# Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

Authors: Mohamed Jleli; Bessem Samet
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We provide sufficient conditions for the nonexistence of global positive solutions to the nonlocal evolution equationutt(x,t)=(J ∗ u-u)(x,t)+up(x,t),(x,t)∈RN×(0,∞),(u(x,0),ut(x,0))=(u0(x),u1(x)), x∈RN, where J:RN→R+, p>1, and (u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). Next, we deal with global nonexistence for certain nonlocal evolution systems. Our method of proof is based on a duality argument.


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## Body

## 1. Introduction

In [1], García-Melián and Quirós considered the nonlocal diffusion problem:(1)utx,t=J∗u-ux,t+upx,t,x,t∈RN×0,∞,ux,0=u0x,x∈RN,where J:RN→R+ is a compactly supported nonnegative function with unit integral, p>1, and u0∈L1(RN;R+)∩L∞(RN;R+). Equation (1) may model a variety of biological, epidemiological, ecological, and physical phenomena involving media with properties varying in space [2, 3]; similar equations appear, for example, in Ising systems with Glauber dynamics [4]. In [1] the authors proved that (1) has a critical exponent: (2)pc=1+2N, which is the Fujita exponent for the classical nonlinear heat equation ut=Δu+up [5]. More precisely, they proved that if 1<p≤pc, the solution blows up in finite time for any nonnegative and nontrivial initial data u0∈L1(RN;R+)∩L∞(RN;R+); if p>pc, there exist global solutions for small initial data u0∈L1(RN;R+)∩L∞(RN;R+). Very recently, Yang [6] considered the nonlinear coupled nonlocal diffusion system:(3)utx,t=J∗u-ux,t+vpx,t,x,t∈RN×0,∞,vtx,t=J∗v-vx,t+uqx,t,x,t∈RN×0,∞,ux,0,vx,0=u0x,v0x,x∈RN,where p,q>1 and (u0,v0)∈L∞(RN;R+)×L∞(RN;R+). Equation (3) can serve as a model for the processes of heat diffusion and combustion in two-component continua with nonlinear heat conduction and volumetric release [7]. In this case, Yang established that the critical Fujita curve is given by (4)pq∗=1+2Nmax⁡p+1,q+1,which is also the Fujita curve for the coupled heat system ut=Δu+vp and vt=Δv+uq, obtained by Escobedo and Herrero [8].In this paper, we are first concerned with the following evolution problem:(5)uttx,t=J∗u-ux,t+upx,t,x,t∈RN×0,∞,ux,0,utx,0=u0x,u1x,x∈RN,where J:RN→R+, p>1, and (u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). We provide a sufficient condition for the nonexistence of global positive solutions to (5). Next, we consider the following two systems:(6)uttx,t=J∗u-ux,t+vpx,t,x,t∈RN×0,∞,vttx,t=J∗v-vx,t+uqx,t,x,t∈RN×0,∞,ux,0,vx,0=u0x,v0x,x∈RN,utx,0,vtx,0=u1x,v1x,x∈RN,(7)uttx,t=J∗v-vx,t+upx,t,x,t∈RN×0,∞,vttx,t=J∗u-ux,t+vqx,t,x,t∈RN×0,∞,ux,0,vx,0=u0x,v0x,x∈RN,utx,0,vtx,0=u1x,v1x,x∈RN,where p,q>1. For each system, we find a bound on N leading to the absence of global nontrivial solutions. Our method of proof is based on a duality argument developed by Mitidieri and Pokhozhaev [9, 10].


