- Email: [email protected]
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
h’onlinw Primdin
Am,,yru. ?-hwy. G&at Britain.
Maho&
& A&$icahm.
Vol. 8. No. 10. pp. 1121-1144,
0x2-s46lc’&r $3.00 + .@I @ 198a Pcrgamoo Rss Ltd.
1W.
SMOOTH SOLUTIONS TO A QUASI-LINEAR SYSTEM OF DIFFUSION EQUATIONS FOR A CERTAIN POPULATION MODEL* Department
of Mathematics, (Receioed
JONG UHN KIM Virginia Polytechnic Institute and State University, Blacksbury. VA 21061, U.S.A. 25 September
1983; received for publication 15 November
1983)
Key words andphrases: System of diffusion equations, population model, smooth nonnegative solutions, Sobolevski’s method, local solutions, energy estimates, global solutions.
0. INTRODUCTION THIS PAPER
deals with the initial-boundary
value problem for the system of equations:
ti, = A(ctr2 + ditifi) + (8, - airi - dia)ti (0.1) ’ I where ci, di, Bi, di and di, i = 1, 2, are nonnegative constants. This system of equations describes a model of two competing species with self- and cross-population pressures. Here, zi and ii denote the population densities of the two competing species. For the derivation of (O.l), the reader is referred to [3]. From the physical consideration, ri and 6 should be nonnegative and (0.1) is subject to the Neumann boundary condition: d,= A(c$ + d#fi)
+ (&-
ti,(r,x)
d2zi - &)ii,
= a,(t,x)
= 0
(t,x)
E[O, 00) x [O,l]
at x = 0,l.
(0.2)
For the case when cl > 0, c2 > 0, di > 0 and d2 = 0, the stationary problem associated with (0.1) was discussed in [4]. Also in its introduction, it was announced that Masuda and Mimura have proved the global existence of nonnegative solutions to (0.1) in the above case. In this paper we shall prove the existence of smooth nonnegative solutions to (0.1) with suitably smooth initial data under the assumption that ci > 0, di > 0, i = 1, 2. In Section 1, we establish the local existence of solutions by the method due to Sobolevski, which is well presented in [2]. We employ the function spaces QJ, s 2 0 (see Section l), which enable us to prove the Cm-regularity of solutions for r > 0. Some properties of a, which are necessary in the development of our arguments are proved in the Appendix. In Section 2, we prove that the local solutions can be extended globally on [0, m) under the additional hypothesis that cl = c2 > 0, but without any restriction on the size of initial data. We shall make some remarks on the structure of (0.1). First of all, we see that (0.1) and (0.2) reduce to u,=a(u+~~)+(E~-alu-blu)u u,=A(yv+uu)+(E~-a2u-b2~)u u,(t, x) = uX(t, x) = 0
1 atx = 0, 1,
l This research was supported by the U.S. Army under contract No. Science Foundation under grant No. MCS-7927062 Mod. 2.
1121
’
(0.3) (0.4)
DRAG 29-80-C-0041 and by the National
JONG UHN KIM
1122
through ic(t,x)
=
+c,r..,, 2
C(t, x) =
2
u(c,t. x)
1
CT1
=
I
(0.5)
1
f,
where y = CZ/C~ > 0, Ei 2 0, ai 2 0 and bi L 0, i = 1,2. Throughout this paper we will consider (0.3), (0.4) instead of (O.l), (0.2). It is interesting to observe some unusual features possessed by (0.3). For simplicity, we shall consider , uf = A(u f uu)
(0.6)
uI = A(yu + UU) and the associated nonlinear operator
9’: (0.7)
Then, for smooth nonnegative functions u and u satisfying the Neumann boundary condition, W(Z)? (;)> L~XLzis not nonnegative in general. In fact, it is strictly negative if we take u = cos nr, for example. Hence, we expect to have difficulty in lOO+ y+6cosnr, u=lOobtaining energy estimates to establish global existence of solutions to (0.6). Now the linear operator associated with (0.7) is -(l
+f)Au
- gAu
(W
-fAu-(y+g)Au
where f and g are assumed to be given nonnegative functions. Then it is easy to see that the right-hand side of (0.8) is not a strongly elliptic system in general. This also suggests that the usual procedure to obtain energy estimates may not be effective. However, if y = 1, i.e. cl = ~2, then we can make use of u - u as an intermediary function to obtain necessary energy estimates. This is illustrated in Section 2. Finally we report that the question of asymptotic stability of solutions remains open. In view of the above remarks, it seems hopeless to get a uniform bound of the solution through energy estimates. In the mean time, the structure of Y discourages us from attempting to construct an invariant set.
1. LOCAL
EXISTENCE
As mentioned above, we shall use the method in [2]. Hence, we write equations (0.3) in the form of an abstract evolution equation and verify all the conditions prescribed in the above reference. Let us define the linear operator A,(t, w) as follows: R,u
-
(1 +feR”)Au
- g eRd Au
R,v-feRs’Alc-(y+geRJ’)Av
(1.1)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
where K, is a positive constant which will be determined u * =U eeRJf, u* =u eeRIr, (0.3) is equivalent to
later on and w =(f).
1123
Writing
(1.2) where 2eRI'u,*uf
+
- bleRdu*)u*
+(Ez-a2eR1’u*
-bbeRdu*)v*
(El - aleRI’u*
$7(;:)I=(2eRdu,*vf
From now on, we shall suppress “*” and use both notations(i) vector. For real S, we define
1 ’
(1.3)
and (a, b) to denote the same
(1.4) m
T
and if f
=
c a, cos nnz and R = 2 b, cos nm lie in Cp,. _, we write n=O n=O
(f, g>, = a& + ;
il (1 + n2n2)%.~
(1.5)
n
and
llflls= w>Y*. Obviously, QSl Cas2and II.lls2=Sll-lls,if ~1> ~2.We also MIS +Ilflls7for all (g,f) E X,, and Lf, =
{
fE LZ[O,l]:
(
&
>
“fE LZ[O,l],
(1.6) define X, = QS x QSr Il(g,f)llx,
k=l,...,m.
I
=
(1.7)
ll*l/r& is defined in an obvious way. When X and Y are Banach spaces, we denote by B(X, Y) the set of all bounded linear operators from X into Y. Let f(x), g(x) be real-valued functions in @$+I, s 3 0, such that
llfllr+17 Iklls+ld M < Co and f(x), g(x) 5 max Denote A,(O, (g,f))
for allx E [0, 11.
by A,. Then we have:
PROPOSITION 1.1. There is a number R (s, M) z 1 depending on s, M such that if R, 2 R (s, M), (;U - A,)-’ is a bounded linear operator on X, for all A E C with Re A s 0, and
(1.8)
1124
JONG UH?VKIM
holds where C(s, M) is a positive constant which depends only on s, M and is independent of A, R,. Furthermore, (ti - A,)-’ is a bounded linear operator from X, into XJ+: with il(j_Z- A,)-*~je~xs,x,,,~ s C(s, M),
for all A E C.
Re is
0.
(1.9)
Proof. First we prove the above assertion in the case where s = m is a nonnegative integer with f, g E Q,, where a={tn’ff”m=. Assume Ilfl/a, llglj .C M and f(x), g(x) 1 max(-l/4, x -y/4) for all x E [0, 11. Now we will follow the well-known procedure. Suppose $ =nzo $,, I P I cos nm E Qm, 77=Ezo q, cos nm E O,,,, u =nzo u, cos nm E Qm+l and v = c v, cos njrx n=O E am+2 satisfy the equations: (A - R,)u
+ (1 + f,,)Au + g,,Av = s’
(A-R,)u+foAu+(y+go)Av=r~
(1.10)
1>
where A is a complex number with Re ,J G 0, R, 3 1, and fo, go are constants such that max(-l/4, -y/4). Then, it is easily seen that for all n 2 0,
go2
{ ,I -
R, -
(y + go)n2.z2}s”, +
” = (A - R,)2 - (/I - Rm)(l + y + f.
gG2n2q,,
+ go)n2n2
+ (ygo
+ yfo)n”n”
fo,
(1.11)
and f&c%& ”
= (A -
RJ2
R, - (1 + fo)n%‘}qn + y + f. + go)n2rc2 + (y
+ {A -
- (A - R,)(l
+ go + yfo)n’x”
(1.12)
Now we will estimate 1u,I and I v,/. Setting A = -,u f iv, ,~a 0, vE R, we can rewrite (l.ll), (1.12): (-#u - R, + iv) En - ( y + go)n’n2~n + gon’n2q, u, = ) (l.ll)* (p + R,)2 + (p + R,)(l + y + fo + go)n2d + ( y + go + v, =
yf~)n”n” - Y2- iv{2(p + R,) + (1 + y+
fon2dtn
+ (-p
R, + iv) q,, - (1 +
-
(P + R,)‘+ + (y + go +
(u f R,)(l
fo+
go)n%‘}
fo)n2n2q.
+ y + f.
(1.12)* + go)n2x2
rf)n4n4 - lJ2- iv{2(y + R,,,) + (1 + y + fo
+ g~)nzxz}
In this case, we use the inequality: {(p + R,)
+ (1 + y+
fo + go)n2x2}2
+ (p + R,,,)(l + y + fo to derive
+ g&h2
5 (p + +
R,)’
( y + go + yfo)n4d
r (1.13)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
1125
and (1.14) where C is a positive constant independent
and 1vi2 2 2{(~ + R,)’
Case 2. (p+RR,)+ld(v12 (Y +
g0 +
of A, fo, go, t,,, q,,. n and R,. + (u +R,)(l
+ y+ f. +go)n’d
+
vfo>n’n’l.
Case 3. 1(,~ + R,) s 1VI d p + R,,,. Case 4. 1VI s &(,u + R,). For Cases 2,3,4, it is easy to obtain (1.13), (1.14). Therefore, we conclude that (1.13), (1.14) hold for all A E C, Re A s 0 and all R m 3 1 where C is a positive constant independent of A, fo, go, g,,, qn, n and R,. Next we shall estimate &%‘Iu,l and &‘I ~~1,for n # 0. From (l.ll)*, (1.12)*, we get iiczn2u, =
(-x + iY>En - (Y + go) 5Cn+ gorln
(1.15)
x2+(1+Y+fo+go)x+(Y+go+yfo)-Y’-~y(~+1+y+fo+go)
and ,7’n2v
n
=
foG
+ c-x
+ iY)Vn - (1 +fo)%
(1.16)
x2+(1+Y+fo+go)x+(Y+go+yfo)-Y2-~Y(~+1+y+fo+go)
where x = (p + R,)/n22
> 0, y = v/n22.
Cu.se1.x2+I(y+go+yjo)~y2
“2~x2+(1+Y+fo+go)x+Y+go+~o~.
In this case, we use the y + go + do to derive that
inequality:
(~+l+y+f~+g~)~>
x2+(1+y+fo+go)x+
n2n21unl s C(lEnl+ lqnl>
(1.17)
~n21unI s C(lEnl+ IsI>,
(1.18)
and
where C is a positive constant independent
of A, fo, go, j,, qn, n and R,,,.
Case 2. 2(x2 +(l + y + f. + go)x + y + go + yJo} s y2. Case 3. J S x2 + 1( y + go + tie).
In Cases 2, 3, it is easy to get (1.17), (1.18). Therefore,
we conclude that
llUllm + II”llm s 1~1 +cR,(ll5llm + Iltlllm>
(1.19)
1126
JONG CHN KIM
and II&+?
+ b/l mr? s c(l1511m + II )Illm)
(1.20)
hold for all A E C, all R, 3 1, where C is a positive constant independent of I.. Ii, g,), E, q and Next we suppose u E @‘,_ ?, u E a,,,+*, e E a,,, and 17E (0, satisfy the following equations:
R,.
(n - R,)u
+ (1 +f)Au
+ gAu = s’
(A - R,)o
+fArc + (y+ g)Au
(1.21)
= 77 I
where AEC, ReASO, R,,,sl, and f, g E D0 are real-valued functions satisfying /]f]]D, -y/4) for all x E [0, 11. Let us choose a partition of llugn/lYy;&f and f(x), g( x ) 2 max(-l/4, as follows: , 1, ’ . . 7 Gs} (9
@ia
@iEC”([O,
l]),
,i
on [0, l] and (d/dx)k$l(x)
@iE1
= 0 at x = 0, 1, for all
1;
ka
(ii) SUpp $i C [0, l] fl [Xi - di, xi + di], x, E [0,11, for some di > 0 and If(xl) -f(x) I G E. ]g(xi) - g(x) I s E hold for all x E [0, l] n [xi - 2di, xi + 2di]. Note that the choice of N. {@I, . , @,v} depends on E and M. Next we define a set of functions {vl, . . . , v.~} such that foreachi=l,...,N, (i)* 0 S vi S 1, IJJiECz((ii)* 7jIiE 1 on
[Xi -
di,
E > 0 will be determined
x, m);
Xi + di],
IjJi s
0 on
(-
3c,
m,
-
[Xi -
2di,
Xi +
2dij.
later on. By multiplying (1.21) by @it we see that
(A - R,)(u$i)
+ (1 + fi)A(u#i)
= j$i + (1 + f)uA#i + (fi -
+ glA(u@i)
+ 2(1 + f)~x+k + g~A$i + 2g~.@u (1.22)
f)A(u@O+ (gi - g)A(U@O
and
(A - Rm)(u$i)+ fA(u#i> + (Y+ gi)A.(uGi> = VPi + fuA$i + 2fud’ti + + (fi -f>NWd
(Y
+ g)uA$i + 2(y + g)Ux@u (1.23)
+ (gi - g)A(u@i),
where fi =f(xi), g; = g(xi), i = 1,. . . , N. From the Appendix, it is easy to see that the right-hand sides of (1.22) and (1.23) lie in a,,, with the following estimates: ii(fi -f)A(W II(fi -f)A(u+i)lIl
s Cll~i(fi
-f>
s &CIIA(u$i)IIl c 24lW&)ll1
(1.24)
110 c c]]A(@J 110; Il~~ll~(~@i)lI~
+ Cll~i(fi
-f>
II~IIA(uQ,) l/j,3
+ ~II~~(f~~f)II~II~(~~~)ll~‘411~(~~~)~~~~ + 5 II vi(fi
-0
ll:b(44
Ilo,
(1.24)’
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
where C denotes positive constants independent II(fi - f)A(u@i)
IIm 6
1127
of E, u, p,, vi and f;
Cm(Il~t(fi -f) /I~~IIA(u@i)lIm + ll~l(fi-f) ll~~llA(u@i) llm-I), (1.24)**
for m 2 2, where C, a positive constant which depends only on m;
(1.25)
IIE$311m s &II 511mll @illa,forallmz=0; forallm20; II0 +f)uWillm~ Cm111 +flloll~llmll~~~ll~,
fordim II(1+f)w#41m s Cm111 +flloll~llmr~ll~illo+~,
20.
