Nonlinear Analysis 48 (2002) 349 – 362 www.elsevier.com/locate/na Convergence of solutions of reaction–di usion systems with time delays C.V Pao ∗ Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA Received 18 July 1999; accepted 22 March 2000 Keywords: Reaction–di usion equations; Time delays; Asymptotic behavior; Global attractor; Upper and lower solutions; Volterra–Lotka models Introduction Di erential equations with discrete or continuous time delays are traditionally formulated in the framework of ordinary di erential systems and much discussions are devoted to the qualitative analysis of the systems In recent years attention has been given to parabolic systems where the e ect of di usion and convection is taken into consideration In this paper we investigate the asymptotic behavior of solutions for a class of reaction–di usion–convection systems with time delays in a bounded domain in RP under Neumann boundary condition The system of equations under consideration is given in the form @ui =@t − Li ui = fi (u; J ∗ u) (t¿0; x ∈ ); @ui =@ = (t¿0; x ∈ @ ); ui (t; x) = Ái (t; x) (t ∈ Ii ; x ∈ ); i = 1; : : : ; N; (1.1) where u = (u1 ; : : : ; uN ); J ∗u = (J1 ∗u1 ; : : : ; JN ∗uN ) and @ui =@ denotes the outward normal derivative of ui on the boundary @ of For each i = 1; : : : ; N; Li is a uniformly ∗ Tel.: 919-515-2382; fax: 919-515-3798 E-mail address: cvpao@math.ncsu.edu (C.V Pao) 0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd All rights reserved PII: S - X ( 0 ) 0 - 350 C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 elliptic operator in the form P L i ui ≡ j; k = (i) ajk (x)@2 ui =@xj @xk + P bj(i) (x)@ui =@xj ; j=1 fi (u; J ∗ u) is, in general, a nonlinear function of u and J ∗ u, and the interval Ii and the components Ji ∗ ui of J ∗ u, which represent the time delays, are given by J i ∗ ui = t −∞ Ji (t − s)ui (s; x) ds; Ji ∗ ui = ui (t − ri ; x); Ii = [ − ri ; 0] Ii = (−∞; 0] for i = 1; : : : ; n0 ; for i = n0 + 1; : : : ; N; (1.2) where ri is a positive constant and n0 ≤ N is a nonnegative integer The special case n0 = represents that the time delays in the system are all of discrete type In the general case n0 ≥ we allow Ji ∗ ui to be the ÿnite integral Ji ∗ ui = t −ri Ji (t − s)ui (s; x) ds (1.3) for some or all i ≤ n0 , where 0¡ri ¡∞ In this situation the continuous time delay is ÿnite and the corresponding interval, Ii in (1.2) is replaced by [ − ri ; 0] The above consideration of Ji ∗ ui includes various combination of continuous and discrete time delays, and in the case of continuous delays it may be either ÿnite or inÿnite It is assumed that the function Ji (t) is piecewise continuous in R+ ≡ [0; ∞) and possesses the property Ji (t) ≥ 0; ∞ Ji (t) dt = 1; (t ≥ 0); i = 1; : : : ; n0 : (1.4) For ÿnite continuous delays the above condition is replaced by Ji (t) ≥ 0; ri Ji (t) dt = 1; (0 ≤ t ≤ ri ): (1.5) Reaction–di usion systems in the form of (1.1) have been treated by many investigators and di erent methods have been used for the qualitative analysis (cf [1–13,15 –19]) The discussions in earlier works are mostly in the framework of semigroup theory and the theory of dynamical systems (cf [4,5,8,16 –18] and the references therein) More recently, the method of upper and lower solutions and its associated monotone iterations have been used to investigate the dynamic property of the system (cf [6,10 –13]) An advantage of this method is that it can lead to various qualitative information of the solution as well as computational algorithm for numerical solutions (cf [6,14]) Although the above methods are useful for obtaining invariant regions of reaction–di usion systems, the determination of the precise asymptotic limit of the time-dependent solution is, in general, more di cult especially when the system possesses multiple steady-state solutions One of the di culties is the lack of explicit information about the steady-state solutions of the corresponding elliptic boundary-value problem when the boundary condition is of Dirichlet or Robin type On the other hand, if the boundary condition is of Neumann type as that in (1.1) then constant steady-state C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 351 solutions can often be found from the nonlinear reaction function in the system The purpose of this paper is to investigate the asymptotic behavior of the time-dependent solution of (1.1) in relation to constant steady-state solutions, including regions of attraction of the stable steady solutions These results are given in Section Speciÿcally, we obtain su cient conditions on the nonlinear function (f1 ; : : : ; fN ) for the global attractor and the convergence of the time-dependent solution It turns out that these conditions are independent of the di usion–convection coe cients and the time delays In Section we apply the general results in Section to two Volterra–Lotka models