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Asymptotic behavior of global solutions for some reaction-diffusion systems
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ASYMPTOTIC BEHAVIOR OF GLOBAL SOLUTIONS FQR SOME REACTION-DIFFUSION SYSTEMS Hmoxr HosRINot and Yosmo YAMADA$ fD+utment of Ap$icd Mathematics, Fukuoka Univ@sity, 8-19-1 Nanakuma, Jonan-kJ. Fukuoka 81401; and h
$Dcparhicnt of Mathematics, Waseda University, 341 Ohkubo. shinjuku-ku, Tbkyo 169, Japan
(Received 23 June 1992; received in revised form 29 Aprii 1993; received for publicatbon 4 August 1993) Key words and p/mwes: Reaction-diffusion system, global solution, asymptotic bhavior, convergence.
rate of
1. INTRODUCTION LET n c RN be a bounded’domain with smooth boundary X2. In this paper, we are concerned with the asymptotic behavior of the global solution for the following reaction-diffusion system
u, = d, Au - u”v”,
X E 62, t > 0,
(l-l)
v, = d,Av - umv”,
XESI, t>o,
(1.2)
at4 aa
(P) 1 - = - = 0, av av
xEaa,t>o,
i(.,O) = ug, (v(*,O)
= v,,
x E 0, 8
x E n,
where di , dz > 0, m, n r 1,and a/av denotes the outward normal derivative’on X2. This system stems from a ‘model for an irreversible chemical reaction, for example, neutralization (a reaction between an acid and a base which produces a salt and water).’ In’the case where we consider the model of the simple irreversible chemical reaction mX + nY + I2 ‘(m, n and I E N),’ if U (resp. V) represents the concentration of X (resp. Y), then the system which (U, V) satisfies is as follows U, = dI AU - KmUmV”,
xen,
q = $AV
xEIca,t>o,
- KnU”V”,
t>o,
(1.1) (1.2)’
xEan,t>o, W-90) = &I, U-90) =
v,,
where K is a positive constant. Ki/(m+n-r)m -(n-l)/(m+n-l)n-(m-l)/(m+n-l)
x
E
62,
x
E
Q,
If we transform u = knW and u = #mV, where k = h , t en it is easily seen that (P)’ is equivalent to (p) with the above transformation. Moreover, we can deal with the case U, = dI AU k yI Urnp (resp. V, = dz A V - K2 U”V”) instead of (1.1)’ (resp. (1 .q)‘) in general, where KI ,.& i 0 and m,n(ER) r 1. 639
H. H-o
640
and Y. YAMADA
We assume that uo, u. E Lm(Q), uo, u. r 0 and uo, tr, + 0. The existence and uniqueness result of a solution (u, u) for (P) on 62 x [0, T& is well known, see e.g. Henry [l] (see also Hollis et al. [2, proposition 11). Because of the assumptions on u. and u,, the maximum principle implies that 0 s U(X,t) s ~~~,,ll~~~a, and 0 s u(x, t) s IIuellLmca,for all x E G and OstsT_, so that there exists a unique global solution (u, u) for(P), i.e. T,, = ao. It is expected that (u(t), u(t)) converges to one equilibrium solution in some sense, say, in L!‘(a) of C(G), as t + ao. Our purpose is to decide a limit of (u(l), u(i)) as I + a0 and to derive a sharp rate of convergence. In the case where ~u&l~e, r ]Iu~~~I~~, , it will be proved that for every P E [O, 2), u(r) --) u, - l~I-‘(lluoll~q~~ - IbolLq~~) and v(t) -, 0 in CWi) as t + =, where 1~1 stands for the volume of Q (in the case of (P)‘, if we suppose jjI.Joj]L~or,2 0, then U(t) -+ U, = Inl-‘([l~~I(~~(u) - (m/n)ll~~ll~l~u,) and V(t) + 0 in WNII vollL’(rl) C“@) as t -+ a). In addition, it will be shown that if u, > 0 and n = 1, then the convergencerate of (u(t), v(t)) is given by an appropriate exponential function and if u, > 0 and n > 1, then (u(l), u(t)) converges with polynomial order. Here it should be observed that the linearized operator around (u,, 0) contains zero as an eigenvalue. Therefore, we cannot apply the existing result (say, Henry [l] or Kielhiifer [3]) to get the exponential convergence. As to the asymptotic behavior of global solutions for a class of reaction-diffusion systems, Conway et al. [4] have given a sufficient condition which assures that every global solution U(x, i) for some parabolic system with homogeneous Neumann boundary condition on XJ tends uniformly and exponentially to its spatial average over 0, i.e. to -+ ao (see also Smaller [5, Chapter 141). Roughly speakD(t) = IGl-‘$e U(x, t)dxinL”(62)asr ing, this convergence holds under the hypothesis that the effect of the diffusion is sufficiently large relative to that of the reaction and there exists an invariant region R in the sense that if U(0) E R, then U(t) E R for all t > 0. However, such conditions are too restrictive and their result does not give us satisfactory information. We state our strategy. We treat integral equations which are equivalent to (P) and use Sobolev’s imbedding theorem in order to investigate the convergence property of (u(t), v(t)) in C*(fi) for every p E [0,2). As regards the rate of convergence when u., > 0, we first consider the decay-rate of Il~ll~~~from the decay-rate of IIu(t) - u,l/c* a) and we make the estimate of Iju(t) - I(,))~,(~~ better with use of the improved estimate of \lu(t)]ltPcaj. In order to study the convergence-rate of I[u(&~~Q, we analyze the integral equation which is equivalent to (1.2) and apply Sobolev’s imbedding theorem. This paper is organized as follows. Section 2 is devoted to assumptions and preliminaries. We state our results in Section 3: theorem 3.1 (convergence properties in C”(fi) for every cr E [O, Z)), theorem 3.2 (the case of the convergence with exponential order) and theorem 3.3 (the case of the convergence with polynomial order). We prove theorem 3.1 in Section 4, theorem 3.2 in Section 5 and theorem 3.3 in Section 6. Finally, we give a few concluding remarks in Section 7. 2. ASSUMPTIONS
AND PRELIMINARIES
In this section we give assumptions, notation, definitions and preliminary lemmas.
Assumption 2.1. (P.l) m, n tz Rand 1)1,n 2 1. (P.2) uo, u. E L=(SQ, uo, u, L 0 and u, + 0, u, + 0.
641
Reaction-diffusionsystem
J
u(x, t) dx -
D
s
u(x, t) dx = E.