(1.26) (1.27)
The estimates for the remaining terms on the right-hand sides of (1.22), (1.23) are similar to (1.24) to (1.27). Thus, (1.20) yields IlU~illm+*
+
II ~~i/lmt*
6
Cm{EIIA(U$i>IIm
+
EIIA(u$i>lIm
i l + $ 1(IIViUi-f)ll”Lk+ IIV&i - g>Ilk + IIWdfi- f) IILL + IIWi(gi-g)II~~)(IIA(U4i)Ilm-l+ IIA(U@i)IIm-1) +
+ (IIEllm + IIrlIlm)ll~illo+ (l + Y+ llfllo+Il~llo)(ll”llm+l + IIuIIm+1)(IIA@iIIo+ IIGiIIo+l)),
(1.28)
for all m 3 0, it is being understood that I( .llm_ l= 0 1 ‘f m = 0.C, is a positive constant which depends only on m and is independent of E, ,I, R,, U, V, #i, vi, g, 7, f and g. Hence, we could have taken E so small that EC, c 1 at the outset. This, in turn, determines N, $t, . . . , qs and QN, depending only on E, M. Now we suppose that such E was fixed and that the Vl,..., corresponding set of functions et, . . . , c#I~,ql, . . . , T)JNwere chosen for given M. Then from (1.28), we find that
IIU@illm+* + II~dJillm+* s C(mdW(llHm + lldlm+ l141m+~ + l141m-~)~
(1.29)
where C(m, M) denotes a positive constant depending only on m and M. By summing (1.29) over i= 1,. . . , N, we deduce that
+ II~IIPPI+* s ChW(II% II4 m+Z
+ Ildlm+ IIUL+*+ II~lL+~),
(1.30)
which, combined with the inequality
II4 m+l
d
Gll~ll~+2ll~ll!t,
(1.31)
gives
Ilullm+Z + IIuIIm+2 s c(m7 M)(llgllm + IIrlllm+ llUllm + IIUllm)~
(1.32)
where C(m, M) denotes positive constants depending only on m, M. Now (1.21) is equivalent to (A - R,)u + Au = c-- fAu - gAv (A-R,)v+yAv=n-fAu-ghu
’
(1.33)
1128
JONGUHNKIM
Combined with (1.32), (1.19). applied to (1.33). yields
cz;;;
;) (IIfilm+ IIIjllm- /I4, + II4m). m
for all j. E C, Re k G 0 and all R, Z- 1. Here, C(m, M) is independent _ With this particular C(m, M). we define: may take C(m. M) 2 ;1.
(1.34) of A and R,.
and we
R(m, M) = 2C(m, iU>.
(1.35)
So for all R ,?=R(m,M)andall~EC,Reh~O,wehave (1.36) which, together with (1.32), implies
II44 where C(m, M) is independent by the following lemma:
m-2s C(m.M)(lj$?I+ /Iqllm), + II011 of A and R,.
(1.37)
Now the proof of the case s = m is completed
LEMMA 1.2. Suppose f, g are real-valued functions in Q,,, IlfllO, //g//, s .M and f(x). g(x) > max(-l/4, - y/4) for all x E [0, 11. Let A E C, Re A c 0 and R, 2 R(m. M). Then. for each t, 71E cD~, there exist unique u, u E Q,,,+? such that (1.21) holds.
Proof. We will use the method parameter cc:
of continuity.
Consider
the following equations
(A - R,)u
+ (1 + r_lf)Au - ,ugAu = s”
(A - R&I
+ ,ufAu + (y + .ug)Au = 77I ’
with
(1.38)
Let us define S = {p E [0,1): for each E, r] E Qm, there exist unique u. u E Qp,-? such that (1.38) holds}. It is obvious that 0 E S. Suppose ~0 E S and consider the mapping T;.?.= from X,,,,: into X m+? defined by (U, 6) k-+ (u, u).
(1.39)
where (u, u) is the unique solution of (A - R,)u
+ (1 + u,,f)Au + ,uagAu = E+- (,uO- u)fAU + (PO- ,u)gAlj
(A - R,)u
+ ,uofAu + (y+
(1.40)
j~ag)Au = rj + (,uO- u)fAU + (PO- ,u)gAti I
With the aid of (1.37), we can choose 6 > 0 independent of 5, 17such that j ,u,)- ,u~< 6 implies T,-,rl.u is a contraction for all f, 71.The fixed point of T:.,,, is the unique solution of (1.38). Hence, S is open. It is easy to see that S is also closed. Therefore, S = [0, I]. We proceed to consider the case where s 5 0 is not an integer. Let k 2 1 be an integer such that k - 1
1129
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
-_://4) for all x E [0, 11. Then, g(x) 2 max(-l/4, R(k - 1, M) by (1.35). Let
we can determine
f(x),
R(s, M) = max(R(k,
M), R(k
R(k,
- 1,M))
M) and
(1.41)
and R, be any positive number such that R, a R(s, M). By taking RL_~ = Rk = R,, we define def A,
=
A/c-@,
(g,f))
=
A&‘,
W)).
en, we have proved that for all 1 E C, Re 1. s 0, and
Th
for all R, 2 R (s, M), (2 - A,)-’
E B(Xk-*,
By interpolation,
Xk-1)
n B(Xk-I,
X
wxk,xk)
k-tl)
nwk,xk-l).
w42)
we can conclude that (I.1 - A,)-’
(1.43)
E B(X,, X,) n B(X,, Xs+z)
and that
ll4+2
(1.44)
s c(s7 p, +w 1 (II EIIJ + II rlllA
IM + IM + II4Lt2
s cww(llEll~
(1.45)
+ IIdlA
where C(s, M) denotes positive constants which depend only on S, M. Now the proof of proposition 1.1 is complete. Next we shall discuss some properties
concerning A,(t, (g,f))
and FS(t, (g,f)).
LEMMA 1.3. Let (gi,fi) E XJ_r, i = 1, 2, 3, s 2 0, such that jlgi]]r+t, Ilfi]lS+rs M/2 and gi(x), .fi(x) 2 4 max(-l/4, -y/4) for all x E [0, 11, i = 1, 2, 3. Let R, a R(s, M) which is determined by (1.35) if s is an integer and by (1.41) if s is not an integer. Using this R,, we define A,(t, (gi,fi)). Let T, be a positive number such that eRSrr~2. Then for all ti E [0, T,], i = 1, 2, 3, it holds that I]{&(ft, (gtJi)) s C(J,
M, R,)(
--A&z, (gzJ2))
b‘&3,
lh - t21 + l/g2 - g2h
k3d3))
-‘~x,,xJ
+ llf~ -f211s+d,
(1.46)
where C(s, M, R,) is a positive constant which depends only on s, M and R, Proof. B(X,,
First of all, by (1.1) and proposition
X,) n B(X,,
X,+2).
Set (u, u) = A,(h,
1.1, we observe
(g3,f3))-’
that As(t3, (g;. fj))-’
E
(El II). Then,
R,u - (1 + f3 eRIQ)Au - g3 eRIOAu = 5 R,v - f3 eRsr3Au - (y+g3eRJc3)Au
= 77 I
(1.47)
and {&(tr, hfd)
-A&
(g2,fd)
}(u, 4
fi eRSrl)Au + (g2 eRSq- gr eRSrl)Ao fi eRSrl)Au + (g2 eR1&- gr eR1’l)Au
(1.48)
1130
JONG C’HY KIM
Using
the inequalities
in the Appendix,
ihLh
-AJ(b
hfd)
it is easy to see that for all fr, r? E [0, rJ].
(gdd)
Ku74 /IX, + \lgl - g2//,4)(ll~/l,-3 + ll4-1)
c C(s, iv, R,)(lr* - [?I + iIf* -fills-l
s C(s,iv, R,)(it, - t2l+ IIf, -fills-~ + llgl-g211s-d(ll~ll~ + IIr7/lJ). From (1.37) and (1.45), M and R,. Next let us define
where
C(s, M, R,) denotes
f+jS P=
positive
constants
depending
(1.49) only on s,
ifOSs< (1.50)
iS
ifs>
1
and K(t, (g, f) > as in (1.3) using any R, LEMMA 1.4. Suppose (gi,J) E XP+r with Ilgillp+l, Ilfillp+l 6 M, i = 1,~ such that eGRs s 2. Then, for all tt, f2 E [0, T,],it holds that
IMh (g1,fd)-
M2t
Let K be a number
I/&
kZTf2))
s C(S,iM,R,)(lt,-fz/ + l/g1 -gzIIp-l + llfl -f2IIp+1)7 where
C(s, M, R,) is a positive
Proof.
constant
depending
(1.51)
only on s, M and R,.
We write
(g1,fd)- qtz7 k2Tfi))
K(t,,
r(
2 eRJrlgtrfir-
eRs6ggZlfLr) +(El
-aleR”lgl
-bleRrrlffl)gl -
= 2(eR”‘glxfl,-eRsr2g~f~)
+(E~--~~e~~‘~g~
L With the aid of the inequalities //@s’l ghfi,
in the Appendix, - e@z ghftiIls
+ eRrh llgtrfix-
bl eRs’*f2)g2
-bzeRs”fJfl
-
-(EI--uI~~‘~~z] (1.52) -(Ez-aZeGQgz J
bt eRsr2f2) f2
we can estimate
gkft,IL
//eRrrlflgl - eRrr2f~g~/lss C(s, M,R,)(lrl estimates
are similar
side of (1.52):
c I eRJ* - eRrhIlglrfirlls
GC(J,M,RJ(lb - hl + /lgl- g2/lp+l + ilfl The remaining
the right-hand
- t2/ + ilfl
-fdp+d,
-f21ip+//gl-gz/iJ.
(1.53) (1.54)
to (1.54) and the proof is complete.
LEMMA 1.5. Let f, g satisfy the same conditions as in proposition 1.1. Define A, = A,(O, (g, f)) with R, 2 R(s, M). Then ELI(Af) is continuously imbedded into X,-r if ,u > 0 is continuously imbedded into where8=fifsalandf?=j -(s/8)ifO~s ,u and 0 < p < 1. (Eb(Af) is equipped with the graph norm.)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
Proof. First we note that A, is a linear operator in X, with %(A,) = X,-z II -IIX,*2 is equivalent to the norm llAs( *) 11x,.Therefore, it follows that
IIxIIx,-Is
1131
and that the norm
C(s, M)~(A,x~~~,~~x~~~;~, for allx E Xr+?,
(1.55)
where 8 = 1 if s B 1 and 8 = 8 - (s/S) if 0 d s < 1, and C(s, M) depends only on s and M. Combined with lemma 17.1 of [2], (1.55) implies that %(A!) is continuously imbedded into .X,+ 1 if p> 8. For the remaining assertion, we define, the operator
Q= Then, Q is a positive-definite x E a(Q), it holds that
I-A,
0,
(
self-adjoint
0 I-A. 1
operator
in X, with G%(Q) = Xs+z. Then, for all
]]x]]a(~n 6 C(P,S, R,, M)]]A,xItX,]]x]]~Y’~ C(,P,S, R,, M)
IlQx IIX,IlxiIi;y;”
(1.56)
where C(p, s, R,, M) denotes positive constants depending only on p, s, R, and M. Again using lemma 17.1 of [2], we conclude that 9 (Q ‘) is continuously imbedded into Z?(A f) where 6 > p. Hence, X s+26 is continuously imbedded into Et (A?) if 6 > ,Uand 0 c p < 1. Now we are ready to establish the local existence of solutions:
PROPOSITION
1.6. Suppose no(x), uO(x) are that IIuolL+“, IIVOIL+ yd M/4 and UO(X),IJO R(s, M) and using this R,, define As = A,(O, t, > 0 such that (0.3) has a unique solution in the initial condition ~(0, x) = UO(X), ~(0, x) such that min($, v/2) > p > CY>8 and
IICUO, OO)IIX,+., a, P, t7and s.
real-valued functions in QI+ “I v > 3, s 2 0 such 3 t max(-4, -y/4), for all x E [0, 11. Let R, = (UO, ~0)). Then, s((A,) = Xscz and there exists CV( [0, rs];%(AP)) nC( (0, t,]; Cd(&)) satisfying = uo(x), where (Y,p and 77are positive numbers 0 C n < /3 - cy. Here r, depends only on
(Recall that R(s, M) is defined by (1.35) ifs is an integer and by (1.41) ifs is not an integer.)
Proof. Choose (Y,/3 and 17such that min(!I, v/2) > /3 > (Y> %and 0 < 17< /!?any positive number and define
y E Crl( [0, t,]; X,) :y(t) is real-vector
Qs(b, K, rl) =
valued, y(0) = A~(uo, UO)and Ily(r) - Y(T) Ilx, 6 Kit - t/“,
Vt, t E [O, tll
We take fs so small that
a: Let K be
1.
e&t*< 2., Kt: <
(1.57)
(1.58) M
4L(a; J, M)
max(llgll,+1,
, L (a; s, M) being the positive constant in the inequality
Ilhllp+1) s L(w s, M)
IbP(g,
h) h
forall k
h) E %A%
(1.59)
1132
JONG UHN
Kt7
-c -
1
1
KIM
Y
4C(a,s,M)maY i -4’ -4 i
(1.60)
’
C( LY,S, M) being the positive constant in the inequality max(llgll~-, Ilhll~=) s C(GS, 1v)lIA,“(g, h)/k,
forall(g,h)
E %A?).
By virtue of (1.57) and (1.60), we find that for all t E [0, tS], for all y(t) E Qs(ts, K, q), min(p(t, x), q(t, x) ) > Imax [-$ holds for all x E [0, 11, where (p(t,x),
q(t,x))
= AFay(t).
