# Blow-Up and Global Existence for a Degenerate Parabolic System with Nonlocal Sources

Authors: Ling Zhengqiu; Wang Zejia
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

This paper investigates the blow-up and global existence of nonnegative solutions for a class of nonlocal degenerate parabolic system. By using the super- and subsolution techniques, the critical exponent of the system is determined. That is, ifPc=p1q1−(m−p2)(n−q2)<0, then every nonnegative solution is global, whereas if Pc>0, there are solutions that blowup and others that are global according to the size of initial values u0(x)
and v0(x). When Pc=0, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.


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## Body

## 1. Introduction and Description of Results

In this paper, we investigate the blowup and global existence of nonnegative solutions for the following degenerate parabolic system with nonlocal sources:(1.1)ut=Δum+vp1‖u‖αp2,(x,t)∈Ω×(0,T),vt=Δvn+uq1‖v‖βq2,(x,t)∈Ω×(0,T),u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,u(x,t)=0,v(x,t)=0,(x,t)∈∂Ω×(0,T),
where Ω is a bounded domain in ℝN(N≥1) with smooth boundary ∂Ω and u0(x),v0(x) are nonnegative bounded functions in Ω, constants m,n>1,  α,β≥1,  p1,q1,p2,q2>0, where ∥·∥αα=∫Ω|·|αdx.Equation (1.1) constitutes a simple example of a reaction diffusion system exhibiting a nontrivial coupling on the unknowns u(x,t),v(x,t), such as heat propagations in a two-component combustible mixture [1], chemical processes [2], and interaction of two biological groups without self-limiting [3]. And they are worth to study because of the applications to heat and mass transport processes. In addition, there exist interesting interactions among the multi-nonlinearities described by the eight exponents m,n,p1,p2,q1,q2 and α,β in the problem (1.1).In the past two decades, many physical phenomena were formulated into nonlocal mathematical models (see [4–7] and references therein) and studied by many authors. Degenerate parabolic equations involving a nonlocal source, which arise in a population model that communicates through chemical means, were studied in [8, 9]. At the same time, there are many important results that have appeared on blowup problems for nonlinear parabolic system. We will recall some of those results concerning the first initial-boundary problem. For the other related works on the global existence and blowup of solutions of nonlinear parabolic system, we refer the readers to [10, 11] and references therein.In [4], Escobedo and Herrero studied the system(1.2)ut=Δu+vp,vt=Δv+uq,x∈Ω,t>0
in a bounded domain Ω⊂ℝN with null Dirichlet boundary conditions. The authors show that if pq≤1, every solution of (1.2) is global, whereas if pq>1, there are solutions that blowup and others that are global according to the size of initial values u0(x) and v0(x).In [12], Galaktionov et al. considered the system(1.3)ut=Δuν+1+vp,vt=Δvμ+1+uq,(x,t)∈Ω×(0,T)
with homogeneous Dirichlet boundary conditions, and they proved that pc=pq-(ν+1)(μ+1) is the critical exponent. Zheng [13] and Li et al. [14] studied the following systems:(1.4)ut=Δu+up1vq1,vt=Δv+up2vq2,(x,t)∈Ω×R+,ut=Δu+∫Ωum(x,t)vn(x,t)dx,vt=Δv+∫Ωup(x,t)vq(x,t)dx,x∈Ω,t>0,
respectively. They obtained some results on the global solutions, the blowup solutions and the blowup profiles. Lately, Deng et al. in [15] considered the following nonlocal degenerate parabolic system:(1.5)ut=Δum+a‖v‖αp,vt=Δvn+b‖u‖βq,(x,t)∈Ω×(0,T)
with homogeneous Dirichlet boundary conditions. Several interesting results are established as follows.(i)
Ifpq<mn, then every nonnegative solution of (1.5) is global.(ii)
Ifpq=mn, then if the domain is sufficiently small, the nonnegative solution of (1.5) is global, whereas if the domain contains a sufficiently large ball and u0(x),v0(x)>0, the nonnegative solution blows up in finite time.(iii)
Ifpq>mn, then there are solutions of (1.5) that blowup and others that are global according to the size of initial data u0(x) and v0(x).Our present work is motivated by [12–15] mentioned above. The main purpose of this paper is to extend and improve the results in [15]. At the same time, we will prove that pc=p1q1-(m-p2)(n-q2) is also the critical exponent of system (1.1). Our main results are as follows, two theorems concern the global existence and blowup conditions of the solutions.Theorem 1.1.
If one of the following conditions holds, then the nonnegative solution of system (1.1) exists globally.(1)
m>p2,n>q2 and p1q1<(m-p2)(n-q2).(2)
m>p2,n>q2,p1q1=(m-p2)(n-q2) and the domain (|Ω|) is sufficiently small.(3)
m>p2,n>q2,p1q1>(m-p2)(n-q2) and the initial data u0(x),v0(x) are sufficiently small.(4)
m≤p2 or n≤q2 and the initial data u0(x),v0(x) are sufficiently small.Theorem 1.2.
If one of the following conditions holds, then the nonnegative solution of system (1.1) blows up in a finite time.(1)
m>p2,n>q2, p1q1>(m-p2)(n-q2) and the initial data u0(x),v0(x) are sufficiently large.(2)
m>p2,n>q2, p1q1=(m-p2)(n-q2) and the domain contains a sufficiently large ball, moreover, u0(x) and v0(x) are large enough.(3)
m≤p2 or n≤q2 and initial data u0(x),v0(x) are sufficiently large.This paper is organized as follows. In the next Section, we establish the local existence theorem and give some auxiliary lemmas. In Section3, which concerns global existence, we prove Theorem 1.1. In Section 4, which deals with the blowup phenomenon, we prove Theorem 1.2.


