Global existence and global non-existence of solutions to a reaction-diffusion system

Nonlinear Analysis 39 (2000) 327 – 340

www.elsevier.nl/locate/na

Global existence and global non-existence of solutions to a reaction-diusion system1 Sining Zheng a; b;∗ a Department

b Department

of Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, USA of Applied Mathematics, Dalian University of Technology, Dalian 116024, P.R. China

Keywords: Reaction-diusion system; Global existence; Global non-existence; Blow-up

1. Introduction This paper deals with the global existence and the global non-existence of positive solutions to the following weakly coupled reaction-diusion system ut =
u + up1 vq1 ; vt =
v + up2 vq2 ;

(x; t) ∈ × R+ ;

(1.1)

with initial and boundary value conditions u(x; 0) = (x); u = v = 0;

v(x; 0) = (x); x ∈ ;

(x; t) ∈ @ × R+ ;

(1.2) (1.3)

where ⊂ Rn is a bounded domain with C 2 boundary; (x); (x) ≥ 0; x ∈ ; constants pi ; qi ≥ 0; i = 1; 2: System (1.1) models such as heat propagations in a two-component combustible mixture [5]; chemical processes [12]; interaction of two biological groups without selflimiting [2, 15, 19], etc. Recently, a number of works have been contributed to the study of (1.1) [3, 5, 6, 19], especially its special cases p1 = q2 = 0 (variational) or q1 = p2 = 0 (uncoupled single equation). For the case p1 = q2 = 0, Escobedo and Herrero analyzed the boundedness ∗ Correspondence address: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P.R. China. E-mail: [email protected] 1

Supported by National Natural Science Foundation of China.

0362-546X/99/$ - see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 1 7 1 - 0

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S. Zheng / Nonlinear Analysis 39 (2000) 327 – 340

and blow-up of solutions [5]; Caristi and Mitidieri obtained the blow-up estimates of solutions [3]. Wu and Yuan studied the uniqueness of generalized solutions with degenerate diusion [19]. It is well known that there have been much more results for the uncoupled single equation case q1 = p2 = 0, including necessary and sucient conditions for nite blow-up [7], estimates of blow-up time [4, 9] and blow-up rates [18], shapes and measures of blow-up sets [10, 13, 14], blow-up behavior [4, 11, 14], etc. In particular, the following global existence and global non-existence results are basic and well known (see e.g. [7]) concerning the single parabolic equation ut − Lu = f(u); u(x; 0) = (x); u = 0;

(x; t) ∈ × R+ ; x ∈ ;

(1.4) (1.5)

(x; t) ∈ @ × R+ ;

(1.6)

where Lu :=

n X

(aij (x)uxi )xj +

i; j=1

n X

bi (x)uxi + c(x)u;

i=1

(aij (x)) is uniformly positive de nite matrix, the coecients of L are suciently smooth  × [0; ∞);  is Holder continuous in ,  and f(u) is Lipschitz continuous in R. in

R∞ du=f(u) = ∞ Theorem 1.1. If f(u) ≥ 0; f0 (u)¿0 for u¿0; f is concave with and  ≥ 0; then all the solutions of Eqs. (1.4) – (1.6) are known to be global. R∞ 0 Theorem 1.2. du=f(u)¡∞ R If f(u) ≥ 0; f (u)¿0 for u¿0; f is convex with and  ≥ 0;  is suciently large; then there is a T ¿0 such that the solution of Eqs. (1.4)–(1.6) exists in QT := × (0; T ) but does not exist in QT +” for any ”¿0. Remark 1. The typical example of f(u) in Theorems 1:1 or 1:2 is f(u) = up ; 0¡p ≤ 1 or p¿1; respectively. These two theorems with f = up and L =
consist of our foundation of this paper. The general form of (1.1) was systematically studied by Escobedo and Levine [6]. They give a complete analysis on the critical blow-up and the global existence numbers for the Cauchy problem of (1.1), where the introduced parameters  and satisfying      q1  1 p1 − 1 = (1.7) p2 q2 − 1 1 played important roles in their framework. In a previous work [20] we established the sucient conditions for the uniform boundedness of positive solutions to Eqs. (1.1) – (1.3) by the following method: rst got the uniform L1 estimates for u and v, and then obtained the uniform L∞ estimates

