Steady State Solutions for Balanced Reaction Di usion Systems on

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Steady State Solutions for Balanced Reaction Diffusion Systems on Heterogeneous Domains W.E. Fitzgibbon * , S.L. Hollis and J.J. Morgan * Abstract We consider a class of semilinear diffractive-diffusion systems of the form (P) -d A i Δu i = f A i (u) x A -d B i Δ˜ u i = f B i u) x B u i u i on ∂A d A i ∂u i ∂η A = d B i ˜ u i ∂η A on ∂A ˜ u i = g i on ∂B\∂A where A and B are smooth bounded domains in < n such that there exists a smooth bounded domain Ω ⊆< n so that A is a strict subdomain of Ω and ¯ A B = Ω. We assume that d A i , d B i > 0, each g i is nonnegative and smooth, and f A =(f A i ) and f B =(f B i ) are locally Lipschitz vector fields which are quasi-positive, nearly balanced, and polynomial bounded. We prove that these conditions guarantee the existence of a nonnegative solution of (P) for the case of n = 2. In addition, for the case of n = 3, we show that nonnegative solutions of (P) exist provided that f A , f B satisfy a quadratic intermediate sum property. In particular, our results imply that, for space dimensions n =2, 3, if (P) arises from standard balanced quadratic mass action kinetics, then nonnegative solutions of (P) are guaranteed. We apply our results to two multicomponent chemical models. AMS (MOS) classification numbers: 35B45, 35J55, 35K57, 35R05 Key Words: steady state solution, reaction diffusion system, heterogeneous domain, semi- linear elliptic system, balanced system, diffractive diffusion, a priori estimates, Lyapunov function, chemical model, mass action kinetics. 1

Transcript of Steady State Solutions for Balanced Reaction Di usion Systems on

Page 1: Steady State Solutions for Balanced Reaction Di usion Systems on

Steady State Solutions for Balanced

Reaction Diffusion Systems on Heterogeneous Domains

W.E. Fitzgibbon∗, S.L. Hollis and J.J. Morgan∗

Abstract

We consider a class of semilinear diffractive-diffusion systems of the form

(P)

−dAi∆ui = fAi(u) x ∈ A−dBi∆ui = fBi(u) x ∈ B

ui = ui on ∂AdAi

∂ui∂ηA

= dBi∂ui∂ηA

on ∂A

ui = gi on ∂B\∂A

where A and B are smooth bounded domains in <n such that there exists a smooth

bounded domain Ω ⊆ <n so that A is a strict subdomain of Ω and A∪B = Ω. We assume

that dAi , dBi > 0, each gi is nonnegative and smooth, and fA = (fAi) and fB = (fBi) are

locally Lipschitz vector fields which are quasi-positive, nearly balanced, and polynomial

bounded. We prove that these conditions guarantee the existence of a nonnegative solution

of (P) for the case of n = 2. In addition, for the case of n = 3, we show that nonnegative

solutions of (P) exist provided that fA, fB satisfy a quadratic intermediate sum property.

In particular, our results imply that, for space dimensions n = 2, 3, if (P) arises from

standard balanced quadratic mass action kinetics, then nonnegative solutions of (P) are

guaranteed. We apply our results to two multicomponent chemical models.

AMS (MOS) classification numbers: 35B45, 35J55, 35K57, 35R05

Key Words: steady state solution, reaction diffusion system, heterogeneous domain, semi-

linear elliptic system, balanced system, diffractive diffusion, a priori estimates, Lyapunov

function, chemical model, mass action kinetics.

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1. Introduction

Many reactive diffusive processes feature a pronounced dependence of the reaction kinetics

and diffusive structure upon the heterogeneity of the domain structure. Systems having

reaction kinetics that depend upon domain structure arise in a variety of applications,

including geochemistry [15] and experimental chemical reactors [21]. Diffusion through

highly heterogeneous media can produce diffractive diffusion. Diffractive diffusion arises

in nuclear reactor physics and has been studied in the American and Russian mathemat-

ical literature ([11], [12], [17], [18], [19], [20]). However, there is a marked absence of

mathematical literature that treats these types of systems and focuses on the presence of

complex kinetic structure.

In the work at hand, we are concerned with steady state solutions to systems featuring

reaction kinetics and diffusive structure dependent upon the heterogeneity of the domain.

We focus on a highly idealized physical scenario whereby m chemical species diffuse and

react in a region Ω. We assume that Ω contains an inner core or subvolume, which we

shall designate by A, and that A is surrounded by one connected region B that separates

∂A from ∂Ω as depicted in Figure 1. We assume that reaction and diffusion transpire

throughout Ω, but we assume that the reaction diffusion systems describing the processes

differ across the interface ∂A of A and B, and that the distinct systems are coupled by

requiring continuity of the state variables (concentration densities) and fluxes across the

interface.

The types of systems to be considered are of the form

(P)

−dAi∆ui = fAi(u) x ∈ A−dBi∆ui = fBi(u) x ∈ B

ui = ui on ∂AdAi

∂ui∂ηA

= dBi∂ui∂ηA

on ∂A

ui = gi on ∂B\∂A.

Here ui and ui represent the concentrations of the ith chemical species in A and B, re-

spectively, for i = 1, . . . ,m. Specific hypotheses concerning the coefficients and functions

present in (P) are given in the next section. However, we take some time here to motivate

our principle assumptions concerning the vector fields fA and fB .

Recall that our primary interest is in models that arise from complex chemistry. To

this end, suppose (P) models the steady state behavior for the reaction and diffusion of

a chemical process that takes place on the domain Ω. Then the unknowns u, u represent

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chemical concentrations that are subject to the same basic chemical reactions regardless

of whether they take place in A or B, with the only difference being the acceleration or

deceleration of reaction rates due to domain properties. In general, if z1, . . . , zm represent

m chemical species that interact in k reaction steps on Ω, then the laws of mass action

kinetics imply that there exists an m × k matrix N (termed the Stoichiometric Matrix)

and reaction rates Rq1(z), . . . , Rqm(z) (which may also depend upon domain properties)

such that

fq(z) = N

Rq1(z)...

Rqm(z)

, for q = A,B,

with N independent of z, reaction rates, etc. Now, for the moment, consider the associated

kinetic equations (with diffusion ignored). These equations have the form

v′(t) = fA(v(t)) on A

or

v′(t) = fB(v(t)) on B.

