ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL REACTION-DIFFUSION SYSTEM

2003,23B(2):261-273

.At'athemd(~,9?cientia

1~~JJ1~m ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL REACTION-DIFFUSION SYSTEM 1 Li Fucai ( *--itt)

Chen Youpeng ( J';1<;1f HIl)

Xie Chunhong (

*.. . ) ~

Department of Mathematics, Nanjing University, Nanjing 210093, China

Abstract This paper deals with reaction-diffusion system with nonlocal source. It is proved that there exists a unique classical solution and the solution either exists globally or blows up in finite time. Furthermore, its blow-up set and asymptotic behavior are obtained provided that the solution blows up in finite time. Key words Nonlocal source, global existence, blow-up, blow-up set, asymptotic behavior of solution 2000 MR Subject Classification

1

35K57, 35K60, 35B40

Introduction and Main Results In this paper, we consider the following system with nonlocal source

Ut - Au

= In f(v(y, t))dy,

Vt - Av =

In g(u(y, t))dy,

u(x, t) = v(x, t) = 0, u(x,O)

= uo(x), v(x, 0) = vo(x),

x.E D, t

> 0,

xED, t > 0,

(1)

x E aD, t > 0, xED,

where D c R N is a bounded domain with boundary aD E aHa, a E (0,1) and For the scalar problem

In f(u(y, t))dy,

u(x, t) = 0,

> 0, x E aD,t > 0,

u(x,O) = Uo(X),

xED,

Ut - Au =

IDI > 0.

xED, t

(2)

it has been studied by many authors, see [1,2,3]. Candam and Yin[2] proved that (2) admits a unique classical solution. Candam, Peire and Yin[l] showed that the solution u blows up in the whole domain if

In uo(x)dx > °and f(8)

satisfies

f(8) :2: 0, f'(8) :2: 0, f(8) is convex, 1

Received July 6, 2001; revised November 8, 2001

IX! 1jf(8)d8 <

00.

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In case of f (s) = s", P > 1 or f (s) = eS , Souplet[3] obtained the asymptotic behavior of solution (denote T* the blow-up time), for the former the limit lim (T* - t)l/(P-l)u(x, t) = [(p -

t-tT'

1)lolt1/ (P- l )

converges uniformly on compact subsets of 0; and for the latter the limit lim Ilog(T* - t)(lu(x, t) = 1

i-vr-

converges uniformly on compact subsets of O. To the authors' best knowledge problem (1) has not been studied, so we consider it in this paper. Before stating our main results, we make some hypotheses as follows: (HI) Uo, Vo E CoCn)

n C2+,,(O), Uo ;:::

0, Vo ;::: 0,

In uo(x)dx > 0, In vo(x)dx > 0;

(H2) uo, Vo satisfy the compatibility condition

-~uo =

In

f(vo(x))dx,

°

-~vo =

In

g(uo(x))dx;

(H3) f(s), g(s), f' (s), g'(s) ;::: for s ;::: a, and f(s), g(s) are convex functions. Now let us state our main results: Theorem 1.1 Suppose that (Hl)-(H3) hold. Then there exists a constant to > Osuch that problem (1) admits a unique classical solution (u,v) and u,v E C2+",l+~(IT x [a, to)). Theorem 1.2 Suppose that (Hl)-(H3) hold, (u, v) is the solution of problem (1), f(s) = sP,g(s) = sq,p,q;::: l,pq > 1. Then (i) (u, v) exists globally if Uo :::; a(1 + 1P(x)), Vo :::; b(1 + 1P(x)); (ii) (u,v) blows up in finite time provided that uo(x) ;::: Mrp(x),vo(x) ;::: Mrp(x), where a, b, M, 1P(x) and rp(x) are to be determined in Section 3. Theorem 1.3 Suppose that (Hl)-(H3) hold, (u, v) is the solution of problem (1), f(s) = g(s) = e". Then (u,v) blows up in finite time provided that uo(x) ;::: M1rp(x),vo(x) ;::: M1rp(x) for some M 1 > 0, where rp(x) is the same function as in Theorem 1.2. Theorem 1.4 Suppose that the solution of problem (1) blows up in finite time. Then the set of all blow-up points is the whole domain IT. Theorem 1.5 Suppose that the solution of problem (1) blows up in finite time. Then (i) In case of f(s) = sP,g(s) = sq,p;::: l,q;::: l,pq > 1, the limits +1

lim (T* - t)~u(x, t) =

t-tT'

