# A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations

Authors: Jiebao Sun; Dazhi Zhang; Boying Wu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We consider a cooperating two-species Lotka-Volterra model of degenerate
parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.


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## Body

## 1. Introduction

In this paper, we consider the following two-species cooperative system:(1.1)ut=Δum1+uα(a-bu+cv),(x,t)∈Ω×R+,(1.2)vt=Δvm2+vβ(d+eu-fv),(x,t)∈Ω×R+,(1.3)u(x,t)=0,v(x,t)=0,(x,t)∈∂Ω×R+,(1.4)u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,
where m1,m2>1, 0<α<m1, 0<β<m2, 1≤(m1-α)(m2-β), a=a(x,t), b=b(x,t), c=c(x,t), d=d(x,t), e=e(x,t), f=f(x,t) are strictly positive smooth functions and periodic in time with period T>0 and u0(x) and v0(x) are nonnegative functions and satisfy u0m1,v0m2∈W01,2(Ω).In dynamics of biological groups, the system (1.1)-(1.2) can be used to describe the interaction of two biological groups. The diffusion terms Δum1 and Δvm2 represent the effect of dispersion in the habitat, which models a tendency to avoid crowding and the speed of the diffusion is rather slow. The boundary conditions (1.3) indicate that the habitat is surrounded by a totally hostile environment. The functions u and v represent the spatial densities of the species at time t and a,d are their respective net birth rate. The functions b and f are intra-specific competitions, whereas c and e are those of interspecific competitions.As famous models for dynamics of population, two-species cooperative systems like (1.1)-(1.2) have been studied extensively, and there have been many excellent results, for detail one can see [1–6] and references therein. As a special case, men studied the following two-species Lotka-Volterra cooperative system of ODEs:(1.5)u′(t)=u(t)(a(t)-b(t)u(t)+c(t)v(t)),v′(t)=v(t)(d(t)+e(t)u(t)-f(t)v(t)).
For this system, Lu and Takeuchi [7] studied the stability of positive periodic solution and Cui [1] discussed the persistence and global stability of it.Whenm1=m2=α=β=1, from (1.1)-(1.2) we get the following classical cooperative system:(1.6)ut=Δu+u(a-bu+cv),vt=Δv+v(d+eu-fv).For this system, Lin et al. [5] showed the existence and asymptotic behavior of T- periodic solutions when a,b,c,e,d,f are all smooth positive and periodic in time with period T>0. When a,b,c,e,d,f are all positive constants, Pao [6] proved that the Dirichlet boundary value problem of this system admits a unique solution which is uniformly bounded when ce<bf, while the blowup solutions are possible when the two species are strongly mutualistic (ce>bf). For the homogeneous Neumann boundary value problem of this system, Lou et al. [4] proved that the solution will blow up in finite time under a sufficient condition on the initial data. When c=e=0 and α=β=1, from (1.1) we get the single degenerate equation (1.7)ut=Δum+u(a-bu).
For this equation, Sun et al. [8] established the existence of nontrivial nonnegative periodic solutions by monotonicity method and showed the attraction of nontrivial nonnegative periodic solutions.In the recent years, much attention has been paid to the study of periodic boundary value problems for parabolic systems; for detail one can see [9–15] and the references therein. Furthermore, many researchers studied the periodic boundary value problem for degenerate parabolic systems, such as [16–19]. Taking into account the impact of periodic factors on the species dynamics, we are also interested in the existence of the nontrivial periodic solutions of the cooperative system (1.1)-(1.2). In this paper, we first show the existence of the global generalized solution of the initial boundary value problem (1.1)–(1.4). Then under the condition that (1.8)blfl>cMeM,
where fM=sup{f(x,t)∣(x,t)∈Ω×ℝ},fl=inf{f(x,t)∣(x,t)∈Ω×ℝ}, we show that the generalized solution is uniformly bounded. At last, by the method of monotone iteration, we establish the existence of the nontrivial periodic solutions of the system (1.1)-(1.2), which follows from the existence of a pair of large periodic supersolution and small periodic subsolution. At last, we show the existence and the attractivity of the maximal periodic solution.Our main efforts center on the discussion of generalized solutions, since the regularity follows from a quite standard approach. Hence we give the following definition of generalized solutions of the problem (1.1)–(1.4).Definition 1.1.
