# Existence of Solutions for a Quasilinear Reaction Diffusion System

Authors: Canrong Tian
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The degenerate reaction diffusion system has been applied to a variety of physical and engineering problems. This paper is extended the existence of solutions from the quasimonotone reaction functions (e.g., inhibitor-inhibitor mechanism) to the mixed quasimonotone reaction functions (e.g., activator-inhibitor mechanism). By Schauder fixed point theorem, it is shown that the system admits at least one positive solution if there exist a coupled of upper and lower solutions. This result is applied to a Lotka-Volterra predator-prey model.


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## Body

## 1. Introduction

We consider a quasilinear reaction diffusion system in a bounded domain under coupled nonlinear boundary conditions. The system of equations is given in the form(1.1)∂ui∂t-∇⋅(aiDi(ui)∇ui)+bi⋅(Di(ui)∇ui)=fi(t,x,u)(t>0,x∈Ω),Di(ui)∂ui∂ν=gi(t,x,u)(t>0,x∈∂Ω),ui(0,x)=ψi(x)(x∈Ω),i=1,…,N,
where u≡(u1,…,uN),  Ω is a bounded domain in ℝn with boundary ∂Ω, ∂/∂ν denotes the outward normal derivative on ∂Ω. It is assumed that the boundary ∂Ω is of class C1+α. It is also assumed that, for each i=1,…,N, the functions ai≡ai(t,x), bi≡bi(t,x)≡(bi(1),…,bi(n)), fi(t,x,·) and gi(t,x,·) are Hölder continuous in [0,∞)×Ω¯. The density-dependent diffusion coefficient Di(ui) may have the property Di(0)=0, which means that the elliptic operators are degenerate.The quasilinear reaction diffusion system has been investigated extensively in the literature [1–3]. Recently by use of upper and lower solutions and its associated monotone iterations, [4, 5] deal with the scalar equation and the system endowed with the nonlinear Neumann-Robin boundary conditions, respectively. The paper in [6] is concerned with the existence, uniqueness, and asymptotic behavior for the quasilinear parabolic systems with the Dirichlet boundary condition. However, the requirement of the reaction functions in [4–6] are monotone nondecreasing. This paper relaxed the condition to mixed quasimonotone reaction functions, which leads to the difficult point that the ordered upper and lower solutions do not exist. To overcome it, we construct the coupled upper and lower solutions.The purpose of this paper is to study the existence for the system (1.1) by the Schauder fixed point theorem. The rest of this paper is organized as follows. In Section 2 we show the existence by the method of upper and lower solutions and the Schauder fixed point theorem. An application is given in Section 3 to the Lotka-Volterra predator-prey model. The paper ends with Section 4 for some discussions.


## 2. Existence of Solutions

To the simplicity, throughout this paper, we denote(2.1)Q=(0,T]×Ω,S=(0,T]×∂Ω,Q¯=[0,T]×Ω¯,
and let Cm(Q) and Cα(Q) be the respective space of m-times differentiable and Hölder continuous functions in Q, where Q represents a domain or a section between two functions. For vector functions with N-components we denote the above function space by 𝒞m(Q) and 𝒞α(Q), respectively.In this paper, we make the following hypothesis.(H) For each i=1,…,N, the following conditions hold:(i)
ai(t,x), bi(l)(t,x)(l=1,…,n) and fi(t,x,·) are in Cα/2,α(Q¯) with ai≥ai*>0, gi(t,x,·)∈C1+α/2,2+α(Q¯);(ii)
Di(ui)∈C1+α/2,1+α(Λi) and Di(ui)>0 for ui>0 and Di(0)≥0;(iii)
f(·,u),g(·,u) are mixed quasimonotone 𝒞1-functions in Λ.In the above hypothesis,Λ and Λi are the sectors between a pair of coupled upper and lower solutions given by (2.8) below. It is allowed that Di(0)=0  for some i and Di(0)>0 for a different i. Particularly, if Di(u) is a positive constant for all i then system (1.1) becomes the standard coupled system of semilinear parabolic equations. Recall that a vector function f(·,u) is said to be mixed quasimonotone in Λ if for each i=1,…,N, there exist nonnegative integers ai and bi with ai+bi=N-1 such that the function fi(·,u)≡fi(·,ui,[u]ai,[u]bi) is nondecreasing with respect to all component [u]ai and is nonincreasing with respect to all component [u]bi, where u≡(ui,[u]ai,[u]bi)∈Λ. Similarly, gi(·,u)≡gi(·,ui,[u]ci,[u]di). Our approach to the existence problem is by the method of coupled upper and lower solutions which are defined as follows.Definition 2.1.