S. Zheng / Nonlinear Analysis 39 (2000) 327 – 340

329

by multiplying the equations with powers of u and v and applying the interactive procedure, motivated by Alikakos [1] and Rothe [16]. In this paper, by means of a detail classi cation of parameters pi ; qi ; i = 1; 2, we give several criteria for the global existence and global non-existence of positive solutions to problem (1.1)–(1.3). The global existence conditions obtained in Section 2 are 0¡pi + qi ≤ 1; pi ; qi ≥ 0; i = 1; 2. In Section 3 we prove that if pi + qi ¿1; pi ; qi ≥ 0; i = 1; 2; max(p1 − p2 ; q2 − q1 )¡1, or p1 + q1 ¿p2 + q2 = 1; p1 − p2 ¡1; p2 q1 ¿0, then there is no positive global solution to Eqs. (1.1) – (1.3) with large initial data. We propose more precise criteria in Section 4 that the global existence may hold even for, e.g. p1 + q1 ¿1, and the global non-existence may be true even for, e.g. p2 + q2 ¡1. 2. Global existence of solutions Corresponding to Theorem 1.1 for the single equation case we have the following global existence result to Eqs. (1.1)–(1.3): Theorem 2.1. Assume 0¡pi + qi ≤ 1; pi ; qi ≥ 0; i = 1; 2. Then all the solutions of Eqs. (1.1)–(1.3) are nonnegative and global. Proof. Clearly, the nonlinear reaction terms in Eq. (1.1) are quasimonotonically increasing. Put U (t) = Cet ;

V (t) = Cet

with C ≥ max(sup (x); sup (x)) ≥ min(sup (x); sup (x)) ≥ 1. Then, we can consider (0; 0) and (U (t); V (t)) as a pair of sub-supersolutions [17] for problem (1.1) – (1.3), where it is easy to check that, e.g., U (t) and V (t) satisfy Ut (t) ≥
U (t) + U (t)p1 V (t)q1 = U (t)p1 +q1 ; Vt (t) ≥
V (t) + U (t)p2 V (t)q2 = V (t)p2 +q2 with pi + qi ≤ 1; i = 1; 2. Moreover (U (t); V (t)) ≥ (u(x; t); v(x; t)) ≥ (0; 0) on the parabolic boundary. Thus 0 ≤ u(x; t) ≤ U (t);

0 ≤ v(x; t) ≤ V (t)

for (x; t) ∈ QT = × (0; T ). On the other hand, the (uncoupled) ODE system U 0 (t) = U (t)p1 +q1 ; V 0 (t) = V (t)p2 +q2 has global solutions provided pi + q1 ≤ 1; i = 1; 2. We get immediately the conclusions of the theorem. Remark 2. Obviously; for the special case q1 = p2 = 0; p1 = q2 = p ≤ 1; Theorem 2:1 is equivalent to Theorem 1:1 with L =
; f = up ; p ≤ 1.

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The following two lemmas provide the critical property of parameters  and corresponding to the global existence. Lemma 2.2. If 0¡pi + qi ¡1; pi ; qi ≥ 0; i = 1; 2; then max(; )¡0: Proof. We know from Eq. (1.7) that =

1 + q 1 − q2 ; p2 q1 − (1 − p1 )(1 − q2 )

(2.1)

=

1 + p2 − p1 : p2 q1 − (1 − p1 )(1 − q2 )

(2.2)