I)
Proof. It is sufficient to note d U(X,t) dx - ~*vy)dx~=0. z Is a
n
j
Assumption 2.2. E r 0. We denote by 1 the smallest positive eigenvalue of -A boundary condition on 6X
with homogeneous
Neumann
Definition 2.1. For p E (1, a): A, : the realization of -di A with homogeneous Neumann boundary condition in ~?(a), i.e. D(A,) = (w E WZsP(Q); &Gh~l~~ = O] and A,w = -dtAw for w ED(A& BP: the realization of -d, A + a with homogeneous Neumann boundary condition in P(Q) with a fiied positive real number a, i.e. D(B,) = (w E W2*“(C3);&~/&[,a = 0) and BPw = -(d2A - a)w for w E D(B,). Remark 2.1. For p E (1, oo), it is well known that -A, (resp. -BP) becomes a sectorial operator (see Henry [l] for the sectorial operator); so that -A,, (resp. -BP> generates an analytic semigroup (e-“p)t rO (resp. ]e -‘Bp]Iz,J. We can also define the fractional ipower BP of B,, for every y r 0 (see, e.g. Henry [l]). Definition 2.2.
Q+ = I-
Qo-
Remark 2.2. The operator Q. (resp. Q+) is the projection operator onto the eigenspace corresponding to the principal eigenvahte 0 (resp. positive eigenvahtes) for -d,A with homogeneous Neumann boundary condition on a. Definition 2.3. Ap+ = A,,lQ+rpcn,, i.e. the restriction of A, onto Q+P(n)l. Remark 2.3. It is well known that Ap+ generates an analytic semigroup (e7fA,+Jtz0 in Q+P(@ and it is possible to define the operator A;+ for every y r 0.
642
H. Hoim~o and Y. YAMADA
LEMMA2.2. For every y E [0, l), there exists a constant C&J) > 0 (i = 1,2) such that for w E L!‘(a) with p E (1, co),
!A:+ e-tip+Q+ d~cn,
e-dlxtiIQ+ wlbcn,,
-< Ch9dO-Y
lb,’e-‘B41~toj 5 G(Mt)-Y
e-‘7I 4Lptoj9
t,> 0, t > 0,
(2.1) (2.2)
where q(t) = min(t, 1). For the proof, see Henry [l] or Rothe [a]. Before stating the following lemma, we introduce the space C*@): Co@) (= C(a)) is the Banach space of continuous functions in 0, C’@) is the Banach space of continuous functions whose first-order derivatives are continuous in a and for iu E (0,l) (resp. p E (1,2)) Cr@) means the Banach space of Holder continuous functions with exponent p in fi (resp. the Banach space of C’(fi) functions whose first-derivatives belong to C”-‘(a)). LEMMA2.3. For given ,UE [0,2), let p E (1, ~0) satisfy /A< 2 - N/p
(2.3)
/I < 2o - N/p.
(2.4)
and let a! E (0,l) satisfy
Then the following imbedding relations hold WA,“,) c CW),
(2.5)
D(B,) c Cfi(G).
(2.6)
For the proof of this lemma, see Hemy [ 11. LEMMA2.4. For 8 E [0, 1) and p E R, there exists a constant C,(& p) > 0 such that t q(s)-’ epsds 5
C&% P) eP’,
ifp>O
C&3, O)(t+ l),
ifp=O
Cd&
ifp
s0 i
P),
for all t > 0. The proof is elementary; refer to Hoshino and Yamada [7]. 3. MAIN As stated in the Introduction, solution (u, u) with
RESULTS
under assumption 2.1 we know that there exists a unique global
0 s M-K t) 5 Il~ol]~co,
and
0 5 v(x, 0 5
Il~ollL-(a,
(3.1)
for all x E fi and t 2 0, by the maximum principle. .The following theorem assures the convergence of u and v in C’@) as t + 00 for every cc E to, 2).
643
Reaction-diffusion system THgoRBla
3.1. Under assumptions 2.1 and 2.2, for every p e [O, 2) the global solution (u; u) of (P) satisfies in Cr(&9) as t + m. I@),+ 0 and u(t) + u, Theorem 3.2 (resp. theorem 3.3) asserts the rate of convergence of u and u in the case of n = 1 (resp. n > 1)~and u, > 0. T~BOREM3.2. If n = 1 and u, > 0, then
llw - ‘40ll,,a,5 ‘and
I
if dlA # ~2:
0,
C=P(+m
if d,l
Ct exp(--Ad),
= u,”
Ilvllccc~, 5 Cexp(-dft) as t + ao, where fl,,,
= min(d,A, u,“].
TEKJREM 3.3. If n > 1 and rrk > 0, then
IIu(t>llcqn, 5a lluw - U.&C(~),
ts
ts
4. -PROOF OF THEOREM
First we rewrite (1.2) as u(t) = e-‘%,
+a
e-(t-WJ~~)
&
+ o-“(“-*)
3.1
e-(t-s)Bp(umu”)(s)ds,
-
0
(4.1)
0
where I$, is defined by deftition 2.1 with any fixed a > 0. Let /3 E (0,l) a&d apply Bj to both sides of (4.1). By virtue of (3.1), it is seen, using lemmas 2.2 and 2.4, that
11#40II,,,, 5 a& A 49
t 2 6,
(4.2)
with any 6 > 0. We can also show that t - B$u(t) is uniformly
continuous on [a, 00)
(4.3)
by lemma 3.1 in [7]. Obviously, 1( also satisfies analogous properties to (4.2) and (4.3). Multiply (1.2) by u and integrate the resulting expression over SJ x (0, t); then
II~wIIL~(n,+
26,
t s 0
llw~)l&~*) ds + 2 f ss0
umun+‘(x,s) dxds = ‘llvoll&*).
