IOP PUBLISHING NONLINEARITY

Nonlinearity 21 (2008) 71–96 doi:10.1088/0951-7715/21/1/004

Calcium waves in systems with immobile buffers as alimit of waves for systems with nonzero diffusion

Bogdan Kazmierczak1 and Vitaly Volpert2

1 Institute of Fundamental Technological Research of PAS Swietokrzyska 21, 00-049 Warsaw,Poland2 Department of Mathematics, UMR 5208 CNRS, Universite Lyon 1, 69622 Villeurbanne, France

E-mail: bkazmier@ippt.gov.pl and volpert@maply.univ-lyon1.fr

Received 5 February 2007, in final form 3 October 2007Published 7 December 2007Online at stacks.iop.org/Non/21/71

Recommended by L Ryzhik

AbstractWe study the existence and properties of calcium waves in the presence ofbuffers. The model represents a reaction–diffusion system of equations withsome diffusion coefficients equal to zero. They correspond to immobile buffers.The proof of the existence of travelling waves is carried out by passing to zeroin the diffusion coefficients.

Mathematics Subject Classification: 34A34, 35K57

1. Introduction

One of the most important mechanisms by which cells control their activity and coordinate itwith their neighbours are calcium oscillations and waves. It is known that their propagation issignificantly influenced by the presence of buffers. Buffers are big proteins (e.g. parvalbumin,calsequestrin, calretinin or EGTA) which can bind a large amount of free calcium inside cells.The amount of Ca2+ ions, which can be bound to different kinds of buffers may reach 99%.Most of the buffers are almost immobile; however, the diffusion coefficient of some of themis not negligible and is in the range between one-tenth to one-half of the diffusion coefficientfor free Ca2+ [3]. The reaction of binding Ca2+ to the ith protein Bi to form the ith bindingcomplex Ca2+Bi (i = 1, . . . , n) can be written as

Bi + Ca2+ � Ca2+Bi.

We assume that the kinetic constant of the ith binding reaction is equal to ki+ and the kinetic

constant of the reverse reaction (release of calcium by the ith buffer) is equal to ki−. The

action of buffers is to stabilize the concentration of the intracellular calcium in an appropriaterange [3].

In this paper we analyse the problem of existence and properties of travelling wavesolutions to the system of equations describing the evolution of the concentration of the free

0951-7715/08/010071+26$30.00 © 2008 IOP Publishing Ltd and London Mathematical Society Printed in the UK 71

72 B Kazmierczak and V Volpert

cytosolic calcium inside cells. The results of the paper are twofold. First we prove the existenceand some properties of the travelling wave solutions for the system of equations with nonzerodiffusion coefficients of n buffers. To do this we use the theory of travelling waves for parabolicsystems contained in [11].

Then by applying appropriate limiting procedure we obtain the existence of waves for thesystem with nonzero diffusion coefficients of M � 0 buffers and zero diffusion coefficientsfor (n − M) � 0 buffers. The existence of travelling waves for the system with nondiffusingbuffers, i.e. for M = 0, was proved recently in a straightforward way in the paper [10] andthe asymptotic in time properties of solutions to this system were analysed in [7]. The methodused in this paper allows us to consider the system with nondiffusing buffers as a limit ofequations with small diffusing coefficients and provides some additional information aboutthe behaviour of solutions to the considered family of systems. Moreover, as noted above, thegeneralization to the case M � 1 seems sensible due to the fact that some of the buffers mayhave nonnegligible diffusion coefficients.

The evolution of the concentration of the free cytosolic calcium can be described by thesystem of reaction–diffusion equations of the form

∂u

∂t= D

∂2u

∂x2+ f (u) +

n∑i=1

[ki−(bi

0 − vi ) − ki+uvi],

∂vi

∂t= Di

∂2vi

∂x2+ ki

−(bi0 − vi ) − ki

+uvi, i = 1, . . . , n,

(1)

(see [10]), where u is the free calcium concentration, f (u) is the function describing calciumtransport into and out of the cytosol. Here vi denotes the concentration of those moleculesof the ith buffer which are not bound to calcium, whereas bi

0 denotes the total concentrationof the ith buffer, ki

−, ki+ are the kinetic coefficients. We have thus bi

0 = [Bi] + [Ca2+ Bi] withvi = [Bi]. D denotes the diffusion coefficient of calcium and Di , i = 1, . . . , n, denotes thediffusion coefficient of the ith kind of buffer particles. It is convenient to make an appropriatechange of variables. Namely, we define vi = bi

0 − vi . That is to say vi = [Ca2+ Bi]. System(1) can thus be written in the form

∂u

∂t= D

∂2u

∂x2+ f (u) +

n∑i=1

[ki−vi − ki

+u(bi0 − vi)],

∂vi

∂t= Di

∂2vi

∂x2− [ki

−vi − ki+u(bi

0 − vi)], i = 1, . . . , n.

(2)

In fact, this kind of variable is used in the paper [8] analysing travelling waves for buffersystems with very fast kinetics.

Assumption 1. Assume that

1. The function f (·) ∈ C2(R) is of bistable type, i.e. that the equation f (u) = 0 has exactlythree solutions: u1 � 0, u3 > u1 and u2 ∈ (u1, u3). The zeros u1 and u3 are stable, i.e.f ′(u1) < 0, f ′(u3) < 0, whereas u2 is unstable, i.e. f ′(u2) > 0.

2. ki−, ki

+ > 0. �

A simple example of a function satisfying condition 1 of the above assumption is a cubicpolynomial f (u) = (u − s)(1 − u)(u − u0) with s ∈ [0, 1) and u0 ∈ (s, 1).

Remark. Let us note that system (2) satisfies so called monotonicity conditions. To bemore precise the derivative of the nonlinearity terms in the equation for u with respectto vi , i = 1, . . . , n, is positive. Likewise, the derivative of the nonlinearity term in the

Calcium waves in systems with immobile buffers 73

equation for vi with respect to u is positive whereas its derivative with respect to vj , j �= i, isidentically zero. Thus it is possible to use the theory contained in [11] (see section 3 for moredetails). �

There are exactly three constant steady states of the system. They correspond to the threesolutions of the equation f (u) = 0. So they are of the form

Pk = (uk, vk1, . . . , v

kn), k = 1, 2, 3, (3)

where

vkj = uk

kj+b

j

0

(kj− + k

j+uk)

. (4)

It is easy to note that

v1j < v2

j < v3j , j = 1, . . . , n, (5)

thus component wise

P1 < P2 < P3. (6)

In this paper we are interested in the travelling wave solutions to system (2) joining the constantsteady states P1 and P3. To be more precise we are looking for solutions being functions of ascalar variable ξ = x − qt , i.e.

u(x, t) = u(ξ), vi(x, t) = vi(ξ), (7)

satisfying the following conditions:

limξ→−∞

(u(ξ), v1(ξ), . . . , vn(ξ)) = (u1, v11, . . . , v

1n) = P1,

limξ→∞

(u(ξ), v1(ξ), . . . , vn(ξ)) = (u3, v31, . . . , v

3n) = P3,

lim|ξ |→∞

(u′(ξ), v′1(ξ), . . . , v′

n(ξ)) = (0, 0, . . . , 0).

(8)

Thus the heteroclinic travelling wave connects the state of low concentration of calcium (bothfree and bound to the buffers) with the state of large concentration of calcium (both free andbound to the buffers).

Assumption (7) changes equations (2) to the system

Du′′ + qu′ + f (u) +n∑

i=1

Gi(u, vi) = 0, (9)

Div′′i + qv′

i − Gi(u, vi) = 0, i = 1, . . . , n, (10)

where ′ denotes differentiation with respect to the variable ξ and

Gi(u, vi) = ki−vi − ki

+u(bi0 − vi). (11)

Remark. Let us note that the functions Gi given by (11) are of C∞ class of theirarguments. �As we mentioned above our main aim is to prove the existence of heteroclinic solutionsfor a partially degenerate version of system (9) and (10) obtained by setting Di to zero fori = M + 1, . . . , n, i.e. the system

Du′′ + qu′ + f (u) +n∑

i=1

Gi(u, vi) = 0, (12)

Div′′i + qv′

i − Gi(u, vi) = 0, i = 1, . . . , M, (13)

qv′i − Gi(u, vi) = 0, i = M + 1, . . . , n. (14)

74 B Kazmierczak and V Volpert

The method of the proof is to pass to the limit in the family of solutions indexed by thediffusion coefficients of system (9) and (10) with Di > 0, i = M + 1, . . . , n. The existence ofheteroclinic solutions to system (9) and (10) with Di > 0 is guaranteed by the theory containedin [11].

In the proof we will refer to the properties of the subsystem describing the evolution ofcalcium and only these buffers for which the diffusion coefficients stay positive in the limitingprocess. This system has the following form:

Du′′ + qu′ + f (u) +M∑i=1

Gi(u, vi) = 0, (15)

Div′′i + qv′

i − Gi(u, vi) = 0, i = 1, . . . , M. (16)

It is easy to note that the constant states for system (15) and (16), which will be denotedby P1, P2, P3, satisfy the relations (3), (5), (6) with j ∈ {1, . . . , M}. Similarly to theheteroclinic solutions joining the states P1 and P3 for system (9) and (10) one can consider theheteroclinic solutions of system (15) and (16) joining its constant states P1 and P3. Accordingto theorem 2.1, p 15 in [11] and the results of section 2 (see the remark after theorem 2), thereexists a unique (up to a translation in ξ ) heteroclinic solution (qr , Ur) for system (15) and (16)satisfying conditions (8) with P1, P3 replaced by P1, P3 and n replaced by M . System (15)and (16) is used to determine the sign of q for system (9) and (10) (see theorems 3 and 6 andlemma 18).

The results of the paper are the following. Some auxiliary results about the principaleigenvalues of the matrices used below are given in section 2. The existence of monotonictravelling wave solutions for system (2) with D > 0, Di > 0, i = 1, . . . , n is shown in section 3and formulated in theorem 2. It is known that nonmonotonic travelling waves for system (2)with nonzero diffusion coefficients are unstable (theorem 6.2 and corollary, p 254 in [11]),whereas the monotonic ones are stable (theorem 6.1, p 250 in [11]). In section 4 we obtaina priori estimates of the modulus of the speed q and the C2(R) norms of the profiles formonotonic waves. The crucial fact is that these estimations are independent of the value ofthe diffusion coefficients Di > 0, i = M + 1, . . . , n, which may be arbitrarily small. Bymeans of these estimations, on every compact subset of R, we are able to pass to the limit inthe sequence of heteroclinic solutions corresponding to a decreasing to zero sequence of thediffusion coefficients Di , i = M + 1, . . . , n. The existence result for monotonic travellingwave solutions to system (2) with D > 0, Di > 0 for i = 1, . . . , M , and Di = 0 fori = M + 1, . . . , n is proved in section 5 and formulated in theorem 3 (see also theorem 6). Insection 6 we prove the uniqueness of monotonic travelling waves (theorem 4). Their structuralstability is proved in section 7 (theorem 5). In fact all the results of the paper are valid formore general forms of the functions Gi . This generalization is given in the final remark of thepaper. In this paper we do not examine the question of stability of the travelling waves withrespect to small perturbations of the initial conditions for system (2). For M = 0 the stabilityof solutions and their attractivity properties is proved in [10]. The problem of stability forM > 0 will be undertaken in a future paper.