+1

lim (T* - t)'/Frv(x, t) =

t-tT'

1] (p + 1) -pj(pq-l) -+, [pq _ 1] -(q+l)/(pq-l) (q + 1) -q/(pq-l) 101---

[pq _

101--1 p+

-(p+l)/(pq-l)

q

q+1

1

p+1

converge uniformly on compact subsets of 0; (ii) In case of f(s) = g(s) = e", the limits lim Ilog(T* - t)I-1u(x, t) = 1,

i-vr-

converge uniformly on compact subsets of O.

lim [Iog (Z" - t)I-1v(x, t) = 1

t-tT'

No.2

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In case of f == g, Uo == Vo, our results are consistent with [1,3], therefore, we generalize their results. This paper is organized as follows. In Section 2, we establish the local existence and uniqueness of solution. The result of global existence and finite time blow-up will be shown in Section 3, and in Section 4, we give the asymptotic behavior of the solution.

2

Local Existence and Uniqueness of Solution

it -

For convenience, we denote L = 6., D T = 12 x (0, T), DT = n x [0, T] and I' an x (0, T], 0< T < +00. We start from the following comparison principle Lemma 2.1 Suppose Ui,Vi E CZ,l(DT) n C(D T), h i j E C(DT),i = 1,2 and (i) h i j :s 0, (x, t) E D T , i, j = 1,2;

In In

In In

In In

In In

(ii) Luz + hlluzdy + hlzvzdy 2: LUl + hlluldy + hlzvldy, Lvz + hai uzdy + hzzvzdy 2: LVl + hZl Uldy + hZZVl dy, (x, t) EDT; (iii) uzlr 2: udr, vzlr 2: Vl!r; (iv) uz(x,a) 2: Ul(X,a),vz(x,a) 2: Vl(X,a),x E n. Then (uz,vz) 2: (ul,vd, (x,t) E DT . Proof Since h i j E C (D T ), we can choose sufficient large constant c > a such that

c/I121 + h ll + h 12 > 0, c/I121 + hZl + h zz > a. Let Wl = Uz - Ul + ce ct , Wz = Vz - Vl calculate directly to obtain

LWl Lwz

k +k +

+ cect , where c is an

k +k

hllwldy +

h 12w zdy 2: ce ct

hZ1Wldy

hzzwzdy 2: ce

ct

arbitrary positive number. One can

k(c/I121 + k(c/I121 +

hll

+ hlZ)dy > 0,

(3)

hZl

+ hzz)dy > a.

(4)

Since Wl(X,a) 2: c > a,wz(x,a) 2: c > a,x En, there exists a () > a satisfies Wl(X,t) > a,wz(x,t) > a for x E n,a:s t:s e. Let

A = {tlt:s T, Wl(X,S) > a,wz(x,s) > 0, for any x E n,):s s:S t}, then there exists a constant I such that I = sup A and a < I :s T. 1ft < T, by Wl(X,a) > a,wz(x,a) > a there exists (x,I) E DT such that wl(x,I) = 001' Wz (x, I) = a. Without loss of generality, we can assume that Wl (x, I) = a. Since Wl (x, t) [r 2: ce ct , we have x E n. Wl gets its minimum at (x, I) in Dr, hence

aawli t

_:s 0, 6.wll«ct) 2: a.

(~~

,

Together with Wl(X,t) 2: a,wz(x,t) 2: a in Dr and hll(x,t):s a,hlZ(x,t):s 0, we obtain

aawli t

_ (~,t)

,inr lii: (x, I)Wl(x, I)dy + inr hl Z(x,l)wz (x, I)dy :s a.

6.wll(~ t) +

It is in contradiction to (3).

