IIDA, MASATO

YAMADA, YOSHIO

YOTSUTANI, SHOJI

Osaka City University Repository

lida, M. Yamada, Y. and Yotsutani, S.
Osaka J. Math.
29 (1992), 483-495
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR
A MATHEMATICAL MODEL ON CHEMICAL
INTERFACIAL REACTIONS
Dedicated to Professor Hiroki Tanabe on his sixtieth birthday
MASATO ΠDA, YOSHIO YAMADA AND SHOJI YOTSUTANI
(Received September 13, 1991)
1. Introduction
In the present paper we investigate the asymptotic behavior of solutions
for parabolic systems closely related with a chemical interfacial reaction model
which is considered in Yamada and Yotsutani [7]. Let / and / denote the in-
tervals (0, 1) and [0, 1] respectively. Consider an initial boundary value problem
for ui=ui(x, z) (*=1, 2, 3 and (x, #)e/χ[0, oo)):
(P) dx
*e(0,oo),
where at(x) (/=!, 2, 3) are given functions, φt (#) (/=!, 2, 3) are given nonnega-
tive initial data and
1> U2, U3) = —k3 RQ(uly U2, U3) ,
I, U2, U3) = (ttji ^2— lίja) β(Ul9 U2y U3)
with positive constants ki(i=l9 2, 3), positive integers ni(i=\J 2, 3) and a suitable
positive function /?.
The initial boundary value problem (P) models chemical reactions on in-
terfaces. Such a model has been proposed by Kawano et al. [3]. They put
α,(«) = c{(\-#) (t=l,2,3),
484 M. IIDA, Y. YAMADA AND S. YOTSUTANI
. ,
1+Uz+UiUz
i.e., β(ul9 u2y u3) =
l
where Cf(i=l9 2, 3) are positive constants. As to the derivation of (P), see also
Appendix of [7].
Taking account of the chemical background, we impose the following con-
ditions on a{ and Jf?f.(ί=l, 2, 3):
(A) αt.eC°°(/), tf,.>0 on / and af>0 on [0, 1) .
(R.I) There exist an open subset U of R3 and a positive constant S
such that
,00)3,
β(ul, u2, u3) ε C~(C7) and β(uly u2, «3)>0 on U .
(R.2) There exists a positive constant CR such that
for all («!, w2, κ3)e[0, oo)3
These conditions assure the existence and uniqueness of nonnegative global
solutions for (P) by the results of [7].
The purpose of the present paper is to obtain the uniform convergence of
solutions for (P), together with all their derivatives, as %-*<*>. Recently, a
related problem has been discussed by [2], [4] and [6] in a simpler interfacial
reaction model with two components (Rι(uly u2)= kλ ifa u"2, R2(u1,u2)=k2uΐιu22).
We have established a method of constructing infinitely many Lyapunov func-
tions systematically in [2]. We will develop such an energy method. A new
ingredient is to introduce a Lyapunov function of special form fitting in with
the nonlinearity peculiar to chemical reactions.
The organization of this paper is as follows. In §2 we state main results
(Theorems 1, 2 and 3). In §3 we prepare a fundamental lemma which will
be used throughout the paper. In §4 we show the existence and uniqueness
of solutions for (P) and the corresponding stationary problem (SP). Finally,
§5 contains the proof of the uniform convergence of solutions and all their
derivatives as #->°o.
NOTATION
We use the following notation throughout this paper. For any vectors
u=(uι, U2y u3) and v=(vl9 v2, v3), we simply write
CHEMICAL INTERFACIAL REACTIONS 485
u>v if and only if ul>vlίu2>v2 and U3>v3,
u>v if and only if u^v^ U2>v2 and u3>v3,
and abbreviate Rfai, u2, u3) as R^u). For a vector-valued function u=u(x, z)=
(u^x, 0), u2(x, 5:), u3(x, ^)), we denote its derivatives by
/ 9ί+; u Qi+J u fii+J u '
— I 1 2 ^ 3
\dz{dxr WQxt' d
For any vector-valued functions u=u(x)=(u1(x)ί u2(x), u3(x)) and p=p(x)=(pι(x),
P&)>pfc)) withp>0,
ess sup I Uf(x) 1 , 11«|(«, = ess sup | u(x) \
*el «eJ
2
ί
p
2. Main results
Our first theorem is concerned with the existence, uniqueness, regularity
and positivity of solutions for (P).