We will concentrate on the first of these. If we expect some sort of conservation of total

mass, or a weighted conservation of total mass, then we expect something like

m∑i=1

biv(t) =m∑i=1

biv(0) for all t ≥ 0,

where b1, . . . , bm > 0. However, if we integrate the kinetic equations, then we obtain

m∑i=1

biv(t) =m∑i=1

(biv(0) +

∫ t

0

bifAi(v(s))ds)

for all t ≥ 0.

Consequently, we can probably expect the conservation above to hold only if

(A)m∑i=1

bifAi(z) = 0 for all admissable z.

From the form of fA above, we would expect this to be a consequence of (b1, · · · , bm)T ∈ker(NT ); that is, we would expect conservation of mass to be determined by stoichiometry

and not by reaction rates. As a result, we would also have

(B)m∑i=1

bifBi(z) = 0 for all admissable z.

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The assumptions (A) and (B) are typically referred to as balancing assumptions (cf. [5],

[7], [8], [9], [10], [13], [14] and the references therein). Interestingly, many chemical systems

give rise to a balancing structure. In fact, all single step reversible chemical reactions give

rise to a balancing structure ([8]), and many multiple step chemical reactions give rise to

similar structure. We will build our analysis around related assumptions.

Our results are quite natural for the case of space dimensions n = 2 and n = 3,

although at first glance, the extra hypothesis that we place on the reaction vector field for

space dimension n = 3 might seem a little odd. In either case, our results guarantee the

existence of nonnegative solutions to systems arising from standard balanced quadratic

mass action kinetics.

Our work relies upon estimates developed by localized duality arguments applied to

a nonstandard scalar steady state diffractive diffusion problem. The scalar diffractive

diffusion problem arises from an assumption of near balancing upon the reaction vector

field in our model. However, due to the structure of our system, the solutions of the

scalar problem are not necessarily continuous across the interface, although their fluxes

are continuous. The lack of continuity in the solutions at the interface causes some difficulty

in obtaining necessary a priori estimates.

Our subsequent work is organized as follows. We give a precise statement of our

problem in Section 2. In Section 3 we develop some estimates for the related, nonstandard

scalar equation. We give the proofs of our main results in Section 4. Finally, Section 5

contains two examples to illustrate the applicability of our results.

2. Statements of Results

We assume throughout that n ≥ 2 and Ω, A and B are given as in the introduction.

More specifically, we assume Ω is a bounded domain in <n whose boundary ∂Ω is a C2+α

manifold such that Ω lies locally on one side of ∂Ω, and there exist domains A,B ⊆ Ω such

that A ⊆ Ω and Ω = A ∪B. In addition, we assume that for each P ∈ ∂A there exists an

ε > 0 such that if we denote Bε(P ) = x | |x− P | < ε then there exist local coordinate

functions φ1, . . . , φn ∈ C3(Bε(P ),<) so that

∂A ∩Bε(P ) = x | φ1(x) = 0,

|∇φi(x)| ≥ δ > 0 for all x ∈ Bε(P ),

and

∇φi(x) · ∇φj(x) = 0 for all x ∈ Bε(P ) if i 6= j.

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Finally, we let ηA denote the unit normal vector on ∂A pointing out of A. We consider

the system (listed here again for easy reference)

(P)

−dAi∆ui = fAi(u) x ∈ A−dBi∆ui = fBi(u) x ∈ B

ui = ui on ∂AdAi

∂ui∂ηA

= dBi∂ui∂ηA

on ∂A

ui = gi on ∂B\∂A

where fA, fB : <m+ → <m are locally Lipschitz, satisfy the quasi-positivity condition:

(QP) fqi(z) ≥ 0 for all z ∈ <m+ with zi = 0 and q = A,B

and are nearly balanced with the same balancing law:

(BAL) There exist bi > 0 and Kq ≥ 0 such that

m∑i=1

bifqi(z) ≤ Kq for all z ∈ <m+ and q = A,B.

Furthermore, we assume that each component of fA and fB is polynomial bounded, i.e.,

(POLY) There exists a polynomial P : <m → < such that

|fqi(z)| ≤ |P (z)| for all z ∈ <m+ , q = A,B, and i = 1, . . . ,m.

In addition, we assume that gi ≥ 0 is smooth and dqi > 0 for each i = 1, . . . ,m, q =

A,B. A classical solution of (P) will be defined to be a pair of functions (u, u) such that

u ∈ C2(A,<m) ∩ C1(A,<m), u ∈ C2(B\∂A,<m) ∩ C1(B,<m) and the functions u and u

satisfy (P). The above hypotheses are sufficient to obtain the following existence result.

Theorem 2.1. Suppose that fA, fB : <m+ → <m are locally Lipschitz and satisfy (QP),

(BAL), and (POLY). In addition, suppose that each gi is nonnegative and smooth, and

dqi > 0 for each i = 1, . . . ,m, q = A,B. Then, if n = 2, there exists a componentwise

nonnegative classical solution of (P).

The problem becomes somewhat more complicated for space dimension n = 3. In this

case, our analysis requires some additional assumptions. Adopting the terminology from

[13] and [14], we impose the following quadratic intermediate sums condition:

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(QUAD) There exist a lower triangular matrix A = (ai,j) with positive diagonal entries

and a quadratic polynomial Q: <m → < such that

i∑j=1

ai,jfqj (z) ≤ Q(z) for all i = 1, . . . ,m, q = A,B and z ∈ <m+ .

We note that (QUAD) does not require each component of fA and fB to be bounded by a

quadratic polynomial. It simply requires that there is an ordering of the components of fAand fB such that the first component is bounded above by a quadratic polynomial and that

certain linear combinations of successive components of fA and fB can be formed so that

higher order terms either cancel each other or are only present with negative coefficients.

It should be clear that (QUAD) need only be stated for i = 1, . . . , (m− 1) due to (BAL).

We can now state our result for the case n = 3.

Theorem 2.2. Suppose that n = 3 and that (QUAD) and the hypotheses of Theorem 2.1

are satisfied. Then there exists a componentwise nonnegative classical solution of (P).

It is also possible to give a number of useful adaptations of the results above by giving

variations on (BAL) and (QUAD). For example, consider the following assumptions and

the subsequent theorem.

