# Blowup Properties for a Semilinear Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

Authors: Dengming Liu; Chunlai Mu
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We investigate the blowup properties of the positive solutions for a semilinear
reaction-diffusion system with nonlinear nonlocal boundary condition. We obtain some sufficient conditions for global existence and blowup by utilizing the method of subsolution and supersolution.


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## Body

## 1. Introduction

In this paper,we deal with the following semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions and nontrivial nonnegative continuous initial data:(1.1)ut=Δu+vp,vt=Δv+uq,x∈Ω,t>0,u(x,t)=∫Ωφ(x,y)um(y,t)dy,x∈∂Ω,t>0,v(x,t)=∫Ωψ(x,y)vn(y,t)dy,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω¯,
where Ω is a bounded domain in ℝN for N≥1 with a smooth boundary ∂Ω,p,q,m,n>0, the weight functions φ(x,y) and ψ(x,y) are nonnegative continuous defined in ∂Ω×Ω¯, and ∫Ωφ(x,y)dy, ∫Ωψ(x,y)dy>0 on ∂Ω. Moreover, for x∈∂Ω, the initial data u0(x), v0(x) satisfy the compatibility conditions u0(x)=∫Ωφ(x,y)u0m(y)dy and v0(x)=∫Ωψ(x,y)v0n(y)dy, respectively.System (1.1) has been formulated from physical models arising in various fields of applied sciences. For example, it can be interpreted as a heat conduction problem with nonlocal nonlinear sources on the boundary of the material body (see [1, 2]). In this case, u(x,t) and v(x,t) represent the temperatures of the interacting components in the evolution processes.The local (in time) existence of classical solutions of system (1.1) can be derived easily by standard parabolic theory. We say that the solution (u(x,t),v(x,t)) of problem (1.1) blows up in finite time if there exists a positive constant T<∞ such that (1.2)limt→T-(|u(·,t)|L∞(Ω)+|v(·,t)|L∞(Ω))=+∞.
In this case, T is called the blowup time. We say that the solution (u(x,t),v(x,t)) exists globally if (1.3)supt∈(0,+∞)(|u(·,t)|L∞(Ω)+|v(·,t)|L∞(Ω))<+∞.In the last few years, a lot of efforts have been devoted to the study of properties of solutions to the semilinear parabolic equationut=Δu+up with homogeneous Dirichlet boundary condition (see, e.g., the classical works in [3, 4]) and to the heat equation ut=Δu with Neumann boundary condition ∂u/∂ν=up  (see, e.g., [5]).Blowup properties for the problem of systems(1.4)ut=Δu+vp,vt=Δv+uq,x∈Ω,t>0,u(x,t)=v(x,t)=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω¯,ut=Δu+vp,vt=Δv+uq,x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω¯,ut=Δu,vt=Δv,x∈Ω,t>0,∂u∂ν=vp,∂v∂ν=uq,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω¯
have been studied very extensively over past years by many researchers. Here p,q>0,ν denotes the unit outer normal vector on ∂Ω. They were concerned with the existence, uniqueness, and regularity of solutions. Furthermore, they investigated the global and nonglobal existence, the blowup set, and the blowup rate for the above systems (see, e.g., [1, 2, 6–9] and the references cited therein). For blowup results for other parabolic systems, we refer the readers to [10–13] and the references cited therein.Moreover, in recent years, many authors (see studies such as those in  [14, 15] and the references cited therein) considered semilinear reaction-diffusion systems with nonlocal Dirichlet boundary conditions of the form(1.5)ut=Δu+f(u,v),vt=Δv+g(u,v),x∈Ω,t>0,u=∫Ωφ(x,y)u(y,t)dy,v=∫Ωψ(x,y)v(y,t)dy,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω¯.
