# Critical Blow-Up and Global Existence for Discrete Nonlinearp-Laplacian Parabolic Equations

Authors: Soon-Yeong Chung
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

The goal of this paper is to investigate the blow-up and the global existence of the solutions to the discretep-Laplacian parabolic equation utx,t=Δp,wux,t+λux,tp-2ux,t, x,t∈S×0,∞, ux,t=0, x,t∈∂S×0,∞, ux,0=u0, depending on the parameters p>1 and λ>0. Besides, we provide several types of the comparison principles to this equation, which play a key role in the proof of the main theorems. In addition, we finally give some numerical examples which exploit the main results.


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## Body

## 1. Introduction

A discretep-Laplacian parabolic equation (a reaction-diffusion system) has found many applications in chemical reactions and biological phenomena. A typical example is an autocatalytic chemical reaction between several chemicals in which the concentration of each chemical grows (or decays) due to diffusion and difference of concentration. In general, many of such phenomena are modeled by
(1)utx,t=Δp,wux,t+λux,tq-1ux,t,x∈S,
with some boundary and initial conditions where S is the set of chemicals and p>1, q>0. Here, Δp,w is the discrete p-Laplace operator on a network S, defined by
(2)Δp,wfx∶=∑y∈S¯‍fy-fxp-2fy-fxωx,y.
For the continuous case, solutions to the initial value problem of (1) may blow up in a finite time or exist globally for all time, depending on the parameters p>1, q>0, and λ∈R (see [1–7] and see the book [2] for more information about the various blow-up phenomena). In particular, the case where p=q-1>1 has been known as the critical case at which solutions may blow up or exist globally, depending on the parameter λ.The goal of this paper is to investigate the blow-up and the global existence of the solutions to the initial value problem of the discretep-Laplacian parabolic equation
(3)utx,t=Δp,wux,t+λux,tp-2ux,t,x∈S,
on a network S with boundary, which is the critical case of (1).The continuous case of (3) has been studied by many authors. For example, the case where 1<p<2 was studied in [3, 6] and the case where p>2 was studied in [3, 4].For the discrete case of (1), the authors studied the general theory of the equation for the case p=2 in [8] and Xin et al. studied the blow-up solutions and global solutions for the case where p=2 and λ=1 in [9]. On the other hand, for the case λ≤0 in (1), Chung et al. [10, 11] investigate the positive solutions and extinctive solutions, depending on the parameters p and q. But, the case where p>1 and λ>0 in (3) has not been studied yet. Thus, the purpose of this paper is to give a complete solution to such a problem about (3).The main results of this paper are summarized as follows.Theorem 1.
(i) Forp>2 and λ>λ0, the solution to
(4)utx,t=Δp,wux,t+λux,tp-2ux,t,kkkkkkkkkkkllx,t∈S×0,∞,ux,t=0,x,t∈∂S×0,∞,ux,0=u0
blows up in a finite time T, for every nonnegative and nontrivial initial data u0.
Here,λ0 is the first eigenvalue of the p-Laplace operator Δp,w on S. Moreover, it is shown that the blow-up rate of the solution is estimated by
(5)max⁡x∈Sux,t≥p-2λ-λ0T-t-1/p-2,t>0.
(ii) Forp>2 and λ≤λ0, the solution to (4) is global, for every nonnegative initial data u0.
(iii) For1<p≤2, the nonnegative solution to (4) is global for every nonnegative initial data u0. In particular, when 1<p<2 and λ<λ0, there exists T>0 (extinction time) such that ux,t≡0 for all t≥T.Besides, we provide several types of the comparison principles to (4) in Section 2, which play a key role in the proof of the main theorems and we give some numerical examples in the final section which exploit the main results.