(1.61)
-$]
We write (0.3) as (1.62)
-$ z(t) + A,(& Z(t)>z(t) = K(r, z(t)), where z(t) =(eeRsr u(t, x), emRfiu(t, x)) 2 on Q(t,, K, 7) as follows:
and R, = R(s, M) as above. Let us define the mapping (1.63)
w(t)++ A%v(r), where z,(t)
is the unique solution of +t z(r) + A,(& A;Q(t))z(t)
= F,(t, A;‘Yt))
z(0) = (uo, uo). By virtue of (1.58), (1.59) and (1.61), it follows from proposition for all w E Qs(ts, K, 7) and all t E [0, t4], A,(t, A;%(t)) B(A,(t, A;%(t))) = XScz and satisfies: ]](a - Ask A;%(t)))
sC(s,M)(]t*-tzl
-A,&,
I 1.1 and lemma 1.3 that is well-defined with
(1.65)
-%(x,.x,) s $$,
for all A E C, Re A 6 0, where C(s, M) is independent ]/{Ax(k A;%(tt))
(1.64)
A;%(@)
of t, w(t) and A;
}A,(ts, A?‘&))
-%(x,,x,, (1.66)
fKL(cu,s,M)lt,-tzl?
for all ti E [0, tS], all w, E Q(t,, K, q), i = 1, 2, 3, j = 1, 2. From lemma 1.4, we see that for all w E Q(t,, K, 7) and all ti E [0, tS], i = 1, 2, I]F&t,
A;‘Q(tt ) > -
K(&, Ax-%(tz) > ]]x,
=GC(s, M)( ItI - t21 + KL(a, s, M) ItI - tzi”).
(1.67)
Since v/2 > /3, it is obvious that (uo, oa) E 9(A!) by lemma 1.5. Now we can follow the procedure in [2]. Let us denote by U,(t, t) the fundamental solution corresponding to A,(& A;%(t) > for w E Qs(ts, K, q). Then the solution z,(t) of (1.64) is given by z,(t) = U,(t, O)(no, uo) +
’ Uw(t, t> F,( r, A-W
t> ) d t
(1.68)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
1133
and hence, (1.69) Using (1.65), (1.66) and (1.67), we can derive all the necessary estimates (see pp. 172-174 of [2]) to conclude that L!! maps Q,(t,, K, 7) into itself and has a unique fixed point G in Qs(fl, K, 77) by taking ts smaller if necessary. Hence A;” w- is a solution of (1.62) and the same calculation that shows 3 is continuous yields the uniqueness of solution in the function class C’([O, t,]; B(AP))
(1.70)
n C((0, lx]; Q(A,)).
Next we shall show that the solution gains regularity for t > 0, starting from the case s = 0: 1.7. Suppose uo(x), U&Y) are real-valued functions in aV, v > 4 such that 11 uollo, and u&), us(x) Z=0 f or all x E [0, 11. Using R. = R(0, M), we define Aa = Ao(0, (~0, ~0)). Then there exists fo > 0 such that (0.3) has a unique solution in Crl( [0, to]; D(M)) rl C((0, ro]; S(b)) satisfying the initial condition u(O,x) = u&z), u(O,x) = u&z), where cyand n are the numbers in the above proposition. COROLLARY
IIuollv s M/4
We take to so small that M
II44 4 IIs/~, II4 4 IIs/~ c -7 2 1 u(r, x), u(r, x) 3 ) max --4,-i i Now let go(r) = (u(r, x), u(r,x)) be the is. a solution to 4(t) = (a(6 4, fi(4 -4) 0 < 6 < ro, satisfying b(S) = fa(8). H ere eeROftO(r)and f(r) =eRor io(r), I‘t is easily
>
,forallrE[O,r~]andallxE[O,1].
=
(1.72)
solution to (0.3) in the above corollary. Suppose (0.3) in C?([S, ro]; 9(&J)) nC((b, ro]; 9(b)), A0 is the same as in the corollary. Writing z(r) = seen that both z(r) and f(r) are solutions of
-&y(t) +&(o+)bO) ~(6)
(1.71)
eeROs 50(S),
= FO(J y(r))
1 J
(1.73)
where Ao(r, .) and Fo(r, a) are defined with Ro = R(0, M) as in the above corollary. By virtue of (1.71), (1.72) and the fact that fo(r) lies in C?([S, to]; 9(M)) cl C((6, to]; %J(Ao)), we can use the argument in [2] to derive that z(r) = i(r) on [ 6, 6 + E] for some E > 0. By repetition of the argument, we conclude that z(r) = i(r) on [a, ro]. Next we observe that AoEo(r) is uniformly Holder continuous in X0 on each compact subset of (0, to]. Fix any r* E (0, to]. Then, co(r*/2) E X2. We take r*/2 as the initial time and to(r*/2) as the initial data, noting that E0 f. In order to apply proposition 1.6 to the case s = 4, let us with R@ = R(1, MG) where M@ is a positive number such ;lef&rre Au2 =Ap(O, E&*/2)) 11 u(t, x)//2 c M&4. From proposition 1.6, there exists a s:F II~(W)ll2~ $E rE 11 2JOl . to] number 6i > 0 depending only on M@ (a, t5 and r,~are fixed) such that (0.3) has a unique solution cl(r) in P([r*/2, r*/2 + d,]; Eb(A$)) nC((r*/2, r*/2 + &I; Eb(Aq)) satisfying
f&‘/2)
= &(t*/2). S’me %(A$) is continuously imbedded into S(A;). f,(r) = &(t) on [r*/2, r*/2 + &] n [r*/2, ra] by the uniqueness of solution. Now if we take any other point of [f*/2, f’] as our initial time and the corresponding &(f) as our initial data. then 6,. the length of the time interval of existence, remains the same in view of the above argument. Therefore if (f*/2) + d1 < fg, we can extend fl(f) on the whole interval [r-/2, fo] such that et(t) E C( (t*/2, ro]; X1/2+2) and Et(t) = &b(t) on [t*/2, to]. Consequently, $(r) E C((t*/2, ro]; XLJ~_2). Next define ri = t*/2 + . + t*/Zk and suppose that $0(r) E C( (t;, lo]; xkjZ+?) has been proved. Then, $~(fki_~) EXCk-l);Z_v. u = 8 > 2. Take fi_ I as the initial time, 50(rkt+l) as the initial data, and define A(kir),? =A(O. fo(f;_,)) with R(k+l)/Z = R((k + 1)/2, iv(kd)/?), where iu(kfI);z is a positive number such that SUP rE[rr-,,r!j @‘~k+r),?. Applying proposition 1.6 to the case ii”h x)ll(k/2,A2~ $;;; lol lb(t,x)ll(klZ)+? s
*.
s = (k+ 1)/2, we obtain a local solution
where bk+t > 0 depends only on iv(kft);Z. By the uniqueness of solution and the fact that is continuously imbedded into %(A$, tk-l(f) = 50(f) on [fki-~.fc-1 ‘6k-1! n [f$+l, to]. As above, we can extend Ek_*(f) on the whole interval [fZel. fo] to arrive at to(f) E C( (G+ 1, to1; X(kTI)j2_Z). By induction, we conclude that Eo(t) E C([f’, t,)]; xk) for all k 2 0, and consequently, u(f, x), u(t, x) E Cl((O, to] x [0, 11) from (0.3) and the fact that t* was chosen arbitrarily. Finally we shall prove that u(r, x), u(f, s) are nonnegative. We may write (0.3) as
9(A(ak+1jp)
u, =
(1 + u)Au + 2u,u, f {Au f (El - alu - bru)}u,
(1.74)
uI
(y + u)Au + 2uXux+ {Au + (E? - azu - bzu) )u.
(1.75)
=
Since Au(f,x) and Au(f,x) may not be bounded near t = 0. we cannot apply the classical maximum principle directly to (1.74) or (1.75) to prove that u(f, x), u(r, x) b 0. However, the maximum principle can be used on the interval [ 6, to] for any 6 > 0, since u(r, x), u(t, X) E C’“( (0, to] x [0, 11). Thus, it is enough to prove that u(f, x), u(r, x) 5 0 for all x E [0, 11 and all f E [0, 61, where 6 is some positive number. For this purpose, let us denote by (un(t, x), un(t,x)) the solution to (0.3) with the initial condition u,,(O, x) = UO(X)+ (l/n), u,(O,x) = uo(x) + (l/n), n Z=1. We choose Ra = R(0, M), where M is the number such that 1 + ]]ug]lV, and write & = Ao(O, (UO,UO)), d,, = 1 + ]]uo]]~G M/4. Using this R o, we define Ao(f;), u. + (l/n))). Now all the constants in the estimates to establish the local Ao(O, (ub + (l/n), existence of solutions(u(f, x), u(f, x)), (un(f, x), u,(t, x)), n Z=1, can be taken uniformly with respect to n (recall the proof of proposition 1.6). Thus, there exists 6 > 0 independent of n 2 1 such that z(t) E CV( [0, 61; Eli(=$g)) f~ C( (0, 61; 9(&o)) is the solution of +f z(t) + Ao(f, z(~))z(f)
= Fo(f, z(f) > ’
(1.76) I
z(O)= (uo,uo)
J
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model and z,(t)
E
P([O,
61; 9(.dZ))
rl
C((0,61; S(d”))
tfz,(f)+Ao(t,
1135
isthe solution of
z,(f))zn(t) = Fo(4zn@>>
(1.77)
where z(t) =eeRd(u(t, x), u(t,x)) and z,(r) =e-%(u,(t,x), o,(t,x)), n 2 1. Choose any Z such that CY>&> 8. Then, ~(.JA:) is continuously imbedded into GJ(.&dg) and consequently, z(t), z~(I) E Cq([O, 61; 9((saf)) cl C( (0, 61; ‘i3(.&)). Subtracting (1.77) from (1.76), we write $ (z(r) - z&)> + Aok z(r))(z(f) = {Ao(r, z,(t)) Let U(t, t) be the fundamental in [2], we can write
- z&)>
- Ao(r, z(f)) }zn(f) + Fo(r, z(t)) -Fdr, solution associated with Ao(f, z(f)).
z(t) - zn(f) = U(& r)(z(r) ’ W,4{Ao(s, Ir
zn(4> -Ao(L
+
I’
4s) > - Fo(J,z&l >1ds
U(f, sHFo(s,
for all 0 < t G f s 6, and subsequently,
(1.78)
Following the argument
- z,(r))
+
I
z&D
-44) I..&) ds
(1.79)
arrive at the inequality: for all 0 < 6 G 6, (1.80)
where C is a positive constant independent of n and 6. Hence, for some 0 < 6 s 8, ]]z”(t) z(f)jjscda + 0 uniformly on [0, $1, as n + =, from which it follows that un(f, x) 7 u(t, x) and o,(f, x)_+ u(f, x) uniformly on [0, S] X [0, 11. Since u,(f, x), un(f, x) E Cx( (0, S] X [0, 11) n C( [0, 61 x [0, 11) and u,(O, x), ~“(0, x) 3 l/n for allx E [0, 11, it is easily deduced that u,(f, x), u,(t, x) a 0 for all (t, x) E [0, 61 x [0, l] with the aid of the maximum principle. Therefore we conclude that u(t, x), u(f,x) a 0 for all (t,x) E [0, 81 x [0, 11. We have completed the proof of the main theorem: 1.8. Suppose Q(X), Q(X) are nonnegative, real-valued functions in 4. Then, exists fo > 0 such that (0.3) has a unique solution in C”( [0, to]; 9(Ao”)) rl [C=( (0, to] x [0, l])]’ satisfying ~(0, x) = UO(X),~(0, x) = UO(X) and uX(f, x) = u,(f, x) = 0 at x = 0, 1 for all t E (0, to]. Furthermore, u(t,x), o(f,x) 3 0 for all (f,x) E [0, to] X [0, 11. (71, LYand Aa are the same as in corollary 1.7.) THEOREM
there
2. GLOBAL
EXISTENCE
OF SOLUTION
In the previous section, we obtained a unique solution (u(t,x), ~(0, x) = uo(x), u(O,x) = uo(x). Let [0, 7’) be the maximal
u(f,x))
time
to (0.3) satisfying interval to which
JONG Urn
KIM
, v(t,x)) can be extended so that (u(t,x), u(r.x)) lie T) X [0,l])]. Our purpose in this section is to prove that T In view of the local existence theorem, it is enough I]u(f, x)]]~ are bounded near t = T, assuming T < =. Assuming y=
u*+ UC) +
u, = A(rc +
o, = A(u + u’;(=
A<+
(EL -
in = to 1,
CZc([O, T); S(&“)) fl x: under the hypothesis prove that i]u(t, x)]iz. we write (0.3) as
bp)u,
(2.1)
ui’) + (EZ - alu - &u)u,
(2.2)
U$
-
(2.3)
G,
where < = u - u and G = (E? - a+ - &o)u - (El - alu - 6ro)u. obtained through three steps.
The
estimates
will
be
Step 1. Multiplying (2.1), (2.2), (2.3) by u, u, -A<, respectively and integrating over [0, 11, we get, using the fact that u(r.x), u(r,x) 2 0 and ux(r, 0) = uX(r, 1) = ~,~(f, 0) = u,(t, 1) = 0, (2.1)
i(l+u)uidr+;
u’dxS-
u?dxc-
+ u)u:dx
-;~‘(A;)u’dx
(2.5)
+ j’E&dx. 0
;;dx=
- 6’ (At)*
dx - /@f)
GW
G dx,
0
from which it follows
that
(u*+u~+~;)~xs-
i’ (1 + u)ut dx - !;‘(I
+ u)uI dx - 4j’L(A:)’
dx
0
U’ + u’) dx + 1
40j
where K is a positive [0, 11, we obtain
constant
’ (CL”+ o’) dx,
depending
(2.7)
for all r E (0, T),
only on E,,
E:. Integrating
(2.1) and (2.2) over
1 u
dx =S El
and
u dx,
for all f E (0, T),
(2.8)
u dx,
for all t E (0, T)
(2.9
I udxsE2
from which follows [udx+[udx~~Vo, where /MOis a positive and the inequality:
constant
Ilf*I/L=S
depending
only on the initial data,
Ijfll$.z+Ellf,ll~*, forallE>O,
1 +i !
!
(2.10)
forallrE[O,T) El, E, and T. From
allfEL.~[O,l],
(2.10)
(2.11)
1137
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
we find that 1 II~311L’~
(I+% s
i
1
uu; dx
u3dx+c I0
0
1+ ;
for all E > 0,
Mo]juz]]L- + El1 uu: dx,
1
all t E [0, T).
(2.12)
0
and hence, 1 -
uu,2dx s;
i0
(1 + ;)Mo]]u]]:=
- ;]]u]]:=,
for all s > 0,
allrE[O,
T).
(2.13)
-+I&,
for all s > 0,
all t E [0, 7).