## 2. Main Results

Through this paper,R+=[0,∞), Q=RN×(0,∞), and J:RN→R+ is a continuous function satisfying the following conditions:(J1)
J is symmetric; that is, J(z)=J(-z), for every z∈RN.(J2)
∫RN‍J(z)dz=1.(J3)
A(J):=∫RN‍J(z)|z|2dz<∞.The following lemmas will be used later.Lemma 1.
Leta,b,ɛ>0 and p>1. Then (8)ab≤ɛap+cɛbp/p-1, where cɛ=(p-1)p-1(ɛp)-1/(p-1).Lemma 2 (see [11]).
LetX, Y, A, B, C, and D be nonnegative functions and let αi and θi, i=1,2,3, be positive reals such that α2<α1, θ2<θ1, θ1,θ3≥1, and α3θ3<α1θ1. If (9)Xα1≤AXα2+BYθ3,Yθ1≤CXα3+DYθ2,then (10)Xα1θ1≤LAα1θ1/α1-α2+Dθ1θ3/θ1-θ2Bθ1+Bθ1Cθ3α1θ1/α1θ1-α3θ3, for some constant L>0.


### 2.1. A Nonexistence Result for (5)

The definition of solutions we adopt for (5) is as follows.Definition 3.
Let(u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). We say that u is a global weak solution to (5) if u∈Llocp(Q;R+), J∗u∈Lloc1(Q;R+), and(11)∫Qupφdxdt+∫RNu1xφx,0dx=∫Quφttdxdt-∫QJ∗u-uφdxdt+∫RNu0xφtx,0dx,for every regular test function φ≥0 with φ(·,t≥T)≡0.Our first main result is given by the following theorem.Theorem 4.
Suppose that one of the following conditions hold:(12)N=1<por(13)N≥2,1<p<N+1N-1.Then (5) admits no global weak solutions other than the trivial one.Proof.
Suppose thatu is a nontrivial global weak solution to (5). As a test function, we take(14)φx,t=ξRxϕωtR,x,t∈Q,where R>0 is large enough, (15)ξRx=exp⁡-x2R2,x∈RN,ω≫1, and ϕ:R+→[0,1] is given by (16)ϕσ=1if0≤σ≤1,0ifσ≥2. From the definition of ϕ, clearly we have (17)φtx,0=0,x∈RN, which yields(18)∫RNu0xφtx,0dx=0.Writing (19)∫Quφttdxdt=∫Quφ1/pφ-1/pφttdxdt and applying Lemma 1, we obtain(20)∫Quφttdxdt≤ɛ∫Qupφdxdt+cɛ∫Qφ-p′/pφttp′dxdt,for some ɛ>0, where p′=p/(p-1). On the other hand, (21)∫Qφ-p′/pφttp′dxdt=ωp′R-2p′∫Qexp⁡-x2R2ϕω-2p/p-1tRhtRp′dxdt, where (22)hσ≔ω-1ϕ′σ+ϕσϕ′′σ,σ≥0. Using the change of variable x=Ry and t=Rs, we obtain(23)∫Qφ-p′/pφttp′dxdt=ωp′RN+1-2p′∫Qexp-y2ϕω-2p/p-1shsp′dyds.The above equality with (20) yields(24)∫Quφttdxdt≤ɛ∫Qupφdxdt+c1RN+1-2p′,for some constant c1>0. Next, we have (25)∫QJ∗uφdxdt=∫0∞∫RN∫RNJx-yuy,tdyφx,tdxdt. Using the symmetry of J and Fubini’s theorem, we obtain(26)∫QJ∗uφdxdt=∫0∞∫RN∫RNJx-yuy,tdyφx,tdxdt=∫0∞∫RN∫RNJy-xuy,tdyφx,tdxdt=∫0∞∫RN∫RNJy-xφx,tdxuy,tdy=∫QJ∗φudxdt. Therefore, (27)∫QJ∗u-uφdxdt=∫QJ∗φ-φudxdt. Using the property (J1), we obtain (28)J∗φ-φx,t=ϕωtR∫RNJx-yξRydy-ξRx=ϕωtR∫RNJy-xξRydy-ξRx=ϕωtR∫RNJzξRx+zdz-ξRx. By the property (J2) and the definition of φ, we have(29)J∗φ-φx,t=ϕωtR∫RNJzξRx+z-ξxdz=ϕωtR∫RNJzexp⁡-x+z2R2-exp⁡-x2R2dz=ϕωtRexp⁡-x2R2∫RNJzexp⁡-x+z2+x2R2-1dz=φx,t∫RNJzexp⁡-1R22∑i=1Nxizi+z2-1dz.
The property (J3) and the inequalityex≥x+1 yield(30)J∗φ-φx,t≥-φx,tR2∫RNJz2∑i=1Nxizi+z2dz=-φx,tR2∫RNJzz2dz=-AJR-2φx,t.
From this, we have(31)-∫QJ∗u-uφdxdt≤AJR-2∫Quφdxdt. Writing (32)∫Quφdxdt=∫Quφ1/pφ1/p′dxdt, using Hölder’s inequality and Lemma 1, we obtain(33)-∫QJ∗u-uφdxdt≤AJR-2∫Qupφdxdt1/p∫Qφdxdt1/p′≤δ∫Qupφdxdt+cδAJp′R-2p′∫Qφdxdt≤δ∫Qupφdxdt+cδAJp′R-2p′∫RNexp⁡-x2R2dx∫0∞ϕωtRdt≤δ∫Qupφdxdt+cδAJp′RN+1-2p′∫RNexp⁡-y2dx∫0∞ϕωsds, for some δ>0. We get(34)-∫QJ∗u-uφdxdt≤δ∫Qupφdxdt+c2RN+1-2p′,for some constant c2>0. Consequently, it follows from (11), (18), (24), and (34) that (35)1-ɛ-δ∫Qupφdxdt≤c1+c2RN+1-2p′.For ɛ=δ=1/4, we obtain (36)∫Qupφdxdt≤cRN+1-2p′, where c=2(c1+c2). Observe that if one of conditions (12) or (13) is satisfied, then N+1-2p′<0. In this case, letting R→∞ in the above inequality and using the monotone convergence theorem, we obtain (37)∫Qupdxdt=0, which is a contradiction. The proof is finished.


