUNDEDNESS OF SOLUTIONS

Let us first apply standard theory for parabolic systems to make sure that (7) admits a uniquelocal-in-time nonnegative solution and that the maximally extended solution can only cease toexist in finite time if at least one of its components u, v and w blows up with respect to the normin L∞(�). Here and in the sequel, we frequently use abbreviations such as u(t) for the functionu(·, t) depending on x only when t�0 is fixed.

Lemma 2.1There exists a maximal existence time Tmax∈(0,∞] and a unique classical solution triple (u,v,w)

of (7) in �×(0,Tmax). Moreover, we have u�0,v�0 and w�0 in �×(0,Tmax), and

if Tmax<∞ then limsupt↗Tmax

(‖u(t)‖L∞(�)+‖v(t)‖L∞(�)+‖w(t)‖L∞(�))=∞ (9)



ProofBy standard existence results for parabolic systems [9], the problem⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ut =Du�u− uv+w+, x ∈�, t>0

vt =Dv�v− uv+lw+, x ∈�, t>0

wt =Dw�w+ u+v+−lw, x ∈�, t>0

�u��

= �v

��= �w

��=0, x ∈��, t>0

u|t=0=u0, v|t=0=v0, w|t=0=w0, x ∈�

(10)

admits a maximally extended classical solution (u, v, w). The maximum principle for scalarparabolic equations guarantees that each of the components u, v and w remains nonnegative, beinga solution of an inhomogeneous linear parabolic equation with nonnegative inhomogeneity. Hence,the subscript ‘+’ in (10) is abundant and (u, v, w) actually solves (7). Now uniqueness and thecharacterization (9) of Tmax again are due to well-known results [9]. �

Unfortunately, it seems that (7) does not admit bounded invariant rectangles, which wouldprovide at least a certain set of initial data yielding global bounded solutions. We, therefore, needto apply more sophisticated arguments in order to prove boundedness of all solutions. Facing afirst step towards this, as a simple but nonetheless important qualitative feature of (7), we obtaintwo identities asserting that some linear combinations of the components of the solution to (7)exhibit certain mass-conservation or mass-dissipation properties. From these we infer an a prioriestimate for solutions in L∞((0,Tmax); L1(�)).

Lemma 2.2We have

d

dt

∫�(v(x, t)+w(x, t))dx=0 for all t ∈(0,Tmax) (11)

and

d

dt

∫�(u(x, t)−v(x, t))dx=−(l−1)

∫�

w(x, t)dt for all t ∈(0,Tmax) (12)

In particular, there exists a constant M>0 such that

‖u(t)‖L1(�)�M, ‖v(t)‖L1(�)�M and ‖w(t)‖L1(�)�M for all t ∈[0,Tmax) (13)

ProofAdding the second to the third equation in (7) and integrating, in view of the boundary condition weobtain that (11) holds. Similarly, (12) follows upon subtracting the second from the first equationin (7). Finally, the inequalities in (13) result upon integrating (11) and (12) and observing thatl>1. �

We next intend to use the above estimates (13) in order to improve themselves so as to end upwith a priori bounds for u and v in L∞((0,Tmax); L p(�)) for arbitrarily large p<∞. This willbe the main step towards the proof of the boundedness of (u,v,w) in �×(0,Tmax). Our proof


1910 M. WINKLER

combines semigroup techniques with standard testing procedures and strongly relies on the explicitform of the nonlinearities in (7).

Let us mention here also that in the study of parabolic systems with two equations, massdissipation has successfully been used as a starting point for the proof of boundedness of solutionsunder various additional conditions (see Section 33 in [6], for instance).Lemma 2.3For all p∈(n,∞) there exists C(p)>0 such that

‖u(t)‖L p(�)�C(p) for all t ∈(0,Tmax) (14)

and

‖v(t)‖L p(�)�C(p) for all t ∈(0,Tmax) (15)

ProofMultiplying the first equation in (7) by u p−1 and integrating by parts, we obtain

1

p

d

dt

∫�u p(x, t)dx=−(p−1)Du

∫�u p−2|∇u|2−

∫�u pv+

∫�u p−1w (16)

for t ∈(0,Tmax). Unfortunately, it seems not adequate here to estimate the positive term on the rightdirectly, for instance, according to

∫� u p−1w�c1

∫� u p+c2

∫� w p, because in a straightforward

attempt to bound the latter term by using the equation for w in (7) one needs to deal with thesource term u ·v that exhibits superlinear growth with respect to (u,v,w). Therefore, in order tocope with the last term in (16) appropriately, we need to make use of the structure of the fullsystem (7) more thoroughly.