They studied how the weight functions φ(x,y) and ψ(x,y) in the nonlocal boundary conditions affect the blowup properties of the solutions of (1.5).However, reaction-diffusion problems coupled with nonlocal nonlinear boundary conditions, to our knowledge, have not been well studied. Recently, Gladkov and Kim [16] considered the following problem for a single semilinear heat equation:(1.6)ut=Δu+c(x,t)up,x∈Ω,t>0,u(x,t)=∫Ωf(x,y,t)ul(y,t)dy,x∈∂Ω,t>0,u(x,0)=u0(x),x∈Ω¯,
where p, l>0. They obtained some criteria for the existence of the global solution as well as for blowup of the solution in finite time.The main purpose of this paper is to understand how the reaction terms, the weight functions and the nonlinear terms in the boundary conditions affect the blowup properties for problem (1.1). We will show that the weight functions φ(x,y),ψ(x,y) and the nonlinear terms um(y,t),vn(y,t) in the boundary conditions of (1.1) play substantial roles in determining blowup or not of solutions.Before starting the main results, we introduce some useful symbols. Throughout this paper, we letλ be the first eigenvalue of the eigenvalue problem(1.7)-Δϕ(x)=λϕ,x∈Ω;ϕ(x)=0,x∈∂Ω,
and ϕ(x) the corresponding eigenfunction with ∫Ωϕ(x)dx=1, ϕ(x)>0 in Ω. In addition, for convenience, we denote that L=supΩ¯ϕ(x) and (1.8)M1=sup∂Ω×Ω¯φ(x,y),M2=inf∂Ω×Ω¯φ(x,y),K1=sup∂Ω×Ω¯ψ(x,y),K2=inf∂Ω×Ω¯ψ(x,y).The main results of this paper are stated as follows.Theorem 1.1.
Assume that0<pq≤1 and m,n≤1. Then the solution of problem (1.1) exists globally for any positive initial data.Theorem 1.2.
Assume thatpq>1 or m,n>1. Then for any φ(x,y),ψ(x,y)>0, the solution of problem (1.1) blows up in finite time for sufficiently large initial data.Theorem 1.3.
Assume thatp>1,  q>1,  m>1, and n>1. Then for any nonnegative continuous φ(x,y) and ψ(x,y), the solution of problem (1.1) exists globally for sufficiently small initial data.Remark 1.4.
Whenp=q,m=n,φ(x,y)=ψ(x,y), and u0(x)=v0(x), system (1.1) is then reduced to a single equation ut=Δu+up with nonlocal nonlinear boundary condition u(x,t)=∫Ωφ(x,y)um(y,t)dy. In this case, our above results are still true and consistent with those in [16].The rest of this paper is organized as follows. In Section2, we establish the comparison principle for problem (1.1). In Sections 3 and 4, we will give the proofs of Theorems 1.1 and 1.2, respectively. Finally, Theorem 1.3 will be proved in Section 5.


## 2. Preliminaries

In this section, we will give a suitable comparison principle for problem (1.1). Let ΩT=Ω×(0,T), ST=∂Ω×(0,T), and Ω¯T=Ω¯×[0,T). We begin with the precise definitions of a subsolution and supersolution of problem (1.1).Definition 2.1.
