Asymptotic behavior of multigroup diffusion equations with Xe feedback

Vol, 17, No. 8, pp. 415~.25, 1990 Printed in Great Brilain. All rights reserved

0306-4549/'90 $ 3 . 0 0 + 0 . 0 0 C o p y r i g h t :~ 1990 P e r g a m o n Press pie

A n n . n u cl. Energ3",

A S Y M P T O T I C B E H A V I O R OF M U L T I G R O U P D I F F U S I O N EQUATIONS WITH Xe FEEDBACK G.-S. CnEN Department of Nuclear Engineering, National Tsing Hua University. Hsinchu, Taiwan, Republic of China 30043

(Received 20 December 1989; received for publication 2 April 1990) Abstraet--II is well known that fission product 135Xehas a very large absorption for thermal neutrons. A small fraction of this nuclear species is formed directly in fission and a major part results from the decay of ~35I. The behavior of t35Xe density can affect neutron flux distribution and hence reactor operation in a thermal reactor. The aim of this paper is to give a nonlinear study of time dependent multigroup diffusion equations coupled with Xe feedback through the notion of the upper lower solution and the monotone iterative method. The study includes the construction of neutron flux, explicitly upper and lower bounds of the neutron flux, and the asymptotic behavior of neutron flux under Xe feedback. Finally, we also obtain sufficient conditions for supercriticality and subcriticality of the coupled system.

I. I N T R O D U C T I O N

Certain fission products in the reactor core possess extremely large thermal neutron absorption cross sections so that they can influence the temporal behavior of a reactor. Some of the most important of these are t~5I and '35Xe. The buildup of ~35Xe can appreciably affect core flux distribution and hence reactor operation. However, the cross sections of Xe fall off quite rapidly for neutron energies above 1 eV; we assume that the Xe absorption cross section is zero except in the thermal group. In this paper, we consider the multigroup diffusion equations coupled with I and Xe balance relations as follows : ]

~;

~,

(4~,,),--V" Do(r)Vc~v(r, t) + {£.u(r, t) +Y~s~(r, t)}qS.(r, t) = ~ Z.~¢v(r. t)~Sg,(r, I) +Zq ~ vcE,~,(r, t)q~..(r, t). l'~l

g" I .q' g-q

q'= I

g = 1,2 . . . . . G - - l . l

(q~a),-V" D~;(r)Vcb(;(r, t) + {E~a(r, t) +E~G + ~x(r, t)} qS~(r, t)

l" G G

G

= ~

Z~¢o(r, t)(oq(r, t) +X~ ~

g'--I

vcY,u'(r, t)(o,,(r, t),

q'~l

G

(I),+2ff(r,t) = 7, ~ £ru'q~u(r.t), g--I G

( X), + 2,X (r, t) = --a,X (r, t)ffa~(r, t) + 2iI(r, t)+Tx ~ £ru'( r, t)d?,~,(r,t), g"

(t > 0, r e ~ ) .

(1)

I

Here, the quantity q~g is the neutron flux in group g, and I and X are the I and Xe density, respectively. The quantity D u is the group coefficient in group g, while £,u, Y>q, ~Trv represent the macroscopic absorption, scattering and fission cross sections in group g, respectively. The quantity vg is the average speed of a neutron in group g, and v,¢ is the average number of neutrons produced in a fission induced by a neutron in group g'. The probability for scattering a neutron from group g' to group g is denoted by E,g'u. The quantity ax is the microscopic absorption cross section of Xe for thermal neutron (9 = G), 7 and 2 denote yields and decay constants. The fraction of the neutrons produced by fission that have energy within group g is denoted by ZuThe notation ~ denotes a bounded and convex reactor domain in R a (d = 1,2, 3) with smooth boundary F. Associated with (1) is a boundary condition of form: 415

416

G.-S. CHiN

C~g(r,t) = 0,

(reF, t > 0)

(2)

and the initial condition is given in the usual form : ~bg(r, 0) = q~go(r), l(r,O) = Io(r),

X(r,O) = Xo(r),

(re~).

(3)