A nonnegative and continuous vector-valued function(u,v) is said to be a generalized solution of the problem (1.1)–(1.4) if, for any 0≤τ<T and any functions φi∈C1(Q¯τ) with φi|∂Ω×[0,τ)=0(i=1,2),∇um1,∇vm2∈L2(Qτ), ∂um1/∂t,∂vm2/∂t∈L2(Qτ) and
(1.9)∬Qτu∂φ1∂t-∇um1∇φ1+uα(a-bu+cv)φ1dxdt=∫Ωu(x,τ)φ1(x,τ)dx-∫Ωu0(x)φ1(x,0)dx,∬Qτv∂φ2∂t-∇vm2∇φ2+vβ(d+eu-fv)φ2dxdt=∫Ωv(x,τ)φ2(x,τ)dx-∫Ωv0(x)φ2(x,0)dx,
where Qτ=Ω×(0,τ).
Similarly, we can define a weak supersolution(u¯,v¯) (subsolution (u̲,v̲)) if they satisfy the inequalities obtained by replacing “=” with “≤” (“≥”) in (1.3), (1.4), and (1.9) and with an additional assumption φi≥0(i=1,2).Definition 1.2.
A vector-valued function(u,v) is said to be a T-periodic solution of the problem (1.1)–(1.3) if it is a solution in [0,T] such that u(·,0)=u(·,T), v(·,0)=v(·,T) in Ω. A vector-valued function (u¯,v¯) is said to be a T-periodic supersolution of the problem (1.1)–(1.3) if it is a supersolution in [0,T] such that u¯(·,0)≥u¯(·,T),v¯(·,0)≥v¯(·,T) in Ω. A vector-valued function (u̲,v̲) is said to be a T-periodic subsolution of the problem (1.1)–(1.3), if it is a subsolution in [0,T] such that u̲(·,0)≤u̲(·,T),v̲(·,0)≤v̲(·,T) in Ω.This paper is organized as follows. In Section2, we show the existence of generalized solutions to the initial boundary value problem and also establish the comparison principle. Section 3 is devoted to the proof of the existence of the nonnegative nontrivial periodic solutions by using the monotone iteration technique.


## 2. The Initial Boundary Value Problem

To solve the problem (1.1)–(1.4), we consider the following regularized problem:(2.1)∂uε∂t=div⁡((muεm1-1+ε)∇uε)+uεα(a-buε+cvε),(x,t)∈QT,(2.2)∂vε∂t=div⁡((mvεm2-1+ε)∇vε)+vεβ(d+euε-fvε),(x,t)∈QT,(2.3)uε(x,t)=0,vε(x,t)=0,(x,t)∈∂Ω×(0,T),(2.4)uε(x,0)=u0ε(x),vε(x,0)=v0ε(x),x∈Ω,
where QT=Ω×(0,T), 0<ɛ<1,u0ɛ,v0ɛ∈C0∞(Ω) are nonnegative bounded smooth functions and satisfy (2.5)0≤u0ɛ≤‖u0‖L∞(Ω),0≤v0ɛ≤‖v0‖L∞(Ω),u0ɛm1⟶u0m1,v0ɛm2⟶v0m2,inW01,2(Ω)asɛ⟶0.
The standard parabolic theory (cf. [20, 21]) shows that (2.1)–(2.4) admits a nonnegative classical solution (uɛ,vɛ). So, the desired solution of the problem (1.1)–(1.4) will be obtained as a limit point of the solutions (uɛ,vɛ) of the problem (2.1)–(2.4). In the following, we show some important uniform estimates for (uɛ,vɛ).Lemma 2.1.
Let(uɛ,vɛ) be a solution of the problem (2.1)–(2.4).(1)
If1<(m1-α)(m2-β), then there exist positive constants r and s large enough such that
(2.6)1m2-β<m1+r-1m2+s-1<m1-α,(2.7)‖uɛ‖Lr(QT)≤C,‖vɛ‖Ls(QT)≤C,
where C is a positive constant only depending on m1,m2,α,β,r,s, |Ω|, and T.(2)
If1=(m1-α)(m2-β), then (2.7) also holds when |Ω| is small enough.Proof.
Multiplying (2.1) by uɛr-1(r>1) and integrating over Ω, we have that
(2.8)∫Ω∂uɛr∂tdx=-4r(r-1)m1(m1+r-1)2∫Ω|∇uɛ(m1+r-1)/2|2dx+r∫Ωuɛα+r-1(a-buɛ+cvɛ)dx.
By Poincaré’s inequality, we have that
(2.9)K∫Ωuɛm1+r-1dx≤∫Ω|∇uɛ(m1+r-1)/2|2dx,
where K is a constant depending only on |Ω| and N and becomes very large when the measure of the domain Ω becomes small. Since α<m1, Young's inequality shows that
(2.10)auɛα+r-1≤Kr(r-1)m1(m1+r-1)2uɛm1+r-1+CK-(α+r-1)/(m1-α),cuɛα+r-1vɛ≤Kr(r-1)m1(m1+r-1)2uɛm1+r-1+CK-(α+r-1)/(m1-α)vɛ(m1+r-1)/(m1-α).
