Steady State Solutions for Balanced Reaction Di usion Systems on
Transcript of 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
12
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 .
13
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.
14
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,
15
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
16
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,
17
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.
18
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].
19
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,
20
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).
21
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.
24