To this end, we utilize the variation-of-constants formula for solutions to scalar semilinearparabolic problems, according to which w is represented as

w(t)=eDw t�w0+∫ t

0eDw(t−s)�u(s)v(s)ds−l

∫ t

0eDw(t−s)�w(s)ds (17)

for all t ∈(0,Tmax). Here, for s>0, es� denotes the semigroup associated with the NeumannLaplacian in �.

From the first equation in (7) we gain the identity uv=Du�u−ut +w so that (17) leads to

w(t) = eDw t�w0+Du

∫ t

0eDw(t−s)��u(s)ds−

∫ t

0eDw(t−s)�ut (s)ds

−(l−1)∫ t

0eDw(t−s)�w(s)ds (18)

Integrating by parts with respect to time, we obtain

−∫ t

0eDw(t−s)�ut (s)ds=−Dw

∫ t

0eDw(t−s)��u(s)ds−u(t)+eDw t�u0

so that, as l>1 and eDw(t−s)�w is nonnegative by the maximum principle, we infer the pointwiseestimate

w(t)�eDw t�(u0+w0)−u(t)+(Du−Dw)

∫ t

0eDw(t−s)��u(s)ds, t ∈(0,Tmax) (19)



Now multiplying this by u p−1(t) and integrating over � we see that

∫�u p−1(t)w(t) �

∫�u p−1(t) ·eDw t�(u0+w0)−

∫�u p(t)

+(Du−Dw) ·∫

�u p−1(t) ·

(∫ t

0eDw t��u(s)ds

)(20)

Using that the es� commutes with its generator −�, we may integrate by parts with respect to xin the last term to obtain∫

�u p−t (t) ·

(∫ t

0eDw t��u(s)ds

)=−(p−1) ·

∫�u p−2(t)∇u(t) ·

(∫ t

0∇eDw(t−s)�u(s)ds

)

In combination with (20) and (16), this yields

1

p

d

dt

∫�u p(x, t)dx � −(p−1)Du

∫�u p−2|∇u|2−

∫�u pv+

∫�u p−1(t) ·eDw t�(u0+w0)−

∫�u p

−(p−1)(Du−Dw) ·∫

�u p−2(t)∇u(t) ·

(∫ t


)(21)

Here, the last term can be estimated using Young’s inequality according to

∣∣∣∣(p−1)(Du−Dw) ·∫

�u p−2(t)∇u(t) ·

(∫ t


)∣∣∣∣�(p−1)Du ·

∫�u p−2|∇u|2+ (p−1)(Du−Dw)2

4·∫

�u p−2(t) ·

∣∣∣∣∫ t


∣∣∣∣2

Once again by Young’s inequality, applied with exponents p/(p−2) and p/2, we find

(p−1)(Du−Dw)2

4·∫

�u p−2(t) ·

∣∣∣∣∫ t


∣∣∣∣2

�1

2

∫�u p(t)+c3

∫�

∣∣∣∣∫ t


∣∣∣∣p

with some constant c3>0. Observing that the second term on the right of (21) is nonpositive, wetherefore obtain

1

p

d

dt

∫�u p(x, t)dx �

∫�u p−1(t) ·eDw(t−s)�(u0+w0)− 1

2

∫�u p

+c3

∫�

∣∣∣∣∫ t


∣∣∣∣p

(22)


1912 M. WINKLER

for t ∈(0,Tmax). By known results on stabilization and L p−Lq estimates for the NeumannLaplacian semigroup (see [10], for instance) and the fact that p>n, we have

‖∇eDw��‖L p(�)�c4�−1/2−n/2(1/n−1/p)e−��‖�‖Ln(�) for all �∈Ln(�) (23)

with a constant c4 and the first positive eigenvalue �1 of −Dw� in � under Neumann boundaryconditions. Hence,

c3

∫�

∣∣∣∣∫ t


∣∣∣∣p

�c3cp4 ·(∫ t

0(t−s)−1/2−n/2(1/n−1/p)e−�1(t−s)‖u(s)‖Ln(�) ds

)p

�c5 ·(

sups∈(0,t)

‖u(s)‖Ln(�)

)p

(24)

where c5 :=c3cp4 ·(∫∞

0 �−1/2−n/2(1/n−1/p)e−�1� d�)p is finite because −1/2−n/2(1/n−1/p)=−1+(n/2p)>−1 for each p<∞.

