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Asymptotic Behavior of Global Solutions for a Chemical Reaction Model
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
220, 640]656 Ž1998.
AY975871
Asymptotic Behavior of Global Solutions for a Chemical Reaction Model* Chengshun Jiang Department of Mathematics, Shandong Uni¨ ersity, Jinan, 250100 People’s Republic of China
Mingxin Wang Department of Applied Mathematics, Southeast Uni¨ ersity, Nanjing, 210018, People’s Republic of China
and Chunhong Xie Department of Mathematics, Nanjing Uni¨ ersity, Nanjing, 210008, People’s Republic of China Submitted by Chia Ven Pao Received July 7, 1995
This paper deals with a system of reaction-diffusion equations modeling a chemical reaction process involving three substances. The mathematical system consists of three semilinear partial differential equations of parabolic type with homogenous Neumann boundary conditions. The goal is to determine whether each substance has a positive asymptotic limit by the method of upper and lower solutions. Q 1998 Academic Press
1. INTRODUCTION In this paper, we consider a chemical reaction model in the transport of ground water. This model is related to three kinds of chemical substances * Project supported by the National Natural Science Foundation of China and Natural Science Foundation of Jiangsu Povince. 640 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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641
M1 , M2 , and M3 which can react to each other to produce two new kinds of chemical compounds Ž M2 . nŽ M1 . m and Ž M3 . r Ž M1 . k at the same time. An example is given by the irreversible reactions NH4qq NOy 3 ª NH4 NO 3 , Ca2qq 2 NOy 3 ª Ca Ž NO 3 . 2 . The above two reactions are neutralization. Generally speaking, the neutralization among one kind of negative ions Žsay M1 . and two kinds of positive ions Žsay M2 and M3 . is irreversible with the following forms nM2 q mM1 ª Ž M2 . n Ž M1 . m , rM3 q kM1 ª Ž M3 . r Ž M1 . k , where m, n, k, r are chemical combination valencies w1]5x. Let u, ¨ , w represent the concentrations of M1 , M2 , and M3 , respectively. By disregarding convection phenomena and considering only diffusion processes, Ž u, ¨ , w . satisfies the following semilinear reaction-diffusion system u t y d1 D u s yu m ¨ n y u k w r ,
Ž 1.1.
¨ t y d 2 D¨ s yu m ¨ n ,
Ž 1.2.
x g V , t ) 0,
wt y d 3 Dw s yu k w r ,
Ž 1.3.
where V is a bounded domain in R N with smooth boundary V, and the d i Ž i s 1, 2, 3. are positive constants. If we assume that there is no exchange between the substances and the outside field, then the boundary condition and initial condition are
urh s ¨ rh s wrh s 0, u Ž x, 0 . s u 0 Ž x . ,
x g V , t ) 0,
¨ Ž x, 0 . s ¨ 0 Ž x . ,
Ž 1.4.
w Ž x, 0 . s w 0 Ž x . , x g V ,
Ž 1.5. where rh denotes the outward normal derivative on V. The goal of this paper is to study the asymptotic behavior of solutions of Ž1.1. ] Ž1.5.. For this purpose, we assume that ŽH 1 . ŽH 2 .
m, n, k, r G 1 are constants. u 0 , ¨ 0 , w 0 g L`Ž V ., u 0 , ¨ 0 , w 0 G 0, and u 0 k 0, ¨ 0 k 0, w 0 k 0.
The existence and uniqueness of solutions, and the asymptotic behavior of global solutions for various reaction-diffusion systems have been studied by many investigators Žsee w6]13x and the references therein .. In w7x, H.
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JIANG, WANG, AND XIE
Hoshino and Y. Yamada studied the asymptotic behavior of solutions of the irreversible reaction model with two kinds of chemical substances, i.e., the case w ' 0 in Ž1.1. ] Ž1.5.. In document w12x, C. V. Pao discussed the asymptotic limit of solutions of the problem
¡u y L u s yawf Ž x, u, ¨ . , t
¨t y
U 1 LU2 ¨
s ybwf Ž x, u, ¨ . ,
x g V, t ) 0
~w s cwf Ž x, u, ¨ . , t
urn 1U s ¨ rn U2 s 0, u Ž x, 0 . s u 0 Ž x . G 0,
¢w Ž x, 0. s w Ž x . G 0,
x g V, t ) 0 ¨ Ž x, 0 . s ¨ 0 Ž x . G 0,
0
x g V,
Ž 1.6. where LUi and rn iU are in self-adjoint form, i s 1, 2, f is a C 1-function satisfying f Ž x, 0, ¨ . s f Ž x, u, 0 . s 0, f u Ž x, u, ¨ . G 0,
f Ž x, u, ¨ . ) 0 when u ) 0, ¨ ) 0 f¨ Ž x, u, ¨ . G 0 when u G 0, ¨ G 0,
and a, b, and c are positive constants. The differences between our problems Ž1.1. ] Ž1.5. and Ž1.6. are the properties of the nonlinear reaction terms. For example, if we denote f i Ž x, u, ¨ , w . s Di wf Ž x, u, ¨ . with D 1 s ya, D 2 s yb, and D 3 s c in Ž1.6., then there exist constants c i such that c i f i s c j f j , i, j s 1, 2, 3. But the nonlinear reaction terms of Ž1.1. ] Ž1.3. do not have these properties. Therefore, we need more detailed analysis to obtain the limit of the solution of Ž1.1. ] Ž1.5.. It is easy to infer that 0 F u F 5 u 0 5 ` , 0 F ¨ F 5 ¨ 0 5 ` , and 0 F w F 5 w 0 5 ` for any x g V and t G 0 by the maximum principle. So Ž1.1. ] Ž1.5. has a unique global solution Ž u, ¨ , w ., see w6]13x. Moreover, we have that uŽ x, t . ª u` , ¨ Ž x, t . ª ¨` , and w Ž x, t . ª w` as t ª q` be the same method as that of w7, 12x, where u` , ¨` , and w` are nonnegative constants. Define E s 5 u 0 5 1 y 5 ¨ 0 5 1 y 5 w 0 5 1 , where 5 z 5 1 is the L1 Ž V . norm of the function z Ž x .. When E G 0, the asymptotic behavior of Ž u, ¨ , w . can be studied by the method of w7, 12x; we omit the details here. When E - 0, we have u` s 0 and ¨` q w` ) 0 by analogous arguments to those in w12x. A natural and important question is whether ¨` ) 0 or ¨` s 0 and whether w` ) 0 or w` s 0. It is our goal to determine the positivities of ¨` and w` when E - 0 by the method of upper and lower solutions. It is important in applications to give conditions which assure ¨` ) 0 or ¨` s 0 and w` ) 0 or w` s 0. For example, if ¨ Žor w . represents the concentration of the cancer causing substance, such as nitrate Ž NO3 .,
A CHEMICAL REACTION MODEL
643
nitrile Ž NO2 ., etc., nitrate will be changed into nitrile with redox processes, i.e., q y NOy 3 q 2 H q 2 e l NO 2 q H 2 O,
where it is assumed that the water contains free Hq ions and free electrons. We must do something to control the initial data to get ¨` s 0 Žor w` s 0.. Conversely, if f Žor w . represents the concentration of the beneficial mineral substance for human body or human life, we hope to control the initial data to get ¨` ) 0 Žor w` ) 0.. Therefore, the discussions and proofs in this paper will help create some conditions to develop and protect the ground water resource for mankind. For further physical meanings of the equations, we refer to w1, 2x and the references therein.
2. PRELIMINARIES By the manners and arguments similar to those used in w7, 12x, we have the following propositions. PROPOSITION 1.
For all t G 0, HV Ž u y ¨ y w . dx s E.
PROPOSITION 2. Assume that the conditions ŽH 1 ., ŽH 2 . hold. Then there exists a unique global solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. which satisfies uŽ x, t . ª u` , ¨ Ž x, t . ª ¨` , and w Ž x, t . ª w` as t ª q`. Moreo¨ er, Ž1. Ž2. Ž3.
u` s Er< V < ) 0, ¨` s w` s 0 if E ) 0. u` s ¨` s w` s 0 if E s 0. u` s 0, ¨` q w` s yEr< V < ) 0 if E - 0.
In order to determine the values of ¨` and w` in the case Ž3. of Proposition 2, we need to prove the following critical lemma. LEMMA.
Assume that z Ž x, t . is a solution of the problem
¡z y dD z s yhŽ t . f Ž x, t . z ~ zrh s 0, ¢z Ž x, 0. s z Ž x . G 0, k 0, t
0
s
,
x g V , t ) 0, x g V , t ) 0, x g V,
Ž 2.1.
where d G 0, s G 1 are constants, hŽ t . G 0, hŽ t . ª 0 as t ª q`, f Ž x, t . G 0, and z Ž x, t . ª z` as t ª q`. Then we ha¨ e Ž1. z` ) 0, if f Ž x, t . F M - q` and H0q` hŽ t . dt - q`. Ž2. z` s 0, if there exists T ) 0 such that f Ž x, t . G M ) 0 for all x g V and t G T, and H0q` hŽ t . dt s q`.
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JIANG, WANG, AND XIE
Proof of the Lemma. Ž1. Assume that f Ž x, t . F M - q` and K J H0q` hŽ t . dt - q`. Because z 0 Ž x . G 0, k 0, by the maximum principle we have z Ž x, t . ) 0 on V for any t ) 0. Without loss of generality we can assume that z 0 Ž x . ) 0 on V. Hence, there exists « ) 0 such that z 0 Ž x . G « on V. Ž18. When s s 1, define z Ž x, t . s z Ž t . s « exp yMH0t hŽ s . ds4 . Then we have
¡z y dD z s yMhŽ t . z F yhŽ t . f Ž x, t . z, ~ zrh s 0, ¢z Ž x, 0. s « F z Ž x . ,
x g V , t ) 0,
t
x g V , t ) 0, x g V.
0
By the comparison principle Žsee w10, 12x. it follows that z Ž x, t . G z Ž x, t ., and hence, z` s lim z Ž ?, t . G lim z Ž ?, t . s « exp yMK 4 ) 0. tªq`
tªq`
Ž28. When s ) 1, define 1.H0t hŽ s . ds41rŽ1y s .. Then we have
¡z y dD z s yMhŽ t . z ~ zrh s 0, ¢z Ž x, 0. s « F z Ž x . , t
s
z Ž x, t . s z Ž t . s « 1y s q M Ž s y
F yh Ž t . f Ž x, t . z s ,
x g V , t ) 0, x g V , t ) 0, x g V.