(4.4)
0
Hence, we see Vu E L2(0, a; L?(n)). In general, we recall that if a function f(t) is uniformly continuous in t and jzf(t) dt is finite, then lim,,,f(t) = 0 follows. Since t - IlVv
H. HOSEINOand Y. YAMADA
644
where v* is independent of x on account of the PoincarC inequality (see [4,8])
Similar results are valid for u. So we may consider that u(Q) + u* in C.(n) asj --, 00, where u* is a constant function in Q. Since umv”” E L’(0, a; L’(G)) by (4.4) and t - (u”v”‘*)(t) is uniformly continuous in L’(Q) on [S,m) by (4.3) (and the corresponding result for u), then lim,,, u”(t)v”+‘(t) = 0 in L’(n). Now the convergence properties shown above imply u+~v*~+~= 0 and lemma 2.1 yields u* - v* = In(-Q = t(,,,. Thus, we see that U* = u, and v* = 0. Because these limits are uniquely determined, we obtain u(t) + u,
and
in C(Q)
v(t) + 0
(4.5)
ast+oo. It is well known that for 0 < (Y< jl < 1
s c 1I~,ti0llg-a,IIv
IIqmllLP(,,
+ 0
as t + 00
(4.6) of IIB~u(t)\~~P,n, by (4.7)
for every CYE (0, 1). Therefore, if we take p E (1,oo) satisfying (2.3) and 01E (0,l) satisfying (2.4), then we obtain limt+aol(u(t)j/Cfi(~j= 0 bet ause of (4.7) and the imbedding relation (2.6). We can also show IIu(t) - u_Jcc(~) + 0 as t -, 00 in the same manner. 5. PROOF
OF THEOREM
3.2
We concentrate upon the case of m = 1. The case m > 1 can be treated almost in the same manner. Because limr__ u(t) = u, in C(a) from theorem 3.1, for any E > 0 there exists a constant T > 0 such that UC0- & I u(t) 5 24, + &
V-1)
for all t L T. Now we take E with 0 < E < u, and we restrict E later. Step 1. Lp(~)-decayof u(t). For p E [l , OD),we multiply both sides of (1.2) by I#‘-’ and integrate over 0. We have (5.1) and it follows from Green’s formula that
s
up-‘Au d.x 5 0;
n
so that it holds that
litit)ll~~c, s IItiT)Ilti~uj expl-(~o. - e)(t - T)] for t z T.
(5.2)
(5.3)
Reaction-diffusion
system
MS
Step 2. Cc@)-decay of u(t) - u,. Note that u satisfies u(t) = e -(‘-Wu(
t
T) -
trT
e-(t-‘M~(uu)(s)ds,
(5.4)
sT from (1.1). Since it holds that QoW
= +,
s
o 4x,
t) cix=
&
s
o v(x, t) dx + u,
from lemma 2.1, we see that (5.5) for t z T by means of (5.3). Now we study the decay of Q+ u(t). If we apply Ai+ Q+ to both sides of (5.4), then we get ApQ+ Q, u(t) = A;+ e-ct-W~+ Q, u(T) -
for t 2 T, where Q is any number in (0,l). we see from (2.1) that
2
:A;+
e-(t--sMp+Q+(uu)(.s)ds
Since it holds that Q, u(T) E qA,“+) (T > 0), then
5 G(O)llA,“+ Q+u(T)bcn,
llA,“+e -(t-T)AP+Q+ u(T)bm for t
s
expI-4l(t
- VI
(5.6)
T. It follows from (2.1), (3.1) and (5.3) that
t ]]A,“+e-(t-s)A p+Q+ (uN) sT
tbca, cb 5 G(dQ+
+tp(o)it uobcn, tb(T)~i~p,,
s t-T
X
thn
q(t - T - s)-” e -dlX(t-T-s)e-(u,-6s *
0
for t
2
T. Here we put t-T IJf)
=
s0
40
-
for t 2 T. When dllZ < u,, we take E with 0 c
T - s)-~ e-
E
d,X(t-T-s)
e-(u,-e)s
h
d,l. Then we see from lemma 2.4 that for
t 2 T, b(t) 5 C&Y, &a - dlL - &)exp{-diL(t - T)].
When dlA
2
u,,
it also follows from lemma 2.4 that I,(t) s C&r, u, - dlA’-
for t
2
T. Therefore,
e) expl-(u..
- e)(t - T)J
combining the above two estimates with (5.6), we get I&+ Q+ u(t)hm
5 Cexpt-Bo(t
- VI,
t
2
T,
(5.7)
H. HOSEINOand Y. YAMADA
644
where /I0 =
:“Y OD- e,
if d,l < u,, if dll z u,.
If we take p and u satisfying the assumptions of lemma 2.3, then we obtain Ilu(t) - U.&~(Q) 5 G(P) expl-M
t 2 T
- 7%
(5.8)
from (5.5), (5.7) and (2.5). Step 3. Sharper estimate of IIv(t)l(fl(oj. We repeat the procedure in step 1 by using (5.8) in place of (5.1) $ IIv(t)ll%(u, 5 -PL
- G ewW%(t
-
t 2 T,
T)llbtO~~G~~~~
with C, = C,(O). Integration of this differential inequality gives
expW& - TM IIWllLqa,s exp(C,/Bo)llu(T)llLPcn,
(5.9)
for t z T. Step 4. Sharper estimate of [[u(t) - u,llcrc~, . Since we have a sharper estimate (5.9) than (5.3), we can repeat the procedure in step 2 to get a better decay-rate for /u(t) - ~,/,~,a,. Clearly, we have only to estimate t-T
&t _ T _ s)-” e-dlW-T-s) e-W &
I(t) = s for t
2
0
T. For dlL # u,, lemma 2.4 yields I(t) 5 C&, %¶ - dlA) exp[-min(d,A,
t
u,j(t - T)],
2
T.
When dl A = u, , we see from lemma 2.4 that I(t) I C&Y, O)(t - T + 1) exp(-u&t
t 1 T.
- T)],
Thus, we obtain 11 u(t) - ‘&?@,
5
c
;~p’;~(;)-e;;~~
(t _ T))
1
,
if u, z dll if u, = d,L
for t 2 T on account of (2.5) with (2.3), (2.4). Step 5. C’(fi)-decay of v(t). We fii a constant a > u,, and write (1.2) in the integral form t
v(t) = e-(‘-nEw(T)
+ a sT
t
e-V-“% v(s) & -
e-(t-s)E$uv)(s) ds
(5.10)
sT
for t 2 T. We apply B; to (5.10). In order to study the decay-rate of ~~v(t)~~cr~~~,it is sufficient to estimate
+-(a) ~I1)(s)~~~~~~ & J(t) = t b,” e-(t-s)EPIIIp(o) sT
t
2
T.
Reaction-diffusionsystem
647
Since a > u,, then (5.9), (2.2) and lemma 2.4 imply that. J(t) 5 G(o)Cs(o,
rk. --a) exp(C,/Bo)IIv(T)II~to,
expl- u-0 - T)J
for t r T. Thus, we obtain [~(t)((c,~a) 4 Cexp(-k&.-
T)),
t r T, ‘.
on account of (2.6) with (2.3), (2.4). Finally, we consider the case m > 1. It follows from theorem 3.1 that .for any e > 0, there exists a positive number T such that
s -P(C - 4Ilu(m&J,~
~llWll?P(~, Therefore,
trT.
the same argument as in the case m = 1 remains valid for m > 1. 6. PROOF
OF THEOREM
3.3
Step 1. L!‘(i&decay of u(t). As in Section 5, it is derived from (1.2), (5.1), (5.2) and Holder’s inequality that
II~
$ for t
2
- ~)mHv(mq.i~l
T, where 0 < E c u,,,. It is standard to see that (6.1) yields
IIw IILW s ((n - l)(u, - e)mlluoll~-~o,
2
(6.1)
(6.2)
l]r’@-‘)
T.