For convenience, here and below, C denotes a positive constant which is independent of ɛ and may take different values on different occasions. Complying (2.8) with (2.9) and (2.10), we obtain
(2.11)∫Ω∂uɛr∂tdx≤-2Kr(r-1)m1(m1+r-1)2∫Ωuɛm1+r-1dx+CK-(α+r-1)/(m1-α)∫Ωvɛ(m1+r-1)/(m1-α)dx+CK-(α+r-1)/(m1-α).
As a similar argument as above, for vɛ and positive constant s>1, we have that
(2.12)∫Ω∂vɛs∂tdx≤-2Ks(s-1)m2(m2+s-1)2∫Ωvɛm2+s-1dx+CK-(β+s-1)/(m2-β)∫Ωuɛ(m2+s-1)/(m2-β)dx+CK-(β+s-1)/(m2-β).
Thus we have that
(2.13)∫Ω(∂uɛr∂t+∂vɛs∂t)dx≤-2Kr(r-1)m1(m1+r-1)2∫Ωuɛm1+r-1dx+CK-(β+s-1)/(m2-β)∫Ωuɛ(m2+s-1)/(m2-β)dx-2Ks(s-1)m2(m2+s-1)2∫Ωvɛm2+s-1dx+CK-(α+r-1)/(m1-α)∫Ωvɛ(m1+r-1)/(m1-α)dx+CK-(α+r-1)/(m1-α)+CK-(β+s-1)/(m2-β).
For the case of1<(m1-α)(m2-β), there exist r,s large enough such that
(2.14)1m1-α<m2+s-1m1+r-1<m2-β.
By Young's inequality, we have that
(2.15)∫Ωuɛ(m2+s-1)/(m2-β)dx≤r(r-1)m1K(m2+s-1)/(m2-β)C(m1+r-1)2∫Ωuɛm1+r-1dx+CK-γ1,∫Ωvɛ(m1+r-1)/(m1-α)dx≤s(s-1)m2K(m1+r-1)/(m1-α)C(m2+s-1)p2∫Ωvɛm2+s-1dx+CK-γ2,
where
(2.16)γ1=(m2+s-1)2[m2-β][(m2-β)(m1+r-1)-(m2+s-1)],γ2=(m1+r-1)2[m1-α][(m1-α)(m2+s-1)-(m1+r-1)].
Together with (2.13), we have that
(2.17)∫Ω(∂uɛr∂t+∂vɛs∂t)dx≤-K∫Ω(uɛm1+r-1+vɛm2+s-1)dx+C(K-θ1+K-θ2)+CK-(α+r-1)/(m1-α)+CK-(β+s-1)/(m2-β),
where
(2.18)θ1=(m2+s-1)+(m1+r-1)(β+s-1)(m2-β)(m1+r-1)-(m2+s-1),θ2=(m1+r-1)+(m2+s-1)(α+r-1)(m1-α)(m2+s-1)-(m1+r-1).
Furthermore, by Hölder's and Young's inequalities, from (2.17) we obtain
(2.19)∫Ω(∂uɛr∂t+∂vɛs∂t)dx≤-K∫Ω(uɛr+vɛs)dx+C(K-θ1+K-θ2)+2K|Ω|+CK-(α+r-1)/(m1-α)+CK-(β+s-1)/(m2-β).
Then by Gronwall's inequality, we obtain
(2.20)∫Ω(uɛr+vɛs)dx≤C.
Now we consider the case of1=(m1-α)(m2-β). It is easy to see that there exist positive constants r,s large enough such that
(2.21)1m1-α=m2+s-1m1+r-1=m2-β.
Due to the continuous dependence of K upon |Ω| in (2.9), from (2.13) we have that
(2.22)∫Ω(∂uɛr∂t+∂vɛs∂t)dx≤-K∫Ω(uɛm1+r-1+vɛm2(p2-1)+s-1)dx+C
when |Ω| is small enough. Then by Young's and Gronwall's inequalities we can also obtain (2.20), and thus we complete the proof of this lemma.Takinguɛm1, vɛm2 as the test functions, we can easily obtain the following lemma.Lemma 2.2.
Let(uɛ,vɛ) be a solution of (2.1)–(2.4); then
(2.23)∬QT|∇uɛm1|2dxdt≤C,∬QT|∇vɛm2|2dxdt≤C,
where C is a positive constant independent of ɛ.Lemma 2.3.
Let(uɛ,vɛ) be a solution of (2.1)–(2.4), then
(2.24)‖uɛ‖L∞(QT)≤C,‖vɛ‖L∞(QT)≤C,
where C is a positive constant independent of ɛ.Proof.