The effect of Xe on the kinetics of a reactor has been widely discussed in the technical literature by many investigators, and various methods for the control of Xe-induced flux spatial oscillations, stability analysis and simulation of spatial Xe-power transients have been developed (cf. Canosa and Brooks, 1966; Christie and Poncelet, 1973; E1-Bassioni and Poncelet, 1974; Cho and Grossman, 1983; Onega and Kisner, 1978; Schultz and Lee, 1980; Teachman and Onega, 1983; Yoo and No, 1985; Stacey, 1968, 1969), In particular, Cho and Grossman have applied a monotone iteration method to construct the steady state solution in a onegroup diffusion equation with Xe feedback. Most of the investigation simplifies equations (1-3) by considering only a few aspects of the complete system. The majority of analyses usually linearize the system of equations (1) or neglect the time derivative of neutron flux in (1). On the other hand, there is an interesting problem concerning the asymptotic behavior of the time-dependent solution of system (1-3) as t --* o% which is related to the subcriticality and supercriticality of the time-dependent system (1-3) under the effect of Xe. The purpose of this paper is to introduce an iterative method, which is more general than that used by Cho and Grossman, for the construction of the neutron flux, upper and lower bounds of neutron flux and the asymptotic behavior of neutron flux in the system through a nonlinear approach without approximation treatment on the system (1-3). The basic tool in this paper is the notion of the upper-lower solution and the monotone iterative method (cf. Ladde et al., 1985; PaD, 1987; Chen and Leung, 1989; Chen, 1989). In Section 2, we use the monotone iterative method and upper and lower solutions for the system (1-3) to establish an existence-comparison theorem and to construct some explicitly upper and lower bounds of the solution. Through suitable construction of upper and lower solutions in Section 3, we obtain some sufficient conditions for supercriticality and subcriticality of the system (1-3) without explicit knowledge of the solution.

2. CONSTRUCTION OF SOLUTION AND THE MONOTONE ITERATIVE METHOD

In this section, we apply the monotone iterative method and the upper and lower solutions to construct the solution of the system (1-3). The monotone iterative method yields monotone sequences that converge to a solution of (1-3). Each member of these sequences happens to be the solution of a certain linear differential equation which can explicitly be solved. Furthermore, the method can be successfully employed to generate the upper and lower bounds of solution of the system (1-3) from which asymptotic behavior of neutron flux can be investigated. Consider a time interval [0, T] for the time variable t, where T is finite but can be arbitrarily large, and let = ~ ~ F, Q r = ~ × (0, T], Q r = ~ x [0, T]. For notational convenience, we write (~bg,/, X) to represent vector function (~bj, 42 . . . . . ~bc, I, X) and: Lg[q~g] = V'DgVd?g-{Y.ag(r,t ) +Y~su(r,t)}d?g,

g = 1,2 . . . . . G.

Throughout the paper we assume that Dg, Zag, Zsg,g, Y~roare continuous positive functions on ~ and Qr, respectively ; ax, 2~, 2x, 7x, 7~, va, Zg, vg are positive constants ; the initial functions
~

G

G

vg (q~g)'-- V" Dg (r) V~g (r, t) + {Z ,g (r, t) + Z ~g(r, t) }~, (r, t) ~>g'~, Z sg'g(r, t)~¢ (r, t) + 7~0g'=Z,v¢E fg.(r, t) ~g, (r, t), g'~g

g= 1,2,...,G-l,

Behavior of diffusion equations with feedback 1

417

(4S~,I , - V " D.(r)Vq~(,', t)+ {E.~(r, t ) + Es~; + cr.X(r, t)}~a(r, t) I

G

G

> 2 z,..+~(r.:)~..(r,t)+z~ Z ,'oz,-, (r. n4~+,. ,/

I

(t

I

G

(f),+2~[(r.t) >~;,i ~ £ t ~ f L ( r , t ) . #/:= I

(X), +)..X(r, t) >>. - a..~(r, t)~c(r, t) +2~/(r. t) +7. ~ E,., (r. t)~e(r, t), (I

1

(~b,,),

1

V. D q ( r l V ~ ( r , t) + {Y.g(r, t) +Y..g(r, tl}~.(r, t) G

(;

~< }2 ~,~.~(r, tl4~.(r,t)+z~ ~ ,',%,.O',t)~<(r.O. ,q'= I

I

~t= 1.2 ..... •-I.

I

q=

(0<..), .- V . D,;(r) V (o<;O', t) + { E.c;(r, t) +£s,; + o-,)~(r, t) }qSo.(r, t) f;

I

<~ ~ z,,,.G(,'.tl_~.(r, tl+z,,. ~ ,',,Z,.,,(r.t)~,,. (i'~ I

,1= I

(/),+2i/(r, t) ~< ~,, ~ Xr.¢~,,(r, t). q'=l G

(X), +.i..X(r, t) ~< - a . X ( r , tl6,;(r, t) +)Q(r. t) +7. 2 Z,.(r. t)~.(t', t). q

(4)

I

the boundary inequalities : ~0(r,t)>0>~(r,t), and the initial inequalities

. q = 1 , 2 . . . . . G,

( r e F , te[0, T]),

(51

:

9 = 1,2 . . . . . G,

(~(r,O)>~c~o(r)>~,(r,O),

l(r,O)>~lo(r)>~l(r,O).