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ACTA MATHEMATICA SCIENTIA

Therefore t = T, i.e., Wi(X,t)

Ul,V2 ~

> O,(x,t) E DT,i

Vo1.23 Ser.B

= 1,2. letting e --+ 0+ we get U2 ~

in (x,t) E DT. Remark 2.1 For comparison principle of scalar equation, a similar result of Lemma 2.1 has been proved by Wang and Wang[4]. In [5], the authors obtained a more general comparison principle for Cauchy problem. Definition 2.1 A couple of functions (u(x, t), v(x, t)) is called an upper solution of (1) if U, v E C 2 ,1 (DT) n C(D T) and satisfy VI

u: ~ In !(v(y, t))dy, Lv ~ In g(u(y, t))dy, (x, t) EDT, u(x, t)lr ~ 0,

v(x, t)lr ~ 0,'

u(x,O) ~ uo(x),v(x,O) ~ vo(x), x

En.

Similarly, we can define the lower solution (ft, v) if ft, v satisfy the above inequalities in inverse way. Obviously, (ft,v) = (0,0) is a lower solution of (1). Let (u,v) be the solution of (1), then (u, v) ~ (0,0) by Lemma 2.1. Hence we need only to consider the nonnegative solution. Proof of Theorem 1.1

Let

°<

T

< 1, M >

°

(to be determined later), and set

uu E CHc>(DT), IWi!ct+"(DT) :::; M,Wl(X,O) = uO(X),W2(X,0)

}.

= vo(x),x E n'Wilr = O,i = 1,2

Considering the following problem

= In !(W2(y, t))dy, LZ2 = Ing(wl(y,t))dy,

(x, t) EDT,

Zl(X,t) = Z2(X,t) = 0,

(x,t) E r,

Zl(X,O) = uo(x), Z2(X,0) = vo(x),

xE

LZ1

(x, t) EDT,

(5)

n,

(5) is a linear system, by standard parabolic theory, there exists a unique solution (Zl' Z2) and E C2+c>,H~(DT), i = 1,2.

z;

Denote W = (Wl,W2), Z = (Zl,Z2) and define a transformation Z = :Jw on EM, we shall show that :J admits a fixed point by employing the Schauder fixed point theorem. We first claim that :J maps EM into itself if T is small enough. Let Zl (x, t) = Zl (x, t) uo(x) and Z2(X, t) = Z2(X, t) - vo(x), then Zl, Z2 satisfy

In !(W2(y, t))dy LZ2 = In g(Wl(y, t))dy LZ1 =

Zl(X,t)

~uo,

(x, t) EDT,

~vo,

(x, t) EDT,

= Z2(X,t) = 0,

Zl (x, 0) = Z2(X, 0) = 0,

r, E n.

x E x

(6)

(6) is a linear system, using Lemma A.7 in [6], we obtain (7)

No.2

Li et al: ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL SYSTEM

IZ

2IC I H

( D T ) :::;

for all J E (0,1), where

K,U) = sup{]

K~) =

supj]

In

IvOIC1H(Q)

+ KT"(KCZ:) + l~voILoo(n))

265 (8)

f(W2(Y' t))dYILoo(D T) IIW2ICl+a(DT) :::; M},

L

g(Wl(Y, t))dY/Loo(DT) IlwlICl+a(DT) :::; M}.

Choose J = a and

M = 1 + max{/uoIC1+a(o), Ivolc1+a(o)}'

T < min{l, [K(K,U)

+ l~uoILOO(O)rl/", [K(Ki;) + l~voILOO(O)rl/"},

by (7) and (8), we have Z E EM. Applying (7) and (8) with J E (a, 1), by the result C1+O(Dr) embedding into CHa(Dr) is compact, it follows that J(EM) is a pre-compact subset of EM' m), w~m) E EM and wi m) -+ To prove the continuity of J, we assume that WI, W2, wi Wl,W~m) -+ W2 as m -+ 00. Denote w(m) == (wim),w~m)), (zim),z~m)) == J(wim),w~m))­

:J(Wl,W2), then (zim),z~m)) satisfies zi m) -

~zim)

=

z~m) - ~z~m)

=

In (J(w~m)(y,t)) In

- f(w2(y,t)))dy,

(x, t) E Dr,

(g(wi m)(y, t)) - g(Wl(y, t)))dy,

(x, t) E Dr,

zi m) (x, t) = z~m) (x, t) = 0, zi m)(x, 0) = z~m) (x, 0) = 0,

(x,t) x E

E

r,

n.