Theorem 1. In addition to (A), (R.I) and (R.2), assume that φ=(φι, Φ2, Φa)
φ<ΞL~(/)3 0rcJ φ>0onl.
Then the initial boundary value problem (P) to a unique solution u=(u1, u2, u3
C°°(/X(0, oo))3 satisfying the initial condition in L2-sense, i.e.,
Moreover u has the following properties :
(i) u is nonnegative and bounded on I X [0, oo)^
(ii) u satisfies the law of "mass conservation" , i.e.,
(2.1) -\\ui(^z)ai\\1+\\u3(^z)a3\\1 = EiJz^[09oo) (ί = 1,2),
#,- ^3
where
E,. = -f IIΦ. «, llι+-f IIΦ3 βslli (* =1,2),
486 M. I IDA, Y. YAMADA AND S. YOTSUTANI
(iii) the posiΐivity of ulf u2 and u3 is assured by that of E1 and E2, i.e.,
, u2>0 and u3>Q on /X(0, oo) if E^O and E2>0 ,
and u2=u3=0 on /X(0, oo) if j^X) and E2 = 0 ,
and ^=^=0 on 7x(0, oo) if E1 = 0
= 1/3=0 0 Λ / x ( 0 , oo) z/ E1 = Q
In the study of asymptotic properties of (P), we should note (2.1). There-
fore, the stationary problem associated with (P) will be written in the form of
algebraic equations for u°°=(uT, u2,
(SP) = 0 ,
Concerning (SP), we get the following theorem.
Theorem 2. Suppose that (A) and (R.I) hold. Then there exists a uni-
que solution u°°=(uT, u2> u%) of (SP). More precisely,
ι%>0 and ί/Γ>0 if £Ί>0 and \
uT = HΦι*ιllι>o and u2=u,= 0 if ^>0 and E2 = Q,IkilL
"
2
 " Ikl l l > ^ « l-«3- */ I" 2> ,
Wf = u2 = Us = 0 if E! = 0 tfrad £3=0.
Now we are ready to state results on the asymptotic behavior of solutions
for (P)ass->oo.
Theorem 3. Suppose that (A), (R.I) and (R.2) hold. It follows that
(2.3) lim||Z)ίZ)ίιι|U =
£->°°
/or α// nonnegatίve integers iyj with (^7*)Φ(0, 0).
REMARK 2.1. It is possible to derive the rates of the convergence stated
in Theorem 3. Indeed, in the case when £Ί>0 and
\\u( ,z)—u~\\~ = O(exp(-λ+*)) as z -> oo ,
λ+* as » ~> oo ,
CHEMICAL INTERFACIAL REACTIONS 487
where λ+ is the least positive eigenvalue for the linearized operator around u°°
associated with (P). We can also obtain analogous rates of convergence in
the remaining cases. We will prove this fact in the forthcoming paper.
3. Preliminary lemma
The following lemma is very useful to derive several uniform convergence
properties from various estimates of solutions.
Lemma 3.1. For a positive integer m, let {pk(z)}tek*mΛfk(x)}ι<sLkZm,
{?*(#)} o^k^m and {<Jk(z)}^k^m-\ be sequences of nonnegatίve functions of class
Cx([l, °°))J and let SQ(Z) be a nonnegatίve function of class C([l, °°)). Suppose
that there exists a positive constant -η such that
dqk-ι
dz •η
-( *
•η y-i
ΣPj+ Σ <?,.) (k = 2, -, m);y=ι j=o
! ?;) (^ = 2, •"> >W) ,
7.) (A =1,2, •»,«),
(A =1,2, ••,»).
Then it holds that
l k=l
J ew m(Σ1 «r = 0 dp*da Σ*=o dx dz< oo ,
= 0 (*=l,2, , f») ,
= 0 (A = 0, 1,—,«—!).
ar-^oo
Proof. See Lemma 3.2 in lida, Yamada and Yotsutani [2].
4. Existence and uniqueness of solutions for (P) and (SP)
In this section we give the proof of Theorems 1 and 2.