Since min(1 + q1 − q2 ; 1 + p2 − p1 )¿0 due to 0¡pi + qi ¡1; pi ; qi ≥ 0; i = 1; 2, it suces to show  := p2 q1 − (1 − p1 )(1 − q2 )¡0; which is obviously true because 1 − p1 ¿q1 ≥ 0; 1 − q2 ¿p2 ≥ 0. Lemma 2.3. If p2 q1 ¿0; 0¡p2 + q2 ¡p1 + q1 = 1 or 0¡p1 + q1 ¡p2 + q2 = 1; then max(; )¡0: Proof. Suppose 0¡p2 + q2 ¡p1 + q1 = 1. Since p2 q1 ¿0, we know that 0¡p2 ¡1 − q2 ; 0¡q1 = 1 − p1 , hence  = p2 q1 − (1 − p1 )(1 − q2 )¡p2 q1 − p2 (1 − p1 ) = 0. On the other hand, 1 + p2 − p1 ¿p2 ¿0; 1 + q1 − q2 ¿1 − q2 ¿p2 ¿0. So, we get max(; ) ¡0. The following key proposition will play some important roles, especially for establishing the global non-existence of solutions in this paper. Proposition 2.4. Let ; be deÿned by (2:1); (2:2). Assume  ¿0 and max(p1 − p2 ; q2 − q1 )¡1. Then the solution (u; v) of (1.1) – (1.3) satisÿes either u(x; t) ≥ k1 v(x; t)=

if

p2 + q2 ≤ p1 + q1

(2.3)

u(x; t) ≤ k2 v(x; t)=

if

p1 + q1 ≤ p2 + q2

(2.4)

or

for all (x; t) ∈ QT \Q with positive constants k1 ; k2 and  ∈ (0; T ); T is the maximum existence time for solution (u; v).

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331

Proof. Denote H1 := q1 − q2 −

(p2 − p1 )(1 + q1 − q2 ) ; 1 + p2 − p1

H2 := p2 − p1 −

(q1 − q2 )(1 + p2 − p1 ) : 1 + q1 − q2

(2.5) (2.6)

We claim that max(H1 ; H2 ) ≥ 0:

(2.7)

Since max(p1 − p2 ; q2 − q1 )¡1 implies min(1 + p2 − p1 ; 1 + q1 − q2 )¿0, we have that if p2 + q2 ≤ p1 + q1 , then H1 ≥ 0 is true because (q1 − q2 )(1 + p2 − p1 ) − (p2 − p1 )(1 + q1 − q2 ) = q1 − q2 − (p2 − p1 ) ≥ 0; if p1 + p2 ≤ p2 + q2 , then H2 ≥ 0 is true because (p2 − p1 )(1 + q1 − q2 ) − (q1 − q2 )(1 + p2 − p1 ) = p2 − p1 − (q1 − q2 ) ≥ 0: Motivated by Friedman and Giga [8], construct k(u) =

u1+p2 −p1 ; 1 + p2 − p1

h(v) =

v1+q1 −q2 ; 1 + q1 − q2

and set J = M1 k(u) − h(v);

I = M2 h(v) − k(u):

Proceeding as in the proof of Lemma 3.2 in [8], we get for simplicity of presentation with n = 1 that Jt − Jxx − b1 Jx − c1 J = (M1 − 1)up2 vq1   (p2 − p1 )(1 + q1 − q2 ) 2 q1 −q2 −1 +v q1 − q2 − vx ; 1 + p2 − p1 It − Ixx − b2 Ix − c2 I = (M2 − 1)up2 vq1   (q1 − q2 )(1 + p2 − p1 ) 2 p2 −p1 −1 p2 − p1 − ux +u 1 + q1 − q2

(2.8)

(2.9)

for suitable bi ; ci ; i = 1; 2. We know from Eqs. (2.7) – (2.9) that either Jt − Jxx − b1 Jx − c1 J ≥ 0

(2.10)

It − Ixx − b2 Ix − c2 I ≥ 0

(2.11)

or

holds for Mi ≥ 1; i = 1; 2. On the other hand, J = I = 0 for (x; t) ∈ @ × (0; T ) due to the boundary condition (1.3). Fix a small 1 ¿0 and choose M1 ≥ 1 such that

