Similarly, our assumptions (H2) of Holder continuity properties for the functions Fj with respect LO the variables t and z with Cjfixed, imply similar Holder continuity properties for Ej and � with respect to

the variables t and

with (;.k and

fixed and so there are constants

and

Kz,

such that

Proof

- - - . - - .a/2

1F;(t, z,

(;.k ' C[)- F;(t,

�k ' C[

�

K, (F;)lt - t I ,

- _• -

• a

IEj(t, z,

�k ' C,) - Ej (t, z , �k'

C,)I

�

Kz(E;)lIz - z II ,

I�(t, Z,

�k '

C;)- �(t, z:

�k '

C;)I

�

Kz(�)lIz - z·lIa .

(3.6.6)

Since

f.k'

�k

and C; are fixed, lhis lemma follows direclly from the definitions of

L, ];,

Ej and

F;

in (3.2.9)-(3.2. 1 2) and assumplion (H2).O

For the purposes of our existence proof, we may henceforth assume our coupling functions

fj

and

Fj

are monotone nondecreasing in the variables

Cj

and

Cj,

respectively, for all).

Remark 3.6.4

CARL and GROSSMAN [47 1 have an analogous definit ion for the funct ions

L

and

l;.

It is shown that i f

h(t, x, u)

are

Caratheodory type

functions, thul i s for almosl all

x E

[2, the funclions ii arc conlinuous on

Rm, and for all

u E

Rm, the functions Ii are measurable on n, then so are the functions

L

and

l;.

This is proved using a measurable seleclion theorem which is usually known from optimisation theory.

3.6.2 Upper and Lower Solutions and Monotone Iteration

We shall now introduce the concepts of upper and lower solutions relative to the monotone system

Sn' E

n. Definition 3.6.1 .

Assume that

(i) For components

i E I,

where

Dj

> 0, fj and

Cj

are continuous functions in [0, T] x

Q

x A with continuous firsl order

Xj

derivatives in (0, T] x n x A , continuous second order Xj derivatives in (0, T] x n x A and conlinuous first order

t

derivati ves in (0, T] x n x A ;

(ii) For componenls

i E I,

where

Dj

H,

= 0, fj and

Cj

are continuous functions in [0, T] x n x A with continuous first order

t

derivatives in (0, T] x n x A ;

(iii) For components

i E J,

where

9Jj

> 0, C and Cj are continuous functions in [0, T] x A , with continuous first order

Zj

derivatives in (0, T] x if , continuous second order

Zj

derivatives in

(0, Tl x A and continuous first order

t

derivatives in ;

(iv) For components

i E

J, where

9Jj

= 0, U · V' C , U · V' Cj $0,

�j

and

Cj

are continuous functions in [0, T] x A , with continuous first order

Zj

derivatives in (0, T) x A and continuous first order

t

derivatives in (0, T] x A ;

(v) For components

i

E J where

9Jj

= 0,

u ' V'�j , u

V'

Cj == 0, C and Cj are continuous functions in

3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 63

The ordered pair of functions (fi ' qi) and (ci ' Ei ) with

fi

� ci on [0, T] x .Q x A and qi � Ei on [0, T] x if are said to be

lower

and

upper solutions

of Sn' Bn respectively, if they satisfy:

tlnd afi ₂ _.

Tt- D/Vx fi

� h Ct, x,

f)

In (0, l1x.QxA, aCi �

_°

on (0, T]xaQ]xA, aco

Di

� Hi (qi - fi ) on (0, l1xaQ2xA, aC

i

f

�- g)V C + u ·VC+ H

(C - c ) < F (t z C -) in (0 11xA at 1 _I _I 1 Bil2 _I 1 - 1 ' , -J ' , VI (;i + !lli � v]Ci.] on (0, l1xaA] ,

aCi 0

- �

°

on (0, 71xaAa, a = 2, 3,

fi CO, x, z) � ci,O in QxA,

D

V�

Ci"? h (t, X, Cj) in (0, T]x.QxA, "? 0 on (0, T]xrJQ]xA,

D

aCj

>

Ho(E -c ) on (0 TJx , an - l "

,

aE ₂

-

- f -

- -' - 91\1 C + u o V C + l-I (C- c » F(t z C -) in (0 l1xA at 1 1 1 1

Bil2

1 1 - 1 ' ' J ' , - VI Ci + !lli v]Cj ] , on (0, l1xaAI , "? 0 on (0, T]xaAa, a = 2, 3 , respectively.