Again employing Lemma A.7 in [6], we obtain

Izi m)/:::; KT"

Iz~m)l:::; KT"

lin (J(w~m)(y,t)) lin

- f(W2(y,t))) dyl-+ 0

as

m

-+ 00,

(g(wim)(y,t)) - g(Wl(y,t)))dyl-+ 0

as

m

-+ 00.

It follows that J is continuous in EM. Since EM is a closed set in the Banach space cHa (Dr), by the previous properties of :J and the Schauder fixed point theorem, there exists a fixed point (u, v) of map J, which is a solution of (1). Suppose (Ul' VI) and (U2' V2) are two solutions of (1). Let Zl = U2 - Ul, Z2 = V2 - VI, then

LZ1

-

LZ2 -

In In

= 0,

(x, t) E Dr,

g('f/(Y, t))Zl (y, t)dy = 0,

(x, t) E Dr,

f(((y, t))Z2(Y' t)dy

Zl(X,t) =Z2(X,t) =0, Zl (x, 0) = 0, Z2(X, 0) = 0,

(x, t) E x E

r,

n.

where ((y,t), 'f/(y,t) are functions on y, t, and ((y,t) lies between Vl(y,t) and V2(y,t), 'f/(y,t) lies between Ul(y,t) and U2(y,t). Using Lemma 2.1 we get Zl ~ 0,Z2 ~ 0, i.e., U2 ~ Ul,V2 ~ VI. Similarly, we can obtain U2 :::; Ul, V2 :::; VI. Therefore Ul = U2, VI = V2' We complete our proof by choosing to = T.

266

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Global Existence and Blow-up in Finite Time In this section, we consider I, 9 of special form and denote

= ~ j(v(y, t»dy,

h1(t)

h2(t)

= ~ g(u(y, t»dy.

(9)

°

Definition 3.1 Let (u,v) be the classical solution of (1), we say (u,v) blows up in finite time if there exists a positive constant T*, < T* < +00 such that lim m~[u(x, t) ",En

t-tT*

Remark 3.1

+ v(x, t)]

By max"'EO[u(x, t) lim maxu(x,t)

HP",~

Proof of Theorem 1.2

+ v(x, t)] = 00.

= 00

~ max"'EOu(x, t)

and lim maxv(x,t) HP",~

+ max"'EOv(x, t),

= 00.

(i) Consider the elliptic problem

-.6.lJ1 = 1, lJ1 = 0,

x E 0, x E 80,

°

which has a unique nonnegative solution 'ljJ(x) satisfing ~ 1/J(x) ~ m, m By pq > 1, there exist two positive constants a, b satisfy

+ 'ljJ(x»

> 0, x E 0.

°

In fact, we can choose a, b >

Let u(x, t) = a(l

we have

sufficiently small such that

, v(x, t) = b(l + 'ljJ(x», then Ut - .6.u

= a ~ 101lJ'(1 + m)P ~ ~ vPdx,

Vt -.6.v

= b ~ 101aq(1 + m)q ~ ~ uqdx.

We obtain that u(x, t)

~

a(l

+ 'ljJ(x» , v(x, t)

~

b(l

+ 'ljJ(x»

by using Lemma 2.1 provided

that uo, Vo satisfy uo(x) ~ a(l + 'ljJ(x» , vo(x) ~ b(l + 'ljJ(x». (ii) To prove that (u,v) blows up in finite time, we use an argument as done in Soupletl'". Since (1) does not a priori make sense of for negative values of u and v, we actually consider the problem Ut

= In v~(y, t)dy, .6.v = In u~(y, t)dy,

-.6.u

Vt -

u(x, t) = v(x, t) = 0, u(x,O)

°

= uo(x),v(x,O) = vo(x),

x E O,t > 0, x E O,t > 0, x E 80,t

> 0,

(10)

xE 0,

Let ip E C 1(O),cp(x) ~ O,cp(x) ~ 0 and cp 18n=O. By translation, we may assume without loss of generality that 0 E

and cp(O)

> O.