A pair of functions(u̲,v̲)∈C2,1(ΩT)∩C(Ω¯T)×C2,1(ΩT)∩C(Ω¯T) is called a subsolution of problem (1.1) in ΩT if
(2.1)u̲t≤Δu̲+v̲p,v̲t≤Δv̲+u̲q,(x,t)∈ΩT,u̲(x,t)≤∫Ωφ(x,y)u̲m(y,t)dy,(x,t)∈ST,v̲(x,t)≤∫Ωψ(x,y)v̲n(y,t)dy,(x,t)∈ST,u̲(x,0)≤u0(x),v̲(x,0)≤v0(x),x∈Ω¯.Similarly, a pair of functions(u¯,v¯)∈C2,1(ΩT)∩C(Ω¯T)×C2,1(ΩT)∩C(Ω¯T) is a supersolution of system (1.1) if the reversed inequalities hold in (2.1). We say that (u,v) is a solution of system (1.1) in ΩT if it is both a subsolution and a supersolution of problem (1.1) in ΩT.Letgi(x,t),hi(x,t)∈C2,1(ΩT)∩C(Ω¯T), χi(x,y)≥0 on ∂Ω×Ω, i=1,2. We first give some hypotheses as follows, which will be used in the sequel.(H1)
Forx∈∂Ω, y∈Ω, t>0, χ1(x,y)g1m-1(y,t), χ1(x,y)h1m-1(y,t), χ2(x,y)g2n-1(y,t), and χ2(x,y)h2n-1(y,t) are nonnegative. Further, ∫Ωmχ1(x,y)g1m-1(y,t)dy≤1, ∫Ωmχ1(x,y)h1m-1(y,t)dy≤1, ∫Ωnχ2(x,y)g2n-1(y,t)dy≤1, and ∫Ωnχ2(x,y)h2n-1(y,t)dy≤1.(H2)
Forx∈∂Ω, y∈Ω, t>0, there exists M>0 such that 0≤mχ1(x,y)g1m-1(y,t)≤M, 0≤mχ1(x,y)h1m-1(y,t)≤M, 0≤nχ2(x,y)g2n-1(y,t)≤M, and 0≤nχ2(x,y)h2n-1(y,t)≤M.Lemma 2.2.
Let (H1) hold, andcij+dij(i,j=1,2) be bounded in ΩT and letcij+dij≥0(i≠j,i,j=1,2). Further, assume that wi(x,t)≥si(x,t)(i=1,2). If χi(x,y)≥0 on ∂Ω×Ω; and gi,hi∈C2,1(ΩT)∩C(Ω¯T)(i=1,2) satisfy
(2.2)g1t-(∑k,l=1nak,l(1)∂2g1∂xk∂xl+∑k=1nbk(1)∂g1∂xk)≥∑i=12c1igi-∑i=12d1ihi+w1(x,t),(x,t)∈ΩT,g2t-(∑k,l=1nak,l(2)∂2g2∂xk∂xl+∑k=1nbk(2)∂g2∂xk)≥∑i=12c2igi-∑i=12d2ihi+w2(x,t),(x,t)∈ΩT,h1t-(∑k,l=1nak,l(1)∂2h1∂xk∂xl+∑k=1nbk(1)∂h1∂xk)≤∑i=12c1ihi-∑i=12d1igi+s1(x,t),(x,t)∈ΩT,h2t-(∑k,l=1nak,l(2)∂2h2∂xk∂xl+∑k=1nbk(2)∂h2∂xk)≤∑i=12c2ihi-∑i=12d2igi+s2(x,t),(x,t)∈ΩT,g1(x,t)≥∫Ωχ1(x,y)g1m(y,t)dy,g2(x,t)≥∫Ωχ2(x,y)g2n(y,t)dy,(x,t)∈ST,h1(x,t)≤∫Ωχ1(x,y)h1m(y,t)dy,h2(x,t)≤∫Ωχ2(x,y)h2n(y,t)dy,(x,t)∈ST,g1(x,0)≥h1(x,0),g2(x,0)≥h2(x,0),x∈Ω¯.
Then (g1(x,t),g2(x,t))≥(h1(x,t),h2(x,t)) in Ω¯T.Proof.
For any givenɛ>0, define
(2.3)g̃i=gi+ɛeγt,h̃i=hi-ɛeγt,i=1,2,
where
(2.4)γ>max{c11+c12+d11+d12,c21+c22+d21+d22}.
Then, a direct computation yields
(2.5)g̃1t-(∑l,k=1nal,k(1)∂2g̃1∂xk∂xl+∑k=1nbk(1)∂g̃1∂xk)>∑i=12c1ig̃i-∑i=12d1ih̃i+w1(x,t),(x,t)∈ΩT,g̃2t-(∑l,k=1nal,k(2)∂2g̃2∂xk∂xl+∑k=1nbk(2)∂g̃2∂xk)>∑i=12c2ig̃i-∑i=12d2ih̃i+w2(x,t),(x,t)∈ΩT,h̃1t-(∑l,k=1nal,k(1)∂2h̃1∂xk∂xl+∑k=1nbk(1)∂h̃1∂xk)<∑i=12c1ih̃i-∑i=12d1ig̃i+s1(x,t),(x,t)∈ΩT,h̃2t-(∑l,k=1nal,k(2)∂2h̃2∂xk∂xl+∑k=1nbk(2)∂h̃2∂xk)<∑i=12c2ih̃i-∑i=12d2ig̃i+s2(x,t),(x,t)∈ΩT.