For a positive constantk>∥u0ɛ∥L∞(Ω), multiplying (2.1) by (uɛ-k)+m1χ[t1,t2] and integrating the results over QT, we have that
(2.25)1m1+1∬QT∂(uɛ-k)+m1+1χ[t1,t2]∂tdxdt+∬QT|∇(uɛ-k)+m1χ[t1,t2]|2dxdt≤∬QTauɛα+m1(a+cvɛ)dxdt,
where s+=max{0,s} and χ[t1,t2] is the characteristic function of [t1,t2](0≤t1<t2≤T). Let
(2.26)Ik(t)=∫Ω(uɛ-k)+m1+1dx;
then Ik(t) is absolutely continuous on [0,T]. Denote by σ the point where Ik(t) takes its maximum. Assume that σ>0, for a sufficient small positive constant ϵ. Taking t1=σ-ϵ, t2=σ in (2.25), we obtain
(2.27)1(m1+1)ϵ∫σ-ϵσ∫Ω∂(uɛ-k)+m1+1∂tdxdt+1ϵ∫σ-ϵσ∫Ω|∇(uɛ-k)+m1|2dxdt≤1ϵ∫σ-ϵσ∫Ωuɛα+m1(a+cvɛ)dxdt.
From
(2.28)∫σ-ϵσ∫Ω∂(uɛ-k)+m1+1∂tdxdt=Ik(σ)-Ik(σ-ϵ)≥0,
we have that
(2.29)1ϵ∫σ-ϵσ∫Ω|∇(uɛ-k)+m1|2dxdt≤1ϵ∫σ-ϵσ∫Ωuɛα+m1(a+cvɛ)dxdt.
Letting ϵ→0+, we have that
(2.30)∫Ω|∇(uɛ(x,σ)-k)+m1|2dx≤∫Ωuɛα+m1(x,σ)(a+cvɛ(x,σ))dx.
Denote Ak(t)={x:uɛ(x,t)>k} and μk=supt∈(0,T)|Ak(t)|; then
(2.31)∫Ak(σ)|∇(uɛ-k)+m1|2dx≤∫Ak(σ)uɛα+m1(a+cvɛ)dx.
By Sobolev's theorem,
(2.32)(∫Ak(σ)((uɛ-k)+m1)pdx)1/p≤C(∫Ak(σ)|∇(uɛ-k)+m1|2dx)1/2,
with
(2.33)2<p<{+∞,N≤2,2NN-2,N>2,
we obtain
(2.34)(∫Ak(σ)((uɛ-k)+m1)pdx)2/p≤C∫Ak(σ)|∇(uɛ-k)+m1|2dx≤C∫Ak(σ)uɛα+m1(a+vɛ)dx≤C(∫Ak(σ)uɛrdx)(m1+α)/r(∫Ak(σ)(a+vɛ)r/(r-m1-α)dx)(r-m1-α)/r≤C(∫Ak(σ)(a+vɛ)r/(r-m1-α)dx)(r-m1-α)/r≤C(∫Ak(σ)(a+vɛ)sdx)1/s|Ak(σ)|(s(r-m1-α)-r)/sr≤Cμk(s(r-m1-α)-r)/sr,
where r>p(m1+α)/(p-2), s>pr/(p(r-m1-α)-2r) and C denotes various positive constants independent of ɛ. By Hölder’s inequality, it yields
(2.35)Ik(σ)=∫Ω(uɛ-k)+m1+1dx=∫Ak(σ)(uɛ-k)+m1+1dx≤(∫Ak(σ)(uɛ-k)+m1pdx)(m1+1)/m1pμk1-(m1+1)/m1p≤Cμk1+[sp(r-m1-α)-pr-2sr](m1+1)/2psrm1.
Then
(2.36)Ik(t)≤Ik(σ)≤Cμk1+[sp(r-m1-α)-pr-2sr](m1+1)/2psrm1,t∈[0,T].
On the other hand, for any h>k and t∈[0,T], we have that
(2.37)Ik(t)≥∫Ak(t)(uɛ-k)+m1+1dx≥(h-k)m1+1|Ah(t)|.
Combined with (2.35), it yields
(2.38)(h-k)m1+1μh≤Cμk1+[sp(r-m1-α)-pr-2sr](m1+1)/2psrm1,
that is,
(2.39)μh≤C(h-k)m1+1μk1+[sp(r-m1-α)-pr-2sr](m1+1)/2psrm1.
It is easy to see that
(2.40)γ=1+[sp(r-m1-α)-pr-2sr](m1+1)2psrm1>1.