488 M. IIDA, Y. YAMADA AND S. YOTSUTANI
Proof of Theorem 1. It is easy to see that the boundary functions
Ri(uι9 u2y u3) (i=ly 2, 3) satisfy the assumption in Theorem 2 of [7] (see also Ap-
pendix of [7]). Thus the existence of a unique solution and the facts (i), (ii) are
assured by [7] and [8j. Among them, we can show the fact u^C°°(ϊx(09 oo))3
by proving u<Ξ ΠΓ.o Hm~l(δ, γ; H2i+\I)z) for any γ>δ>0 and all integers m
(see, e.g., [8]).
Now we will prove (iii). It follows from Friedman's lemma (a parabolic
version of Hopf's lemma) on the maximum principle, see [1], that
(4.1) Ui(x, *)>0 on 7x(0, oo) or w,.(#,#) = 0 on 7x(0, oo)
for each i=l, 2, 3. In fact, by (R.I) and the nonnegativity of solutions, */,-((},#)
—0 implies D
x
 «,.(0, z)=Rf(u(0, *))^0.
Consider the case where EΊ>0 and E2>0. We will show u(x, #)>0 for
all (xy #) e7x (0, oo) by contradiction. Suppose that ^  = 0 on 7x (0, oo). Then
it follows that
D
x
 1 (^0, z) = 0 for all z e (0, oo ) ,
which, together with the boundary condition at x— 0, implies
1*3(0, *) = 0 for all z e (0, oo ) .
Therefore, by (4.1), we get
u3(x, z) = 0 for all (#, z) e Jx (0, oo) ,
which contradicts £Ί>0; so that (4.1) assures
«ι(#> -sr)>0 on 7χ (0, oo ) .
Similarly we see that
and u3(x, ^)>0 on 7x(0, oo) .
Consequently £Ί>0 and E2>0 imply u>0 on 7x(0, oo). The rest of (iii) is
straightforward from (2.1) and (4.1).
Proof of Theorem 2. Let us consider the case where £Ί>0 and E2>0.
We set N(UI, u2y u3)=u"ι u22—u^. Note that N(ulί u2, u3) satisfies
( N(ul9 0, 0) = 0 for any
N(0,w2,0) = 0 for any u2>0 ,
9Λ/\
 π
 9N r
>0, >0 for any
9^ 9z/2
9Λ7"
——<0 for any W!>0, t/2>0,
9%
CHEMICAL INTERFACIAL REACTIONS 489
We will show the existence and uniqueness of solutions of
/(£) = 0 on [OΛ||«3IIΓlmin£,.],
where
m = * (A (*>-¥* * )• A (E*-Ψ O ?)Nkll i v *3 ' INIi v k2 / /
We see from the above properties of ΛΓ^, w2> ^3) that/(?) satisfies
/(0)>0,
'min £,)<(),
1 = 1-2
<0 for .
dξ f-1.2
Hence there exists a unique real number £„ such that
/(£„) = 0 and ^^(O.^
Consequently the solution of (SP) is given by
~ _ fei (
 F \\OB\\I t\ fy~ — ^Ml
 ~ Γi — ΓΓ V ^ """T — ^o / > U2 — 7T —IkllΛ &3 / Ik
which implies «°°>0.
If £
x
=0 or E2— 0, then (SP) clearly has a unique solution
, , .IkilL Iklli
Thus we complete the proof.
5. Proof of Theorem 3
The following Lyapunov functional plays a crucial role. For vector-
valued functions u=u(x)=(u1(x),u2(x),u3(x)) and v=v(x)=(vl(x), v2(x), v3(x))
with tt>0 and v>0, set
where
Ψ(«, »): = ΣJ ^ Γ1 n,. ( ΨK , »,) α, ^  ,
i = ι J7
, fo): = f t (log f-log fβ) Λ = f (log e-log fβJδo
We give some estimates of solutions for (P) which are essential for the proof of
(2.2).