On the other hand, for (x,t)∈ST, we have
(2.6)g̃1(x,t)≥ɛeγt+∫Ωχ1(x,y)g1m(y,t)dy=∫Ωχ1(x,y)g̃1m(y,t)dy+ɛeγt-∫Ωχ1(x,y)(g̃1m(y,t)-g1m(y,t))dy=∫Ωχ1(x,y)g̃1m(y,t)dy+ɛeγt-ɛeγt∫Ωmχ1(x,y)θ1m-1(y,t)dy,
where θ1 is an intermediate value between g1 and g̃1. From (H1), it follows that
(2.7)g̃1(x,t)>∫Ωχ1(x,y)g̃1m(y,t)dyfor(x,t)∈ST.
Likewise, for any (x,t)∈ST, we have
(2.8)g̃2(x,t)>∫Ωχ2(x,y)g̃2n(y,t)dy,h̃1(x,t)<∫Ωχ1(x,y)h̃1m(y,t)dy,h̃2(x,t)<∫Ωχ2(x,y)h̃2n(y,t)dy.
In addition, it is obvious that
(2.9)g̃1(x,0)-ɛ≥h̃1(x,0)+ɛ,g̃2(x,0)-ɛ≥h̃2(x,0)+ɛ,
and hence, we know that
(2.10)g̃1(x,0)>h̃1(x,0),g̃2(x,0)>h̃2(x,0).
Put(2.11)h(x,t)=g̃1(x,t)-g̃2(x,t).
Next, our task is to show that
(2.12)(η1(x,t),η2(x,t))>(0,0).
Actually, if (2.12) is true; then we can immediately get
(2.13)gi(x,t)+ɛeγt≥hi(x,t)-ɛeγt(i=1,2),∀(x,t)∈Ω¯T,
which means that (g1(x,t),g2(x,t))≥(h1(x,t),h2(x,t)) in Ω¯T as desired.
In order to prove (2.12), we set (2.14)η̃1=η1e-σt,η̃2=η2e-σt
with σ>max{supΩ¯T(c11+d11),supΩ¯T(c22+d22)}. Then from (2.5)–(2.10), we have
(2.15)η̃1t-(∑l,k=1nal,k(1)∂2η̃1∂xk∂xl+∑k=1nbk(1)∂η̃1∂xk)>(c11+d11-σ)η̃1+(c12+d12)η̃2,(x,t)∈ΩT,η̃2t-(∑l,k=1nal,k(1)∂2η̃2∂xk∂xl+∑k=1nbk(1)∂η̃2∂xk)>(c21+d21)η̃1+(c22+d22-σ)η̃2,(x,t)∈ΩT,η̃1(x,t)>∫Ωmχ1(x,y)θ2m-1η̃1(y,t)dy,(x,t)∈ST,η̃2(x,t)>∫Ωnχ2(x,y)θ3n-1η̃2(y,t)dy,(x,t)∈ST,η̃1(x,0)>0,η̃2(x,0)>0,x∈Ω¯,
where θ2 is an intermediate value between g̃1 and h̃1, θ3 is an intermediate value between g̃2 and h̃2.
Sinceη̃1(x,0),η̃2(x,0)>0, there exists δ>0 such that η̃1,η̃2>0 for (x,t)∈Ω¯×(0,δ). Suppose a contradiction that(2.16)t¯=sup{t∈(0,T):η̃1,η̃2>0onΩ¯×[0,t]}<T.