Then by the De Giorgi iteration lemma [22], we have that
(2.41)μl+d=sup⁡|Al+d(t)|=0,
where d=C1/(m1+1)μl(γ-1)/(m1+1)2γ/(γ-1). That is,
(2.42)uɛ≤l+da.e.inQT.
It is the same for the second inequality of (2.24). The proof is completed.Lemma 2.4.
The solution(uɛ,vɛ) of (2.1)–(2.4) satisfies the following:
(2.43)∬QT|∂uɛm1∂t|2dxdt≤C,∬QT|∂vɛm2∂t|2dxdt≤C,
where C is a positive constant independent of ɛ.Proof.
Multiplying (2.1) by (∂/∂t)uɛm1 and integrating over Ω, by (2.3), (2.4) and Young's inequality we have that
(2.44)4m1(m1+1)2∬QT|∂∂tuɛ(m1+1)/2|2dxdt=∬QT∂uɛ∂t∂uɛm1∂tdxdt=12∫Ω|∇uɛm1(x,0)|2dx-12∫Ω|∇uɛm1(x,T)|2dx+∬QTm1uɛα+m1-1(a-buɛ+cvɛ)∂uɛ∂tdxdt=12∫Ω|∇uɛm1(x,0)|2dx-12∫Ω|∇uɛm1(x,T)|2dx+∬QT2m1m1+1uɛ(2α+m1-1)/2(a-buɛ+cvɛ)∂uɛ(m1+1)/2∂tdxdt≤12∫Ω|∇uɛm1(x,0)|2dx+2m1∬QTuɛ2α+m1-1(a-buɛ+cvɛ)2dxdt+2m1(m1+1)2∬QT|∂∂tuɛ(m1+1)/2|2dxdt,
which together with the bound of a,b,c,uɛ,vɛ shows that
(2.45)∬QT|∂uɛ(m1+1)/2∂t|2dxdt≤C,
where C is a positive constant independent of ɛ. Noticing the bound of uɛ, we have that
(2.46)∬QT|∂uɛm1∂t|2dxdt=4m12(m1+1)2∬QTuɛm1-1|∂∂tuɛ(m1+1)/2|2dxdt≤C.
It is the same for the second inequality. The proof is completed.From the above estimates ofuɛ,vɛ, we have the following results.Theorem 2.5.
The problem (1.1)–(1.4) admits a generalized solution.Proof.
By Lemmas2.2, 2.3, and 2.4, we can see that there exist subsequences of {uɛ},{vɛ} (denoted by themselves for simplicity) and functions u,v such that
(2.47)uɛ⟶u,vɛ⟶v,a.einQT,∂uɛm1∂t⟶∂um1∂t,∂vɛm2∂t⟶∂vm2∂t,weaklyinL2(QT),∇uɛm1⟶∇um1,∇vɛm2⟶∇vm2,weaklyinL2(QT),
as ɛ→0. Then a rather standard argument as [23] shows that (u,v) is a generalized solution of (1.1)–(1.4) in the sense of Definition 1.1.In order to prove that the generalized solution of (1.1)–(1.4) is uniformly bounded, we need the following comparison principle.Lemma 2.6.
Let(u̲,v̲) be a subsolution of the problem (1.1)–(1.4) with the initial value (u̲0,v̲0) and (u¯,v¯) a supersolution with a positive lower bound of the problem (1.1)–(1.4) with the initial value (u¯0,v¯0). If u̲0≤v¯0, u̲0≤v¯0, then u̲(x,t)≤u¯(x,t), v̲(x,t)≤v¯(x,t) on QT.Proof.
Without loss of generality, we might assume that∥u̲(x,t)∥L∞(QT), ∥u¯(x,t)∥L∞(QT),∥v̲(x,t)∥L∞(QT), ∥v¯(x,t)∥L∞(QT)≤M, where M is a positive constant. By the definitions of subsolution and supersolution, we have that
(2.48)∫0t∫Ω-u̲∂φ∂t+∇u̲m1∇φdxdτ+∫Ωu̲(x,t)φ(x,t)dx-∫Ωu̲0(x)φ(x,0)dx≤∫0t∫Ωu̲α(a-bu̲+cv̲)φdxdτ,∫0t∫Ω-u¯∂φ∂t+∇u¯m1∇φdxdτ+∫Ωu¯(x,t)φ(x,t)dx-∫Ωv¯0(x)φ(x,0)dx≥∫0t∫Ωu¯α(a-bu¯+cv¯)φdxdτ.
Take the test function as
(2.49)φ(x,t)=Hɛ(u̲m1(x,t)-u¯m1(x,t)),
where Hɛ(s) is a monotone increasing smooth approximation of the function H(s) defined as follows:
(2.50)H(s)={1,s>0,0,otherwise.