(BAL1) There exist Kq ≥ 0 for q = A,B, k ∈ 1, . . . ,m, bi > 0, and a polynomial

P1: <k → < such that

k∑i=1

bifqi(z) ≤ Kq andm∑

i=k+1

bifqi(z) ≤ P1(z1 . . . , zk)

for all z ∈ <m+ and q = A,B.

(QUAD1) There exists a lower triangular matrixA = (ai,j) with positive diagonal entries

and polynomials Q1:<k → <, Q2:<m → < such that

i∑j=1

ai,jfqj (z) ≤ Q1(z1, . . . , zk) ∀ i = 1, . . . , k, q = A,B, z ∈ <m+ ,

andi∑

j=k+1

ai,jfqj (z) ≤ Q2(z) ∀ i = k + 1, . . . ,m, q = A,B, z ∈ <m+ ,

where Q1 is quadratic and Q2 is quadratic in zk+1, . . . , zm.

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As we will see below, the first inequality in (BAL1) allows us to obtain a priori

estimates for ui and ui for i = 1, . . . , k. Once that is done, the second inequality in

(BAL1) becomes a balancing condition which allows us to obtain a priori estimates for the

remaining components. (QUAD1) is used in a similar manner.

Theorem 2.3. Suppose that fA, fB: <m+ → <m are locally Lipschitz and satisfy (QP),

(BAL1) and (POLY). In addition, suppose that gi is nonnegative and smooth and that

dqi > 0 for each i = 1, . . . ,m, q = A,B. Then, if n = 2 or if n = 3 and (QUAD1) is

satisfied, there exists a componentwise nonnegative classical solution of (P).

We give an example in the last section to illustrate Theorem 2.3.

3. Preliminary Estimates

In this section, we develop some preliminary scalar estimates. We assume g is a fixed

nonnegative smooth function, GA ∈ Lq(A) and GB ∈ Lq(B) with 1 ≤ q < ∞, d > 0, and

ηA (respectively ηB) denotes the unit normal vector on ∂A (respectively ∂B) pointing out

of A (respectively B). Throughout this section we assume U ∈ C2(A,<) ∩ C1(A,<) and

W ∈ C2(B\∂A,<) ∩ C1(B,<) are nonnegative functions satisfying the system

(SC)

−∆U ≤ GA in A−∆W ≤ GB in B∂U∂ηA

= ∂W∂ηA

on ∂A1dW ≤ U ≤ dW on ∂AW = g on ∂B\∂A.

Remark. At first glance, it might appear as though estimates for solutions of (SC) are a

simple matter. However, this is simply not the case.

We begin by obtaining an L1 estimate for solutions of (SC).

Lemma 3.1. There exists a constant C dependent upon the L1 norms of GA and GB,

but independent of U and W , such that∫∂A

(U +W )dσ +∫A

U dx+∫B

W dx ≤ C.

Proof: Let φ and ψ solve, respectively,−∆φ = 1 in Aφ = 0 on ∂A and

−∆ψ = Θ in Bψ = 1 on ∂Aψ = 0 on ∂B\∂A.

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Note that φ ≥ 0 in A and ∂φ∂ηA

< 0 on ∂A by virtue of the maximum principle. Also, we

have the estimates ∫A

U ≤∫A

U(−∆φ) ≤ −∫∂A

U∂φ

∂ηA+∫A

GAφ(INT3.1)

≤ −d∫∂A

W∂φ

∂ηA+∫A

GAφ

and

(INT3.2)∫B

WΘ ≤∫∂A

−W ∂ψ

∂ηB+ ‖GA‖1,A −

∫∂B\∂A

g∂ψ

∂ηB+∫B

GBψ

(in the first estimate we used the algebraic sign property of the normal derivative). We

begin by taking Θ = 0 above. Noting that ∂ψ∂ηB

> 0 on the compact set ∂A, we obtain a

bound for the L1 norm of W over ∂A. We now return to (INT3.2) and take Θ = 1. By

combining (INT3.1) and (INT3.2) we obtain our result.

In lieu of the result above, we know that U and W can be bounded a priori on subsets

of A and B which lie away from ∂A. We state this result in the following lemma. A proof

establishing these interior estimates can be found in either [1] or [6].

Lemma 3.2. Suppose that GA ∈ Lq(A) and GB ∈ Lq(B) with q > n2 . For every ε > 0

there exists a constant Cε > 0, independent of U andW , such that if ΩA ⊆ A and ΩB ⊆ B

with dist(ΩA, ∂A), dist(ΩB, ∂A) > ε, then U(x), W (y) ≤ Cε for all x ∈ ΩA and y ∈ ΩB.

Consequently, we need to concentrate on obtaining estimates near ∂A. Our first set of

these estimates is obtained in the proof of the following lemma. The lemma is stated in

terms of the resulting estimates for U and W in all of A and B respectively.

Lemma 3.3. If 1 < p < ∞ is given such that pp−1

> n2

, then there exists a constant

Cp > 0 dependent upon the L1 norms of GA and GB , but independent of U and W , such

that ‖U‖p,A, ‖W‖p,B < Cp.

Proof: We need only develop estimates in a neighborhood of ∂A. To this end, let P ∈ ∂Aand suppose φ1, . . . , φn is a smooth, orthogonal, local coordinate system for a neighbor-

hood of P such that φ1 = 0 determines ∂A locally; i.e. there exists an ε > 0 such that

φ1, . . . , φn are smooth on Bε(P ) = x | |x−P | < ε, ∂A∩Bε(P ) = x | φ1(x) = 0∩Bε(P ),

|∇φi(x)| ≥ δ > 0 on Bε(P ), and ∇φi(x) · ∇φj(x) = 0 on Bε(P ) if i 6= j. Assume without

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loss of generality that ∇φ1(x) points out of A at each point x ∈ ∂A∩Bε(P ) (see Figure 2).

Then there exist αi > 0 such that

x∣∣ |φi(x)| ≤ αi, i = 1, . . . , n ⊆ Bε(P ) and x | φ1(x) ≤ 0 ∩Bε(P ) ⊆ A,

and near P we can treat U and W as functions of φ1, . . . , φn.

If we apply this change of variables to U and W then we obtain

∆xU =n∑i=1

(Uφiφi |∇φi|2 + Uφi∆xφi

)and ∆xW =

n∑i=1

(Wφiφi |∇φi|2 +Wφi∆xφi

).