### 2.2.  A Nonexistence Result for System (6)

The definition of solutions we adopt for (6) is as follows.Definition 5.
Let(ui,vi)∈Lloc1(RN;R+)×Lloc1(RN;R+), i=0,1. We say that the pair (u,v) is a global weak solution to (6) if (u,v)∈Llocq(Q;R+)×Llocp(Q;R+), (J∗u,J∗v)∈Lloc1(Q;R+)×Lloc1(Q;R+), and(38)∫Qvpφdxdt+∫RNu1xφx,0dx=∫Quφttdxdt-∫QJ∗u-uφdxdt+∫RNu0xφtx,0dx;(39)∫Quqφdxdt+∫RNv1xφx,0dx=∫Qvφttdxdt-∫QJ∗v-vφdxdt+∫RNv0xφtx,0dx,for every regular test function φ≥0 with φ(·,t≥T)≡0.We have the following result.Theorem 6.
Letp,q>1. Suppose that(40)1≤N<1+2pq-1max⁡p+1,q+1.Then (6) admits no global weak solutions other than the trivial one.Proof.
Suppose that(u,v) is a nontrivial global weak solution to (6). As a test function, we take the function φ defined by (14). From the definition of φ, we have(41)∫RNu0xφtx,0dx=∫RNv0xφtx,0dx=0.Writing (42)∫Quφttdxdt=∫Quφ1/qφ-1/qφttdxdt and using Hölder’s inequality, we obtain (43)∫Quφttdxdt≤∫Quqφdxdt1/q∫Qφ-q′/qφttq′dxdt1/q′, where q′=q/(q-1). On the other hand, from (23), we have (44)∫Qφ-q′/qφttq′dxdt=ωq′RN+1-2q′∫Qexp⁡-y2ϕω-2q/q-1shsq′dyds, which yields(45)∫Quφttdxdt≤c1RN+1/q′-2∫Quqφdxdt1/q,for some constant c1>0. Using (33), we obtain (46)-∫QJ∗u-uφdxdt≤AJR-2∫Quqφdxdt1/q∫Qφdxdt1/q′≤AJR-2∫Quqφdxdt1/q∫RNexp⁡-x2R2dx1/q′∫0∞ϕωtRdt1/q′≤AJR-2+N+1/q′∫RNexp⁡-y2dx1/q′∫0∞ϕωsds1/q′∫Quqφdxdt1/q, which yields(47)-∫QJ∗u-uφdxdt≤c2R-2+N+1/q′∫Quqφdxdt1/q,for some constant c2>0. As consequence, from (38), (45), and (47), it follows that(48)∫Qvpφdxdt≤C1R-2+N+1/q′∫Quqφdxdt1/q,where C1=c1+c2. Similarly, we have(49)∫Quqφdxdt≤C2R-2+N+1/p′∫Qvpφdxdt1/p,for some constant C2>0. Combining (48) with (49), we obtain(50)∫Qvpφdxdt1-1/pq≤CRλ1,∫Quqφdxdt1-1/pq≤CRλ2,for some constant C>0, where (51)λ1=-2-2q+N+1pqqp-1,λ2=-2-2p+N+1pqqp-1.Observe that (40) is equivalent to λi<0, i=1,2. Under this condition, letting R→∞ in (50), we get (52)∫Qvpdxdt=∫Quqdxdt=0, which is a contradiction.Remark 7.
Takingu=v and p=q in Theorem 6, we obtain the result given by Theorem 4 for (5).