Again as p>n, we may interpolate the Ln(�) norm of u(s) and use the a priori estimate toslightly reduce the power on the right-hand side of (24); that is, from Holder’s inequality we find

‖u(s)‖Ln(�)�‖u(s)‖�L p(�)

·‖u(s)‖1−�L1(�)

for s∈(0,Tmax)

with

�= 1−1/n

1−1/p∈(0,1)

Thus, by (24) and (13), (22) entails

1

p

d

dt

∫�u p(x, t)dx �

∫�u p−1(t) ·eDw t�(u0+w0)− 1

2

∫�u p

+c5M(1−�)p · sup

s∈(0,t)

(∫�u p(s)

)�

(25)

for t ∈(0,Tmax). Finally, as to the first term on the right, we have∣∣∣∣∫

�u p−1(t) ·eDw t�(u0+w0)

∣∣∣∣� 1

4

∫�u p(t)+c6

∫�

|eDw t�(u0+w0)|p

� 1

4

∫�u p(t)+c6|�|·‖u0+w0‖p

L∞(�)

for t ∈(0,Tmax) with some c6>0, where we have used the maximum principle in the pointwiseestimate es��‖�‖L∞(�) for all �∈L∞(�) and s>0.



Altogether, (25) turns into the ‘nonlocal’ (in time) differential inequality

1

p

d

dt

∫�u p(x, t)dx�− 1

4

∫�u p(t)+c7 · sup

s∈(0,t)

(∫�u p(s)

)�

+c8 for t ∈(0,Tmax) (26)

where c7 and c8 only depend on p and � and the initial data.Writing y(t) :=∫� u p(t) and S(t) :=sups∈(0,t)

∫� u p(s) for t ∈(0,Tmax) and observing that S is

nondecreasing with respect to t , we thus have

1

py′(t)�− 1

4y(t)+c7S

�(T )+c8 for 0<t<T<Tmax

which after an integration gives

y(t) � e−p/4t y(0)+4(c7S�(T )+c8) ·(1−e−p/4t )

� y(0)+4(c7S�(T )+c8) for 0<t<T<Tmax

Hence,

S(T )�y(0)+4c7S�(T )+4c8 for all T<Tmax

and as �<1, this implies that

S(T )�max{1, [y(0)+4(c7+c8)]1/1−�} for all T<Tmax

and thereby shows (14), because all constants appearing on the right-hand side are independentof T .

The proof of (15) can be run quite similarly, with u replaced by v in the above reasoning;the only marginal difference is that in (18) the last term does not appear, for the factor (l−1)changes to (l−l)=0. From (19) on, all formulae remain untouched, except for an exchange ofu and v. �

In light of standard L p theory for scalar parabolic equations, it is not surprising that with theabove estimates at hand, we can find appropriate upper bounds for the nonlinearities in each ofthe three partial differential equations (PDEs) in (7) separately and thereby obtain a priori boundsfor u,v and w with respect to the norm in L∞(�×(0,T )) for any finite T�Tmax. It turns out,however, that if one proceeds sufficiently carefully then the resulting estimates can be establishedin such a way that they are even independent of T . Beyond excluding blow-up in finite time, wethus can also infer that all solutions to (7) remain globally bounded.

Theorem 2.4Assume that l>1, and let u0,v0 and w0 be continuous and nonnegative in �. Then (7) possessesa unique global solution (u,v,w). This solution is nonnegative and bounded in �×(0,∞).