(2.14)
In the same way, 1 -
I0 Substituting
uu:dx+
+$‘&]]u]]:-
(2.13), (2.14) into (2.7) and using (2.10), we have r --dl (u’ + u* + {f) dx c - o*(u: + u:) dx - ; i,’ (A{)’ dx dt2 i ,, i + K o1(u’ + u*) dx + Wo( /]u]]‘L-+ /Iv ]I:=, i +;
(1 + ;)
which can be rewritten,
~o(ll42L-+ Il4lL->- ~(ll4b + ll4bL
after taking E = 2/Mo,
l --dl (u*+u*+~:)dx%dt2 i o +
K
(2.15)
i
o1 (u’+
u*)
i0
l( uf + uf) dx - f I o1(Ac)‘dx
dx - W,(]]u]l:-
+ ]iu]]:-)
+ 1Mi(l + @40)(]]u]]t- + ]]u]lt~),
for all t 3 0, we can apply Gronwall’s inequality
Since -#for3 +1M$(l + @fo)t* < C(M0) to deduce that i0
(2.16)
for allt E (0, T).
’ (u* + u* + ct) dx s Ml,
for all r E [0, T),
(2.17)
where Mi is a positive constant depending on Er, Ez, T, I]uoIIL:,j]uolI~~and ]/uox -UUJLL Step 2. Multiplying (2.1), (2.2) by -Au, obtain Id l -u;dxs-2dt Io u*lAul
respectively,
~1(Au)2dx-b1~(A~)2dx
1
+K
-Au,
dx + K
1 uulhul
I 0
and integrating over [0, 11, we
- j#-,r(uA:+2u,&)Au
dx
1
dx + K
I 0
uidx,
forallt
~(0, T)
(2.18)
1138
JONG Urn KIM
and Id ’ u:dxS2dt j O
--
6’ (Au)‘dx
1
+K
I
- i1 u(Au)‘dx
0
I
21~,i’,~)Audx
I
1
021Aul dx + K
+ L1 (uA;+
uulhuj
dx + K
I
foralllE(O,
uzdx,
T)
(2.19)
0
0
where K denotes positive constants depending only on Ei, a,, bi, i = 1, 2. Applying the Laplacian A to both sides of (2.3), multiplying by AE and integrating over [0, 11. we have ;$~t(A~)*dxc
-;br(Ai,)‘dx
for all t E (0, T).
+ ;l’G;dx,
(2.20)
Now we will estimate each term on the right-hand sides of (2.18), (2.19) and (2.20): 1 u(Att>(Au>
lj 0
dx
s =s /IA~~~L"/I~//L~~IAu~~L~
s
~jlAul/:z
+
QllAuII:2 + 2I/A[11~=Ilujl&
2Mt { (1 + ;) IIAilitl + eilAT.li:~}
s 9iiAul& + &I/A&/; 2 + C(Ml)jlA~/l~~,
for all t E (0, T),
(2.21)
which follows from (2.11) with E= 1/16Mt. 1
ii 0
2u,&Au dx
I--
/jl@Oh 0
+4lh=[ddx
= iiACll~= i,l(-wxI
dx s IIA~I/~-II~I/~~/~A~/IL~
s gllAull~2 + QllA[,jlf.z
+
C(Mr)jlA[/~2,
for all t E (0, 7).
(2.22)
+Q[(Au)*dx,
forallt
(2.23)
1 K
‘uuIAuldxSK*
j0
j0
u”dx i K* [u’dx
E(0, T)
1
j0
u4dxSI/uII&
j1u2dx
s MI ( 1 + 1E) llull:2 + M~EIJu,II~z, for all E> 0
and all t E (0, T).
0
(2.24) Similar inequalities
hold for K Jh u’j Au / d x and Jb u’ d x. Combining these inequalities,
we
get I
uuIAuIdx
4 ol(Au)ldr j
+ C(M1, K)( llu$.~ + II~~lli.2) + C(MI, K),
for allt E (0, 73.
The right-hand side of (2.19) can be estimated analogously to (2.21), (2.22) and (2.25). 1 I0
GfdxSK
‘(u:+u;)dx+K I0
I( uz + u?)(u; I0
+ vf) dx,
(2.25)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
c
1139
K o1(uz + ul) dx + $(IIAullZz + i/Au//i?) + C(M1, K)(llu,ll~~ I + /Iuxljt2),
(2.26)
for all t E (0, T),
where K denotes positive constants depending (2.18), (2.19) and (2.20), we find that
only on E,, a,, bi, i = 1, 2. Now summing
~b1{~:+ui+(Ar)‘}dx4C(Ml.Ei,a,,b,)I1{~:+u~+(A~)’)dx 0 +
C(M1, Ei, a,, bi),
for allt E (0, T),
(2.27)
from which we derive, using Gronwall’s inequality, I0
for all t E [6, T),
1{u,2 + uz + (Ac)2} dx s Mz,
where 0 < 6 < T and M2 is a positive constant 11 uo(x) [Iv. Here we fk 6 and proceed to the last step.
depending
(2.28)
on 6, T and
IIuo(x)lly +
Step 3. Apply the Laplacian
A to both sides of (2.1), (2.2), (2.3), multiply the resulting equations by Au, Au, -A2{, respectively, and integrate over [0, 11: ;$1,1(Au)2dx~
-j’(Au,)‘dx-
jru(Au,)‘dx-
j16u,(Au)(Au,)
dx
0
0
0 1 -
%;(Au)(Au,)
I0
dx
-
I0
’ 3ux(Aif)(A.u,)
dx
1 I0
+ 1 ‘(Au)‘dx, i0
u(AS;x)(Aur) dx + 1 ’ (AH)‘dx I0
for all t E (0, T),
(2.29)
where H = (El - alu - blu)u, f$ll(Au)‘dx~
-1’ (Au~)~ dx - /lu(Au,)’
1 +
1 %-x(Au)(Aux)
I0
dx - ~‘6uX(Au)(AuX) dx
0
0
dx
+
I0
3uAAf)(Aux)
dx
1 + i0
for all r E (0, T),
u(A~x)(Auox) dx + 4
(2.30)
where J = (E2 - azu - b2u)u, ;$ll(Acz)‘dx
s -Ij1(A2c)2dx 0
+ &11(AG)2dx.
(2.31)
1110
JONG Uw
KIM
As before, we will estimate each term on the right-hand sides of the above inequalities. (A) Estimate of Jd u,(Au)(Au,)
dx.
First we observe that
(Au(t,x))’
= 2j’:ud,, d
for allx E [0, 11, t E(0, T),
dx,
(2.33)
where a is a point depending on t such that u,(t, a) = 0, and hence, for all t E (0, 7). Using (2.32), (2.33) and the inequality: we obtain
(2.34)
a’!~‘-’ G ,Ia + (1 - ,I)b, for all a, b 2 0, 0 < I. < 1.
1
-21 l .4-pVl;
j0
‘u:,dx+$
j0
u:,,
(2.35)
d x,
for all .s > 0, all t E [6, T)
(B) Estimate of _fdu,(AC)(Au,) We write, by integration
dx.
by parts,
1
j0
uJAs;)(Au,)
dx = -
’ (A[)(Au)*
(AiT>(
Au dx.
0
Since ]/AC/IL- C Ifi AL dx] 6 (J-&AL)* dx)“? / I1 0
dx - ~‘u,(A<~)
dx
/ s
ilA%_=
1’ 0
(-wxxx)
a being the point at which AC = 0, we see that
dx
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
1141
On the other hand,
~.Mr(lolu~=dx)~2(lul(A1;)Zdx)1:‘, From (2.36),
(2.37),
foralltE[&T).
(2.37)
we have 1
1
ux(A~)(Au,)
dx
+~jol{(A;,)‘+ dddx,
for all E > 0,
(2.38)
all t E [6, T). (C) Estimates of J,j &(Au)(Au,)
dx and Jb u(A&)(Au,)
dx.
It is easily seen that
from which follows
=S ~IIAu,ll;r
+ ~~jAu~l&,
for all E> 0 and all r E [6, T).
(2.40)
With the aid of (2.11), we obtain 1 : i
0
u(AL;x)(Au,)
dx
c
W2
II~IIL=IIA~~IIL~IIAu,IIL~~
+ ~MI)~~~A~~~~L~IIAu~II~?
for all E> 0 and all I E [6, T).
s &~~Au.~I~?+~(~MI+~MI)IIA~~/~~~,
(2.41) (D) Estimate of Jb {(AH)2 +(AJ)’ + (AG)*} dx. It is obvious that I0
‘{(AH)*+
+
K
(AJ)‘+
(AG)‘}dxS
Ki’(u’+
~~){(Au)~+(Au)~}dx
1 (uf + u:)2 dx + K i* {(Au)’ + (Au)2} dx, 0
where K denotes E = 1, we have
positive constants
depending
=z (MZ + 2iM1) O’{(A~)’ + (Au)‘}
j
on E,, a,, b,. i = 1. 2. Using (2.11) with
for all f E [ 6. T).
dx,
(2.12)
Since Il4ll& + II&-
s (I,ljAul
d.r)‘+
(b’iAu/
s l1 (Au)* dx + i,’ (Au)‘dx,
cix)’
for all t E (0, T),
we find that j
for all
01(U:+uj)‘dr~M2j1(A~)‘dx+Mlj’(Au)ldX, 0
t E[b,
T).
(3.13)
0
The remaining estimates can be obtained similarly to (A), (B) and (C). Adding (2.29). (2.30). (2.31), and substituting the above inequalities into the right-hand side, we obtain by taking E sufficiently small &ll x
j
{(AU)’ + (Au)’ + (Ai’X)‘} dx 6 C(Ei, a,, b;, M,, iCIz) ol{(A~)’ + (Au)’ + (A~J’} dx,
for all f E [ 6, T).
(7.44)
where C(E,, ai, bi, Ml, Mz) denotes a positive constant depending only on E,, ai, b:. i = 1. 2, and MI, Mz. We deduce by Gronwall’s inequality that
J
‘{(Au)~+(Au)*+(A&)‘}~~~M~,
(2.15)
foralltE[&T),
0
where M3 is a positive constant depending on 6, T, MI, Mz and lluollv- ji ug;Iv. Combining the above estimates and theorem 1.8. we can conclude: 2.1. Suppose y = 1 in (0.3) and Q(X), z&x) are real-valued, nonnegative functions in a”,, Y>?. Then, (0.3) has a unique global solution in C&([O, 7;); S(AS)) n [C’“((O, m) x [0, 1])12 satisfying u(O,x) = Q(X), u(O,x)= UO(X)and u,(t,x) = u,(t,x) = 0 at x = 0, 1 for all t > 0. Furthermore, it holds that u(r,x), u(t, x) a 0 for all (I, x) E 10, =) x 10, 11. THEOREM
APPENDIX A.l.
Multiplication
Proof. Since @a =f.a fact that multiplication theorem 9.4 in [j].
is a continuous
bilinear
map of @,,2_c@ ~1,2_ z into QO provided
E 5 +.
and QL2_r is continuously imbedded into Li2-r for any E s 1, the assertion follows from the is a continuous bilinear map of L$?_,$ IL.I,~_, into Lj for ~6 $, which is a special case of
$nodth
solutions to a quasi-linear system of diffusion equations for a certain population model
A.2 If E> 0 and m is a nonnegative into Q,.
integer. then multiplication is a continuous bilinear map of 0m-r-m 9
x
E oj,Z_c_mand g =“TOb, Proof. Let f =,zoa, cosniCX
cos KU E Q,.
c
1143 $0,
k Define fk =,zO a. cos nj~~ and gk = *
nz,b,cosnzr.
Then, as k+
=, fk-fin
in G,, andftgk E,Q Q, for each k. Now multiplication
@LZ-~-~, gt-g
is a continuous bilinear mapping from Liz-,-, 9Li into L$ by theorem 9.5 in [S]. Thus. fkgk-tfg as k* m in .G since @fl+c_m and @, are continuously imbedded into Lt2-rTm and f.i, respectively. The norm ]I& is equivalent in @,. from which we deduce is a Cauchy sequence to the norm ]].]]~i and hence { fkg4}i_, fg E @m.
A.3. Multiplication is a continuous bilinear mapping from @,,T_~+~ $ @, into @, provided J a 0, E > 0. Proof. The assertion follows from A.2 by interpolation
[I].
A.4. Multiplication is a continuous bilinear mapping from Q, @ @, into Q, provided s 5 1 Proof. If m Z=1 is an integer, the norm 11. Ilrnis equivalent to the norm ]I.I/L; and it is easy to see that multiplication is a continuous bilinear mapping from LL 3 L$ into Li. Since Qrn is continuously imbedded into L$, the assertion follows when s = m, and the general case follows by interpolation. AS. Multiplication is a continuous bilinear mapping from @++(v+ $ @w+(~Q, into @, provided 0 s s C 1 Proof. By interpolation, A.6. Define T: (f, g)e (i)
@514 63 @5i4H
the proof is immediate from A.1 and A.4. f,g,.
Then, T is a continuous bilinear mapping:
00.
(ii) @,+r $ Qm+ii+ Q,, for all integers m 2 1. Proof. (i) Suppose fE 0514and g E @!,r. Since @r/ais continuously imbedded into L$a and @O= La, the assertion follows from theorem 9.4 in [_5]. x k k (ii) Let f = C a, cos njzx E O,-, and g =“zO b. cos nzx E Qm-j. Define fk ==zO a. cos run and gc =“zO b. n-0
cos nxx. Then, fbgh
r E,g
@i for each k, since sin(n;n)
sin(l;rx) = Hcos(n-
&rx - cos(n + I);n}. In the mean time,
m-i and a,,, are continuously imbedded into ,L.i,i and Li, respectively, and the norms 11 .Ilmbl. ii $, are zquivalent to the norms II*III.;,,, /[.I~L;,respectively. Therefore, ftigb+bgr in L$ and (fbgh}K_i is Cauchy sequence in 0,. from which the conclusion follows. A.7. T is a continuous bilinear mapping: (i) @++(v+ @ @r/d+ts/d)l---*a,
provided 0 =ss C 1;
(ii) @r+t@@,+i+Qr
provided s P 1;
(iii) QJ.Q-~$@~+@,J
provided e>O;
(iv) %-,@(Pz+ (v) O.-,@Q,-i-@r
@I provided E > 0; provided s 2 0 and r + 1< k
Proof. (i) and (ii) follow from A.6 by interpolation A.3, A.6, and by interpolation.
Acknowledgement-I
[l], and (iii) to (v) can be proved by the same argument as in
am very grateful to Professor M. Crandall for his invaluable advice and encouragement throughout this work. In particular, he pointed out some serious errors in Section 1 and significantly simplified the original lengthy estimate of the L&norm in Section 2.