A pair of functionsũ=(ũ1,…,ũN),û=(û1,…,ûN)∈𝒞(Q¯)∩𝒞1,2(Q) are called coupled upper and lower solutions of (1.1) if ũ≥û and if
(2.2)∂ũi∂t-∇⋅(aiDi(ũi)∇ũi)+bi⋅(Di(ũi)∇ũi)≥fi(⋅,ũi,[ũ]ai,[û]bi)inQ,∂ûi∂t-∇⋅(aiDi(ûi)∇ûi)+bi⋅(Di(ûi)∇ûi)≤fi(⋅,ûi,[û]ai,[ũ]bi)inQ,Di(ũi)∂ũi∂ν≥gi(⋅,ũi,[ũ]ci,[û]di)on⁡S,Di(ûi)∂ûi∂ν≤gi(⋅,ûi,[û]ci,[ũ]di)on⁡S,ũi(0,x)≥ψi(x),ûi(0,x)≤ψi(x)inΩ,i=1,…,N.Define(2.3)wi=Ii(ui)=∫0uiDi(s)dsforui≥0,i=1,…,N,
it follows from the following Hypothesis (H) that(2.4)Ii′(ui)=dIidui=Di(ui)>0,
then the inverse ui≡qi(wi) exists and is an increasing function of wi>0. In view of(2.5)∂wi∂t=Di(ui)∂ui∂t,∇wi=Di(ui)∇ui,∂wi∂ν=Di(ui)∂ui∂ν
we may write (1.1) in the equivalent form(2.6)(Di(ui))-1∂wi∂t-∇⋅(ai∇wi)+bi⋅∇wi=fi(t,x,ui,[u]ai,[u]bi)inQ,∂wi∂ν=gi(t,x,ui,[u]ci,[u]di)on⁡S,wi(0,x)=ηi(x)inΩ,ui=qi(wi)i=1,…,NinΩ¯,
where ηi(x)=Ii(ψi(x)). Thus the pair (ũ,w̃) and (û,ŵ), where w̃i=Ii(ũi) and ŵi=I(ûi), satisfy the inequalities(2.7)(Di(ũi))-1∂w̃i∂t-∇⋅(ai∇w̃i)+bi⋅(∇w̃i)≥fi(⋅,ũi,[ũ]ai,[û]bi)inQ,(Di(ûi))-1∂ŵi∂t-∇⋅(ai∇ŵi)+bi⋅(∇ŵi)≤fi(⋅,ûi,[û]ai,[ũ]bi)inQ,∂w̃i∂ν≥gi(⋅,ũi,[ũ]ci,[û]di)on⁡S,∂ŵi∂ν≤gi(⋅,ûi,[û]ci,[ũ]di)on⁡S,w̃i(0,x)≥ηi(x)ŵi(0,x)≤ηi(x)inΩ,i=1,…,N,
is referred to as coupled upper and lower solutions of (2.6). For a given pair of coupled upper and lower solutions ũ, û we set(2.8)Λi={ui∈C(Q¯):ûi≤u≤ũi},Λ={u∈C(Q¯):û≤u≤ũ},Λ×Λ¯={(u,w)∈C(Q¯)×C(Q¯):(û,ŵ)≤(u,w)≤(ũ,w̃)}.
In Hypothesis (H)-(ii) we allow Di(0)=0 which leads to a degenerate diffusion coefficient. If Di(0)=0, we set ûi≥δi>0, which ensures that Di(ui) has a positive lower bound. Since (H)-(iii), there exist smooth nonnegative functions ci(l)≡ci(l)(t,x),l=1,2, such that(2.9)ci(1)Di(ui)+∂fi∂ui(⋅,u)≥0,ci(2)Di(ui)+∂gi∂ui(⋅,u)≥0foru∈Λ.
In fact, it suffices to choose any c(1)≡(c1(1),…,cN(1)), c(2)≡(c1(2),…,cN(2)) satisfying(2.10)ci(1)(t,x)≥max⁡{-∂fi/∂ui(t,x,u)Di(ui):u∈Λ},ci(2)(t,x)≥max⁡{-∂gi/∂ui(t,x,u)Di(ui):u∈Λ}.