ProofIn view of Lemma 2.1, we only need to show that Tmax=∞ and that (u,v,w) is bounded. Assumefirst that Tmax<∞. Then by the representation formula (17) and the maximum principle,

w(t)�‖w0‖L∞(�)+∥∥∥∥∫ t

0eDw(t−s)�u(s)v(s)ds

∥∥∥∥L∞(�)

(27)


1914 M. WINKLER

holds throughout � for all t ∈(0,Tmax). We fix some q>n/2 and apply the Lq −L∞ smoothingestimate for the heat semigroup [10] to obtain∥∥∥∥

∫ t


∥∥∥∥L∞(�)

�c1 ·(∫ t

0(t−s)−n/2q ds

)· sups∈(0,Tmax)

‖u(s)v(s)‖Lq (�)

�c1T1−n/2qmax

1−n/2q· sups∈(0,Tmax)

‖u(s)v(s)‖Lq (�) (28)

for t ∈(0,Tmax) with some c1>0. By Holder’s inequality and (14) and (15),∥∥∥∥∫ t


∥∥∥∥L∞(�)

�c1T1−n/2qmax

1−n/2q·(C(2q))1/2(C(2q))1/2 (29)

Therefore, (27) shows that if Tmax<∞ then

‖w(t)‖L∞(�)�c2 for all t ∈(0,Tmax) (30)

with a suitable c2>0. Now the maximum principle applied separately to the first two equationsin (7) yields also that u and v are bounded in �×(0,Tmax), which together with (30) contradicts(9). Consequently, we must have Tmax=∞.

In order to prove boundedness of w in �×(0,∞), we let t j := j for j ∈N and, instead of (17),consider the shifted representation formula:

w(t j +�) = eDw��w(t j )+∫ �

0eDw(�−�)�u(t j +�)v(t j +�)d�

−l ·∫ �

0eDw(�−�)�w(t j +�)d� for �>0 (31)

With q>n/2 as above, the second term on the right can be estimated as in (28) and (29); that is,∥∥∥∥∫ �

0eDw(�−�)�u(t j +�)v(t j +�)d�

∥∥∥∥L∞(�)

�c1 ·21−n/2q

1−n/2q·C(2q) for �<2 (32)

As to the first term on the right-hand side of (31), however, we proceed differently and invoke(13) and L1−L∞ smoothing to see that

‖eDw��w(t j )‖L∞(�) � c3�−n/2‖w(t j )‖L1(�)

� c3�−n/2M

� c3M for �>1 (33)

with some c3>0.



Observing the nonpositivity of the last term in (31), we thus infer from (31)–(33) that‖w(t j +�)‖L∞(�)�c4 is valid for all �∈(1,2) and a certain constant c4. This means that w isbounded in �×( j+1, j+2) for all j ∈N and thereby shows that w is bounded.

Now the proof of boundedness of u and v can be run as a simplified variant of the abovereasoning: As u ·v�0, we have the pointwise estimate:

u(t j +�)�eDu��u(t j )+∫ �

0eDu(�−�)�w(t j +�)d� for �>0 (34)

Thus,

‖u(t j +�)‖L∞(�)�c5�−n/2‖u(t j )‖L1(�)+�· sup

�∈(0,�)‖w(t j +�)‖L∞(�) (35)

for all �>0 and some c5>0. By (13) and the boundedness of w, we obtain a uniform bound for‖u(t j +�)‖L∞(�) for �∈(1,2) and infer also that u is bounded in �×(0,∞). The correspondingL∞ estimate for v can be derived following exactly the same lines. �

We postpone our final result providing global bounded solutions to the full original system (3)and (4) until the following section, because for the proof of boundedness of X� (being equivalentto that of the solution z to (8)) we need to know that w, regarded as inhomogeneity in (8), decayssufficiently fast in time. This will be a by-product of our first result on asymptotic behavior below.

3. ASYMPTOTIC BEHAVIOR OF SOLUTIONS

As in the previous section, we start with some simple integrations of suitable linear combinationsof equations in (7). As a starting point for our further analysis, we thereby obtain some ratherweak but useful information on the large time behavior of solutions, expressed in terms of certainfinite integrals over the whole semiaxis t ∈(0,∞).

This is the only point in this paper where the assumption l>1 is essentially referred to—ratherthan l�1—which would have been sufficient up to now.

