dk) E C2+a,a [.0 x A Rn(l)] I "

by the same argument as above using Theorem

4.1 .2.

In Lhe case of (II), we note that for components i, where

D.

= 1-1. = 0,

(4.3.37)

4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 1 1

Cfk-I)

i s Holder continuous in x and z with exponent a, then

cfk)

is also Holder continuous in x and z with

exponent a, implying that

cfk)

E

Ca,a

x /1 ,

Rn(l»).

To prove (III)-(V) in the general case, we need only observe that F;(z,

C)k-I» +Hi!a.a2 cfk-I)

_E

Ca

[/1 x

_{Rn(J) , Rn(J)]}

_{from (i)-(v).}

In the case of (III), we note that for components i, where

!i'i >

0,

by the same arguments as above using Theorem 4. 1 .2.

In the case of (IV), we note that for components i, where �i = 0, _{u ·}

vcy)

== 0 ,

dk) =

I

F(z I '

_C(k-I»

_J +

H·I c(k-I)

_I a.a2 _I

(4.3.38)

(4.3.39)

exists and is unique since it is uniquely defined. By the same argument as for the proof of (II), we see that

(4.3.40) In the case of (V), we see that by (Hs) and (HiJ , z(z I , z 1 0, zo) ancl Yi(Z I , a, YiO; ZO) are unique solutions of (4.3 .22) and (4.3.24), respectively, on

A I.

Choose YiO = Ci, I (ZO) and note that if Z = Z(Z I , a , zo), then because of uniqueness, Zo = z(a, ZI , z). Also, the solution (Z(ZI , a, zo),

Yi(ZI ,

a,

YiO;

zo» of the systems (4.3.22) and (4.3 .24) is a characteristic equation of (4.3 . 1 9). Hence, for each solution of (4.3 .22) and (4.3.24), we have

(4.3.41) and consequent! y ,

(4.3.42)

Now by using assumptions (Hs) and (H6 ) , it is easy to show that

C_fk

)(z) defined by (4.3 .42) satisfies (4. 3 . 19).

To show uniqueness of solutions of (4. 3 . 1 9), we suppose, that

Ci(lk)

and

cg)

are two solutions of (4.3. 1 9) on

A = A I X An-l.

By Theorem 3.2.9 (Strong Com parison Theorem) for fi rst order partial differential equations, we see that

cflk)

$

cg)

$

cft)

�U1d therefore

cg)

coincides willI

cg).

The Holder continuity of C?)O,z) is obtained by examining the characteristic equations (4.3.22) and

(4.3.24), so that

(4.3.43)

Finally, we show that

(fi' fi)

and (ci' Cj) are lower and upper solutions of

(cfk), CY» .

To show that

(Cj, Cj) is an upper solution of

_{(cfk), C[k» ,}

we need to only observe iliat

4.3 _{EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 1 2}

D.�(c.- C(k» + H

·(

c

·

-c(k»

= > 0 on

dQ2xA

' an

I I ' " ' " -

,

- 9jv2(cj-cfk» + u · V(cj- cfk»

+

Hjs( cj-cfk»

� F;(z, C)-F;(z,

(cj- cfk-1»

� °

in A,

since Ii and

Fj

are monotone in Cj and Cj' respectively.

We may therefore apply Lemma 4. 1.8 (Maximum Principle) for the elliptic operator or Theorem 3.2.6

(Strong Comparison Theorem) for first order partial differential equations or algebraic inequalities to conclude that

(Cj' Cj)� (cfk), cfk» .

Note that in the case for components

i E

1 , _where

Dj

₌

Hj

_{= 0, we}

have

- (k)

> -)

(k-I) - (k-I)

_{> () .}

Cj- Cj

x, Cj

_X.

_Cj

_{+ Cj-Cj}

_ •

and in the case for components

i E

J, where

9)j

=

U

·

V(Cj-Cfk» =

0, we need only observe that

H·S( C·-

_{I I}

C(k»

_I > _-F (z _{I '}

C

_J·) - F(z _{I '}

C(k-l»

_J

+ HJ

_{I (Jil2} (c·-_I

c(k-l»

_I ->

°

in

A

,

since

fj

and

Fj

are monotone in

Cj

and Cj' respectively and therefore

(Cj,

Cj)

�

(

cfk), cfk» .