### 2.3.  A Nonexistence Result for System (7)

The definition of solutions we adopt for (7) is as follows.Definition 8.
Let(ui,vi)∈Lloc1(RN;R+)×Lloc1(RN;R+), i=0,1. We say that the pair (u,v) is a global weak solution to (6) if (u,v)∈Llocp(Q;R+)×Llocq(Q;R+), u≢0, v≢0, (J∗u,J∗v)∈Lloc1(Q;R+)×Lloc1(Q;R+), and(53)∫Qupφdxdt+∫RNu1xφx,0dx=∫Quφttdxdt-∫QJ∗v-vφdxdt+∫RNu0xφtx,0dx;(54)∫Qvqφdxdt+∫RNv1xφx,0dx=∫Qvφttdxdt-∫QJ∗u-uφdxdt+∫RNv0xφtx,0dx,for every regular test function φ≥0 with φ(·,t≥T)≡0.We have the following result.Theorem 9.
Letp,q>1. If(55)1≤N<max⁡Θ1,Θ2,where (56)Θ1=min⁡p+1p-1,q+1q-1,pq+2p+1pq-1,Θ2=min⁡p+1p-1,q+1q-1,pq+2q+1pq-1,then (7) admits nonglobal weak solutions.Proof.
As before, we argue by contradiction. Suppose that(u,v) is a nontrivial global weak solution to (7). As a test function, we take the function φ defined by (14). From (45), we have(57)∫Quφttdxdt≤c1RN+1/p′-2∫Qupφdxdt1/p.From (47), we have(58)-∫QJ∗v-vφdxdt≤c2R-2+N+1/q′∫Qvqφdxdt1/q.Using (53), (57), and (58), we get(59)∫Qupφdxdt≤c1RN+1/p′-2∫Qupφdxdt1/p+c2R-2+N+1/q′∫Qvqφdxdt1/q.Similarly, using (54), (57), and (58), we get(60)∫Qvqφdxdt≤d1RN+1/q′-2∫Qvqφdxdt1/q+d2R-2+N+1/p′∫Qupφdxdt1/p.Here, ci, di, i=1,2, are some positive constants. Set (61)X=∫Qupφdxdt1/p,Y=∫Qvqφdxdt1/q; we obtain from (59) and (60) the following system: (62)Xp≤c1RN+1/p′-2X+c2R-2+N+1/q′Y,Yq≤d2R-2+N+1/p′X+d1RN+1/q′-2Y.Using Lemma 2, we obtain(63)Xpq≤cRλ1+Rλ2+Rλ3,where (64)p-1qλ1=N+1p-1-2p,q-1qλ2=N+1q-1-2q,pq-1pqλ3=-2q+N+1q-1+N+1p-1-2pp.Similarly. we have(65)Ypq≤cRμ1+Rμ2+Rμ3,where (66)q-1pμ1=N+1q-1-2q,p-1pμ2=N+1p-1-2p,pq-1pqμ3=-2p+N+1p-1+N+1q-1-2qq.It is not difficult to observe that condition (55) is equivalent to λi<0, i=1,2,3, or μi<0, i=1,2,3. In both cases, letting R→∞ in (63) or in (65), we obtain XY=0, which is a contradiction.