Define for each i=1,…,N,(2.11)Fi(t,x,u)=ci(1)(t,x)Ii(ui)+fi(t,x,u),Gi(t,x,u)=ci(2)(t,x)Ii(ui)+gi(t,x,u),Liwi=∇⋅(ai∇wi)-bi∇⋅wi-ci(1)(t,x)wi,Biwi=∂wi∂ν+ci(2)(t,x)wi.
Since (2.9), (H) and Ii′(ui)=Di(ui), Fi(·,u) and Gi(·,u) possess the property(2.12)Fi(⋅,vi,[v]ai,[u]bi)≤Fi(⋅,ui,[u]ai,[v]bi)Gi(⋅,vi,[v]ci,[u]di)≤Gi(⋅,ui,[u]ci,[v]di),wheneverû≤v≤u≤ũ.
Moreover, (2.6) is equivalent to(2.13)(Di(ui))-1∂wi∂t-Liwi=Fi(⋅,ui,[u]ai,[u]bi)inQ,Biwi=Gi(⋅,ui,[u]ci,[u]di)on⁡S,wi(0,x)=ηi(x)inΩ,ui=qi(wi),i=1,…,N,inΩ¯.
Thus the pair (ũ,w̃) and (û,ŵ), where w̃i=Ii(ũi) and ŵi=Ii(ûi), satisfies the inequalities(2.14)(Di(ũi))-1∂w̃i∂t-Liw̃i≥Fi(⋅,ũi,[ũ]ai,[û]bi)inQ,(Di(ûi))-1∂ŵi∂t-Liŵi≤Fi(⋅,ûi,[ũ]ai,[û]bi)inQ,Biw̃i≥Gi(⋅,ũi,[ũ]ci,[û]di)on⁡S,Biŵi≤Gi(⋅,ûi,[û]ci,[ũ]di)on⁡S,w̃i(0,x)≥ηi(x)ŵi(0,x)≤ηi(x)inΩ,
are referred to coupled upper and lower solutions of (2.13).The property (2.12) is quite useful for the construction of monotone convergent sequences. To ensure the existence of the sequence to be constructed in the iteration process (2.16) below we assume that either Di(0)>0 or Di(0)=0 for ûi≥δi>0. Define a modified function D¯i(ui) by(2.15)D¯i(ui)={Di(ui)+(ui-ũi)ifui>ũi,Di(ui)ifûi≤ui≤ũi,Di(ui)+(ûi-ui)ifui<ûi.
Then by the above assumption, there exists d0>0 such that D¯i(u)≥d0 for all u∈ℝ.By usingu̲(0)=û and u¯(0)=ũ as the initial iteration we can construct sequences {u̲(m),w̲(m)} and {u¯(m),w¯(m)} from the nonlinear iteration process(2.16)(D¯i(u¯i(m)))-1∂w¯i(m)∂t-Liw¯i(m)=Fi(⋅,u¯i(m-1),[u¯(m-1)]ai,[u̲(m-1)]bi)inQ,(D¯i(u̲i(m)))-1∂w̲i(m)∂t-Liw̲i(m)=Fi(⋅,u̲i(m-1),[u̲(m-1)]ai,[u¯(m-1)]bi)inQ,Biw¯i(m)=Gi(⋅,u¯i(m-1),[u¯(m-1)]ci,[u̲(m-1)]di)on⁡S,Biw̲i(m)=Gi(⋅,u̲i(m-1),[u̲(m-1)]ci,[u¯(m-1)]di)on⁡S,w¯i(m)(0,x)=ηi(x),w̲i(m)(0,x)=ηi(x)inΩ.
The sequences {u̲(m),w̲(m)} and {u¯(m),w¯(m)} are well defined by the existence theorem of [1]. The following lemma gives the monotone property of these sequences.Lemma 2.2.
The sequences{u¯(m),w¯(m)}, {u̲(m),w̲(m)} governed by (2.16) possess the monotone property
(2.17)(û,ŵ)≤(u̲(m),w̲(m))≤(u̲(m+1),w̲(m+1))≤(u¯(m+1),w¯(m+1))≤(u¯(m),w¯(m))≤(ũ,w̃)form=1,2,….