## 2. Preliminaries and Discrete Comparison Principles

In this section, we start with some definitions of graph theoretic notions frequently used throughout this paper (see [8, 12, 13] for more details).For a graphG=G(V,E) we mean finite sets V ofvertices (ornodes) with a set E of two-element subsets of V (whose elements are callededges). The set of vertices and edges of a graph G is sometimes denoted by V(G) and E(G), or simply V and E, respectively. Conventionally, we denote by x∈V or x∈G the facts that x is a vertex in G.A graphG is said to besimple if it has neither multiple edges nor loops, and G is said to beconnected if, for every pair of vertices x and y, there exists a sequence (called apath) of vertices x=x0,x1,…,xn-1,xn=y such that xj-1 and xj are connected by an edge (calledadjacent) for j=1,…,n.A graphS=S(V′,E′) is said to be asubgraph of G(V,E) if V′⊂V and E′⊂E.Aweight on a graph G is a function ω:V×V→[0,+∞) satisfying(i)
ω(x,x)=0,x∈V,(ii)
ω(x,y)=ω(y,x) if x~y,(iii)
ω(x,y)=0 if and only if x≁y.Here,x~y means that two vertices x and y are connected (adjacent) by an edge in E. A graph associated with a weight is said to be aweight graph or anetwork.For a subgraphS of a graph G(V,E), the (vertex)boundary∂S of S is the set of all vertices z∈V∖S but adjacent to some vertex in S; that is,
(6)∂S∶=z∈V∖S∣z~yforsomey∈S.
By S¯ we denote a graph, whose vertices and edges are in both S and ∂S. Throughout this paper, all subgraphs S and S¯ in our concern are assumed to be simple and connected.For a functionu:S¯→R, thediscrete p-LaplacianΔp,ω on S is defined by
(7)Δp,ωu(x)∶=∑y∈S¯‍u(y)-u(x)p-2[u(y)-u(x)]ω(x,y)
for x∈S.The following lemmas are useful throughout this paper.Lemma 2 (see [14, 15]).
For functionsf,g:S¯→R, one has the following: (i)
2∑x∈S¯‍gx[-Δp,ωfx]=∑x,y∈S¯‍fy-fxp-2[fy-fx]·[gy-gx]ω(x,y);(ii)
2∑x∈S¯‍fx[-Δp,ωfx]=∑x,y∈S¯‍fx-fypω(x,y).Lemma 3 (see [14, 15]).
Forp>1, there exist λ0>0 and ϕ0x>0, x∈S, such that
(8)-Δp,ωϕ0x=λ0ϕ0p-1x,x∈S,ϕ0x=0,x∈∂S,∑x∈S‍ϕ0px=1.
Moreover,
(9)λ0=min⁡u∈A,u≢01/2∑x,y∈S¯ux-uypωx,y∑x∈S¯uxp≤∑y∈S¯‍ωx,y,x∈S,
where A∶=u:S¯→R∣u=0on∂S.In the above, the numberλ0 is called the first eigenvalue of Δp,ω on a network S¯ with corresponding eigenfunction ϕ0 (see [12, 13] for the spectral theory of the Laplacian operators).The rest of this section is devoted to proving the comparison principles for the discretep-Laplacian parabolic equation
(10)utx,t=Δp,ωux,t+λux,tp-2ux,t,kkkkkkkkkkkllx,t∈S×0,∞,ux,t=0,x,t∈∂S×0,∞,ux,0=u0,x∈S¯,
where λ>0 and p>1, in order to study the blow-up occurrence and global existence, which we begin in the next section.Now, we state the comparison principles and some related corollaries.Theorem 4.
LetT>0 (T may be +∞), λ>0, and p≥2. Suppose that real-valued functions u(x,·), v(x,·)∈C[0,T) are differentiable in (0,T) for each x∈S¯ and satisfy
(11)utx,t-Δp,ωux,t-λux,tp-2u(x,t)≥vtx,t-Δp,ωvx,t-λvx,tp-2v(x,t),kkkkkkkkkkkkkkkkkkkklkkkl(x,t)∈S×0,T,ux,t≥vx,t,(x,t)∈∂S×[0,T),ux,0≥vx,0,x∈S¯.
Then ux,t≥vx,t for all (x,t)∈S¯×[0,T).Proof.