2. Properties of the constant states

To begin with we will analyse the properties of the matrices obtained by linearization of thesource terms at the right-hand sides of system (2) at the points Pk , k = 1, 2, 3. To be moreprecise, we are interested in the properties of their eigenvalues and eigenvectors.

Calcium waves in systems with immobile buffers 75

Consider the following (n + 1) × (n + 1) matrix:

K =

a −n∑

i=1

ai b1 . . . bn

a1 −b1 . . . 0

.. ... . . ..

an 0 . . . −bn

(17)

where

ai > 0, bi > 0, i = 1, . . . , n. (18)

One can easily check that for u = uk + δu, vj = vkj + δvj the first order Taylor expansion of

the source terms of system (2) has the formf (u) +

n∑i=1

[ki−vi − ki

+u(bi0 − vi)]

−k1−v1 + k1

+u(b10 − v1)

. . .

−kn−vn + kn

+u(bn0 − vn)

∼= K

δu

δv1

. . .

δvn

, (19)

where K has the form defined by (17) with

a = f ′(uk), ai = ki+(b

i0 − vk

i ), bi = ki− + ki

+uk. (20)

Obviously, for ai , bi given by the above expressions the relations (18) are satisfied.Below p � 1 will denote a natural number.

Lemma 1 (see theorem 5 in [4], p 350). Let C be a real p × p-matrix with nonnegativeoff-diagonal entries cik � 0, i �= k. Then all the eigenvalues of C have negative real parts iffthe following inequalities are satisfied:

J1 = c11 < 0, J2 =∣∣∣∣∣c11 c12

c21 c22

∣∣∣∣∣ > 0, . . . , Jp = (−1)p

∣∣∣∣∣∣∣∣∣∣c11 c12 . . . c1p

c21 c22 . . . c2p

.. .. . . . ..

cp1 cp2 . . . cpp

∣∣∣∣∣∣∣∣∣∣> 0.

(21)

�The index PF corresponds to the notion of Perron–Frobenius eigenvalue, which is used to callthe maximal (or principal) eigenvalue in the case of matrices with positive off-diagonal entries.

Lemma 2. Let C be a real p ×p, p � 2, matrix with nonnegative off-diagonal entries, whichis irreducible. Then C has a real simple eigenvalue µPF(C) (so called Perron–Frobenius orprincipal eigenvalue) such that an associated eigenvector can be chosen positive and everyother eigenvalue of C has its real part less than µPF(C) (so called Perron–Frobenius orprincipal eigenvector). Moreover, the principal eigenvalue µPF(C) is a strictly increasingfunction of any of its entries.

Proof. The proof of this lemma follows from a series of theorems from [4] (theorem XIII.4.6,theorem XIII.2.1, theorem XIII.2.2 and theorem XIII.3.3). �According to inequalities (18) the matrix K has positive off-diagonal entries. Using lemma 1we can prove that for a < 0 its maximal eigenvalue µPF(K) is negative.

76 B Kazmierczak and V Volpert

Lemma 3. Let a < 0. Then the eigenvalues µ of the matrix K are contained in the lefthalf-plane Re(µ) < 0. �Proof. Let us note that K11 < 0. Consider the matrix Kk = Kij , i, j = 1, . . . , k,k = 2, . . . , n + 1. Let us compute the determinant of the matrix Kk . To this end, let us add thelast k − 1 rows of the matrix Kk to its first row. Expanding the determinant of this matrix withrespect to the first row, we immediately obtain det Kk = (a − ∑n

j=k aj )(−1)k−1b1 . . . bk−1.Its sign is equal to (−1)k . Also det Kn+1 = det K = (a)(−1)nb1 . . . bn. Thus the sequence ofinequalities from lemma 1 is satisfied. The lemma is proved. �The statement converse to lemma 3 is also true. Namely, the following lemma holds.

Lemma 4. Let a > 0. Then at least one of the eigenvalues µ of the matrix K is contained inthe half-plane Re(µ) > 0. �Proof. Let us consider the matrix K with a = 0. Then (by adding to the first row all the otherrows) we note that det(K) = 0. Now, the claim of the lemma follows from the last statementof lemma 2. �

Now, we will analyse the irreducibility of the matrix K .

Lemma 5. Independently of the value of a the matrix K is irreducible. �Proof. Let us recall that an (n+1)×(n+1) matrix C is called reducible if the set {1, . . . , n+1}can be divided into two disjoint subsets I and J , that is to say {1, . . . , n+1} = I ∪J , I ∩J = ∅such that cij = 0 for all i ∈ I, j ∈ J . The matrix is called irreducible, if it is not reducible.

Suppose to the contrary that K is reducible for some value of a. It follows from thedefinition of reducibility that i ∈ I implies i /∈ J and vice versa j ∈ J implies j /∈ I . Ask1j �= 0 for j �= 1 we note that 1 /∈ I . Hence 1 ∈ J . Consequently there exists 1 �= i ∈ I suchthat ki1 = 0. But ki1 can be equal to zero only for i = 1. We thus arrive at a contradictionwhich proves the irreducibility of K independently of the value of a. �

Using lemmas 3–5 one can prove the validity of the following statement.

Lemma 6. Let K be defined by conditions (17) and (18). Then its principal eigenvector (corre-sponding to the principal eigenvalue) may be chosen positive independently of the value of a. �

3. Existence theorem for the nondegenerate system

The following theorem holds.

Theorem 1 (theorem 2.1, p 15 in [11]). Let us consider the system∂U

∂t= A�U + F(U), (22)

where � = ∂2/∂x21 + · · · + ∂2/∂x2

s , s � 1, U = (U1, . . . , Up), p � 2, is a vector-valuedfunction, A is a diagonal positive-definite matrix and C1 F(·) : R

p → Rp. Let system (22)

be monotonic, i.e.∂Fi

∂Uj

� 0, i, j = 1, . . . , p, i �= j.

Further, let the function F(U) vanish at a finite number of points w−, w+ and Uk ,(k = 1, . . . , m) with w− < Uk < w+. Let us assume that all the eigenvalues of the matrices∂F (w−) and ∂F (w+) lie in the left half-plane, and that the matrices ∂F (Uk) (k = 1, . . . , m)are irreducible and have at least one eigenvalue in the right half-plane. Then there exists aunique monotonic travelling wave, i.e. a constant q and a twice continuously differentiable

Calcium waves in systems with immobile buffers 77

monotonic vector-valued function U(ξ), ξ = x1 − qt , satisfying the system

AU ′′ + qU ′ + F(U) = 0, (23)

such that U ′j (ξ) > 0 for all j = 1, . . . , p, ξ ∈ R, and

limξ→±∞

U(ξ) = w±, limξ→±∞

U ′(ξ) = 0. �

By means of the results of section 2 and theorem 1 one can easily check the validity ofthe existence theorem for the travelling wave solutions to system (2) with positive diffusioncoefficients.

Theorem 2. Let assumption 1 be satisfied. Let D, D1, . . . , Dn be positive. Then there existsa unique monotone heteroclinic solution to system (9) and (10) satisfying conditions (8), thatis a unique speed of propagation q and a unique up to translation vector-valued function(u, v1, . . . , vn) of class C2(R). �

Remark. In exactly the same way one can prove the existence of a unique monotonicheteroclinic solution to system (15)–(16), i.e. a unique speed qr and a unique (up to translation)vector valued function connecting the states P1 and P3.

4. A priori estimates of the solutions

Now using the special structure of the considered system we will estimate the derivatives ofthe monotonic heteroclinic solutions to system (9) and (10) satisfying conditions (8). Ouraim is mainly to examine the properties of the heteroclinic solutions as the coefficients Di ,i = M + 1, . . . , n, tend to zero. The estimations derived in this section will be a basis forobtaining a solution to the system with Di = 0, i = M + 1, . . . , n.

Lemma 7. For any C2 monotonic heteroclinic solution (u, v1, . . . , vn)(·) of system (9) and (10)we have the estimations

‖u‖C3(R) < S1, ‖vi‖C3(R) < S2, i = 1, . . . , M,

‖vi‖C2(R) < S3, i = M + 1, . . . , n.(24)

These estimations are independent of the values of the diffusion coefficients Di ,i = M + 1, . . . , n. �Proof. First let us estimate the first derivative of the function u(·). Let us suppose that thesupremum of u′ is attained for some ξ0 ∈ R. As limξ→±∞ u′(ξ) = 0, we must have u′′(ξ0) = 0and q = −G(u(ξ0), v1(ξ0), . . . , vn(ξ0))(u

′(ξ0)−1) where we have denoted

G(u, v1, . . . , vn) = f (u) +n∑

i=1

Gi(u, vi). (25)

Multiplying equation (9) by u′(ξ) and integrating on (−∞, ξ0) we obtain the equality

1

2Du′2(ξ0) = G(u(ξ0), v1(ξ0), . . . , vn(ξ0))

∫ ξ0

−∞

u′(ξ)

u′(ξ0)u′(ξ) dξ

−∫ ξ0

−∞G(u(ξ), v1(ξ), . . . , vn(ξ))u′(ξ) dξ.

Hence, due to the monotonicity of the functions u, v1, . . . , vn, and the continuity of thefunction G, the right-hand side of this equation can be estimated from above by the expression

C

∫ ξ0

−∞u′(ξ) dξ � Cu(ξ0),

78 B Kazmierczak and V Volpert

where C is a constant independent of the value of ξ0. As D > 0 we obtain from here the globalboundedness of the C1 norm of the function u(·) independently of the value of q. In the sameway we can prove the estimate for v′

i (·), i = 1, . . . , M . Next, note that for all ξ ∈ R we have|qu′(ξ)| � |qu′(ξ0)| (where ξ0 is the point of supremum of u′). Likewise, for i = 1, . . . , M ,|qv′

i (ξ )| � |qv′i (ζi)| for some ζi . The quantities on the right-hand sides of these inequalities

are estimated only by the properties of the functions G and Gi , respectively. Thus using theequations for u and vi , i = 1, . . . , M , we can also estimate the second derivatives of thesefunctions. As a result we have proved that

‖u‖C2(R) < S1, ‖vi‖C2(R) < S2, i = 1, . . . , M. (26)

Now, using what we have obtained above, we will prove that also the C1 norms of thefunctions vk , k = M + 1, . . . , n, are bounded without imposing any lower bounds for thecoefficients Dk > 0. Suppose that v′

k(·) attains its global maximum at some ξ = ξ0. (Thenξ0 must be finite as v′

k(ξ) → 0 for ξ → ±∞.) Differentiating the kth (k = M + 1, . . . , n)equation of system (10) with respect to ξ we obtain the following equation

0 = Dkv′′′k (ξ0) − Gk,vk

(u(ξ0), vk(ξ0))v′k(ξ0) − Gk,u(u(ξ0), vk(ξ0))u

′(ξ0).