~(r,O)>~Xo(r)>~X(r,O),

( ,'~ . )
In the above definition upper and lower solutions are interrelated through the inequalities of neutron flux (~<; and Xe concentration X. A pair of smooth functions in definition mean that (~q, L J~) and (~,,,L ~) are continuous in (r, t) and twice continuously differentiable in r and once continuously differentiable in t for (r. tl in (j~. We suppose that (~,,, I, X), ( ~ , / , X) are upper and lower solutions lor the system (1 3) such that:

(~, L ~?) > (~.~,L _x) > (o. o. o). on Q , (i.e. 9~,, >~ ~, >~ 0, g = 1,2 . . . . . G, [>~ [ >~ 0, )7 >~ X >/0 on Qr) and choose a constant M > 0 such that:

M

>

~,(4~,~+~?)-

(7)

To establish an existence~comparison theorem for the system (1 3) in terms of upper and lower solutions (q~, l, J~) and (4q, I, X), we construct the sequence JZ(m' 1"('~>J?V,,,] with the initial iteration (4~,°k T ''~, ~o,) = (q~o, l, )~') and ~(o>= ~., X(o,= X from the linear equations: I

G

G

( .~. .g. .< " q--~./ . .[..-g(,,,) . . .')(r,t)+Xv . .2 . . . . . 1 = ~ Y,¢o(r, tlq~;~"

vuZ,.u.(r,t)c~y'

"(r,t),

.q= 1 , . . . . G - - I .

q" #~1

I (X(m)~

~ rX(,,.)~~- MY,("')

U¢, G

= M447>-~.X_ ~" ')4~(g' " +

I

~ Z,,.~(,.,t)~;" "(r,t)+Z,; y'= I

,,~.Z,~(r.n~;" "(,',n. q~

I

418

G.-S. CHEN

([(")), + ;j(m)(r,t) = ~ ~ Y'~g'~m '), g'=l G

(£(m)),+(M+L)£(")(r, 0 =--Ox2(m--"C(~m--')(r, 0+~,pm--')+~ ~ ~ r~¢C~m ~)(r,t),

(8)

with the boundary and initial conditions : q~e(")(r, t) = 0,

( r e F , t6 [0, T]),

¢(")(r, 0) = ~bgo(r), )~(")(r, 0) = Xo(r),

f(m)(r,O) = lo(r),

(re~),

for every rn = 1,2,3 . . . . while the sequence ~,6 (m)~ --I(~), X (') } with (¢~ol, i(o)_,_X(°)~ = (¢g,/, X) and q~(o) = q~, t'~_g -)~(o) = )~ is constructed from the equations :

1 (; -Uq- ( ,~--g ~ ( " )I ~ , - L g t't_" [(5(m)] g J = E Z~v'g(r,t)dp(¢m-°(r,t)+Zg -g'=l

~;

~ veZr¢(r,t)O~"-I)(r,t), , -~

l

G

__[A(m)~w.aJ,--LGtwvrd~(m)l-uM~)~') = M ¢ UG .

.

g = 1. . . . . G - l ,

g'=l

g'¢g

l)--ax£(m--I)(~(m--I)"~.

.

Z

l

G

Ysg'c,(r,t)¢ (m ') + Z6 E Vg'Eru'¢(~m 1),

g'=l

g'=l

G

G

(X_(")),+(M+L)x(m)(r,O=MX (" I)-,~X_(" ')5(~"-'~(r,O+~J_ (" " + ~ ~,=,:~¢~_~ ')(r, 0,

(9)

with the b o u n d a r y and initial conditions"

¢(fl)(r,t) = 0,

( r e F , te[0, T]),

~"°(r, 0) = ¢a0(r),

x(m)(r,O) = Xo(r),

l(")(r,O) = Io(r),

(re~),

for every m = 1,2, 3 . . . . The systems (8) and (9) are interrelated since the solution (d~m o, [(.,), )~(~.)) and (,~(m) (" ~),[("-~),)(("-~)) and (O(fl ~),I_ (" ~),X ~)) are known "rg ~ _i(,.) , X(,.)~ -] can be determined only when (q~u _ ('' simultaneously. However, it is clear by starting from any smooth f u n c t i o n (q~(q0),~(0), )~(0)) = ( ~ , ~,.~) and (¢(0),/(0) that (q~(m),~(m),)~(m))and @u ( ( ' ) ,-/ (m), X(,,) _~ . . . X(O)) . = (C~q,I,X) ~ _ ) are well-defined and can be obtained by successively solving the linear scalar initial boundary value problem (8-9) (cf. Friedman, 1964; Pao, 1987). With this construction we can obtain two sequences from (8) and (9) simultaneously and denote them by {(~((~), [(rn), .~(m)} and ~g])(m) I(,.) x(m)~ respectively. Our aim is to show that these two sequences are monotonic and both converge to a unique solution of the system (1-3). We first establish the monotone property of the sequences. To achieve the result, we need the following maximum principle which we merely state (cf. Ladde et al., 1985). Lemma 1. If u = u(r, t) is a smooth function on (~v such that :

u,-cV'D(r)Vu(r,t)+d(r,t)u(r,t)

>~ O,

((r,t)eQT),

u(r,O) >>.0, ( r ~ ) , u(r,t) >10,

(re F, t e [0, T]),

where c > 0, D (r) > 0 and d(r, t) >1 0 for each (r, t) in (~T, then u(r, t) >~ 0 for each (r, t) in Qr. Using the result of Lemma 1, we first establish the monotone property of the sequences. Lemma 2. Suppose ( ~ , I.)~) and (¢g,/, X) be upper and lower solutions as defined in (4-6) such that (Ca, I, X) ~> (¢¢,/, X) ~> (0, 0, 0). Then the sequences { o , and t-~ , : , obtained from (89) possess the m o n o t o n e property :

(d)(m) I(m) X(")) <~ (¢~,+ ~) l(m+ ~) X(,.+ 0) ~< (¢-(u,-+0, i-(,-+ ~), )~(m+ l)) ~< (~(,.) [(m), )~(m)), m = 1,2,3 . . . . .