## 2. Local Existence and Comparison Principle

Similar to the Propositions 2.1 and 2.2 of [15], we give the maximum principle and the comparison principle for the nonlocal parabolic system. For convenience, we denote QT=Ω×(0,T),Q¯T=Ω¯×[0,T],ST=∂Ω×(0,T), where 0<T<+∞.As it is now well known that degenerate equation needs not possess classical solution, we begin by giving a precise definition of a weak solution for system (1.1). To this end, define the class of test functions(2.1)Ψ≡{ψ(x,t)∈C(Q¯T);ψt,Δψ∈C(QT)∩L2(QT);ψ≥0;ψ(x,t)|x∈∂Ω=0}.Definition 2.1.
A pair of vector function(u¯(x,t),v¯(x,t)) defined on Q¯T is called a super-solution of (1.1), if all the following conditions hold:(1)
u¯(x,t),v¯(x,t)∈L∞(QT);(2)
if(x,t)∈ST,u¯(x,t),v¯(x,t)≥0, and for all x∈Ω, u¯(x,0)≥u0(x),v¯(x,0)≥v0(x);(3)
for everyt∈[0,T] and any ψ1,ψ2∈Ψ,
(2.2)∫Ω(u¯(x,t)ψ1(x,t)-u0(x)ψ1(x,0))dx≥∫0t∫Ω(u¯ψ1s+u¯mΔψ1+v¯p1‖u¯‖αp2ψ1)dxds,∫Ω(v¯(x,t)ψ2(x,t)-v0(x)ψ2(x,0))dx≥∫0t∫Ω(v¯ψ2s+v¯nΔψ2+u¯q1‖v¯‖βq2ψ2)dxds.
A subsolution (ũ(x,t),ṽ(x,t)) can be defined in a similar way.Next, we state the maximum principle and comparison principle, and the proofs that are quite standard, we omit them here.Lemma 2.2 (maximum principle).
Suppose thatω1(x,t),ω2(x,t)∈C2,1(QT)∩C(Q¯T) and satisfy
(2.3)M1ω=ω1t-d1Δω1-∑j=1Na1jω1xj-c11ω1-c12ω2-c13∫Ωc14ω1dx≥0,(x,t)∈QT,M2ω=ω2t-d2Δω2-∑j=1Na2jω2xj-c21ω2-c22ω1-c23∫Ωc24ω2dx≥0,(x,t)∈QT,ω1(x,t)≥0,ω2(x,t)≥0,(x,t)∈ST,ω1(x,0)≥0,ω2(x,0)≥0,x∈Ω,
where di(x,t),cij(x,t)(i=1,2;j=1,2,3,4) and aij(x,t),(i=1,2;j=1,2,…,N) are the continuous and the bounded functions on Q¯T, respectively, and
(2.4)di(x,t),ci2(x,t),ci3(x,t),ci4(x,t)≥0,i=1,2,(x,t)∈Ω×(0,T].
Then ωi(x,t)≥0 on Q¯T.Lemma 2.3 (comparison principle).
Let(u¯,v¯) and (ũ,ṽ) be a nonnegative supersolution and a nonnegative subsolution of system (1.1), respectively. Then (ũ,ṽ)≤(u¯,v¯) on QT if (ũ(x,0),ṽ(x,0))≤(u¯(x,0),v¯(x,0)) and either
(2.5)u¯,v¯≥ρ>0orũ,ṽ≥ρ>0
hold.Theorem 2.4 (local existence and continuation).
Assumeu0,v0≥0,u0,v0∈L∞(Ω), there is a T*=T*(u0,v0)>0 such that there exists a nonnegative weak solution (u(x,t),v(x,t)) of (1.1) for each T<T*. Furthermore, either T*=+∞ or
(2.6)limsup⁡t→T*‖u(⋅,t)‖∞=+∞orlimsup⁡t→T*‖v(⋅,t)‖∞=+∞.Proof.
Owing to the degeneracy of equations of (1.1), in order to prove the existence of solution, for k=1,2,…, we first consider the following corresponding regularized system