As ξ0 is the point of the global maximum of v′k we must have v′′′

k (ξ0) � 0. Now, independentlyof ξ0,

Gk,vk(u(ξ0), vk(ξ0)) � gk > 0, (27)

(see point 2 of assumption 1) so the sum of the first two terms at the right-hand side can beannihilated only by the third term as −Gk,u(u(ξ0), vk(ξ0)) > 0. However, the last quantity isbounded for any P1 � (u, v1, . . . , vn) � P3. Thus also v′

k(·) must be bounded in its C1 normfor all k ∈ {M + 1, . . . , n}.

Finally, let us estimate the second derivatives of the functions vk , k = M + 1, . . . , n,independently of the values of Dk > 0. These estimations are also based on inequalities (27).Namely, differentiating twice the equation for vk , k = M + 1, . . . , n (see the remark afterequation (11)), we obtain for ξ = ξ0

0 = Dkv′′′′k (ξ0) + qv′′′

k (ξ0) − Gk,vk(u(ξ0), vk(ξ0))v

′′k (ξ0) − Gk,vk vk

(u(ξ0), vk(ξ0))v′k(ξ0)v

′k(ξ0)

− Gk,u(u(ξ0), vk(ξ0))u′′(ξ0) − 2Gk,u vk

(u(ξ0), vk(ξ0))u′(ξ0)v

′k(ξ0)

− Gk,u u(u(ξ0), vk(ξ0))u′(ξ0)u

′(ξ0).

(Note that due to the specific form of the functions Gk the second derivatives Gk,vkvk(u, vk)

and Gk,uu(u, vk) are zero identically, but it is not essential here.) Suppose that v′′k attains its

global maximum (minimum) at ξ = ξ0. Then v′′′k (ξ0) = 0 and v′′′′

k (ξ0) � 0 (v′′′′k (ξ0) � 0).

Thus using the inequality (27) we infer that |v′′k (ξ0)| can be estimated by |u|C2(R) and |v|C1(R).

Next, differentiating the equations for u, v1, . . . , vM and using the arguments leading to theestimations of the second derivatives of these functions (as before (26)), we obtain the estimatesof the third derivatives of u, v1, . . . , vM . We have thus completed the estimations (24). Thelemma is proved. �

Having the estimations for the derivatives of the monotonic solutions we are able toestimate the absolute value of the wave speed q. We will apply the method used in [2,11]. Tosimplify the notation we define

U := (U1, U2, . . . , Un+1) := (u, v1, . . . , vn), F1(U) := G(u, v1, . . . , vn),

Fj (U) := −Gj−1(u, vj−1), j = 2, . . . , n + 1, F (U) := (F1(U), F2(U), . . . , Fn+1(U)),

(28)

with G given by (25).

Calcium waves in systems with immobile buffers 79

Lemma 8. Let F be defined as above. Let N(P1) and N(P3) denote the eigenvectorscorresponding to the Perron–Frobenius eigenvalues of DF(P1) and DF(P3) respectively.Then there exist r > 0 and ϑ > 0 such that for each i ∈ {1, . . . , n + 1}

dist(U, W0i ) < ϑ �⇒ Fi(U) < 0,

dist(U, W1i ) < ϑ �⇒ Fi(U) > 0,

where

W0i = {U : P1 � U � P1 + rN(P1), Ui = P1i + rNi(P1)},W1i = {U : P3 � U � P3 − rN(P3), Ui = P3i − rNi(P3)}. �

Proof. Let µPF denote the Perron–Frobenius eigenvalue of DF(P1). Then r may betaken so small that F(P1 + rN(P1)) = r DF N(P1) + o(r) < 1

2 rµPFN(P1). By means ofconditions (17), (18) we conclude that Fi(U) < 0 for U ∈ W0i . In consequence there existsϑ = ϑ(r) > 0 such that the first of the above relations is satisfied. In the same way we provethe second relation. �

Lemma 9. Let F and r be the same as in lemma 8. Then for any point U ∈ W0 = {U : P1 �U � P1 + rN(P1), U �= P1} there exists i ∈ {1, . . . , n + 1} such that Fi(U) < 0. Likewise, forany point U ∈ W1 = {U : P3 � U � P3 − rN(P3), U �= P3} there exists i ∈ {1, . . . , n + 1}such that Fi(U) > 0. �

Proof. Let us take an arbitrary point U = (U1, . . . , Un+1) ∈ W0. Let

r = maxj

(Uj − P1j )(rNj (P1))−1 = (Uk − P1k)(rNk(P1))

−1

for some k ∈ {1, . . . , n + 1}. (Let us recall that the components of the Perron–Frobeniuseigenvectors are strictly positive according to lemma 6.) As r � r then it follows from theproof of lemma 8 that it holds with r replaced by r and ϑ(r) replaced by ϑ(r). In consequenceFk(U) < 0 for U ∈ W0k = {U : P1 � U � P1 + rN(P1), Uk = P1k + rNk(P1)}. In the sameway we consider the parallelepiped W1. The lemma is proved. �As a corollary to lemma 9 we have the following lemma.

Lemma 10. Let F be the same as in lemma 8. Then there does not exist a point U with0 <| U − P1 |< δ, δ sufficiently small, such that Fi(U) � 0 for all i ∈ {1, . . . , n}. Likewisethere does not exist a point U, 0 <| P3 − U |< δ, δ sufficiently small, such that Fi(U) � 0 forall i ∈ {1, . . . , n + 1}. �

Proof. It suffices to take δ < r and apply lemma 9. �

Now, we are able to prove a priori estimates for q.

Lemma 11. If (q, U(·)) is a strictly monotonic heteroclinic solution for system (9) and (10),then |q| < Q, where Q is independent of U . �

Proof. As U(ξ) → P1 monotonically as ξ → −∞, then there must exist an index i andξ = ξ0 such that U(ξ) enters the region P1 � U � P1 + rN(P1) through the (closed) setW0i , i.e. U(ξ0) ∈ W0i (see lemma 8). Let us take ξ1 < ξ0 such that Ui(ξ0) − Ui(ξ1) = ϑ

2 .Integrating the ith equation of system (9) and (10) we obtain

R1 + qϑ

2+

∫ ξ0

ξ1

Fi(U(s)) ds = 0, (29)

80 B Kazmierczak and V Volpert

where |R1| ∈ (0, max{S1, S2, S3}) according to lemma 7 and Fi(U(s)) < 0 for s ∈ (ξ1, ξ0)

according to lemma 10. If q � 0 then q � −2R1ϑ−1. If q > 0 then by analysing the behaviour

of a heteroclinic trajectory near P3 we can prove the upper bound for q. �

5. Existence of a solution to the degenerate system

In this section, we prove the existence of a heteroclinic solution to system (9) and (10) withDi = 0 for i = M + 1, . . . , n. The proof will be carried out by passing to the limit Di → 0 inthe family of solutions for the nondegenerate system (with Di > 0).

For Di = 0, i = M + 1, . . . , n, and for q �= 0, system (9) and (10) can be written as a firstorder system of ODEs of the form

u′ = z,

v′i = zi, i = 1, . . . , M,

z′ = 1

D

[−qz − f (u) −

n∑k=1

Gk(u, vk)

],

z′i = 1

Di

[−qzi + Gi(u, vi)], i = 1, . . . , M,

v′i = 1

qGi(u, vi), i = M + 1, . . . , n,

(30)

with the linearization around the point (u2, v21, . . . , v

2M, 0, 0, . . . , 0︸ ︷︷ ︸

M times

, v2M+1, . . . , v

2n) (which

corresponds to P2) having the following form:

N ′ = LN, (31)

where

N = (hu, h∗1, . . . , h∗M, hz, h1, . . . , hn)T

and

L =

0 0 . . . 0 1 0 . . . 0 0 . . . 0

0 0 . . . 0 0 1 . . . 0 0 . . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 . . . 0 0 0 . . . 1 0 . . . 0

1

D

[−a +

n∑i=1

ai

]− 1

Db1 . . . − 1

DbM − 1

Dq 0 . . . 0 − 1

DbM+1 . . . − 1

Dbn

− 1

D1a1

1

D1b1 . . . 0 0 − 1

D1q . . . 0 0 . . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

− 1

DM

aM 0 . . .1

DM

bM 0 0 . . . − 1

DM

q 0 . . . 0

− 1

qaM+1 0 . . . 0 0 0 . . . 0

1

qbM+1 . . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

− 1

qan 0 . . . 0 0 0 . . . 0 0 . . .

1

qbn

,

(32)

Calcium waves in systems with immobile buffers 81

with a, ai and bi satisfying (18) and (20) for k = 2. The block structure of the matrix L is thesame as that of the matrix defined in equation (46).

First we will prove that while decreasing the coefficients Di , i = M +1, . . . , n, to zero theheteroclinic solution joining P1 with P3 cannot split into two waves joining in turn the pointsP1 with P2 and P2 with P3. (For the details see [5], p 478.) Suppose to the contrary that theabove splitting into two heteroclinics is possible. These waves would be a limit of heteroclinicswith positive first derivatives. Then, for a fixed q �= 0 there would exist simultaneously

1. nonnegative eigenvalue λ+ of the matrix (32) with a > 0 and a corresponding eigenvectorN2+ = (hu, h∗1, . . . , h∗M, hz, h1, . . . , hn)

T with nonnegative components and2. nonpositive eigenvalue λ− of the matrix (32) with a > 0 and a corresponding eigenvector

N2− = (hu, h∗1, . . . , h∗M, hz, h1, . . . , hn)T with hu � 0, h∗k � 0, hz � 0, hk � 0 for

k = 1, . . . , M and hk � 0 for k = M + 1, . . . , n.

We note that the existence of such eigenvalues would follow from the existence oftwo nondecreasing heteroclinic trajectories. The first one starts from P2 into the set{(U : U − P2 � 0 componentwise} as ξ increases from (−∞), whereas the second trajectoryachieves P2 from the set {(U : U − P2 � 0 componentwise} as ξ tends to (∞).

Below, we will prove that these conditions cannot occur for the same value of q.