0
By the comparison principle it follows that z Ž x, t . G z Ž x, t ., so that z` s lim z Ž ?, t . G lim z Ž ?, t . s « 1y s q MK Ž s y 1 . tªq`
1r Ž1y s .
tªq`
) 0.
Ž2. Assume that there exists T ) 0 such that f Ž x, t . G M ) 0 for all x g V and t G T, and H0q` hŽ t . dt s q`. Ž18. When s s 1, take z Ž x, t . s z Ž t . s « 1 exp yMH0t hŽ s . ds4 , where « 1 s max V z Ž x, T . ) 0. Then we have lim t ªq` z Ž x, t . s 0 and
¡z y dD z s yMhŽ t . z Ž t . G yhŽ t . f Ž x, t . z Ž x, t . , ¢z Ž x, 0. s « G z Ž x, T . ,
~ zrh s 0, t
1
x g V, t ) T, x g V, t ) T, x g V.
According to the comparison principle, we know z Ž x, t . F z Ž x, t . ,
z` s lim z Ž ?, t . F lim z Ž ?, t . s 0,
and hence z` s 0 because z` G 0.
tªq`
tªq`
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A CHEMICAL REACTION MODEL
Ž28. When s ) 1, take « 2 ) 0, so small that « 21rŽ1y s . ) max V z Ž x, T ., and take a small c ) 0, such that c F Ž s y 1. M. Put z Ž x, t . s z Ž t . s « 2 q c
ž
t
H0
1r Ž1y s .
h Ž s . ds
/
.
Then we have lim t ªq` z Ž x, t . s 0 and
¡z ydD zsy Ž crŽ s y1. . h Ž t . z Gyh Ž t . f Ž x, t . z ~ zrh s 0, ¢z Ž x, T . s « G max z Ž x, T . G z Ž x, T . , s
t
1rŽ1y s . 2
V
s
, xgV, tGT , x g V, t G T, x g V.
So we get z Ž x, t . F z Ž x, t ., 0 F z` s lim t ªq` z Ž?, t . F lim t ªq` z Ž?, t . s 0, and hence z` s 0. The proof is completed. COROLLARY. Assume that the conditions of the lemma hold and f Ž x, t . ' C ) 0 in problem Ž2.1.. Then Ž1. Ž2.
z` ) 0, if z t y dD z G yChŽ t . z s and H0q` hŽ t . dt s K - q`. z` s 0, if z t y dD z F yChŽ t . z s and H0q` hŽ t . dt s q`. 3. MAIN THEOREMS
In this section, we always assume that ŽH 1 ., ŽH 2 . hold. From Proposition 2, the asymptotic behavior of Ž u, ¨ , w . is clear when E G 0. Therefore, we need only to discuss the case E - 0. From Proposition 2Ž3., we know that u` s 0 and ¨` q w` ) 0. So either ¨` or w` is positive. A natural question is whether ¨` ) 0 Ž w` ) 0. or not. Under some conditions, we will give a positive answer. Our main results are the following theorems. THEOREM 1. Ž1. Ž2. Ž3.
Assume that E - 0. Then
¨` ) 0 if m ) k y 1.
w` ) 0 if k ) m y 1. ¨` ) 0 and w` ) 0 if m ) k y 1 and k ) m y 1.
Before stating the following theorems, we denote max V u 0 Ž x . s a1 , min V u 0 Ž x . s b1; max V ¨ 0 Ž x . s a2 , min V ¨ 0 Ž x . s b 2 ; max V w 0 Ž x . s a3 and min V w 0 Ž x . s b 3 . We first state the conclusions for the case w` ) 0. From Theorem 1, we know that ¨` ) 0 when k - m q 1. Hence, in the following theorems, we need only to discuss the case k G m q 1.
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JIANG, WANG, AND XIE
THEOREM 2. Then ¨` ) 0 if
Suppose that E - 0, w` ) 0, k G m q 1, and m s n s 1.
Ž i . a1 ) 1 and a1 b 2 exp 1 y a1 4 G 2;
or
Ž ii . a1 F 1 and b 2 ) 1.
THEOREM 3. Suppose that E - 0, w` ) 0, k G m q 1, m s 1, and n ) 1. Then ¨` ) 0 if n Ži. b 2 ) Ž1rna1 .1rŽ ny1. and Ž na1 . n rŽ1yn. exp nrŽ n y 1. y b1y ra1 2 Ž n y 1.4 ) 1; or Žii. 1 - b 2 F Ž1rna1 .1rŽ ny1..
THEOREM 4. Suppose that E - 0, w` ) 0, k G m q 1, m ) 1, and n s 1. Then ¨` ) 0 if Ži. ma1m G 1 and 2rŽ2 m y 1. F Ž2 m y 1. a12 my1 b 2 exp 1 y a1m 4 ; or Žii. ma1m - 1 and ma1my1 b 2 ) 1. THEOREM 5. Suppose that E - 0, w` ) 0, k G m q 1, m ) 1, and n ) 1. Then ¨` ) 0 if Ži. mna1m - b1yn and 1 - ma1my1 b 2n ; or 2 Žii. mna1m G b1yn and 1 - ma1my1qm n rŽ ny1. Ž mn. n rŽ1yn. expŽ mna1m 2 1y m . m y b2 rmŽ n y 1. a1 4 . Remark 3.1. In the case ¨` ) 0, we get w` ) 0 by Theorem 1 when m - k q 1. When m G k q 1, similar to Theorems 2]5, the conditions on w` ) 0 can be obtained. We omit the details here. Remark 3.2. The Appendix will show that it is necessary to add the conditions upon a i and bi Ž i s 1, 2, 3. in Theorems 2]5 in order to ensure ¨` ) 0. Remark 3.3. It is well known that when the initial average concentration of M1 is less than the total initial average concentrations of M2 and M3 , M1 disappears as t ª q` and the total concentrations of M2 and M3 remain positive. However, in this case, it is possible that one of M2 and M3 disappears ultimately. Our results give the conditions on the initial concentrations of M1 , M2 , and M3 to ensure that both M2 and M3 do not vanish. This is interesting and important in applications.
4. PROOFS OF THEOREMS Before giving the proofs of our theorems, we first state the definitions of upper and lower solutions, and the comparison principle for problem Ž1.1. ] Ž1.5.. Problem Ž1.1. ] Ž1.3. is a quasi-monotone nonincreasing system
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647
for u, ¨ , w G 0. As in w12, 13x, nonnegative functions Ž u, ¨ , w . and Ž u, ¨ , w . are called the coupled upper and lower solutions of Ž1.1. ] Ž1.5., if they satisfy the differential inequalities u t y d1 D u G yu m ¨ n y u k w r ,
u t y d1 D u F yu m ¨ n y u k w r ,
¨ t y d 2 D¨ G yu m ¨ n ,
¨ t y d 2 D¨ F yu m ¨ n , x g V , t ) 0,
wt y d 3 Dw G yu k w r ,
wt y d 3 Dw F yu k w r ,
urh G 0 G urh ,
¨ rh G 0 G ¨ rh ,
wrh G 0 G wrh , x g V , t ) 0 u Ž x, 0 . G u 0 Ž x . G u Ž x, 0 . ,
¨ Ž x, 0 . G ¨ 0 Ž x . G ¨ Ž x, 0 .
w Ž x, 0 . G w 0 Ž x . G w Ž x, 0 . , x g V . By the results of w12x, we know that if Ž u, ¨ , w . and Ž u, ¨ , w . are the coupled upper and lower solutions of Ž1.1. ] Ž1.5., then the solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. satisfies u F u F u, ¨ F ¨ F ¨ , w F w F w. To prove our theorems, it suffices to find a positive lower bound of the unknown function, which ensures the positive limit. With this aim, we will construct the proper couple of upper and lower solutions of Ž1.1. ] Ž1.5. by using solutions of a set of ordinary differential equations. This idea was used by many authors, see w12, 13x and the references therein. We prove only Theorems 1, 2, and 5. Theorems 3 and 4 can be proved in a similar way. Because u 0 Ž x ., ¨ 0 Ž x ., w 0 Ž x . G 0 Žk 0., the maximum principle assures that uŽ x, t ., ¨ Ž x, t ., w Ž x, t . ) 0 on V for any t ) 0. Without loss of generality we may assume that u 0 Ž x ., ¨ 0 Ž x ., w 0 Ž x . ) 0 on V. Proof of Theorem 1. Ž1. Because ¨` q w` ) 0, we have ¨` ) 0 or w` ) 0. If ¨` ) 0, then the conclusion holds. If w` ) 0, then there exists T ) 0 such that w Ž x, t . G w` r2 ) 0 for all x g V and t G T. Hence
¡u y d D u F yu Ž w r2. , ~ urh s 0, ¢uŽ x, t . < s uŽ x, T . ) 0, r
k
t
`
1
tsT
x g V, t G T, x g V, t G T, x g V.
By the maximum principle, we get for some constants a ) 0 and C s C ŽT . ) 0, u Ž x, t . F C exp ya t 4 , u Ž x, t . F C Ž 1 q t .
1r Ž1yk .
x g V , t G 0, if k s 1; ,
x g V , t G 0, if k ) 1,
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JIANG, WANG, AND XIE
see w7x. Define hŽ t . s exp ym a t 4 if k s 1, hŽ t . s Ž1 q t . m rŽ1yk . if k ) 1. Then H0` hŽ t . dt - q` if k s 1, or 1 - k - 1 q m. By Ž1.2., Ž1.4., and Ž1.5. we have
¡¨ y d D¨ G yChŽ t . ¨ , ~ ¨ rh s 0, ¢¨ Ž x, 0. s ¨ Ž x . G 0 Ž k 0. ,
x g V, t ) 0 x g V, t ) 0 x g V.
n
t
2
0
Because ¨ ª ¨` as t ª q`, from the Corollary of Section 2 we know that
¨` ) 0.
Ž2. By the same arguments as in Ž1. it follows that w` ) 0 if m s 1, or 1 - m - 1 q k. Ž3. From Ž1. and Ž2., we know that Ž3. is valid. Theorem 1 is proved.
Proof of Theorem 2. Take 1 - p - 2, so k ) p. Let uŽ t . s a1Ž1 q t .1rŽ1yp., uŽ t . s Beyb t , where 0 - B F b1 and b ) 0 are to be determined later. Let ¨ Ž t ., ¨ Ž t ., w Ž t ., w Ž t . be solutions of the following ordinary differential equations respectively ¨ 9 Ž t . s yu Ž t . ¨ Ž t . s yBeyb t ¨ Ž t . ; ¨ 9 Ž t . s yu Ž t . ¨ Ž t . s ya1 Ž 1 q t .