Similarly,

(fj.

C) may be shown to be a lower solution of

(cfk)

.

C?'»

and the theorem is complete.O

To start off the iterative procedure, we IIced some continu ity properties of

(

fj , �j) and

(Cj' Ci) .

The

properties of the mapping !!J, from

(cfk-I), C?-l»

to

(Cfk), Cfk»

are then given by thefollowing lemma.

Lemma 4.3.6.

Consider the

BVP

(4.3.16)-(4.3.21) and suppose that the assumptions (J-JI), (J-J2)-(J-J�J hold. Let there

exist

(fi ' �i)

and (ci' Ci) which are lower and upper solutions of Sn' En '

Assume that

(i)

For components

j E 1. where

Dj• Hj

> O. £j. Cj E

C2+a.a[Q

X

A. Rn(/)];

(ii)

For components

j

E

1,

_where Dj

=

H

j = 0,

£j' Cj

E

Ca.a[Q

x

A , Rn(/) ] ;

(iii)

For components

j

E

J,

where

9Jj > 0,

(dj' Ej E

C2+a lA,

_{RII(J)] ;}

(iv)

For components

j E J,

_where

9Jj = 0,

U · VCY-I)

$ 0,

(dj'

Cj

_E

C1+a [A,

RII(J)], and assumptions

(Hs)-(H;') hold;

(v)

For components

j

E

J.

where 9)j

= 0, _{U ·}

VC;k-l)

== 0,

(dj' Ej E Ca[A. Rn(J) ] .

Then the mapping

!!J,

from (e;k-I), cjk-I»

to (eV), dk»

possesses the following properties:

4.3 EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM 1 13

Proof

We first consider !.he case when Dj, g}j > 0 for all components j. The natural imbedding of C2+a.a Rn(l)] into C2.a [.Q x A , Rn(l)] and C2+a[A, Rn(J)] into C2 [A, Rn(J)] implies !.hat £j' Cj E

C2.a [.Q X

A,

Rn(l)] and Sj' Ej E C2 [A, Rn(J)] .

The boundedness of .Q and A, together wiLh the fact that their boundaries belong to C2+a, shows that, if c;k-I)(x;

Z) E

Rn(l)l (with A treated as a parameter space) and

C?-I)(Z) E

C2 [A , Rn(J)] , then

c;k-l)(x;

Z) E Wi

Rn(l)l (with A treated as a parameter space) and CY-I)(z)

E Wi

[A, Rn(J)l for q > 1 . From Lemma

4. 1 .5,

we may take q to be identical in both cases.

This, in view of Theorem 4. 1 . 1 (Imbedding Theorem), yields that c;k-I)(x; z)

E

CI+a[.Q, Rn(l)l (wiLh A treated as a parameter spaee) and Cyk-l)(z) E CI+a [A , Rn(J)]. From Lemma 4. 1 .3, a may be chosen to be identical in both cases. From arguments similar to that shown in Lhe proof of Lemma 4.3.5, we see Lhat

C;k-l)(X, z)

E

CI+a•a[.f2 x A , Rn(l)].

It is immediate that the proof of (I) follows from the choices (c;k-l), CY-I» = (£j' 9.j) and ( (k-l) C(k-l» - ( - C- ) ' c j ' j - c j' j In emma . L 4 3 . . 5

All the oLher possible cases are treated similarly.

We have shown that if (c(O) C(O» = _{J ' J} (c· _{J ' J} E ) then (dO) dO» _{, ' ,}

>

_-fI'(c(O) C(O» = (d1) C�l) _{J ' J ' "} and if

(c(O) C(O» = (c -_{J ' J -J ' -J}

C)

then (c(O) C�O» _{" ' ,} < _-fI'(c(O) C(O»

=

(c(1) C(I» _{We have in fact proved that the}

J ' J _{', . , '} •

mapping fl'maps intervals I £j . C:j I and ISj' Ej J onlO themselves.