LetT′>0 be arbitrarily given with T′<T. Then by the mean value theorem, for each x∈S and 0≤t≤T′,
(12)ux,tp-2ux,t-vx,tp-2vx,t=p-1ξx,tp-2ux,t-vx,t
for some ξx,t lying between ux,t and vx,t. Then it follows from (11) that we have
(13)ut-Δp,ωu-λp-1ξx,tp-2ux,t≥vt-Δp,ωv-λ(p-1)ξx,tp-2vx,t
for all (x,t)∈S×(0,T′]. Let u~,v~:S¯×[0,T′]→R be the functions defined by(14)u~x,t∶=e-2λ(p-1)Ltux,t,v~x,t∶=e-2λ(p-1)Ltvx,t,
where L∶=max⁡r≤Mrp-2 and M∶=max⁡x∈S¯,t∈[0,T′]{ux,t,vx,t}.
Then inequality (13) can be written as
(15)u~tx,t-v~tx,t-e2λLp-1p-2t×Δp,ωu~x,t-Δp,ωv~x,t+λ(p-1)×2L-ξx,tp-2u~x,t-v~x,t≥0
for all (x,t)∈S×(0,T′]. Since S¯×[0,T′] is compact, there exists (x0,t0)∈S¯×[0,T′] such that
(16)(u~-v~)(x0,t0)=min⁡x∈S¯min⁡0≤t≤T′(u~-v~)x,t.
Then we have only to show that (u~-v~)(x0,t0)≥0. Suppose that (u~-v~)(x0,t0)<0, on the contrary. Since (u~-v~)(x,t)≥0 on both ∂S×[0,T′] and S¯×{0}, we have (x0,t0)∈S×(0,T′]. Then we have
(17)u~tx0,t0≤v~tx0,t0,Δp,ωu~x0,t0≥Δp,ωv~x0,t0.
Since ξp-2x,t≤max⁡r≤Mrp-2=L, we have
(18)[2L-ξ(x0,t0)p-2](u~-v~)(x0,t0)≤L(u~-v~)(x0,t0)<0.
Combining (17) and (18), we obtain
(19)u~tx0,t0-v~tx0,t0-e2λLp-1p-2t0×Δp,ωu~x0,t0-Δp,ωv~x0,t0+λ(p-1)×2L-ξx0,t0p-2u~x0,t0-v~x0,t0<0
which contradicts (15). Therefore, u~x,t-v~x,t≥0 for all (x,t)∈S×(0,T′] so that we get ux,t≥vx,t for all (x,t)∈S¯×[0,T), since T′<T is arbitrarily given.Now, we obtain a strict comparison principle as a corollary.Corollary 5 (strict comparison principle).
In Theorem4, if u0(x*)>v0(x*) for some x*∈S, then ux,t>vx,t for all (x,t)∈S×(0,T).Proof.
First, note thatu≥v on S¯×[0,T) by Theorem 4. Let T′>0 be arbitrarily given with T′<T and let τ:S¯×[0,T′]→R be a function defined by(20)τx,t∶=ux,t-vx,t,x,t∈S¯×0,T′.
Then τ(x,t)≥0 for all (x,t)∈S¯×[0,T′]. Since |u(x*,t)|p-2u(x*,t)≥|v(x*,t)|p-2v(x*,t) for all 0<t≤T′, and τ(x*,0)>0, we obtain from inequality (15) that
(21)τtx*,t-Δp,ωux*,t-Δp,ωvx*,t≥0,
for all 0<t≤T′. Then by the mean value theorem, for each y∈S¯ and t with 0≤t≤T′, it follows that
(22)uy,t-ux*,tp-2uy,t-ux*,t-vy,t-vx*,tp-2vy,t-vx*,t=p-1ηx*,y,tp-2τy,t-τx*,t,
and ηx*,y,t<2M where M∶=max⁡0<t≤T′{ux*,t,vx*,t}.