Lemma 3.1We have ∫ ∞

0

∫�uv<∞ and

∫ ∞

0

∫�

w<∞ (36)

In particular, there exists a sequence t j →∞ such that∫�u(x, t j )v(x, t j )dx→0 and

∫�

w(x, t j )→0 as j →∞dx (37)

ProofIntegrating (12) with respect to t , we obtain∫

�(u(x, t)−v(x, t))dx+(l−1)

∫ t

0

∫�

w(x,s)dx ds=∫

�(u0(x)−v0(x))dx

for t>0. As v is bounded, this implies that∫∞0

∫� w<∞, because l>1.


1916 M. WINKLER

We now integrate the first equation in (7) with respect to x and t to see that∫�u(x, t)dx+

∫ t

0

∫�u(x,s)v(x,s)dx ds=

∫�u0(x)dx+

∫ t

0

∫�

w(x, t)dx

for t>0. According to what we have just shown, the right-hand side is bounded, so that (36) andhence also (37) follows. �

Transformed back to the original variables, Theorem 2.4 in conjunction with the above lemmaimmediately yields the following.

Corollary 3.2Assume that k1,k−1 and k2 are given positive constants, and let X0,E0,C0 and X�

0 be continuousand nonnegative in �. Then (3) and (4) posses a unique global solution (X,E,C, X�). This solutionis nonnegative and bounded in �×(0,∞).

ProofFirst, Theorem 2.4 says that u,v and w are global and bounded. In particular, the inhomogeneity in(8) is globally bounded, so that standard theory [11] implies that (8) has a unique global classicalsolution z. Integrating the PDE in (8) over �×(0,∞) and using the second inequality in (36)reveals that ‖z(t)‖L1(�) is uniformly bounded for all t>0. Now a reasoning almost identical tothat in the proof of boundedness of u in Theorem 2.4 (cf. (34) and (35)) shows that in fact z isbounded in L∞(�×(0,∞)). Read in the original coordinates, this yields the claim. �

Throughout the remainder of this section, we assume that

Dv =Dw =:Dand that

v0+w0 /≡0 (38)

and let

�(x, t) :=v(x, t)+w(x, t), x ∈�, t�0 (39)

Then adding the second to the third equation in (7) we see that � satisfies the heat equation �t =D�� with homogeneous Neumann boundary conditions and initial data �(x,0)=v0(x)+w0(x).It is well known (cf. also (23)) that � stabilizes to its spatial mean at an exponential rate; moreprecisely,

‖�(t)−‖L∞(�)�c�e−D�1t for all t�0 (40)

with some c�>0 and the first positive eigenvalue �1 of the Neumann Laplacian −� in �. Thenumber

= 1

|�| ·∫

�(v0(x)+w0(x))dx (41)

is positive by (38).We proceed to extract a subsequence of times t j →∞ along which each of the components

u,v and w stabilize to their respective spatially homogeneous limits suggested by (37) and (40).



Owing to parabolic theory, we can of course choose L∞(�) as the topology defining space forthis convergence.

Lemma 3.3Assume that Dv =Dw and that v0+w0 /≡0. Then there exists t j →∞ such that

‖u(t j )‖L∞(�) → 0 (42)

‖v(t j )−‖L∞(�) → 0 (43)

and

‖w(t j )‖L∞(�) →0 (44)

as j →∞, where is given by (41).

ProofAs u,v and w are bounded, we may apply standard regularity theory for scalar parabolic problems[11] to each of the equations in (7) to obtain uniform estimates for u,v and w in the Holder spaceC,/2(�×[1,T ]) for T>1 and some ∈(0,1), where the corresponding bounds are independentof T . Hence, the Arzela–Ascoli theorem ensures that the semiorbits (u(t))t>1, (v(t))t>1 and(w(t))t>1 are precompact in C0(�).

Now let t j →∞ be as in Lemma 3.1, that is, such that∫�

w(x, t j )dx→0 as j →∞ (45)

and ∫�

v(x, t j )dx→0 as j →∞ (46)

Then along a subsequence, again denoted by t j , we have w(t j )→w∞ in C0(�) with somenonnegative continuous w∞. As

∫� w∞(x)dx must be zero by (45), we have w∞ ≡0 and thus

(44) follows. Now (43) results from this and (40), whereas (46) and (43) combined with a similarcompactness argument imply (42). �

Our final result asserts stabilization along the entire net t→∞ at a rate that is at least exponential,the exponent being explicitly bounded from below. Moreover, the corresponding rates of decay ofu and w can be estimated even independently of the size of �.