in ecology for studying the global stability and instability of the various constant steady-state solutions The stability conditions for these model problems are given in terms of the rate constants of the reaction function and is independent of the time delays and the e ect of di usion Global existence and dynamics Let Q0(i) ≡ (−∞; 0] × Q0 ≡ Q0(1) for i = 1; : : : ; n0 ; Q0(i) ≡ [ − ri ; 0] × (i) Q0 ; Q0 Q0(N ) , for i = n0 + 1; : : : ; N and × ··· × and let be the same domains as Q0(i) and Q0 deÿned above when is replaced by ≡ ∪ @ For any domain D we denote by C m+ (D) the set of scalar-valued functions that are Holder continuous in D (with exponent ∈ (0; 1)) and by Cm+ (D) the corresponding set of vector-valued functions in Cm (D) For any vector w = (w1 ; : : : ; wN ) in RN we write |w| = |w1 | + · · · + |wN | It is always assumed that @ is of class C 1+ and for each i = 1; : : : ; N , the coe cients of Li and (i) (i) are in C ( ) and Ái ∈ C (Q0 ) For inÿnite time delays we the ÿrst derivatives of ajk also assume that −∞ |Ái (t; x)| dt¡∞ In addition to the above general assumptions for parabolic equations we impose the following basic hypothesis on the vector-function f : × ∗ → RN , where and ∗ N are subsets of R and f(u; v) ≡ (f1 (u; v); : : : ; fN (u; v)): (H) The function f(u; v) is mixed quasimonotone in a subset × for each i = 1; : : : ; N; fi (u; v) satisÿes the Lipschitz condition ∗ of RN × RN , and |fi (u; v) − fi (u ; v )| ≤ Ki (|u − u | + |v − v |) for (u; v); (u ; v ) in × ∗ ; (2.1) where Ki is a positive constant Recall that by writing the vectors u; v in the split form u ≡ (ui ; [u]ai ; [u]bi ); v ≡ ([v]ci ; [v]di ); where [w] denotes a vector with number of components of w, the function f(u; v) is said to be mixed quasimonotone in × ∗ if for each i = 1; : : : ; N , there exist nonnegative integers ; bi ; ci and di satisfying the relations + bi = N − and ci + d i = N (i = 1; : : : ; N ) (2.2) 352 C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 such that the function fi (u; v) ≡ fi (ui ; [u]ai ; [u]bi ; [v]ci ; [v]di ) is monotone nondecreasing in [u]ai and [v]ci , and is monotone nonincreasing in [u]bi and [v]di for all (u; v) ∈ × ∗ (cf [11,13]) The subsets ; ∗ in hypothesis (H) are ˆ c˜ , where taken as = ∗ = c; ˆ c˜ ≡ {c ∈ RN ; cˆ ≤ c ≤ c} ˜ c; (2.3) N ˆ c˜ are a pair of constant vectors in R and c; such that cˆ ≤ c˜ and fi (c˜i ; [c˜ ]ai ; [cˆ ]bi ; [c˜ ]ci ; [cˆ ]di ) ≤ 0; fi (cˆi ; [cˆ ]ai ; [c˜ ]bi ; [cˆ ]ci ; [c˜ ]di ) ≥ (i = 1; : : : ; N ): (2.4) Under hypothesis (H) we have the following global existence–uniqueness result ˜ cˆ be a pair of constant vectors satisfying c˜ ≥ cˆ and condition Theorem 2.1 Let c; ˆ c˜ Then problem (1:1) has a (2:4); and let hypothesis (H) holds with = ∗ = c; unique global solution u(t; x) such that cˆ ≤ u(t; x) ≤ c˜ for all t¿0; x ∈ ; (2.5) whenever cˆ ≤ Á(t; x) ≤ c˜ in Q0 ; where Á = (Á1 ; : : : ; ÁN ) Proof It is known that if problem (1.1) has a pair of coupled upper and lower solutions ˜ uˆ and hypothesis (H) holds in the sector u; ˆ u˜ then there exists a unique solution u; ˆ u˜ , where u; ˆ u˜ is deÿned in the same form as that in u(t; x) to (1:1) and u ∈ u; (2.3) (cf [12,13]) For the present system (1.1), coupled upper and lower solutions u˜ ≡ (u˜ ; : : : ; u˜ N ); uˆ ≡ (uˆ ; : : : ; uˆ N ) are required to satisfy u˜ ≥ uˆ and the relation ˜ ; [u] ˆ bi ; [J ∗ u] ˜ ci ; [J ∗ u] ˆ di ); @u˜ i =@t − Li u˜ i ≥ fi (u˜ i ; [u] ˆ ; [u] ˜ bi ; [J ∗ u] ˆ ci ; [J ∗ u] ˜ di ) @uˆ i =@t − Li uˆ i ≤ fi (u˜ i ; [u] @u˜ i =@ ≥ ≥ @uˆ i =@ in D; on S; u˜ i (t; x) ≥ Ái (t; x) ≥ uˆ i (t; x) in Q0(i) (i = 1; : : : ; N ); (2.6) where D ≡ (0; ∞) × and S ≡ (0; ∞) × @ It is easy to verify from condition (2:4) and J ∗ c = c for every constant vector c that all the inequalities in (2.6) are satisÿed ˜ x) = c˜ and u(t; ˆ x) = cˆ whenever c˜ ≥ Á(t; x) ≥ cˆ in Q0 The by the constant vectors u(t; existence of a unique solution u(t; x) and relation (2.5) follows from Theorem 2.2 of [13] (see also [12]) To investigate the dynamics of the system we deÿne two sequences of constant (m) (m) (m) } ≡ {c(m) vectors {c(m) } ≡ {c(m) ; : : : ; cN }; {c ; : : : ; cN } from the recursion relation c(m) = c(m−1) + fi (c(m−1) ; [c(m−1) ]ai ; [c(m−1) ]bi ; [c(m−1) ]ci ; [ c(m−1) ]di ); i i i Ki C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 c(m) = c(m−1) + i i 353 fi (c(m−1) ; [c(m−1) ]ai ; [c(m−1) ]bi ; [ c(m−1) ]ci ; [c(m−1) ]di ) i Ki for m = 1; 2; : : : (i = 1; : : : ; N ); (2.7) ˜ c(0) = cˆ and Ki is the Lipschitz constant in (2.1) It is clear that these sewhere c(0) = c; quences of constant vectors are well deÿned The following lemma gives the monotone property of these sequences Lemma 2.1 The sequences {c(m) }; {c(m) } given by (2:7) with c(0) = c˜ and c(0) = cˆ possess the monotone property ˜ cˆ ≤ c(m) ≤ c(m+1) ≤ c(m+1) ≤ c(m) ≤ c; m = 1; 2; : : : : (2.8) ˜ c) ˆ we have