(2.7)ukt=Δfk(uk)+(gk′(vk))p1‖gk(uk)‖αp2,(x,t)∈QT,vkt=Δfk′(vk)+(gk(uk))q1‖gk′(vk)‖βq2,(x,t)∈QT,uk(x,t)=vk(x,t)=1k,(x,t)∈ST,uk(x,0)=u0i(x)+1k,vk(x,0)=v0i(x)+1k,x∈Ω,
where
(2.8)fk(uk)={ukm,uk≥1k,(1k)m,uk<1k,fk′(vk)={vkn,vk≥1k,(1k)n,vk<1k,gk(uk)={uk,uk≥1k,1k,uk<1k,gk′(vk)={vk,vk≥1k,1k,vk<1k,
and u0i(x),v0i(x) are smooth approximation of u0(x),v0(x) with suppu0i⊂Ω and suppv0i⊂Ω, respectively. It is known that the system (2.7) has a unique classical solution (uki,vki)∈C(Ω¯×[0,Ti(k)))∩C2,1(Ω×(0,Ti(k))) for 0<Ti(k)≤∞ by the classical theory for parabolic equations, where Ti(k) is the maximal existence time. By a direct computation and the classical maximum principle, we have uki,vki≥1/k. Hence (uki,vki) satisfies
(2.9)(uki)t=Δ(uki)m+(vki)p1‖uki‖αp2,(vki)t=Δ(vki)n+(uki)q1‖vki‖βq2,(x,t)∈QTi(k)
with the corresponding initial and boundary conditions. At the same time, if k1>k2, according to Lemma 2.2, we get
(2.10)(uk1i(x,t),vk1i(x,t))≤(uk2i(x,t),vk2i(x,t)),(x,t)∈Ω¯×[0,Ti(k2)),
and Ti(k1)≥Ti(k2). On the other hand, passing to the limit i→∞, it follows that
(2.11)uk(x,t)=lim⁡i→∞uki(x,t),vk(x,t)=lim⁡i→∞vki(x,t),
and (uk,vk) is a weak solution of
(2.12)(uk)t=Δ(uk)m+(vk)p1‖uk‖αp2,(vk)t=Δ(vk)n+(uk)q1‖vk‖βq2,(x,t)∈QT(k)
with the corresponding initial and boundary conditions on QT(k), where T(k)=limi→∞Ti(k) is the maximal existence time. Here a weak solution of (2.12) is defined in a manner similar to that for (1.1), only the equalities for u and v; (2.2) may be replaced with
(2.13)∫Ω(uk(x,t)ψ1(x,t)-(u0(x)+1k)ψ1(x,0))dx=∫0t∫Ω(ukψ1s+ukmΔψ1+vkp1‖uk‖αp2ψ1)dxds+1k∫0t∫∂Ω(∂ψ1∂ν)dσds,∫Ω(vk(x,t)ψ2(x,t)-(v0(x)+1k)ψ2(x,0))dx=∫0t∫Ω(vkψ2s+vknΔψ2+ukq1‖vk‖βq2ψ2)dxds+1k∫0t∫∂Ω(∂ψ2∂ν)dσds,
respectively. Then, passing to the limit i→∞, it happens that (uk1,vk1)≤(uk2,vk2) and T(k1)≥T(k2) if k1>k2.
Therefore, the limitT*=limk→∞T(k) exists, and the pointwise limit
(2.14)u(x,t)=lim⁡k→∞uk(x,t),v(x,t)=lim⁡k→∞vk(x,t)
exists for any (x,t)∈Ω¯×[0,T*). Furthermore, as the convergence of the sequence is monotone, passing to the limit k→∞ in (2.13), we get that (u(x,t),v(x,t)) is a nonnegative weak solution of (1.1). Thus the proof is completed.Denote(2.15)A=(m-p2-p1-q1n-q2),L=(l1l2).We give Lemmas2.5 and 2.6 that will be used in the following; please see [16] for their proofs.Lemma 2.5.
Ifm>p2,n>q2 and p1q1<(m-p2)(n-q2), then there exist two positive constants l1,l2 such that AL>(0,0)T.Lemma 2.6.
Ifm≤p2 or n≤q2 or p1q1>(m-p2)(n-q2), then there exist two positive constants l1,l2 such that AL<(0,0)T.