Step 2. Cp(@-decrry of u(t). In steps 2 and 3, we takep E (1, 00) satisfyingl(23) satisfying (2.4). We fm a constant a > 0 and consider, in place of (1.2), r v(t) = e-(‘-~%v(T)
+ a
r e-(‘-“%v(s)&
-
sT
’
e-(t-s%(~m@)(~)& sT
for t 2 T. We apply BP to (6.3). Since we have (3.1) and(6.2),
sT
and Q E (0, 1)
(6.3)
it follows from (2.2) that
IIB,”e+bVlfl(Oj +fi2) IIu(s)IILp(0) do s
C,(ol)lall’Pllv,l(,m~,,(~(t - T) + l)-l’o-*)
s s r-T
X
4(t
_
T
_
s)-a
e-Mr-T-J)
I”‘,:‘:
1)1’@-‘)*
0
for t 2 T, where b = (n - l)(u_ - ~)~llz)~llt;&, . We will prove the uniform boundedness in t 2 T for the following function r-T
K(t) =’
4(t _ T _ &-a e-Mr-T-a)
0
H. H-
648
and Y. YAMADA
For this purpose, we now consider the function -&-T-s)
Y(S)= e
w - n +1 bs
+1
for 0 I s s t - T,where w is any positive number. Because y”(s) > 0 on (0, t - T), we get y(s) 5 mar@(O), y(t - T)] = max[e-““-Q{b(t
- T) + l), 1) 5 C,,
where C, is a positive constant which is independent of t and T. Let 0 c o < (n - 1)a. Then the above calculations yieId Z-T K(t) I cy-1)
q(t -
T - s)-‘* exp[(w/(n - 1) - a)(t - T - s)] ds
s0
by making use of lemma 2.4. Therefore, IIB;tit)ll Lpc-,j5 C[b(t - T) + l)-l’(n-l) for t
2
T by (6.3), which means that Ilv(t)l&a,
I C(b(t - T) + I)-l’(n-l)
for t
2
Step
3. CP(E+decuy of u(t) - u,. If we express
T by means of (2.6).
u(t) - U, = (Qo WI - d
+ Q, W
and we refer to steps 2 and 4 of Section 5 and the calculations in step 2 of this section, then we can obtain the same decaying order of I/u(t) - u,)lcc(~) as for u. Thus, the proof of theorem 3.3 is complete. 7. CONCLUDING
REMARKS
We state two remarks in the case of d, = d2 (= d). In theorems 3.2 and 3.3 the case uQ = 0 is excluded and in theorem 3.2 the rate of convergence of ZJis not known in the critical case d,l = u,“. However, when d, = d,, we can overcome this difficulty. We can get the rate of convergence of (u(t), v(t)) in the case U, = 0 as well as the case U, > 0 with dl A = u,“. In this case w = u - o satisfies the linear heat equation w, = dAw with &v/&&a = 0, so that we have
IIW - 4~~~~5 Ce-&
(7.1)
for t 2 0 on account of j. w(x, 0) dx = E. (1) When a, = 0, it holds that for t > 0 Ilu(t)JI&j:, 5 C(1 + t)-l’(m+“-‘)
(7.2)
((v(t)((.*(Q., 5 C(1 + t)-l’(m+n-l!
(7.3)
and
Reaction-diffusion system
649
In fact, in the case m E N we have n* = (U + w)” = E * u”-kwk, k=o 0 k so from (1.2), (7.1), (5.2) and Holders inequality it is derived that there is a constant C, > 0 such that (7.4) for t > 0. Put g(t) = expt-Gil
-
exp~-~~~l~~~~~lllu(~~ll~~~~.
Then from (7.4) g(t) satisfies the following inequality
dg z
exp[C,(m + n - I)(1 - exp(-d~t)j/(~~~)]g(P+m+“-l)/p 22-PIal -(m+n-l)‘p
for t > 0. Therefore,
(7.5)
by solving (7.9, we get
5 Il~oll~pc~, mW,U IIv(OllLqrI)
exp(-dAt)]/(d@)](bt
+ l)-l’(m+n-l),
(7.6)
where b = (m + n - ~)l~l-~“l”-“‘~llu~~~~~~~~. Next, if we consider (4.1), then we obtain (7.3) from the analogous calculations as in step 2 of Section 6. Furthermore, if we refer to the procedure in step 3 of Section 6 with (7.1) and (7.6), then we get (7.2). Even in the case of general m (2 l), we can show that (7.2) and (7.3) hold because of u = t) + w r 0 and the Taylor expansion. (2) When dA = u,” in theorem 3.2, it holds that
IluW- k&6cij, 5 Cexp(-d!O, for t > 0. Let m = 1 for simplicity. Then we see that
s Cew(--k0, IIv(&qn) 9Ilw - ~JLP~o~ for t r 0 by calculating E’-energy of u(t) and making use of (7.1) (see step 3 of Section 5). Put u, and fm a positive constant a as a > u,. Then zi satisfies u’=u& = (dA - a)fi + au”- u,v - z?v,
t > 0,
so it is enough to estimate fit,
1 q(t _ S)-o e-e-s)
=
e-u”s &,
t>o
s in order to obtain the decay-ra:e of Il~(t)ll - = IIu(t> - uJcr(uB in the same way as in Section 5. Since f(t) 5 C&X, u, - a) exp(-iJ$) for t > 0, we can show IIu(t> - u,IJcr(uj I Cexp(-24-t). REFERENCES 1. HENRYD., Geometric theory of semilinear parabolic equations, Lecture Notes in Muthemutics, Vol. 840. Springer, Berlin (1981). 2. Hours S., MAam R. & F%RRBM.. Global existence and boundedness in reaction-diffusion systems, SIAM .T. math. Analysk 18, 744-761 (1987).
650
H. ma
and Y. YAMADA
3. KIELH&~RH., On the Lyapunov-stability of stationary solutions for semilineq parabolic differential equations, J. diff. Eqns 22, 193-208 (1976). 4. CONWAY
E.. HOFP D. & SYOL~BRJ., Large time behavior of solutions of systems of nonlinear reaction-diffusion
equations, SIAM J. appt. Math. 35, l-18(1978). 5. Tmlrea H.. Eauations crfEvolution. Pitman. London (1979). 6. R~TBB F., Gldbal soluths of reaction-cliffkion syst&s, lecture Notes in Mathematics. Vol. 1072. Springer, Berlin (1984).