Li et al: ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL SYSTEM

No.2

267

Set

w(x t) 1 V ( Ixl ) Z(x t) _ 1 V ( Ixl ) , - (T - t)'Yl (T - t)Ul' , - (T - t)'Y2 (T - t)U2 with V(y)

where Ii, CTi

> 0 (i

= 1, 2),A

y2

A

= 1 + "2 -

2A' y ~ 0,

> 1 and 0 < T < 1 are to be determined. Note that

= B(O, R(T -

en,

(11)

= B(O, R(T - t)U2) C B(O, RT(2) c n > 0 with R = (A(2 + A) )1/2.

(12)

supp w+(', t)

t)Ul) C B(O, RTul)

supp z., (-, t) for sufficiently small T

tt

Denote Yi = Ixl/ (T -

i

,

i = 1,2. Calculating directly we obtain

-~w(x t) =

N/A -~z(x t) = NjA . ' ( T - t)'Y2+ 2u2 ' (T - t)'Yl +2Ul '

,

and

{

in

z~(x, t)dx =

Zt -

A

~z

-

{ M1 V+ [ Ixl ] dx (T - t)'Y2P iB(O,R(T-t)"2) (T - t)U2

qd X < "Y2(1+A) w+ - (T - t)'Y2+l

If u. ~ A, we have V(Yi) A

Wt -

~W -

Zt -

~z

A

such that

=

M2 + (T - N/A - ---'::""--,-t)'Y2+2 U2 (T - t)'Ylq-Nul .

s 1 and V'(Yi) ::; -1, i = 1,2.

-

1 1

By pq > 1, we can choose 0 II

M1 (T - t)'Y2P- Nu2'

(13)

1 n

=

Z Pd x

n +

Then

< I1- CT1 A + -,-,----'-----,,-N/A - (T - t)'Yl +l (T - t)'Yl +2Ul '

CT2 A Wqd x < 12- (T - t)'Y2+l

n +

N/A + -,-,----'-----,,-(T - t)'Y2+ 2u2 .

< CT1, CT2 < 1/2, () > 0 and

(1 + ())(1 + p) + NCT1P + NCT2 pq _ 1 ' 12

=

(14)

(1 + ())(1 + q) + NCT2q + NCT1 pq _ 1

(15) (16)

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ACTA MATHEMATICA SCIENTIA

Vol.23 Ser.B

°

> max:{1, 'Yl!al' 'Y2/ a2}, then, for T > sufficiently small, (13)-(16) imply that Wt

-.6.w -

°

In z~dx s 0,

Zt

-.6.z -

In w+dx ::; 0,

(x, t) EO x (0, T),

Since and ip is continuous, there exist two positive constants p and E > Osuch that 1 and' (ii) of Theorem 1.2, we can draw the conclusion. To prove Theorem 1.4, we give the following definition.

°

Definition 3.2 A point Xo E :IT is called a blow-up point of (u, v) if there exists a sequence {(xm,tm)},x m E O,t m < T*,(xm,tm) -+ (xo,T*),m -+ 00 such that

Lemma 3.1 Then

Suppose that (u,v) blows up in finite time, h 1(t),h2(t) are defined as (9).

Proof Let G(x,~; t, T) be the Green's function associated with the operator L along with null Dirichlet boundary condition in 0 x (0, T*). Then for any T < T* the solution (u, v) can be written as

In G(x,~;t,O)uo(~)d~+ hIn G(x,~;t,T)hl(T)d~dT, t

u(x,t) =

In G(x,~; t, O)vo(~)d~ + hIn G(x,~; t, T)h2(T)d~dT,

(x,t) EDT,

(17)

(z, t) EDT·

(18)

t

v(x, t) =

For the Green function

G(X,~;t,T),

we have the estimate (see [7])

(19) Therefore,

(20) where C is independent of t. Let Xl be a blow-up point of (u,v), then there exists a sequence {(xm,tm)},x m E O,t m < T*,(xm,t m) -+ (xl,T*),m -+ 00 such that (21) By (17)-(21), we obtain