Moreover, for each m=1,2,…,u¯(m) and u̲(m) are coupled upper and lower solutions of (1.1).Proof.
Letz̲i(1)=w̲i(1)-w̲i(0), i=1,…,N. Then by (2.14) and (2.16), z̲i(1) satisfies
(2.18)(D¯i(u̲i(1)))-1∂z̲i(1)∂t-Liz̲i(1)=Fi(⋅,u̲i(0),[u̲(0)]ai,[u¯(0)]bi)-[(D¯i(u̲i(1)))-1∂w̲i(0)∂t-Liw̲i(0)]=Fi(⋅,u̲i(0),[u̲(0)]ai,[u¯(0)]bi)-[(D¯i(u̲i(0)))-1∂w̲i(0)∂t-Liw̲i(0)]-[(D¯i(u̲i(1)))-1-(D¯i(u̲i(0)))-1]∂w̲i(0)∂t≥-[(D¯i(u̲i(1)))-1-(D¯i(u̲i(0)))-1]∂w̲i(0)∂t.
Since by the mean value theorem,
(2.19)(D¯i(u̲i(1)))-1-(D¯i(u̲i(0)))-1=-[Di′¯(ξ(0))(D¯i(ξ(0)))2](u̲i(1)-u̲i(0))=-[Di′¯(ξ(0))(D¯i(ξ(0)))3](w̲i(1)-w̲i(0)),
for some intermediate value ξ(0)≡ξ(0)(t,x) between u̲i(0) and u̲i(1), we have
(2.20)(D¯i(u̲i(1)))-1∂z̲i(1)∂t-Liz̲i(1)+γ(0)z̲i(1)≥0,
where
(2.21)γ(0)=-[Di′¯(ξ(0))(D¯i(ξ(0)))3]∂w̲i(0)∂t.
Since (2.14), the boundary and initial inequalities
(2.22)Biz̲i(1)=Gi(⋅,u̲i(0),[u̲(0)]ci,[u¯(0)]di)-Biŵi≥0on⁡S,z̲i(1)(0,x)=ηi(x)-ηi(x)=0inΩ.
In view of the definition of D¯i in (2.15), the function D¯i(u̲i(1))γ(0) of (2.20) is bounded. From the weak maximum principle, it follows z̲i(1)≥0 on Q¯. This gives w̲i(1)≥w̲i(0) and thus u̲i(1)≥u̲i(0). A similar argument yields w¯i(1)≤w¯i(0) and u¯i(1)≤u¯i(0).
Moreover, lettingzi(1)=w¯i(1)-w̲i(1), by (2.12), (2.16), and after the similar above argument
(2.23)(D¯i(u¯i(1)))-1∂zi(1)∂t-Lizi(1)+γi(0)zi(1)=Fi(⋅,u¯i(0),[u¯(0)]ai,[u̲(0)]bi)-Fi(⋅,u̲i(0),[u̲(0)]ai,[u¯(0)]bi)≥0inQ,Bizi(1)=Gi(⋅,u¯i(0),[u¯(0)]ci,[u̲(0)]di)-Gi(⋅,u̲i(0),[u̲(0)]ci,[u¯(0)]di)≥0on⁡S,zi(1)(0,x)=ηi(x)-ηi(x)=0inΩ,
where
(2.24)γi(0)=-[Di′¯(ξi(0))(D¯i(ξi(0)))3]∂w̲i(0)∂t,
for some intermediate value ξi(0)≡ξi(0)(t,x) between u̲i(0) and   u¯i(1). It follows again from the weak maximum principle that w¯(1)≥w̲(1) and thus u¯(1)≥u̲(1). The above conclusions show that
(2.25)(u̲(0),w̲(0))≤(u̲(1),w̲(1))≤(u¯(1),w¯(1))≤(u¯(0),w¯(0)).