Consequently, if we define (see Figures 2 and 3)

φ = (φ1, φ2, . . . , φn), φ = (−φ1, φ2, . . . , φn),

R = (φ1, φ2, . . . , φn) | −α1 < φ1 < 0 and |φj | < αj for j = 2, . . . , n,

Γ = φ | φ ∈ R and φ1 = 0,

and

Z(φ) = U(φ) +W (φ) for (φ1, . . . , φn) ∈ R,

then routine calculations yield

(ZEQ)

−n∑i=1

ai(φ)Zφiφi(φ) ≤ F (φ) φ ∈ R

Zφ1(φ) = 0 φ ∈ Γ

where each ai ∈ C2(R) with ai > 0 and

F (φ) =n∑i=1

(c1,i(φ)Wφiφi(φ) + c2,iUφi(φ) + c3,i(φ)Wφi(φ)

)+H,

where c1,i ∈ C2(R) with c1,i(φ) = 0 for all φ ∈ Γ, c2,i, c3,i ∈ C1(R), andH is an L1 function

dependent upon GA and GB. We remark that since Z(φ) = U(φ) + W (φ), Lemma 3.1

yields a bound for Z in each of L1(R) and L1(Γ). We also note that from Lemma 3.2 and

the compactness of the boundary, it is sufficient to obtain bounds for Z in Lq(R) to prove

the lemma.

We obtain these estimates for Z as follows. Begin by taking 0 < βi < αi and defining

µ: R → [0, 1] such that µ is smooth, ∂µ∂ηR

= 0 on ∂R, µ(φ) = 1 for all φ such that

−βi2 < φ1 < 0 and |φj| < βj2 for j = 2, . . . , n, and µ(φ) = 0 for all φ such that either

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−α1 ≤ φ1 ≤ − 3β14 or |φj | ≥ 3βj

4 for j = 2, . . . , n (see Figure 3). Now take X to be a

smooth bounded domain such that

φ | −α1 < φ1 < 0 and |φj| < 3αj/4, j = 2, . . . , n ⊆ X ⊆ R

(see Figure 3). In addition, suppose that n2< p < ∞ and Θ ∈ Lp(R) is such that Θ ≥ 0

a.e. and ‖Θ‖p,Ω = 1. Note that if ηXi is the ith component of the unit normal vector on

∂X pointing out of X and M > 0 is chosen sufficiently large, then there exists a unique,

nonnegative solution y ∈W 2p (X) of

(DUAL)

n∑i=1

(aiy)φiφi +My = Θ φ ∈ Xn∑i=1

∂(aiy)∂ηXi

= 0 φ ∈ ∂X.

Furthermore, with M chosen in this manner, there exists Cp > 0 independent of Θ such

that ‖y‖2,p,X ≤ Cp, where ‖·‖2,p,X denotes the standard Sobolev norm on W 2p (X) ([1], [2],

[6]). Since p > n2 , we can apply Sobolev imbedding results [6] to show y ∈ C(X)∩W 2

p (X)

and obtain a constant Cp > 0 independent of Θ so that ‖y‖∞,X ≤ Cp. In addition, in

the case when p > n, we have y ∈ C1(X) ∩ W 2p (X) and Cp can be chosen [6] so that

‖yφi‖∞,X ≤ Cp for each i = 1, . . . , n.

We proceed via duality. Integration over X with respect to φ yields∫X

µZΘ =∫X

µZ(−

n∑i=1

(aiy)φiφi +My)

=∫X

µZMy −∫X

n∑i=1

aiy(µZ)φiφi

= M

∫X

µZy −∫X

n∑i=1

aiy(µφiφiZ + 2µφiZφi + µZφiφi

)≤M

∫X

µZy −∫X

n∑i=1

(aiyµφiφi − 2(aiyµφi)φi

)Z −

∫Γ∩∂X

2a1yµφiZ +∫X

µyF.

Also, ∫X

µyF =∫X

µy( n∑i=1

(c1,iWφiφi(φ) + c2,iUφi + c3,iWφi(φ)

)+H

)

=∫X

(n∑i=1

(((µyc1,i)φiφi − (µyc3,i)φi

)W (φ)− (µyc2,i)φiU(φ)

)+ µyH

)

+∫

Γ∩∂X

(− (µyc1,1)φ1W (φ) + µyc2,1U(φ) + µyc3,1W (φ)

).

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If we combine these computations with the boundary conditions satisfied by y on Γ and

the definition of Z then we can show there exist functions g1, g2, g3, g4 ∈ C(X) with g2

independent of the choice of βi for i = 1, . . . , n so that

(INT3.3)∫X

µZΘ ≤∫Γ

Zy g1 +∫X

(µZg2

n∑i=1

|c1,i yφiφi |+ (Z +H)y g3 + µZg4

n∑i=1

|yφi |).

From the conditions that g1, . . . , g4 satisfy, there exist K,Kβ > 0 so that ‖g2‖∞ < K and

‖gi‖∞ < Kβ for i = 1, 3, 4. Now, suppose for the moment that p > n. Then we know from

our discussion above that y ∈ C1(X) and there exists Cp > 0 independent of Θ so that

‖y‖∞,X , ‖yφi‖∞,X , ‖y‖2,p,X < Cp.

Consequently, if we let C be the constant given in Lemma 3.1 then (INT3.3) yields

(INT3.4)∫X

µZΘ ≤ (n + 2 + ‖H‖1,X)CCpKβ +CpK

∫X

(µZ

n∑i=1

|c1,i|) pp−1p−1p

.

Now recall that c1,i ∈ C(X) with c1,i = 0 on Γ, and that the values βi determine the

support of µ. Combining this information with (INT3.4) we obtain (through duality) the

existence of ε > 0 such that if 0 < βi < ε for each i = 1, . . . , n and 0 < δ < 1n−1

, then

there exists Cβ,δ > 0 independent of Z such that

(INT3.5) ‖µZ‖ nn−1−δ,Ω ≤ Cβ,δ.