Proof By (2.7), (2:4) and (c(0) ; c(0) ) = (c; (1) (0) = c(0) c(0) i − ci i − ci + =− fi (c˜i ; [c˜ ]ai ; [cˆ ]bi ; [c˜ ]ci ; [cˆ ]di ) ≥ 0; Ki (0) c(1) = c(0) i − ci i + = (0) (0) (0) (0) fi (c(0) i ; [c ]ai ; [ c ]bi ; [c ]ci ; [ c ]di ) Ki (0) (0) (0) (0) (0) fi (c(0) i ; [c ]ai ; [c ]bi ; [ c ]ci ; [c ]di ) − ci Ki fi (cˆi ; [cˆ ]ai ; [c˜ ]bi ; [cˆ ]ci ; [c˜ ]di ) ≥ Ki (i = 1; : : : ; N ): (1) (0) and c(1) for i = 1; : : : ; N Similarly by (2.7), (2:1) and the This gives c(0) i ≥ ci i ≥ ci quasimonotone property of f(u; v), (1) (0) (0) (0) (0) (0) (0) (0) Ki (c(1) i − ci ) = Ki (ci − ci ) + fi (ci ; [c ]ai ; [ c ]bi ; [c ]ci ; [ c ]di ) (0) (0) (0) (0) −fi (c(0) i ; [c ]ai ; [c ]bi ; [ c ]ci ; [c ]di ) ≥ 0: (1) This yields c(1) for i = 1; : : : ; N The above conclusions show that c(0) ≤ c(1) ≤ i ≥ ci (1) (0) c ≤ c Assume, by induction, that c(m−1) ≤ c(m) ≤ c(m) ≤ c(m−1) for some m¿1 Then by (2:7) and hypothesis (H), − c(m+1) ) Ki (c(m) i i (m−1) − c(m) ; [c(m−1) ]ai ; [ c(m−1) ]bi ; [c(m−1) ]ci ; [ c(m−1) ]di ) = (c(m−1) i i ) + fi (ci (m) −fi (c(m) ]ai ; [c(m) ]bi ; [c(m) ]ci ; [c(m) ]di ) i ; [c ≥ 0: (m+1 (m+1) for i =1; : : : ; N A similar argument gives c(m) ≤ c(m+1) This leads to c(m) i ≥ ci i ≤ci i for i = 1; : : : ; N The monotone property (2.8) follows by the principle of induction 354 C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 In view of the monotone property (2.8) the constant limits lim c(m) = c m→∞ and lim c(m) = c (2.9) m→∞ exist and satisfy the relation ˜ cˆ ≤ c(m) ≤ c(m+1) ≤ c ≤ c ≤ c(m+1) ≤ c(m) ≤ c: (2.10) Letting m → ∞ in (2.7) shows that the limits c ≡ (c1 ; : : : ; cN ); c ≡ (c1 ; : : : ; cN ) satisfy the equations fi (ci ; [c ]ai ; [c ]bi ; [c ]ci ; [c ]di ) = 0; fi (ci ; [ c ]ai ; [c ]bi ; [c ]ci ; [c ]di ) = (i = 1; : : : ; N ): (2.11) It is clear that the constant vectors c; c are not necessarily steady-state solutions of (1.1) unless c = c In the latter case, c (or c) is the unique steady-state solution in ˆ c˜ (cf [13]) In the following theorem we show the asymptotic behavior of the c; time-dependent solution u(t; x) in relation to c and c Theorem 2.2 Let the conditions in Theorem 2:1 hold, and let c; c be the limits in (2:9) Then for any initial function Á(t; x) in c; ˆ c˜ the solution u(t; x) of (1:1) satisÿes the relation c ≤ u(t; x) ≤ c as t → ∞ (x ∈ ): ∗ (2.12) ∗ ˆ c˜ If, in addition, c = c( ≡ c ) then c is the unique steady-state solution of (1:1) in c; and lim u(t; x) = c∗ ; t→∞ (x ∈ ): (2.13) Proof Consider the steady-state problem −Li ui = fi (u; u) in @ui =@ = ; on @ : (2.14) ˜ uˆ in C ( )∩C( ) are called coupled In analogy to relation (2.6), a pair of functions u; upper and lower solutions of (2.14) if u˜ ≥ uˆ on and they satisfy the inequalities in (2.6) without the time derivative terms and the initial conditions (cf [12,13]) This ˜ cˆ are coupled upper and lower solutions of (2.14) implies that the constant vectors c; Let u(0) = c˜ and u(0) = cˆ be a pair of initial iterations in the linear iteration process (m) −Li u(m) i + Ki u i = Ki u(m−1) + fi (u(m−1) ; [u(m−1) ]ai ; [u(m−1) ]bi ; [u(m−1) ]ci ; [ u(m−1) ]di ); i i (m) −Li u(m) i + Ki u i = Ki u(m−1) + fi (u(m−1) ; [u(m−1) ]ai ; [u(m−1) ]bi ; [ u(m−1) ]ci ; [u(m−1) ]di ); i i (m) @u(m) i =@ = @u i =@ = 0; m = 1; 2; : : : (i = 1; : : : ; N ): (2.15) C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 355 Since for each i = 1; : : : ; N; fi (u; v) is a constant whenever u; v are constant vectors we see from Ki ¿0 and the uniqueness of the soltuion for linear scalar boundary-value (m) (m) } ≡ {u(m) problems that the sequence of iterations {u(m) } ≡ {u(m) ;:::; ; : : : ; u N }; {u (m) u N } governed by (2.15) coincide, respectively, with the constant sequences {c(m) } and {c(m) } given by (2.7) In view of Lemma 2.1 lim u(m) = c m→∞ and lim u(m) = c m→∞ and c and c, called quasisolutions of (2.14), satisfy the equations in (2.11) Moreover, ˆ c˜ then c ≤ u(x) ≤ c on The conclusion if u(x) is a solution of (2.14) in the sector c; of the theorem follows from Theorem 3.2 of [13] (see also [12]) When the time delay Ji ∗ ui is either discrete or ÿnite continuous (or a combination of these two types) we have the following global result Corollary 2.1 Let the conditions in Theorem 2:2 hold and let u(t; x) ≡ (u1 (t; x); : : : ; uN (t; x)) be the solution of (1:1) with an arbitrary intial function Á(t; x) Assume that Ji ∗ ui is given by (1:3) for i = 1; : : : ; n0 and by (1:2) for i = n0 + 1; : : : ; N If there exists t ∗ ¿0 such that cˆi ≤ ui (t; x) ≤ c˜i for t ∗ − ri ≤ t ≤ t ∗ ; x∈ (i = 1; : : : ; N ); (2.16) ∗ then u(t; x) satisÿes (2:12) Moreover (2:13) holds if c = c( ≡ c ) Proof This is a consequence of Theorem 3.3 in [13] Details are omitted If, for each i; fi ≡ fi (u) is independent of J ∗u then the main condition (2.4) becomes fi (c˜i ; [c˜ ]ai ; [cˆ ]bi ) ≤ ≤ fi (cˆi ; [cˆ ]ai ; [c˜ ]bi ) (i = 1; : : : ; N ) (2.17) and hypothesis (H) is reduced to the following: ˆ c˜ and satisÿes the Lipschitz condition (H) : f(u) is mixed quasimonotone in ≡ c; |fi (u) − fi (u )| ≤ Ki |u − u | for u; u ∈ (i = 1; : : : ; N ): (2.18) As a consequence of the above theorems we have the following conclusion for system (1.1) without time delay ˜ cˆ be constant vectors Corollary 2.2 Let fi ≡ fi (u) be independent of J ∗ u, and let c; satisfying c˜ ≥ cˆ and relation (2:17) Assume that hypothesis (H ) holds Then all the conclusions in Theorems 2:1 and 2:2 remain true for system (1:1) without time delays Moreover, the result in Corollary 2:1 holds if condition (2:16) is satisÿed It is obvious that a similar conclusion in Corollary 2.2 holds if f ≡ f(J ∗ u) is independent of u Applications It is seen from the results of the previous section that under the mixed quasimonotone condition on the nonlinear function f(u; J ∗ u), an invariant region and the 356 C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 convergence of the time-dependent solution to a uniform steady-state solution can be ˜ cˆ satisfying relation obtained through the construction of a pair of constant vectors c; (2.4) The construction of these vectors depends only on the nonlinear function f, and a suitable construction can sometimes lead to a characterization of the stability or instability of the various steady-state solutions To demonstrate this possibility we consider two model problems arising from ecology where the nonlinear reaction function f is ˜ cˆ yields some simple condition mixed quasimonotone Our construction of the vectors c; on the reaction rate constants of f so that the global asymptotic stability (or instability) of a constant steady-state solution can be determined It is assumed for physical reasons that the initial function Á(t; x) in these model problems is nonnegative in Q0 3.1 A prey-predator system The well-known Volterra–Lotka prey-predator reaction–di usion system with discrete or continuous time delays is given by ut − L1 u = u(a1 − b1 u − c1 J2 ∗ v); vt − L2 v = v(a2 + b2 J1 ∗ u − c2 v) in D; @u=@ = 0; @u=@ = on S; u(t; x) = Á1 (t; x); v(t; x) = Á2 (t; x) in Q0 ; (3.1) where ; bi and ci ; i = 1; 2, are positive constants (which are di erent from the subscripts in (2.4)), and J1 ∗ u and J2 ∗ v are given by (1.2) with either n0 = (continuous delays) or n0 = (continuous and discrete delays) or n0 = (discrete delays) Problem (3:1) is a special case of (1.1) with N = 2; (u1 ; u2 ) = (u; v) and f1 (u; J ∗ u) ≡ u(a1 − b1 u − c1 J2 ∗ v); f2 (u; J ∗ u) ≡ v(a2 + b2 J1 ∗ u − c2 v): (3.2) It is obvious that the steady-state problem of (3.1) possesses the trivial solution (0,0) and the semitrivial solutions (a1 =b1 ; 0) and (0; a2 =c2 ) If a2 =a1 ¡c2 =c1 then it has also the positive constant steady-state solution c∗ ≡ (c1∗ ; c2∗ ) with c1∗ = (a1 c2 − a2 c1 )=(b1 c2 + b2 c1 ); c2∗ = (a1 b2 + a2 b1 )=(b1 c2 + b2 c1 ): (3.3) In view of (3.2) the function f = (f1 ; f2 ) is mixed quasimonotone in × ∗ with = ∗ = R+ × R+ , and the requirement of c˜ ≡ (c˜1 ; c˜2 ), cˆ ≡ (cˆ1 ; cˆ2 ) in (2.4) becomes c˜1 (a1 − b1 c˜1 − c1 J2 ∗ cˆ2 ) ≤ 0; cˆ1 (a1 − b1 cˆ1 − c1 J2 ∗ c˜2 ) ≥ 0; c˜2 (a2 + b2 J1 ∗ c˜1 − c2 c˜2 ) ≤ 0; cˆ2 (a2 − b2 J2 ∗ cˆ1 − c1 cˆ2 ) ≥ 0: (3.4) Since J1 ∗c = J2 ∗c = c for every constant c (see (1.4) and (1.5)), the above inequalities are all satisÿed by (cˆ1 ; cˆ2 ) = (0; 0) and (c˜1 ; c˜2 ) = (M1 ; M2 ), where M1 and M2 are any constants satisfying M1 ≥ a1 =b1 , M2 ≥ (a2 + b2 M1 )=c2 By choosing Mi ≥ Ái , i = 1; 2; we see from Theorem 2.1 that a unique global solution (u; v) to (3.1) exists and satisÿes C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 357 the relation (0; 0) ≤ (u; v) ≤ (M1 ; M2 ) The maximal principle for standard parabolic boundary-value problems implies that (u; v)¿(0; 0) in D when Ái (0; x) ≡ for i = 1; It is obvious that if Á1 (0; x) ≡ and Á2 (0; x) ≡ then v(t; x) = 0, u(t; x)¿0 in D and (u; v) → (a1 =b1 ; 0) as t → ∞ (cf [11, p 201]) Similarly, if Á1 (0; x) ≡ and Á2 (0; x) = then u(t; x) = 0, v(t; x)¿0 in D and (u; v) → (0; a2 =c2 ) as t → ∞ To investigate the asymptotic behavior of the solution when Ái (0; x) = for i = 1; 2; we ÿrst consider the case a2 =a1 + b2 =b1 ¡c2 =c1 : (3.5) This ensures that a unique positive constant steady-state solution exists and is given by (c1∗ ; c2∗ ) in (3.3) Let ; be some su ciently small positive constants such that ≤ (c1 =b1 ) , and let (c˜1 ; c˜2 ) = (a1 =b1 + ; a1 =c1 − ); (cˆ1 ; cˆ2 ) = ( ; ): (3.6) It is easy to verify from J1 ∗ c1 = c1 , J2 ∗ c2 = c2 and condition (3.5) that the above pair (m) (m) satisfy relation (3.4) This implies that the sequences {c(m) } = {c(m) }= ; c2 }, {c (0) (0) (0) (0) (m) (m) {c1 ; c2 }, governed by (2.7) with (c1 ; c2 ), (c1 ; c2 ) given by (3.6) and (f1 ; f2 ) given by (3.2) possess the monotone property (2.8) and converge to some limits c ≡ (c1 ; c2 ) and c ≡ (c1 ; c2 ) respectively Moreover, these limits satisfy the relation (c1 ; c2 ) ≥ (c1 ; c2 ) ≥ ( ; ) and the equation (see Eq (2.11)) a1 − b1 c1 − c1 c2 = 0; a1 − b1 c1 − c1 c2 = 0; a2 + b2 c1 − c2 c2 = 0; a2 + b2 c1 − c2 c2 = 0: (3.7) Subtraction of the corresponding equations (3.7) gives b1 (c1 − c1 ) − c1 (c2 − c2 ) = 0; b2 (c1 − c1 ) − c2 (c2 − c2 ) = 0: By condition (3.5) we have c1 = c1 , c2 = c2 , and therefore (c1 ; c2 ) (or (c1 ; c2 )) is a positive steady-state solution of (3.1) The uniqueness of the positive solution (c1∗ ; c2∗ ) ensures that (c1 ; c2 ) = (c1 ; c2 ) = (c1∗ ; c2∗ ) By an application of Theorem 2.1 the timedependent solution (u; v) of (3.1) converges to (c1∗ ; c2∗ ) as t → ∞ when Á1 ≤ a1 =b1 and Á1 ¡a1 =c1 in Q0 For arbitrary (Á1 ; Á2 ) ≥ (0; 0) with Ái (0; x) = 0, i = 1; 2; we observe from (u; v)¿(0; 0) in D that there exist positive constants , t0 such that (u; v) ≥ ( ; ) on [t0 ; ∞) × Moreover, the standard comparison theorem for parabolic boundary-value problems implies that u(t; x) ≤ U (t; x) in D, where U is the positive solution of the scalar boundary-value problem Ut − L1 U = U (a1 − b1 U ) in D; @U=@ = on S; U (0; x) = Á1 (0; x) in (cf [11,12]) Since U (t; x) → a1 =b1 as t → ∞ we see that there exists t1 ¿0 such that u(t; x) ≤ a1 =b1 on [t1 ; ∞) × This property also implies that v(t; x) ≤ V (t; x) in D, where V is the solution of the scalar boundary-value problem Vt − L V = V ( − c2 V ) in D; @V= = on S; V (0; x) = Á2 (0; x) in 358 C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 with = a2 +a1 (b2 =b1 ) Since V (t; x) → =c2 as t → ∞ we see that there exists t2 ¿0 such that v(t; x) ≤ =c2 on [t2 ; ∞) × In view of =c2 = (a1 =c2 )(a2 =a1 + b2 =b1 ) and condition (3.5), we have v(t; x)¡a1 =c1 on [t2 ; ∞) × The above conclusions show that there exists t ∗ ¿0 such that ≤ u ≤ a1 =b1 and ≤ v¡a1 =c1 on [t ∗ ; ∞) × By an application of Corollary 2.1 we conclude that if the time delay is ÿnite then the solution (u; v) converges to (c1∗ ; c2∗ ) for any nontrivial (Á1 ; Á2 ) To summarize the above results we have the following Theorem 3.1 Let (Á1 ; Á2 ) ≥ (0; 0) with Ái (0; x) = for i = 1; 2; and let condition (3:5) be satisÿed Then a unique global positive solution (u; v) to problem (3:1) exists and lim (u(t; x); v(t; x)) = (c1∗ ; c2∗ ) (x ∈ ); (3.8) t→∞ when Á1 ≤ a1 =b1 ; Á2 ¡a1 =c1 ; where c1∗ and c2∗ are given by (3:3) Moreover, the relation (3:8) holds for any (Á1 ; Á2 ) ≥ (0; 0) with Ái (0; x) = 0, i = 1; 2; if the time-delays J1 ∗ u and J2 ∗ v are either discrete or ÿnite continuous We next consider the case a2 =a1 ≥ c2 =c1 Let (c˜1 ; c˜2 ) = (M1 ; M2 ), (cˆ1 ; cˆ2 ) = (0; ) for some positive constants M1 ; M2 and It is easy to verify that this pair satisfy all the inequalities in (3.4) if is su ciently small and M1 ≥ a1 =b1 ; M2 ≥ (a2 + b2 M1 )=c2 : (0) (0) (0) Using (c(0) ; c2 ) = (M1 ; M2 ) and (c1 ; c2 ) = (0; ) in the iteration process (2.7), where (m) (m) (m) (f1 ; f2 ) is given by (3.2), we obtain two sequences of constants {c(m) ; c2 }, {c1 ; c2 } which converge monotonically to some limits (c1 ; c2 ), (c1 ; c2 ), respectively, such that (c1 ; c2 ) ≥ (c1 ; c2 ) ≥ (0; ) Since by (2.7) and (3.2), = c(m−1) + f1 (c(m−1) ; c(m−1) ) c(m) 1 K1 + = c(m−1) 1 (m−1) c (a1 − b1 c(m−1) − c1 c(m−1) ); K1 we see from c(0) = that c(m) = for every m = 1; 2; : : : This implies c1 = while c1 ; c2 and c2 satisfy the equations c1 (a1 − b1 c1 − c1 c2 ) = 0; By the relation c2 ≥ c2 ≥ c2 = a2 =c2 ; c2 (a1 + b2 c1 − c2 c2 ) = 0; c2 (a2 − c2 c2 ) = 0: we have c2 = (a2 + b2 c1 )=c2 and either c1 = or c1 = (a1 − c1 c2 )=b1 = [a1 − a2 (c1 =c2 )]=b1 : However, by the condition a2 =a1 ≥ c2 =c1 and c1 ≥ we must have c1 = This shows that c1 = c1 = and c2 = c2 = a2 =c2 As a consequence of Theorem 2.1 and the arbitrariness of (M1 ; M2 ) we have the following conclusion C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 359 Theorem 3.2 Let (Á1 ; Á2 ) ≥ (0; 0) with Á2 (0; x) = 0; and let a2 =a1 ≥ c2 =c1 Then a unique global positive solution (u; v) to (3:1) exists and lim (u(t; x); v(t; x)) = (0; a2 =c2 ); (x ∈ ): t→∞ (3.9) Remark 3.1 Theorems 3.1 and 3.2 imply that under condition (3.5) the positive constant steady-state solution (c1∗ ; c2∗ ) is asymptotically stable while the trivial solution (0,0) and the semitrivial solutions (a1 =b1 ; 0) and (0; a2 =c2 ) are all unstable (in the sense of Lyapunov stability) Moreover, for ÿnite continuous or discrete time delays, including the case r1 = r2 = without time delays, the constant (c1∗ ; c2∗ ) is globally asymptotically stable This implies that there can exist no nonuniform positive steady-state solution On the other hand, if the reversed inequality in (3.5) is satisÿed then the semitrivial solution (0; a2 =c2 ) is globally asymptotically while the trivial solution (0,0) and the semitrivial solution (a1 =b1 ; 0) are unstable This is true for