1144
JONG UH;VKIM
REFERENCES 1. CALDERONA. P., Intermediate spaces and interpolation, the complex method, Srudin Marh. 24, 1 G-190 (1964). 2. FRIED~~V A., Partial Diflerenfiaial Equations, R. E. Krieger Publishing Co., Huntington, N.Y. (1976). 3. KAWASAKIK., SHIGESADA,N. & TERAMOTO,E., Spatial segregation of interacting species, /. rheoret. Biol. 79, 83-99 (1979). 1. MI,MURAM., Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. /. 11, 621-635 (1981). 5. PALAISR. S., Foundations of Global Non-Linear Analysis, Benjamin Inc., New York (1968).
Am,,yru. ?-hwy. G&at Britain.
Maho&
& A&$icahm.
Vol. 8. No. 10. pp. 1121-1144,
0x2-s46lc’&r $3.00 + .@I @ 198a Pcrgamoo Rss Ltd.
1W.
SMOOTH SOLUTIONS TO A QUASI-LINEAR SYSTEM OF DIFFUSION EQUATIONS FOR A CERTAIN POPULATION MODEL* Department
of Mathematics, (Receioed
JONG UHN KIM Virginia Polytechnic Institute and State University, Blacksbury. VA 21061, U.S.A. 25 September
1983; received for publication 15 November
1983)
Key words andphrases: System of diffusion equations, population model, smooth nonnegative solutions, Sobolevski’s method, local solutions, energy estimates, global solutions.
0. INTRODUCTION THIS PAPER
deals with the initial-boundary
value problem for the system of equations:
ti, = A(ctr2 + ditifi) + (8, - airi - dia)ti (0.1) ’ I where ci, di, Bi, di and di, i = 1, 2, are nonnegative constants. This system of equations describes a model of two competing species with self- and cross-population pressures. Here, zi and ii denote the population densities of the two competing species. For the derivation of (O.l), the reader is referred to [3]. From the physical consideration, ri and 6 should be nonnegative and (0.1) is subject to the Neumann boundary condition: d,= A(c$ + d#fi)
+ (&-
ti,(r,x)
d2zi - &)ii,
= a,(t,x)
= 0
(t,x)
E[O, 00) x [O,l]
at x = 0,l.
(0.2)
For the case when cl > 0, c2 > 0, di > 0 and d2 = 0, the stationary problem associated with (0.1) was discussed in [4]. Also in its introduction, it was announced that Masuda and Mimura have proved the global existence of nonnegative solutions to (0.1) in the above case. In this paper we shall prove the existence of smooth nonnegative solutions to (0.1) with suitably smooth initial data under the assumption that ci > 0, di > 0, i = 1, 2. In Section 1, we establish the local existence of solutions by the method due to Sobolevski, which is well presented in [2]. We employ the function spaces QJ, s 2 0 (see Section l), which enable us to prove the Cm-regularity of solutions for r > 0. Some properties of a, which are necessary in the development of our arguments are proved in the Appendix. In Section 2, we prove that the local solutions can be extended globally on [0, m) under the additional hypothesis that cl = c2 > 0, but without any restriction on the size of initial data. We shall make some remarks on the structure of (0.1). First of all, we see that (0.1) and (0.2) reduce to u,=a(u+~~)+(E~-alu-blu)u u,=A(yv+uu)+(E~-a2u-b2~)u u,(t, x) = uX(t, x) = 0
1 atx = 0, 1,
l This research was supported by the U.S. Army under contract No. Science Foundation under grant No. MCS-7927062 Mod. 2.
1121
’
(0.3) (0.4)
DRAG 29-80-C-0041 and by the National
JONG UHN KIM
1122
through ic(t,x)
=
+c,r..,, 2
C(t, x) =
2
u(c,t. x)
1
CT1
=
I
(0.5)
1
f,
where y = CZ/C~ > 0, Ei 2 0, ai 2 0 and bi L 0, i = 1,2. Throughout this paper we will consider (0.3), (0.4) instead of (O.l), (0.2). It is interesting to observe some unusual features possessed by (0.3). For simplicity, we shall consider , uf = A(u f uu)
(0.6)
uI = A(yu + UU) and the associated nonlinear operator
9’: (0.7)
Then, for smooth nonnegative functions u and u satisfying the Neumann boundary condition, W(Z)? (;)> L~XLzis not nonnegative in general. In fact, it is strictly negative if we take u = cos nr, for example. Hence, we expect to have difficulty in lOO+ y+6cosnr, u=lOobtaining energy estimates to establish global existence of solutions to (0.6). Now the linear operator associated with (0.7) is -(l
+f)Au
- gAu
(W
-fAu-(y+g)Au
where f and g are assumed to be given nonnegative functions. Then it is easy to see that the right-hand side of (0.8) is not a strongly elliptic system in general. This also suggests that the usual procedure to obtain energy estimates may not be effective. However, if y = 1, i.e. cl = ~2, then we can make use of u - u as an intermediary function to obtain necessary energy estimates. This is illustrated in Section 2. Finally we report that the question of asymptotic stability of solutions remains open. In view of the above remarks, it seems hopeless to get a uniform bound of the solution through energy estimates. In the mean time, the structure of Y discourages us from attempting to construct an invariant set.
1. LOCAL
EXISTENCE
As mentioned above, we shall use the method in [2]. Hence, we write equations (0.3) in the form of an abstract evolution equation and verify all the conditions prescribed in the above reference. Let us define the linear operator A,(t, w) as follows: R,u
-
(1 +feR”)Au
- g eRd Au
R,v-feRs’Alc-(y+geRJ’)Av
(1.1)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
where K, is a positive constant which will be determined u * =U eeRJf, u* =u eeRIr, (0.3) is equivalent to
later on and w =(f).
1123
Writing
(1.2) where 2eRI'u,*uf
+
- bleRdu*)u*
+(Ez-a2eR1’u*
-bbeRdu*)v*
(El - aleRI’u*
$7(;:)I=(2eRdu,*vf
From now on, we shall suppress “*” and use both notations(i) vector. For real S, we define
1 ’
(1.3)
and (a, b) to denote the same
(1.4) m
T
and if f
=
c a, cos nnz and R = 2 b, cos nm lie in Cp,. _, we write n=O n=O
(f, g>, = a& + ;
il (1 + n2n2)%.~
(1.5)
n
and
llflls= w>Y*. Obviously, QSl Cas2and II.lls2=Sll-lls,if ~1> ~2.We also MIS +Ilflls7for all (g,f) E X,, and Lf, =
{
fE LZ[O,l]:
(
&
>
“fE LZ[O,l],
(1.6) define X, = QS x QSr Il(g,f)llx,
k=l,...,m.
I
=
(1.7)
ll*l/r& is defined in an obvious way. When X and Y are Banach spaces, we denote by B(X, Y) the set of all bounded linear operators from X into Y. Let f(x), g(x) be real-valued functions in @$+I, s 3 0, such that
llfllr+17 Iklls+ld M < Co and f(x), g(x) 5 max Denote A,(O, (g,f))
for allx E [0, 11.
by A,. Then we have:
PROPOSITION 1.1. There is a number R (s, M) z 1 depending on s, M such that if R, 2 R (s, M), (;U - A,)-’ is a bounded linear operator on X, for all A E C with Re A s 0, and
(1.8)
1124
JONG UH?VKIM
holds where C(s, M) is a positive constant which depends only on s, M and is independent of A, R,. Furthermore, (ti - A,)-’ is a bounded linear operator from X, into XJ+: with il(j_Z- A,)-*~je~xs,x,,,~ s C(s, M),
for all A E C.
Re is
0.
(1.9)
Proof. First we prove the above assertion in the case where s = m is a nonnegative integer with f, g E Q,, where a={tn’ff”m=. Assume Ilfl/a, llglj .C M and f(x), g(x) 1 max(-l/4, x -y/4) for all x E [0, 11. Now we will follow the well-known procedure. Suppose $ =nzo $,, I P I cos nm E Qm, 77=Ezo q, cos nm E O,,,, u =nzo u, cos nm E Qm+l and v = c v, cos njrx n=O E am+2 satisfy the equations: (A - R,)u
+ (1 + f,,)Au + g,,Av = s’
(A-R,)u+foAu+(y+go)Av=r~
(1.10)
1>
where A is a complex number with Re ,J G 0, R, 3 1, and fo, go are constants such that max(-l/4, -y/4). Then, it is easily seen that for all n 2 0,
go2
{ ,I -
R, -
(y + go)n2.z2}s”, +
” = (A - R,)2 - (/I - Rm)(l + y + f.
gG2n2q,,
+ go)n2n2
+ (ygo
+ yfo)n”n”
fo,
(1.11)
and f&c%& ”
= (A -
RJ2
R, - (1 + fo)n%‘}qn + y + f. + go)n2rc2 + (y
+ {A -
- (A - R,)(l
+ go + yfo)n’x”
(1.12)
Now we will estimate 1u,I and I v,/. Setting A = -,u f iv, ,~a 0, vE R, we can rewrite (l.ll), (1.12): (-#u - R, + iv) En - ( y + go)n’n2~n + gon’n2q, u, = ) (l.ll)* (p + R,)2 + (p + R,)(l + y + fo + go)n2d + ( y + go + v, =
yf~)n”n” - Y2- iv{2(p + R,) + (1 + y+
fon2dtn
+ (-p
R, + iv) q,, - (1 +
-
(P + R,)‘+ + (y + go +
(u f R,)(l
fo+
go)n%‘}
fo)n2n2q.
+ y + f.
(1.12)* + go)n2x2
rf)n4n4 - lJ2- iv{2(y + R,,,) + (1 + y + fo
+ g~)nzxz}
In this case, we use the inequality: {(p + R,)
+ (1 + y+
fo + go)n2x2}2
+ (p + R,,,)(l + y + fo to derive
+ g&h2
5 (p + +
R,)’
( y + go + yfo)n4d
r (1.13)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
1125
and (1.14) where C is a positive constant independent
and 1vi2 2 2{(~ + R,)’
Case 2. (p+RR,)+ld(v12 (Y +
g0 +
of A, fo, go, t,,, q,,. n and R,. + (u +R,)(l
+ y+ f. +go)n’d
+
vfo>n’n’l.
Case 3. 1(,~ + R,) s 1VI d p + R,,,. Case 4. 1VI s &(,u + R,). For Cases 2,3,4, it is easy to obtain (1.13), (1.14). Therefore, we conclude that (1.13), (1.14) hold for all A E C, Re A s 0 and all R m 3 1 where C is a positive constant independent of A, fo, go, g,,, qn, n and R,. Next we shall estimate &%‘Iu,l and &‘I ~~1,for n # 0. From (l.ll)*, (1.12)*, we get iiczn2u, =
(-x + iY>En - (Y + go) 5Cn+ gorln
(1.15)
x2+(1+Y+fo+go)x+(Y+go+yfo)-Y’-~y(~+1+y+fo+go)
and ,7’n2v
n
=
foG
+ c-x
+ iY)Vn - (1 +fo)%
(1.16)
x2+(1+Y+fo+go)x+(Y+go+yfo)-Y2-~Y(~+1+y+fo+go)
where x = (p + R,)/n22
> 0, y = v/n22.
Cu.se1.x2+I(y+go+yjo)~y2
“2~x2+(1+Y+fo+go)x+Y+go+~o~.
In this case, we use the y + go + do to derive that
inequality:
(~+l+y+f~+g~)~>
x2+(1+y+fo+go)x+
n2n21unl s C(lEnl+ lqnl>
(1.17)
~n21unI s C(lEnl+ IsI>,
(1.18)
and
where C is a positive constant independent
of A, fo, go, j,, qn, n and R,,,.
Case 2. 2(x2 +(l + y + f. + go)x + y + go + yJo} s y2. Case 3. J S x2 + 1( y + go + tie).
In Cases 2, 3, it is easy to get (1.17), (1.18). Therefore,
we conclude that
llUllm + II”llm s 1~1 +cR,(ll5llm + Iltlllm>
(1.19)
1126
JONG CHN KIM
and II&+?
+ b/l mr? s c(l1511m + II )Illm)
(1.20)
hold for all A E C, all R, 3 1, where C is a positive constant independent of I.. Ii, g,), E, q and Next we suppose u E @‘,_ ?, u E a,,,+*, e E a,,, and 17E (0, satisfy the following equations:
R,.
(n - R,)u
+ (1 +f)Au
+ gAu = s’
(A - R,)o
+fArc + (y+ g)Au
(1.21)
= 77 I
where AEC, ReASO, R,,,sl, and f, g E D0 are real-valued functions satisfying /]f]]D, -y/4) for all x E [0, 11. Let us choose a partition of llugn/lYy;&f and f(x), g( x ) 2 max(-l/4, as follows: , 1, ’ . . 7 Gs} (9
@ia
@iEC”([O,
l]),
,i
on [0, l] and (d/dx)k$l(x)
@iE1
= 0 at x = 0, 1, for all
1;
ka
(ii) SUpp $i C [0, l] fl [Xi - di, xi + di], x, E [0,11, for some di > 0 and If(xl) -f(x) I G E. ]g(xi) - g(x) I s E hold for all x E [0, l] n [xi - 2di, xi + 2di]. Note that the choice of N. {@I, . , @,v} depends on E and M. Next we define a set of functions {vl, . . . , v.~} such that foreachi=l,...,N, (i)* 0 S vi S 1, IJJiECz((ii)* 7jIiE 1 on
[Xi -
di,
E > 0 will be determined
x, m);
Xi + di],
IjJi s
0 on
(-
3c,
m,
-
[Xi -
2di,
Xi +
2dij.
later on. By multiplying (1.21) by @it we see that
(A - R,)(u$i)
+ (1 + fi)A(u#i)
= j$i + (1 + f)uA#i + (fi -
+ glA(u@i)
+ 2(1 + f)~x+k + g~A$i + 2g~.@u (1.22)
f)A(u@O+ (gi - g)A(U@O
and
(A - Rm)(u$i)+ fA(u#i> + (Y+ gi)A.(uGi> = VPi + fuA$i + 2fud’ti + + (fi -f>NWd
(Y
+ g)uA$i + 2(y + g)Ux@u (1.23)
+ (gi - g)A(u@i),
where fi =f(xi), g; = g(xi), i = 1,. . . , N. From the Appendix, it is easy to see that the right-hand sides of (1.22) and (1.23) lie in a,,, with the following estimates: ii(fi -f)A(W II(fi -f)A(u+i)lIl
s Cll~i(fi
-f>
s &CIIA(u$i)IIl c 24lW&)ll1
(1.24)
110 c c]]A(@J 110; Il~~ll~(~@i)lI~
+ Cll~i(fi
-f>
II~IIA(uQ,) l/j,3
+ ~II~~(f~~f)II~II~(~~~)ll~‘411~(~~~)~~~~ + 5 II vi(fi
-0
ll:b(44
Ilo,
(1.24)’
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
where C denotes positive constants independent II(fi - f)A(u@i)
IIm 6
1127
of E, u, p,, vi and f;
Cm(Il~t(fi -f) /I~~IIA(u@i)lIm + ll~l(fi-f) ll~~llA(u@i) llm-I), (1.24)**
for m 2 2, where C, a positive constant which depends only on m;
(1.25)
IIE$311m s &II 511mll @illa,forallmz=0; forallm20; II0 +f)uWillm~ Cm111 +flloll~llmll~~~ll~,
fordim II(1+f)w#41m s Cm111 +flloll~llmr~ll~illo+~,
20.