The strong comparison theorem shows that if (fj, q; ) and (Cj, Ei) are

lower

and

upper

solutions of Sn, Bn and (Cj, Cj) is a solution of Sn' Bn , then

fi

� Cj � Cj and (;j � Ci � Ei .

The existence of monotone sequences depend therefore on a suitable pair of lower and upper solutions. This is by no means ensured without addition restrictions on the nonlinear reaction functions. PAO

[222] gives sufficient conditions for the existence of lower and upper solutions for parabolic equations.

. , aF .

These will reqUIre thatli or to be umformly bounded and Fi or _, to be umformly bounded. CARL

3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY ST ATE PROBLEM 64

[46] gives a method to show how these lower and upper solutions can be constructed. TAM [278-28 1 ] used comparison theorems to construct upper and lower solutions for parabol ic equations and compared these solutions with the exact numerical solutions. It is important to note that these lower and upper solutions provide lower and upper bounds for solutions and that these bounds can be improved by monotone i terative techniques. We shall for the purposes of an existence theorem. assume tJ1at upper and lower solutions exist.

In order to establish an existence llieorem for Sn . Bn in terms of upper and lower solutions. we define a transformation fi, by

(dk) dk» = ff(c<.k-I) c<.k-I» _J

' I I J ' J '

and consider the sequences ((c(k). C(k» } _" where c(k) _, is obtained from the linear system

ac(k) D·V2C<k)

=

x c<.k-I» in (0 l1xQxA at J x J , , ac(k) ₌

_°

_on_{(0. 'J'jxoQ1xA.} aC<k) D· I an + HI I c(k)

=

I-f.C(k-l) J , on (0 l1xaf22xA ' , C(k)(O. _, x. z) = C, _.0 in Q xA.

and C<k) is obtained from the linear system _,

C<k\O. _, z) = Cj _.0 in A. 'th < (k-l) < - _{[0 T]} _A _d_C_<_C₍_k_-

_l

_l_<_C- _{[0 T] A} _f _k_- ₁ WI

£j

_ Cj _ Cj on • X an _j _ j _ j on • x or - ... (3.6.7) (3.6.8) (3.6.9) (3.6.10) (3.6. 1 1 ) (3.6. 1 2) (3.6.13) (3.6.14) (3.6.15)

For each k. the system (3 .6.8) consists of n(J) linear. completely uncoupled initial value problems with boundary and initial conditions given by (3.6.9)-(3 .6. 1 1 ) and this system is uncoupled from the system (3.6. l 2) which consists of

n(J)

linear. completely uncoupled initial value problems willi initial and boundary conditions given by (3.6. 1 3)-(3.6. 1 5).

Since cfk)(I, x, z) is not differentiated with respect to z in (3.6.8), A may be considered to be a parameter space in (3 .6.8)-(3.6. 1 1 ). For functions dk)(t, x, z) where

D j

H j

= 0, f2 may also be

considered to be a parameter space and for functions C?)(t. z). where

9)j

= 0, U ·

vcfk-l)

= O. we may

similarly treat A as a parameter space. The existence and uniqueness of sequences ( C

f

k) , Cj(k» } may therefore follow from solving standard scalar systems of linear parabolic equations (LADYSHENKAYA [ \55] or FRIEDMAN [94]) which may or may not depend on parameters, systems of ordinary differential equations which depend on parameters (HARTMAN [ 1 19, p. 93]) and systems of first order partial differential equations (LAKSHMIKANTHAM el al. [ 1 60]).

3.6 EXISTENCE OF SOLUTIONS TO THE UNSTEADY STATE PROBLEM 65

All these theorems will require H6Ider continuity properties on the functions h(t, x,

C)k-I»

and

F (t z

C<.k-I» + Hf

which are satisfied if either

c�k-1)

C(I+cx)I2,I+a,cx [(0 T] x Q x A Rn(l)]

, , ' J ' aQ2 J " ,

c(k-I)

₎ E

Ccx/2,cx,a[(0 T] x .Q x A

C<.k-I)

J E

C(1+CX)/2,I+CX[(0 T] x A Rn(J)]

" or

C<.k-l)

j E

Cal2,a

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 70-73)