Now we show thatu¯(1) and u̲(1) are coupled upper and lower solutions of (1.1). Since (2.25), D¯i(u¯i(1))=Di(u¯i(1)) for i=1,…,N. It suffices to show that u¯(1) and u̲(1) satisfy (2.14). Since (2.12) and (2.16), we have
(2.26)(Di(u¯i(1)))-1∂w¯i(1)∂t-Liw¯i(1)=Fi(⋅,u¯i(0),[u¯(0)]ai,[u̲(0)]bi)≥Fi(⋅,u¯i(1),[u¯(1)]ai,[u̲(1)]bi),(Di(u̲i(1)))-1∂w̲i(1)∂t-Liw̲i(1)=Fi(⋅,u̲i(0),[u̲(0)]ai,[u¯(0)]bi)≤Fi(⋅,u̲i(1),[u̲(1)]ai,[u¯(1)]bi),Biu¯i(1)=Gi(⋅,u¯i(0),[u¯(0)]ci,[u̲(0)]di)≥Gi(⋅,u¯i(1),[u¯(1)]ci,[u̲(1)]di),Biu̲i(1)=Gi(⋅,u̲i(0),[u̲(0)]ci,[u¯(0)]di)≤Gi(⋅,u̲i(1),[u̲(1)]ci,[u¯(1)]di),u¯i(1)(0,x)=ηi(x)u̲i(1)(0,x)=ηi(x).
Next we use an induction method. We assume thatu¯(m) and u̲(m) are coupled upper and lower solutions of (1.1) and satisfying the following relation:
(2.27)(û,ŵ)≤(u̲(m),w̲(m))≤(u¯(m),w¯(m))≤(ũ,w̃).
Then by choosing u¯(m) and u̲(m) as the coupled upper and lower solutions ũ and û, after the similar above argument, we have
(2.28)(u̲(m),w̲(m))≤(u̲(m+1),w̲(m+1))≤(u¯(m+1),w¯(m+1))≤(u¯(m),w¯(m)),u¯(m+1) and u̲(m+1) are coupled upper and lower solutions of (1.1). The conclusion of the lemma follows from the induction principle.Theorem 2.3.
Letũ, û be a pair of coupled upper and lower solutions of (1.1), and let hypothesis (H) hold. Assume that either Di(0)>0 for some i or ûi≥δi>0. Then the problem (1.1) has at least one solution u∈Λ.Proof.
We first consider the problem (2.13), where Di is replaced by D¯i. For each i=1,…,N, we define operators ℒi:𝒟i×𝒟¯i→ℛi and ℒ:𝒟×𝒟¯→ℛ by
(2.29)Li(ui,wi)=(D¯i(ui))-1∂wi∂t-Liwi(i=1,…,N),L(u,w)=(L1(u1,w1),…,LN(uN,wN)),
where
(2.30)Di={ui∈C1+α/2,2+α(Q);ui(0,x)=ψi(x)inΩ},Ri={ui∈Cα/2,α(Q)},D=D1×…×DN,R=R1×…×RN.
Define also
(2.31)F(u)=(F1(⋅,u1,[u]ai,[u]bi),…,FN(⋅,uN,[u]ai,[u]bi)).
Then the system (2.13), in which Di is replaced by D¯i, may be written in the form
(2.32)L(u,w)=F(u),B(w)=G(u),((u,w)∈D×D¯),
where B=(B1,…,BN) and G=(G1,…,GN) are given in (2.11). Given any v∈Λ and any i=1,…,N, we consider the scalar problem
(2.33)Li(ui,wi)=Fi(v)inQ,Biwi=Gi(v)on⁡S,ui(0,x)=ψi(x)inΩ.
It follows from the existence theorem of [1] (Chapter V, Section 7) that (2.33) has a unique solution (ui*,wi*)∈𝒟i×𝒟¯i. In fact, the inverse ℒi-1:ℛi→𝒟i×𝒟¯i exists and is a positive compact operator on ℛi. This implies that the equation
(2.34)L(u,w)=F(v)B(w)=G(v),
has a unique solution (u,w)=ℒ-1[ℱ(v)],w=B-1G(v). Let χ be the closed bounded convex sunset given by
(2.35)χ={(u,w)∈R×R¯:û≤u≤ũ,ŵ≤w≤w̃}.
By the compact property on ℒ-1 and the hypothesis on f the operator ℒ-1ℱ is compact on χ. We show that ℒ-1ℱ maps χ to itself.
Let(v,z)∈χ be given, and (u,w)=ℒ-1[ℱ(v)]. After the similar argument of the proof of Lemma 2.2, we conclude û≤u≤ũ, therefore ℒ-1ℱ maps χ to itself. It follows from the Schauder fixed point theorem that (2.13) with Di being replaced by D¯i has at least one solution u∈χ. Since û≤u≤ũ, it follows from (2.15) that D¯i(ui)=Di(ui) for i=1,…,N. Thus u is also the solution of (2.13). Therefore the existence of the solution to (1.1) is proved.