Then inequality (21) gives
(23)τtx*,t≥∑y∈S¯‍p-1ηx*,y,tp-2kkl×τy,t-τx*,tωx*,y≥-dp-12Mp-2τx*,t,
where d=∑y∈S¯ω(x*,y). This implies
(24)τx*,t≥τx*,0e-dp-12Mp-2t>0,t∈[0,T′].
Now, suppose there exists (x0,t0)∈S×(0,T′] such that
(25)τx0,t0=min⁡x∈S,0<t≤T′⁡τx,t=0.
Then
(26)τtx0,t0≤0,Δp,ωux0,t0≥Δp,ωvx0,t0.
Hence, inequality (21) gives
(27)0≤τtx0,t0-Δp,ωux0,t0-Δp,ωvx0,t0≤0.
Therefore,
(28)Δp,ωux0,t0=Δp,ωvx0,t0;
that is,
(29)∑y∈S¯‍uy,t0-ux0,t0p-2uy,t0-ux0,t0kkkk-vy,t0-vx0,t0p-2vy,t0-vx0,t0×ωx0,y=0,
which implies that τy,t0=0 for all y∈S¯ with y~x0. Now, for any x∈S¯, there exists a path
(30)x0~x1~⋯~xn-1~xn=x,
since S¯ is connected. By applying the same argument as above inductively we see that τ(x,t0)=0 for every x∈S¯. This gives a contradiction to (24).For the case1<p<2, it is well known that (10) may not have unique solution, in general, and the comparison principle in usual form as in Theorem 4 may not hold. Instead, with a strict condition on the parabolic boundary we obtain a similar comparison principle as follows.Theorem 6.
LetT>0  (T may be +∞), λ>0, and p>1. Suppose that real-valued functions u(x,·), v(x,·)∈C0,T are differentiable in (0,T) for each x∈S¯ and satisfy
(31)utx,t-Δp,ωux,t-λux,tp-2u(x,t)≥vtx,t-Δp,ωvx,t-λvx,tp-2v(x,t),kkkkkkkkkkkkkkkkkkkkkkkk(x,t)∈S×0,T,ux,t>vx,t,x,t∈∂S×0,T,ux,0>vx,0,x∈S¯.
Then ux,t≥vx,t for all (x,t)∈S×0,T.Proof.
LetT′>0 and δ>0 be arbitrarily given with T′<T and 0<δ<min⁡(x,t)∈Γ[u(x,t)-v(x,t)], respectively, where Γ∶={(x,t)∈S¯×[0,T′]∣t=0orx∈∂S} (called a parabolic boundary).
Now, let a functionτ:S¯×(0,T′]→R be a function defined by
(32)τx,t∶=ux,t-vx,t-δ,x,t∈S¯×0,T′.
Then τ(x,t)>0 on Γ. Now, we suppose that min⁡x∈S,0<t≤T′τ(x,t)<0. Then there exists (x0,t0)∈S×(0,T′] such that(i)
τ(x0,t0)=0,(ii)
τ(y,t0)≥τ(x0,t0)=0, y∈S,(iii)
τ(x,t)>0, (x,t)∈S×(0,t0).
Then(33)τt(x0,t0)≤0,Δp,ωux0,t0≥Δp,ωvx0,t0,
since uy,t0-ux0,t0≥vy,t0-vx0,t0. Hence, inequality (31) gives
(34)0≥τtx0,t0≥λux0,t0p-2ux0,t0-vx0,t0p-2kkkk×ux0,t0p-2ux0,t0-vx0,t0p-2vx0,t0=λvx0,t0+δp-2vx0,t0+δ-vx0,t0p-2vx0,t0kkkk-vx0,t0p-2vx0,t0>0,
which leads to a contradiction. Hence, τ(x,t)≥0 for all (x,t)∈S×(0,T′] so that we have u(x,t)≥v(x,t) for all (x,t)∈S×(0,T), since δ and T′ are arbitrary.