## 2.1. A Nonexistence Result for (5)

The definition of solutions we adopt for (5) is as follows.Definition 3.
Let(u0,u1)∈Lloc1(RN;R+)×Lloc1(RN;R+). We say that u is a global weak solution to (5) if u∈Llocp(Q;R+), J∗u∈Lloc1(Q;R+), and(11)∫Qupφdxdt+∫RNu1xφx,0dx=∫Quφttdxdt-∫QJ∗u-uφdxdt+∫RNu0xφtx,0dx,for every regular test function φ≥0 with φ(·,t≥T)≡0.Our first main result is given by the following theorem.Theorem 4.
Suppose that one of the following conditions hold:(12)N=1<por(13)N≥2,1<p<N+1N-1.Then (5) admits no global weak solutions other than the trivial one.Proof.
Suppose thatu is a nontrivial global weak solution to (5). As a test function, we take(14)φx,t=ξRxϕωtR,x,t∈Q,where R>0 is large enough, (15)ξRx=exp⁡-x2R2,x∈RN,ω≫1, and ϕ:R+→[0,1] is given by (16)ϕσ=1if0≤σ≤1,0ifσ≥2. From the definition of ϕ, clearly we have (17)φtx,0=0,x∈RN, which yields(18)∫RNu0xφtx,0dx=0.Writing (19)∫Quφttdxdt=∫Quφ1/pφ-1/pφttdxdt and applying Lemma 1, we obtain(20)∫Quφttdxdt≤ɛ∫Qupφdxdt+cɛ∫Qφ-p′/pφttp′dxdt,for some ɛ>0, where p′=p/(p-1). On the other hand, (21)∫Qφ-p′/pφttp′dxdt=ωp′R-2p′∫Qexp⁡-x2R2ϕω-2p/p-1tRhtRp′dxdt, where (22)hσ≔ω-1ϕ′σ+ϕσϕ′′σ,σ≥0. Using the change of variable x=Ry and t=Rs, we obtain(23)∫Qφ-p′/pφttp′dxdt=ωp′RN+1-2p′∫Qexp-y2ϕω-2p/p-1shsp′dyds.The above equality with (20) yields(24)∫Quφttdxdt≤ɛ∫Qupφdxdt+c1RN+1-2p′,for some constant c1>0. Next, we have (25)∫QJ∗uφdxdt=∫0∞∫RN∫RNJx-yuy,tdyφx,tdxdt. Using the symmetry of J and Fubini’s theorem, we obtain(26)∫QJ∗uφdxdt=∫0∞∫RN∫RNJx-yuy,tdyφx,tdxdt=∫0∞∫RN∫RNJy-xuy,tdyφx,tdxdt=∫0∞∫RN∫RNJy-xφx,tdxuy,tdy=∫QJ∗φudxdt. Therefore, (27)∫QJ∗u-uφdxdt=∫QJ∗φ-φudxdt. Using the property (J1), we obtain (28)J∗φ-φx,t=ϕωtR∫RNJx-yξRydy-ξRx=ϕωtR∫RNJy-xξRydy-ξRx=ϕωtR∫RNJzξRx+zdz-ξRx. By the property (J2) and the definition of φ, we have(29)J∗φ-φx,t=ϕωtR∫RNJzξRx+z-ξxdz=ϕωtR∫RNJzexp⁡-x+z2R2-exp⁡-x2R2dz=ϕωtRexp⁡-x2R2∫RNJzexp⁡-x+z2+x2R2-1dz=φx,t∫RNJzexp⁡-1R22∑i=1Nxizi+z2-1dz.
The property (J3) and the inequalityex≥x+1 yield(30)J∗φ-φx,t≥-φx,tR2∫RNJz2∑i=1Nxizi+z2dz=-φx,tR2∫RNJzz2dz=-AJR-2φx,t.
From this, we have(31)-∫QJ∗u-uφdxdt≤AJR-2∫Quφdxdt. Writing (32)∫Quφdxdt=∫Quφ1/pφ1/p′dxdt, using Hölder’s inequality and Lemma 1, we obtain(33)-∫QJ∗u-uφdxdt≤AJR-2∫Qupφdxdt1/p∫Qφdxdt1/p′≤δ∫Qupφdxdt+cδAJp′R-2p′∫Qφdxdt≤δ∫Qupφdxdt+cδAJp′R-2p′∫RNexp⁡-x2R2dx∫0∞ϕωtRdt≤δ∫Qupφdxdt+cδAJp′RN+1-2p′∫RNexp⁡-y2dx∫0∞ϕωsds, for some δ>0. We get(34)-∫QJ∗u-uφdxdt≤δ∫Qupφdxdt+c2RN+1-2p′,for some constant c2>0. Consequently, it follows from (11), (18), (24), and (34) that (35)1-ɛ-δ∫Qupφdxdt≤c1+c2RN+1-2p′.For ɛ=δ=1/4, we obtain (36)∫Qupφdxdt≤cRN+1-2p′, where c=2(c1+c2). Observe that if one of conditions (12) or (13) is satisfied, then N+1-2p′<0. In this case, letting R→∞ in the above inequality and using the monotone convergence theorem, we obtain (37)∫Qupdxdt=0, which is a contradiction. The proof is finished.