(1.26) (1.27)
The estimates for the remaining terms on the right-hand sides of (1.22), (1.23) are similar to (1.24) to (1.27). Thus, (1.20) yields IlU~illm+*
+
II ~~i/lmt*
6
Cm{EIIA(U$i>IIm
+
EIIA(u$i>lIm
i l + $ 1(IIViUi-f)ll”Lk+ IIV&i - g>Ilk + IIWdfi- f) IILL + IIWi(gi-g)II~~)(IIA(U4i)Ilm-l+ IIA(U@i)IIm-1) +
+ (IIEllm + IIrlIlm)ll~illo+ (l + Y+ llfllo+Il~llo)(ll”llm+l + IIuIIm+1)(IIA@iIIo+ IIGiIIo+l)),
(1.28)
for all m 3 0, it is being understood that I( .llm_ l= 0 1 ‘f m = 0.C, is a positive constant which depends only on m and is independent of E, ,I, R,, U, V, #i, vi, g, 7, f and g. Hence, we could have taken E so small that EC, c 1 at the outset. This, in turn, determines N, $t, . . . , qs and QN, depending only on E, M. Now we suppose that such E was fixed and that the Vl,..., corresponding set of functions et, . . . , c#I~,ql, . . . , T)JNwere chosen for given M. Then from (1.28), we find that
IIU@illm+* + II~dJillm+* s C(mdW(llHm + lldlm+ l141m+~ + l141m-~)~
(1.29)
where C(m, M) denotes a positive constant depending only on m and M. By summing (1.29) over i= 1,. . . , N, we deduce that
+ II~IIPPI+* s ChW(II% II4 m+Z
+ Ildlm+ IIUL+*+ II~lL+~),
(1.30)
which, combined with the inequality
II4 m+l
d
Gll~ll~+2ll~ll!t,
(1.31)
gives
Ilullm+Z + IIuIIm+2 s c(m7 M)(llgllm + IIrlllm+ llUllm + IIUllm)~
(1.32)
where C(m, M) denotes positive constants depending only on m, M. Now (1.21) is equivalent to (A - R,)u + Au = c-- fAu - gAv (A-R,)v+yAv=n-fAu-ghu
’
(1.33)
1128
JONGUHNKIM
Combined with (1.32), (1.19). applied to (1.33). yields
cz;;;
;) (IIfilm+ IIIjllm- /I4, + II4m). m
for all j. E C, Re k G 0 and all R, Z- 1. Here, C(m, M) is independent _ With this particular C(m, M). we define: may take C(m. M) 2 ;1.
(1.34) of A and R,.
and we
R(m, M) = 2C(m, iU>.
(1.35)
So for all R ,?=R(m,M)andall~EC,Reh~O,wehave (1.36) which, together with (1.32), implies
II44 where C(m, M) is independent by the following lemma:
m-2s C(m.M)(lj$?I+ /Iqllm), + II011 of A and R,.
(1.37)
Now the proof of the case s = m is completed
LEMMA 1.2. Suppose f, g are real-valued functions in Q,,, IlfllO, //g//, s .M and f(x). g(x) > max(-l/4, - y/4) for all x E [0, 11. Let A E C, Re A c 0 and R, 2 R(m. M). Then. for each t, 71E cD~, there exist unique u, u E Q,,,+? such that (1.21) holds.
Proof. We will use the method parameter cc:
of continuity.
Consider
the following equations
(A - R,)u
+ (1 + r_lf)Au - ,ugAu = s”
(A - R&I
+ ,ufAu + (y + .ug)Au = 77I ’
with
(1.38)
Let us define S = {p E [0,1): for each E, r] E Qm, there exist unique u. u E Qp,-? such that (1.38) holds}. It is obvious that 0 E S. Suppose ~0 E S and consider the mapping T;.?.= from X,,,,: into X m+? defined by (U, 6) k-+ (u, u).
(1.39)
where (u, u) is the unique solution of (A - R,)u
+ (1 + u,,f)Au + ,uagAu = E+- (,uO- u)fAU + (PO- ,u)gAlj
(A - R,)u
+ ,uofAu + (y+
(1.40)
j~ag)Au = rj + (,uO- u)fAU + (PO- ,u)gAti I
With the aid of (1.37), we can choose 6 > 0 independent of 5, 17such that j ,u,)- ,u~< 6 implies T,-,rl.u is a contraction for all f, 71.The fixed point of T:.,,, is the unique solution of (1.38). Hence, S is open. It is easy to see that S is also closed. Therefore, S = [0, I]. We proceed to consider the case where s 5 0 is not an integer. Let k 2 1 be an integer such that k - 1
1129
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
-_://4) for all x E [0, 11. Then, g(x) 2 max(-l/4, R(k - 1, M) by (1.35). Let
we can determine
f(x),
R(s, M) = max(R(k,
M), R(k
R(k,
- 1,M))
M) and
(1.41)
and R, be any positive number such that R, a R(s, M). By taking RL_~ = Rk = R,, we define def A,
=
A/c-@,
(g,f))
=
A&‘,
W)).
en, we have proved that for all 1 E C, Re 1. s 0, and
Th
for all R, 2 R (s, M), (2 - A,)-’
E B(Xk-*,
By interpolation,
Xk-1)
n B(Xk-I,
X
wxk,xk)
k-tl)
nwk,xk-l).
w42)
we can conclude that (I.1 - A,)-’
(1.43)
E B(X,, X,) n B(X,, Xs+z)
and that
ll4+2
(1.44)
s c(s7 p, +w 1 (II EIIJ + II rlllA
IM + IM + II4Lt2
s cww(llEll~
(1.45)
+ IIdlA
where C(s, M) denotes positive constants which depend only on S, M. Now the proof of proposition 1.1 is complete. Next we shall discuss some properties
concerning A,(t, (g,f))
and FS(t, (g,f)).
LEMMA 1.3. Let (gi,fi) E XJ_r, i = 1, 2, 3, s 2 0, such that jlgi]]r+t, Ilfi]lS+rs M/2 and gi(x), .fi(x) 2 4 max(-l/4, -y/4) for all x E [0, 11, i = 1, 2, 3. Let R, a R(s, M) which is determined by (1.35) if s is an integer and by (1.41) if s is not an integer. Using this R,, we define A,(t, (gi,fi)). Let T, be a positive number such that eRSrr~2. Then for all ti E [0, T,], i = 1, 2, 3, it holds that I]{&(ft, (gtJi)) s C(J,
M, R,)(
--A&z, (gzJ2))
b‘&3,
lh - t21 + l/g2 - g2h
k3d3))
-‘~x,,xJ
+ llf~ -f211s+d,
(1.46)
where C(s, M, R,) is a positive constant which depends only on s, M and R, Proof. B(X,,
First of all, by (1.1) and proposition
X,) n B(X,,
X,+2).
Set (u, u) = A,(h,
1.1, we observe
(g3,f3))-’
that As(t3, (g;. fj))-’
E
(El II). Then,
R,u - (1 + f3 eRIQ)Au - g3 eRIOAu = 5 R,v - f3 eRsr3Au - (y+g3eRJc3)Au
= 77 I
(1.47)
and {&(tr, hfd)
-A&
(g2,fd)
}(u, 4
fi eRSrl)Au + (g2 eRSq- gr eRSrl)Ao fi eRSrl)Au + (g2 eR1&- gr eR1’l)Au
(1.48)
1130
JONG C’HY KIM
Using
the inequalities
in the Appendix,
ihLh
-AJ(b
hfd)
it is easy to see that for all fr, r? E [0, rJ].
(gdd)
Ku74 /IX, + \lgl - g2//,4)(ll~/l,-3 + ll4-1)
c C(s, iv, R,)(lr* - [?I + iIf* -fills-l
s C(s,iv, R,)(it, - t2l+ IIf, -fills-~ + llgl-g211s-d(ll~ll~ + IIr7/lJ). From (1.37) and (1.45), M and R,. Next let us define
where
C(s, M, R,) denotes
f+jS P=
positive
constants
depending
(1.49) only on s,
ifOSs< (1.50)
iS
ifs>
1
and K(t, (g, f) > as in (1.3) using any R, LEMMA 1.4. Suppose (gi,J) E XP+r with Ilgillp+l, Ilfillp+l 6 M, i = 1,~ such that eGRs s 2. Then, for all tt, f2 E [0, T,],it holds that
IMh (g1,fd)-
M2t
Let K be a number
I/&
kZTf2))
s C(S,iM,R,)(lt,-fz/ + l/g1 -gzIIp-l + llfl -f2IIp+1)7 where
C(s, M, R,) is a positive
Proof.
constant
depending
(1.51)
only on s, M and R,.
We write
(g1,fd)- qtz7 k2Tfi))
K(t,,
r(
2 eRJrlgtrfir-
eRs6ggZlfLr) +(El
-aleR”lgl
-bleRrrlffl)gl -
= 2(eR”‘glxfl,-eRsr2g~f~)
+(E~--~~e~~‘~g~
L With the aid of the inequalities //@s’l ghfi,
in the Appendix, - e@z ghftiIls
+ eRrh llgtrfix-
bl eRs’*f2)g2
-bzeRs”fJfl
-
-(EI--uI~~‘~~z] (1.52) -(Ez-aZeGQgz J
bt eRsr2f2) f2
we can estimate
gkft,IL
//eRrrlflgl - eRrr2f~g~/lss C(s, M,R,)(lrl estimates
are similar
side of (1.52):
c I eRJ* - eRrhIlglrfirlls
GC(J,M,RJ(lb - hl + /lgl- g2/lp+l + ilfl The remaining
the right-hand
- t2/ + ilfl
-fdp+d,
-f21ip+//gl-gz/iJ.
(1.53) (1.54)
to (1.54) and the proof is complete.
LEMMA 1.5. Let f, g satisfy the same conditions as in proposition 1.1. Define A, = A,(O, (g, f)) with R, 2 R(s, M). Then ELI(Af) is continuously imbedded into X,-r if ,u > 0 is continuously imbedded into where8=fifsalandf?=j -(s/8)ifO~s
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
Proof. First we note that A, is a linear operator in X, with %(A,) = X,-z II -IIX,*2 is equivalent to the norm llAs( *) 11x,.Therefore, it follows that
IIxIIx,-Is
1131
and that the norm
C(s, M)~(A,x~~~,~~x~~~;~, for allx E Xr+?,
(1.55)
where 8 = 1 if s B 1 and 8 = 8 - (s/S) if 0 d s < 1, and C(s, M) depends only on s and M. Combined with lemma 17.1 of [2], (1.55) implies that %(A!) is continuously imbedded into .X,+ 1 if p> 8. For the remaining assertion, we define, the operator
Q= Then, Q is a positive-definite x E a(Q), it holds that
I-A,
0,
(
self-adjoint
0 I-A. 1
operator
in X, with G%(Q) = Xs+z. Then, for all
]]x]]a(~n 6 C(P,S, R,, M)]]A,xItX,]]x]]~Y’~ C(,P,S, R,, M)
IlQx IIX,IlxiIi;y;”
(1.56)
where C(p, s, R,, M) denotes positive constants depending only on p, s, R, and M. Again using lemma 17.1 of [2], we conclude that 9 (Q ‘) is continuously imbedded into Z?(A f) where 6 > p. Hence, X s+26 is continuously imbedded into Et (A?) if 6 > ,Uand 0 c p < 1. Now we are ready to establish the local existence of solutions:
PROPOSITION
1.6. Suppose no(x), uO(x) are that IIuolL+“, IIVOIL+ yd M/4 and UO(X),IJO R(s, M) and using this R,, define As = A,(O, t, > 0 such that (0.3) has a unique solution in the initial condition ~(0, x) = UO(X), ~(0, x) such that min($, v/2) > p > CY>8 and
IICUO, OO)IIX,+., a, P, t7and s.
real-valued functions in QI+ “I v > 3, s 2 0 such 3 t max(-4, -y/4), for all x E [0, 11. Let R, = (UO, ~0)). Then, s((A,) = Xscz and there exists CV( [0, rs];%(AP)) nC( (0, t,]; Cd(&)) satisfying = uo(x), where (Y,p and 77are positive numbers 0 C n < /3 - cy. Here r, depends only on
(Recall that R(s, M) is defined by (1.35) ifs is an integer and by (1.41) ifs is not an integer.)
Proof. Choose (Y,/3 and 17such that min(!I, v/2) > /3 > (Y> %and 0 < 17< /!?any positive number and define
y E Crl( [0, t,]; X,) :y(t) is real-vector
Qs(b, K, rl) =
valued, y(0) = A~(uo, UO)and Ily(r) - Y(T) Ilx, 6 Kit - t/“,
Vt, t E [O, tll
We take fs so small that
a: Let K be
1.
e&t*< 2., Kt: <
(1.57)
(1.58) M
4L(a; J, M)
max(llgll,+1,
, L (a; s, M) being the positive constant in the inequality
Ilhllp+1) s L(w s, M)
IbP(g,
h) h
forall k
h) E %A%
(1.59)
1132
JONG UHN
Kt7
-c -
1
1
KIM
Y
4C(a,s,M)maY i -4’ -4 i
(1.60)
’
C( LY,S, M) being the positive constant in the inequality max(llgll~-, Ilhll~=) s C(GS, 1v)lIA,“(g, h)/k,
forall(g,h)
E %A?).
By virtue of (1.57) and (1.60), we find that for all t E [0, tS], for all y(t) E Qs(ts, K, q), min(p(t, x), q(t, x) ) > Imax [-$ holds for all x E [0, 11, where (p(t,x),
q(t,x))
= AFay(t).