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M1 k(u(x; 1 ))¿h(v(x; 1 )); x a small 2 ¿0 and choose M2 ≥ 1 such that M2 h(v(x; 2 )) ¿k(u(x; 2 )). Let  := max(1 ; 2 ), then by the maximum principle we have either J = M1 k(u) − h(v) ≥ 0

or

I = M2 h(v) − k(u) ≥ 0

and, since  ¿0, equivalently, either u(x; t) ≥ k1 v(x; t)(1+q1 −q2 )=(1+p2 −p1 ) = k1 v(x; t)= or u(x; t) ≤ k2 v(x; t)(1+q1 −q2 )=(1+p2 −p1 ) = k2 v(x; t)= holds for all (x; t) ∈ QT \Q . Remark 3. Theorem 2:1 can be obtained by means of Proposition 2:4 as well. For example 0¡pi + qi ¡1; pi ; qi ≥ 0; i = 1; 2 or 0¡p2 + q2 ¡p1 + q1 = 1 with p2 q1 ¿0; pi ; qi ≥ 0; i = 1; 2; we know from Lemmas 2:2 and 2:3 that max(; )¡0. By using Proposition 2:4 we have v ≤ cu =

for (x; t) ∈ × [; ∞):

(2.12)

Substituting Eq. (2:12) into the ÿrst equation of Eq. (1:1); we get ut −
u ≤ cq1 up1 uq1 = =: 1 up :

(2.13)

It follows from Eq. (1:7) and Lemma 2:2 that 0¡p =

1 p1  + q1  + 1 = = 1 + ¡1:   

By using the comparison principle to Eqs. (2:13); (1:4) and Theorem 1:1 with L =
; f =1 up we obtain that the ÿrst component u of the solution (u; v) of Eqs. (1:1)– (1:3) exists globally. This implies the global existence of the second component v as well since v ≤ cu = . We will use Proposition 2.4 to establish the global non-existence results in the next section. 3. Global non-existence of solutions We describe some conditions under which there is no global solution to Eqs. (1.1)–(1.3) with large initial data. Theorem 3.1. Assume (i) pi + qi ¿1; pi ; qi ≥ 0; i = 1; 2; (ii) max(p1 − p2 ; q2 − q1 )¡1;

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333

R R (iii) ; large enough. Then there is no positive global solution to Eqs. (1.1) – (1.3). Theorem 3.2. Assume (i) p1 + q1 ¿p2 + q2 = 1; (ii) p1 − p2 ¡1; (iii) p R R 2 q1 ¿0; (iv) ; large enough. Then there is no positive global solution to Eqs. (1.1) – (1.3). Remark 4. The assumption p2 q1 ¿0 in Theorem 3:2 means that the system (1:1) is completely coupled. This is just the nontrivial case we are interested in. Corresponding to Lemmas 2.2 and 2.3, the following two lemmas describe the critical property of parameters  and needed for the blow-up case: Lemma 3.3. Under conditions (i), (ii) of Theorem 3.1 we have min(; )¿0: Lemma 3.4. Under the conditions (i)–(iii) of Theorem 3.2 we have min(; )¿0: Proof of Lemma 3.3. We know from Eqs. (2.1) and (2.2) that it suces to show p2 q1 − (p1 − 1)(q2 − 1)¿0 is true. If (p1 − 1)(q2 − 1) = 0, then p1 = 1 and hence p2 ¿p1 − 1 = 0 by condition (ii), q1 ¿0 due to p1 +q1 ¿1; or q2 = 1 and hence q1 ¿q2 −1 = 0; p2 ¿0 due to p2 +q2 ¿1. We have p2 q1 − (p1 − 1)(q2 − 1)¿0: If (p1 − 1)(q2 − 1)¡0, then p2 q1 − (p1 − 1)(q2 − 1) ≥ − (p1 − 1)(q2 − 1)¿0. If (p1 − 1)(q2 − 1)¿0, then for p1 − 1¿0 and q2 − 1¿0, we have p2 ¿p1 − 1¿0 and q1 ¿q2 − 1¿0, and so p2 q1 − (p1 − 1)(q2 − 1)¿0; for p1 − 1¡0 and q2 − 1¡0, we have p2 ¿1 − q2 ¿0; q1 ¿1 − p1 ¿0 due to pi + qi ¿1; i = 1; 2, and so p2 q1 − (p1 − 1)(q2 − 1)¿0. Remark 5. If we assume min(p1 − p2 ; q2 − q1 )¿1 instead of max(p1 − p2 ; q2 − q1 )¡1 in Lemma 3:3; then it is easy to check p2 q1 −(p1 −1)(q2 −1)¡0 and so min(; )¿0 holds as well. Proof of Lemma 3.4. p2 ; q1 ¿0 with condition (i) imply q2 = 1 − p2 ¡1 and q2 − q1 ¡1 − q1 ¡1. So we have max(p1 − p2 ; q2 − q1 )¡1. On the other hand, it follows from 1 − q2 = p2 ¿0 and p1 + q1 ¿1 that p2 q1 − (p1 − 1)(q2 − 1) = p2 (q1 + p1 − 1) ¿0.