## 2.2.  A Nonexistence Result for System (6)

The definition of solutions we adopt for (6) is as follows.Definition 5.
Let(ui,vi)∈Lloc1(RN;R+)×Lloc1(RN;R+), i=0,1. We say that the pair (u,v) is a global weak solution to (6) if (u,v)∈Llocq(Q;R+)×Llocp(Q;R+), (J∗u,J∗v)∈Lloc1(Q;R+)×Lloc1(Q;R+), and(38)∫Qvpφdxdt+∫RNu1xφx,0dx=∫Quφttdxdt-∫QJ∗u-uφdxdt+∫RNu0xφtx,0dx;(39)∫Quqφdxdt+∫RNv1xφx,0dx=∫Qvφttdxdt-∫QJ∗v-vφdxdt+∫RNv0xφtx,0dx,for every regular test function φ≥0 with φ(·,t≥T)≡0.We have the following result.Theorem 6.
Letp,q>1. Suppose that(40)1≤N<1+2pq-1max⁡p+1,q+1.Then (6) admits no global weak solutions other than the trivial one.Proof.
Suppose that(u,v) is a nontrivial global weak solution to (6). As a test function, we take the function φ defined by (14). From the definition of φ, we have(41)∫RNu0xφtx,0dx=∫RNv0xφtx,0dx=0.Writing (42)∫Quφttdxdt=∫Quφ1/qφ-1/qφttdxdt and using Hölder’s inequality, we obtain (43)∫Quφttdxdt≤∫Quqφdxdt1/q∫Qφ-q′/qφttq′dxdt1/q′, where q′=q/(q-1). On the other hand, from (23), we have (44)∫Qφ-q′/qφttq′dxdt=ωq′RN+1-2q′∫Qexp⁡-y2ϕω-2q/q-1shsq′dyds, which yields(45)∫Quφttdxdt≤c1RN+1/q′-2∫Quqφdxdt1/q,for some constant c1>0. Using (33), we obtain (46)-∫QJ∗u-uφdxdt≤AJR-2∫Quqφdxdt1/q∫Qφdxdt1/q′≤AJR-2∫Quqφdxdt1/q∫RNexp⁡-x2R2dx1/q′∫0∞ϕωtRdt1/q′≤AJR-2+N+1/q′∫RNexp⁡-y2dx1/q′∫0∞ϕωsds1/q′∫Quqφdxdt1/q, which yields(47)-∫QJ∗u-uφdxdt≤c2R-2+N+1/q′∫Quqφdxdt1/q,for some constant c2>0. As consequence, from (38), (45), and (47), it follows that(48)∫Qvpφdxdt≤C1R-2+N+1/q′∫Quqφdxdt1/q,where C1=c1+c2. Similarly, we have(49)∫Quqφdxdt≤C2R-2+N+1/p′∫Qvpφdxdt1/p,for some constant C2>0. Combining (48) with (49), we obtain(50)∫Qvpφdxdt1-1/pq≤CRλ1,∫Quqφdxdt1-1/pq≤CRλ2,for some constant C>0, where (51)λ1=-2-2q+N+1pqqp-1,λ2=-2-2p+N+1pqqp-1.Observe that (40) is equivalent to λi<0, i=1,2. Under this condition, letting R→∞ in (50), we get (52)∫Qvpdxdt=∫Quqdxdt=0, which is a contradiction.Remark 7.
Takingu=v and p=q in Theorem 6, we obtain the result given by Theorem 4 for (5).