Li et al: ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL SYSTEM

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269

Proof of Theorem 1.4(This proof uses some ideas of [4]) For any given Xo E IT, to show Xo is a blow-up point, it needs only to show that for some c > 0, u(x, t), v(x, t) is unbounded in (B(xo, c) nO) x (0, T*), where B(xo, c) is a ball of R N centered at Xo with the radius c. Denote 0' = B(xo, e) nO. On the contrary, we suppose that there exists a constant M > a such that u(x,t) :::; M,v(x,t) :::; Min 0'. Let G'(X,~;t,T) be the Green's function associated with the operator L along with null Dirichlet boundary condition in 0' x (0, T*). Then we have

u(x,t) =

01

-

v(x,t) =

i: r i:

t

1r G'(x,~;t,O)uo(~)d~+ ~r lrw G'(x,~;t,T)hi(T)d~dT o

8G'(X,~;t,T) S ( 8~ u (~,T )ddT, x,) t

ao'

n

-

*) ,

o

1

8G'(X,~;t,T) ( )ddT, S ( 8~ VeT x,) t

ao'

n

> a and G'laol

t

_

ao

rr

E 0' H

a on 80'.

= 0, we have 8G'/8ii:::;

_r r 8G'(~~;t,T)u(~,T)dSdT ~ 0, 1 l l n 0

X ( 0, T

G'(x,~;t,T)h2(T)d~dT 10 G'(x,~;t,O)vo(~)d~+ 1r0 1r 0 1

Since G'(X,~;t,T)

E 0' H

10 l aol

X ( 0, T *) .

Hence,

8G'(~~;t,T)V(~,T)dSdT ~ O. n

These estimates give

u(x,t)+v(x,t)~

r(h (T) + h2(T)) 1r G'(x,~;t,T)d~dT, 1 i t

0/

0

(x,t)EO'X(t,T*).

(22)

Choosing a compact subset 0" cc 0', by using the strong maximum principle, we obtain that there exists a 8 = 8(0") > a such that

1r0 / G'(x,~;t,T)d~ ~ 8(0"),

V(x,t) E 0" x (T,T*),T > O.

(23)

From (22) to (23) it follows that for all (x, t) E 0" x [0, T*),

u(x, t)

+ v(x, t)

~ 8(0")

Therefore, lim (u(x, t)

t-+T*

I

t

(hi(T)

+ h 2(T))dT.

+ v(x, t)) = 00.

It is in contradiction to the assumptions of u(x, t), v(x, t) ::::; M in 0 x (0, T*). Therefore, Xo is a blow-up point. By the arbitrariness of xo, we get the blow-up set is IT.

4

Asymptotic Behavior of Solution By Theorem 1.4, we know that (u, v) is global blow-up. Furthermore, by Remark 3.1, we

have u and v blow up in the same time. Denote T* the blow-up time, we give the followin s definition for convenience. Definition 4.1 Let T > 0, Wi, W2: 0 x [0, T) --+ (0,00) satisfy lim t-+T x E O. Then we say Wi, W2 are equivalent and denote

WI ((X'tt)) W2 x,

= 1 for any

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ACTA MATHEMATICA SCIENTIA

Vol.23 Ser.B

The main task of this section is to prove Theorem 1.5. We use some ideas of [3] and generalize his results. Denote H1(t) = I~ h1(s)ds, H 2(t) = I~ h2(s)ds and C1,C2,C3 , ' " be the various positive constants (possibly depending on the solution (u, v)). We have the following Theorem 4.1 Suppose that u,v E C 2,1(D x [O,T*)) satisfy (1) and blow up at T*. Then the limits lim u(x, t) = 1, lim v(x, t) = 1 (24) HT* H1(t) HT* H 2(t) converge uniformly on compact subsets of n. To prove Theorem 4.1, we start from some Lemmas. Lemma 4.1 Suppose that u, v satisfy the hypotheses 'in Theorem 4.1. Then (25)

Proof

+ HI (t),

u(x, t)

~

C3

(x, t) E (n x [0, T*)),

(26)

v(x,t)

~

C4 +H2(t), (x,t) E (n x [O,T*)).