## 3. Applications

As an application of the results obtained in the previous section we consider a Lotka-Volterra predator model. This model involves two speciesu1 and u2 that are governed by the system(3.1)∂u1∂t-∇⋅(D1(u1)∇u1)=u1(a1-b11u1-b12u2)(t>0,x∈Ω)∂u2∂t-∇⋅(D2(u2)∇u2)=u2(a2+b21u1-b22u2)(t>0,x∈Ω),∂u1∂ν+β1u1=0,∂u2∂ν+β2u2=0(t>0,x∈∂Ω),ui(0,x)=ψi(x)i=1,2,(x∈Ω),
where ai,bij are positive constants βi≡βi(x)≥0∈∂Ω, the initial functions ψi(x) for i=1,2 have a positive lower bound. The density-dependent diffusion coefficients D1(0)=D2(0)=0.It is easy to verify that if(ũ1,ũ2) and (û1,û2) satisfy (ũ1,ũ2)≥(û1,û2) and the following inequalities:(3.2)∂ũ1∂t-∇⋅(D1(ũ1)∇ũ1)≥ũ1(a1-b11ũ1-b12û2),∂ũ2∂t-∇⋅(D2(ũ2)∇ũ2)≥ũ2(a2+b21ũ1-b22ũ2),∂û1∂t-∇⋅(D1(û1)∇û1)≤û1(a1-b11û1-b12ũ2),∂û2∂t-∇⋅(D2(û2)∇û2)≤û2(a2+b21û1-b22û2),∂ũ1∂ν+β1ũ1≥0,∂ũ2∂ν+β2ũ2≥0,∂û1∂ν+β1û1≤0,∂û2∂ν+β2û2≤0,ûi(0,x)≤ψi(x)≤ũi(0,x),i=1,2,
then the pair (ũ1,ũ2),(û1,û2) are coupled upper and lower solutions of (3.1).To guarantee (3.2), we seek such a pair in the form(3.3)(ũ1,ũ2)=(M1,M2),(û1,û2)=(q1(δ1ϕ1),q2(δ2ϕ2)),
where for each i=1,2,  Mi and δi are positive constants to be chosen, qi is the inverse of (2.3), and ϕi is the (normalized) positive eigenfunction corresponding to the smallest eigenvalue of the eigenvalue problem(3.4)∇2ϕi+λiϕ=0inΩ,∂ϕi∂ν+γiϕi=0on⁡∂Ω(i=1,2).
The constant γi>0 will be determined in the following discussion. If we set(3.5)M1=a1b11M2=(a2+b21M1)b22,
then the first and second inequalities of (3.2) are satisfied. The third and fourth inequalities become(3.6)-∇2(δ1ϕ1)≤q1(δ1ϕ1)(a1-b11q1(δ1ϕ1)-b12M2),-∇2(δ2ϕ2)≤q2(δ2ϕ2)(a2+b21q1(δ1ϕ1)-b22q2(δ2ϕ2)).
By (3.4) and qi(δiϕi)>0, the above inequalities are satisfied by some sufficiently small δi>0 if(3.7)λ1<(q1(δ1ϕ1)δ1ϕ1)(a1-b12M2),λ2<(q2(δ2ϕ2)δ2ϕ2)a2.
Since Di(0)=0, by L’Hopital’s rule,(3.8)lim⁡w→0+[qi(w)w]=lim⁡w→0+qi′(w)=lim⁡z→0+1Di(z)=∞,
we see that there exists δi*>0 such that the inequalities in (3.7) are satisfied by every δi≤δi* if we impose the condition(3.9)M2<a1b12.