1r Ž1yp .
¨ Ž t. ;
¨ Ž 0 . s a2 ,
Ž 4.1.
¨ Ž 0 . s b 2 ) 0,
Ž 4.2.
w9 Ž t . s yu k Ž t . w r Ž t . s yB k exp ykbt 4 w r Ž t . ; w9 Ž t . s yu k Ž t . w r Ž t . s ya1k Ž 1 q t .
kr Ž1yp .
w Ž 0 . s a3 , Ž 4.3. w Ž 0 . s b 3 ) 0,
wrŽ t. ;
Ž 4.4. where 9 s drdt. It is obvious that ¨ Ž t . F a2 and w Ž t . F a3 . From p - 2, p - k, and the Corollary of Section 2 we have ¨ Ž t . ª ¨ Ž q` . ) 0,
w Ž t . ª w Ž q` . ) 0 as t ª q`.
¨ Ž t . G ¨ Ž q` . ,
w Ž t . G w Ž q` . .
It follows from Ž4.1. that ¨ Ž t . s ¨ Ž 0 . exp B Ž eyb t y 1 . rb 4 s a2 exp B Ž eyb t y 1 . rb 4 .
Let a s Ž p y 1.rŽ p y 2. - 0. From Ž4.2. we have ¨ Ž t . s b 2 exp aa1 1 y Ž 1 q t .
½
1ra
5.
Ž 4.5.
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A CHEMICAL REACTION MODEL
We first verify that for some suitable b ) 0 and B ) 0, u satisfies u9 F yu¨ y u k w r .
Ž 4.6.
Direct computation shows that Ž4.6. holds if and only if b G a2 exp B Ž eyb t y 1 . rb 4 q B ky1 eybŽ ky1.t w r .
Ž 4.7.
Because w Ž t . F a3 and eyb t y 1 F 0, it is obvious that if b G a2 q B ky 1 a3r ,
Ž 4.8.
then Ž4.7. holds and so does Ž4.6.. Since k G 2, it follows that for any given 1 - p - 2, there exist 0 - B F b1 and b ) 0 such that Ž4.8. holds. Therefore, Ž4.6. holds. Secondly, we verify that for some suitable 1 - p - 2. u satisfies uX G yu¨ y u k w r .
Ž 4.9.
By direct computation we have that Ž4.9. holds if and only if 1 py1
F b 2 Ž 1 qt . exp aa1 1 yŽ 1 qt .
½
1ra
5 qa
ky1 1
Ž 1 qt .
Ž pyk .r Ž py1 .
w r.
Ž 4.10. Therefore, Ž4.10. holds if 1r Ž p y 1 . F b 2 Ž 1 q t . exp aa1 1 y Ž 1 q t .
½
1ra
5 J f Ž y. ,
Ž 4.11.
where y s Ž1 q t .
1r Ž py1 .
G 1,
f Ž y . s b 2 y py1 exp aa1 Ž 1 y y py2 . 4 .
By direct computation we have f 9 Ž y . s Ž p y 1 . b 2 y py 2 exp aa1 Ž 1 y y py 2 . 4 y b 2 y py 1 exp aa1 Ž 1 y y py 2 . 4 aa1 Ž p y 2 . y py 3 s Ž p y 1 . y py 2 exp aa1 Ž 1 y y py2 . 4 b 2 1 y a1 y py2 . It follows that f 9Ž y . - 0 when y py 2 G 1ra1 , and f 9Ž y . ) 0 when y py 2 1ra1. This shows that f Ž a11rŽ2yp. . is the minimum of f Ž y .. If the condition Ži. of Theorem 2 holds, i.e., a1 ) 1 and a1 b 2 exp 1 y a14 G 2, then f Ž y . G f Ž a1rŽ2yp. . s b2 a1ya exp aŽ a1 y 1. 4 . 1
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JIANG, WANG, AND XIE
From this we see that if 1r Ž p y 1 . F b 2 a1ya exp a Ž a1 y 1 . 4 ,
Ž 4.12.
then Ž4.11. holds. Choosing p s 3r2, we have a s Ž p y 1.rŽ p y 2. s y1 and 1rŽ p y 1. s 2. The condition a1 b 2 exp 1 y a14 G 2 implies that Ž4.12. holds. Thus Ž4.9. follows. If the condition Žii. of Theorem 2 holds, i.e., a1 F 1 and b 2 ) 1, then f 9Ž y . G 0 because y G 1. Therefore f Ž y . G f Ž1. s b 2 . It follows that by b 2 ) 1, there exists 1 - p - 2 such that 1rŽ p y 1. F b 2 . Hence Ž4.11. and, therefore, Ž4.9. hold true. Equations Ž4.1. ] Ž4.4. and inequalities Ž4.6., Ž4.9. show that Ž u, ¨ , w ., and Ž u, ¨ , w . are coupled upper and lower solutions of Ž1.1. ] Ž1.5.. So the solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. satisfies u Ž t . F u Ž x, t . F u Ž t . ,
¨ Ž t . F ¨ Ž x, t . F ¨ Ž t . ,
and
w Ž t . F w Ž x, t . F w Ž t . . Because ¨ Ž t . ª b 2 exp aa14 ) 0 as t ª q` Žsee Ž4.5.., it yields ¨` ) 0. The proof is completed. Proof of Theorem 5. Take p g Ž m, m q 1., then m - p - k. Define u Ž t . s a1 Ž 1 q t .
1r Ž1yp .
uŽ t . s B Ž 1 q t .
,
1r Ž1ym .
,
where 0 - B F b1 is to be determined. Since m - p, we have uŽ t . G uŽ t .. Let ¨ Ž t ., ¨ Ž t ., w Ž t ., w Ž t . be solutions of the following initial value problems, respectively, ¨ 9 s yu m ¨ n s yB m Ž 1 q t .
mr Ž1ym .
¨ Ž 0 . s a2 ,
Ž 4.13.
¨ n;
¨ Ž 0. s b2 ,
Ž 4.14.
wr;
w Ž 0 . s a3 ,
Ž 4.15.
w Ž 0. s b3 ,
Ž 4.16.
¨ 9 s yu m ¨ n s ya1m Ž 1 q t .
mr Ž1yp .
w9 s yu k w r s yB k Ž 1 q t .
kr Ž1ym .
w9 s yu k w r s ya1k Ž 1 q t .
kr Ž1yp .
¨ n;
wr;
where 9 s drdt. It is obvious that ¨ Ž t . F a2 and w Ž t . F a3 . By Ž4.14. we have ¨ Ž t . s b1yn q Ž n y 1 . u a1m 1 y Ž 1 q t . 2
½
y1r u
5
1r Ž1yn .
,
where u s Ž p y 1.rŽ m q 1 y p .. The following proof is similar to that of Theorem 2. We first verify that for some suitable 0 - B F b1 , u satisfies u9 F yu m ¨ n y u k w r .
Ž 4.17.
651
A CHEMICAL REACTION MODEL
By direct computation we know that Ž4.17. is equivalent to 1r Ž m y 1 . G B my1 ¨ n Ž t . q B ky1 Ž 1 q t .
Ž myk .r Ž my1 .
w r Ž t . . Ž 4.18.
Since 1 - m - k, n, r G 1, ¨ Ž t . F a2 , and w Ž t . F a3 , it follows that Ž4.18. holds if 1r Ž m y 1 . G B my 1a2n q B ky 1a3r .
Ž 4.19.
It is obvious that Ž4.19. holds if B ) 0 is small since 1 - m - k. Hence Ž4.17. holds for small B ) 0. In the following we will verify that for some suitable m - p - m q 1, u satisfies u9 G yu m ¨ n y u k w r .
Ž 4.20.
By direct computation we see that Ž4.20. holds if and only if 1r Ž p y 1 . F a1my1 Ž 1 q t .
a
½b
q a1ky1 Ž 1 q t .
1yn 2
q Ž n y 1 . u a1m 1 y Ž 1 q t .
Ž pyk .r Ž py1 .
y1r u
wr,
5
b
Ž 4.21.
where a s Ž p y m.rŽ p y 1., b s nrŽ1 y n.. Because w Ž t . G 0, it follows that Ž4.21. holds if 1r Ž p y 1 . F a1my1 Ž 1 q t .
a
½b
1yn 2
q Ž n y 1 . u a1m 1 y Ž 1 q t .
y1r u
J f Ž y. ,
5
b
Ž 4.22.
where y s Ž1 q t .1rŽ py1. G 1 and f Ž y . s a1my1 y pym b1yn q Žn y 2 1.u a1m Ž1 y y py my1 .4 b. After some computations, we have f 9 Ž y . s a1my1 y pymy1 b1yn q Ž n y 1 . u a1m Ž 1 y y pymy1 . 2
by1
n = Ž p y m . b1y q Ž n y 1 . u a1m Ž 1 y y pymy1 . 2
yn Ž p y 1 . a1m y py my1 4 , Ž 4.23. and n q Ž n y 1 . u a1m Ž 1 y y pymy1 . y n Ž p y 1 . a1m y pymy1 Ž p y m . b1y 2 n s Ž p y m . b1y q Ž p y m . Ž n y 1 . u a1m 2
y Ž p y m . Ž n y 1 . u a1m q n Ž p y 1 . a1m y pymy1 .
Ž 4.24.
652
JIANG, WANG, AND XIE
If Ži. of Theorem 5 holds, there exists p g Ž m, m q 1. such that n G n Ž p y 1 . a1m , Ž p y m . b1y 2
1 F Ž p y 1 . a1my 1 b 2n .
Ž 4.25.