7. Hosm~o H. & Y-A Y., Solvability and smoothing effect for semilinear parabolic equations, Funkcicrliqi Ekvacioj 34, 475-494 (1991). 8. SMOG J., Shock Waves and Reaction-Dvfusion Equations, Springer, Berlin (1983).
Pefgamoa
ASYMPTOTIC BEHAVIOR OF GLOBAL SOLUTIONS FQR SOME REACTION-DIFFUSION SYSTEMS Hmoxr HosRINot and Yosmo YAMADA$ fD+utment of Ap$icd Mathematics, Fukuoka Univ@sity, 8-19-1 Nanakuma, Jonan-kJ. Fukuoka 81401; and h
$Dcparhicnt of Mathematics, Waseda University, 341 Ohkubo. shinjuku-ku, Tbkyo 169, Japan
(Received 23 June 1992; received in revised form 29 Aprii 1993; received for publicatbon 4 August 1993) Key words and p/mwes: Reaction-diffusion system, global solution, asymptotic bhavior, convergence.
rate of
1. INTRODUCTION LET n c RN be a bounded’domain with smooth boundary X2. In this paper, we are concerned with the asymptotic behavior of the global solution for the following reaction-diffusion system
u, = d, Au - u”v”,
X E 62, t > 0,
(l-l)
v, = d,Av - umv”,
XESI, t>o,
(1.2)
at4 aa
(P) 1 - = - = 0, av av
xEaa,t>o,
i(.,O) = ug, (v(*,O)
= v,,
x E 0, 8
x E n,
where di , dz > 0, m, n r 1,and a/av denotes the outward normal derivative’on X2. This system stems from a ‘model for an irreversible chemical reaction, for example, neutralization (a reaction between an acid and a base which produces a salt and water).’ In’the case where we consider the model of the simple irreversible chemical reaction mX + nY + I2 ‘(m, n and I E N),’ if U (resp. V) represents the concentration of X (resp. Y), then the system which (U, V) satisfies is as follows U, = dI AU - KmUmV”,
xen,
q = $AV
xEIca,t>o,
- KnU”V”,
t>o,
(1.1) (1.2)’
xEan,t>o, W-90) = &I, U-90) =
v,,
where K is a positive constant. Ki/(m+n-r)m -(n-l)/(m+n-l)n-(m-l)/(m+n-l)
x
E
62,
x
E
Q,
If we transform u = knW and u = #mV, where k = h , t en it is easily seen that (P)’ is equivalent to (p) with the above transformation. Moreover, we can deal with the case U, = dI AU k yI Urnp (resp. V, = dz A V - K2 U”V”) instead of (1.1)’ (resp. (1 .q)‘) in general, where KI ,.& i 0 and m,n(ER) r 1. 639
H. H-o
640
and Y. YAMADA
We assume that uo, u. E Lm(Q), uo, u. r 0 and uo, tr, + 0. The existence and uniqueness result of a solution (u, u) for (P) on 62 x [0, T& is well known, see e.g. Henry [l] (see also Hollis et al. [2, proposition 11). Because of the assumptions on u. and u,, the maximum principle implies that 0 s U(X,t) s ~~~,,ll~~~a, and 0 s u(x, t) s IIuellLmca,for all x E G and OstsT_, so that there exists a unique global solution (u, u) for(P), i.e. T,, = ao. It is expected that (u(t), u(t)) converges to one equilibrium solution in some sense, say, in L!‘(a) of C(G), as t + ao. Our purpose is to decide a limit of (u(l), u(i)) as I + a0 and to derive a sharp rate of convergence. In the case where ~u&l~e, r ]Iu~~~I~~, , it will be proved that for every P E [O, 2), u(r) --) u, - l~I-‘(lluoll~q~~ - IbolLq~~) and v(t) -, 0 in CWi) as t + =, where 1~1 stands for the volume of Q (in the case of (P)‘, if we suppose jjI.Joj]L~or,2 0, then U(t) -+ U, = Inl-‘([l~~I(~~(u) - (m/n)ll~~ll~l~u,) and V(t) + 0 in WNII vollL’(rl) C“@) as t -+ a). In addition, it will be shown that if u, > 0 and n = 1, then the convergencerate of (u(t), v(t)) is given by an appropriate exponential function and if u, > 0 and n > 1, then (u(l), u(t)) converges with polynomial order. Here it should be observed that the linearized operator around (u,, 0) contains zero as an eigenvalue. Therefore, we cannot apply the existing result (say, Henry [l] or Kielhiifer [3]) to get the exponential convergence. As to the asymptotic behavior of global solutions for a class of reaction-diffusion systems, Conway et al. [4] have given a sufficient condition which assures that every global solution U(x, i) for some parabolic system with homogeneous Neumann boundary condition on XJ tends uniformly and exponentially to its spatial average over 0, i.e. to -+ ao (see also Smaller [5, Chapter 141). Roughly speakD(t) = IGl-‘$e U(x, t)dxinL”(62)asr ing, this convergence holds under the hypothesis that the effect of the diffusion is sufficiently large relative to that of the reaction and there exists an invariant region R in the sense that if U(0) E R, then U(t) E R for all t > 0. However, such conditions are too restrictive and their result does not give us satisfactory information. We state our strategy. We treat integral equations which are equivalent to (P) and use Sobolev’s imbedding theorem in order to investigate the convergence property of (u(t), v(t)) in C*(fi) for every p E [0,2). As regards the rate of convergence when u., > 0, we first consider the decay-rate of Il~
AND PRELIMINARIES
In this section we give assumptions, notation, definitions and preliminary lemmas.
Assumption 2.1. (P.l) m, n tz Rand 1)1,n 2 1. (P.2) uo, u. E L=(SQ, uo, u, L 0 and u, + 0, u, + 0.
641
Reaction-diffusionsystem
J
u(x, t) dx -
D
s
u(x, t) dx = E.