Lemma 12. For a fixed q �= 0 conditions 1 and 2 cannot be fulfilled simultaneously. �

Proof. First suppose that q > 0. Let us assume that there exists a nonnegative eigenvalueλ+ � 0 and a nonzero associated eigenvector which is nonnegative and has the formN2+ = (hu, h∗1, . . . , h∗M, hz, h1, . . . , hn)

T. We have

LN2+ =

hz

h1

. . .

hM

1

D

[−a +

n∑i=1

ai

]hu − q

Dhz − 1

D

(M∑

k=1

bkh∗k +n∑

k=M+1

bkhk

)

− 1

D1a1hu +

1

D1b1h∗1 − q

D1h1

. . .

− 1

DM

aMhu +1

DM

bMh∗M − q

DM

hM

− 1

qaM+1hu +

1

qbM+1hM+1

. . .

− 1

qanhu +

1

qbnhn

. (33)

Let Hk = h∗k for k = 1, . . . , M and Hk = hk for k = M + 1, . . . , n. Then the nonnegativityof the last n components of LN2+ implies that

n∑k=1

(−huak + Hkbk) � q

M∑j=1

hj .

82 B Kazmierczak and V Volpert

On the other hand, the nonnegativity of the (M + 2)-component impliesn∑

k=1

(huak − Hkbk) − qhz − hua � 0, k = 1, . . . , n.

Thus for q > 0 and a > 0 we have either hu = 0, hz = 0, Hk = 0, k = 1, . . . , n, together withhj = 0, j = 1, . . . , M , which means that N2+ ≡ 0 or hu < 0, contradicting our assumptions,which implies that the last two inequalities cannot be satisfied simultaneously.

Consider now the case q < 0. Suppose that there exist a nonpositive eigenvalue λ− andthe associated eigenvector N2− = (hu, h∗1, . . . , h∗M, hz, h1, . . . , hn)

T with hu � 0, h∗k � 0,hz � 0, hk � 0 for k = 1, . . . , M and and hk � 0 for k = M + 1, . . . , n. Then

(LN2−)l+M+2 = λ−hl � 0, l = 1, . . . , n.

These relations can be written as

−alhu + blh∗l = qhl + Dlλ−hl � 0, l = 1, . . . , M,

−alhu + blhl = qλ−hl, l = M + 1, . . . , n.

Hence, with Hk defined as above we haven∑

k=1

(huak − Hkbk) � 0.

As (LN2−)M+2 = λ−hz � 0, we must have

− 1

Dahu − q

Dhz +

1

D

n∑k=1

(huak − Hkbk) � 0.

As a > 0, this inequality may be fulfilled only for hu = hz = 0 and∑n

k=1(huak −Hkbk) = 0.Consequently each Hk must be equal to zero and N2− ≡ 0. This finishes the proof of thelemma. �

Remark. Let us note that the linearization matrices of system (30) at the points P1 and P3 havealso the form (32) with a, ai and bi determined by (18) and (20) for k = 1, 3, respectively.

Lemma 13. Assume that a < 0 and ak, bk > 0. Then no eigenvalue of the matrix (32) canhave its real part equal to zero. �

Proof. Let D := diag(D, D1, . . . , DM, 0, . . . , 0). Suppose to the contrary that there exists aneigenvector N ∈ R

n+M+2 of matrix (32) corresponding to a purely imaginary pair of eigenvaluesλ = ±iκ . Now, let N := (hu, h∗1, . . . , h∗M, hM+1, . . . , hn)

T (in the notation of equation (31)),where hu, h∗1, . . . , h∗M, hM+1, . . . , hn are the components of the eigenvector N . Then it is easyto see that the eigenvalue iκ and N would satisfy the relation (−Dκ2 + iqκ +∂F )N = 0, whichcan be written as

(−Dκ2 + ∂F )N = −iqκN.

This means that the matrix (−Dκ2 + ∂F ) has a purely imaginary eigenvalue and thecorresponding eigenvector N . But this is impossible due to lemma 3 and the last statement oflemma 2. �

When D > 0, Di > 0, i = 1, . . . , n, every component of the heteroclinic solutionsU(ξ) provided by theorem 2 are monotonically increasing. Consequently the solutions tosystem (30), if they exist, obtained by passing to the limit Di → 0 are nondecreasing. Asξ → ±∞ the trajectories of these solutions in the (n + 2 + M)-dimensional phase-space are

Calcium waves in systems with immobile buffers 83

tangent to the eigenvectors of the linearization matrices for system (30) at the starting andending points of the heteroclinic solution, respectively, i.e. at the points

(u, v1, . . . , vM, z, z1, . . . , zM, vM+1, . . . , vn) = (u1, v11, . . . , v

1M, 0, 0, . . . , 0, v1

M+1, . . . , v1n)

and

(u, v1, . . . , vM, z, z1, . . . , zM, vM+1, . . . , vn) = (u3, v31, . . . , v

3M, 0, 0, . . . , 0, v3

M+1, . . . , v3n),

respectively. According to lemma 13, the eigenvalues corresponding to these eigenvectorsmust be positive (as ξ → −∞) and negative (as ξ → ∞). Moreover, the solutions arenondecreasing, so if the components of these eigenvectors corresponding to (u, v1, . . . , vn)

are chosen nonnegative, then the components corresponding to (z, z1, . . . , zM) should benonnegative for ξ = −∞ and nonpositive for ξ = ∞. The following additional propertiescan be proved.

Lemma 14. Let q �= 0, a �= 0 and ak, bk > 0. Let N = (hu, h∗1, . . . , h∗M, hz, h1, . . . , hn)T

be an eigenvector of the linearization matrix (32) corresponding to an eigenvalue λ such that

1. if λ > 0, then N has nonnegative components and2. if λ < 0, then hu � 0, h∗k � 0, hz � 0, hk � 0 for k = 1, . . . , M and hk � 0 for

k = M + 1, . . . , n.

Then the inequalities concerning λ and the components of the vector N are strict. �

Proof. In the proof we will exploit equation (33). Suppose that λ > 0 and that the associatedeigenvector N is nonnegative. Then either hu = 0 or hu, hz > 0. Suppose that hu = 0. Thenalso hz = 0. It follows that all Hk = 0 and consequently that also all hk = 0, k = 1, . . . , M .(For the definition of Hk see the proof of lemma 12.) We thus arrive at a contradiction whichproves that hu, hz > 0. Now, if we suppose that hk = 0 for some k = M + 1, . . . , n, we willhave 1

qbkhk = 1

qakhu �= 0. Hence we must have hk > 0 for all k = M + 1, . . . , n. Thus

also all h∗k > 0 (according to the first M + 1 equations). Now, let us suppose that λ < 0 andhz � 0 whereas hu, hk � 0. Then λhu = hz. As before, this implies that either hu = hz = 0or hu > 0, hz < 0. It is easily seen that we can repeat the rest of the proof almost verbatim toconclude that also for λ < 0 the claim of the lemma is true. �

Using lemma 13 we can easily see the validity of the following lemma.

Lemma 15. Let L1 and L3 denote the linearization matrices of system (30) at the points P1

and P3, respectively. Then all the eigenvalues of both L1 and L3 have nonzero real parts. Thenumber of eigenvalues with positive and negative real parts is the same for L1 and L3.

Proof. The first part follows from lemma 13.Let a[1], a

[1]k , b

[1]k and a[3], a

[3]k , b

[3]k denote the parameters of the matrices L1 and L3,

respectively. Let us define the homotopy a(s) = a[1]+s(a[3]−a[1]), ak(s) = a[1]k +s(a

[3]k −a

[1]k ),

bs = b[1]k + s(b

[3]k − b

[1]k ), with s ∈ [0, 1]. During the homotopy the eigenvalues of the matrix

determined by parameters corresponding to s ∈ [0, 1] change continuously. However theirreal parts cannot be equal to zero according to lemma 13. The proof is complete. �

Below, we will also need the following lemma.

Lemma 16. Consider system (9) and (10) (with Di > 0). If one of the functions z =u′, zi = v′

i , i ∈ {1, . . . , n} attains a minimum equal to zero for some ξ = ξ0 thenz(·), z1(·), . . . , zn(·) ≡ 0. The same is true for system (12)–(14). �

84 B Kazmierczak and V Volpert

Proof. Let us suppose that for some i ∈ {1, . . . , n} the function zi(ξ0) attains a minimum.Then z′

i (ξ0) = 0 and z′′i (ξ0) � 0. Thus differentiating the equation for vi we obtain the relation

Diz′′i (ξ0) = Gi,u(u(ξ0), vi(ξ0))z + Gi,vi

(u(ξ0), vi(ξ0))zi .

As Gi,u(u(ξ0), vi(ξ0)) < 0 hence z′′i (ξ0) = 0 and z(·) must attain a minimum equal to zero at

ξ = ξ0. By differentiating the equation for u we infer that also zj (ξ0) = 0 for all j = 1, . . . , n

and the source terms of all the equations are equal to zero. Hence the lemma is proved. If thevalue zero is attained by the function z(·) then the proof is the same. For system (12)–(14) theproof can be carried out along the same lines. �

For α, β = 0, 1, 2, 3, let

Bαβ(R) = Sα × Sα × . . . × Sα︸ ︷︷ ︸M times

× Sβ × . . . × Sβ︸ ︷︷ ︸n−M times

, (34)

where, for γ = 1, 2, 3,

Sγ (R) ={f ∈ Cγ (R) : lim

ξ→±∞f (ξ) exist, lim

ξ→±∞f (j)(ξ) = 0, 1 � j � γ

},

whereas

S0(R) ={f ∈ C0(R) : lim

ξ→±∞f (ξ) exist

}.

Here, by f (j) we mean the j th derivative of f . Bαβ are the Banach spaces under the supremumnorm. To be more precise

‖(u, v1, . . . , vn)‖Bαβ= ‖u‖Sα

+M∑

k=1

‖vk‖Sα+

n∑k=M+1

‖vk‖Sβ, ‖ · ‖Sγ

= ‖ · ‖Cγ (R).

Finally, for α, β = 0, 1, 2, 3, and K ⊂ R an arbitrary compact set, let

Bαβ(K) = {f ∈ Bαβ(R)|K}.The considered heteroclinic solutions of system (12)–(14) are determined only up to a shift inthe ξ -space. To get rid of the translational symmetry we impose an additional condition

u(0) = (u1 + u2)/2, (35)

where u1, u2 are defined in assumption 1.Let us recall that qr denoted the speed of the unique (in the sense of profile) monotonically

increasing heteroclinic solution for system (15) and (16) joining the states P1 and P3.