(10)

((r,t) e Q r ) .

dS(O Proof. Let u~ = dS(O)_ .~ y.q = q~g _q~(~), .q g = 1,2 . . . . , G, y = / ( o ) _ ~ ( o = ~ _ [ ( o and w = ~(0)_)~(,) = )~_~(~). Then using (8-9) and (4-6), (u¢, y, w) satisfies the relation :

1

~)g(~g)t--Lg[uy]

=l

~q(~g)t--Lg[¢g]-•

~ Z~g.a(r,t)~o.(r,t)+Ea ~ Vg.Zr¢(r,t)~¢(r,t)>~O , g = 1,2 . . . . . G - - l , g'=l

g'~g

g'=l

Behavior of diffusion equations with feedback

I (u~,),_L(;[u,;]+Mu ~ = 1 (~p+;),-La[d~]+a~X(r,t)d~+;1"(,

U(;

419

~+ E~,,,,;(r,t)<~,,(r,t)-Z+; g'~

I

v,,.Y+,.~.(r.t) >~ O. .q'- I

(T),+j-;,, ~ L+.4+(,-,t) >/0.

y,+;.~y=

q.= I

q

[

with the initial and boundary conditions : u,(r,O) = q~q(r,0)-qS,(r, 0) >/0,

(re 9 ) ,

y(r, O) = "[(r, O ) - lo(r) >~ 0,

(re O ),

w(r, 0) = )~(r, 0)--X0(r) ~> 0,

(re~),

u~(r,t)=~,~(r,t)-O~(r,t)~>O,

g = 1,2 . . . . . G,

( r ~ F , te[O,T]).

(12)

The above relation implies that : (u,~),-vqV" D+(r)V'uq +v+(~+~, +Z~.+)u~f ~> 0,

,q = 1. . . . . G - I

and (uf;), - c+;V" D~(r)V • u~ 0.

(13)

The relations (12 13) and Lemma 1 ensure that Uq >1-O, g = 1. . . . . G, for each (r,t) in Q.~. This shows that .~(o0) ~t r , t), .q = I . . . . . G, for each (r, t) in Qr. The inequalities in (I 1-12) : ~l '~(r, t) <~ u, y,+20'~>0

and

y(r, 0)~>0,

imply that : y(r,t) >l y(r,O) e x+, >~O and hence Pu(r,t)<~Pm(r,t)

((r,t) eO~.).

Similarly, the inequalities in (11 12) : w,+(Xx+M)w~>0

and

w(r, 0) ~>0,

imply t h a t w(r,t) >~ w(r, 0) e -~x~+gl' ~> 0 and then Xtll(r,t) <~ X¢°~(r,t),

((r,t)zQl).

This proves : (c~u,p,L£+~)

<< ,++~ ~g,~o~, -r~o~,xt°>), -

((,',t)~O,.).

A similar argument using relation (11-12) and the property of lower solution (5) shows that: (~5~o> po) X~O~) ~(rb~) p u X~U),

((r,t) eO, r).

Now let u+ = 0u-tu_ ~ ) , 3, = jr<.)- _ P ° , w = +Y~ - X ~u_ . Then by (7-9) and the relation: (G2.['°',~

'°>) ~ (?+,~,2) >i (4,+,Z,x) - ~ ' " '

~<"' ,,.~o,~

we have : I (Uq)t--Lu[uo]

Y'sg'g(r,t)(~v--4u')q-~q

= ,q'=l g'+9

ANE

17:8-C

2 q'= I

vo,Eru,(~¢--4¢) >~ O, ,q = 1,2 . . . . . G - l ,

420

G.-S. CHEN

1

a i

VG

g'= I

--(Ua),--La[UGI+ Mua = M(d~a-~a)-ax(X_~a-X~a)+ Z E~.o'o(d~¢-~g') G

G--I

g'=l

g'=l

G

+•c,

Z vcZr¢(4S.¢-~u') I> 0,

g'=]

(14)

G g'=l G

w,+(M+2x)w

M(Y--XI--ax(X~a--Z~G)+).i([--I)+7.