both ÿnite and inÿnite continuous delays 3.2 A competition model As a second application we consider the Volterra–Lotka competition model with time delays: ut − L1 u = u(a1 − b1 u − c1 J2 ∗ v) vt − L2 v = v(a2 − b2 J1 ∗ u − c2 v) in D; (3.10) where ; bi ; ci ; J1 ∗ u and J2 ∗ v as well as the boundary and initial conditions are the same as that in (3.1) It is obvious that this system possesses the same trivial and semitrivial steady-state solutions as that in (3.1) If the rate constants in (3.10) satisfy the relation b2 =b1 ¡a2 =a1 ¡c2 =c1 ; (3.11) then it has also the positive steady-state solution (c1∗ ; c2∗ ), where c1∗ = (a1 c2 − a2 c1 )=(b1 c2 − b2 c1 ); c2∗ = (a2 b1 − a1 b2 )=(b1 c2 − b2 c1 ): (3.12) By the monotone property of the reaction function in (3.10) the requirement of (c˜1 ; c˜2 ), (cˆ1 ; cˆ2 ) in (2.4) becomes c˜1 (a1 − b1 c˜1 − c1 J2 ∗ cˆ2 ) ≤ 0; cˆ1 (a1 − b1 cˆ1 − c1 J2 ∗ c˜2 ) ≥ 0; c˜2 (a2 − b2 J1 ∗ cˆ1 − c2 c˜2 ) ≤ 0; cˆ2 (a2 − b2 J1 ∗ c˜1 − c2 cˆ2 ) ≥ 0: (3.13) The above inequalities are satisÿed by (c˜1 ; c˜2 ) = (M1 ; M2 ) and (cˆ1 ; cˆ2 ) = (0; 0) for any constants M1 ; M2 such that M1 ≥ a1 =b1 ; M2 ≥ a2 =c2 The arbitrary largeness of M1 ; M2 implies that for any (Á1 ; Á2 ) ≥ (0; 0) system (3.10) (under the boundary and initial conditions in (3.1)) has a unique global solution (u; v), and (u; v) is positive in (0; ∞)× if Ái (0; x) = for i = 1; To investigate the asymptotic behavior of the solution (u; v) we ÿrst consider the case where the condition (3.11) holds Our aim is to show that (u; v) converges to the 360 C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 positive constant (c1∗ ; c2∗ ) as t → ∞ Choose positive constants (c˜1 ; c˜2 ); (cˆ1 ; cˆ2 ) with (cˆ1 ; cˆ2 ) su ciently small and a1 =b1 ¡c˜1 ¡a2 =b2 ; a2 =c2 ¡c˜2 ¡a1 =c1 : (3.14) This is possible because of condition (3.11) It is easy to verify from (1.4) and (1.5) that the pair (c˜1 ; c˜2 ) and (cˆ1 ; cˆ2 ) satisfy all the inequalities in (3.13) Hence, (0) (0) (m) (m) (m) the sequences {c(m) ; c2 }; {c1 ; c2 } given by (2.7) with (c1 ; c2 ) = (c˜1 ; c˜2 ) and (0) (c(0) ; c2 ) = (cˆ1 ; cˆ2 ) (and with (f1 ; f2 ) given by the right-hand side of (3.10)) converge monotonically to some positive constants (c1 ; c2 ); (c1 ; c2 ) that satisfy the equations c1 (a1 − b1 c1 − c1 c2 ) = 0; c1 (a1 − b1 c1 − c1 c2 ) = 0; c2 (a2 − b2 c1 − c2 c2 ) = 0; c2 (a2 − b2 c1 − c2 c2 ) = 0: (3.15) Since c1 ≥ cˆ1 ¿0; c2 ≥ cˆ2 ¿0, a subtraction of the corresponding equations in (3.15) leads to b1 (c1 − c1 ) − c1 (c2 − c2 ) = 0; b2 (c1 − c1 ) − c2 (c2 − c2 ) = 0: It follows from condition (3.11) and the uniqueness of the positive solution (c1∗ ; c2∗ ) that c1 = c1 = c1∗ and c2 = c2 = c2∗ , where c1∗ and c2∗ are given by (3.12) By Theorem 2.1 the time-dependent solution (u; v) of (3.10) converges to (c1∗ ; c2∗ ) as t → ∞ whenever (cˆ1 ; cˆ2 ) ≤ (Á1 ; Á2 ) ≤ (c˜1 ; c˜2 ) We show that if the time delays are discrete or ÿnite continuous then the above convergence property holds true for any (Á1 ; Á2 ) ≥ (0; 0) with Ái (0; x) = 0; i = 1; In view of (u; v)¿(0; 0) in D and (cˆ1 ; cˆ2 ) can be chosen arbitrarily small there exists t1 ¿0 such that (u; v) ≥ (cˆ1 ; cˆ2 ) for t ≥ t1 ; x ∈ On the other hand, by the standard comparison theorem for scalar parabolic boundary-value problems, there exists t2 ¿0 such that (u; v) ≤ (c˜1 ; c˜2 ) for t ≥ t2 ; x ∈ (cf [11,12]) This implies that there exists t ∗ ¿0 such that (cˆ1 ; cˆ2 ) ≤ (u; v) ≤ (c˜1 ; c˜2 ) for t ≥ t ∗ ; x ∈ By an application of Corollary 2.1 we have the following conclusion Theorem 3.3 Let condition (3:11) be satisÿed Then for any nonnegative (Á1 ; Á2 ) ≤ (c˜1 ; c˜2 ) with Ái (0; x) = 0; i = 1; 2; where c˜1 and c˜2 are any constants satisfying (3:14); a unique global positive solution (u; v) to (3:10) exists and lim (u(t; x); v(t; x)) = (c1∗ ; c2∗ ); t→∞ c1∗ and (x ∈ ); (3.16) where c2∗ are given by (3:12) Moreover; if the time delays are ÿnite continuous or discrete then (u; v) satisÿes the relation (3:16) for every nonnegative (Á1 ; Á2 ) with Ái (0; x) = 0; i = 1; Theorem 3.3 implies that under condition (3.11) the positive steady-state solution (c1∗ ; c2∗ ) is asymptotically stable with a stability region (0; c˜1 ] × (0; c˜2 ], while the trivial solution (0,0) and the semitrivial solutions (a1 =b1 ; 0); (0; a2 =c2 ) are all unstable For ÿnite continuous or discrete time delays, including the case without time delays, the stability property of (c1∗ ; c2∗ ) is global with respect to nontrivial nonnegative initial perturbations In this situation, there exists no nonuniform steady-state solution despite the fact that the coe cients of the elliptic operators L1 and L2 may depend on x in C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 361 We next show that if condition (3.11) is replaced by either a2 =a1 ¡b2 =b1 ¡c2 =c1 or a2 =a1 ¿b2 =b1 ¿c2 =c1 ; (3.17) then there exists no positive