Then η̃1,η̃2≥0 on Ω¯t¯, and at least one of η̃1, η̃2 vanishes at (x¯,t¯) for some x¯∈Ω¯. Without loss of generality, suppose that η̃1(x¯,t¯)=0=infΩ¯t¯η̃1. If (x¯,t¯)∈Ωt¯, by virtue of the first inequality of (2.15), we find that
(2.17)η̃1t-(∑k,l=1nal,k(1)∂2η̃1∂xk∂xl+∑k=1nbk(1)∂η̃1∂xk)>(a11+b11-σ)η̃1,(x,t)∈Ωt¯.
This leads us to conclude that η̃1≡0 in Ωt¯ by the strong maximum principle, a contradiction. If (x¯,t¯)∈St¯, this also results in a contradiction, that is
(2.18)0=η̃1(x¯,t¯)>∫Ωmφ(x¯,y)θ2m-1η̃1(y,t¯)dy≥0.
This proves that η̃1,η̃2>0, and in turn (g1(x,t),g2(x,t))≥(h1(x,t),h2(x,t)) in Ω¯T. The proof of Lemma 2.2 is complete.Lemma 2.3.
Letthe hypotheses of Lemma2.2, with (H1) replaced by (H2), be satisfied. Then
(2.19)(g1(x,t),g2(x,t))≥(h1(x,t),h2(x,t))inΩ¯T.Proof.
Choose a positive functionf satisfying f|x∈∂Ω=1 and ∫Ωf(y)dy<1/M. Set
(2.20)gi(x,t)=f(x)ρi(x,t),hi(x,t)=f(x)ζi(x,t),i=1,2.
Then from (2.2), we have
(2.21)ρ1t-ℒ1ρ1≥∑i=12c1ifρi-∑i=12d1ifζi+w1f,(x,t)∈ΩT,ρ2t-ℒ2ρ2≥∑i=12c2ifρi-∑i=12d2ifζi+w2f,(x,t)∈ΩT,ζ1t-ℒ1ζ1≤∑i=12c1ifζi-∑i=12d1ifρi+s1f,(x,t)∈ΩT,ζ2t-ℒ2ζ2≤∑i=12c2ifζi-∑i=12d2ifρi+s2f,(x,t)∈ΩT,ρ1(x,t)≥∫Ωχ1(x,y)fm(y)ρ1m(y,t)dy,(x,t)∈ST,ρ2(x,t)≥∫Ωχ2(x,y)fn(y)ρ2n(y,t)dy,(x,t)∈ST,ζ1(x,t)≤∫Ωχ1(x,y)fm(y)ζ1m(y,t)dy,(x,t)∈ST,ζ2(x,t)≤∫Ωχ2(x,y)fn(y)ζ2n(y,t)dy,(x,t)∈ST,ρ1(x,0)≥ζ1(x,0),ρ2(x,0)≥ζ2(x,0),x∈Ω¯,
where
(2.22)ℒs=∑l,k=1nal,k(s)∂2∂xk∂xl+∑k=1n(∑l=1n2ak,l(s)∂f∂xl+bk(s)f)1f∂∂xk,s=1,2,
is a uniformly elliptic operator. By (H2), it is easy to see that
(2.23)∫Ωmχ1(x,y)ρ1m-1(y,t)fm(y)dy≤M∫Ωf(y)dy≤1,∫Ωnχ2(x,y)ρ2n-1(y,t)fn(y)dy≤M∫Ωf(y)dy≤1.
Similarly, we have
(2.24)∫Ωmχ1(x,y)ζ1m-1(y,t)fm(y)dy≤1,∫Ωnχ2(x,y)ζ2n-1(y,t)fn(y)dy≤1.
Therefore, in view of Lemma 2.2, we have
(2.25)ρi(x,t)≥ζi(x,t),i=1,2,
which implies that
(2.26)gi(x,t)≥hi(x,t),i=1,2.
The proof of Lemma 2.3 is complete.On the basis of the above lemmas, we obtain the following comparison principle for problem (1.1).Proposition 2.4 (Comparison principle).