It is easy to see that Hɛ′(s)→δ(s) as ɛ→0. Since ∂u̲m1/∂t,∂u¯m1/∂t∈L2(QT), the test function φ(x,t) is suitable. By the positivity of a,b,c we have that
(2.51)∫Ω(u̲-u¯)Hɛ(u̲m1-u¯m1)dx-∫0t∫Ω(u̲-u¯)∂Hɛ(u̲m1-u¯m1)∂tdxdτ+∫0t∫ΩHɛ′(u̲m1-u¯m1)|∇(u̲m1-u¯m1)|2dxdτ≤∫0t∫Ωa(u̲α-u¯α)Hɛ(u̲m1-u¯m1)+c(u̲αv̲-u¯αv¯)Hɛ(u̲m1-u¯m1)dxdτ,
where C is a positive constant depending on ∥a(x,t)∥C(Qt),∥c(x,t)∥C(Qt). Letting ɛ→0 and noticing that
(2.52)∫0t∫ΩHɛ′(u̲m1-u¯m1)|∇(u̲m-u¯m)|2dxdτ≥0,
we arrive at
(2.53)∫Ω[u̲(x,t)-u¯(x,t)]+dx≤C∫0t∫Ω(u̲α-u¯α)++v̲(u̲α-u¯α)++u¯α(v̲-v¯)+dxdτ.
Let (u¯,v¯) be a supsolution with a positive lower bound σ. Noticing that
(2.54)(xα-yα)+≤C(α)(x-y)+,forα≥1,(xα-yα)+≤xα-1(x-y)+≤yα-1(x-y)+,forα<1,
with x,y>0, we have that
(2.55)∫0t∫Ω(u̲α-u¯α)++v̲(u̲α-u¯α)++u¯α(v̲-v¯)+dxdτ≤C∫0t∫Ω(u̲-u¯)++(v̲-v¯)+dxdτ,
where C is a positive constant depending upon α,σ,M.
Similarly, we also have that(2.56)∫Ω[v̲(x,t)-v¯(x,t)]+dx≤C∫0t∫Ω(u̲-u¯)++(v̲-v¯)+dxdτ.
Combining the above two inequalities, we obtain
(2.57)∫Ω[u̲(x,t)-u¯(x,t)]++[v̲(x,t)-v¯(x,t)]+dx≤C∫0t∫Ω(u̲-u¯)++(v̲-v¯)+dxdτ.
By Gronwall's lemma, we see that u̲≤u¯,v̲≤v¯. The proof is completed.Corollary 2.7.
Ifblfl>cMeM, then the problem (1.1)–(1.4) admits at most one global solution which is uniformly bounded in Ω¯×[0,∞).Proof.
The uniqueness comes from the comparison principle immediately. In order to prove that the solution is global, we just need to construct a bounded positive supersolution of (1.1)–(1.4).
Letρ1=(aMfl+dMcM)/(blfl-cMeM) and ρ2=(aMeM+dMbl)/(blfl-cMeM), since blfl>cMeM; then ρ1,ρ2>0 and satisfy
(2.58)aM-blρ1+cMρ2=0,dM+eMρ1-flρ2=0.
Let (u¯,v¯)=(ηρ1,ηρ2), where η>1 is a constant such that (u0,v0)≤(ηρ1,ηρ2); then we have that
(2.59)u¯t-Δu¯m1=0≥u¯α(a-bu¯+cv¯),v¯t-Δv¯m2=0≥v¯β(d+eu¯-fv¯).
That is, (u¯,v¯)=(ηρ1,ηρ2) is a positive supersolution of (1.1)–(1.4). Since u¯,v¯ are global and uniformly bounded, so are u and v.


## 3. Periodic Solutions

In order to establish the existence of the nontrivial nonnegative periodic solutions of the problem (1.1)–(1.3), we need the following lemmas. Firstly, we construct a pair of T-periodic supersolution and T-periodic subsolution as follows.Lemma 3.1.
In case ofblfl>cMeM, there exists a pair of T-periodic supersolution and T-periodic subsolution of the problem (1.1)–(1.3).Proof.
We first construct aT-periodic subsolution of (1.1)–(1.3). Let λ be the first eigenvalue and ϕ be the uniqueness solution of the following elliptic problem:
(3.1)-Δϕ=λϕ,x∈Ω,ϕ=0,x∈∂Ω;
then we have that
(3.2)λ>0,ϕ(x)>0inΩ,|∇ϕ|>0on⁡∂Ω,M=max⁡x∈Ω¯ϕ(x)<∞.