steady-state solution, and the time dependent solution (u; v) converges to the semitrivial solution (a1 =b1 ; 0) To prove this we choose (c1 ; c2 ) = (M1 ; M2 ) and (cˆ1 ; cˆ2 ) = ( ; 0) for a su ciently small ¿0, where M1 and M2 are any constants satisfying M1 ≥ a1 =b1 ; M2 ¿max{a1 =c1 ; a2 =c2 }: (3.18) It is easy to verify that the pair (M1 ; M2 ) and ( ; 0) satisfy all the inequalities in (3.13) (0) (0) (m) (m) (m) Hence the sequences {c(m) ; c2 }, {c1 ; c2 } governed by (2.7) with (c1 ; c2 ) = (0) (0) (M1 ; M2 ), (c1 ; c2 ) = ( ; 0) converge monotonically to some limits (c1 ; c2 ), (c1 ; c2 ) that satisfy the equations in (3.15) Since by (2.7) and the monotone property of the function (f1 ; f2 ) in (3.10), c(m) = c(m−1) + f2 (c(m−1) ; c(m−1) ) 2 K2 + = c(m−1) (m−1) − c2 c(m−1) ); c (a2 − b2 c(m−1) K2 (m) we see from c(0) = that c2 = for every m = 1; 2; : : : : This implies that c2 = It follows from (3:15) and c1 ≥ c1 ¿ that c1 = a1 =b1 , a1 − b1 c1 − c1 c2 = and c2 (a2 − b2 c1 − c2 c2 ) = 0: This ensures that c2 = 0, for if c2 = then by solving the above two equations for c2 and using either one of the conditions in (3.17) we obtain c2 = (a1 b2 − a2 b1 )=(b2 c1 − b1 c2 )¡0 which contracts the fact that c2 ≥ c2 = Hence (c1 ; c2 ) = (c1 ; c2 ) = (a1 =b1 ; 0) By Theorem 2.1 and the arbitrariness of (M1 ; M2 ) we conclude that for any nonnegative (Á1 ; Á2 ) with Á1 (0; x) ≡ the solution (u; v) of (3.10) converges to (a1 =b1 ; 0) This implies that the semitrivial solution (a1 =b1 ; 0) is globally asymptotically stable (with respect to nontrivial nonnegative initial perturbations) while the trivial and semitrivial solutions (0; 0) and (0; a2 =c2 ) are unstable On the other hand, if either b2 =b1 ¿c2 =c1 ¿a2 =a1 or a2 =a1 ¿c2 =c1 ¿b2 =b1 ; (3.19) then the same argument as above shows that for any nonnegative (Á1 ; Á2 ) with Á2 (0; x) ≡ the solution (u; v) of (3.10) converges to (0; a2 =c2 ) as t → ∞ This implies that (0; a2 =c2 ) is globally asymptotically stable while (0; 0) and (a1 =b1 ; 0) are unstable To summarize the above conclusions we have the following results Theorem 3.4 Let (u; v) be the solution of (3:10) under the boundary–initial conditions in (3:1) with (Á1 ; Á2 ) ≥ (0; 0) and Ái (0; x) = for i = 1; Then lim (u(t; x); v(t; x)) = (a1 =b1 ; 0) t→∞ 362 C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 if one of the conditions in (3:17) holds; and lim (u(t; x); v(t; x)) = (0; a2 =c2 ) t→∞ if one of the conditions in (3:19) holds Remark 3.2 The results in Theorems 3.3 and 3.4 for the competition model (3:10) have been obtained in [16] using the approach of order-preserving semi ows (see also [18]) In this work the time delays are of continuous type and the initial function (Á1 ; Á2 ) is required to satisfy some additional conditions References [1] J.M Cushing, Integrodi erential Equations and Delay Models in Population Dynamics, Lecture Notes in Biolmath., Vol 20, Springer, New York, 1977 [2] K Gopalsamy, B.D Aggarwala, Limit cycles in two species competition with time delays, J Austral Math Soc 22 B (1980) 148–160 [3] Y Kuang, Delay Di erential Equations with Applications in Population Dynamics, Academic Press, New York, 1993 [4] Y Kuang, H.L Smith, Global stability in di usive delay Lotka–Volterra systems, Di erential Integral Equations (1991) 117–128 [5] K Kunisch, W Schappacher, Necessary conditions for partial di erential equations with delay to generate C0 -semigroups, J Di erential Equation 50 (1983) 49–79 [6] X Lu, Persistence and extinction in a competition-di usion system with time delays, Canad Appl Math Quart (1994) 231–246 [7] W MacDonald, Time Lags in Biological Models, Lecture Notes in Biolmath., Vol 27, Springer, New York, 1979 [8] R.H Martin, H.L Smith, Reaction–di usion systems with time delays: monotonicity, invariance, comparison and convergence, J Reine Angew Math 413 (1991) 1–35 [9] M.C Memory, Bifurcation and asymptotic behavior of solutions of a delay-di erential equation with di usion, SIAM J Math Anal 20 (1989) 533–546 [10] C.V Pao, On a coupled reaction di usion system with time delays, SIAM J Math Anal 18 (1987) 1026–1039 [11] C.V Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992 [12] C.V Pao, Dynamics of nonlinear parabolic systems with time delays, J Math Anal Appl 198 (1996) 751–779 [13] C.V Pao, Systems of parabolic equations with continuous and discrete delays, J Math Anal Appl 205 (1997) 157–185 [14] C.V Pao, Numerical analysis of coupled systems of nonlinear parabolic equations, SIAM J Numer Anal 36 (1999) 393–416 [15] R Redlinger, On Volterra’s population equation with di usion, SIAM J Math Anal 16 (1985) 135–142 [16] S Ruan, J Wu, Reaction–di usion equations with inÿnite delay, Canad Appl Math Quart (1994) 485–550 [17] C.C Travis, G.F Webb, Existence stability and compactness in the -norm for partial functional di erential equations, Trans Amer Math Soc 200 (1974) 395–418 [18] J Wu, Theory and Applications of Partial Functional Di erential Equations, Springer, New York, 1996 [19] Y Yamada, Asymptotic behavior of solutions for semilinear di usions with time delay arising in ecology, Hiroshima Math J 12 (1982) 321–348 [...]