(1.61)
-$]
We write (0.3) as (1.62)
-$ z(t) + A,(& Z(t)>z(t) = K(r, z(t)), where z(t) =(eeRsr u(t, x), emRfiu(t, x)) 2 on Q(t,, K, 7) as follows:
and R, = R(s, M) as above. Let us define the mapping (1.63)
w(t)++ A%v(r), where z,(t)
is the unique solution of +t z(r) + A,(& A;Q(t))z(t)
= F,(t, A;‘Yt))
z(0) = (uo, uo). By virtue of (1.58), (1.59) and (1.61), it follows from proposition for all w E Qs(ts, K, 7) and all t E [0, t4], A,(t, A;%(t)) B(A,(t, A;%(t))) = XScz and satisfies: ]](a - Ask A;%(t)))
sC(s,M)(]t*-tzl
-A,&,
I 1.1 and lemma 1.3 that is well-defined with
(1.65)
-%(x,.x,) s $$,
for all A E C, Re A 6 0, where C(s, M) is independent ]/{Ax(k A;%(tt))
(1.64)
A;%(@)
of t, w(t) and A;
}A,(ts, A?‘&))
-%(x,,x,, (1.66)
fKL(cu,s,M)lt,-tzl?
for all ti E [0, tS], all w, E Q(t,, K, q), i = 1, 2, 3, j = 1, 2. From lemma 1.4, we see that for all w E Q(t,, K, 7) and all ti E [0, tS], i = 1, 2, I]F&t,
A;‘Q(tt ) > -
K(&, Ax-%(tz) > ]]x,
=GC(s, M)( ItI - t21 + KL(a, s, M) ItI - tzi”).
(1.67)
Since v/2 > /3, it is obvious that (uo, oa) E 9(A!) by lemma 1.5. Now we can follow the procedure in [2]. Let us denote by U,(t, t) the fundamental solution corresponding to A,(& A;%(t) > for w E Qs(ts, K, q). Then the solution z,(t) of (1.64) is given by z,(t) = U,(t, O)(no, uo) +
’ Uw(t, t> F,( r, A-W
t> ) d t
(1.68)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
1133
and hence, (1.69) Using (1.65), (1.66) and (1.67), we can derive all the necessary estimates (see pp. 172-174 of [2]) to conclude that L!! maps Q,(t,, K, 7) into itself and has a unique fixed point G in Qs(fl, K, 77) by taking ts smaller if necessary. Hence A;” w- is a solution of (1.62) and the same calculation that shows 3 is continuous yields the uniqueness of solution in the function class C’([O, t,]; B(AP))
(1.70)
n C((0, lx]; Q(A,)).
Next we shall show that the solution gains regularity for t > 0, starting from the case s = 0: 1.7. Suppose uo(x), U&Y) are real-valued functions in aV, v > 4 such that 11 uollo, and u&), us(x) Z=0 f or all x E [0, 11. Using R. = R(0, M), we define Aa = Ao(0, (~0, ~0)). Then there exists fo > 0 such that (0.3) has a unique solution in Crl( [0, to]; D(M)) rl C((0, ro]; S(b)) satisfying the initial condition u(O,x) = u&z), u(O,x) = u&z), where cyand n are the numbers in the above proposition. COROLLARY
IIuollv s M/4
We take to so small that M
II44 4 IIs/~, II4 4 IIs/~ c -7 2 1 u(r, x), u(r, x) 3 ) max --4,-i i Now let go(r) = (u(r, x), u(r,x)) be the is. a solution to 4(t) = (a(6 4, fi(4 -4) 0 < 6 < ro, satisfying b(S) = fa(8). H ere eeROftO(r)and f(r) =eRor io(r), I‘t is easily
>
,forallrE[O,r~]andallxE[O,1].
=
(1.72)
solution to (0.3) in the above corollary. Suppose (0.3) in C?([S, ro]; 9(&J)) nC((b, ro]; 9(b)), A0 is the same as in the corollary. Writing z(r) = seen that both z(r) and f(r) are solutions of
-&y(t) +&(o+)bO) ~(6)
(1.71)
eeROs 50(S),
= FO(J y(r))
1 J
(1.73)
where Ao(r, .) and Fo(r, a) are defined with Ro = R(0, M) as in the above corollary. By virtue of (1.71), (1.72) and the fact that fo(r) lies in C?([S, to]; 9(M)) cl C((6, to]; %J(Ao)), we can use the argument in [2] to derive that z(r) = i(r) on [ 6, 6 + E] for some E > 0. By repetition of the argument, we conclude that z(r) = i(r) on [a, ro]. Next we observe that AoEo(r) is uniformly Holder continuous in X0 on each compact subset of (0, to]. Fix any r* E (0, to]. Then, co(r*/2) E X2. We take r*/2 as the initial time and to(r*/2) as the initial data, noting that E0
f&‘/2)
= &(t*/2). S’me %(A$) is continuously imbedded into S(A;). f,(r) = &(t) on [r*/2, r*/2 + &] n [r*/2, ra] by the uniqueness of solution. Now if we take any other point of [f*/2, f’] as our initial time and the corresponding &(f) as our initial data. then 6,. the length of the time interval of existence, remains the same in view of the above argument. Therefore if (f*/2) + d1 < fg, we can extend fl(f) on the whole interval [r-/2, fo] such that et(t) E C( (t*/2, ro]; X1/2+2) and Et(t) = &b(t) on [t*/2, to]. Consequently, $(r) E C((t*/2, ro]; XLJ~_2). Next define ri = t*/2 + . + t*/Zk and suppose that $0(r) E C( (t;, lo]; xkjZ+?) has been proved. Then, $~(fki_~) EXCk-l);Z_v. u = 8 > 2. Take fi_ I as the initial time, 50(rkt+l) as the initial data, and define A(kir),? =A(O. fo(f;_,)) with R(k+l)/Z = R((k + 1)/2, iv(kd)/?), where iu(kfI);z is a positive number such that SUP rE[rr-,,r!j @‘~k+r),?. Applying proposition 1.6 to the case ii”h x)ll(k/2,A2~ $;;; lol lb(t,x)ll(klZ)+? s
*.
s = (k+ 1)/2, we obtain a local solution
where bk+t > 0 depends only on iv(kft);Z. By the uniqueness of solution and the fact that is continuously imbedded into %(A$, tk-l(f) = 50(f) on [fki-~.fc-1 ‘6k-1! n [f$+l, to]. As above, we can extend Ek_*(f) on the whole interval [fZel. fo] to arrive at to(f) E C( (G+ 1, to1; X(kTI)j2_Z). By induction, we conclude that Eo(t) E C([f’, t,)]; xk) for all k 2 0, and consequently, u(f, x), u(t, x) E Cl((O, to] x [0, 11) from (0.3) and the fact that t* was chosen arbitrarily. Finally we shall prove that u(r, x), u(f, s) are nonnegative. We may write (0.3) as
9(A(ak+1jp)
u, =
(1 + u)Au + 2u,u, f {Au f (El - alu - bru)}u,
(1.74)
uI
(y + u)Au + 2uXux+ {Au + (E? - azu - bzu) )u.
(1.75)
=
Since Au(f,x) and Au(f,x) may not be bounded near t = 0. we cannot apply the classical maximum principle directly to (1.74) or (1.75) to prove that u(f, x), u(r, x) b 0. However, the maximum principle can be used on the interval [ 6, to] for any 6 > 0, since u(r, x), u(t, X) E C’“( (0, to] x [0, 11). Thus, it is enough to prove that u(f, x), u(r, x) 5 0 for all x E [0, 11 and all f E [0, 61, where 6 is some positive number. For this purpose, let us denote by (un(t, x), un(t,x)) the solution to (0.3) with the initial condition u,,(O, x) = UO(X)+ (l/n), u,(O,x) = uo(x) + (l/n), n Z=1. We choose Ra = R(0, M), where M is the number such that 1 + ]]ug]lV, and write & = Ao(O, (UO,UO)), d,, = 1 + ]]uo]]~G M/4. Using this R o, we define Ao(f;), u. + (l/n))). Now all the constants in the estimates to establish the local Ao(O, (ub + (l/n), existence of solutions(u(f, x), u(f, x)), (un(f, x), u,(t, x)), n Z=1, can be taken uniformly with respect to n (recall the proof of proposition 1.6). Thus, there exists 6 > 0 independent of n 2 1 such that z(t) E CV( [0, 61; Eli(=$g)) f~ C( (0, 61; 9(&o)) is the solution of +f z(t) + Ao(f, z(~))z(f)
= Fo(f, z(f) > ’
(1.76) I
z(O)= (uo,uo)
J
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model and z,(t)
E
P([O,
61; 9(.dZ))
rl
C((0,61; S(d”))
tfz,(f)+Ao(t,
1135
isthe solution of
z,(f))zn(t) = Fo(4zn@>>
(1.77)
where z(t) =eeRd(u(t, x), u(t,x)) and z,(r) =e-%(u,(t,x), o,(t,x)), n 2 1. Choose any Z such that CY>&> 8. Then, ~(.JA:) is continuously imbedded into GJ(.&dg) and consequently, z(t), z~(I) E Cq([O, 61; 9((saf)) cl C( (0, 61; ‘i3(.&)). Subtracting (1.77) from (1.76), we write $ (z(r) - z&)> + Aok z(r))(z(f) = {Ao(r, z,(t)) Let U(t, t) be the fundamental in [2], we can write
- z&)>
- Ao(r, z(f)) }zn(f) + Fo(r, z(t)) -Fdr, solution associated with Ao(f, z(f)).
z(t) - zn(f) = U(& r)(z(r) ’ W,4{Ao(s, Ir
zn(4> -Ao(L
+
I’
4s) > - Fo(J,z&l >1ds
U(f, sHFo(s,
for all 0 < t G f s 6, and subsequently,
(1.78)
Following the argument
- z,(r))
+
I
z&D
-44) I..&) ds
(1.79)
arrive at the inequality: for all 0 < 6 G 6, (1.80)
where C is a positive constant independent of n and 6. Hence, for some 0 < 6 s 8, ]]z”(t) z(f)jjscda + 0 uniformly on [0, $1, as n + =, from which it follows that un(f, x) 7 u(t, x) and o,(f, x)_+ u(f, x) uniformly on [0, S] X [0, 11. Since u,(f, x), un(f, x) E Cx( (0, S] X [0, 11) n C( [0, 61 x [0, 11) and u,(O, x), ~“(0, x) 3 l/n for allx E [0, 11, it is easily deduced that u,(f, x), u,(t, x) a 0 for all (t, x) E [0, 61 x [0, l] with the aid of the maximum principle. Therefore we conclude that u(t, x), u(f,x) a 0 for all (t,x) E [0, 81 x [0, 11. We have completed the proof of the main theorem: 1.8. Suppose Q(X), Q(X) are nonnegative, real-valued functions in
there
2. GLOBAL
EXISTENCE
OF SOLUTION
In the previous section, we obtained a unique solution (u(t,x), ~(0, x) = uo(x), u(O,x) = uo(x). Let [0, 7’) be the maximal
u(f,x))
time
to (0.3) satisfying interval to which
JONG Urn
KIM
, v(t,x)) can be extended so that (u(t,x), u(r.x)) lie T) X [0,l])]. Our purpose in this section is to prove that T In view of the local existence theorem, it is enough I]u(f, x)]]~ are bounded near t = T, assuming T < =. Assuming y=
u*+ UC) +
u, = A(rc +
o, = A(u + u’;(=
A<+
(EL -
in = to 1,
CZc([O, T); S(&“)) fl x: under the hypothesis prove that i]u(t, x)]iz. we write (0.3) as
bp)u,
(2.1)
ui’) + (EZ - alu - &u)u,
(2.2)
U$
-
(2.3)
G,
where < = u - u and G = (E? - a+ - &o)u - (El - alu - 6ro)u. obtained through three steps.
The
estimates
will
be
Step 1. Multiplying (2.1), (2.2), (2.3) by u, u, -A<, respectively and integrating over [0, 11, we get, using the fact that u(r.x), u(r,x) 2 0 and ux(r, 0) = uX(r, 1) = ~,~(f, 0) = u,(t, 1) = 0, (2.1)
i(l+u)uidr+;
u’dxS-
u?dxc-
+ u)u:dx
-;~‘(A;)u’dx
(2.5)
+ j’E&dx. 0
;;dx=
- 6’ (At)*
dx - /@f)
GW
G dx,
0
from which it follows
that
(u*+u~+~;)~xs-
i’ (1 + u)ut dx - !;‘(I
+ u)uI dx - 4j’L(A:)’
dx
0
U’ + u’) dx + 1
40j
where K is a positive [0, 11, we obtain
constant
’ (CL”+ o’) dx,
depending
(2.7)
for all r E (0, T),
only on E,,
E:. Integrating
(2.1) and (2.2) over
1 u
dx =S El
and
u dx,
for all f E (0, T),
(2.8)
u dx,
for all t E (0, T)
(2.9
I udxsE2
from which follows [udx+[udx~~Vo, where /MOis a positive and the inequality:
constant
Ilf*I/L=S
depending
only on the initial data,
Ijfll$.z+Ellf,ll~*, forallE>O,
1 +i !
!
(2.10)
forallrE[O,T) El, E, and T. From
allfEL.~[O,l],
(2.10)
(2.11)
1137
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
we find that 1 II~311L’~
(I+% s
i
1
uu; dx
u3dx+c I0
0
1+ ;
for all E > 0,
Mo]juz]]L- + El1 uu: dx,
1
all t E [0, T).
(2.12)
0
and hence, 1 -
uu,2dx s;
i0
(1 + ;)Mo]]u]]:=
- ;]]u]]:=,
for all s > 0,
allrE[O,
T).
(2.13)
-+I&,
for all s > 0,
all t E [0, 7).
(2.14)
In the same way, 1 -
I0 Substituting
uu:dx+
+$‘&]]u]]:-
(2.13), (2.14) into (2.7) and using (2.10), we have r --dl (u’ + u* + {f) dx c - o*(u: + u:) dx - ; i,’ (A{)’ dx dt2 i ,, i + K o1(u’ + u*) dx + Wo( /]u]]‘L-+ /Iv ]I:=, i +;
(1 + ;)
which can be rewritten,
~o(ll42L-+ Il4lL->- ~(ll4b + ll4bL
after taking E = 2/Mo,
l --dl (u*+u*+~:)dx%dt2 i o +
K
(2.15)
i
o1 (u’+
u*)
i0
l( uf + uf) dx - f I o1(Ac)‘dx
dx - W,(]]u]l:-
+ ]iu]]:-)
+ 1Mi(l + @40)(]]u]]t- + ]]u]lt~),
for all t 3 0, we can apply Gronwall’s inequality
Since -#for3 +1M$(l + @fo)t* < C(M0) to deduce that i0
(2.16)
for allt E (0, T).