Since y py my1 F 1, it yields f 9Ž y . G 0 Žby Ž4.23.. and the first inequality of Ž4.25.. Thus f Ž y . G f Ž1. s a1my 1 b 2n. This inequality and the second one of Ž4.25. imply that Ž4.22. holds, and hence Ž4.20. holds. If Žii. of Theorem 5 holds, we set Ms s
n q Ž p y m . Ž n y 1 . u a1m Ž p y m . b1y 2 n Ž p y 1 . a1m q Ž p y m . Ž n y 1 . u a1m n a Ž m q 1 y p . b1y q Ž n y 1 . Ž p y 1 . a1m 2
Ž n q m y p . a1m
.
n It follows from the first inequality of Žii. that nŽ p y 1. a1m ) Ž p y m. b1y 2 Ž Ž .. Ž . and M - 1 because a - 1rm for all p g m, m q 1 . By 4.23 and Ž4.24. we have that f 9Ž y . - 0 if y py my1 ) M. f 9Ž y . ) 0 if y py my1 - M. This shows that f Ž M 1rŽ pymy1. . is the minimum of f Ž y .. Therefore,
f Ž y . G f Ž M 1rŽ pymy1. . s a1my 1 M Ž pym.rŽ pymy1. b
= b1yn q Ž n y 1 . u a1m Ž 1 y M . 4 . 2 By the expression of M, we have n q Ž n y 1 . u a1m Ž 1 y M . . nMa1mra s n Ž p y 1 . Ma1mr Ž p y m . s b1y 2
So f Ž M 1rŽ pymy1. . s a1my 1 M Ž pym.rŽ pymy1. nMa1m ra 4 b. It follows that Ž4.22. holds provided that 1 py1
b
F a1mq1qm b M Ž pym.rŽ pymy1.q b Ž nra . .
Ž 4.26.
By using the relation M Ž pym.rŽ pymy1. s Ž p y m. =
½
Ž pym .r Ž pymy1 .
q Ž n y 1 . Ž p y 1 . a1m Ž m q 1 y p . b1yn 2 Ž p y 1 . Ž m q n y p . a1m
Ž pym .Ž pymy1 .
5
653
A CHEMICAL REACTION MODEL
and the L’Hopital rule we see that lim M
Ž pym.rŽ pymy1.
pymq1
s e exp
½
n ma1m y b1y 2
m Ž n y 1 . a1m
5
.
From this and lim py mq1 M s 1, we see that the right hand side of Ž4.26. converges to
½
a1my1qm b exp 1 q
n ma1m y b1y 2
m Ž n y 1 . a1m
s a1my1qm b exp
½
5
Ž mn .
n mna1m y b1y 2
m Ž n y 1 . a1m
5
b
b Ž mn . .
From Žii. of Theorem 5, it follows that there exists p: p - m q 1 and near to m q 1, such that Ž4.26. holds, so do Ž4.22. and Ž4.20.. Equations Ž4.13. ] Ž4.16. and the inequalities Ž4.17. and Ž4.20. show that Ž u, ¨ , w . and Ž u, ¨ , w . are coupled upper and lower solutions of Ž1.1. ] Ž1.5.. So the solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. satisfies u Ž t . F u Ž x, t . F u Ž t . ,
¨ Ž t . F ¨ Ž x, t . F ¨ Ž t . ,
w Ž t . F w Ž x, t . F w Ž t . . n Because ¨ Ž t . ª b1y q wŽ p y 1.Ž n y 1.rŽ m q 1 y p .x a1m 41rŽ1yn. ) 0 as 2 t ª q`, we get ¨` ) 0. Theorem 5 is proved.
Remark. Theorems 3 and 4 can be proved in the same way as that of Theorem 2. For simplicity, we give an outline of the proofs here. For the proof of Theorem 3, we choose 1 - p - 2 and let u Ž x, t . s u Ž t . s a1 Ž 1 q t .
1r Ž1yp .
u Ž x, t . s u Ž t . s Beyb t .
,
Also let ¨ Ž t ., ¨ Ž t ., w Ž t ., and w Ž t .a be the solutions of the following initial value problems respectively, ¨ 9 Ž t . s yu Ž t . ¨ n Ž t . s yBeyb t ¨ n Ž t . ; ¨ 9 Ž t . s yu Ž t . ¨ n Ž t . s ya1 Ž 1 q t .
1r Ž1yp .
¨ Ž 0 . s a2 ,
¨ nŽ t. ;
w9 Ž t . s yu k Ž t . w r Ž t . s yB k exp ykbt 4 w r Ž t . ; w9 Ž t . s yu k Ž t . w r Ž t . s ya1k Ž 1 q t .
kr Ž1yp .
wrŽ t. ;
¨ Ž 0. s b2 ,
Ž 4.27. Ž 4.28.
w Ž 0 . s a3 , Ž 4.29. w Ž 0. s b3 .
Ž 4.30.
654
JIANG, WANG, AND XIE
Then Ž uŽ t ., ¨ Ž t ., w Ž t .. and Ž uŽ t ., ¨ Ž t ., w Ž t .. are the coupled upper and lower solutions of Ž1.1. ] Ž1.5. for a suitable b ) 0 and 0 - B F b1. Define a s Ž p y 1.rŽ p y 2. - 0. Then by Ž4.28., we have ¨ Ž t . s b1yn y Ž n y 1 . aa1 1 y Ž 1 q t . 2
½
ª b1yn y Ž n y 1 . aa1 4 2
1r Ž1yn .
ya y1
5
1r Ž1yn .
)0
as t ª q`. This yields ¨` ) 0. For the proof of Theorem 4, we choose p: p - m q 1 and near m q 1, and let u s a1 Ž 1 q t .
1r Ž1yp .
u ss B Ž 1 q t .
,
1r Ž1ym .
.
Also let ¨ Ž t ., ¨ Ž t ., w Ž t . be the solutions of the following initial value problems respectively, mr 1ym
¨ Ž t. ;
¨ Ž 0 . s a2 ,
Ž 4.31.
mr Ž1yp .
¨ Ž t. ;
¨ Ž 0. s b2 ,
Ž 4.32.
¨ 9 Ž t . s yu m Ž t . ¨ Ž t . s yB m Ž 1 q t . ¨ 9 Ž t . s yu m Ž t . ¨ Ž t . s ya1m Ž 1 q t .
w Ž t . s yu k Ž t . w r Ž t . s yB k Ž 1 q t .
kr Ž1ym .
wrŽ t. ;
w Ž 0 . s a3 ,
Ž 4.33. w9 Ž t . s yu k Ž t . w r Ž t . s ya1k Ž 1 q t .
kr Ž1yp .
wrŽ t. ;
w Ž 0. s b3 .
Ž 4.34. Then Ž uŽ t ., ¨ Ž t ., w Ž t .. and Ž uŽ t ., ¨ Ž t ., w Ž t .. are the coupled upper and lower solutions of Ž1.1. ] Ž1.5. for the suitable 0 - B F b1 and m - p m q 1. Denote c s Ž m q 1 y p .rŽ p y 1. ) 0. Then we have, by Ž4.32., ¨ Ž t . s b 2 exp a1m Ž 1rc .
½
c
Ž1 q t . y 1
5 ªb
2
exp ya1m Ž 1rc . 4 ) 0
as t ª q`. It yields ¨` ) 0. APPENDIX In this appendix, we will demonstrate that when k G m q 1, even ŽH 2 . is valid and E - 0, u` ) 0 cannot still be assured without any other additional conditions on the initial data u 0 Ž x ., ¨ 0 Ž x ., and w 0 Ž x .. For example, when m s n s r s 1 and k s 2, choose a g Ž0, 1. such that
655
A CHEMICAL REACTION MODEL
ae - 1, a2 e - a, and let n 0 : 0 - ¨ 0 F a y a2 e. Consider the Cauchy problem
¡u9Ž t . s yu¨ y u w, 2
~¨ 9 Ž t . s yu¨ ,
x g V , t ) 0,
Ž A1.
w9 Ž t . s yu w, u Ž 0 . s a, ¨ Ž 0 . s ¨ 0 ) 0, 2
¢
w Ž 0 . s ae.
It is easy to see that Ž u, ¨ , w . and Ž u, ¨ , w . are the coupled upper and lower solutions of ŽA1., respectively, with the expressions a a uŽ t . s , uŽ t . s , 2 1 q at 1qa t ¨ Ž t . s ¨ 0 Ž 1 q at .
y1
¨ Ž t . s ¨ 0 Ž 1 q a2 t .
,
w Ž t . s w 0 exp a Ž 1 q at . w Ž t . s w 0 exp Ž 1 q a2 t .
½
y1
y1
y1 ra
F ¨ Ž t. ,
y a4 ,
y 1 F wŽ t. .
5
So the solution Ž u, ¨ , w . of problem ŽA1. satisfies u F u F u,
¨ F¨ F¨,
w F w F w.
Because ¨ ª 0 as t ª q`, and ¨ Ž t . G 0 we have ¨ Ž t . ª 0 as t ª q`. If we take the initial values u 0 Ž x . ' a, ¨ 0 Ž x . ' ¨ 0 , and w 0 Ž x . ' ae in Ž1.1. ] Ž1.5., then the solution of Ž1.1. ] Ž1.5. is also the solution of problem ŽA1. and vice versa. So the solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. satisfies ¨ Ž x, t . ª 0 as t ª q`, i.e., ¨` s 0. ACKNOWLEDGMENT We are very grateful to Professor C. V. Pao for his many helpful suggestions and comments which improved our presentation.
REFERENCES 1. E. O. Frind et al., Simulation of nitrate and sulfate transport and transformation in the Fuhrberger Feld Aquifer, in ‘‘Contaminant Transport in Ground Water’’ ŽH. E. Kobus and W. Kinzelbach, Eds.., pp. 97]104, Proceedings of the International Symposium on Contaminant Transport in Ground Water, Stuttgart, April 4]6, 1989. 2. J. Grossman and B. Merkel, One-dimensional simulation of the impact of nitrogen fertilizers on the Carbonate Equilibrium, in ‘‘Contaminant Transport in Ground Water’’ ŽH. E. Kobus and W. Kinzelbach, Eds.., pp. 163]169, Proceedings of the International Symposium on Contaminant Transport in Ground Water, Stuttgart, April 4]6, 1989.
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JIANG, WANG, AND XIE
3. G. Pinder and S. Stathoff, Can the sharp interface salt fresh water model capture transient behavior, in Developments in Water science, Vol. 35, pp. 217]224, Elsevier, Amsterdam, 1988. 4. W. Brey, ‘‘Physical Chemistry and Its Biological Applications,’’ Academic Press, San Diego, 1978. 5. X. Y. Zhu and C. H. Xie, ‘‘Modeling of Transport in Subsurface Flow,’’ China Architecture and Building Press, Beijing, 1990. wIn Chinesex 6. W. E. Fitzgibbon, C. B. Martin, and J. J. Morgan, A diffusive epidemic model with criss-cross dynamics, J. Math. Anal. Appl. 184 Ž1994., 399]414. 7. H. Hoshino and Y. Yamada, Asymptotic behavior of global solutions for some reactiondiffusion systems, Nonlinear Anal. 23, No. 5 Ž1994., 639]650. 8. A. Parshotam, A. McNabb, and G. Wake, Comparison theorems for coupled reactiondiffusion equation in chemical reactor analysis, J. Math. Anal. Appl. 178 Ž1993., 196]220. 9. W. H. Ruan and C. V. Pao, Asymptotic behavior and positive solutions of a chemical reaction diffusion system, J. Math. Anal. Appl. 169 Ž1992., 157]178. 10. G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, ‘‘Monotone Iterative Techniques for Nonlinear Differential Equations,’’ Pitman, New York, 1985. 11. C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 87 Ž1982., 165]198. 12. C. V. Pao, ‘‘Nonlinear Parabolic and Elliptic Equations,’’ Plenum, New York, 1992. 13. M. X. Wang, Global solution, asymptotic behavior and blow-up problems for a model of nuclear reactors, Sci. China Ser. A 35, No. 2 Ž1992., 129]141.