=

~ Zr¢(q~g.-~.,), g'= I G

= (M-~x~I(2-X)+~2(~-_@~)+~,(7-.S)+~

y~ ~,~.(47¢-_~g.) 1> 0, (15) g'~l

with the initial and b o u n d a r y c o n d i t i o n s : u~(r,O)=O,

y(r,O)=O,

w(r,O)=O,

(re~),

uu (r, t) = 0,

(r e F, t • [0, T]).

U s i n g L e m m a I again, we c o n c l u d e that Hence :

(16)

(u~,y,w) >1 0 on (~r (i.e. 4~ ~ ~> ~g d~(1) ~ [(o >~ _i ( o , j~(u ~> X(U on (~T). -

@o~, i_~o~,x_~o~)<. (~_~,~,i, ,,, x~ ,~) <~ (&,,, r~'~,2~ ,~) ~ (&,, r ,o~,~o,). A s s u m e for s o m e m > 1 : .

.

.

!:"~ i < ~

.

(qT~,f~m>,2<'>)

~m~

p" '~,2("-~).

de"), y = -I (m+ u _ -l(m) and w = _iX(m+ u T h e n using (7 9), the functions Uq = ~b _e('+ ~) - ~_.q relation : l

a

a

])g

g'= I g'~g

g'= I

_x(m) satisfy the respective

--(u~)t-L,q[U,o] = E Z~.q'g(r't){~_(q'°-~_~m-'~}+Zg Z vu'Zf¢{~_~"')-~_~~- ')} I>0, 1

--

=

(UG)t--LG[ucJ+Mua

Al[ f,#i(rn) ,.A(m-- l)'l. "'*I.W_G - - ~ G }--~x

~.~(m) d)(m )

<

_

r--

__2(m

_

I)(~bm

DG

g=

1,2 . . . . . G - l ,

~ I)}~_ E Y~,g.~(r, t){~¢(m) --~..( m - I) } .q'= I

G

+z~ g 'E= l ~.z,~.{~.~.",-_~,~ ',} = {M-~e'~'}{_~': '-_~5~-''} +~x_¢:-"{~"° "-~<'°~} G

+~

E v.o.z,.,.{_,;"'-0,

G

yt+YXi = 7i E Yr,'{_~ m' --_~5"-l)} ~> O, g'=l

w,+ (M+2~)w

= M { ) ( (m> --_._X(m i)} _ oxg~f~v ( l ) / lWG i _ Av(m-_ I>~(m~a I)~j i~],~>,(--fl(l)"__ [(~) } G

+~x ~

v

S~(r")__~g(m--I>}

~f,q'tW_g"

-

= St M - - a x"k'G ~T~(rn)'((x(rn)--.~(m-1))-~¢~xX(m It_ _ _

l'{~(m-l'i~

n,}

g'=l

+,~,{s<")-s("-"} +~,x__

G

Y z,~.~_,.'4~,~,-~m-',} >1 o. g'=

1

In addition, u~, y and w also satisfy the same initial and b o u n d a r y c o n d i t i o n s as (16). U s i n g L e m m a 1 again we c o n c l u d e that (ug, y , y ) / > (0, O, 0). This s h o w s the relation:

(d?u('), 1 (') X("% ~< (~(,,~+ i) i(m+l) T h e s a m e a r g u m e n t gives the result :

x(m+l).

Behavior of diffusion equations with feedback

(4>~"+ '),_/('+ ~), ~('+ '>) ~< (d.("+ '), P°+

42 I

~?("+ '~)

'L

and

(~,,,+ '), ~ + ,). 2(.,+ ,>) ~< (4~m), p'=). >?(,,,). The conclusion in (10) follows by the mathematical induction principle. This completes the proof of the lemma. in view of relation (10) the pointwise : r '°' , x '°'') = (4L, f, x ) , m

z ("',

= (_0,,, _I, _x),

exist and satisfy the relation : (~7),[(m',x_ (m') ~ (~g,/, _X) ~< (q~.,LR') ~< ( ~ . ' . / " = ' , A"(").

m = 0.1,2 . . . .

Letting m --. oo in (8-9) and using the standard argument (cf. Ladde et al.. 1985), we conclude that both (4)~.LX) and (~.q,_/, _)O satisfy all equations in (1 3) except the equations for thermal group O. and Xe concentration X replaced, respectively, by: l

(; g"

I?C:

c; I

q'

I

q=

I

G g,-

I

tbr (4)., 1, X) and (;

_l (~,;)~-L.[~.] : -o.£0.+ UG

.

.

.

.