I)
Proof. It is sufficient to note d U(X,t) dx - ~*vy)dx~=0. z Is a
n
j
Assumption 2.2. E r 0. We denote by 1 the smallest positive eigenvalue of -A boundary condition on 6X
with homogeneous
Neumann
Definition 2.1. For p E (1, a): A, : the realization of -di A with homogeneous Neumann boundary condition in ~?(a), i.e. D(A,) = (w E WZsP(Q); &Gh~l~~ = O] and A,w = -dtAw for w ED(A& BP: the realization of -d, A + a with homogeneous Neumann boundary condition in P(Q) with a fiied positive real number a, i.e. D(B,) = (w E W2*“(C3);&~/&[,a = 0) and BPw = -(d2A - a)w for w E D(B,). Remark 2.1. For p E (1, oo), it is well known that -A, (resp. -BP) becomes a sectorial operator (see Henry [l] for the sectorial operator); so that -A,, (resp. -BP> generates an analytic semigroup (e-“p)t rO (resp. ]e -‘Bp]Iz,J. We can also define the fractional ipower BP of B,, for every y r 0 (see, e.g. Henry [l]). Definition 2.2.
Q+ = I-
Qo-
Remark 2.2. The operator Q. (resp. Q+) is the projection operator onto the eigenspace corresponding to the principal eigenvahte 0 (resp. positive eigenvahtes) for -d,A with homogeneous Neumann boundary condition on a. Definition 2.3. Ap+ = A,,lQ+rpcn,, i.e. the restriction of A, onto Q+P(n)l. Remark 2.3. It is well known that Ap+ generates an analytic semigroup (e7fA,+Jtz0 in Q+P(@ and it is possible to define the operator A;+ for every y r 0.
642
H. Hoim~o and Y. YAMADA
LEMMA2.2. For every y E [0, l), there exists a constant C&J) > 0 (i = 1,2) such that for w E L!‘(a) with p E (1, co),
!A:+ e-tip+Q+ d~cn,
e-dlxtiIQ+ wlbcn,,
-< Ch9dO-Y
lb,’e-‘B41~toj 5 G(Mt)-Y
e-‘7I 4Lptoj9
t,> 0, t > 0,
(2.1) (2.2)
where q(t) = min(t, 1). For the proof, see Henry [l] or Rothe [a]. Before stating the following lemma, we introduce the space C*@): Co@) (= C(a)) is the Banach space of continuous functions in 0, C’@) is the Banach space of continuous functions whose first-order derivatives are continuous in a and for iu E (0,l) (resp. p E (1,2)) Cr@) means the Banach space of Holder continuous functions with exponent p in fi (resp. the Banach space of C’(fi) functions whose first-derivatives belong to C”-‘(a)). LEMMA2.3. For given ,UE [0,2), let p E (1, ~0) satisfy /A< 2 - N/p
(2.3)
/I < 2o - N/p.
(2.4)
and let a! E (0,l) satisfy
Then the following imbedding relations hold WA,“,) c CW),
(2.5)
D(B,) c Cfi(G).
(2.6)
For the proof of this lemma, see Hemy [ 11. LEMMA2.4. For 8 E [0, 1) and p E R, there exists a constant C,(& p) > 0 such that t q(s)-’ epsds 5
C&% P) eP’,
ifp>O
C&3, O)(t+ l),
ifp=O
Cd&
ifp
s0 i
P),
for all t > 0. The proof is elementary; refer to Hoshino and Yamada [7]. 3. MAIN As stated in the Introduction, solution (u, u) with
RESULTS
under assumption 2.1 we know that there exists a unique global
0 s M-K t) 5 Il~ol]~co,
and
0 5 v(x, 0 5
Il~ollL-(a,
(3.1)
for all x E fi and t 2 0, by the maximum principle. .The following theorem assures the convergence of u and v in C’@) as t + 00 for every cc E to, 2).
643
Reaction-diffusion system THgoRBla
3.1. Under assumptions 2.1 and 2.2, for every p e [O, 2) the global solution (u; u) of (P) satisfies in Cr(&9) as t + m. I@),+ 0 and u(t) + u, Theorem 3.2 (resp. theorem 3.3) asserts the rate of convergence of u and u in the case of n = 1 (resp. n > 1)~and u, > 0. T~BOREM3.2. If n = 1 and u, > 0, then
llw - ‘40ll,,a,5 ‘and
I
if dlA # ~2:
0,
C=P(+m
if d,l
Ct exp(--Ad),
= u,”
Ilv
= min(d,A, u,“].
TEKJREM 3.3. If n > 1 and rrk > 0, then
IIu(t>llcqn, 5a lluw - U.&C(~),
ts
ts
4. -PROOF OF THEOREM
First we rewrite (1.2) as u(t) = e-‘%,
+a
e-(t-WJ~~)
&
+ o-“(“-*)
3.1
e-(t-s)Bp(umu”)(s)ds,
-
0
(4.1)
0
where I$, is defined by deftition 2.1 with any fixed a > 0. Let /3 E (0,l) a&d apply Bj to both sides of (4.1). By virtue of (3.1), it is seen, using lemmas 2.2 and 2.4, that
11#40II,,,, 5 a& A 49
t 2 6,
(4.2)
with any 6 > 0. We can also show that t - B$u(t) is uniformly
continuous on [a, 00)
(4.3)
by lemma 3.1 in [7]. Obviously, 1( also satisfies analogous properties to (4.2) and (4.3). Multiply (1.2) by u and integrate the resulting expression over SJ x (0, t); then
II~wIIL~(n,+
26,
t s 0
llw~)l&~*) ds + 2 f ss0
umun+‘(x,s) dxds = ‘llvoll&*).
(4.4)
0
Hence, we see Vu E L2(0, a; L?(n)). In general, we recall that if a function f(t) is uniformly continuous in t and jzf(t) dt is finite, then lim,,,f(t) = 0 follows. Since t - IlVv
H. HOSEINOand Y. YAMADA
644
where v* is independent of x on account of the PoincarC inequality (see [4,8])
Similar results are valid for u. So we may consider that u(Q) + u* in C.(n) asj --, 00, where u* is a constant function in Q. Since umv”” E L’(0, a; L’(G)) by (4.4) and t - (u”v”‘*)(t) is uniformly continuous in L’(Q) on [S,m) by (4.3) (and the corresponding result for u), then lim,,, u”(t)v”+‘(t) = 0 in L’(n). Now the convergence properties shown above imply u+~v*~+~= 0 and lemma 2.1 yields u* - v* = In(-Q = t(,,,. Thus, we see that U* = u, and v* = 0. Because these limits are uniquely determined, we obtain u(t) + u,
and
in C(Q)
v(t) + 0
(4.5)
ast+oo. It is well known that for 0 < (Y< jl < 1
s c 1I~,ti0llg-a,IIv
IIqmllLP(,,
+ 0
as t + 00
(4.6) of IIB~u(t)\~~P,n, by (4.7)
for every CYE (0, 1). Therefore, if we take p E (1,oo) satisfying (2.3) and 01E (0,l) satisfying (2.4), then we obtain limt+aol(u(t)j/Cfi(~j= 0 bet ause of (4.7) and the imbedding relation (2.6). We can also show IIu(t) - u_Jcc(~) + 0 as t -, 00 in the same manner. 5. PROOF
OF THEOREM
3.2
We concentrate upon the case of m = 1. The case m > 1 can be treated almost in the same manner. Because limr__ u(t) = u, in C(a) from theorem 3.1, for any E > 0 there exists a constant T > 0 such that UC0- & I u(t) 5 24, + &
V-1)
for all t L T. Now we take E with 0 < E < u, and we restrict E later. Step 1. Lp(~)-decayof u(t). For p E [l , OD),we multiply both sides of (1.2) by I#‘-’ and integrate over 0. We have (5.1) and it follows from Green’s formula that
s
up-‘Au d.x 5 0;
n
so that it holds that
litit)ll~~c, s IItiT)Ilti~uj expl-(~o. - e)(t - T)] for t z T.