220, 640]656 Ž1998.
AY975871
Asymptotic Behavior of Global Solutions for a Chemical Reaction Model* Chengshun Jiang Department of Mathematics, Shandong Uni¨ ersity, Jinan, 250100 People’s Republic of China
Mingxin Wang Department of Applied Mathematics, Southeast Uni¨ ersity, Nanjing, 210018, People’s Republic of China
and Chunhong Xie Department of Mathematics, Nanjing Uni¨ ersity, Nanjing, 210008, People’s Republic of China Submitted by Chia Ven Pao Received July 7, 1995
This paper deals with a system of reaction-diffusion equations modeling a chemical reaction process involving three substances. The mathematical system consists of three semilinear partial differential equations of parabolic type with homogenous Neumann boundary conditions. The goal is to determine whether each substance has a positive asymptotic limit by the method of upper and lower solutions. Q 1998 Academic Press
1. INTRODUCTION In this paper, we consider a chemical reaction model in the transport of ground water. This model is related to three kinds of chemical substances * Project supported by the National Natural Science Foundation of China and Natural Science Foundation of Jiangsu Povince. 640 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
A CHEMICAL REACTION MODEL
641
M1 , M2 , and M3 which can react to each other to produce two new kinds of chemical compounds Ž M2 . nŽ M1 . m and Ž M3 . r Ž M1 . k at the same time. An example is given by the irreversible reactions NH4qq NOy 3 ª NH4 NO 3 , Ca2qq 2 NOy 3 ª Ca Ž NO 3 . 2 . The above two reactions are neutralization. Generally speaking, the neutralization among one kind of negative ions Žsay M1 . and two kinds of positive ions Žsay M2 and M3 . is irreversible with the following forms nM2 q mM1 ª Ž M2 . n Ž M1 . m , rM3 q kM1 ª Ž M3 . r Ž M1 . k , where m, n, k, r are chemical combination valencies w1]5x. Let u, ¨ , w represent the concentrations of M1 , M2 , and M3 , respectively. By disregarding convection phenomena and considering only diffusion processes, Ž u, ¨ , w . satisfies the following semilinear reaction-diffusion system u t y d1 D u s yu m ¨ n y u k w r ,
Ž 1.1.
¨ t y d 2 D¨ s yu m ¨ n ,
Ž 1.2.
x g V , t ) 0,
wt y d 3 Dw s yu k w r ,
Ž 1.3.
where V is a bounded domain in R N with smooth boundary V, and the d i Ž i s 1, 2, 3. are positive constants. If we assume that there is no exchange between the substances and the outside field, then the boundary condition and initial condition are
urh s ¨ rh s wrh s 0, u Ž x, 0 . s u 0 Ž x . ,
x g V , t ) 0,
¨ Ž x, 0 . s ¨ 0 Ž x . ,
Ž 1.4.
w Ž x, 0 . s w 0 Ž x . , x g V ,
Ž 1.5. where rh denotes the outward normal derivative on V. The goal of this paper is to study the asymptotic behavior of solutions of Ž1.1. ] Ž1.5.. For this purpose, we assume that ŽH 1 . ŽH 2 .
m, n, k, r G 1 are constants. u 0 , ¨ 0 , w 0 g L`Ž V ., u 0 , ¨ 0 , w 0 G 0, and u 0 k 0, ¨ 0 k 0, w 0 k 0.
The existence and uniqueness of solutions, and the asymptotic behavior of global solutions for various reaction-diffusion systems have been studied by many investigators Žsee w6]13x and the references therein .. In w7x, H.
642
JIANG, WANG, AND XIE
Hoshino and Y. Yamada studied the asymptotic behavior of solutions of the irreversible reaction model with two kinds of chemical substances, i.e., the case w ' 0 in Ž1.1. ] Ž1.5.. In document w12x, C. V. Pao discussed the asymptotic limit of solutions of the problem
¡u y L u s yawf Ž x, u, ¨ . , t
¨t y
U 1 LU2 ¨
s ybwf Ž x, u, ¨ . ,
x g V, t ) 0
~w s cwf Ž x, u, ¨ . , t
urn 1U s ¨ rn U2 s 0, u Ž x, 0 . s u 0 Ž x . G 0,
¢w Ž x, 0. s w Ž x . G 0,
x g V, t ) 0 ¨ Ž x, 0 . s ¨ 0 Ž x . G 0,
0
x g V,
Ž 1.6. where LUi and rn iU are in self-adjoint form, i s 1, 2, f is a C 1-function satisfying f Ž x, 0, ¨ . s f Ž x, u, 0 . s 0, f u Ž x, u, ¨ . G 0,
f Ž x, u, ¨ . ) 0 when u ) 0, ¨ ) 0 f¨ Ž x, u, ¨ . G 0 when u G 0, ¨ G 0,
and a, b, and c are positive constants. The differences between our problems Ž1.1. ] Ž1.5. and Ž1.6. are the properties of the nonlinear reaction terms. For example, if we denote f i Ž x, u, ¨ , w . s Di wf Ž x, u, ¨ . with D 1 s ya, D 2 s yb, and D 3 s c in Ž1.6., then there exist constants c i such that c i f i s c j f j , i, j s 1, 2, 3. But the nonlinear reaction terms of Ž1.1. ] Ž1.3. do not have these properties. Therefore, we need more detailed analysis to obtain the limit of the solution of Ž1.1. ] Ž1.5.. It is easy to infer that 0 F u F 5 u 0 5 ` , 0 F ¨ F 5 ¨ 0 5 ` , and 0 F w F 5 w 0 5 ` for any x g V and t G 0 by the maximum principle. So Ž1.1. ] Ž1.5. has a unique global solution Ž u, ¨ , w ., see w6]13x. Moreover, we have that uŽ x, t . ª u` , ¨ Ž x, t . ª ¨` , and w Ž x, t . ª w` as t ª q` be the same method as that of w7, 12x, where u` , ¨` , and w` are nonnegative constants. Define E s 5 u 0 5 1 y 5 ¨ 0 5 1 y 5 w 0 5 1 , where 5 z 5 1 is the L1 Ž V . norm of the function z Ž x .. When E G 0, the asymptotic behavior of Ž u, ¨ , w . can be studied by the method of w7, 12x; we omit the details here. When E - 0, we have u` s 0 and ¨` q w` ) 0 by analogous arguments to those in w12x. A natural and important question is whether ¨` ) 0 or ¨` s 0 and whether w` ) 0 or w` s 0. It is our goal to determine the positivities of ¨` and w` when E - 0 by the method of upper and lower solutions. It is important in applications to give conditions which assure ¨` ) 0 or ¨` s 0 and w` ) 0 or w` s 0. For example, if ¨ Žor w . represents the concentration of the cancer causing substance, such as nitrate Ž NO3 .,
A CHEMICAL REACTION MODEL
643
nitrile Ž NO2 ., etc., nitrate will be changed into nitrile with redox processes, i.e., q y NOy 3 q 2 H q 2 e l NO 2 q H 2 O,
where it is assumed that the water contains free Hq ions and free electrons. We must do something to control the initial data to get ¨` s 0 Žor w` s 0.. Conversely, if f Žor w . represents the concentration of the beneficial mineral substance for human body or human life, we hope to control the initial data to get ¨` ) 0 Žor w` ) 0.. Therefore, the discussions and proofs in this paper will help create some conditions to develop and protect the ground water resource for mankind. For further physical meanings of the equations, we refer to w1, 2x and the references therein.
2. PRELIMINARIES By the manners and arguments similar to those used in w7, 12x, we have the following propositions. PROPOSITION 1.
For all t G 0, HV Ž u y ¨ y w . dx s E.
PROPOSITION 2. Assume that the conditions ŽH 1 ., ŽH 2 . hold. Then there exists a unique global solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. which satisfies uŽ x, t . ª u` , ¨ Ž x, t . ª ¨` , and w Ž x, t . ª w` as t ª q`. Moreo¨ er, Ž1. Ž2. Ž3.
u` s Er< V < ) 0, ¨` s w` s 0 if E ) 0. u` s ¨` s w` s 0 if E s 0. u` s 0, ¨` q w` s yEr< V < ) 0 if E - 0.
In order to determine the values of ¨` and w` in the case Ž3. of Proposition 2, we need to prove the following critical lemma. LEMMA.
Assume that z Ž x, t . is a solution of the problem
¡z y dD z s yhŽ t . f Ž x, t . z ~ zrh s 0, ¢z Ž x, 0. s z Ž x . G 0, k 0, t
0
s
,
x g V , t ) 0, x g V , t ) 0, x g V,
Ž 2.1.
where d G 0, s G 1 are constants, hŽ t . G 0, hŽ t . ª 0 as t ª q`, f Ž x, t . G 0, and z Ž x, t . ª z` as t ª q`. Then we ha¨ e Ž1. z` ) 0, if f Ž x, t . F M - q` and H0q` hŽ t . dt - q`. Ž2. z` s 0, if there exists T ) 0 such that f Ž x, t . G M ) 0 for all x g V and t G T, and H0q` hŽ t . dt s q`.
644
JIANG, WANG, AND XIE
Proof of the Lemma. Ž1. Assume that f Ž x, t . F M - q` and K J H0q` hŽ t . dt - q`. Because z 0 Ž x . G 0, k 0, by the maximum principle we have z Ž x, t . ) 0 on V for any t ) 0. Without loss of generality we can assume that z 0 Ž x . ) 0 on V. Hence, there exists « ) 0 such that z 0 Ž x . G « on V. Ž18. When s s 1, define z Ž x, t . s z Ž t . s « exp yMH0t hŽ s . ds4 . Then we have
¡z y dD z s yMhŽ t . z F yhŽ t . f Ž x, t . z, ~ zrh s 0, ¢z Ž x, 0. s « F z Ž x . ,
x g V , t ) 0,
t
x g V , t ) 0, x g V.