## 3. Blow-Up and Global Existence

In this section, we discuss the blow-up and global existence of the equation(35)utx,t=Δp,ωux,t+λux,tp-2ux,t,kkkkkkkkkkkkx,t∈S×0,∞,ux,t=0,x,t∈∂S×0,∞,ux,0=u0≥0,x∈S¯,
where λ>0, p>1, and the initial data u0 is nontrivial on S. According to the comparison principle in the previous section, the nonhomogeneous term λux,tp-2u(x,t) can be written as a simpler form as λup-1x,t, for the case where p≥2.Definition 7 (blow-up).
We say that a solutionu to an equation defined on a network S¯ blows up in finite time T, if there exists x∈S such that |u(x,t)|→+∞ as t↗T-.Theorem 8.
Forp>2 and λ>0, the solution to
(36)utx,t=Δp,ωux,t+λup-1x,t,x,t∈S×0,∞,ux,t=0,x,t∈∂S×0,∞,ux,0=u0x≥0(nontrivial)
blows up at some T*, provided that
(37)-12∑x,y∈S¯‍u0x-u0ypωx,y+λ∑x∈S¯‍u0px>0.Proof.
First, we note thatu(x,t)>0 on S×(0,∞), by the strict comparison principle (Corollary 5). Now, we define a functional by
(38)Jt∶=-12p∑x,y∈S¯‍ux,t-uy,tpωx,y+λp∑x∈S¯‍ux,tp,t≥0.
Then by (37),
(39)J0=-12p∑x,y∈S¯‍u0x-u0ypωx,y+λp∑x∈S¯‍u0xp>0.
Multiplying (36) by u and summing up over S¯, we obtain from Lemma 2(40)ddt∑x∈S¯‍u2x,t=-∑x,y∈S¯‍ux,t-uy,tpωx,y+2λ∑x∈S¯‍upx,t.
Multiplying (36) by ut and summing up over S¯, we obtain from Lemma 2(41)∑x∈S¯‍ut2x,t=-∑x∈S¯‍utx,t-Δp,ωux,tkkkkkkkkkkkkk+ddtλp∑x∈S¯‍upx,t,∑x∈S¯‍utx,t-Δp,ωux,t=12∑x,y∈S¯‍uy,t-ux,tp-2×uy,t-ux,tuty-utx×ωx,y=ddt12p∑x,y∈S¯‍uy,t-ux,tpωx,y.
Then it follows that
(42)∑x∈S¯‍ut2x,t=-ddt12p∑x,y∈S¯‍uy,t-ux,tpωx,y+ddtλp∑x∈S¯‍upx,t.
Moreover, it follows from (42) that
(43)J′t=∑x∈S¯‍ut2x,t,Jt=∫0t‍J′sds+J0=∫0t‍∑x∈S¯‍ut2x,sds+J0.
Now, we introduce a new function
(44)It=∫0t‍∑x∈S¯‍u2x,sds+M,t≥0,
where M>0 is a constant to be determined later. Then by (40) we have
(45)I′t=∑x∈S¯‍u2x,t,(46)I′′t=ddt∑x∈S¯‍u2x,t=-∑x,y∈S¯‍ux,t-uy,tpωx,y+2λ∑x∈S¯‍upx,t=2pJ0+∑x∈S¯‍∫0t‍ut2x,sds.
Moreover,
(47)I′t=∑x∈S¯‍u2x,t=∑x∈S¯‍∫0t‍2ux,sutx,sds+∑x∈S¯‍u02x.
Using the Schwarz inequality, we obtain
(48)I′t2≤41+ϵ∑x∈S¯‍∫0t‍ux,sutx,sds2+1+1ϵ∑x∈S¯‍u02x2≤41+ϵ∑x∈S¯‍∫0t‍u2x,sds1/2∫0t‍ut2x,sds1/22+1+1ϵ∑x∈S¯‍u02x2≤41+ϵ∑x∈S¯‍∫0t‍u2x,sds∑x∈S¯‍∫0t‍ut2x,sds+1+1ϵ∑x∈S¯‍u02x2,
where ϵ>0 is arbitrary. Combining the above estimates (46), (47), and (48), we obtain that, for α=ϵ=p/2-1>0,
(49)I′′tIt-1+αI′t2≥2pJ0+∑x∈S¯‍∫0t‍ut2x,sds×∑x∈S¯‍∫0t‍u2x,sds+M-41+α1+ϵ∑x∈S¯‍∫0t‍u2x,sds×∑x∈S¯‍∫0t‍ut2x,sds-1+α1+1ϵ∑x∈S¯‍u02x2≥2pM·J0-1+α1+1ϵ∑x∈S¯‍u02x2.