(5.2)
(5.3)
Reaction-diffusion
system
MS
Step 2. Cc@)-decay of u(t) - u,. Note that u satisfies u(t) = e -(‘-Wu(
t
T) -
trT
e-(t-‘M~(uu)(s)ds,
(5.4)
sT from (1.1). Since it holds that QoW
= +,
s
o 4x,
t) cix=
&
s
o v(x, t) dx + u,
from lemma 2.1, we see that (5.5) for t z T by means of (5.3). Now we study the decay of Q+ u(t). If we apply Ai+ Q+ to both sides of (5.4), then we get ApQ+ Q, u(t) = A;+ e-ct-W~+ Q, u(T) -
for t 2 T, where Q is any number in (0,l). we see from (2.1) that
2
:A;+
e-(t--sMp+Q+(uu)(.s)ds
Since it holds that Q, u(T) E qA,“+) (T > 0), then
5 G(O)llA,“+ Q+u(T)bcn,
llA,“+e -(t-T)AP+Q+ u(T)bm for t
s
expI-4l(t
- VI
(5.6)
T. It follows from (2.1), (3.1) and (5.3) that
t ]]A,“+e-(t-s)A p+Q+ (uN) sT
tbca, cb 5 G(dQ+
+tp(o)it uobcn, tb(T)~i~p,,
s t-T
X
thn
q(t - T - s)-” e -dlX(t-T-s)e-(u,-6s *
0
for t
2
T. Here we put t-T IJf)
=
s0
40
-
for t 2 T. When dllZ < u,, we take E with 0 c
T - s)-~ e-
E
d,X(t-T-s)
e-(u,-e)s
h
d,l. Then we see from lemma 2.4 that for
t 2 T, b(t) 5 C&Y, &a - dlL - &)exp{-diL(t - T)].
When dlA
2
u,,
it also follows from lemma 2.4 that I,(t) s C&r, u, - dlA’-
for t
2
T. Therefore,
e) expl-(u..
- e)(t - T)J
combining the above two estimates with (5.6), we get I&+ Q+ u(t)hm
5 Cexpt-Bo(t
- VI,
t
2
T,
(5.7)
H. HOSEINOand Y. YAMADA
644
where /I0 =
:“Y OD- e,
if d,l < u,, if dll z u,.
If we take p and u satisfying the assumptions of lemma 2.3, then we obtain Ilu(t) - U.&~(Q) 5 G(P) expl-M
t 2 T
- 7%
(5.8)
from (5.5), (5.7) and (2.5). Step 3. Sharper estimate of IIv(t)l(fl(oj. We repeat the procedure in step 1 by using (5.8) in place of (5.1) $ IIv(t)ll%(u, 5 -PL
- G ewW%(t
-
t 2 T,
T)llbtO~~G~~~~
with C, = C,(O). Integration of this differential inequality gives
expW& - TM IIWllLqa,s exp(C,/Bo)llu(T)llLPcn,
(5.9)
for t z T. Step 4. Sharper estimate of [[u(t) - u,llcrc~, . Since we have a sharper estimate (5.9) than (5.3), we can repeat the procedure in step 2 to get a better decay-rate for /u(t) - ~,/,~,a,. Clearly, we have only to estimate t-T
&t _ T _ s)-” e-dlW-T-s) e-W &
I(t) = s for t
2
0
T. For dlL # u,, lemma 2.4 yields I(t) 5 C&, %¶ - dlA) exp[-min(d,A,
t
u,j(t - T)],
2
T.
When dl A = u, , we see from lemma 2.4 that I(t) I C&Y, O)(t - T + 1) exp(-u&t
t 1 T.
- T)],
Thus, we obtain 11 u(t) - ‘&?@,
5
c
;~p’;~(;)-e;;~~
(t _ T))
1
,
if u, z dll if u, = d,L
for t 2 T on account of (2.5) with (2.3), (2.4). Step 5. C’(fi)-decay of v(t). We fii a constant a > u,, and write (1.2) in the integral form t
v(t) = e-(‘-nEw(T)
+ a sT
t
e-V-“% v(s) & -
e-(t-s)E$uv)(s) ds
(5.10)
sT
for t 2 T. We apply B; to (5.10). In order to study the decay-rate of ~~v(t)~~cr~~~,it is sufficient to estimate
+-(a) ~I1)(s)~~~~~~ & J(t) = t b,” e-(t-s)EPIIIp(o) sT
t
2
T.
Reaction-diffusionsystem
647
Since a > u,, then (5.9), (2.2) and lemma 2.4 imply that. J(t) 5 G(o)Cs(o,
rk. --a) exp(C,/Bo)IIv(T)II~to,
expl- u-0 - T)J
for t r T. Thus, we obtain [~(t)((c,~a) 4 Cexp(-k&.-
T)),
t r T, ‘.
on account of (2.6) with (2.3), (2.4). Finally, we consider the case m > 1. It follows from theorem 3.1 that .for any e > 0, there exists a positive number T such that
s -P(C - 4Ilu(m&J,~
~llWll?P(~, Therefore,
trT.
the same argument as in the case m = 1 remains valid for m > 1. 6. PROOF
OF THEOREM
3.3
Step 1. L!‘(i&decay of u(t). As in Section 5, it is derived from (1.2), (5.1), (5.2) and Holder’s inequality that
II~
$ for t
2
- ~)mHv(mq.i~l
T, where 0 < E c u,,,. It is standard to see that (6.1) yields
IIw IILW s ((n - l)(u, - e)mlluoll~-~o,
2
(6.1)
(6.2)
l]r’@-‘)
T.