Let
(3.3)(u̲,v̲)=(ɛϕ2/m1(x),ɛϕ2/m2(x)),
where ɛ>0 is a small constant to be determined. We will show that (u̲,v̲) is a (time independent, hence T-periodic) subsolution of (1.1)–(1.3).
Taking the nonnegative functionφ1(x,t)∈C1(Q¯T) as the test function, we have that
(3.4)∬QT(u̲∂φ1∂t+Δu̲m1φ1+u̲α(a-bu̲+cv̲)φ1)dxdt+∫Ωu̲(x,0)φ1(x,0)-u̲(x,T)φ1(x,T)dx=∬QT(u̲α(a-bu̲+cv̲)+Δu̲m1)φ1dxdt=∬QTu̲α(a-bu̲+cv̲)φ1dxdt-∬QT∇u̲m1∇φ1dxdt=∬QTu̲α(a-bu̲+cv̲)φ1dxdt-2ɛm1∬QTϕ∇ϕ⋅∇φ1dxdt=∬QTu̲α(a-bu̲+cv̲)φ1dxdt-2ɛm1∬QT∇ϕ∇(ϕφ1)-|∇ϕ|2φ1dxdt=∬QTu̲α(a-bu̲+cv̲)φ1dxdt-2ɛm1∬QT-div(∇ϕ)ϕφ1-|∇ϕ|2φ1dxdt=∬QTu̲α(a-bu̲+cv̲)φ1dxdt-2ɛm1∬QT(λϕ2-|∇ϕ|2)φ1dxdt.
Similarly, for any nonnegative test function φ2(x,t)∈C1(Q¯T), we have that
(3.5)∬QT(v̲∂φ2∂t+Δv̲m2φ2+v̲β(d+eu̲-fv̲)φ2)dxdt+∫Ωv̲(x,0)φ2(x,0)-v̲(x,T)φ2(x,T)dx=∬QTv̲β(d+eu̲-fv̲)φ2dxdt-2ɛm2∬QT(λϕ2-|∇ϕ|2)φ2dxdt.
We just need to prove the nonnegativity of the right-hand side of (3.4) and (3.5).
Sinceϕ1=ϕ2=0,|∇ϕ1|,|∇ϕ2|>0 on ∂Ω, then there exists δ>0 such that
(3.6)λϕ2-|∇ϕ|2≤0,x∈Ω¯δ,
where Ω¯δ={x∈Ω∣dist(x,∂Ω)≤δ}. Choosing
(3.7)ɛ≤min⁡{albMM2/m1,dlfMM2/m2},
then we have that
(3.8)2ɛm1∫0T∫Ωδ(λϕ2-|∇ϕ|2)φ1dxdt≤0≤∫0T∫Ωδu̲α(a-bu̲+cv̲)φ1dxdt,2ɛm2∫0T∫Ωδ(λϕ2-|∇ϕ|2)φ2dxdt≤0≤∫0T∫Ωδv̲β(d+eu̲-fv̲)φ2dxdt,
which shows that (u̲,v̲) is a positive (time independent, hence T-periodic) subsolution of (1.1)–(1.3) on Ω¯δ×(0,T).
Moreover, we can see that, for someσ>0,
(3.9)ϕ(x)≥σ>0,x∈Ω∖Ω¯δ.
Choosing
(3.10)ɛ≤min⁡{al2bMM2/m1,(al4λM2(m1-α)/m1)1/(m1-α),dl2fMM2/m2,(dl4λM2(m2-β)/m2)1/(m2-β)},
then
(3.11)ɛαϕ2α/m1a-bɛα+1ϕ2(α+1)/m1+cɛαϕ2α/m1ɛϕ2/m2-2ɛm1λϕ2≥0,ɛβϕ2β/m2d+eɛϕ2/m1ɛβϕ2β/m2-fɛβ+1ϕ2(β+1)/m2-2ɛm2λϕ2≥0
on QT, that is
(3.12)∬QTu̲α(a-bu̲+cv̲)φ1dxdt-2ɛm1∬QT(λϕ2-|∇ϕ|2)φ1dxdt≥0,∬QTv̲β(d+eu̲-fv̲)φ2dxdt-2ɛm2∬QT(λϕ2-|∇ϕ|2)φ2dxdt≥0.
These relations show that (u̲,v̲)=(ɛϕ12/m1(x),ɛϕ22/m2(x)) is a positive (time independent, hence T-periodic) subsolution of (1.1)–(1.3).
Letting(u¯,v¯)=(ηρ1,ηρ2), where η,ρ1,ρ2 are taken as those in Corollary 2.7, it is easy to see that (u¯,v¯) is a positive (time independent, hence T-periodic) subsolution of (1.1)–(1.3).