0
By the comparison principle Žsee w10, 12x. it follows that z Ž x, t . G z Ž x, t ., and hence, z` s lim z Ž ?, t . G lim z Ž ?, t . s « exp yMK 4 ) 0. tªq`
tªq`
Ž28. When s ) 1, define 1.H0t hŽ s . ds41rŽ1y s .. Then we have
¡z y dD z s yMhŽ t . z ~ zrh s 0, ¢z Ž x, 0. s « F z Ž x . , t
s
z Ž x, t . s z Ž t . s « 1y s q M Ž s y
F yh Ž t . f Ž x, t . z s ,
x g V , t ) 0, x g V , t ) 0, x g V.
0
By the comparison principle it follows that z Ž x, t . G z Ž x, t ., so that z` s lim z Ž ?, t . G lim z Ž ?, t . s « 1y s q MK Ž s y 1 . tªq`
1r Ž1y s .
tªq`
) 0.
Ž2. Assume that there exists T ) 0 such that f Ž x, t . G M ) 0 for all x g V and t G T, and H0q` hŽ t . dt s q`. Ž18. When s s 1, take z Ž x, t . s z Ž t . s « 1 exp yMH0t hŽ s . ds4 , where « 1 s max V z Ž x, T . ) 0. Then we have lim t ªq` z Ž x, t . s 0 and
¡z y dD z s yMhŽ t . z Ž t . G yhŽ t . f Ž x, t . z Ž x, t . , ¢z Ž x, 0. s « G z Ž x, T . ,
~ zrh s 0, t
1
x g V, t ) T, x g V, t ) T, x g V.
According to the comparison principle, we know z Ž x, t . F z Ž x, t . ,
z` s lim z Ž ?, t . F lim z Ž ?, t . s 0,
and hence z` s 0 because z` G 0.
tªq`
tªq`
645
A CHEMICAL REACTION MODEL
Ž28. When s ) 1, take « 2 ) 0, so small that « 21rŽ1y s . ) max V z Ž x, T ., and take a small c ) 0, such that c F Ž s y 1. M. Put z Ž x, t . s z Ž t . s « 2 q c
ž
t
H0
1r Ž1y s .
h Ž s . ds
/
.
Then we have lim t ªq` z Ž x, t . s 0 and
¡z ydD zsy Ž crŽ s y1. . h Ž t . z Gyh Ž t . f Ž x, t . z ~ zrh s 0, ¢z Ž x, T . s « G max z Ž x, T . G z Ž x, T . , s
t
1rŽ1y s . 2
V
s
, xgV, tGT , x g V, t G T, x g V.
So we get z Ž x, t . F z Ž x, t ., 0 F z` s lim t ªq` z Ž?, t . F lim t ªq` z Ž?, t . s 0, and hence z` s 0. The proof is completed. COROLLARY. Assume that the conditions of the lemma hold and f Ž x, t . ' C ) 0 in problem Ž2.1.. Then Ž1. Ž2.
z` ) 0, if z t y dD z G yChŽ t . z s and H0q` hŽ t . dt s K - q`. z` s 0, if z t y dD z F yChŽ t . z s and H0q` hŽ t . dt s q`. 3. MAIN THEOREMS
In this section, we always assume that ŽH 1 ., ŽH 2 . hold. From Proposition 2, the asymptotic behavior of Ž u, ¨ , w . is clear when E G 0. Therefore, we need only to discuss the case E - 0. From Proposition 2Ž3., we know that u` s 0 and ¨` q w` ) 0. So either ¨` or w` is positive. A natural question is whether ¨` ) 0 Ž w` ) 0. or not. Under some conditions, we will give a positive answer. Our main results are the following theorems. THEOREM 1. Ž1. Ž2. Ž3.
Assume that E - 0. Then
¨` ) 0 if m ) k y 1.
w` ) 0 if k ) m y 1. ¨` ) 0 and w` ) 0 if m ) k y 1 and k ) m y 1.
Before stating the following theorems, we denote max V u 0 Ž x . s a1 , min V u 0 Ž x . s b1; max V ¨ 0 Ž x . s a2 , min V ¨ 0 Ž x . s b 2 ; max V w 0 Ž x . s a3 and min V w 0 Ž x . s b 3 . We first state the conclusions for the case w` ) 0. From Theorem 1, we know that ¨` ) 0 when k - m q 1. Hence, in the following theorems, we need only to discuss the case k G m q 1.
646
JIANG, WANG, AND XIE
THEOREM 2. Then ¨` ) 0 if
Suppose that E - 0, w` ) 0, k G m q 1, and m s n s 1.
Ž i . a1 ) 1 and a1 b 2 exp 1 y a1 4 G 2;
or
Ž ii . a1 F 1 and b 2 ) 1.
THEOREM 3. Suppose that E - 0, w` ) 0, k G m q 1, m s 1, and n ) 1. Then ¨` ) 0 if n Ži. b 2 ) Ž1rna1 .1rŽ ny1. and Ž na1 . n rŽ1yn. exp nrŽ n y 1. y b1y ra1 2 Ž n y 1.4 ) 1; or Žii. 1 - b 2 F Ž1rna1 .1rŽ ny1..
THEOREM 4. Suppose that E - 0, w` ) 0, k G m q 1, m ) 1, and n s 1. Then ¨` ) 0 if Ži. ma1m G 1 and 2rŽ2 m y 1. F Ž2 m y 1. a12 my1 b 2 exp 1 y a1m 4 ; or Žii. ma1m - 1 and ma1my1 b 2 ) 1. THEOREM 5. Suppose that E - 0, w` ) 0, k G m q 1, m ) 1, and n ) 1. Then ¨` ) 0 if Ži. mna1m - b1yn and 1 - ma1my1 b 2n ; or 2 Žii. mna1m G b1yn and 1 - ma1my1qm n rŽ ny1. Ž mn. n rŽ1yn. expŽ mna1m 2 1y m . m y b2 rmŽ n y 1. a1 4 . Remark 3.1. In the case ¨` ) 0, we get w` ) 0 by Theorem 1 when m - k q 1. When m G k q 1, similar to Theorems 2]5, the conditions on w` ) 0 can be obtained. We omit the details here. Remark 3.2. The Appendix will show that it is necessary to add the conditions upon a i and bi Ž i s 1, 2, 3. in Theorems 2]5 in order to ensure ¨` ) 0. Remark 3.3. It is well known that when the initial average concentration of M1 is less than the total initial average concentrations of M2 and M3 , M1 disappears as t ª q` and the total concentrations of M2 and M3 remain positive. However, in this case, it is possible that one of M2 and M3 disappears ultimately. Our results give the conditions on the initial concentrations of M1 , M2 , and M3 to ensure that both M2 and M3 do not vanish. This is interesting and important in applications.
4. PROOFS OF THEOREMS Before giving the proofs of our theorems, we first state the definitions of upper and lower solutions, and the comparison principle for problem Ž1.1. ] Ž1.5.. Problem Ž1.1. ] Ž1.3. is a quasi-monotone nonincreasing system
A CHEMICAL REACTION MODEL
647
for u, ¨ , w G 0. As in w12, 13x, nonnegative functions Ž u, ¨ , w . and Ž u, ¨ , w . are called the coupled upper and lower solutions of Ž1.1. ] Ž1.5., if they satisfy the differential inequalities u t y d1 D u G yu m ¨ n y u k w r ,
u t y d1 D u F yu m ¨ n y u k w r ,
¨ t y d 2 D¨ G yu m ¨ n ,
¨ t y d 2 D¨ F yu m ¨ n , x g V , t ) 0,
wt y d 3 Dw G yu k w r ,
wt y d 3 Dw F yu k w r ,
urh G 0 G urh ,
¨ rh G 0 G ¨ rh ,
wrh G 0 G wrh , x g V , t ) 0 u Ž x, 0 . G u 0 Ž x . G u Ž x, 0 . ,
¨ Ž x, 0 . G ¨ 0 Ž x . G ¨ Ž x, 0 .
w Ž x, 0 . G w 0 Ž x . G w Ž x, 0 . , x g V . By the results of w12x, we know that if Ž u, ¨ , w . and Ž u, ¨ , w . are the coupled upper and lower solutions of Ž1.1. ] Ž1.5., then the solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. satisfies u F u F u, ¨ F ¨ F ¨ , w F w F w. To prove our theorems, it suffices to find a positive lower bound of the unknown function, which ensures the positive limit. With this aim, we will construct the proper couple of upper and lower solutions of Ž1.1. ] Ž1.5. by using solutions of a set of ordinary differential equations. This idea was used by many authors, see w12, 13x and the references therein. We prove only Theorems 1, 2, and 5. Theorems 3 and 4 can be proved in a similar way. Because u 0 Ž x ., ¨ 0 Ž x ., w 0 Ž x . G 0 Žk 0., the maximum principle assures that uŽ x, t ., ¨ Ž x, t ., w Ž x, t . ) 0 on V for any t ) 0. Without loss of generality we may assume that u 0 Ž x ., ¨ 0 Ž x ., w 0 Ž x . ) 0 on V. Proof of Theorem 1. Ž1. Because ¨` q w` ) 0, we have ¨` ) 0 or w` ) 0. If ¨` ) 0, then the conclusion holds. If w` ) 0, then there exists T ) 0 such that w Ž x, t . G w` r2 ) 0 for all x g V and t G T. Hence
¡u y d D u F yu Ž w r2. , ~ urh s 0, ¢uŽ x, t . < s uŽ x, T . ) 0, r
k
t
`
1
tsT
x g V, t G T, x g V, t G T, x g V.
By the maximum principle, we get for some constants a ) 0 and C s C ŽT . ) 0, u Ž x, t . F C exp ya t 4 , u Ž x, t . F C Ž 1 q t .
1r Ž1yk .
x g V , t G 0, if k s 1; ,
x g V , t G 0, if k ) 1,
648
JIANG, WANG, AND XIE
see w7x. Define hŽ t . s exp ym a t 4 if k s 1, hŽ t . s Ž1 q t . m rŽ1yk . if k ) 1. Then H0` hŽ t . dt - q` if k s 1, or 1 - k - 1 q m. By Ž1.2., Ž1.4., and Ž1.5. we have
¡¨ y d D¨ G yChŽ t . ¨ , ~ ¨ rh s 0, ¢¨ Ž x, 0. s ¨ Ž x . G 0 Ž k 0. ,
x g V, t ) 0 x g V, t ) 0 x g V.
n
t
2
0
Because ¨ ª ¨` as t ª q`, from the Corollary of Section 2 we know that
¨` ) 0.