By (3.3), the fifth inequalities of (3.2) are trivially satisfied, and the sixth inequalities of (3.2) become(3.10)D1(q1(δ1ϕ1))∂q1(δ1ϕ1)∂ν+β1D1(q1(δ1ϕ1))q1(δ1ϕ1)≤0,D2(q2(δ2ϕ2))∂q2(δ2ϕ2)∂ν+β2D2(q2(δ2ϕ2))q2(δ2ϕ2)≤0,
Substituting (3.4) into (3.10) yields(3.11)-γ1(δ1ϕ1)≤-β1D1(q1(δ1ϕ1))q1(δ1ϕ1),-γ2(δ2ϕ2)≤-β2D2(q2(δ2ϕ2))q2(δ2ϕ2),
It is obvious that the above relations hold for any γi≥0 if βi(x)≡0. In the general case βi(x)≢0 the relations δ1ϕ1=I1(û1) and δ2ϕ2=I2(û2), where Ii is defined in (2.3), implies that (3.11) is satisfied if(3.12)γ1≥β1û1D1(û1)I1(û1),γ2≥β2û2D2(û2)I2(û2).
Since Ii′(z)=Di(z),(3.13)lim⁡z→0+[zDi(z)Ii(z)]=lim⁡z→0+[Di(z)+zDi′(z)Di(z)]=1+lim⁡z→0+[zDi′(z)Di(z)].
If we impose the condition(3.14)lim⁡z→0+zDi′(z)Di(z)=ρi,
then by setting(3.15)γi>βi(1+ρi),(3.12) is satisfied. If the below (3.16) holds, then (3.5) and (3.9) are satisfied. Thus all inequalities of (3.2) are satisfied. Directly applying Theorem 2.3, we have the following theorem.Theorem 3.1.
Suppose the initial functionsψi(x)≤Mi in (2.8) for i=1,2. Let
(3.16)a1a2>b11b12b11b22-b12b21,
and let βi(x)≥0 and D1(u1),D2(u2) satisfy (H)-(ii) with D1(0)=D2(0)=0. Assume that either βi(x)≡0 or
(3.17)lim⁡z→0+[zDi′(z)Di(z)]=ρi,i=1,2,
for some constants ρi. Then the system (3.1) admits at least one positive solution.Remark 3.2.
Pao and Ruan [5] have considered a Lotka-Volterra competition model with density-dependent diffusion, where the coefficient b21 of the system (3.1) is negative. The difference between them is that our method does not require that the reaction functions possess the monotone nondecreasing property. The condition for the existence for the solutions of the competition model is b12/b22<a1/a2<b11/b21, while the condition for the existence for the solutions of the predator model is a1/a2>b11b12/(b11b22-b12b21).Remark 3.3.
In a special caseD1(u1)=(m-1)u1m-1, D2(u2)=(m-1)u2m-1 for m>1, (3.1) becomes
(3.18)∂u1∂t-∇2u1m=u1(a1-b11u1-b12u2)(t>0,x∈Ω),∂u2∂t-∇2u2m=v(a2+b21u1-b22u2)(t>0,x∈Ω),∂u1∂ν+β1u1=0,∂u2∂ν+β2u2=0(t>0,x∈∂Ω).
Then the condition (3.17) is trivially satisfied. The conclusions in Theorem 3.1 hold true for (3.18). In fact, if D1(u1)=D2(u2)=d, the condition (3.17) is also trivial true, hence Theorem 2.3 is also valid. After the similar proof as Theorem 3.1, we conclude that Theorem 3.1 holds true for semilinear parabolic system.


## 4. Discussions

The intension of the present paper is to demonstrate the existence of solutions for the degenerate diffusion reaction system with nonlinear boundary condition. Our method is to look for the positive solution by constructing the coupled upper and lower solutions. The virtue of the technique is that it helps to extend the results for the scalar equation to the coupled system. Our existence theorem of Theorem2.3 in this paper is applicable to various Lotka-Volterra models, such as competition, predator-prey, or mutualism model, while the method in [6] is not applicable to predator-prey model.Since Levin and Segel illuminated the important role of the diffusion on the patterns in [7], a number of Lotka-Volterra models with constant diffusion have been investigated in the past three decades. In fact the concern of the density-dependent diffusion is also reasonable in animal disperse model (see [8] for a review). Our study is a starting attempt to consider the role of the density-dependent diffusion on Lotka-Volterra model. In biological terms, the results of Theorems 3.1 imply that if the rate of intraspecific competition of the predator is large, the two species are coexistent. The results also have applicability to 3 species model. Note that for Lotka-Volterra predator-prey model with constant diffusion, when the rate of intraspecific competition of the prey is large, the two species are both extinct. When the density-dependent diffusion is taken into account, it is an open problem whether there exist the extinct phenomena.



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*Source: 368425-2012-06-24.xml*