’ (u* + u* + ct) dx s Ml,
for all r E [0, T),
(2.17)
where Mi is a positive constant depending on Er, Ez, T, I]uoIIL:,j]uolI~~and ]/uox -UUJLL Step 2. Multiplying (2.1), (2.2) by -Au, obtain Id l -u;dxs-2dt Io u*lAul
respectively,
~1(Au)2dx-b1~(A~)2dx
1
+K
-Au,
dx + K
1 uulhul
I 0
and integrating over [0, 11, we
- j#-,r(uA:+2u,&)Au
dx
1
dx + K
I 0
uidx,
forallt
~(0, T)
(2.18)
1138
JONG Urn KIM
and Id ’ u:dxS2dt j O
--
6’ (Au)‘dx
1
+K
I
- i1 u(Au)‘dx
0
I
21~,i’,~)Audx
I
1
021Aul dx + K
+ L1 (uA;+
uulhuj
dx + K
I
foralllE(O,
uzdx,
T)
(2.19)
0
0
where K denotes positive constants depending only on Ei, a,, bi, i = 1, 2. Applying the Laplacian A to both sides of (2.3), multiplying by AE and integrating over [0, 11. we have ;$~t(A~)*dxc
-;br(Ai,)‘dx
for all t E (0, T).
+ ;l’G;dx,
(2.20)
Now we will estimate each term on the right-hand sides of (2.18), (2.19) and (2.20): 1 u(Att>(Au>
lj 0
dx
s =s /IA~~~L"/I~//L~~IAu~~L~
s
~jlAul/:z
+
QllAuII:2 + 2I/A[11~=Ilujl&
2Mt { (1 + ;) IIAilitl + eilAT.li:~}
s 9iiAul& + &I/A&/; 2 + C(Ml)jlA~/l~~,
for all t E (0, T),
(2.21)
which follows from (2.11) with E= 1/16Mt. 1
ii 0
2u,&Au dx
I--
/jl@Oh 0
+4lh=[ddx
= iiACll~= i,l(-wxI
dx s IIA~I/~-II~I/~~/~A~/IL~
s gllAull~2 + QllA[,jlf.z
+
C(Mr)jlA[/~2,
for all t E (0, 7).
(2.22)
+Q[(Au)*dx,
forallt
(2.23)
1 K
‘uuIAuldxSK*
j0
j0
u”dx i K* [u’dx
E(0, T)
1
j0
u4dxSI/uII&
j1u2dx
s MI ( 1 + 1E) llull:2 + M~EIJu,II~z, for all E> 0
and all t E (0, T).
0
(2.24) Similar inequalities
hold for K Jh u’j Au / d x and Jb u’ d x. Combining these inequalities,
we
get I
uuIAuIdx
4 ol(Au)ldr j
+ C(M1, K)( llu$.~ + II~~lli.2) + C(MI, K),
for allt E (0, 73.
The right-hand side of (2.19) can be estimated analogously to (2.21), (2.22) and (2.25). 1 I0
GfdxSK
‘(u:+u;)dx+K I0
I( uz + u?)(u; I0
+ vf) dx,
(2.25)
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
c
1139
K o1(uz + ul) dx + $(IIAullZz + i/Au//i?) + C(M1, K)(llu,ll~~ I + /Iuxljt2),
(2.26)
for all t E (0, T),
where K denotes positive constants depending (2.18), (2.19) and (2.20), we find that
only on E,, a,, bi, i = 1, 2. Now summing
~b1{~:+ui+(Ar)‘}dx4C(Ml.Ei,a,,b,)I1{~:+u~+(A~)’)dx 0 +
C(M1, Ei, a,, bi),
for allt E (0, T),
(2.27)
from which we derive, using Gronwall’s inequality, I0
for all t E [6, T),
1{u,2 + uz + (Ac)2} dx s Mz,
where 0 < 6 < T and M2 is a positive constant 11 uo(x) [Iv. Here we fk 6 and proceed to the last step.
depending
(2.28)
on 6, T and
IIuo(x)lly +
Step 3. Apply the Laplacian
A to both sides of (2.1), (2.2), (2.3), multiply the resulting equations by Au, Au, -A2{, respectively, and integrate over [0, 11: ;$1,1(Au)2dx~
-j’(Au,)‘dx-
jru(Au,)‘dx-
j16u,(Au)(Au,)
dx
0
0
0 1 -
%;(Au)(Au,)
I0
dx
-
I0
’ 3ux(Aif)(A.u,)
dx
1 I0
+ 1 ‘(Au)‘dx, i0
u(AS;x)(Aur) dx + 1 ’ (AH)‘dx I0
for all t E (0, T),
(2.29)
where H = (El - alu - blu)u, f$ll(Au)‘dx~
-1’ (Au~)~ dx - /lu(Au,)’
1 +
1 %-x(Au)(Aux)
I0
dx - ~‘6uX(Au)(AuX) dx
0
0
dx
+
I0
3uAAf)(Aux)
dx
1 + i0
for all r E (0, T),
u(A~x)(Auox) dx + 4
(2.30)
where J = (E2 - azu - b2u)u, ;$ll(Acz)‘dx
s -Ij1(A2c)2dx 0
+ &11(AG)2dx.
(2.31)
1110
JONG Uw
KIM
As before, we will estimate each term on the right-hand sides of the above inequalities. (A) Estimate of Jd u,(Au)(Au,)
dx.
First we observe that
(Au(t,x))’
= 2j’:ud,, d
for allx E [0, 11, t E(0, T),
dx,
(2.33)
where a is a point depending on t such that u,(t, a) = 0, and hence, for all t E (0, 7). Using (2.32), (2.33) and the inequality: we obtain
(2.34)
a’!~‘-’ G ,Ia + (1 - ,I)b, for all a, b 2 0, 0 < I. < 1.
1
-21 l .4-pVl;
j0
‘u:,dx+$
j0
u:,,
(2.35)
d x,
for all .s > 0, all t E [6, T)
(B) Estimate of _fdu,(AC)(Au,) We write, by integration
dx.
by parts,
1
j0
uJAs;)(Au,)
dx = -
’ (A[)(Au)*
(AiT>(
Au dx.
0
Since ]/AC/IL- C Ifi AL dx] 6 (J-&AL)* dx)“? / I1 0
dx - ~‘u,(A<~)
dx
/ s
ilA%_=
1’ 0
(-wxxx)
a being the point at which AC = 0, we see that
dx
Smooth solutions to a quasi-linear system of diffusion equations for a certain population model
1141
On the other hand,
~.Mr(lolu~=dx)~2(lul(A1;)Zdx)1:‘, From (2.36),
(2.37),
foralltE[&T).
(2.37)
we have 1
1
ux(A~)(Au,)
dx
+~jol{(A;,)‘+ dddx,
for all E > 0,
(2.38)
all t E [6, T). (C) Estimates of J,j &(Au)(Au,)
dx and Jb u(A&)(Au,)
dx.
It is easily seen that
from which follows
=S ~IIAu,ll;r
+ ~~jAu~l&,
for all E> 0 and all r E [6, T).
(2.40)
With the aid of (2.11), we obtain 1 : i
0
u(AL;x)(Au,)
dx
c
W2
II~IIL=IIA~~IIL~IIAu,IIL~~
+ ~MI)~~~A~~~~L~IIAu~II~?
for all E> 0 and all I E [6, T).
s &~~Au.~I~?+~(~MI+~MI)IIA~~/~~~,
(2.41) (D) Estimate of Jb {(AH)2 +(AJ)’ + (AG)*} dx. It is obvious that I0
‘{(AH)*+
+
K
(AJ)‘+
(AG)‘}dxS
Ki’(u’+
~~){(Au)~+(Au)~}dx
1 (uf + u:)2 dx + K i* {(Au)’ + (Au)2} dx, 0
where K denotes E = 1, we have
positive constants
depending
=z (MZ + 2iM1) O’{(A~)’ + (Au)‘}
j
on E,, a,, b,. i = 1. 2. Using (2.11) with
for all f E [ 6. T).
dx,
(2.12)
Since Il4ll& + II&-
s (I,ljAul
d.r)‘+
(b’iAu/
s l1 (Au)* dx + i,’ (Au)‘dx,
cix)’
for all t E (0, T),
we find that j
for all
01(U:+uj)‘dr~M2j1(A~)‘dx+Mlj’(Au)ldX, 0
t E[b,
T).
(3.13)
0
The remaining estimates can be obtained similarly to (A), (B) and (C). Adding (2.29). (2.30). (2.31), and substituting the above inequalities into the right-hand side, we obtain by taking E sufficiently small &ll x
j
{(AU)’ + (Au)’ + (Ai’X)‘} dx 6 C(Ei, a,, b;, M,, iCIz) ol{(A~)’ + (Au)’ + (A~J’} dx,
for all f E [ 6, T).
(7.44)
where C(E,, ai, bi, Ml, Mz) denotes a positive constant depending only on E,, ai, b:. i = 1. 2, and MI, Mz. We deduce by Gronwall’s inequality that
J
‘{(Au)~+(Au)*+(A&)‘}~~~M~,
(2.15)
foralltE[&T),
0
where M3 is a positive constant depending on 6, T, MI, Mz and lluollv- ji ug;Iv. Combining the above estimates and theorem 1.8. we can conclude: 2.1. Suppose y = 1 in (0.3) and Q(X), z&x) are real-valued, nonnegative functions in a”,, Y>?. Then, (0.3) has a unique global solution in C&([O, 7;); S(AS)) n [C’“((O, m) x [0, 1])12 satisfying u(O,x) = Q(X), u(O,x)= UO(X)and u,(t,x) = u,(t,x) = 0 at x = 0, 1 for all t > 0. Furthermore, it holds that u(r,x), u(t, x) a 0 for all (I, x) E 10, =) x 10, 11. THEOREM
APPENDIX A.l.
Multiplication
Proof. Since @a =f.a fact that multiplication theorem 9.4 in [j].
is a continuous
bilinear
map of @,,2_c@ ~1,2_ z into QO provided
E 5 +.
and QL2_r is continuously imbedded into Li2-r for any E s 1, the assertion follows from the is a continuous bilinear map of L$?_,$ IL.I,~_, into Lj for ~6 $, which is a special case of
$nodth
solutions to a quasi-linear system of diffusion equations for a certain population model
A.2 If E> 0 and m is a nonnegative into Q,.
integer. then multiplication is a continuous bilinear map of 0m-r-m 9
x
E oj,Z_c_mand g =“TOb, Proof. Let f =,zoa, cosniCX
cos KU E Q,.
c
1143 $0,
k Define fk =,zO a. cos nj~~ and gk = *
nz,b,cosnzr.
Then, as k+
=, fk-fin
in G,, andftgk E,Q Q, for each k. Now multiplication
@LZ-~-~, gt-g
is a continuous bilinear mapping from Liz-,-, 9Li into L$ by theorem 9.5 in [S]. Thus. fkgk-tfg as k* m in .G since @fl+c_m and @, are continuously imbedded into Lt2-rTm and f.i, respectively. The norm ]I& is equivalent in @,. from which we deduce is a Cauchy sequence to the norm ]].]]~i and hence { fkg4}i_, fg E @m.
A.3. Multiplication is a continuous bilinear mapping from @,,T_~+~ $ @, into @, provided J a 0, E > 0. Proof. The assertion follows from A.2 by interpolation
[I].
A.4. Multiplication is a continuous bilinear mapping from Q, @ @, into Q, provided s 5 1 Proof. If m Z=1 is an integer, the norm 11. Ilrnis equivalent to the norm ]I.I/L; and it is easy to see that multiplication is a continuous bilinear mapping from LL 3 L$ into Li. Since Qrn is continuously imbedded into L$, the assertion follows when s = m, and the general case follows by interpolation. AS. Multiplication is a continuous bilinear mapping from @++(v+ $ @w+(~Q, into @, provided 0 s s C 1 Proof. By interpolation, A.6. Define T: (f, g)e (i)
@514 63 @5i4H
the proof is immediate from A.1 and A.4. f,g,.
Then, T is a continuous bilinear mapping:
00.
(ii) @,+r $ Qm+ii+ Q,, for all integers m 2 1. Proof. (i) Suppose fE 0514and g E @!,r. Since @r/ais continuously imbedded into L$a and @O= La, the assertion follows from theorem 9.4 in [_5]. x k k (ii) Let f = C a, cos njzx E O,-, and g =“zO b. cos nzx E Qm-j. Define fk ==zO a. cos run and gc =“zO b. n-0
cos nxx. Then, fbgh
r E,g
@i for each k, since sin(n;n)
sin(l;rx) = Hcos(n-
&rx - cos(n + I);n}. In the mean time,
m-i and a,,, are continuously imbedded into ,L.i,i and Li, respectively, and the norms 11 .Ilmbl. ii $, are zquivalent to the norms II*III.;,,, /[.I~L;,respectively. Therefore, ftigb+bgr in L$ and (fbgh}K_i is Cauchy sequence in 0,. from which the conclusion follows. A.7. T is a continuous bilinear mapping: (i) @++(v+ @ @r/d+ts/d)l---*a,
provided 0 =ss C 1;
(ii) @r+t@@,+i+Qr
provided s P 1;
(iii) QJ.Q-~$@~+@,J
provided e>O;
(iv) %-,@(Pz+ (v) O.-,@Q,-i-@r
@I provided E > 0; provided s 2 0 and r + 1< k
Proof. (i) and (ii) follow from A.6 by interpolation A.3, A.6, and by interpolation.
Acknowledgement-I
[l], and (iii) to (v) can be proved by the same argument as in
am very grateful to Professor M. Crandall for his invaluable advice and encouragement throughout this work. In particular, he pointed out some serious errors in Section 1 and significantly simplified the original lengthy estimate of the L&norm in Section 2.
1144
JONG UH;VKIM
REFERENCES 1. CALDERONA. P., Intermediate spaces and interpolation, the complex method, Srudin Marh. 24, 1 G-190 (1964). 2. FRIED~~V A., Partial Diflerenfiaial Equations, R. E. Krieger Publishing Co., Huntington, N.Y. (1976). 3. KAWASAKIK., SHIGESADA,N. & TERAMOTO,E., Spatial segregation of interacting species, /. rheoret. Biol. 79, 83-99 (1979). 1. MI,MURAM., Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. /. 11, 621-635 (1981). 5. PALAISR. S., Foundations of Global Non-Linear Analysis, Benjamin Inc., New York (1968).