I

Y.z....O<+x,~ q'~

~.Y.~.,4)..,

_,

I

,

G

ix_), + ~ , x _ :

- ~=x_~,; + ~,_r+.l. ~

z,,.~_,.

for ( ~ , / , X). Hence, (~q. L ~') and (@.,/, X) are not necessarily a solution of (1 3) unless (hc,(r, t) : O.(r. t) and X(r. t) = X(r, t) ['or (r, t) ~ QT. Using the similar argument as in the references (for example. Pao. 1987 ; Chan, 1974) shows (4)../, X) : (@u,/, X), including ~c,, : ~c;. X = >( and (4~. L )~) is the unique solution. This is given in the following existence-comparison theorem. Theorem 1. Let (~u, I, )~) and (4)0,1, X) be upper and lower solutions such that (q~q I. X) >/(4). I, X) >i (0,0,0). Then the sequences t.~q~(-,),~/(-,7 ~ . . .solution .. g , ~(,=)~,, j {4)~'=),/("), _X(m)} converge monotonically ' to ' a unique (4).. 1, X) of (1-3) from above and below, respectively, and

(0,, L, _x) ~< (O_~'),l]').x ('~) < . . . . < . (_4)~-,>,_/(°,),_x(-,)) ~< ... = (4>,,z,x) ~ . . .

~<

(4)_,,L ~_') - (4),,~,x)

~ ((,~,,,Y(,,,,2 `,,,)) <~ ... <~ (~5,~,),y(,).£ (,)) <~ (~,. ~..i~).

(~7)

It is seen from Theorem 1 that problem (1-3) has a unique solution if there is a pair of upper-lower solutions (4)~, L X) and (~.0,~/, X) with (4)~, I, X) >~ (~.,/, X) ~> (0, 0, 0). To construct some explicitly upper and lower solutions so that a solution is guaranteed, we choose the functions ( ~ . , / , X ) = (0,0,0) and (4).. I, X) = ( A e p', A e p~, A eP~), where A and p are some positive constants satisfying : A ~4)g0(r),

A>~to(r),

A>~Xo(r),

(r~)

(18)

and p>~v.

~ Z,¢o(r. q ' = l

L iq ,'~#

~.Zr~(r,, , ,t)

+,;.+(',,,+~,.)~

E,.u(r,t).

"

"

"=

q' =

g=l,2.

G,

((r.t)~O.r).

"''"

I

According to Definition 1. the pair (A e p', A e v', A e p') and (0, 0, 0) fulfil the requirements in (4) if:

(19)

422

G.-S. CHEN

[p/Vq+(Zag+Zs~)]Ae"'>~ ~.')eq;7"lqf ~ Z,¢g+~g. .o'=, ~ Vq.Zf¢}A e"',

g'=l

[p+XxlAe~'~>

g :

1,2 . . . . . G,

_J

[ °]

,t~+yx F, 2f~, g'= l

J

(20)

Ae".

Then by Z.g i> 0, Z~g ~> 0, 2j/> 0, 2~ ~> 0 the above inequalities hold when : 2 Vg,~.~fg, -~-(7i-~7x ) 2 Xfg '-~-/~i' i f = 1,2 . . . . . a . (21) ,= 1 g 1 • g'= I Lg'~g Then initial and boundary requirements in (5) and (6) are also satisfied by the choice of ,4 as (18). Hence the function (A e p', A e p~,A e p~) and (0, 0, 0) satisfy all the requirements in (4-6) and then are upper and lower solutions. Moreover, (A &', A e ~', A e p') /> (0, 0, 0). In view of Theorem 1, we have the following theorem. Theorem 2. There exist positive constants A and p such that a unique solution (¢,, I, X) to the problem (1-3) exists and satisfies the relation : p)Vq

(0, 0) ~< (¢,(r, t), l(r, t), X(r, t)) < (A e p', A e p', A eP'),

((r, t) e Qr).

Moreover, the constants A and p can be determined from (18) and (19).

3. ASYMPTOTIC BEHAVIOR OF NEUTRON FLUX

The existence-comparison Theorem 1 for the reactor system (1-3) can be applied to investigate the asymptotic behavior of the neutron flux with a suitable construction of the upper and lower solutions. The aim of this section is to construct explicitly upper and lower solutions so that either the neutron flux converges to zero or it grows unbounded as t -~ ~ . This decay and growth property of the neutron flux ~bg depends on the parameters of the reactor system without explicit knowledge of the solution of (1-3). Let p i ~> 0 denote the first eigenvalue and co the corresponding eigenfunction of the eigenvalue problem : V2co(r)+lqco = 0 in @,

co = 0 in boundary F.

(22)

It is clear that p~ is real, positive and co is positive in 9 . We normalize the eigenfunction co with max {co(r)[reL~} = 1. The first eigenvalue p~ and the corresponding eigenfunction co for different reactor geometries can be found in most of the nuclear reactor theory textbooks (cf. Duderstadt and Hamilton, 1976). We assume D~ is independent of r in Theorems 3 and 4, and define the maximum and minimum of ~ag o n OT as follows, respectively : ~aq : max {Zag(r, t)l(r, t) ~ (~T}, E ag = min {Zag(r, t)l(r, t) ~ Or},

(23)

for g = 1,2 . . . . G. Similarly, define Zsg,g, Z~,g, l~fg, Zfg to be the corresponding maximum and minimum of Zsa,o and Ere on Or, respectively. Theorem 3. (Subcriticality) let ego ~ B, Io ~< B, X0 ~< B for some constant B > 0,g = 1. . . . . G. If: G