Let(u¯,v¯) and (u̲,v̲) be a nonnegative supersolution and a nonnegative subsolution of problem (1.1) in ΩT, respectively. Suppose that (u̲,v̲)>(0,0) and (u¯,v¯)>(0,0) in Ω¯T if min{p,q,m,n}<1. If (u¯(x,0),v¯(x,0))≥(u̲(x,0),v̲(x,0)) for x∈Ω¯, then (u¯,v¯)≥(u̲,v̲) in Ω¯T.Proof.
It is easy to check thatu̲, v̲, u¯, v¯ and φ, ψ satisfy hypotheses (H2).Next, we state the local existence theorem, and its proof is standard; hence we omit it.Theorem 2.5 (Local existence).
For any nonnegative nontrivialu0(x),v0(x)∈C(Ω¯), there exists a constant T*=T*(u0,v0)>0 such that problem (1.1) admits nonnegative solution (u(x,t),v(x,t))∈C2,1(ΩT)∩C(Ω¯T)×C2,1(ΩT)∩C(Ω¯T) for each T<T*. Furthermore, either T*=∞ or
(2.27)limt→T*sup(∥u(x,t)∥∞+∥v(x,t)∥∞)=∞.Remark 2.6.
From maximum principle, we know that the solution of system (1.1) is positive when u0(x) and v0(x) are positive. Indeed, since ut-Δu-vp≥0 and vt-Δv-uq≥0, the minimum of (u,v) in Ω¯T should be obtained at a parabolic boundary point by maximum principle. Furthermore, ∫Ωφ(x,y)dy, ∫Ωψ(x,y)dy>0 on ∂Ω imply that φ(x,t)≢0 and ψ(x,t)≢0, then we have (u,v)>(0,0) for (x,t)∈∂Ω×(0,T]. Thus (u,v)>(0,0) provided that u0(x) and v0(x) are positive. In the rest of this paper, we assume that (u0(x),v0(x))>(0,0).Remark 2.7.
Ifpq≥1,  m≥1, and n≥1, we could obtain the uniqueness of the solution easily by comparison principle.


## 3. Proof of Theorem1.1

In this section, by constructing special supersolution, we will give the sufficient condition for the existence of global solution of problem (1.1) under the hypotheses 0<pq≤1 and m,n<1.Proof of Theorem1.1.
Since0<pq≤1, there exist α, β∈(0,1) such that
(3.1)1p≥βα,1q≥αβ.
Define k=1/α+1/β. Let Φ(x,y)≥max{φ(x,y),ψ(x,y)} be a continuous function defined for (x,y)∈∂Ω×Ω¯, and
(3.2)a(x)=(∫ΩΦ(x,y)dy)(1-α)/α,b(x)=(∫ΩΦ(x,y)dy)(1-β)/β,x∈∂Ω.
Suppose that ω is the solution of the following problem:
(3.3)ωt=Δω+kω,x∈Ω,t>0,ω(x,t)=(a(x)+b(x)+1)∫Ω(Φ(x,y)+1|Ω|)ω(y,t)dy,x∈∂Ω,t>0,ω(x,0)=u01/α(x)+v01/β(x)+1,x∈Ω,
where |Ω| denotes the measure of Ω. By Theorem 2.5 in [16], we know that ω is a global solution of (3.3). Moreover, ω(x,t)>1 in (x,t)∈Ω¯×[0,+∞) by the maximum principle.
Set(u¯,v¯)=(ωα,ωβ). A simple computation shows that(3.4)u¯t=αωα-1ωt=αωα-1Δω+(1+αβ)ωα,Δu¯=αωα-1Δω+α(α-1)ωα-2|∇ω|2,
and thus
(3.5)u¯t-Δu¯=(1+αβ)ωα-α(α-1)ωα-2|∇ω|2≥(1+αβ)ωα≥(ωβ)α/β≥v¯p.
here, we used p≤α/β and ω>1.