Step 2. Cp(@-decrry of u(t). In steps 2 and 3, we takep E (1, 00) satisfyingl(23) satisfying (2.4). We fm a constant a > 0 and consider, in place of (1.2), r v(t) = e-(‘-~%v(T)
+ a
r e-(‘-“%v(s)&
-
sT
’
e-(t-s%(~m@)(~)& sT
for t 2 T. We apply BP to (6.3). Since we have (3.1) and(6.2),
sT
and Q E (0, 1)
(6.3)
it follows from (2.2) that
IIB,”e+bVlfl(Oj +fi2) IIu(s)IILp(0) do s
C,(ol)lall’Pllv,l(,m~,,(~(t - T) + l)-l’o-*)
s s r-T
X
4(t
_
T
_
s)-a
e-Mr-T-J)
I”‘,:‘:
1)1’@-‘)*
0
for t 2 T, where b = (n - l)(u_ - ~)~llz)~llt;&, . We will prove the uniform boundedness in t 2 T for the following function r-T
K(t) =’
4(t _ T _ &-a e-Mr-T-a)
0
H. H-
648
and Y. YAMADA
For this purpose, we now consider the function -&-T-s)
Y(S)= e
w - n +1 bs
+1
for 0 I s s t - T,where w is any positive number. Because y”(s) > 0 on (0, t - T), we get y(s) 5 mar@(O), y(t - T)] = max[e-““-Q{b(t
- T) + l), 1) 5 C,,
where C, is a positive constant which is independent of t and T. Let 0 c o < (n - 1)a. Then the above calculations yieId Z-T K(t) I cy-1)
q(t -
T - s)-‘* exp[(w/(n - 1) - a)(t - T - s)] ds
s0
by making use of lemma 2.4. Therefore, IIB;tit)ll Lpc-,j5 C[b(t - T) + l)-l’(n-l) for t
2
T by (6.3), which means that Ilv(t)l&a,
I C(b(t - T) + I)-l’(n-l)
for t
2
Step
3. CP(E+decuy of u(t) - u,. If we express
T by means of (2.6).
u(t) - U, = (Qo WI - d
+ Q, W
and we refer to steps 2 and 4 of Section 5 and the calculations in step 2 of this section, then we can obtain the same decaying order of I/u(t) - u,)lcc(~) as for u. Thus, the proof of theorem 3.3 is complete. 7. CONCLUDING
REMARKS
We state two remarks in the case of d, = d2 (= d). In theorems 3.2 and 3.3 the case uQ = 0 is excluded and in theorem 3.2 the rate of convergence of ZJis not known in the critical case d,l = u,“. However, when d, = d,, we can overcome this difficulty. We can get the rate of convergence of (u(t), v(t)) in the case U, = 0 as well as the case U, > 0 with dl A = u,“. In this case w = u - o satisfies the linear heat equation w, = dAw with &v/&&a = 0, so that we have
IIW - 4~~~~5 Ce-&
(7.1)
for t 2 0 on account of j. w(x, 0) dx = E. (1) When a, = 0, it holds that for t > 0 Ilu(t)JI&j:, 5 C(1 + t)-l’(m+“-‘)
(7.2)
((v(t)((.*(Q., 5 C(1 + t)-l’(m+n-l!
(7.3)
and
Reaction-diffusion system
649
In fact, in the case m E N we have n* = (U + w)” = E * u”-kwk, k=o 0 k so from (1.2), (7.1), (5.2) and Holders inequality it is derived that there is a constant C, > 0 such that (7.4) for t > 0. Put g(t) = expt-Gil
-
exp~-~~~l~~~~~lllu(~~ll~~~~.
Then from (7.4) g(t) satisfies the following inequality
dg z
exp[C,(m + n - I)(1 - exp(-d~t)j/(~~~)]g(P+m+“-l)/p 22-PIal -(m+n-l)‘p
for t > 0. Therefore,
(7.5)
by solving (7.9, we get
5 Il~oll~pc~, mW,U IIv(OllLqrI)
exp(-dAt)]/(d@)](bt
+ l)-l’(m+n-l),
(7.6)
where b = (m + n - ~)l~l-~“l”-“‘~llu~~~~~~~~. Next, if we consider (4.1), then we obtain (7.3) from the analogous calculations as in step 2 of Section 6. Furthermore, if we refer to the procedure in step 3 of Section 6 with (7.1) and (7.6), then we get (7.2). Even in the case of general m (2 l), we can show that (7.2) and (7.3) hold because of u = t) + w r 0 and the Taylor expansion. (2) When dA = u,” in theorem 3.2, it holds that
IluW- k&6cij, 5 Cexp(-d!O, for t > 0. Let m = 1 for simplicity. Then we see that
s Cew(--k0, IIv(&qn) 9Ilw - ~JLP~o~ for t r 0 by calculating E’-energy of u(t) and making use of (7.1) (see step 3 of Section 5). Put u, and fm a positive constant a as a > u,. Then zi satisfies u’=u& = (dA - a)fi + au”- u,v - z?v,
t > 0,
so it is enough to estimate fit,
1 q(t _ S)-o e-e-s)
=
e-u”s &,
t>o
s in order to obtain the decay-ra:e of Il~(t)ll - = IIu(t> - uJcr(uB in the same way as in Section 5. Since f(t) 5 C&X, u, - a) exp(-iJ$) for t > 0, we can show IIu(t> - u,IJcr(uj I Cexp(-24-t). REFERENCES 1. HENRYD., Geometric theory of semilinear parabolic equations, Lecture Notes in Muthemutics, Vol. 840. Springer, Berlin (1981). 2. Hours S., MAam R. & F%RRBM.. Global existence and boundedness in reaction-diffusion systems, SIAM .T. math. Analysk 18, 744-761 (1987).
650
H. ma
and Y. YAMADA
3. KIELH&~RH., On the Lyapunov-stability of stationary solutions for semilineq parabolic differential equations, J. diff. Eqns 22, 193-208 (1976). 4. CONWAY
E.. HOFP D. & SYOL~BRJ., Large time behavior of solutions of systems of nonlinear reaction-diffusion
equations, SIAM J. appt. Math. 35, l-18(1978). 5. Tmlrea H.. Eauations crfEvolution. Pitman. London (1979). 6. R~TBB F., Gldbal soluths of reaction-cliffkion syst&s, lecture Notes in Mathematics. Vol. 1072. Springer, Berlin (1984).
7. Hosm~o H. & Y-A Y., Solvability and smoothing effect for semilinear parabolic equations, Funkcicrliqi Ekvacioj 34, 475-494 (1991). 8. SMOG J., Shock Waves and Reaction-Dvfusion Equations, Springer, Berlin (1983).