G

~llDg+~-agq-~-sg > Z £sg'g-~-~g Z g'=l g'¢g

Vg,~fg,,

g = 1,2 . . . . . G,

(24)

g'--I

then there exist positive constants p, s such that a unique solution (~bq,I, X) exists and satisfies :

O<-.. Og(r,t) <,.Bco(r)e -p', g = 1,2 . . . . . G, 0 <~ I(r, t) <~ B e ' , 0 <~ X(r, t) <-%B e",

((r, t) e Or),

(25)

Behavior of diffusion equations with feedback

423

where co is eigenfunction corresponding to the first eigenvalue pj of (22). Moreover, qSo(r, t) --, 0 as t -* ~ for eachrin~andg= 1,2 . . . . ,G. Proq/i Using (24), we can choose a small positive constant p such that :

{ p . D , + E a ¢ + E ~ u } >I

+Zu ~ v~.~.q

+p/'c,,

g = 1. . . . . G.

(26)

Let s be such that : G

s > 2i+(7~+7i) ~ Zf,,. g'

(27)

I

By Theorem 1 it suffices to show that (qSg,I, X) = (Boo(r) e ~', B e ~~,B e") and (qSu,/, X) = (0, 0, 0) are upper and lower solution pairs. Since all the initial and boundary requirements in (5) and (6) are trivially satisfied by the assumption of B and the choice ofoo as (22), we only need to show that (4S,~,I, )?) and ( ~ , , / , X) satisfy the differential inequalities in (4), this is :

Be

"'~-pco/~,.-D.V~oo+(Z~,.(r.O+~.)co}

>1 B e

'"

•

Y~ ~ . ~ c o + Z , q'= [ ~g' ¢ q

"

~ q'~l

%Z,~'co

.

,q = I . . . . . ~;.

"

B e " ( s + 2 i ) >/"~'i ~ El, "Be~t" g'- I

B e " ( s + f i x ) >~ fliBe~+7~ ~ Z~.~.Be ~'' In view of the eigenfunction equation (22) and the normalized property ofoo. the above inequalities are satisfied by: G

Dul~.+52~u+~g>~

G

~ Y.~g,u+X,, ~ vuYr¢+p/v.~, g'--I

,q= 1,2 . . . . . G,

g'=l

g'~9 G

s+2i >1 yi ~ ~q, e i~.+p, q'=] G

s+2~ ~ L + T x

~ ~fg, e ~,.+~,1,. g'

(28)

I

By (23) it is clear that the first inequality in (28) is satisfied by inequality (26). Using 2, > 0, 2~ > (L and e ~ ~'" ~ 1, the second and third inequalities are also satisfied when : S/>)'i

~ g=l

Et~q', G

s >~ )-~+7~ ~ 52f,/.

(29)

The above relation follows immediately from (27). This shows that (B e ~'co(r), B e ~', B e") and (0, 0, 0) are upper and lower solutions, respectively. By Theorem 1, a unique solution (0u, X, I) of (1-3) exists and satisfies : O~O~(r,t)<~Be 0 <~ l(r, t) <<.B e"', 0 <~ X(r, t) <~ B e"'. This completes the proof of the theorem.

"'co(r),

g = 1,2 . . . . . G,

424

G.-S. O-my

Theorem 4. (Supercriticality) assume ~bgo >/re), g = 1. . . . . G - 1, qSGo >/0, Io/> 0, Xo ~> 0 for some constant 6 > 0, where o) is eigenfunction corresponding to the first eigenvalue #t of (22). If: G--I

#,Du+£,a+F-sg <

G

~

I

E sa,g+Zg ~

g'~l g' ~ g

vg,_Er¢, g = 1,2 . . . . . O - l ,

(30)

g'=l

then there exists a constant ~ > 0 such that the system 0 - 3 ) has a unique solution which satisfies :

Og(r, t) ~ ~

g ----- 1,2 . . . . . G - 1,

Cetto(r),

dpa(r,t) >~O,I(r,t) >lO, X(r,t) >~O, ((r,t)~Qr).

(31)

Moreover, C~g(r,t) ~ oo as t ~ ~ for each r in ~ and g = 1,2 . . . . . G - 1. Proof. Using (30), we can choose a small positive constant ~ such that : G

1

G

I~.Og+Y,,g+Zs~+e/v, <~ ~ E sg,,+~t ~ ~ v,Ea, ' g : g'=l

1,2 . . . . . G - 1 .