Ž2. By the same arguments as in Ž1. it follows that w` ) 0 if m s 1, or 1 - m - 1 q k. Ž3. From Ž1. and Ž2., we know that Ž3. is valid. Theorem 1 is proved.
Proof of Theorem 2. Take 1 - p - 2, so k ) p. Let uŽ t . s a1Ž1 q t .1rŽ1yp., uŽ t . s Beyb t , where 0 - B F b1 and b ) 0 are to be determined later. Let ¨ Ž t ., ¨ Ž t ., w Ž t ., w Ž t . be solutions of the following ordinary differential equations respectively ¨ 9 Ž t . s yu Ž t . ¨ Ž t . s yBeyb t ¨ Ž t . ; ¨ 9 Ž t . s yu Ž t . ¨ Ž t . s ya1 Ž 1 q t .
1r Ž1yp .
¨ Ž t. ;
¨ Ž 0 . s a2 ,
Ž 4.1.
¨ Ž 0 . s b 2 ) 0,
Ž 4.2.
w9 Ž t . s yu k Ž t . w r Ž t . s yB k exp ykbt 4 w r Ž t . ; w9 Ž t . s yu k Ž t . w r Ž t . s ya1k Ž 1 q t .
kr Ž1yp .
w Ž 0 . s a3 , Ž 4.3. w Ž 0 . s b 3 ) 0,
wrŽ t. ;
Ž 4.4. where 9 s drdt. It is obvious that ¨ Ž t . F a2 and w Ž t . F a3 . From p - 2, p - k, and the Corollary of Section 2 we have ¨ Ž t . ª ¨ Ž q` . ) 0,
w Ž t . ª w Ž q` . ) 0 as t ª q`.
¨ Ž t . G ¨ Ž q` . ,
w Ž t . G w Ž q` . .
It follows from Ž4.1. that ¨ Ž t . s ¨ Ž 0 . exp B Ž eyb t y 1 . rb 4 s a2 exp B Ž eyb t y 1 . rb 4 .
Let a s Ž p y 1.rŽ p y 2. - 0. From Ž4.2. we have ¨ Ž t . s b 2 exp aa1 1 y Ž 1 q t .
½
1ra
5.
Ž 4.5.
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A CHEMICAL REACTION MODEL
We first verify that for some suitable b ) 0 and B ) 0, u satisfies u9 F yu¨ y u k w r .
Ž 4.6.
Direct computation shows that Ž4.6. holds if and only if b G a2 exp B Ž eyb t y 1 . rb 4 q B ky1 eybŽ ky1.t w r .
Ž 4.7.
Because w Ž t . F a3 and eyb t y 1 F 0, it is obvious that if b G a2 q B ky 1 a3r ,
Ž 4.8.
then Ž4.7. holds and so does Ž4.6.. Since k G 2, it follows that for any given 1 - p - 2, there exist 0 - B F b1 and b ) 0 such that Ž4.8. holds. Therefore, Ž4.6. holds. Secondly, we verify that for some suitable 1 - p - 2. u satisfies uX G yu¨ y u k w r .
Ž 4.9.
By direct computation we have that Ž4.9. holds if and only if 1 py1
F b 2 Ž 1 qt . exp aa1 1 yŽ 1 qt .
½
1ra
5 qa
ky1 1
Ž 1 qt .
Ž pyk .r Ž py1 .
w r.
Ž 4.10. Therefore, Ž4.10. holds if 1r Ž p y 1 . F b 2 Ž 1 q t . exp aa1 1 y Ž 1 q t .
½
1ra
5 J f Ž y. ,
Ž 4.11.
where y s Ž1 q t .
1r Ž py1 .
G 1,
f Ž y . s b 2 y py1 exp aa1 Ž 1 y y py2 . 4 .
By direct computation we have f 9 Ž y . s Ž p y 1 . b 2 y py 2 exp aa1 Ž 1 y y py 2 . 4 y b 2 y py 1 exp aa1 Ž 1 y y py 2 . 4 aa1 Ž p y 2 . y py 3 s Ž p y 1 . y py 2 exp aa1 Ž 1 y y py2 . 4 b 2 1 y a1 y py2 . It follows that f 9Ž y . - 0 when y py 2 G 1ra1 , and f 9Ž y . ) 0 when y py 2 1ra1. This shows that f Ž a11rŽ2yp. . is the minimum of f Ž y .. If the condition Ži. of Theorem 2 holds, i.e., a1 ) 1 and a1 b 2 exp 1 y a14 G 2, then f Ž y . G f Ž a1rŽ2yp. . s b2 a1ya exp aŽ a1 y 1. 4 . 1
650
JIANG, WANG, AND XIE
From this we see that if 1r Ž p y 1 . F b 2 a1ya exp a Ž a1 y 1 . 4 ,
Ž 4.12.
then Ž4.11. holds. Choosing p s 3r2, we have a s Ž p y 1.rŽ p y 2. s y1 and 1rŽ p y 1. s 2. The condition a1 b 2 exp 1 y a14 G 2 implies that Ž4.12. holds. Thus Ž4.9. follows. If the condition Žii. of Theorem 2 holds, i.e., a1 F 1 and b 2 ) 1, then f 9Ž y . G 0 because y G 1. Therefore f Ž y . G f Ž1. s b 2 . It follows that by b 2 ) 1, there exists 1 - p - 2 such that 1rŽ p y 1. F b 2 . Hence Ž4.11. and, therefore, Ž4.9. hold true. Equations Ž4.1. ] Ž4.4. and inequalities Ž4.6., Ž4.9. show that Ž u, ¨ , w ., and Ž u, ¨ , w . are coupled upper and lower solutions of Ž1.1. ] Ž1.5.. So the solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. satisfies u Ž t . F u Ž x, t . F u Ž t . ,
¨ Ž t . F ¨ Ž x, t . F ¨ Ž t . ,
and
w Ž t . F w Ž x, t . F w Ž t . . Because ¨ Ž t . ª b 2 exp aa14 ) 0 as t ª q` Žsee Ž4.5.., it yields ¨` ) 0. The proof is completed. Proof of Theorem 5. Take p g Ž m, m q 1., then m - p - k. Define u Ž t . s a1 Ž 1 q t .
1r Ž1yp .
uŽ t . s B Ž 1 q t .
,
1r Ž1ym .
,
where 0 - B F b1 is to be determined. Since m - p, we have uŽ t . G uŽ t .. Let ¨ Ž t ., ¨ Ž t ., w Ž t ., w Ž t . be solutions of the following initial value problems, respectively, ¨ 9 s yu m ¨ n s yB m Ž 1 q t .
mr Ž1ym .
¨ Ž 0 . s a2 ,
Ž 4.13.
¨ n;
¨ Ž 0. s b2 ,
Ž 4.14.
wr;
w Ž 0 . s a3 ,
Ž 4.15.
w Ž 0. s b3 ,
Ž 4.16.
¨ 9 s yu m ¨ n s ya1m Ž 1 q t .
mr Ž1yp .
w9 s yu k w r s yB k Ž 1 q t .
kr Ž1ym .
w9 s yu k w r s ya1k Ž 1 q t .
kr Ž1yp .
¨ n;
wr;
where 9 s drdt. It is obvious that ¨ Ž t . F a2 and w Ž t . F a3 . By Ž4.14. we have ¨ Ž t . s b1yn q Ž n y 1 . u a1m 1 y Ž 1 q t . 2
½
y1r u
5
1r Ž1yn .
,
where u s Ž p y 1.rŽ m q 1 y p .. The following proof is similar to that of Theorem 2. We first verify that for some suitable 0 - B F b1 , u satisfies u9 F yu m ¨ n y u k w r .
Ž 4.17.
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A CHEMICAL REACTION MODEL
By direct computation we know that Ž4.17. is equivalent to 1r Ž m y 1 . G B my1 ¨ n Ž t . q B ky1 Ž 1 q t .
Ž myk .r Ž my1 .
w r Ž t . . Ž 4.18.
Since 1 - m - k, n, r G 1, ¨ Ž t . F a2 , and w Ž t . F a3 , it follows that Ž4.18. holds if 1r Ž m y 1 . G B my 1a2n q B ky 1a3r .
Ž 4.19.
It is obvious that Ž4.19. holds if B ) 0 is small since 1 - m - k. Hence Ž4.17. holds for small B ) 0. In the following we will verify that for some suitable m - p - m q 1, u satisfies u9 G yu m ¨ n y u k w r .
Ž 4.20.
By direct computation we see that Ž4.20. holds if and only if 1r Ž p y 1 . F a1my1 Ž 1 q t .
a
½b
q a1ky1 Ž 1 q t .
1yn 2
q Ž n y 1 . u a1m 1 y Ž 1 q t .
Ž pyk .r Ž py1 .
y1r u
wr,
5
b
Ž 4.21.
where a s Ž p y m.rŽ p y 1., b s nrŽ1 y n.. Because w Ž t . G 0, it follows that Ž4.21. holds if 1r Ž p y 1 . F a1my1 Ž 1 q t .
a
½b
1yn 2
q Ž n y 1 . u a1m 1 y Ž 1 q t .
y1r u
J f Ž y. ,
5
b
Ž 4.22.
where y s Ž1 q t .1rŽ py1. G 1 and f Ž y . s a1my1 y pym b1yn q Žn y 2 1.u a1m Ž1 y y py my1 .4 b. After some computations, we have f 9 Ž y . s a1my1 y pymy1 b1yn q Ž n y 1 . u a1m Ž 1 y y pymy1 . 2
by1
n = Ž p y m . b1y q Ž n y 1 . u a1m Ž 1 y y pymy1 . 2
yn Ž p y 1 . a1m y py my1 4 , Ž 4.23. and n q Ž n y 1 . u a1m Ž 1 y y pymy1 . y n Ž p y 1 . a1m y pymy1 Ž p y m . b1y 2 n s Ž p y m . b1y q Ž p y m . Ž n y 1 . u a1m 2
y Ž p y m . Ž n y 1 . u a1m q n Ž p y 1 . a1m y pymy1 .
Ž 4.24.