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Proof of Theorem 3.1. Observe that min(; )¿0 by Lemma 3.3 and max(p1 − p2 )(q2 −q1 )¡1 by condition (ii). Without loss of generality, suppose p2 +q2 ≤ p1 +q1 . According to Proposition 2.4 we know that the solution (u; v) of Eqs. (1.1) – (1.3) satis es u(x; t) ≥ k1 v(x; t)=

for (x; t) ∈ QT \Q :

(3.1)

Substitute Eq. (3.1) into the second equation of Eq. (1.1) to get vt −
v ≥ k1p2 vp2 = vq2 = k1p2 v(p2 +q2 )= =: 1 vq :

(3.2)

We claim that q¿1 in Eq. (3.2). Indeed q − 1=

1+ 1 p2  + q2 − 1= − 1 = ¿0

by Eq. (1.7) and Lemma 3.3. Denote by T1 the maximum existence time for the single equation Vt −
V = 1 V q

(3.3)

with initial and boundary conditions (1.5) and (1.6). Then, it follows from the comparison principle with Eqs. (3.2)–(3.3) and Theorem 1.2 with L =
; f = V q that there exists T ≤ T1 such that lim sup v(x; t) = ∞:

t→T

Thereby we get the blow-up result for u(x; t) as well because of Eq. (3.1). The case of p1 + q1 ≤ p2 + q2 can be treated in the same way. Proof of Theorem 3.2. Observing min(; )¿0 by Lemma 3.4; max(p1 − p2 ; q2 − q1 )¡1 by conditions (i), (ii); and p2 + q2 ¡p1 + q1 by condition (i), we conclude from Proposition 2.4 that Eq. (3.1) is true. We can repeat the procedure as that in the proof of Theorem 3.1 to get the nite blow-up conclusion for the positive solution (u; v) of Eqs. (1.1)–(1.3). For the variational case p1 = q2 = 0; q1 = p¿0; p2 = q¿0, i.e., ut =
u + vp ; vt =
v + uq ;

(x; t) ∈ × R+

(3.4)

with initial and boundary data (1.2) and (1.3), we have max(p1 − p2 ; q2 − q1 )¡0, i.e. the condition (ii) in Theorems 3.1 and 3.2 is satis ed automatically. Combining Theorems 3.1 and 3.2 with Theorem 2.1, we have the following result: Theorem 3.5. If p ≤ 1; q ≤ 1; then all the solutions of Eqs. (3.4), (1.2) and (1.3) are global. If p¿1; q ≥ 1 or p ≥ 1; q¿1; then there is no global solution with large initial data.