Theorem 3. Suppose that assumption 1 is satisfied. Then there exists a heteroclinic solutionto system (12)–(14), i.e. a speed q and a vector function (u, v1, . . . , vn) ∈ B21(R) satisfyingconditions (8) and monotonic in each of its components. If qr �= 0 then also q �= 0. �

Proof. For ε ∈ (0, 1], i = M + 1, . . . , n, Di = Di(ε), with

Di(ε) → 0 as ε → 0. (36)

Let us consider a sequence {εl}l=∞l=1 such that εl > 0 and liml→∞ εl = 0. For simplicity,

we will use below the notation introduced in (28). Thus, we will denote by U[l] the uniqueheteroclinic solution (u, v1, . . . , vn)[l] of system (12)–(14) satisfying conditions (8) and (35)(which now reads U1(0) = (u1 + u2)/2) assigned to ε = εl and its (unique) speed by ql .By means of lemma 7 we infer that (ql, U[l]) ∈ R × B32 and the quantity (‖U[l]‖B32 + |ql|)is bounded uniformly with respect to l. Using the Arzela–Ascoli lemma we conclude that

Calcium waves in systems with immobile buffers 85

on every compact interval Iy = [−y, y], y > 0, we may find a subsequence {kl}l=∞l=1 and

the corresponding subsequence {εkl}l=∞l=1 such that both U[kl ] and qkl

are converging in B21(Iy)

and R, respectively. Next out of this subsequence we can choose another subsequence (denotedfor simplicity in the same way) such that the sequences U[kl ] and qkl

are convergent in the normof the space B21(Iy+1) and R respectively. This procedure can be continued. It follows thaton every compact subset of the form I = [− , ] with natural arbitrarily large, we canfind subsequences {U[kl ]}l=∞

l=1 and {qkl}l=∞l=1 converging to the solutions of system (12)–(14).

By differentiation of the equations of this system we easily conclude that the limiting solution(Q, �) belongs to the space R × B32. One can prove that the function �(·) connects theconstant states P1 and P3 as was desired. So, according to the fact that the first derivativesof the functions U[l], l ∈ {1, 2, . . .}, are positive in R, and tend to zero at infinities, � ′(ξ)

must tend to 0 for |ξ | → ∞. Moreover, as ξ → ±∞, due to the monotonicity, the functions�1(·), �2(·), . . . , �n+1(·) must attain their limits. Due to condition (35), limξ→−∞ �(ξ) = P1.Now, if it was not true that limξ→∞ �(ξ) = P3, we would have �(ξ) → P2 as ξ → ∞. Butit is easy to note that then we would have another wave � such that �(ξ) → P2 and P3 asξ → ±∞, respectively, that is to say there would exist two waves (of the same speed q)joining the states P1 with P2 and P2 with P3 consecutively. (Let us note that every U[l](·)is a C1(R) function of its argument and limξ→∞ U[l](ξ) = P3. For detailed considerationsthe reader is referred to [5] p 478.) When Q �= 0 then this possibility should be howeverexcluded due to lemma 12. If Q = 0, the first M + 1 equations decouple from the rest. Forthis system an analogous (to lemma 12) property holds also for q = 0 (see [2, 11]). Thuslimξ→∞(�1(ξ), . . . , �M+1(ξ)) = P3 (compare the remarks after system (15) and (16)) andconsequently limξ→∞ �(ξ) = P3 also in this case.

Now, we only need to show the last statement concerning the relations between q and qr .This follows from lemma 17, which is formulated and proved below. �

Lemma 17. Assume that qr �= 0. Then for ε � 0 sufficiently small the heteroclinic solutions(q, U) to system (12)–(14) described in theorem 3 must satisfy the condition q �= 0. �

Proof. Suppose that there exists a subsequence {εkl}l=∞l=1 such that {qkl

}l=∞l=1 → 0 as l → ∞.

Thus, according to lemma 7 for every ρ > 0 arbitrarily small there exists lρ such that|qkl

v′kl i

(ξ) + Di(εkl)v′′

kl i(ξ)| < ρ, i = M + 1, . . . , n, for all l > lρ (and all ξ ∈ R). Using

these estimations we obtain system (15) and (16) perturbed by terms of the order O(ρ).For ρ = 0 (or l = ∞) we obtain exactly system (15) and (16) (as from the last n − M

equations it would follow that Gi(u(ξ), vi(ξ)) ≡ 0, i = M + 1, . . . , n). According totheorem 1 this system has a unique monotonically increasing heteroclinic solution (qr , ur)

(with qr �= 0 by the assumption of the lemma). Hence we arrive at a contradiction with the factthat qkl

→ 0. �

Let us note that for qr = 0 the existence of a solution is straightforward.

Lemma 18. Suppose that qr = 0. Then there exists a unique heteroclinic solution(u, v1, . . . , vn) ∈ B21(R) satisfying conditions (8) of system (12)–(14) corresponding to thespeed q = 0. �

Proof. Take q = 0. Then, the first M + 1 equations separate from the rest. Accordingto our assumptions and using theorem 2.1, p 15 in [11] (theorem 1) there exists a uniquemonotonically increasing heteroclinic solution joining the corresponding constant states P1

and P3. The remaining n − M equations have the form

Gi(u(ξ), vi(ξ)) = 0, i = M + 1, . . . , n.

86 B Kazmierczak and V Volpert

Given u(ξ), according to the form of the functions, Gi equations can be solved explicitly withrespect to vi(ξ). �

6. Uniqueness of heteroclinic solutions

In this section we will prove the uniqueness of the monotonically increasing heteroclinicsolutions of system (12)–(14).

Suppose that there exist two heteroclinic solutions (q1, W1) and (q2, W2) tosystem (12)–(14) satisfying conditions (8). We know from lemma 16 that both W1 and W2 arestrictly monotone. Let us note that the difference z = W2 − W1 satisfies the following systemof equations:

Diz′′i (ξ ) + q1z

′i (ξ ) +

n+1∑j=1

Fi,j [W1(ξ) + θi(ξ)(W2(ξ) − W1(ξ))]zj (ξ) = −(q2 − q1)W′2i (ξ ),

where, for all ξ ∈ R, 0 � θi(ξ) � 1 and Di = 0 for i = M + 2, . . . , n + 1. Here for simplicitywe denoted

D1 = D, Di+1 := Di, i = 1, . . . , M. (37)

Both W1 and W2 tend to P1 (P3) as ξ tend to −∞ (∞). According to lemma 14 they attain theirlimits exponentially, so the matrices Fi,j [W1(ξ) + θi(ξ)(W2(ξ)−W1(ξ))], i, j = 1, . . . , n + 1,also tend to their asymptotic limits exponentially. To proceed we will need the followinglemma.

Lemma 19. Let the function z : R → Rn+1 satisfy the system

Lz � −δφ(ξ), (38)

where φ(ξ) > 0, ξ ∈ R, δ � 0, and, for i = 1, . . . , n + 1,

(Lz)i(ξ) = Diz′′i (ξ ) + qz′

i (ξ ) +n+1∑j=1

Xij (ξ)zj (ξ), (39)

with Di = const � 0. Suppose that X(ξ) is (n + 1)× (n + 1) matrix function with nonnegativeoff-diagonal elements, continuous in ξ and such that there exist its limits at infinities. Supposethat X± = limξ→±∞ X(ξ) are such that X± are irreducible and have their principal eigenvaluenegative. Suppose that limξ→±∞ z(ξ) = 0. Then there exist numbers r+ and r− < r+ such thatif z(r±) > 0 (component wise) then z(ξ) > 0 for all ξ > r+ and ξ < r−. �

Proof. By a natural modification of the proof of lemma 5.3, p 213 in [11] to the systems whichmay contain equations of the first order, one concludes that z(ξ) � 0 for all ξ ∈ R\[r−, r+],where r± are such that z(r±) > 0 and the principal eigenvalues of X(ξ) are negative and X(ξ)

is irreducible for ξ > r+ and ξ < r−. Using this property, we will prove that z(ξ) > 0 for allsuch ξ . Let us consider the case ξ > r+. Suppose that for some ξ0 ∈ (r+, ∞) the inequalityz(ξ0) > 0 is not true. Let I (ξ0) = {j : zj (ξ0) = 0} and J (ξ0) = {j : zj (ξ0) > 0}. ObviouslyI (ξ0) ∪ J (ξ0) = {1, . . . , n + 1}. Suppose that J = ∅. Hence zi(ξ0) = 0 for all i. If δ > 0,then the left-hand side of the equation for zi is nonnegative whereas the right-hand side isnegative—a contradiction. If δ = 0, then from the uniqueness of solutions of the initial valueproblem, it follows that z(·) ≡ 0 contradicting the assumption z(r+) > 0. So, there mustexist at least one j ∈ J (ξ0) and at least one i ∈ I (ξ0) such that Xij (ξ0) > 0 as otherwise thematrix B would be reducible contrary to our assumptions. Thus we must have z′

i (ξ0) = 0 and

Calcium waves in systems with immobile buffers 87

z′′i (ξ0) � 0, if Di > 0. But then equation (38) could not be satisfied since zj (ξ0) > 0. The

same considerations can be carried out for ξ < −r−. The lemma is proved. �

Now, we are in a position to prove the following uniqueness result.

Theorem 4. Any two strictly monotonic solutions W1 and W2 of system (12)–(14)corresponding to the speeds q1 and q2, respectively, and satisfying conditions (8) coincideup to a shift and q1 = q2. �

Proof. Without losing generality we can suppose that q2 � q1. Let us recall that, accordingto lemma 5, ∂F (W1(ξ)) and ∂F (W2(ξ)) are irreducible for all ξ . Their principal eigenvaluesare negative at ±∞.

For fixed W1 and W2 let r > 0 be such that Wi(ξ) belong to sufficiently smallneighbourhoods of the points P1 and P3 if |ξ | > r . Then the principal eigenvalues of both∂F (W1(ξ)) and ∂F (W2(ξ)) are negative for |ξ | � r .

Let us consider the function W2h(ξ) := W2(ξ + h). It is obvious that taking sufficientlylarge h > 0 we can achieve that W2(r + h) > W1(r). Moreover, for ξ = −r − h we have

W2h(−r − h) = W2(−r − h + h) = W2(−r) > W1(−r − h),

for h sufficiently large. So, increasing the value of h if necessary, we can guarantee that, takingr+ = r and r− = −r − h, we have z(ξ, h) := W2h(ξ) − W1(ξ) > 0 for all ξ ∈ [r−, r+].