Roughly speaking, in the proof we wait until some sufficiently large t= t j from Lemma 3.3 inorder to be sure that at this time the solution is captured in a suitably small neighborhood of thesteady state to be approached. Beyond this time, we employ a comparison technique for a two-component cooperative parabolic system associated with (7), using explicit comparison functionshaving the desired asymptotics.

Theorem 3.4Assume that Dv =Dw ≡D and v0+w0 /≡0. Let =1/|�|∫�(v0(x)+w0(x))dx and the positivenumber �0 be given by

�0 := +l−√(+l)2−4(l−1)

2(47)


1918 M. WINKLER

Then for all �∈(0,�0) there exists C�>0 such that

‖u(t)‖L∞(�) �C�e−�t (48)

‖v(t)−‖L∞(�) �C�e−min{�,D�1}·t (49)

and

‖w(t)‖L∞(�)�C�e−�t (50)

for all t>0, where �1>0 is as in (40).

ProofUsing �=v+w, we can eliminate v from (7) and thereby reduce the problem to the system oftwo equations given by{

ut =Du�u−�(x, t)u+uw+w, x ∈�, t>0

wt =D�w+�(x, t)u−uw−lw, x ∈�, t>0(51)

Recalling that u�0 and v�0 and hence w��, this can equivalently be rewritten as{ut =Du�u−�(x, t)u+u+w+w, x ∈�, t>0

wt =D�w+(�(x, t)−w)+·u−lw, x ∈�, t>0(52)

As f (x, t,u,w) :=−�(x, t)u+u+w+w and g(x, t,u,w) :=(�(x, t)−w)+·u+lw satisfy � f /�w

=u++1>0 and �g/�u=(�(x, t)−w)+�0 for all (x, t,u,w)∈�×(0,∞)×R×R, it follows thatsystem (52) is cooperative and hence allows for a comparison principle; a short proof for this canbe organized following Section 52.7 in [6], for instance.

In order to construct an appropriate supersolution pair (u,w), let �∈(0,�0) be given. As �0 isthe smaller root of �(�′) :=(−�′)(l−�′)−, we can fix ε>0 such that (−ε)/(l−�)<−ε−�and then a>0 such that

−ε

l−�<

−ε−�

1+a

It is then possible to choose b>0 fulfilling

+ε

l−�a<b<

−ε−�

1+aa (53)

According to Lemma 3.3 and (40), we can now pick some large t0>0 such that

‖u(t0)‖L∞(�) � a (54)

‖w(t0)‖L∞(�) � b (55)

and

‖�(t)−‖L∞(�)�ε for all t�t0 (56)

Let

u(x, t) :=ae−�(t−t0) and w(x, t) :=be−�(t−t0)



for x ∈ � and t�t0. Then, clearly, �u/��=�w/��=0 on ��, and at t=0 we have u�u as well asw�w. Using (56), we find

ut −Du�u+�u−u+w−w = ut +�u−uw−w

� ut +(−ε)u−uw−w

= −�ae−�(t−t0)+(−ε)ae−�(t−t0)

−abe−2�(t−t0)−be−�(t−t0)

� (−�a+(−ε)a−ab−b)e−�(t−t0)

> 0 in �×(t0,∞)

in view of the second inequality in (53). Moreover, as w�b<−ε�� in �×(t0,∞) by (53) and(56), we have

wt −D�w−(�−w)+u+lw = wt −�u+uw+lw

� wt −(+ε)u+uw−lw

= −�be−�(t−t0)−(+ε)ae−�(t−t0)

+abe−2�(t−t0)+lbe−�(t−t0)

� (−�b−(+ε)a+lb)e−�(t−t0)

> 0 in �×(t0,∞)

according to the first inequality in (53). Consequently, (u,w) is a supersolution to (52) in�×(t0,∞)

and thus both u�u and w�w hold in �×(t0,∞). These majorizations imply (48) and (50),whereas (49) results from (50), (40) and the fact that �=v+w. �

Lemma 3.5Suppose that l ′>0, Dv =Dw ≡D, v0+w0 /≡0, and let �0 and �1 be as defined in (47) and (40),respectively. Then for all �∈(0,min{�0,Dz�1}) there exists C�>0 such that the solution z to (8)satisfies

‖z(t)− ‖L∞(�)�C�e−�t for all t>0 (57)

where the positive constant is given by

:= 1

|�|∫

�z0+ l ′

|�|∫ ∞

0

∫�

w (58)

ProofThe proof consists of a straightforward analysis of (8) in light of the knowledge on w providedby (50); for completeness, we outline the basic steps, however, in order to show why we need torestrict � in the above way.