Since J0>0 by assumption, we can choose M>0 to be large enough so that
(50)I′′(t)It-1+αI′t2>0.
Thus inequality (50) implies that, for t≥0,
(51)ddtI′tIα+1t>0i.e.,I′t≥I′0Iα+10Iα+1t.
Therefore, it follows that It cannot remain finite for all t>0. In other words, the solutions ux,t blow up at some time T*.Remark 9.
(i) Condition (37) implies that
(52)λ>1/2∑x,y∈S¯u0x-u0ypωx,y∑x∈S¯u0px≥λ0,
where λ0 is the first eigenvalue of Δp,ω.
(ii) The initial datau0 with J0>0 always exists. In fact, consider an eigenvalue λ0>0 and eigenfunction ϕ0x>0 in Lemma 3. Taking u0x=ϕ0x, then we have
(53)J0=-1p∑x∈S¯‍ϕ0x-Δp,ωϕ0x+λp∑x∈S¯‍ϕ0px=λ-λ0p∑x∈S¯‍ϕ0px>0,
where λ>λ0.
(iii) When the solution to (37) is global, then we must have
(54)Jt=-12p∑x,y∈S¯‍ux,t-uy,tpωx,y+λp∑x∈S¯‍ux,tp<0
for all t>0.
(iv) The blow-up time in the above can be estimated roughly. Taking(55)M∶=p/p-21+p/2∑x∈S¯u02x2-∑x,y∈S¯u0x-u0ypωx,y+2λ∑x∈S¯u0xp
we see that
(56)I′t≥∑x∈S¯u02xMα+1Iα+1t,t>0,I0=M,
which implies
(57)It≥1Mα-α∑x∈S¯u02xMα+1t-1/α,
where α=p/2-1>0. Then the blow-up time T* satisfies
(58)0<T*≤Mα∑x∈S¯u02x.Now, we state the main theorem as follows.Theorem 10.
The solution to (36) with p>2 and λ>λ0 blows up in finite time, for every nonnegative and nontrivial initial data u0.Proof.
First, we note that (36) has a unique solution such that
(59)ux,t>0,x∈S,t>0.
Take t0>0, k>0 arbitrarily and consider an equation
(60)vtx,t=Δp,ωvx,t+λvp-1x,t,x∈S,t>t0,vx,t=0,x∈∂S,t≥t0vx,t0=kϕ0x,x∈S,
where ϕ0 is the eigenfunction corresponding to the first eigenvalue λ0>0. Then the initial data vx,t0 satisfies
(61)-12∑x,y∈S¯‍vx,t0-vy,t0pωx,y+λ∑x∈S¯‍vpx,t0=kpλ-λ0∑x∈S¯‍ϕ0px>0.
Then, by Theorem 8, the solution vx,t is positive and blows up in a finite time. Since
(62)ux,t=0=vx,t,x∈S,t≥t0,ux,t0>vx,t0=kϕ0x,x∈S
for a small k>0, the comparison principle enables us to see
(63)0<vx,t≤ux,tx∈S,t≥t0,
which completes the proof.Remark 11.
According to Remark9 (iv) to Theorem 8, the blow-up time for the solution to (36) is estimated by
(64)0<T*≤p1+p/22kp-2p-22λ-λ0∑x∈S¯‍ϕ02x2.We now derive the lower bound for the maximum function of blow-up solutions.Theorem 12.
Letu be the solutions to (36) blowing up at finite time T. Then it follows that
(65)max⁡x∈S¯ux,t≥p-2λ-λ0T-t-1/p-2
for all t>0.Proof.