Since the principal eigenvalue of the matrix ∂F is negative in some neighbourhoods ofthe points P1 and P3 and due to the choice of r , the principal eigenvalue of the matrix with theelements Fi,j [(1 − θi(ξ))W1(ξ) + θi(ξ)W2h(ξ)], i, j = 1, . . . , n + 1 is also negative for ξ > r

and ξ < −r − h and for all h � 0.Thus, we may take r− = −r − h and r+ = r in lemma 19. Consequently, z(ξ, h) > 0 for

all ξ ∈ R. Now, let us decrease h and find its first value, h = h0, for which the above conditiondoes not hold. Let us note that if ω0 = {ξ0 : zi(ξ0, h0) = 0 for some i = 1, . . . , n + 1},then there exists at least one x0 ∈ ω0 ∩ [r−, r+] (according to lemma 19 and the conditionsimposed on the value of r). Let ξ0 be such that the inequality z(ξ0, h0) > 0 is not true.Let I (ξ0, h0) = {i : zi(ξ0, h0) = 0} and J (ξ0, h0) = {j : zj (ξ0, h0) > 0}. ObviouslyI (ξ0, h0) ∪ J (ξ0, h0) = {1, . . . , n + 1}. Suppose that J (ξ0, h0) = ∅. Hence zi(ξ0, h0) = 0 forall i and for all i = 1, . . . , n + 1 the equation for zi(ξ, h0) at point ξ = ξ0 would have the form

Diz′′i (ξ0, h0) = −(q2 − q1)W

′2h0i

(ξ0),

due to the fact that Fi(W2h0(ξ0)) − Fi(W1(ξ0)) = 0. As Diz′′i (ξ0) � 0 we arrive at a

contradiction, unless q2 = q1. If the latter condition is satisfied, then from the uniqueness ofsolutions of the initial value problem it follows that z(ξ) ≡ 0. Suppose that there exists at leastone i ∈ I (ξ0, h0) and at least one j ∈ J (ξ0, h0) such that Fi,j (W) > 0 for W ∈ [P1, P3]. Sucha pair must exist as otherwise the matrix B would be reducible contrary to our assumptions.But then the equation for zi(ξ, h0):

Diz′′i (ξ0, h0) + q1z

′i (ξ, h0) + Fi(W2h0(ξ0)) − Fi(W1(ξ0)) = −(q2 − q1)W

′2h0 i (ξ0),

which at point ξ = ξ0 reads

Diz′′i (ξ0, h0) + Fi(W2h0(ξ0)) − Fi(W1(ξ0)) = −(q2 − q1)W

′2h0i

(ξ0)

cannot be satisfied since z′′i (ξ0, h0) � 0, Fi(W2h0(ξ0)) − Fi(W1(ξ0)) > 0 (due to the fact

that zj (ξ0) > 0) and by assumption (q2 − q1) � 0 and W ′2h0

(ξ) > 0. As h0 with the aboveproperties must exist, then we conclude that W2h0(ξ) = W1(ξ) for all ξ ∈ R and (q2 −q1) = 0.The lemma is proved. �

88 B Kazmierczak and V Volpert

7. Linearized operator and structural stability

In this section we will analyse the linearization of the operator generated by the left-handsides of the equations of system (12)–(14). The main objective of this analysis is to provethe structural stability of the heteroclinic solutions to system (12)–(14) whose existence wasproved in section 5. To be more precise, we will consider the perturbed system of the form:

DiU′′i + qU ′

i + Fi(U) + �i(τ, q, U) = 0, i = 1, . . . , M + 1,

qU ′i + Fi(U) + �i(τ, q, U) = 0, i = M + 2, . . . , n + 1,

(40)

where Rl τ is sufficiently close to 0. (Let us note that in fact �i may depend on ξ .)

We will prove that, under some differentiability conditions imposed on the terms �i (seeassumption 2) system (40) has a unique heteroclinic solution Uτ (corresponding to a uniquevalue of qτ ) such that Uτ → U0 and qτ → q0, where by (q0, U0) we have denoted the solutionto the unperturbed system (τ = 0), which is in fact system (12)–(14). The precise result isformulated in theorem 5.

In the appendix we show the existence of a bounded solution to the conjugate ofsystem (12)–(14) linearized around the solution U0. Then we prove that this solution isunique (lemma 21) and apply the results of [6], which are based on the notion of exponentialdichotomies.

Let us consider the operator M generated by the left-hand sides of equations ofsystem (12)–(14). This operator can be considered as acting from the Banach space R × B21

consisting of the pairs (q, U) to the space B00. Here B21 denotes the space B21 of vectorfunctions U(·) satisfying the condition

U1(0) = (U1(−∞) + U1(∞))/2. (41)

(The spaces Bαβ are defined in section 5, page 84.) The aim of introducing condition (41) isto pin the solutions in the ξ -space, i.e. to get rid of the translational symmetry of solutions toautonomous systems. Let (q0, U0) denote the heteroclinic solution to system (12)–(14). Thelinearized operator acting between the same spaces (being in fact the Frechet derivative of Mwith respect to U and q at the point (q, U) = (q0, U0)) has the form

DM(q0, U0) : (q, U) → DiU′′i + q0U

′i + (∂F (U0(ξ))U)i + qU ′

0i .

with Di = 0 for i = M + 2, . . . , n + 1. Now, it is obvious that, according to the translationalinvariance, the function U ′

0 satisfies the system

DiU′′i + q0U

′i +

n+1∑j=1

Fi,j (U0(ξ))Uj = 0. (42)

i = 1, . . . , n + 1. Using a method similar to the proof of lemmas 19 and 4 we can prove thevalidity of the following lemma.

Lemma 20. Suppose that q0 �= 0. Then there is no solution to system (42) bounded in B21

other than Z1 = U ′0. �

Proof. Suppose to the contrary that such a solution Z2 exists. According to the fact that allof the eigenvalues of ∂F (U0(±∞)) have their real part different from zero (see lemma 13),the solutions to equation (42) either grow to infinity (in their absolute value) or tend to zero as|ξ | → ∞. It is clear that there exists a number µ > 0 such that µZ1(ξ) < Z2(ξ) < −µZ1(ξ)

for all ξ ∈ [−r, r]. Thus one can prove (repeating the proof of lemma 19 with δ = 0) that, for|ξ | � r ,

�+µ(ξ) = (µZ1(ξ) − Z2(ξ)) > 0 �−

µ(ξ) = (−µZ1(ξ) − Z2(ξ)) < 0. (43)

Calcium waves in systems with immobile buffers 89

Now, let us decrease the value of µ and find the first value, µ = ν for which the aboveinequalities do not hold. If ων± = {ξ0 : �±

νi(ξ0) = 0, for some i = 1, . . . , n + 1}, then thereexists at least one x0 ∈ ων± ∩ [r−, r+], as otherwise, according to what we said before, theinequalities (43) would remain valid for all ξ ∈ R.

Suppose that �+νi(ξ0) = 0 for some ξ0 ∈ [−r, r] and some i ∈ {1, . . . , n + 1}. Then

�+νi(ξ) = 0 attains a minimum at ξ = ξ0, hence (�+

νi)′′(ξ0) � 0, if Di > 0. Let I (ξ0) =

{i : �+νi(ξ) = 0} and J (ξ0) = {j : �+

νj (ξ) > 0}. Obviously I (ξ0) ∪ J (ξ0) = {1, . . . , n + 1}.Suppose that J (ξ0) = ∅. Then �+

νj (ξ) = 0 for all j and all ξ due to the uniqueness of the initialvalue problem. So suppose there exists at least one i ∈ I (ξ0) and at least one j ∈ J (ξ0) suchthat Fi,j (U) > 0 for U ∈ [P1, P3]. Such a pair must exist as otherwise the matrix B wouldbe reducible contrary to our assumptions. But then we arrive at a contradiction as �+

νi(ξ0)

satisfies the equation

Di (�+νi)

′′(ξ0) +∑j �=i

Fi,j (ξ0)�+νj (ξ0) = 0.

The same considerations may be carried out for the function (−�−µ). As a conclusion the

only situation which does not lead to a contradiction is the existence of real constant ν suchthat Z2 ≡ νZ1 and the thesis of the lemma follows. �

In the appendix we show the existence and positivity of a bounded solution V to theadjoint of system (42) if q0 �= 0. Concerning this solution one can prove a lemma analogousto lemma 20.

Lemma 21. There is no solution to the adjoint of system (42) bounded in B21 otherthan V . �

Proof. The proof can be done along the lines of the proof of lemma 20. �

Assumption 2. Suppose that, for i = 1, . . . , n + 1, the mappings �i : Rl × R × B21 → B00,

l � 1 are continuous and �i(0, q, U) ≡ 0. Assume that for τ = 0 there exists a unique (up totranslation in ξ ) heteroclinic solution for system (40) joining the states P1 and P3 with U0(·)monotonically increasing. Suppose that � is continuously Frechet differentiable with respectto (q, U) in some open neighbourhood of the point (τ, q, U) = (0, q0, U0). �

Now, we are able to formulate a theorem expressing the structural stability of theheteroclinic solutions for system (12)–(14).

Theorem 5. Suppose that qr �= 0 and that assumptions 1 and 2 are fulfilled. Then forall 0 � |τ | < τ0, with τ0 sufficiently small, there exists a unique heteroclinic solution(qτ , Uτ ) ∈ R × B21 for system (40) such that

|qτ − q0| + ‖Uτ − U0‖B21 → 0

as |τ | → 0. �

Proof. Note that q0 �= 0 due to lemma 17. To prove the above result it is convenient to write(40) as a first order system of n + M + 2 equations:

U ′i − Zi = 0 i = 1, . . . , M + 1,

Z′i +

1

Di

[qZi + Fi(U) + �i(τ, q, U)] = 0, i = 1, . . . , M + 1,

U ′i +

1

q[Fi(U) + �i(τ, q, U)] = 0, i = M + 2, . . . , n + 1,

(44)

90 B Kazmierczak and V Volpert

where we use notation (28) and (37). The left-hand sides of the equations of system (44) definethe operator Pτ acting on (n + M + 2 + 1)-tuples

(q, U(·)) = (q, U1, . . . , UM+1, Z1, . . . , ZM+1, UM+2 . . . , Un+1)(·)from the space

R × Bn+M+21 (R) := R × S1 × S1 × . . . × S1︸ ︷︷ ︸

n+M+2 times

to the space

Bn+M+20 (R) := S0 × S0 × . . . × S0︸ ︷︷ ︸

n+M+2 times

.