(27)

Since the right side of equations in (1) is independent of x, one can easily deduce (-~u)t

-

~(-~u)

= 0, (x, t) E (n x [0, T*)),

(-~v)t

-

~(-~v)

= 0, (x, t) E (n

-~u(x, t)

= hI (t),

-~v(x, t)

= h2(t),

x [0, T*)),

an x [0, T*), x E an x [0, T*). xE

Let C 1 = l~uoILOO(n)' C2 = l~voILOO(n)' (25) follows from h1(t) 2: 0,h 2(t) 2: O,i maximum principle. Integrate the equations in (1) over (0, t)(t < T*) to yield u(x, t) = u(x, 0) + HI (t) v(x, t)

+

= v(x, 0) + H 2(t) +

I ~u(x, I ~v(x,

= 1,2 and

the

t

s)ds,

t

s)ds.

Hence, by (25) we get u(x, t)

~

v(x, t)

~

sup uo(x) + HI (t) + C1T*, x E

xEn

sup uo(x)

xEn

+ H 2(t) + C2T*,

xE

n, n.

This completes the proof. Let Al > 0 be the first eigenvalue of -~ in n with null Dirichlet condition and ¢ be the corresponding eigenfunction such that ¢(x) > 0 in n and ¢(x)dx = 1. Define

In

Zl(X,t) = H1(t) - u(x,t), Z2(X,t) = H 2(t) - v(x,t), f31(t) =

L

Zl(y,t)¢(y)dy, f32(t) =

L

Z2(y,t)¢(y)dy,

No.2

Li et al: ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL SYSTEM

and

M 1(t ) =

I

t

HI (s)ds, M 2(t ) =

I

t

271

H 2(s)ds.

By Green's formula, we have

,8f (t) = I n (h1(t ) = -

Ut(y, t))¢(y)dy

In u(y, t)Ll¢(y)dy =

= -Al,81(t) + AIHl(t),

Al In u(y, t)¢(y)dy

,8Ht) = I n (h2(t ) - Vt(y, t)¢(y)dy = -

In v(y, t)Ll¢(y)dy =

= -Al,82(t)

= - In Llu(y, t)¢(y)dy (28)

= - In Llv(y, t)¢(y)dy

Al In v(y, t)¢(y)dy

(29)

+ AIH2(t).

Integrate (28) ,(29) over (0, t) to yield

,81 (t) = ,81 (0)e- A1t + A1 e- A1t ,82(t) = ,82(0)e- A1t + Ale-A1t

I I

t t

eA1SH 1(s )ds

s C5 (1 + M 1(t)),

(30)

eA1SH 2(s)ds

sC

(31)

6(1

+ M 2(t )).

On the other hand, by (26) and (27) we have

which combines with (30) and (31) implies

In I In

Z 1 (y,

+ M 1(t)),

(33)

s Cs(1 + M 2(t )).

(34)

t)I¢(y)dy ::; C7(1

IZ 2(y, t)I¢(y)dy

By (25) we have (35) Lemma 4.2

Let K Pi = {y E nldist(y, an) ::; pd,O

< Pi < diamift), i = 1,2. Then we

have

Proof By using (32)-(35), the proof is the same as Lemma 4.5 in (3) except that we replace Z by Zl, Z2' Proof of Theorem 4.1 By Lemma 4.2, (26) and (27), we obtain

_ ~ < 1 _ u(x, t) < H 1(t) -

_~

~ 1 + M 1 (t)

H 1(t ) - pf+l

H 1(t )

< 1- v(x, t) < ClO 1 + M 2 (t )

H 2(t) -

H 2(t) - pf+l

H 2(t)

, x E K Pll

, x E K p2 '

272

ACTA MATHEMATICA SCIENTIA

Vol.23 Ser.B

Since HI (t), H 2(t) are nondecreasing, it follows that for all c

Using Lemma 3.1, we deduce that lim Mi(t)/ Hi(t) t--4T*

Proof of Theorem 1.5

> 0,

= 0, i = 1,2.

Therefore (24) holds.