## 3. Proof of Theorem1.1

According to Lemma2.3, we only need to construct bounded super-solutions for any T>0. Let φ(x) be the unique position solution of the following linear elliptic problem:(3.1)-Δφ(x)=1,x∈Ω;φ(x)=0,x∈∂Ω.
Denote C=maxx∈Ωφ(x), then 0≤φ(x)≤C. We define the functions u¯,v¯ as follows:(3.2)u¯(x,t)=(k(φ(x)+1))l1,v¯(x,t)=(k(φ(x)+1))l2,
where l1,l2<1 such that ml1,nl2<1, and k>0 will be fixed later. Clearly, for any T>0, (u¯,v¯) is a bounded function and u¯≥kl1>0,v¯≥kl2>0. Then, we have
(3.3)u¯t-Δu¯m=-kml1ml1(ml1-1)(φ+1)ml1-2|∇φ|2+kml1ml1(φ+1)ml1-1≥kml1ml1(φ+1)ml1-1≥kml1ml1(C+1)ml1-1,v¯p1‖u¯‖αp2=kp1l2(φ+1)p1l2‖(k(φ+1))l1‖αp2≤kp1l2+p2l1(C+1)p1l2+p2l1|Ω|p2/α,v¯t-Δv¯n≥knl2nl2(C+1)nl2-1,u¯q1‖v¯‖βq2≤kq1l1+q2l2(C+1)q1l1+q2l2|Ω|q2/β.
Denote(3.4)C1=(|Ω|p2/αml1(C+1)p1l2+p2l1-ml1+1)1/(ml1-p1l2-p2l1),C2=(|Ω|q2/βnl2(C+1)q1l1+q2l2-nl2+1)1/(nl2-q1l1-q2l2).(1) Ifm>p2,n>q2 and p1q1<(m-p2)(n-q2), by Lemma 2.5, there exist two positive constants l1,l2<1 such that(3.5)p1l2+p2l1<ml1,q1l1+q2l2<nl2,ml1,nl2<1.
Therefore, we can choose k sufficiently large that k>max{C1,C2} and(3.6)(k(φ+1))l1≥u0(x),(k(φ+1))l2≥v0(x).
Now, it follows from (3.3)–(3.6) that (u¯,v¯) is a positive super-solution of (1.1).(2) Ifm>p2,n>q2 and p1q1=(m-p2)(n-q2), then there exist two positive constants l1,l2<1 such that(3.7)p1l2+p2l1=ml1,q1l1+q2l2=nl2,ml1,nl2<1.
Without loss of generality, we may assume that Ω⊂⊂B, where B is a sufficiently large ball. And we denote φB(x) is the unique positive solution of the following linear elliptic problem:(3.8)-Δφ(x)=1,x∈B;φ(x)=0,x∈∂B.
Let C0=maxx∈BφB(x), then C≤C0. Therefore, as long as Ω is sufficiently small and such that(3.9)|Ω|<min⁡⁡{(ml1C0+1)α/p2,(nl2C0+1)β/q2}.
Furthermore, choose k large enough to satisfy (3.6).Then, it follows from (3.3) and (3.6)–(3.9) that (u¯,v¯) is a positive super-solution of (1.1).(3) Ifm>p2,n>q2 and p1q1>(m-p2)(n-q2), by Lemma 2.6, there exist two positive constants l1,l2<1 such that(3.10)p1l2+p2l1>ml1,q1l1+q2l2>nl2,ml1,nl2<1.
Hence, we can choose k sufficiently small that k<min{C1,C2}, and provided u0(x),v0(x) are also sufficiently small to satisfy (3.6). Then, from (3.3) and (3.6), (3.10), we know that (u¯,v¯) is a positive super-solution of (1.1).(4) Finally, ifm≤p2 or n≤q2, there exist also positive constants l1,l2<1 such that (3.10) and ml1,nl2<1. Similar to the proof of (3), we get that (u¯,v¯) is a positive super-solution of (1.1).Thus the proof of Theorem1.1 is completed.