Obviously, we may assume thatu̲(x,t)≤u¯(x,t), v̲(x,t)≤v¯(x,t) by changing η, ɛ appropriately.Lemma 3.2 (see [24, 25]).
Letu be the solution of the following Dirichlet boundary value problem
(3.13)∂u∂t=Δum+f(x,t),(x,t)∈Ω×(0,T),u(x,t)=0,(x,t)∈∂Ω×(0,T),
where f∈L∞(Ω×(0,T)); then there exist positive constants K and α∈(0,1) depending only upon τ∈(0,T) and ∥f∥L∞(Ω×(0,T)), such that, for any (xi,ti)∈Ω×[τ,T](i=1,2),
(3.14)|u(x1,t1)-u(x2,t2)|≤K(|x1-x2|α+|t1-t2|α/2).Lemma 3.3 (see [26]).
Define a Poincaré mapping(3.15)Pt:L∞(Ω)×L∞(Ω)⟶L∞(Ω)×L∞(Ω),Pt(u0(x),v0(x))∶=(u(x,t),v(x,t))(t>0),
where (u(x,t),v(x,t)) is the solution of (1.1)–(1.4) with initial value (u0(x),v0(x)). According to Lemmas 2.6 and 3.2 and Theorem 2.5, the map Pt has the following properties:(i)
Pt is defined for any t>0 and order preserving;(ii)
Pt is order preserving;(iii)
Pt is compact.Observe that the operatorPT is the classical Poincaré map and thus a fixed point of the Poincaré map gives a T-periodic solution setting. This will be made by the following iteration procedure.Theorem 3.4.
Assume thatblfl>cMeM and there exists a pair of nontrivial nonnegative T-periodic subsolution (u̲(x,t),v̲(x,t)) and T-periodic supersolution (u¯(x,t),v¯(x,t)) of the problem (1.1)–(1.3) with u̲(x,0)≤u¯(x,0); then the problem (1.1)–(1.3) admits a pair of nontrivial nonnegative periodic solutions (u*(x,t),v*(x,t)),(u*(x,t),v*(x,t)) such that
(3.16)u̲(x,t)≤u*(x,t)≤u*(x,t)≤u¯(x,t),v̲(x,t)≤v*(x,t)≤v*(x,t)≤v¯(x,t),inQT.Proof.
Takingu¯(x,t), u̲(x,t) as those in Lemma 3.1 and choosing suitable B(x0,δ),B(x0,δ′),Ω′, k1, k2, and K, we can obtain u̲(x,0)≤u¯(x,0). By Lemma 2.6, we have that  PT(u̲(·,0))≥u̲(·,T). Hence by Definition 1.2 we get PT(u̲(·,0))≥u̲(·,0), which implies P(k+1)T(u̲(·,0))≥PkT(u̲(·,0)) for any k∈ℕ. Similarly we have that PT(u¯(·,0))≤u¯(·,T)≤u¯(·,0), and hence P(k+1)T(u¯(·,0))≤PkT(u¯(·,0)) for any k∈ℕ. By Lemma 2.6, we have that PkT(u̲(·,0))≤PkT(u¯(·,0)) for any k∈ℕ. Then
(3.17)u*(x,0)=lim⁡k→∞PkT(u̲(x,0)),u*(x,0)=lim⁡k→∞PkT(u¯(x,0))
exist for almost every x∈Ω. Since the operator PT is compact (see Lemma 3.3), the above limits exist in L∞(Ω), too. Moreover, both u*(x,0) and u*(x,0) are fixed points of PT. With the similar method as [26], it is easy to show that the even extension of the function u*(x,t), which is the solution of the problem (1.1)–(1.4) with the initial value u*(x,0), is indeed a nontrivial nonnegative periodic solution of the problem (1.1)–(1.3). It is the same for the existence of u*(x,t). Furthermore, by Lemma 2.6, we obtain (3.16) immediately, and thus we complete the proof.Furthermore, by De Giorgi iteration technique, we can also establish a prior upper bound of all nonnegative periodic solutions of (1.1)–(1.3). Then with a similar method as [18], we have the following remark which shows the existence and attractivity of the maximal periodic solution.Remark 3.5.
Ifblfl>cMeM, the problem (1.1)–(1.3) admits a maximal periodic solution (U,V). Moreover, if (u,v) is the solution of the initial boundary value problem (1.1)–(1.4) with nonnegative initial value (u0,v0), then, for any ɛ>0, there exists t depending on u0, v0, and ɛ, such that
(3.18)0≤u≤U+ɛ,0≤v≤V+ɛ,forx∈Ω,t≥t.



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*Source: 714248-2011-04-14.xml*