Proposition 5.1. Let u be the solution of (P). Suppose that £j>0 and
490
E2>0. Then
(5.1)
(5.2)
M. I IDA, Y. YAMADA AND S. YOTSUTANI
, u\\ΐ+€{R0(u(0,
— II-*--* *|| <i\\U
χ
U\\dz
I I A w l l ϊ .^^idlo.ttlli^
' M1 are positive constants independent of z and
g = g(x) = (gι(x)> g2(x), &(*))> k = h(x) = (*!(#), h2(x), kz(x))
i(x) = ai(x)gi(x) ( f = l , 2 , 3 ) .
/or
with
Proposition 5.2. L ί^ w be the solution of (P). Suppose that £Ί>0
£2>0. Then it holds for #e(0, oo)
K2 are positive constants independent of z.
REMARK 5.1. This kind of Lyapunov functional Ψ was employed in Rothe
[5]. He investigated the asymptotic behavior of solutions for the following
reaction-diffusion system
- = D1 Δ Wj+^3—^! w2 >
= D2 Δ W2 +
 W3~^1 ^2 >
= A Δ i
8ί
"97
Quz
Qt
with the homogeneous Neumann boundary condition. This system models
the situation where chemical substances are reacting in a bounded domain.
REMARK 5.2. Equality (5.2) is employed in Shinomiya [6]. The impor-
tant point in (5.2) is that it does not include a term related to the boundary con-
ditions jR, (tt(0, *)) (i= 1, 2, 3).
The following proposition essentially shown in [2] plays a crucial role for
CHEMICAL INTERFACIAL REACTIONS
the proof of (2.3).
Proposition 5.3. Let u be the solution of (P). Then
491
-f
dz
A.
dz
\mu\\l.,
a
^Mk(\\Dlu\\l.,h+\\D*D,u\\2),
for k=2, 3, ••• tf/zJ #^[1, °°), s<;A£r£ M
Λ
 w a positive constant independent of %
and gy h are the functions used in Proposition 5.1.
Before proving the above propositions, we give the proof of Theorem 3
by using them.
Proof of (2.2). Consider the case where £Ί>0 and £2>0. Observe that
the estimates in Proposition 5.1 correspond to the inequalities (with m=l)
in the assumption of Lemma 3.1. Therefore,
(5.3)
in particular, {u( , #)}
z
;>ι is uniformly bounded and equi-continuous. By virtue
of (2.1) and (5.3), Ascoli-Arzela's theorem assures that there exist a sequence
{ar .J yϋ and a consatnt U=(Hl, U2, tt3) such that
= 0 ,
=
 Ei (, =ι,2),
and
It follows from Theorem 2 that U=u°°^(Q, oo)3, which implies
(5.4) lim| |tt( ,*y)-fr||oo = 0.j+°» J
On the other hand, (5.1) in Proposition 5.1 yields
492 M. I IDA, Y. YAMADA AND S. YOTSUTANI
A
ψ(M)ίdz
which leads to
lim Ψ(u( , #), u°°) = Ψ(w°°, u°°) = 0
by (5.4). Hence it follows from Proposition 5.2 that
Since {w( , #)K ι^ is uniformly bounded and equi-continuous, we can derive
by (5.5).
We consider the case where E1= 0 or £"2=0. Without loss of generality we
may assume E1=0. We see from (iii) of Theorem 1 that
u^x, z) = 0 , u3(x, z) = Q on / X [0, oo )
and M2 satisfies
' a2(x) Dz u2 = D2xu2, (x, x) e/χ (0, oo) ,
), w2(0, )^ - D, U2(l, z) = 0,
Thus we can show
lim ==o
in the standard manner for parabolic equations. In fact, we have only to ob-
serve that
-
2 dz
, oo).
Proof of (2.3). As in the proof of (2.2), Propositions 5.1 and 5.3 combined
with Lemma 3.1 yield
]fa\\D>-lDtu\\ = Q (A =1,2,-),
lim||Z)iu||
ί
;. = 0 (A =1,2,-).
CHEMICAL INTERFACIAL REACTIONS 493
In the case £Ί=0 or £"2=0, we can easily get the same results by slight modifica-
tions (e.g., use \\u— u°°\\2
 ; a instead of Ψ(u9 u°°) in (5.1)). Thus we can derive the
uniform convergence of any derivative Z)J D{ u by using imbedding theorems.