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335

4. Global existence with p1 + q1 ¿1 and global non-existence with p2 + q2 ¡1 We begin with a more precise extension of Theorem 3.5. Theorem 4.1. If pq ≤ 1, then all the solutions of Eqs. (3.4), (1.2) and (1.3) are global. If pq¿1; then there is no global solution with large initial data. Proof. (i) Assume pq¿1. We know from Eqs. (2.1) and (2.2) that =

1+p ¿0; pq − 1

=

1+q ¿0: pq − 1

Without loss of generality, suppose q ≤ p. Then max(p1 − p2 ; q2 − q1 ) = max(−q; −p) = −q¡0¡1; p2 + q2 = q ≤ p1 + q1 = p: We have by using Proposition 2.4 that u ≥ k1 v= = k1 v(1+p)=(1+q)

for (x; t) ∈ QT \Q :

(4.1)

Substituting (4.1) into the second equation of (3.4), we get vt −
v ≥ k1q vq(1+p)=(1+q) =: 1 v 1 :

(4.2)

Since ; ¿0, it is easy to check that 1 − 1 =

pq − 1 1 q(1 + p) − 1= = ¿0: 1+q 1+q

We can repeat the procedures as those in the proof of Theorem 3.1 to get the global non-existence conclusion here. (ii) Assume pq ≤ 1. Without loss of generality, suppose q ≤ p. Then there exists

≥ 1 such that q ≤ 1 and p= ≤ 1. Set U (t) = Ce t ; V (t) = Cet with C ≥ max(sup (x); sup (x)). Then, as a pair of supersolution of (3.4), (1.2), (1.3), U (t) and V (t) satisfy Ut (t) ≥
U (t) + C1 V (t)p = C2 U (t)p= ;

(4.3)

Vt (t) ≥
V (t) + C3 U (t)q = C4 V (t) q :

(4.4)

The global existence result follows directly from (4.3), (4.4) with q ≤ 1; p= ≤ 1. Remark 6. The conclusion of Theorem 4:1 is interesting that the solution of Eqs. (3:16); (1:2) and (1:3) may globally exist even if max(p; q)¿1; and may blow up in a ÿnite time even with min(p; q)¡1. As the result of the interaction between the two components; the only criterion is pq ≤ 1 or pq¿1.

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Let us extend Theorem 4.1 to the more general cases furthermore. We will give blow-up conditions even p2 + q2 ¡1 and global existence conditions even p1 + q1 ¿1 to (1.1) –(1.3). Lemma 4.2. Assume either (i) p2 + q2 ¡1¡p1 + q1 ; 1 − p2 1 =2 ¡p1 ¡1 + p2 with 1 := p1 + q1 − 1; 2 := 1 − p2 − q2 ; or (ii) p2 + q2 = 1¡p1 + q1 ; p1 ¡1 + p2 with p2 q1 ¿0: Then min(; )¿0: Proof. It is easy to check from condition (i) or (ii) that max(p1 − p2 ; q2 − q1 )¡1: In view of Eqs. (2.1) and (2.2), we just need to show p2 q1 − (1 − p1 )(1 − q2 )¿0. For case (i), denote  := (p1 + q1 )(p2 + q2 ): Then (p1 + q1 )(p2 + q2 ) −  = p1 (p2 + q2 ) + q2 (p1 + q1 ) + p2 q1 − p1 q2 −  = 0:

(4.5)

By using Eq. (4.5) we have p2 q1 − (1 − p1 )(1 − q2 ) = p2 q1 − p1 q2 + p1 + q2 − 1 − [p1 (p2 + q2 ) + q2 (p1 + q1 ) + p2 q1 − p1 q2 − ] = p1 (1 − p2 − q2 ) − q2 (p1 + q1 − 1) − 1 +  = p1 2 − q2 1 − 1 + ¿0 provided p1 2 ¿q2 1 + 1 − : Noticing  = (p1 + q1 )(p2 + q2 ) = (1 + 1 )(p2 + q2 ); so q2 1 + 1 −  = 1 − p2 − q2 − p2 1 = 2 − p2 1 ; and hence Eq. (4.6) is equivalent to p1 2 ¿2 − p2 1 ;

(4.6)