We first multiply (8) by −�z and integrate by parts to obtain

1

2

d

dt

∫�

|∇z|2=−Dz

∫�

|�z|2−l ′∫

�w�z (59)


1920 M. WINKLER

for t>0. Now given �∈(0,min{�0,Dz�1}), we fix ε>0 such that �<(1−ε)Dz�1, then some �1>�satisfying �1<(1−ε)Dz�1 and finally m∈N large enough fulfilling (1−n/2m)�1��. Then from(50) we know that

‖w(t)‖L∞(�)�c1e−�1t (60)

holds for some c1>0 and all t>0. Using this, the Young inequality and the Poincare inequality∫� |�z|2��1

∫� |∇z|2, we derive from (59) that

1

2

d

dt

∫�

|∇z|2 � −(1−ε)Dz

∫�

|�z|2+ l ′2

4εDz

∫�

w2

� −(1−ε)Dz�1

∫�

|∇z|2+c2e−2�1t for all t>0

with a certain constant c2>0. Integrating this ordinary differential inequality for y(t) :=∫� |∇z(t)|2gives, as (1−ε)Dz�1>�1,∫

�|∇z(t)|2 � y(0) ·e−2(1−ε)Dz�1t + c2

(1−ε)Dz�1−�1(e−2�1t −e−2(1−ε)Dz�1t )

�(y(0)+ c2

(1−ε)Dz�1>�1

)e−2�1t for all t>0

Again using the Poincare inequality, we infer that ‖z(t)−z(t)‖L2(�)�c3e−�1t with some c3>0,where z(t) :=1/|�|∫� z(x, t)dx . This together with the fact that z(t) remains bounded in Wm,2(�)

uniformly in t in view of parabolic regularity results; invoking a Gagliardo–Nirenberg interpolationwe obtain

‖z(t)−z(t)‖L∞(�) � c4‖z(t)−z(t)‖n/2mWm,2(�)

‖z(t)−z(t)‖1−n/2mL2(�)

� c5e(1−n/2m)�1t for t>0 (61)

with positive constants c4 and c5.In order to obtain information about z(t), we integrate (8) to find z(t)= z(0)+l ′/|�|∫ t

0

∫� w.

Recalling (58) and again using (60), we thus have

0� −z(t) = l ′

|�|∫ ∞

t

∫�

w

� l ′c1�1

e−�1t for t>0

Combining this with (61) yields the claim. �

Passing to the original variables, we have without further comment our final result on theasymptotic behavior of solutions to (3) and (4).

Corollary 3.6Let k1,k−1 and k2 be given positive constants and assume that DE =DC =D. Moreover,suppose that X0,E0,C0 and X�

0 are continuous and nonnegative functions in � with E0+C0 /≡0.



Let l=(k−1+k2)/k−1, =k1/|�|k−1∫�(E0+C0) and

�0 := +l−√(+l)2−4(l−1)

2·k−1

Then for all ε>0 there exists Cε>0 such that the solution (X,E,C, X�) to (3) and (4) satisfies

‖X (t)‖L∞(�) �Cεe−(�0−ε)t

‖E(t)−E∞‖L∞(�) �Cεe−(min{�0,DE�1}−ε)t

‖C(t)‖L∞(�) �Cεe−(�0−ε)t

and

‖X�(t)−X�∞‖L∞(�)�Cεe−(min{�0,DX��1}−ε)t

for all t>0, where E∞ and X�∞ are given by (5) and (6), respectively.

RemarkTaking spatially constant initial data in (3) and (4) as a by-product we obtain from Corollary 3.6also that all solutions of the ODE system (2) converge to their respective equilibria at exponentialrates.

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Boundedness and exponential stabilization in a signal transduction model with diffusion