We now return to (INT3.3) and suppose p > n2 . Then y ∈ C(X) ∩W 2

p (X) and there

exists Cp > 0 independent of Θ such that ‖y‖∞,X , ‖y‖(2)p,X < Cp. Then Sobolev imbedding

results imply there exists Cn,γ > 0 so that ‖yφi‖n+γ,Ω ≤ Cn,γ for all γ > 0 sufficiently

small. These estimates and (INT3.3) yield

(INT3.6)∫X

µZΘ ≤ (2 + ‖H‖1,X)CCpKβ +CpK

∫X

(µZ

n∑i=1

|c1,i|) pp−1

p−1p

+∫X

µZg4

n∑i=1

|yφi |.

Therefore, if we apply (INT3.5) and the estimate above for yφi above with appropriate

choices for δ, γ > 0, then we find

∫X

µZΘ ≤ (2 + ‖H‖1,X)CCpKβ + CpK

∫X

(µZ

n∑i=1

|c1,i|) pp−1p−1p

+ nKβCβ,δCn,γ.

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Thus, in a manner similar to that used to obtain (INT3.5), we find that there exists kp > 0

independent of Z so that ‖µZ‖ pp−1 ,X

≤ kp for all p > n2 . That is, there exists Kq > 0

independent of Z so that ‖µZ‖q,X ≤ Kq for all 1 ≤ q < nn−2 (where n

n−2 is interpreted to

be ∞ when n = 2). Consequently, if we recall our remark at the beginning of the proof,

then we have our result.

We remark that when n = 2, Lemma 3.3 implies that U ∈ Lp(A) and W ∈ Lp(B)

for all 1 ≤ p < ∞, with bounds independent of U and W . This will ultimately give

us the necessary estimates to obtain Theorem 2.1 and the first portion of Theorem 2.3.

Unfortunately, it falls short of what we need to obtain Theorem 2.2 and the final portion of

Theorem 2.3. The following lemma will be used in conjunction with Lemma 3.3 to obtain

these latter results.

Lemma 3.4. Suppose that n = 3 and there exists C > 0 such that ‖GA‖2,A, ‖GB‖2,B,

‖U‖2,∂A, ‖W‖2,∂A ≤ C . Then there exists Kq > 0 dependent upon C , but otherwise

independent of GA, GB , U , and W , such that ‖U‖q,A, ‖W‖q,A ≤ Cq for all 1 ≤ q <∞.

Proof: Proceed as in the proof of Lemma 3.3, but disregard the assumption that p > n2 .

We have the a priori estimate ‖Z‖2,Γ ≤ 2C (from the hypothesis). Furthermore, for every

1 ≤ q < 3 there exists Cq > 0 such that ‖Z‖q,G ≤ Cq from Lemma 3.3. Using the a priori

bound on y in W 2,p(X), we can be assured that for every δ > 0 there exists 1 < p < 1 + δ

and Kp > 0 such that ‖y‖3+δ,X , ‖yφi‖ 32 +δ,X , ‖y‖2+δ,Γ ≤ Kp. If we employ these estimates

and the hypothesis in (INT3.3) then we obtain our result by proceeding analogously to the

final portion of the proof of Lemma 3.3.

The proofs of our main results require some fundamental results for scalar diffractive-

diffusion problems. To this end, suppose that dA, dB > 0, gA ∈ Lp(A), gB ∈ Lp(B), and

h ∈ H1/2(∂B\∂A). We consider the problem of determining functions φA, φB defined on

A,B respectively such that

(SCAL)

−dA∆φA = gA x ∈ A−dB∆φB = gB x ∈ B

φA = φB on ∂AdA

∂φA∂ηA

= dB∂φB∂ηA

on ∂A

φB = h on ∂B\∂A

in some appropriate sense. The notion of a weak solution of (SCAL) is defined in the usual

manner. One produces a function ψ ∈ H1(B) so that ψ = h on ∂B\∂A and the support of

ψ does not intersect ∂A. Then the substitution φB = φB + ψ is made in (SCAL) to yield

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a system similar to (SCAL) with a homogeneous boundary condition on ∂B\∂A. Now

define the sets

HAB = (vA, vB) ∈ H1(A) ×H1(B) | vA = vB a.e. on ∂A

and

HAB,0 = (vA, vB) ∈ H1(A)×H1(B) | vB = 0 a.e. on ∂B\∂A and vA = vB a.e. on ∂A

equipped with the inner products 〈·, ·〉: HAB ×HAB → < and 〈·, ·〉0: HAB,0×HAB,0 → <defined by⟨

(vA, vB), (wA, wB)⟩

= dA

∫A

∇vA · ∇wA +∫B

(dB∇vB · ∇wB + vBwB)

for all ((vA, vB), (wA, wB)) ∈ HAB ×HAB and⟨(vA, vB), (wA, wB)

⟩0

= dA

∫A

∇vA · ∇wA + dB

∫B

∇vB · ∇wB

for all ((vA, vB), (wA, wB)) ∈ HAB,0 × HAB,0. It is a simple matter to show that HABand HAB,0 are Hilbert spaces when they are equipped with these inner products. We say

(φA, φB) ∈ HAB is a weak solution of (SCAL) provided (φA, φB) ∈ HAB,0 and

(WK3.7)∫A

dA∇φA · ∇vA +∫B

dB∇φB · ∇vB =∫A

gAvA +∫B

(gBvB − dB∇ψ · ∇vB)

for all (vA, vB) ∈ HAB,0.

Proposition 3.5. Let p > 2nn+2 , gA ∈ Lp(A), gB ∈ Lp(B), and h ∈ H1/2(∂B\∂A). Then

there exists a unique weak solution (φA, φB) ∈ HAB to (SCAL). Furthermore, there exists

Kp > 0 independent of gA and gB such that

‖(φA, φB)‖HAB ≤ Kp

(‖gA‖p,A + ‖gB‖p,B + ‖h‖H1/2(∂B\∂A)

).

Proof: Set q = pp−1 . Note that there exist K1 > 0 independent of h such that the selection

of ψ can be made in the manner above with ‖ψ‖1,2,B ≤ K1‖h‖H1/2(∂B\∂A). Furthermore,

if (vA, vB) ∈ HAB,0 then (vA, vB) ∈ Lq(A) × Lq(B) and there exists Cq > 0 independent

of (vA, vB) such that

‖vA‖q,A + ‖vB‖q,B ≤ Cp ‖(vA, vB)‖HAB,0 .