When(x,t)∈∂Ω×(0,+∞), according to Hölder's inequality, we have that (3.6)u¯(x,t)≥aα(x){∫ΩΦ(x,y)ω(y,t)dy}α={∫ΩΦ(x,y)dy}1-α{∫ΩΦ(x,y)ω(y,t)dy}α≥{∫Ωφ(x,y)dy}1-α{∫Ωφ(x,y)ω(y,t)dy}α≥∫Ωφ(x,y)ωα(y,t)dy=∫Ωφ(x,y)u¯(x,t)dy≥∫Ωφ(x,y)u¯m(x,t)dy.
Likewise, we also have for v¯ that
(3.7)v¯t-Δv¯≥u¯q,x∈Ω,t>0,v¯(x,t)≥∫Ωψ(x,y)v¯n(x,t)dy,x∈∂Ω,t>0.
On the other hand, since α<1, we have
(3.8)u¯(x,0)=ωα(x,0)=(u01/α(x)+v01/β(x)+1)α≥u0(x),
and similarly,
(3.9)v¯(x,0)=ωβ(x,0)=(u01/α(x)+v01/β(x)+1)β≥v0(x).
Therefore, (u¯,v¯) is a global supersolution of (1.1); by Proposition 2.4, the solution of (1.1) exists globally. The proof of Theorem 1.1 is complete.


## 4. Proof of Theorem1.2

In this section, we will establish that the solution of system (1.1) blows up in finite time for the case pq>1 or m,n>1. We employ a variant of Kaplan's method (see [17] for more details) to obtain our blowup conclusion.Proof of Theorem1.2.
Let(4.1)J1(t)=∫Ωu(x,t)ϕ(x)dx,J2(t)=∫Ωv(x,t)ϕ(x)dx,
where ϕ(x) is defined in (1.7). Taking the derivative of J1(t) with respect to t, we could obtain
(4.2)J1′(t)=∫Ωut(x,t)ϕ(x)dx=∫Ω(Δu+vp)ϕdx=-∫ΩDu·Dφdx+∫∂Ω∂u∂νϕdS+∫Ωvpϕ(x)dx=∫ΩuΔϕdx-∫∂Ω∂ϕ∂νudS+∫Ωvpϕdx=-∫Ωλuϕdx-∫∂Ω∂ϕ∂ν(∫Ωφ(x,y)um(y,t)dy)dS+∫Ωvpϕdx.
Applying the equality ∫∂Ω(∂ϕ/∂ν)dS=-λ to (4.2), we find that
(4.3)J1′(t)≥-λ∫Ωuϕdx+λM2L∫Ωϕumdx+∫Ωvpϕdx.
Symmetrically, we deduce that
(4.4)J2′(t)≥-λ∫Ωvϕdx+λK2L∫Ωϕvndx+∫Ωuqϕdx.Case 1.
For the casepq>1; we first prove the assertion under the stronger assumption p,q>1. Without loss of generality, we assume that q=max{p,q}>1. Then using Jensen's inequality to (4.3) and (4.4), we see that
(4.5)J1′(t)≥-λJ1+J2p,J2′(t)≥-λJ2+J1q.
Therefore J=J1+J2 satisfies
(4.6)J'=J1′+J2′≥-λJ+J1q+J2p≥-λJ+J1p+J2p-J1.
In view of the inequality J1p+J2p≥21-pJp, we discover that
(4.7)J'≥-(λ+1)J+21-pJp.
It follows that J blows up in finite time whenever
(4.8)J(0)=∫Ω(u0(x)+v0(x))ϕ(x)dx
is sufficiently large. Furthermore, the solution of system (1.1) blows up in finite time.
Ifp<1 or q<1, in order to obtain our conclusion, we consider system (1.1) with zero Dirichlet boundary condition; then in light of Theorem 2 in [1], we obtain our result immediately.Case 2.
Consider now the case thatm,n>1. Since m,n>1, Jensen's inequality can be applied to (4.3) and (4.4) like Step 1 to get
(4.9)J1′(t)≥-λJ1+λM1LJ1m,J2′(t)≥-λJ2+λM2LJ2n.
Then the left arguments are the same as those for Case 1, we omit the details. The proof of Theorem 1.2 is complete.