## 4. Proof of Theorem1.2

Due to the requirement of the comparison principle, we will construct blowup subsolutions in some subdomain ofΩ in which u,v>0. We use an idea from Souplet [17] and apply it to degenerate parabolic equation. By translation, one may assume without loss of generality that 0∈Ω. Let B=B(0,R)⊂Ω be an open ball with radius R, and ψ(x) is a nontrivial nonnegative continuous function, vanished on ∂B and ψ(0)>0. Set(4.1)ũ(x,t)=1(T-t)l1V[|x|(T-t)σ],ṽ(x,t)=1(T-t)l2V[|x|(T-t)σ],
with(4.2)V(r)=R36-R2r2+13r3,r=|x|(T-t)σ,0≤r≤R,
where l1,l2,σ>0 and 0<T<1 are to be determined later. Clearly, 0≤V(r)≤R3/6 and V(r) is nonincreasing since V′(r)=r(r-R)≤0. Note that, for T small enough,(4.3)Suppũ(⋅,t)=Suppṽ(⋅,t)=B¯(0,R(T-t)σ)⊂B¯(0,RTσ)⊂Ω,0≤t<T.Obviously,(ũ,ṽ) becomes unbounded as t→T- at the point x=0. Calculating directly, we obtain(4.4)ũt-Δũm=l1V(r)+σrV′(r)(T-t)l1+1-m(m-1)Vm-2(r)(V′(r))2(T-t)ml1+2σ-mVm-1(r)(T-t)ml1+2σ(-NR+(N+1)r)≤l1R3/6(T-t)l1+1+mVm-1(r)(T-t)ml1+2σ(NR-(N+1)r),(x,t)∈B×(0,T),ṽt-Δṽn≤l2R3/6(T-t)l2+1+nVn-1(r)(T-t)nl2+2σ(NR-(N+1)r),(x,t)∈B×(0,T).
notice T<1 is sufficiently small. Therefore, If 0≤r≤r0=NR/(N+1), we have V(r)≥R3(3N+1)/(6(N+1)3); then(4.5)ṽp1‖ũ‖αp2=Vp1(r)(T-t)p1l2(∫BVα(r)(T-t)αl1)p2/α≥|B|p2/α(T-t)p1l2+p2l1(R3(3N+1)6(N+1)3)p1+p2,ũq1‖ṽ‖βq2=Vq1(r)(T-t)q1l1(∫BVβ(r)(T-t)βl2)q2/β≥|B|q2/β(T-t)q1l1+q2l2(R3(3N+1)6(N+1)3)q1+q2.
Hence,(4.6)ũt-Δũm-ṽp1‖ũ‖αp2≤l1R3/6(T-t)l1+1+mVm-1(r)(T-t)ml1+2σ(NR-(N+1)r)-|B|p2/α(T-t)p1l2+p2l1(R3(3N+1)6(N+1)3)p1+p2,ṽt-Δṽn-ũq1‖ṽ‖βq2≤l2R3/6(T-t)l2+1+nVn-1(r)(T-t)nl2+2σ(NR-(N+1)r)-|B|q2/β(T-t)q1l1+q2l2(R3(3N+1)6(N+1)3)q1+q2.
Similarly, if NR/(N+1)<r≤R, then(4.7)ũt-Δũm-ṽp1‖ũ‖αp2≤l1R3/6(T-t)l1+1+mVm-1(r)(T-t)ml1+2σ(NR-(N+1)r),(4.8)ṽt-Δṽn-ũq1‖ṽ‖βq2≤l2R3/6(T-t)l2+1+nVn-1(r)(T-t)nl2+2σ(NR-(N+1)r).(1) Ifm>p2,n>q2 and p1q1>(m-p2)(n-q2), by Lemma 2.6, there exist two positive constants l1,l2 large enough that(4.9)p1l2+p2l1>ml1,q1l1+q2l2>nl2,(m-1)l1>1,(n-1)l2>1.Then, we can chooseσ>0 sufficiently small such that(4.10)p1l2+p2l1>ml1+2σ>ml1>l1+1,q1l1+q2l2>nl2+2σ>nl2>l2+1.