... system with time delays, Canad Appl Math Quart 2 (1994) 231–246 [7] W MacDonald, Time Lags in Biological Models, Lecture Notes in Biolmath., Vol 27, Springer, New York, 1979 [8] R.H Martin, H.L Smith, Reaction–di usion systems with time delays: monotonicity, invariance, comparison and convergence, J Reine Angew Math 413 (1991) 1–35 [9] M.C Memory, Bifurcation and asymptotic behavior of solutions of a... equation with di usion, SIAM J Math Anal 20 (1989) 533–546 [10] C.V Pao, On a coupled reaction di usion system with time delays, SIAM J Math Anal 18 (1987) 1026–1039 [11] C.V Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992 [12] C.V Pao, Dynamics of nonlinear parabolic systems with time delays, J Math Anal Appl 198 (1996) 751–779 [13] C.V Pao, Systems of parabolic equations with. .. 0); (0; a2 =c2 ) are all unstable For ÿnite continuous or discrete time delays, including the case without time delays, the stability property of (c1∗ ; c2∗ ) is global with respect to nontrivial nonnegative initial perturbations In this situation, there exists no nonuniform steady-state solution despite the fact that the coe cients of the elliptic operators L1 and L2 may depend on x in C.V Pao / Nonlinear... of the positive solution (c1∗ ; c2∗ ) that c1 = c1 = c1∗ and c2 = c2 = c2∗ , where c1∗ and c2∗ are given by (3.12) By Theorem 2.1 the time- dependent solution (u; v) of (3.10) converges to (c1∗ ; c2∗ ) as t → ∞ whenever (cˆ1 ; cˆ2 ) ≤ (Á1 ; Á2 ) ≤ (c˜1 ; c˜2 ) We show that if the time delays are discrete or ÿnite continuous then the above convergence property holds true for any (Á1 ; Á2 ) ≥ (0; 0) with. .. asymptotically stable while the trivial solution (0,0) and the semitrivial solutions (a1 =b1 ; 0) and (0; a2 =c2 ) are all unstable (in the sense of Lyapunov stability) Moreover, for ÿnite continuous or discrete time delays, including the case r1 = r2 = 0 without time delays, the constant (c1∗ ; c2∗ ) is globally asymptotically stable This implies that there can exist no nonuniform positive steady-state solution... competition with time delays, J Austral Math Soc 22 B (1980) 148–160 [3] Y Kuang, Delay Di erential Equations with Applications in Population Dynamics, Academic Press, New York, 1993 [4] Y Kuang, H.L Smith, Global stability in di usive delay Lotka–Volterra systems, Di erential Integral Equations 4 (1991) 117–128 [5] K Kunisch, W Schappacher, Necessary conditions for partial di erential equations with delay... parabolic equations with continuous and discrete delays, J Math Anal Appl 205 (1997) 157–185 [14] C.V Pao, Numerical analysis of coupled systems of nonlinear parabolic equations, SIAM J Numer Anal 36 (1999) 393–416 [15] R Redlinger, On Volterra’s population equation with di usion, SIAM J Math Anal 16 (1985) 135–142 [16] S Ruan, J Wu, Reaction–di usion equations with inÿnite delay, Canad Appl Math Quart 2... Moreover; if the time delays are ÿnite continuous or discrete then (u; v) satisÿes the relation (3:16) for every nonnegative (Á1 ; Á2 ) with Ái (0; x) = 0; i = 1; 2 Theorem 3.3 implies that under condition (3.11) the positive steady-state solution (c1∗ ; c2∗ ) is asymptotically stable with a stability region (0; c˜1 ] × (0; c˜2 ], while the trivial solution (0,0) and the semitrivial solutions (a1 =b1... -norm for partial functional di erential equations, Trans Amer Math Soc 200 (1974) 395–418 [18] J Wu, Theory and Applications of Partial Functional Di erential Equations, Springer, New York, 1996 [19] Y Yamada, Asymptotic behavior of solutions for semilinear di usions with time delay arising in ecology, Hiroshima Math J 12 (1982) 321–348 ... asymptotically while the trivial solution (0,0) and the semitrivial solution (a1 =b1 ; 0) are unstable This is true for both ÿnite and inÿnite continuous delays 3.2 A competition model As a second application we consider the Volterra–Lotka competition model with time delays: ut − L1 u = u(a1 − b1 u − c1 J2 ∗ v) vt − L2 v = v(a2 − b2 J1 ∗ u − c2 v) in D; (3.10) where ai ; bi ; ci ; J1 ∗ u and J2 ∗ v as well as ... SIAM J Math Anal 20 (1989) 533–546 [10] C.V Pao, On a coupled reaction di usion system with time delays, SIAM J Math Anal 18 (1987) 1026–1039 [11] C.V Pao, Nonlinear Parabolic and Elliptic Equations,... Equations, Plenum Press, New York, 1992 [12] C.V Pao, Dynamics of nonlinear parabolic systems with time delays, J Math Anal Appl 198 (1996) 751–779 [13] C.V Pao, Systems of parabolic equations with... ci and di satisfying the relations + bi = N − and ci + d i = N (i = 1; : : : ; N ) (2.2) 352 C.V Pao / Nonlinear Analysis 48 (2002) 349 – 362 such that the function fi (u; v) ≡ fi (ui ; [u]ai