(32)

g'~l

g'~g

We consider :

~g(r, t) = &o e", g = 1. . . . . G - 1 ~6(r, t) = 0,

[(r, t) = 0,

X(r, t) = 0

(33)

and

d~q(r,t) = A & ' , /7(r, t) = A e p',

g=

1. . . . . G

)~(r, t) = A e p',

(34)

with p > s > 0 and A > 0 chosen as (18) and (19). We now show that (q~g, ~ ) ? ) and (¢g,/, X) are upper and lower solutions, respectively. Clearly, using (33) and (22) we have for each g = 1 , . . . , G - 1 : 1

c

--{t~,.rg,'~t--D..V2~ g "~-{~'~ag'q-Zsg}~l)g -

=

6e"

a

~ E~¢u(r, t)q~¢--Zg ~ g'~-g

2to-.~-Dg/tlto-k-(Zag-~-Zsg)(D/99

veEfu,(r,t)d?¢

Zsg,g(r '

~

Z Vg,Yfg,(r,t)to

g'=l

'~1

g'¢g

<~6e"~o

I

~--[[email protected]@Zsg

G 1

G-I

--Zg

~fg,

~0,

G

g'=l

.4t"= 1 G

I

G

= -- Z Es¢6(r, t)~g--~ta Z v~'Era'(r, t)~u" = 0 <~ O, g'=l G

g'=l

G

(/),+~.i/(r, t) --?i E Ea'(r, t)~g.(r, t) = --7~ E Erg.(r, t)~¢.(r, t) = 0 4 O, g'=l

g'=l

G

(XI,+2~X(r, t)+a~Xq~c+2,1--Tx

~ g'=l

G

Zr,q,(r, tl~g,(r,t) = --Tx 2 Ero,(r,t)~¢(r,t) = 0 <~ O. g'=l

Similarly, using (34) and with p > 0 chosen as (19) we have f o r g = 1. . . . . G - 1 :

Behavior of diffusion equations with feedback 1 U~I

(;

G '

"

g'=

425

1

"

" g'~

I

,

vqZ,.'~

Z, G 1

~

D

I

C,

~~

•

UG

g'

g'-: I

I

= A e p' P + E : , , + Z ~ ( ; ~ZT(;

Zr¢(r,t)~o¢(r,t)=Ae"'

(]'),+,i~l(r,l)--7~ g'=l

(5,,+).,)7+6,)~c..--2i[--2,

]

vqZ,-,, />0, t¢

]

/

p + 2 , - 7 ~ ~ Z,-~. >~0, k

~

Z~u,<;-~(;

~ .q =

~'

I

Zr¢(r,t)~.o.(r,t,:Ae"'{p+).,-2,-7~

q'=l

~ Z,.¢}~>O, g~

I

Moreover, the initial and boundary inequalities in (5) and (6) are a i m satisfied by the choice of A and ~o as (l 8) and (22), respectively. By Theorem 1, a unique solution of (1--3) exists and satisfies: A¢"~>qb~(r,t)>~be~'~o,

g=

1. . . . . G - l ,

A &' >~ 4~(r,t) > 0,

A e p'>~l(r,t) >~0,

A e ~''>~X(r,t)>lO.

This completes the p r o o f of the theorem. Remark. The differential system (1-3) can be approximated by finite-difference equations with the C r a n k Nicolson implicit method or other methods. Using an analogous definition of upper and lower solutions as in Definition 1 for the resulting finite-difference equations it is possible to construct two monotone sequences with a similar iterative method as defined in (4~). These sequences may converge monotonically to a solution of finite-difference equations. The numerical solution for (1-3) with the m o n o t o n e iterative method will be studied in future investigation.

REFERENCES

Canosa J. and Brooks H. (1966) Nucl. Sci. Engng 2 6 , 237. Chan C. Y. (1974) SIAMJ. Appl. Math. 27, 72. Chen G. S. (1989) Ann nucl. Energy 16, 279. Chert G. S. and Leung A. W. (1989) S1AMJ. Appl. Math. 49, 871. Cho N. Z. and Grossman L. M. (1983) Nucl. Sci. Engng 83, 136. Christie A. M. and Poncelet C. G. (1973) Nucl. Sci. Engng 51, 10. E1-Bassioni A. A. and Poncelet C. G. (1974) Ann. nucl. Sci. Engng 1,529. Duderstadt J. J. and Hamilton L. J. (1976) Nuclear Reactor Analysis. Wiley, New York, Friedman A. (1964) Partial DOferential Equation of Parabolic Type. Prentice-Hall, Englewood Cliffs, New Jersey. Ladde G. S., Lakshmikantham V. and Vatsala A. S. (1985) Monotone Iteration Techniques for Nonlinear D([~,rential Equations. Pitman, Boston. Onega R. J~ and Kisner R. A. (1978) Ann. nucl. Energy 5, 13. Pao C. V. (1987) SIAM J. Math. Anal. 18, 1026. Schultz E. J. and Lee J. C. (1980) Nucl. Sci. Engng 73, 140. Stacey W. M. Jr (1968) Trans Am. Nucl. Soc. II, 227. Stacey W. M. Jr (1969) Space Time Nuclear Reactor Kinetics. Academic Press, New York. Teachman J. D. and Onega R. J. (1983) Nucl. Sci. Enyng 83, 149. Yoo M. H. and No H. C. (1985) Nucl. Sci. Engng90, 203.