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i.e., p1 ¿1 − p2 1 =2 :

(4.7)

For case (ii), we have clearly p2 q1 − (1 − p1 )(1 − q2 ) = p2 (p1 + q1 − 1)¿0. Under the conditions of Lemma 4.2 we know that p2 + q2 ¡p1 + q1 ; max(p1 − p2 ; q2 − q1 )¡1, and min(; )¿0. According to Proposition 2.4, we have u ≥ k1 v=

for (x; t) ∈ QT \Q :

Proceeding as in the proof of Theorem 3.1 we obtain the following blow-up results: Theorem 4.3. If the assumptions of Lemma 4:2 areRsatisÿed; R then there is no global solution to Eqs. (1.1)–(1.3) with large initial data ; . Remark 7. Since p2 + q2 ≤ 1 in Theorem 4:3; we have to have p2 ¿0 for the blowup of the second component v. Indeed; for case (i), if p1 ≥ 1; then 1 ≤ p1 ¡1 + p2 implies p2 ¿0; if p1 ¡1; then 1 − p2 1 =2 ¡p1 ¡1 rules out the possibility of p2 = 0. Moreover; if p1 ≤ 1; then we have also q1 ¿0 due to p1 + q1 ¿1. Either p2 ¿0 or q1 ¿0 describes that the two equations are really coupled. Finally, consider the global existence with p1 + q1 ¿1. Lemma 4.4. Assume p2 + q2 ¡1¡p1 + q1 ; p1 ¡1 − p2 1 =2 . Then max(; )¡0: Proof. Since max(p1 −p2 ; q2 −q1 ) ≤ max(p1 ; q2 )¡1, it suces to prove p2 q1 −(1−p1 ) (1 − q2 )¡0. Corresponding to Eq. (4.6) (or equivalently, Eq. (4.7)) we know that p1 2 ¡q2 1 + 1 − 

(4.8)

or equivalently p1 ¡1 − p2 1 =2

(4.9)

implies p2 q1 − (1 − p1 )(1 − q2 )¡0: It follows from Lemma 4.4 that max(; )¡0; p2 + q2 ¡p1 + q1 ; max(p1 − p2 ; q2 − q1 )¡1, i.e. the conditions of Proposition 2.4 are satis ed, so the comparison relationship (4.1) is true. We can use Remark 3 to get the following global existence theorem: Theorem 4.5. Assume the conditions in Lemma 4:4 hold. Then all the solutions of Eqs. (1.1)–(1.3) are global.

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Through the above discussion, we nd that either max(; )¡0 or min(; )¿0 is essential for the global existence or global non-existence of solutions to Eqs. (1.1)–(1.3). We have a more general theorem as follows. Theorem 4.6. (i) If p2 q1 − (1 − p1 )(1 − q2 ) ≤ 0

(4.10)

with p1 ; q2 ≤ 1; then all the solutions of Eqs. (1.1) – (1.3) are global. (ii) If p2 q1 − (1 − p1 )(1 − q2 )¿0

(4.11)

with min(1 + p2 − p1 ; 1 + q2 − q1 )¿0; then there is no global solution to Eqs. (1.1) – (1.3) with large initial data. Proof. (i) If p1 = 1 or q2 = 1, then p2 q1 = 0 by Eq. (4.10). For p2 = 0 together with q2 ≤ 1, we know by using Theorem 1.1 that now the second (uncoupled) equation of (1.1) vt −
v = vq2 has gobal solution v(x; t), i.e. for any T ¿0, there exists M (T )¿0 such that 0 ≤ v(x; t) ≤ M (T ) for (x; t) ∈ QT , and hence u(x; t) is global as well because ut −
u ≤ M (T )q1 up1 for (x; t) ∈ QT with p1 ≤ 1: For q1 = 0 with p1 ≤ 1, we can get the global existence at rst for u(x; t) and then for v(x; t) as above in the same procedure. If p1 ; q2 ¡1, we know from Eq. (4.10) that 0≤

q1 p2 · ≤ 1: 1 − q2 1 − p1

Thus, there exists constant ≥ 1 such that e.g. 0≤

p2 ≤ 1; 1 − q2

0≤

q1 ≤ 1;