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Finally, from the Poincare inequality there exists K2 > 0 such that for every (vA, vB) ∈HAB,0

‖vA‖1,2,A + ‖vB‖1,2,B ≤ K2‖(vA, vB)‖HAB,0 .

Now define F : HAB,0 → < via

F (vA, vB) =∫A

gAvA +∫B

(gBvB − dB∇ψ · ∇vB)

for all (vA, vB) ∈ HAB,0. Clearly F is well defined and linear. also,

|F (vA, vB)| ≤ ‖gA‖p,A‖vA‖q,A + ‖gB‖p,B‖vB‖q,B + dB‖ψ‖1,2,B‖vB‖1,2,B≤(‖gA‖p,A + ‖gB‖p,B

)Cp‖(vA, vB)‖HAB,0

+ dBK1‖h‖H1/2(∂B\∂A)K2‖(vA, vB)‖HAB,0 .

Therefore, F is a bounded linear functional on HAB,0. As a result, a simple application of

the Riesz Representation Theorem guarantees the existence of a unique (φA, φB) ∈ HAB,0that solves (WK3.7). Moreover, ‖(φA, φB)‖HAB,0 = ‖F‖. The result follows.

The theorem below is well known ([11], [12], [17]). Since our focus does not lie with

the properties of the functions gi in (P), our statement of the result below is given for

smooth h in (SCAL). Certainly, if h is smooth, then we have h ∈ H1/2(∂B\∂A). Also,

recall from the beginning of Section 2 that we are assuming n ≥ 2 throughout this work.

Consequently, n2 ≥2nn+2 , and therefore the results of Proposition 3.5 above can be applied

in the setting of the theorem below.

Theorem 3.6. Let h be a smooth function. If gA ∈ Lp(A) and gB ∈ Lp(B) with p > n2 ,

then there exists 0 < α < 1 such that the unique weak solution (φA, φB) ∈ HAB of (SCAL)

satisfies φA ∈ Cα(A) and φB ∈ Cα(B). If there exists 0 < β < 1 such that gA ∈ Cβ(A)

and gB ∈ Cβ(B), then φA ∈ C2(A) ∩ C1(A), φB ∈ C2(B\∂A) ∩ C1(B), and (φA, φB)

satisfies (SCAL) in the classical sense. In either case, the norm estimates on (φA, φB)

depend only upon norm estimates for h, gA and gB.

We close this section with the statement of a well known fixed point result [22].

Theorem 3.7. Let X be a Banach space and suppose that T : X → X is a completely

continuous map. If there exists a constant K > 0 such that ‖y‖X ≤ K whenever 0 ≤ σ ≤ 1

and y ∈ X such that y = σT (y), then there exists x ∈ X such that T (x) = x.

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4. Proofs of the Main Results

Since our results have a common flavor, our analysis begins with a structure which can be

applied uniformly. We begin by truncating our vector field and forcing nonnegativity. To

this end, we let ψ ∈ C∞0 (<m, [0, 1]) and define Fq(z) = ψ(z)fq(z+), for q = A,B. Note

that Fq is Lipschitz. We now create a natural fixed point problem associated with (P).

Define

T : L2(A,<m)× L2(B,<m)→ L2(A,<m)× L2(B,<m)

via T (v, v) = (u, u) where (u, u) is the unique weak solution of

(PN)

−dAi∆ui = FAi(v) x ∈ A−dBi∆ui = FBi(v) x ∈ B

ui = ui on ∂AdAi

∂ui∂ηA

= dBi∂ui∂ηA

on ∂A

ui = gi on ∂B\∂A

guaranteed by Proposition 3.5. (Note that (FAi(v), FBi(v)) ∈ L∞(A) × L∞(B) for all

(v, v) ∈ L2(A)× L2(B).)

Lemma 4.1. T is a completely continuous map on L2(A)× L2(B). Furthermore, if 0 ≤λ ≤ 1 and (u, u) ∈ L2(A)× L2(B) are such that (u, u) = λT (u, u), then u ∈ C2(A,<m) ∩C1(A), u ∈ C2(B\∂A,<m) ∩ C1(B), and (u, u) = (u+, u+).

Proof: From the definitions of FA and FB we know there exists Mψ > 0 such that

‖FAi(v)‖∞,A, ‖FBi(v)‖∞,B ≤ Mψ for all (v, v) ∈ L2(A) × L2(B). Consequently, from

Proposition 3.5, T maps all of L2(A) × L2(B) into a bounded subset of HAB , which is

clearly compactly imbedded in L2(A)×L2(B). Similarly, the continuity of T can be easily

obtained from Proposition 3.5 since FA and FB are Lipschitz. As a result, T is a completely

continuous map on L2(A) × L2(B).

Now suppose that 0 ≤ λ ≤ 1 and (u, u) ∈ L2(A)×L2(B) is such that (u, u) = λT (u, u).

Then (u, u) is a weak solution of

(λPN)

−dAi∆ui = λFAi(u) x ∈ A−dBi∆ui = λFBi(u) x ∈ B

ui = ui on ∂AdAi

∂ui∂ηA

= dBi∂ui∂ηA

on ∂A

ui = λgi on ∂B\∂A.

Furthermore, from the properties of FA and FB, and the first portion of Theorem 3.6, we

know there exists 0 < α < 1 such that (u, u) ∈ Cα(A,<m) × Cα(B,<m). Consequently,

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if we apply this information to the properties of FA and FB , and the second portion

of Theorem 3.6, then we have u ∈ C2(A,<m) ∩ C1(A), u ∈ C2(B\∂A,<m) ∩ C1(B) and

(u, u) satisfies (λPN) in the classical sense. Therefore, if we multiply the partial differential

equation for ui in (λPN) by u−i and integrate by parts, then we obtain

(INT4.1) −dAi∫A

|∇u−i |2dx+ dBi

∫∂A

u−i∂ui∂ηB

dσ ≥ 0

from the quasi-positivity condition (QP). If we perform a similar calculation with u−i then

after rearranging terms, we are lead to

(INT4.2) −dBi∫∂B

u−i∂ui∂ηB

∂σ − dBi∫B

|∇u−i |2dx ≥ 0.

If we add (INT4.1) and (INT4.2), then we find

(INT4.3) −dBi∫

∂B\∂A

u−i∂ui∂ηB

dσ − dAi∫A

|∇u−i |2dx− dBi∫B

|∇u−i |2dx ≥ 0.