Here S1 is the subspace of S1 consisting of functions satisfying the condition (41).The Frechet derivative DP0 of the operator P0 at a point (q0, U0) (corresponding to

(q0, U0)) by identifying U0 = (U01, . . . , U0(M+1), Z01, . . . , Z0(M+1), U0(M+2), . . . , U0(n+1)),Z0j = U ′

0j , with respect to (q, U) is well defined. It has the following form:

DP0(q, UT ) = UT ′ − P∗UT + qZT , (45)

with

Z(ξ) := (0, . . . , 0︸ ︷︷ ︸M+1 times

, D−11 U ′

01, . . . , D−1M+1U

′0(M+1), −q−2

0 FM+2(U0), . . . , −q−20 Fn+1(U0))(ξ)

and P∗ having the following form:

P∗ =(

O(M+1)×(M+1) I(M+1)×(M+1) O(M+1)×(n−M)

−D∗KM+1 −q0D∗ −D∗Kn−M

), (46)

where D∗ is an (n + 1) × (n + 1) matrix, D∗ = diag(D−11 , . . . , D−1

M+1, q−10 , . . . , q−1

0 ), D∗ isan (n + 1) × (M + 1) matrix such that D∗

ij = D−1i δij for i = 1, . . . , M + 1, and D∗

ij = 0 fori = M + 2, . . . , n + 1, K = ∂F = (∂F )(U0(ξ)) and F is given by (28). KM+1 denotes the(n+1)×(M+1)- matrix composed of the first M+1 columns of K , Kn−M is the (n+1)×(n−M)

matrix composed of the last n − M columns of K . (In fact P∗ has the form of the matrix L

defined after (31) with the coefficients a, ai and bi depending on ξ .)Now, let us note that at P1 and P3 the eigenvalues of the linearization matrices for

system (30) (with respect to U ) have nonzero real part (see lemma 13) and according to theremarks preceding lemma 14 the linearization matrices at P1 and P3 have at least one positiveand one negative eigenvalue, respectively. In conclusion, system (64) has the exponentialdichotomy on both the half lines (see e.g. lemma 3.4 in [6]) and the operator J defined by

(JU)(ξ) = U ′(ξ) − P∗(ξ)U(ξ)

is Fredholm with index zero as acting from C1 to C0 spaces (see [6], lemma 4.2). We haveshown above that, up to a multiplicative constant, there is only one bounded solution U ′

0 tosystem (42) (lemma 20). Likewise, up to a multiplicative constant, the adjoint of this systemhas only one bounded solution V (lemma 21). In consequence there is only one (up to amultiplicative constant) solution to the system (JU)(ξ) = 0, which does not belong to thespace Bn+M+2

1 (R) as U ′01 is everywhere positive and vanishes at infinities, so does not satisfy

condition (41), and only one (up to a multiplicative constant) bounded solution A of the adjointto this system (see lemma 21 and appendix). Hence according to lemma 4.2 in [6], by means ofsome additional considerations using the asymptotic properties of the matrix P∗(·) (in particularlemma 13; see e.g. [5]), the equation

DP0(q, UT ) = χ

Calcium waves in systems with immobile buffers 91

with χ ∈ Bn+M+20 has a unique solution in Bn+M+2

1 iff the following orthogonality condition issatisfied: ∫

R

n+M+2∑i=1

[χi(ξ) − qZi (ξ )]Ai(ξ) dξ = 0. (47)

Using the fact that (q0, U0) satisfies system (40) for τ = 0, we conclude that

Z(ξ) := (0, . . . , 0, D−11 U ′

01(ξ), . . . , D−1M+1U

′0(M+1), q

−10 U ′

0(M+2), . . . , q−10 U ′

0(n+1))(ξ).

Now, we can use the relations between A and V , proved in the appendix (see (66), (69)and (70)). Thus we note that, as for j = 1, . . . , n + 1, Vj (ξ) > 0, hence also Aj+M+1(ξ) > 0for all ξ ∈ R. In consequence, condition (47) may always be satisfied by a proper choiceof q. (U ′

0 vanishes exponentially at infinities according to lemma 13.) Thus the operatorDP0 defines an isomorphism between the spaces R × Bn+M+2

1 and Bn+M+20 , thus its inverse

is bounded (see e.g. theorem 4.2-H, p 180 in [9]). Hence according to the implicit functiontheorem (see e.g. [1]) system (44) is uniquely solvable for (q, U). One notes that Ui, Zi ∈ S1

implies Ui ∈ S2, i = 1, . . . , M + 1. Hence the claim of theorem 5 holds. �

Next, according to relations (66), (69) and (70) in the appendix, we infer that condition (47)for χ such that

χi ≡ 0 for i = 1, . . . , M + 1, (48)

can be written asn+1∑i=1

∫R

[χi(ξ) − qU ′0i (ξ )]Vi(ξ) dξ = 0, (49)

where

χi = Diχi+M+1 i = 1, . . . , M + 1, χi = q0χi+M+1 i = M + 2, . . . , n + 1.

(Note that condition (48) expresses the fact that the perturbations in (44) do not occur in thefirst M + 1 equations.) Hence according to the implicit function theorem and condition (49),δq = qτ − q0 can be determined in the first approximation from the following relation∫

R

n+1∑i=1

{�i(τ, q0, U0)(ξ) + δqU ′0i (ξ )}Vi(ξ) dξ = 0. (50)

For more details the reader is referred to [5] or [6].

Corollary. The speed of the wave is a decreasing function of F(·). To be more precise, letus consider system (12)–(14) with F instead of F such that F (U) � F(U) for U ∈ [P1, P3].Then, if for some j , Fj (U) > Fj (U) for some U ∈ [P1, P3], the inequality q < q holds. Theproof follows from relation (50). �

Remark. Using theorem 5 and the fact that �i may depend on ξ , we can recover and strengthenthe convergence result contained in the proof of theorem 3. Note that for Dk , k = M +1, . . . , n,sufficiently close to zero, the heteroclinic solutions of system (9) and (10) from the spaceR × B21 satisfy system (12)–(14) with the additional nonhomogeneous perturbation Dkv

′′k (ξ)

in the equation for vk , k = M+1, . . . , n, and 0 in the remaining first M+1 equations. Obviouslythese terms belong to the space B00. Now, we may use theorem 5 putting τ ∈ R

l , l = n − M ,τ 2j = Dj+M , j = 1, . . . , n − M , �k(τ, q, U) = τ 2

k−M−1v′′k−1(ξ) for k = M + 2, . . . , n + 1

and �k ≡ 0 for k = 1, . . . , M + 1. Hence we conclude that for |τ | sufficiently small the

92 B Kazmierczak and V Volpert

unique heteroclinic solution of the perturbed system (i.e. system (9) and (10)) differs from(q0, U0) by the terms of the order of τ 2 = ∑n

i=M+1 Di , hence ‖Uτ (·) − U0(·)‖B21 = O(τ 2),|qτ − q0| = O(τ 2) as τ → 0. �

The next lemma may be treated as a refinement of lemma 17.

Lemma 22. Assume that qr > 0 (qr < 0). Let U(·) be a heteroclinic solution to system (9)and (10) satisfying condition (8) and corresponding to the speed q. Then for

∑ni=M+1 Di � 0

sufficiently small, q must satisfy the condition q > 0 (q < 0). �

Proof. Consider the family of systems

Du′′ + qu′ + f (u) +M∑i=1

Gi(u, vi) + λ

n∑i=M+1

Gi(u, vi) = 0, (51)

Div′′i + qv′

i − Gi(u, vi) = 0, i = 1, . . . , M, (52)

qv′i − λGi(u, vi) = 0, i = M + 1, . . . , n. (53)

For each fixed λ ∈ (0, 2) this system has a monotonically increasing heteroclinic solution(qλ, Uλ) satisfying conditions (8). Let us note that for λ sufficiently close to 0 due to themonotonicity of the components of Uλ, the C0(R) norms of the terms λGi(u, vi) could be madearbitrarily small. Treating these terms as given functions from the space S0(R) tending to zeroat infinities, we can apply the implicit function theorem to the system (15) and (16) to concludethat qλ is arbitrarily close to qr if λ > 0 is sufficiently close to zero (see [5]). Now, accordingto theorem 5 we can obtain a unique branch of heteroclinic solutions (qλ, Uλ) ∈ R × B21

for all λ ∈ (0, 2). Due to lemma 17, qλ �= 0. As qλ is of the same sign as qr for λ > 0sufficiently small, and qλ cannot change its sign when λ increases, then we conclude that thelemma holds. �

Now, let d := (DM+1, . . . , Dn) ∈ Rn−M and let Ud(·) denote the unique heteroclinic

solution to system (9) and (10) satisfying conditions (8) and belonging to the space B21(R).Let qd denote the (unique) corresponding speed.

Lemma 22, together with the remark preceding it, can be put into theorem 3 to obtain thefollowing theorem.

Theorem 6. Suppose that qr �= 0 and that assumption 1 is satisfied. Then there exists aunique heteroclinic solution to system (12)–(14), i.e. a speed q0 and unique vector functionU0(·) ∈ B21(R) satisfying conditions (8). Moreover,

‖Ud(·) − U0(·)‖B21 = O(|d|), |qd − q0| = O(|d|)as |d| → 0. If qr > 0 (qr < 0) then also q0 > 0 (q0 < 0). �

Final remark. Let us note that the concrete form of the functions Gi is not essential for thevalidity of the main theorems proved in the paper. Thus, let f satisfy point 1 of assumption 1.Assume that the constant states of system (12)–(14) satisfy conditions (3) and (5) and thatGi = Gi(u, vi) are functions of C3 class such that Gi,vi

(u, vi) > 0, Gi,u(u, vi) < 0,

Calcium waves in systems with immobile buffers 93

i = 1, . . . , n, for all P1 � (u, v1, . . . , vn) � P3. Then the statements of theorem 6 remainvalid. Moreover, if assumption 2 is satisfied then theorem 5 holds. To show it, we can repeatthe proofs of theorems 6 and 5 without any essential changes. �

Acknowledgments

This paper was partially supported by the Polish Ministry of Science and Higher EducationGrant No 1P03A01230.

Appendix. Properties of a bounded solution to the adjoint system

Here, we are interested in the existence and properties of bounded solutions to the adjoint ofthe linearization of system (30) around the heteroclinic solution (U0, U

′0) corresponding to the

speed q0 �= 0 found in section 5. For D, D1, . . . , Dn > 0 system (9) and (10) can be writtenas a first order system of the form

u′ = z,

v′i = zi i = 1, . . . , n,

z′ = 1

D

[−qz − f (u) −

n∑i=1

Gi(u, vi)

],

z′i = 1

Di

[−qzi + Gi(u, vi)] i = 1, . . . , n

(54)

We assume that for k = M + 1, . . . , n and ε ∈ (0, 1], Dk = Dk(ε) > 0 with Dk(ε) → 0, asε → 0, whereas D and Di > 0, i = 1, . . . , M , are independent of ε. According to theorem 2,for every ε > 0 there exists a unique heteroclinic solution [Uε(·), U ′

ε(·)] to system (54)satisfying condition (35) and corresponding to the speed qε. Within the notation (28), forε > 0 the linearization of system (54) around the heteroclinic solution [Uε(·), U ′

ε(·)] has theform

(U, Z)T′ = L∗(U, Z)T ,

where

L∗ =(

0 I

−D−1K −qεD−1

), (55)

where D−1 = diag(D−11 , . . . , D−1

M+1, D−1M+2, . . . , D−1

n+1), where Di are defined in (37) fori = 1, . . . , M + 1, Di (ε) = Di−1(ε), i = M + 2, . . . , n + 1, K = K(ξ) = (∂F )(Uε(ξ)) andF is given by (28). I is the unit (n + 1) × (n + 1) matrix. Solutions (Y, V ), Y, V : R → R

n+1,to the adjoint system satisfy the set of equations

Y ′ = (D−1K)T V

V ′ = −IY + qεD−1V.