(1 − p1 )

(4.12)

or equivalently

≥ p1 + q1 ;

1 ≥ p2 + q2 :

Set U (t) = Ce t ;

V (t) = Cet

(4.13)

S. Zheng / Nonlinear Analysis 39 (2000) 327 – 340

339

with C ≥ max(sup (x); sup (x)). We have Ut (t) ≥
U (t) + C5 U (t)p1 V (t)q1 = C6 U (t)( p1 +q1 )= ; p2

q2

p2 +q2

Vt (t) ≥
V (t) + C7 U (t) V (t) = C8 U (t)

:

(4.14) (4.15)

Since (U (t); V (t)) is a pair supersolution, we conclude the global existence of solutions from (4.13), (4.14) and (4.15). (ii) Clearly, it follows from (2.1) and (2.2) that min(; )¿0 with max(p1 −p2 ; q2 − q1 )¡1. Thus, we can use at rst Proposition 2.4 and then the procedure in the proof of Theorem 3.1 to get the blow-up result. Remark 7. Observe that p1 +q1 ¿1 is assumed in Theorem 4:5; and permited in (i) of Theorem 4:6 as well for the global existence conclusion. In this case of p1 + q1 ¿1; clearly; there should be the condtition q1 ¿0 (really coupled) to ensure the global existence of the ÿrst component u; which is really true because we have p1 ≤ 1 here. For the same reason; if we take p2 + q2 ≤ 1 in (ii) of Theorem 4:6; then there must be p2 ¿0 (really coupled) to make the blow-up for v; which holds also because p2 ¿p1 − 1 ≥ 0 if p1 ≥ 1 or p2 q1 ¿(1 − p1 )(1 − q2 ) ≥ 0 if p1 ¡1. Remark 8. If we take p1 = q2 = 0; q1 = p; p2 = q in Theorems 4:3 and 4:6; then 2 = 1 − p2 = 1 − q; 1 = q1 − 1 = p − 1. So (4:7) becomes 0¿1 − p2 − (q1 − 1)p2 = 1 − p2 q1 ; i.e., pq¿1; Eq. (4:10) becomes pq ≤ 1; and (4:11) becomes pq¿1. Therefore; Theorem 4:1 can be considered as somewhat special case of Theorems 4:3 and 4:6. Acknowledgements The author would like to thank the referee for his helpful suggestion to simplify the proof for the global existence in this paper. References [1] N.D. Alikakos, An application of the invariance principle to reaction-diusion equations, J. Dierential Equations 33 (1979) 201– 225. [2] R.C. Cantrell, C. Cosner, V. Huston, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh 123 A (1993) 535 – 559. [3] G. Caristi, E. Mitidieri, Blow-up estimates of positive solutions of a parabolic system, J. Dierential Equations 113 (1994) 265 – 271. [4] H.W. Chen, Analysis of blowup for a nonlinear degenerate parabolic equation, J. Math. Anal. Appl. 192 (1995) 180 –193. [5] E. Escobedo, M.A. Herrero, Boundedness and blow-up for a semilinear reaction-diusion system, J. Dierential Equations 89 (1991) 176 – 202. [6] E. Escobedo, H.A. Levine, Critical blowup and global existence numbers for weakly coupled system of reaction-diusion equations, Arch. Rational Mech. Anal. 129 (1995) 47–100. [7] A. Friedman, Blow-up of solutions of nonlinear parabolic equations, in: W.M. Ni, P.L. Peletier, J. Serrin (Eds.), Nonlinear Diusion Equations and Their Equilibrium States I, Springer, New York, 1988. [8] A. Friedman, Y. Giga, A single point blow-up for solutions of semilinear parabolic system, J. Fac. Sci. Univ. Tokyo Set. IA Math. 34 (1987) 65 – 71.

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