Note that by hypothesis, we have u−i = 0 on ∂B\∂A. Therefore, we conclude from (INT4.3)

that (u, u) is nonnegative.

Proof of Theorem 2.1. Define U =m∑i=1

bidAiui, W =m∑i=1

bidBi ui and g =m∑i=1

bidBigi.

Then clearly U and W satisfy (SC) with HA = KA and HB = KB . Consequently, if n = 2

then from Lemma 3.3 we have ‖U‖q,A, ‖W‖q,B < Cq for all 1 ≤ q <∞. This estimate, in

turn, implies ‖ui‖q,A, ‖ui‖q,B < Cq/bi for all 1 ≤ q <∞ and i = 1, . . . ,m, and we remark

that these estimates are independent of λ and the cut-off function ψ. If we now recall

(POLY), then we have a bound for λFqi(u) in every Lp space, for q = A,B, i = 1, . . . ,m

and 1 ≤ p <∞. As a result, if we apply Theorem 3.6, then we obtain C∞ > 0 independent

of λ and ψ such that ‖ui‖∞,A, ‖ui‖∞,B < C∞. Therefore, from Theorem 3.7, we have a

fixed point for T , and hence a solution of (NP). In addition, a sup-norm bound for this

fixed point is obtained independent of the cut-off function. Consequently, by choosing ψ

appropriately, we can conclude that (P) has a nonnegative solution when n = 2.

Proof of Theorem 2.2. We proceed as in the proof of Theorem 2.1. First define

U =m∑i=1

bidAiui, W =m∑i=1

bidBi ui, and g =m∑i=1

bidBigi. As above, U and W satisfy

(SC). Consequently, if n = 3 then from Lemma 3.3 we have ‖U‖q,A, ‖W‖q,B < Cq for all

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1 ≤ q < 3. This estimate, in turn, implies ‖ui‖q,A, ‖ui‖q,B < Cq/bi for all 1 ≤ q < 3 and

i = 1, . . . ,m, and we remark that these estimates are independent of λ and the cut-off

function ψ.

We now employ the intermediate sums condition (QUAD) to improve our estimates.

We proceed inductively. First note that

(INT4.4)∫A

d1|∇u1|2 +∫B

dB1|∇u1|2 ≤∫A

u1Q(u) +∫B

u1Q(u).

From Lemma 3.3, there exists C > 0 such that

‖u1‖1,A, ‖u1‖1,B, ‖Q(u)‖4/3,A, ‖Q(u)‖4/3,B < C.

In addition, from Sobolev Imbedding there exists a constant K > 0 such that

‖u1‖4,A + ‖u1‖4,B < K(‖u1‖1,2,A + ‖u1‖1,2,B

)and (

‖u1‖1,2,A + ‖u1‖1,2,B)2 ≤ K(C +

∫A

d1|∇u1|2 +∫B

dB1 |∇u1|2).

Combining these inequalities with (INT4.4) yields

(‖u1‖1,2,A + ‖u1‖1,2,B)2 < CK2(‖u1‖1,2,A + ‖u1‖1,2,B + 1

).

As a result, there exists L > 0 such that

‖u1‖1,2,A, ‖u1‖1,2,B ≤ L,

with L independent of λ and ψ. Now suppose that 1 ≤ k < m and there exists L > 0 such

that ‖ui‖1,2,A, ‖ui‖1,2,B ≤ L for all 1 ≤ i ≤ k, with L given independent of λ and ψ. If we

again employ (QUAD), then we obtain∫A

∇(k+1∑i=1

ui

)· ∇(k+1∑i=1

dAiui

)+∫B

∇(k+1∑i=1

ui

)· ∇(k+1∑i=1

dBi ui

)(INT4.5)

≤∫A

k+1∑i=1

uiQ(u) +∫B

k+1∑i=1

uiQ(u).

Consequently, if we argue similarly as above and apply our induction hypothesis, then we

obtain L > 0 independent of λ and ψ such that ‖uk+1‖1,2,A, ‖uk+1‖1,2,B ≤ L. Therefore,

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we can conclude that there exists L > 0 such that ‖ui‖1,2,A, ‖ui‖1,2,B ≤ L for all 1 ≤ i ≤ m,

with L given independent of λ and ψ.

We can now complete our proof. From the estimates above and trace class imbeddings,

we have ‖U‖3,∂A, ‖W‖3,∂A ≤ C where C > 0 can be chosen independent of λ and ψ.

As a result, Lemma 3.4 implies ‖U‖q,A, ‖W‖q,B < Cq for all 1 ≤ q < ∞, and hence

‖ui‖q,A, ‖ui‖q,B < Cq/bi for all 1 ≤ q < ∞ and i = 1, . . . ,m, with these estimates being

independent of λ and the cut-off function ψ. The proof can now be completed in a manner

similar to that of Theorem 2.1.

Proof of Theorem 2.3. We proceed similarly to the proofs of Theorems 2.1 and 2.2, with

adjustments made to accommodate the differences between (BAL)–(QUAD) and (BAL1)–

(QUAD1). To this end we first let U =k∑i=1

bidAiui, W =k∑i=1

bidBi ui and g =k∑i=1

bidBigi

and follow the arguments above to obtain ‖ui‖q,A, ‖ui‖q,B < Cq for all 1 ≤ q < ∞and 1 ≤ i ≤ k, with Cq independent of λ and the cut-off function ψ. We then take

U =m∑

i=k+1

bidAiui, W =m∑

i=k+1

bidBi ui and g =m∑

i=k+1

bidBigi and note that there exists

d > 0 such that

−∆U ≤ P1(u1, . . . , uk) in A−∆W ≤ P1(u1, . . . , uk) in B

∂U∂ηA

= ∂W∂ηA

on ∂A1dW ≤ U ≤ dW on ∂AW = g on ∂B\∂A.

Since P1 is a polynomial and we have Lq estimates for ui, ui for 1 ≤ i ≤ k, we can employ

the estimates from Section 3 to obtain estimates for U and W dependent upon our Lq

estimates for ui, ui for 1 ≤ i ≤ k. If we continue as in the proofs of Theorem 2.1 and 2.2,

then we obtain Lp estimates for ui, ui for k + 1 ≤ i ≤ m and 1 ≤ p <∞, dependent upon

our earlier estimates for ui, ui for 1 ≤ i ≤ k, with all estimates independent of λ and the

cut-off function ψ. Continuing to follow the proofs above, we obtain our result.