Refer to the proof of Theorem 4 in [2] for the detail.
Now we will prove Propositions 5.1 and 5.2.
Proof of Proposition 5.1. We will show (5.1). The other estimates are
obtained in the same way as the proof of Proposition 5.1 in [2]. Let u=
(uly u2, u3) be the solution of (P). Then we have
* =ι dz
= Σ kT1 Hi { (log Uf— log u7) a; Dt u{ dx
ί = l JI
= Σ kT1 n, ^  (log 11,-log «Γ) Dl u, dx
= Σ ^Γ
1
 n, [(log 11,-log «7) Dt «,]!- Σ ^ Γ1 «, ( |Z)*" |2 dx1=1 i = ι JI Ui
= {log («Γ) ι (M?)»2-log («Γ)H R0(u(0, *))
- {log
 Ml(0, *) ι M2(0, *)i_log «3(0, *)•.} Λp(β(0, *))
ί=ι
which, together with (wΓ)*1 (U2)*2=(u3)**, implies
^ Ψ(u, u~)
+ {log ^ (0, ar) ι w2(0, ^)»2-log %(0, #)«3} Λ0(w(0, *)) = 0 .
We note that the following elementary inequality holds:
£ for f,
Consequently we obtain (5.1) from (i) of Theorem 1.
Proof of Proposition 5.2. Noting (i) of Theorem 1, we can reduce this
propopsition to the following fundamental lemma.
Lemma 5.1* Let ξQ and ξ be constants with 0<ξQ<ξ. Then
^ιll-lol2^^(?,?o)^^ll-?ol2 forall |e(0,f),
where
494 M. IIDA, Y. YAMADA AND S. YOTSUTANI
Proof. Consider a function
°) for
for ί=f
-
Since ζ(ξ)eC1((0, °°)) and (dζ)l(dξ)<0 on (0, <χ>), we get
f o r
 fe
For the proof of Proposition 5.3, see §6 of [2].
References
[1] A. Friedman: Remarks on the maximum principle for parabolic equations and its
applications, Pacific J. Math. 8(1958), 201-211.
[2] M. lida, Y. Yamada and S. Yotsutani: Convergence of solutions of a chemical inter-
facial reaction model (to appear in Funkical. Ekvac.).
[3] Y. Kawano, K. Kusano, K. Kondo and F. Nakashio: Extraction rate of acetic acid
by long-chain alkylamine in horizontal rectangular channel, Kagaku Kogaku Ronbunshu
9(1983), 44-51.
[4] T. Nagasawa: The rate of convergence for chemical interfacial reaction models (to ap-
pear in Funkcial. Ekvac.).
[5] F. Rothe: Global Solutions of Reaction-Diffusion Systems, Lee. Notes in Math.
Vol. 1072, Springer-Verlag, Berlin/New York, 1984.
[6] Y. Sinomiya: On the asymptotic stability as t-^co of diffusion equations with non-
linear boundary conditions, Master Thesis, Osaka Univ. (in Japanese) (1990).
[7] Y. Yamada and S. Yotsutani: A mathematical model on chemical interfacial reac-
tions, Japan J. Appl. Math. 7(1990), 369-398.
[8] Y. Yamada and S. Yotsutani: Regularity of solutions for a mathematical model on
chemical interfacial reactions, in preparation.
M. lida
Department of Mathematics
Faculty of Science
Osaka University
Toyonaka, Osaka 560
Japan
Current address
CHEMICAL INTERFACIAL REACTIONS 495
Department of Information Sciences
Tokyo Institute of Technology
Oh-Okayama, Meguro, Tokyo 152
Japan
Y. Yamada
Department of Mathemaitcs
Waseda University
3-4-1 Ohkubo, Shinjuku, Tokyo 169
Japan
S. Yotsutani
Department of Applied Mathematics and
Informatics
Ryukoku University
Seta, Ohtsu 520-21
Japan



Asymptotic behavior of solutions for a mathematical model on chemical interfacial reactions (Dedicated to Professor Hiroki Tanabe on his sixtieth birthday)

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Asymptotic behavior of solutions for a mathematical model on chemical interfacial reactions (Dedicated to Professor Hiroki Tanabe on his sixtieth birthday)

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