## 5. Proof of Theorem1.3

In this section, we will use an idea from Gladkov and Kim [16] to prove Theorem 1.3.Proof of Theorem1.3.
LetΩ1 be a bounded domain in ℝN satisfying the property that Ω⋐Ω1 and let λ1 be the first eigenvalue of -Δ on Ω1 with null Dirichlet boundary condition which satisfies the inequality 0<λ1<λ.
Sinceφ(x,y) and ψ(x,y) are nonnegative continuous defined in ∂Ω×Ω¯; then there exist some constants 0<A,B<+∞ such that(5.1)∫Ωφ(x,y)dy≤A,∫Ωψ(x,y)dy≤B.
Denote ϕ̃ an eigenfunction corresponding the eigenvalue λ1; then it is obviously that
(5.2)supΩ1ϕ̃infΩ¯ϕ̃≤supΩ1ϕ̃infΩ1ϕ̃<δ,
where δ>1 is some constant. Choosing any ɛ which satisfies the inequality
(5.3)0<ɛ≤min{(Aδm)-1/(m-1),(Bδn)-1/(n-1)},
and taking
(5.4)supΩ1ϕ̃=δɛ,
then, from (5.2), it follows easily that
(5.5)inf∂Ωϕ̃>ɛ.Case 1.
Forq≥p>1, set
(5.6)f(t)=e-λ1t[1+max{(δɛ)p-1,(δɛ)q-1}e-(p-1)λ1tλ1]-1/(p-1).
It is easy to check that f(t) satisfy the following ordinary differential equation:
(5.7)f'(t)+λ1f-max{(δɛ)p-1,(δɛ)q-1}fp=0.
Observe next that f(t)<1, and so fp≥fq under the condition q≥p>1.
Let(5.8)u¯(x,t)=v¯(x,t)=ϕ̃(x)f(t).
A series of computations yields
(5.9)u¯t-Δu¯-v¯q=ϕ̃(f′+λ1f-ϕ̃q-1fq)≥ϕ̃(f′+λ1f-max{(δɛ)p-1,(δɛ)q-1}fp)≥0.
And similarly, we have
(5.10)v¯t-Δv¯-u¯p=ϕ̃(f'+λ1f-ϕ̃p-1fp)≥ϕ̃(f'+λ1f-max{(δɛ)p-1,(δɛ)q-1}fp)≥0.
On the other hand, since ∫Ωφ(x,y)dy≤A, we have on the boundary that
(5.11)u¯(x,t)>ɛf(t)≥A(δɛ)mf(t)≥∫Ωφ(x,y)(ϕ̃(y)f(t))mdy.
Likewise, we have that
(5.12)v¯(x,t)>∫Ωψ(x,y)(ϕ̃(y)f(t))ndy.
Thus, by exploiting (5.9)–(5.12) and comparison principle, the solution of (1.1) exists globally provided that
(5.13)max{u0(x),v0(x)}≤[1+max{(δɛ)p-1,(δɛ)q-1}λ1]-1/(p-1)ϕ̃(x).Case 2.
Forp>q>1, set
(5.14)f(t)=e-λ1t[1+max{(δɛ)p-1,(δɛ)q-1}e-(q-1)λ1tλ1]-1/(q-1).
We can immediately verify that f(t) satisfy the following ordinary differential equation:
(5.15)f'(t)+λ1f-max{(δɛ)p-1,(δɛ)q-1}fq=0.
In addition, it is obvious that f(t)<1. Then we have that fq≥fp under the condition p>q>1.
Let(5.16)u¯(x,t)=v¯(x,t)=ϕ̃(x)f(t).
Similar to the arguments for the case q≥p>1, we can prove that (u¯(x,t),v¯(x,t)) is a global supersolution of problem (1.1) provided that
(5.17)max{u0(x),v0(x)}≤[1+max{(δɛ)p-1,(δɛ)q-1}λ1]-1/(q-1)ϕ̃(x).The proof of Theorem1.3 is complete.



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*Source: 148035-2010-06-24.xml*