## 2.3.  A Nonexistence Result for System (7)

The definition of solutions we adopt for (7) is as follows.Definition 8.
Let(ui,vi)∈Lloc1(RN;R+)×Lloc1(RN;R+), i=0,1. We say that the pair (u,v) is a global weak solution to (6) if (u,v)∈Llocp(Q;R+)×Llocq(Q;R+), u≢0, v≢0, (J∗u,J∗v)∈Lloc1(Q;R+)×Lloc1(Q;R+), and(53)∫Qupφdxdt+∫RNu1xφx,0dx=∫Quφttdxdt-∫QJ∗v-vφdxdt+∫RNu0xφtx,0dx;(54)∫Qvqφdxdt+∫RNv1xφx,0dx=∫Qvφttdxdt-∫QJ∗u-uφdxdt+∫RNv0xφtx,0dx,for every regular test function φ≥0 with φ(·,t≥T)≡0.We have the following result.Theorem 9.
Letp,q>1. If(55)1≤N<max⁡Θ1,Θ2,where (56)Θ1=min⁡p+1p-1,q+1q-1,pq+2p+1pq-1,Θ2=min⁡p+1p-1,q+1q-1,pq+2q+1pq-1,then (7) admits nonglobal weak solutions.Proof.
As before, we argue by contradiction. Suppose that(u,v) is a nontrivial global weak solution to (7). As a test function, we take the function φ defined by (14). From (45), we have(57)∫Quφttdxdt≤c1RN+1/p′-2∫Qupφdxdt1/p.From (47), we have(58)-∫QJ∗v-vφdxdt≤c2R-2+N+1/q′∫Qvqφdxdt1/q.Using (53), (57), and (58), we get(59)∫Qupφdxdt≤c1RN+1/p′-2∫Qupφdxdt1/p+c2R-2+N+1/q′∫Qvqφdxdt1/q.Similarly, using (54), (57), and (58), we get(60)∫Qvqφdxdt≤d1RN+1/q′-2∫Qvqφdxdt1/q+d2R-2+N+1/p′∫Qupφdxdt1/p.Here, ci, di, i=1,2, are some positive constants. Set (61)X=∫Qupφdxdt1/p,Y=∫Qvqφdxdt1/q; we obtain from (59) and (60) the following system: (62)Xp≤c1RN+1/p′-2X+c2R-2+N+1/q′Y,Yq≤d2R-2+N+1/p′X+d1RN+1/q′-2Y.Using Lemma 2, we obtain(63)Xpq≤cRλ1+Rλ2+Rλ3,where (64)p-1qλ1=N+1p-1-2p,q-1qλ2=N+1q-1-2q,pq-1pqλ3=-2q+N+1q-1+N+1p-1-2pp.Similarly. we have(65)Ypq≤cRμ1+Rμ2+Rμ3,where (66)q-1pμ1=N+1q-1-2q,p-1pμ2=N+1p-1-2p,pq-1pqμ3=-2p+N+1p-1+N+1q-1-2qq.It is not difficult to observe that condition (55) is equivalent to λi<0, i=1,2,3, or μi<0, i=1,2,3. In both cases, letting R→∞ in (63) or in (65), we obtain XY=0, which is a contradiction.



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*Source: 102639-2015-12-03.xml*