(56)

Differentiating the equation for V and using the equation for Y we conclude that V satisfiesthe equation

V ′′ = qεD−1V ′ − (D−1K)T V . (57)

94 B Kazmierczak and V Volpert

Hence V satisfies the adjoint of the second order linearized system. We have

(D−1K)T

=

1

D1

[a −

n∑i=1

aεi

]1

D2aε

1 . . .1

DM+1aε

M

1

DM+2(ε)aε

M+1 . . .1

Dn+1(ε)aε

n

1

D1bε

1 − 1

D2bε

1 . . . 0 0 . . . 0

. . . . . . . . . . . . . . . . . . . . .

1

D1bε

M 0 . . . − 1

DM+1bε

M 0 . . . 0

1

D1bε

M+1 0 . . . 0 − 1

DM+2(ε)bε

M+1 . . . 0

. . . . . . . . . . . . . . . . . . . . .

1

D1bε

n 0 . . . 0 0 . . . − 1

Dn+1(ε)bε

n

,

(58)

where we have denoted

aε = aε(ξ) = f ′(Uε(ξ)), bεj = bε

j (ξ) = G1,vj(Uε(ξ)), aε

j = aεj (ξ) = −Gj,u(Uε(ξ)).

Now, according to theorem 4.5.1 in [11] we know that for all ε > 0 there exists a solution(V1, . . . , Vn+1) to system (57) for which Vi(ξ) > 0 for all ξ ∈ R, i = 1, . . . , n+1, and such thatVi(ξ) vanishes exponentially as ξ → ±∞ together with its first and second derivatives. Let usnormalize this solution demanding that supξ∈R

V1(ξ) = 1. It is easy to note that the maximalvalue of the functions Vj , j ∈ {M + 2, . . . , n + 1}, is of the order of O(Dj (ε)). This followsfrom the application of the maximum principle to the equations for Vj , j ∈ {M + 2, . . . , n+ 1}.Likewise we infer that the maximal values of the functions Vi , i ∈ {2, . . . , M + 1} are of theorder of O(1). From this, as in the proof of lemma 7, we conclude that ‖V ′

i ‖C0(R) = O(1)

and ‖V ′′i ‖C0(R) = O(1) for i ∈ {1, 2, . . . , M + 1}. Let us note that for all ε > 0, ‖Uε‖C2

and qε are uniformly bounded. In consequence, thanks to the smoothness of the functionsGi , the coefficients aε

i and bεi are twice boundedly differentiable. Differentiating the j th

equation of system (57) with respect to ξ and using the maximum principle we concludethat ‖V ′

j‖C0(R) = O(Dj (ε)) for j ∈ {M + 2, . . . , n + 1}. Obviously, we can also find anε-independent bound for ‖V ′′′

i ‖C0(R) for i ∈ {1, 2, . . . , M + 1}. Further, the equation for Vj ,j ∈ {M + 2, . . . , n + 1}, can be written in the form

Dj (ε)V′′j − qεV

′j +

1

D1bε

j−1(ξ)V1 − bεj−1(ξ)Vj = 0, (59)

where Vj := D−1j (ε)Vj . Differentiating (59) twice with respect to ξ , we can estimate

V ′′j by a constant independent explicitly of ε. Hence ‖V ′′

j ‖C0(R) = O(Dj (ε)) for j ∈{M + 2, . . . , n + 1}. Assume that there exists a sequence {εl}l=∞

l=1 such that (qε, Uε)

converges in the space R × B21 to (q0, U0), q0 > 0. Let us denote the solution tosystem (57) corresponding to εl by V (εl, ·). It follows that for l → ∞, V (εl, ξ) =(V1(εl, ξ), . . . , VM+1(εl, ξ), VM+2(εl, ξ), . . . , Vn+1(εl, ξ)), where Vk(εl, ξ) = D−1

k Vk(εl, ξ)

for k = 1, . . . , M + 1 and Vk(εl, ξ) = D−1k (εl)Vk(εl, ξ) for k = M + 2, . . . , n + 1, converges

Calcium waves in systems with immobile buffers 95

on every compact subset in the norm of the space B21(R) to the solution of the system

D1V′′

1 − q0V′

1 +n+1∑k=1

K1,j (ξ)Vj = 0, (60)

Dj V′′j − q0V

′j + Kj,1(ξ)V1 + Kj,j (ξ)Vj = 0, j = 2, . . . , M + 1, (61)

−q0V′j + Kj,1(ξ)V1 + Kj,j (ξ)Vj = 0, j = M + 2, . . . , n + 1, (62)

where K is equal to

[a −

n∑i=1

ai

]a1 . . . aM aM+1 . . . an

b1 −b1 . . . 0 0 . . . 0

. . . . . . . . . . . . . . . . . . . . .

bM 0 . . . −bM 0 . . . 0

bM+1 0 . . . 0 −bM+1 . . . 0

. . . . . . . . . . . . . . . . . . . . .

bn 0 . . . 0 0 . . . −bn

, (63)

and

a(ξ) = f ′(U0(ξ)), bj (ξ) = G1,vj(U0(ξ)), aj (ξ) = −Gj,u(U0(ξ)).

The components of V vanish exponentially at infinities. This statement is due to the obviousfact that after writing (60)–(62) as a first order system y ′(ξ) = T (ξ)y(ξ), y : R → R

n+M+2,one notes that all the eigenvalues of the matrices T (±∞) have their real part different fromzero. (To be more precise, we may use the proof of lemma 13, because the matrix (63) isthe transpose of the matrix (17), so lemma 3 holds.) Obviously the components of V arenonnegative. Suppose that Vj (ξ0) = 0 for some ξ0 ∈ R and some j ∈ {1, . . . , n + 1}. ThenV ′

j (ξ0) = 0 and V ′′j (ξ0) � 0 (if j = 1, . . . , M + 1). Hence, thanks to the irreducibility of

the matrix it would follow that Vj (ξ0) = 0, V ′j (ξ0) = 0 for all j ∈ {1, n + 1}. Due to the

uniqueness of solutions, this would imply V ≡ 0. However, this would lead to a contradictionas V1(0) = 1 by definition. Thus Vj (ξ) > 0 for all j and ξ ∈ R.

On the other hand, let us consider the linearization of system (30), around the heteroclinicsolution (q0, U0), where U0 is defined before (45). It has the form

UT ′ = P∗UT , (64)

where U = (u, v1, . . . , vM, z, z1, . . . , zM, vM+1, . . . , vn), where P∗ has the form (46). Hencethe adjoint system AT ′ = −P T

∗ A has the form

A′i =

M+1∑k=1

D−1k Fk,iAk+M+1 +

n+1∑k=M+2

q−10 Fk,iAk+M+1, i = 1, . . . , M + 1,

A′i = −Ai−M−1 + q0D−1

i−M−1Ai, i = M + 2, . . . , 2M + 2,

A′i =

M+1∑k=1

D−1k Fk,i−M−1Ak+M+1 +

n+1∑k=M+2

q−10 Fk,i−M−1Ak+M+1, i = 2M + 3, . . . , n + M + 2.

(65)

96 B Kazmierczak and V Volpert

Differentiating the equation for Ai , i = M + 2, . . . , 2M + 2, and using the equations for Aj

with j = 1, . . . , M + 1 we obtain the equation

A′′i = q0D−1

i−M−1A′i −

(M+1∑k=1

D−1k Fk,i−M−1Ak+M+1 +

n+1∑k=M+2

q−10 Fk,i−M−1Ak+M+1

).

By defining

Wj = Aj+M+1, (66)

we obtain the equations for Wj , j = 1, . . . , M + 1,

W ′′j = q0D−1

j W ′j −

(M+1∑k=1

D−1k Fk,j Wk +

n+1∑k=M+2

q−10 Fk,j Wk

), (67)

and for Wj , j = M + 2, . . . , n + 1, we have

W ′j =

M+1∑k=1

D−1k Fk,j Wk +

n+1∑k=M+2

q−10 Fk,j Wk. (68)

If we further define

Wi = D−1i Wi for i = 1, . . . , M + 1 (69)

and

Wi = q−10 Wi for i = M + 2, . . . , n + 1, (70)

we see that the vector function W(·) = (W1, . . . , WM+1, WM+2, . . . , Wn+1)(·) satisfiesexactly system (60)–(62). On the other hand the vector function F = (F1, . . . , Fn+1) =(D1V1, . . . , DM+1VM+1, q0VM+2, . . . , q0Vn+1) satisfies system (67)–(68). Now, let Ai+M+1 =Fi , i = 1, . . . , n + 1. Then, using the second set of equations in (65) (i.e. fori = M + 2, . . . , 2M + 2), we can straightforwardly determine the functions Ai , i = 1, . . .,M + 1. In consequence, having a positive solution of system (60)–(62) exponentiallyvanishing at ξ = ±∞, we can find an exponentially vanishing at infinities solution to systemAT ′ = −P T

∗ A, with its last n + 1 components positive. In consequence there is a one-to-onecorrespondence between the solutions to the adjoint of system (64) and the solutions obtainedfrom solutions to system (56) by passing to the limit ε → 0 and appropriate rescaling.

References

[1] Crandall M 1977 An introduction to constructive aspects of bifurcation theory and implicit function theoremApplications of Bifurcation Theory ed P Rabinowitz (New York: Academic)

[2] Crooks E C M and Toland J F 1998 Travelling waves for reaction–diffusion–convection systems TopologicalMethods Nonlinear Anal. 11 19–43

[3] Falcke M 2004 Reading the patterns in living cells—the physics of Ca2+ signaling Adv. Phys. 53 255–440[4] Gantmacher F R 1960 The Theory of Matrices (New York: Chelsea)[5] Kazmierczak B and Volpert V 2003 Existence of heteroclinic orbits for systems satisfying monotonicity

conditions Nonlinear Anal. 55 467–91[6] Palmer K J 1984 Exponential dichotomies and transversal homoclinic points J. Diff. Eqns 20 225–56[7] Shenq Guo J and Tsai J 2006 The asymptotic behavior of solutions of the buffered bistable system J. Math. Biol.

53 179–213[8] Sneyd J, Dale P D and Duffy A 1998 Traveling waves in buffered systems: applications to calcium waves SIAM

J. Appl. Math. 58 1178–92[9] Taylor A E 1958 Introduction to Functional Analysis (New York: Wiley)

[10] Tsai J and Sneyd J 2005 Existence and stability of traveling waves in buffered systems SIAM J. Appl. Math. 66237–65

[11] Volpert A, Volpert V and Volpert V 1994 Travelling Wave Solutions of Parabolic Systems (Providence, RI:American Mathematical Society)