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5. Examples

In this section we illustrate the applicability of our results by applying them to two non-

trivial chemical models.

Example 5.1. Perhaps the greatest source of interesting problems in this area is the

modeling of multi-species chemical reactions. For example, let us consider the following,

seemingly simple, reversible reaction in which tin oxide reacts with carbon monoxide to

form form metallic tin and carbon dioxide:

SnO2 + 2CO Sn + 2CO2.

We assume that this process takes place on a heterogeneous domain in <2 of the type

described in the previous sections. Let u1, u2, u3, u4 denote the concentrations of SnO2,

CO, Sn, CO2 respectively in A, and u1, u2, u3, u4 the associated counterparts in B. Then

we might model the reaction diffusion process for this single step chemical process in the

form of the system (P). In this case we have

fA(z) =

12−1−2

(kArz3z24 − kAfz1z

22

)and

fB(z) =

12−1−2

(kBrz3z24 − kBfz1z

22

),

where kqf , kqr are forward, reverse effective rates, respectively, for q = A,B. Here the

terms kqf , kqr are assumed to be smooth, bounded, nonnegative, and possible dependent

upon z. Close examination of the vector fields above show that (QP) and (POLY) are

clearly satisfied. Furthermore, (BAL) holds with Kq = 0 for q = A,B, and

[b1, b2, b3, b4] = [1, 1, 1, 1].

Consequently, we can conclude from Theorem 2.1 that (P) has a componentwise nonneg-

ative classical solution.

Of course, from the comments in Section 1 this should be no surprise. As stated in

that section, all single step reversible chemical processes satisfy (BAL). In addition, it is

easy to show that (QP) and (POLY) are satisfied by these systems [8].

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Example 5.2. In this example we demonstrate the applicability of our results by consid-

ering a somewhat more complex chemical process. More specifically, we consider the inter-

actions of the chemical concentrations of the minimal chlorite-iodite reaction. A somewhat

simplified version of this model was considered by Ouyang, Castets, Boissonade, Roux and

DeKepper [16]. There the researchers considered the one-dimensional Couette reactor in

which chemical concentrations of the minimal chlorite-iodite reaction are fed at both ends

of the reactor. The authors employed a simplifying assumption to reduce the complexity

of the model equations. A number of spatial patterns were found experimentally, and

Ouyang, et al, ran numerical tests on the model to reinforce their findings. The chemical

process is described by the following set of reactions steps:

H+ + ClO−2 + I− HOCl + HOI

H+ + HOI + I− I2 + H2O

ClO−2 + HOI + H+ HIO2 + HOCl

HOCl + I− HOI + Cl−

HIO2 + I− + H+ 2HOI

2HIO2 HOI + IO−3 + H+

HIO2 + HOI I− + IO−3 + H+

HIO2 + HOCl IO−3 + Cl− + 2H+.

If no simplifying assumptions are made, and the laws of mass action kinetics are employed,

then these reaction steps lead to a ten-component reaction-diffusion model. We consider

the scenario in which we have a reaction diffusion process taking place on a smooth bounded

region Ω of <3 which decomposes similarly to the one given in Section 1. Consideration

of the associated steady state problem can lead to a system of the form (P) with fA, fB

described through the following mechanism. First, we denote u1 = [ClO−2 ], u2 = [Cl−],

u3 = [I−], u4 = [HOCl], u5 = [HOI], u6 = [I2], u7 = [H2O], u8 = [HIO2], u9 = [IO−3 ] and

u10 = [H+] in A, with corresponding assignments for u1, . . . , u10 in B. We then denote the

reaction rates by

R1(z) = −k1z1z3z10, R2(z) = k−2z6z7 − k2z3z5z10, R3(z) = −k3z1z5z10,

R4(z) = −k4z3z4, R5(z) = k−5z25 − k5z3z8z10, R6(z) = −k6z

28 ,

R7(z) = −k7z5z8, and R8(z) = −k8z4z8,

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and make analogous assignments for Ri(z) with the only changes being the use of kj , k−j .

Here we assume each kj , k−j , kj , k−j is nonnegative, bounded, and possibly smoothly

dependent upon z. Using standard terminology, we refer to kj , k−j , kj, k−j as forward and

reverse effective rates for the jth reaction step in the regions A and B. We then define the

stoichiometric matrix

N =

1 0 1 0 0 0 0 00 0 0 −1 0 0 0 −11 1 0 1 1 0 −1 0−1 0 −1 1 0 0 0 1−1 1 1 −1 −2 −1 1 0

0 −1 0 0 0 0 0 00 −1 0 0 0 0 0 00 0 −1 0 1 2 1 10 0 0 0 0 −1 −1 −11 1 1 0 1 −1 −2 −2

.

Then fA(z) = NR(z) and fB(z) = NR(z), where R(z) = (Ri(z)) and R(z) = (Ri(z)) are

vector valued functions. It is a simple matter to check that (QP) and (POLY) are satisfied.

Also, if we denoteN9 to be the 9×8 matrix obtained by deleting the tenth row fromN , then

it is easy to see that ( 1, 1, . . . , 1 )T ∈ ker(NT9 ). In addition, fq10 (z) ≤ P1(z4, z5, z6, z7, z8)

for q = A,B, where P1 is a quadratic polynomial. As a result, (BAL1) is satisfied with

bi = 1 for all i = 1, . . . ,m, Kq = 0, k = 9 and P1 as above. Finally, if we define

A =

1 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 00 0 2 0 1 0 0 0 0 00 0 1 0 0 1 0 0 0 00 0 1 0 0 0 1 0 0 01 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 1

,

then it is a simple matter to check (QUAD1). As a result, we can apply Theorem 2.3 to

guarantee the existence of a nonnegative solution to (P).

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Figures

1= –

22

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W.E. Fitzgibbon∗

Department of Mathematics

University of Houston

Houston, TX 77204-3476

S.L. Hollis

Department of Mathematics

Armstrong Atlantic State University

Savannah, GA 31419

J.J. Morgan∗

Department of Mathematics

Texas A&M University

College Station, TX 77843-3368

∗These authors gratefully acknowledge support from NSF Grant DMS 9207064 and DMS 9208046

respectively.

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