To prove (II), let Cjl> Cj2 E [£j' cjl and Cjl, Cj2 E [9.j' Ejl where (Cjl , Cjl)'?,(cj2 , Cj2) for all components j. We want to show that fIf(cjl. Cjl) ? fIf(cj2. Cj2) .

Let

(ui.

Ui)=fI'(Cjl. Cjl)- fI'(Cj2. Cj2). Then the monotone nondecreasing property of Ii and Fi implies tliat au· D· -

' + f-I.u· =

-

C2)

>

0 on aQ2xA

, on

"

" , ,

-

. , 2

f

'

-�V

C1) - F(z C2) + H·

c'l - c2 > O m A

,

"

"

"

•

J " J '

(Jil2 ' , -

, (4.3.44) (4.3.45) (4.3.46)

and from Lemma 4. 1 .8 (Maximum Principle) for Lhe elliptic operator, or Theorem 3.2.6 (Strong Comparison

Theorem) for first order partial differential equations or from algebraic inequalities, we see that CUj, Vj) � 0 or fIf(Cil ,CiI ) ?' fIf(Ci2

.Cj2) ·

This shows that S'is a monotone operator on the intervals [£j' Cj] and Le, Cd .O

The monotone operator fIf will play a central role in Lhe iteration scheme. Remark 4.3.1

As with the unsteady state system Sn, Bn, we see that if Ii and Fj are strictly monotone increasing in Cj and

Cj' respectively, then by Theorem 3.2. 1 3 (Generalised Strong Comparison (Contact) Theorem), we see Lhat fI'(Cjl' Cj1 » fI'(Cj2, Cj2) , (unless S'(Cjl, Cjl )=fI'(Cj2, Cj2) in which case the right hand sides of

(4.3.44)

and

(4.3.46)

are identically zero; but this happens only if (Cjl, Cjl) =(Cj2, Cj2) , from the strict monotone property ofli and F;). We say that the monotone operator fl'is monotone operator in the sense of

4.3 _{EXISTENCE OF SOLUTIONS TO THE STEADY STATE PROBLEM} 1 1 4

Remark

4.3.2

As with the unsleady Slate problem, Sn, Bn, we see that iff; and Fj are monotone nonincreasing in

Cj

and

Cj,

respectively, the operator S' is alternating on the intervals

[fj,

c';] and [C , C\] in the sense that

(Cjb Cj2)

�

(Cj2, Cjl)

_{implies that}

S'(Cjl, Cj2)�S'(Cj2' Cjl).

_{To prove this, let}

(Uj, Uj) = S'(Cjl, Cj2)­

S'(Cj2, Cjl) .

Then the monotone non increasing property off; and Fj implies that

Ju

·

D·_I + H·u· = fJ.(C-I - C-2 ) < O

on

Jfl2XA

' an

" I I I

-

t

-9J..V2U·

I I

+ U · VU

I

+ fJ..%U· =

I I F(z I ' J

C -I)

-F (z I ' J

C -2) + H·f

I aD2

(C·I

I

-C·2)

I -> 0 in

A

, and from the same arguments as in Lemma 4.3.6,

!!kjl � !!kj2

and

PJCjl � PJCj2.

It is necessary to choose a proper initial iteration to ensure that the sequences

{(c�k), Cfk» }

are monotone sequences that converge to a solution of Sn, Bn and are within the intervals [fi ' cd and

[�j,

C\]. From Lemma 4.3.6, it is obvious that the monotonicity of these sequences obviously depend on the monotonicity off; and Fj and the initial iteration is taken to be either an upper or a lower solution which is required to satisfy certain inequalities on the corresponding system.

We may therefore use the initial iteration (Ci(O) ,

G(O» = (Cj,

CI

)

to construct the sequence

{(c?), G(k» }

from the following equations

-!llV2C.(k)

I I

+ U · VC(k) + f-f.S'lC.(k) =

I I I F(z I '

C�k-I» +

J

H·f

I aDz

e.(k-I)

I in

A

'

or the sequence

{(dk), ��k» }

with

(dO), ��O» = (fi'

C) may be determined from the equations

In document A mathematical analysis of reaction diffusion systems in chemical and biological reactors with macro and micro structures :|ba thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy at Massey University (Page 118-122)