652
JIANG, WANG, AND XIE
If Ži. of Theorem 5 holds, there exists p g Ž m, m q 1. such that n G n Ž p y 1 . a1m , Ž p y m . b1y 2
1 F Ž p y 1 . a1my 1 b 2n .
Ž 4.25.
Since y py my1 F 1, it yields f 9Ž y . G 0 Žby Ž4.23.. and the first inequality of Ž4.25.. Thus f Ž y . G f Ž1. s a1my 1 b 2n. This inequality and the second one of Ž4.25. imply that Ž4.22. holds, and hence Ž4.20. holds. If Žii. of Theorem 5 holds, we set Ms s
n q Ž p y m . Ž n y 1 . u a1m Ž p y m . b1y 2 n Ž p y 1 . a1m q Ž p y m . Ž n y 1 . u a1m n a Ž m q 1 y p . b1y q Ž n y 1 . Ž p y 1 . a1m 2
Ž n q m y p . a1m
.
n It follows from the first inequality of Žii. that nŽ p y 1. a1m ) Ž p y m. b1y 2 Ž Ž .. Ž . and M - 1 because a - 1rm for all p g m, m q 1 . By 4.23 and Ž4.24. we have that f 9Ž y . - 0 if y py my1 ) M. f 9Ž y . ) 0 if y py my1 - M. This shows that f Ž M 1rŽ pymy1. . is the minimum of f Ž y .. Therefore,
f Ž y . G f Ž M 1rŽ pymy1. . s a1my 1 M Ž pym.rŽ pymy1. b
= b1yn q Ž n y 1 . u a1m Ž 1 y M . 4 . 2 By the expression of M, we have n q Ž n y 1 . u a1m Ž 1 y M . . nMa1mra s n Ž p y 1 . Ma1mr Ž p y m . s b1y 2
So f Ž M 1rŽ pymy1. . s a1my 1 M Ž pym.rŽ pymy1. nMa1m ra 4 b. It follows that Ž4.22. holds provided that 1 py1
b
F a1mq1qm b M Ž pym.rŽ pymy1.q b Ž nra . .
Ž 4.26.
By using the relation M Ž pym.rŽ pymy1. s Ž p y m. =
½
Ž pym .r Ž pymy1 .
q Ž n y 1 . Ž p y 1 . a1m Ž m q 1 y p . b1yn 2 Ž p y 1 . Ž m q n y p . a1m
Ž pym .Ž pymy1 .
5
653
A CHEMICAL REACTION MODEL
and the L’Hopital rule we see that lim M
Ž pym.rŽ pymy1.
pymq1
s e exp
½
n ma1m y b1y 2
m Ž n y 1 . a1m
5
.
From this and lim py mq1 M s 1, we see that the right hand side of Ž4.26. converges to
½
a1my1qm b exp 1 q
n ma1m y b1y 2
m Ž n y 1 . a1m
s a1my1qm b exp
½
5
Ž mn .
n mna1m y b1y 2
m Ž n y 1 . a1m
5
b
b Ž mn . .
From Žii. of Theorem 5, it follows that there exists p: p - m q 1 and near to m q 1, such that Ž4.26. holds, so do Ž4.22. and Ž4.20.. Equations Ž4.13. ] Ž4.16. and the inequalities Ž4.17. and Ž4.20. show that Ž u, ¨ , w . and Ž u, ¨ , w . are coupled upper and lower solutions of Ž1.1. ] Ž1.5.. So the solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. satisfies u Ž t . F u Ž x, t . F u Ž t . ,
¨ Ž t . F ¨ Ž x, t . F ¨ Ž t . ,
w Ž t . F w Ž x, t . F w Ž t . . n Because ¨ Ž t . ª b1y q wŽ p y 1.Ž n y 1.rŽ m q 1 y p .x a1m 41rŽ1yn. ) 0 as 2 t ª q`, we get ¨` ) 0. Theorem 5 is proved.
Remark. Theorems 3 and 4 can be proved in the same way as that of Theorem 2. For simplicity, we give an outline of the proofs here. For the proof of Theorem 3, we choose 1 - p - 2 and let u Ž x, t . s u Ž t . s a1 Ž 1 q t .
1r Ž1yp .
u Ž x, t . s u Ž t . s Beyb t .
,
Also let ¨ Ž t ., ¨ Ž t ., w Ž t ., and w Ž t .a be the solutions of the following initial value problems respectively, ¨ 9 Ž t . s yu Ž t . ¨ n Ž t . s yBeyb t ¨ n Ž t . ; ¨ 9 Ž t . s yu Ž t . ¨ n Ž t . s ya1 Ž 1 q t .
1r Ž1yp .
¨ Ž 0 . s a2 ,
¨ nŽ t. ;
w9 Ž t . s yu k Ž t . w r Ž t . s yB k exp ykbt 4 w r Ž t . ; w9 Ž t . s yu k Ž t . w r Ž t . s ya1k Ž 1 q t .
kr Ž1yp .
wrŽ t. ;
¨ Ž 0. s b2 ,
Ž 4.27. Ž 4.28.
w Ž 0 . s a3 , Ž 4.29. w Ž 0. s b3 .
Ž 4.30.
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JIANG, WANG, AND XIE
Then Ž uŽ t ., ¨ Ž t ., w Ž t .. and Ž uŽ t ., ¨ Ž t ., w Ž t .. are the coupled upper and lower solutions of Ž1.1. ] Ž1.5. for a suitable b ) 0 and 0 - B F b1. Define a s Ž p y 1.rŽ p y 2. - 0. Then by Ž4.28., we have ¨ Ž t . s b1yn y Ž n y 1 . aa1 1 y Ž 1 q t . 2
½
ª b1yn y Ž n y 1 . aa1 4 2
1r Ž1yn .
ya y1
5
1r Ž1yn .
)0
as t ª q`. This yields ¨` ) 0. For the proof of Theorem 4, we choose p: p - m q 1 and near m q 1, and let u s a1 Ž 1 q t .
1r Ž1yp .
u ss B Ž 1 q t .
,
1r Ž1ym .
.
Also let ¨ Ž t ., ¨ Ž t ., w Ž t . be the solutions of the following initial value problems respectively, mr 1ym
¨ Ž t. ;
¨ Ž 0 . s a2 ,
Ž 4.31.
mr Ž1yp .
¨ Ž t. ;
¨ Ž 0. s b2 ,
Ž 4.32.
¨ 9 Ž t . s yu m Ž t . ¨ Ž t . s yB m Ž 1 q t . ¨ 9 Ž t . s yu m Ž t . ¨ Ž t . s ya1m Ž 1 q t .
w Ž t . s yu k Ž t . w r Ž t . s yB k Ž 1 q t .
kr Ž1ym .
wrŽ t. ;
w Ž 0 . s a3 ,
Ž 4.33. w9 Ž t . s yu k Ž t . w r Ž t . s ya1k Ž 1 q t .
kr Ž1yp .
wrŽ t. ;
w Ž 0. s b3 .
Ž 4.34. Then Ž uŽ t ., ¨ Ž t ., w Ž t .. and Ž uŽ t ., ¨ Ž t ., w Ž t .. are the coupled upper and lower solutions of Ž1.1. ] Ž1.5. for the suitable 0 - B F b1 and m - p m q 1. Denote c s Ž m q 1 y p .rŽ p y 1. ) 0. Then we have, by Ž4.32., ¨ Ž t . s b 2 exp a1m Ž 1rc .
½
c
Ž1 q t . y 1
5 ªb
2
exp ya1m Ž 1rc . 4 ) 0
as t ª q`. It yields ¨` ) 0. APPENDIX In this appendix, we will demonstrate that when k G m q 1, even ŽH 2 . is valid and E - 0, u` ) 0 cannot still be assured without any other additional conditions on the initial data u 0 Ž x ., ¨ 0 Ž x ., and w 0 Ž x .. For example, when m s n s r s 1 and k s 2, choose a g Ž0, 1. such that
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ae - 1, a2 e - a, and let n 0 : 0 - ¨ 0 F a y a2 e. Consider the Cauchy problem
¡u9Ž t . s yu¨ y u w, 2
~¨ 9 Ž t . s yu¨ ,
x g V , t ) 0,
Ž A1.
w9 Ž t . s yu w, u Ž 0 . s a, ¨ Ž 0 . s ¨ 0 ) 0, 2
¢
w Ž 0 . s ae.
It is easy to see that Ž u, ¨ , w . and Ž u, ¨ , w . are the coupled upper and lower solutions of ŽA1., respectively, with the expressions a a uŽ t . s , uŽ t . s , 2 1 q at 1qa t ¨ Ž t . s ¨ 0 Ž 1 q at .
y1
¨ Ž t . s ¨ 0 Ž 1 q a2 t .
,
w Ž t . s w 0 exp a Ž 1 q at . w Ž t . s w 0 exp Ž 1 q a2 t .
½
y1
y1
y1 ra
F ¨ Ž t. ,
y a4 ,
y 1 F wŽ t. .
5
So the solution Ž u, ¨ , w . of problem ŽA1. satisfies u F u F u,
¨ F¨ F¨,
w F w F w.
Because ¨ ª 0 as t ª q`, and ¨ Ž t . G 0 we have ¨ Ž t . ª 0 as t ª q`. If we take the initial values u 0 Ž x . ' a, ¨ 0 Ž x . ' ¨ 0 , and w 0 Ž x . ' ae in Ž1.1. ] Ž1.5., then the solution of Ž1.1. ] Ž1.5. is also the solution of problem ŽA1. and vice versa. So the solution Ž u, ¨ , w . of Ž1.1. ] Ž1.5. satisfies ¨ Ž x, t . ª 0 as t ª q`, i.e., ¨` s 0. ACKNOWLEDGMENT We are very grateful to Professor C. V. Pao for his many helpful suggestions and comments which improved our presentation.
REFERENCES 1. E. O. Frind et al., Simulation of nitrate and sulfate transport and transformation in the Fuhrberger Feld Aquifer, in ‘‘Contaminant Transport in Ground Water’’ ŽH. E. Kobus and W. Kinzelbach, Eds.., pp. 97]104, Proceedings of the International Symposium on Contaminant Transport in Ground Water, Stuttgart, April 4]6, 1989. 2. J. Grossman and B. Merkel, One-dimensional simulation of the impact of nitrogen fertilizers on the Carbonate Equilibrium, in ‘‘Contaminant Transport in Ground Water’’ ŽH. E. Kobus and W. Kinzelbach, Eds.., pp. 163]169, Proceedings of the International Symposium on Contaminant Transport in Ground Water, Stuttgart, April 4]6, 1989.
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