(i) We apply Theorem 4.1 with

By (24), we obtain lim uq(x, t) = 1 HT*

and

H'{(t)

lim vP(x, t) = 1, V x E

,

H~(t)

HT*

o ~ I~:( :; I ~ c, 0 ~ I~:( :; I ~ C, x E n,

n,

t is sufficiently close to T*.

By using Lebesgue's dominated convergence theorem, we infer that

H~(t) = hI(t) = H~(t) =

h2(t) =

In In

vP(y,t)dy '"

InIH~(t),

t -+ T*,

(36)

uq(y,t)dy '" InIH'{(t), t -+ T*.

(37)

Combining (36) with (37), we have

_1_ H q+ I(t) '" _1_ H P+l(t) t -+ T*. q+1 1 p+1 2 , By (36)-(38), we deduce

1) p/(p+I) HI :++ 1) q/(q+I) H +

p(q+1)

H~ (t) '" InIH~(t) '" Inl ( p q+1

H~(t) '" InIHf(t) '" In/ (

p+1

q(p+1)

1

2

q+1

(38)

(t), t -+ T*, (t), t -+ T*.

Hence,

~)'

( H I P + 1 (t)

~)'

( H 2 q + 1 (t)

pq -1 (p+ 1)P/(P+l) '" - p + 1 Inl q + 1 ' t -+ T*, '"

pq -1 (q + 1)q/(q+l) ---Inl -, q+1

p+1

t -+ T*.

Integrating (39),(40) over (t, T*) yields

H (t) '" 1

-1 (1)P/(P+l)

[

Inl~ p+ p+1

q+1

(T* - t)

-1 (1)q/(q+l)

H 2(t)", [ Inl~+1

::1

] -(p+l)/(pq-I) '

t -+ T*,

] -(q+I)/(pq-I) (T*-t)

, t-+T*.

(39) (40)

Li et al: ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL SYSTEM

No.2

We get our conclusion by using (24). (ii) Let hl(t) =

.In ev(y,tJdy,

hz(t) =

.In eu(y,tJdy.

For any given compact subset Ken, by Theorem 4.1, we obtain that u(x, t) v(x, t) 2: Hz (t) /2 if t is sufficiently close to T*. Therefore,

H~ (t)

= hI (t) =

H~(t) =

hz(t) =

273

2: HI (t)/2,

.In ev(y,tJdy 2: InleH2 J!Z,

t -+ T*,

(41)

.In ev(y,tJdy 2: InleH1(tJ!Z,

t -+ T*.

(42)

(t

Combining (41) with (42), we obtain (43) Integrate (43) over (t, T*) to yield

Hence, (44) By the proof of Theorem 4.1, we have

t -+ T*,

(45)

t -+ T*.

(46)

By (44)-(46),(24), we obtain Ilog(T* - t)1 - C7 ~ u(x, t) ~ Ilog(T* - t)1

+ Cs ,

Ilog(T* - t)1

+ C lD .

Ilog(T* - t)l- C g

~

v(x, t)

~

This complete the proof. References

2 3 4 5 6 7

Candam J M, Peire A, Yin H M. The blow-up property of the solutions to some diffusion equations with localized nonlinear reactions. J Math Anal Appl, 1992, 169(2): 313-328 Candam J M, Yin H M. An iteration procedure for a class of integro-differential equations of parabolic type. J Integral equations Appl, 1989, 2(1): 31-47 Souplet P. Uniform blow-up profile and boundary behavior for diffusion equations with nonlocal nonlinear source. J Differential Equations, 1999, 153(2): 374-406 Wang M, Wang Y. Properties of positive solutions for non-local reaction-diffusion problems. Math Meth Appl Sci, 1996, 19(1): 1141-1156 Wang M, Wang S, Xie C H. Critical fujita exponents for nonlocal reaction diffusion systems. J partial Differential Equations, 1999, 12(3): 201-212 Souplet P. Blow-up in nonlocal reaction-diffusion equations. SIAM J Math Anal, 1998, 29(4): 1301-1334 Friedman A. Partial differential equation of parabolic type. New Jersey: Prentice-Hill, Inc, 1964