Hence, for sufficiently small T>0, (4.6)–(4.8) imply that(4.11)ũt-Δũm-ṽp1‖ũ‖αp2≤0,ṽt-Δṽn-ũq1‖ṽ‖βq2≤0,(x,t)∈B×(0,T).Sinceψ(0)>0 and ψ(x) is continuous, there exist two positive constants ρ and ɛ such that ψ(x)≥ɛ for all x∈B(0,ρ)⊂B(0,R). Choose T small enough to insure B(0,RTσ)⊂B(0,ρ)⊂Ω, hence ũ≤0,ṽ≤0 on ∂Ω×(0,T), and from (4.3) it follows that ũ(x,0)≤Kψ(x),ṽ(x,0)≤Kψ(x) for sufficiently large K. By comparison principle, we have (ũ,ṽ)≤(u,v) provided that u0(x)≥Kψ(x) and v0(x)≥Kψ(x). It follows that (u,v) blows up in finite time.(2) Next, we consider the casem>p2,n>q2 and p1q1=(m-p2)(n-q2). Clearly, there exist two positive constants l1,l2 such that(4.12)ml1=p1l2+p2l1,nl2=q1l1+q2l2,(m-1)l1>1,(n-1)l2>1.Denote byλBR>0 and ϕR(r) the first eigenvalue and the corresponding eigenfunction of the following eigenvalue problem:(4.13)-ϕ′′(r)-N-1rϕ′(r)=λϕ(r),r∈(0,R);ϕ′(0)=0,ϕ(R)=0.
It is well known that ϕR(r) can be normalized as ϕR(r)>0 in B and ϕR(0)=maxBϕR(r)=1. By the property (let τ=r/R) of eigenvalues and eigenfunctions we see that λBR=R-2λB1 and ϕR(r)=ϕ1(r/R)=ϕ1(τ), where λB1 and ϕ1(τ) are the first eigenvalue and the corresponding normalized eigenfunction of the eigenvalue problem in the unit ball B1(0). Moreover,(4.14)max⁡B1ϕ1(τ)=ϕ1(0)=ϕR(0)=max⁡BϕR(r)=1.Similar to (4.1), we define the functions ũ(x,t),ṽ(x,t) in the form(4.15)ũ(x,t)=1(T-t)l1ϕRl1(|x|),ṽ(x,t)=1(T-t)l2ϕRl2(|x|).
In the following, we will prove that (ũ,ṽ) blows up in finite time in the ball B=B(0,R). Because of so, (ũ,ṽ) does blow up in the larger domain Ω. Calculating directly, we have(4.16)ũt-Δũm-ṽp1‖ũ‖αp2≤ϕRl1(T-t)l1+1(l1-1(T-t)ml1-l1-1(c1-λBRml1)),ṽt-Δṽn-ũq1‖ṽ‖βq2≤ϕRl2(T-t)l2+1(l2-1(T-t)nl2-l2-1(c2-λBRnl2)),
where(4.17)c1=ϕRp1l2‖ϕRl1‖αp2≤K1RNp2/α,c2=ϕRq1l1‖ϕRl2‖βq2≤K2RNq2/β
and K1,K2 are constants independent of R. Then, in view of λBR=R-2λB1, we may assume that R, that is, the ball B, is sufficiently large that(4.18)λBR<min⁡⁡{c1ml1,c2nl2}.
Hence, for sufficiently small T>0, (4.16) implies that(4.19)ũt-Δũm-ṽp1‖ũ‖αp2≤0,ṽt-Δṽn-ũq1‖ṽ‖βq2≤0.Therefore,(ũ,ṽ) is a positive subsolution of (1.1) in the ball B, which blows up in finite time provided the initial data is sufficiently large that(4.20)ũ(x,0)=T-l1ϕRl1(|x|)≤u0(x),ṽ(x,0)=T-l2ϕRl2(|x|)≤v0(x)
in the ball B.(3) Finally, ifm≤p2 or n≤q2, there also exist two positive constants l1,l2 to satisfy (4.9). Similar to the proof of case (1), we can get that (ũ,ṽ) is a subsolution of (1.1), which blows up in finite time.Thus the proof of Theorem1.2 is completed.



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*Source: 956564-2012-05-21.xml*