For eacht>0, let xt∈S be the node such that
(66)uxt,t=max⁡x∈S¯ux,t.
In fact, we note that uxt,t is continuous on [0,∞) and differentiable for almost all t>0. Then (36) can be written as
(67)utxs,s=Δp,ωuxs,s+λup-1xs,s=∑y∈S¯‍uy,s-uxs,sp-2×uy,s-uxs,sωxs,y+λup-1xs,s
for almost all s>0.
Now define a functionv:S¯×[0,∞)→[0,∞) by
(68)vx,t∶=uxt,t,x∈S,t≥0,0,x∈∂S,t≥0.
Then v(x,t)=v(y,t), t>0, if x,y∈S. Multiplying (67) by u(xs,s) and summing over S¯, we obtain, for almost all t>0,
(69)∑x∈S¯‍vtx,tvx,t=∑x∈S¯‍vx,tΔp,ωvx,t+λ∑x∈S¯‍vpx,t≤λ-λ0∑x∈S¯‍vpx,t
or, equivalently,
(70)vtx,t≤λ-λ0vp-1x,t,x∈S.
Integrating over t,T, we have
(71)vx,t≥p-2λ-λ0T-t-1/p-2,
which is desired.Now, we state the global existence of the solutions.Theorem 13.
Forp>2 and λ≤λ0, the solution to (36) is global for every nonnegative initial data u0.Proof.
Consider an eigenvalueλ0>0 and eigenfunction ϕ0x>0 in Lemma 3 and vx,t∶=Kϕ0x, x∈S¯, t≥0. Then by taking K large enough to be Kϕ0x≥u0x, x∈S, we have
(72)vx,0≥ux,0,x∈S¯,vtx,t-Δp,ωvx,t-λvp-1x,t=λ0-λKp-1ϕ0x≥0
for all t>0 and x∈S. Then by the comparison principle, we see that 0≤ux,t≤Kϕ0x, x∈S, t≥0, which is desired.We have so far discussed the blow-up or the global existence (35) for the case p≥2. Now, we discuss the case 1<p≤2. Here, we note that when 1<p<2, the solution to the equation may not have unique positive solution.Theorem 14.
For1<p≤2, the nonnegative solution to
(73)utx,t=Δp,ωux,t+λup-1x,t,x,t∈S×0,∞,ux,t=0,x,t∈∂S×0,∞,ux,0=u0x≥0
is global for every λ>0. In particular, when 1<p<2 and λ<λ0, then there exists T (extinction time) such that ux,t≡0 for t≥T.Proof.
First, consider ODE(74)ddtzt=λzp-1,t>0,z0=u¯0+1,
where u¯0=max⁡x∈S¯u0x>0. Then
(75)zt=2-pt+λz2-p01/2-p,1<p<2,λz0et,p=2.
Then by the comparison principle (Theorem 6), we see that
(76)0≤ux,t≤zt,x∈S,t≥0,
which implies that ux,t must be global.
Now, assume1<p<2 and λ<λ0. When u is a trivial solution, then we are done. So now we assume that u is nontrivial. Multiplying (73) by u, as done in (40), we have
(77)12ddt∑x∈S¯‍u2x,t=-12∑x,y∈S¯‍ux,t-uy,tpωx,y+λ∑x∈S¯‍upx,t≤-λ0-λ∑x∈S¯‍upx,t≤-λ0-λ∑x∈S¯‍u2x,tp/2,
where the last inequality follows from the elementary inequality
(78)∑j=1n‍tjp1/p≥∑j=1n‍tj21/2,1<p<2
for t1,t2,…,tn≥0. Now suppose yt∶=∑x∈Su2x,t>0 for t≥0, on the contrary. Since ux,t=0, x∈∂S, t≥0, we obtain
(79)y′t≤-2λ0-λyp/2,t>0,y0=∑x∈S‍u02x>0.
Integrating (79) over [0,t], we obtain
(80)0<y1-p/2t≤y1-p/20-2-pλ0-λt,
which is absurd for large t>0. Hence, yT=0 for some T>0 and, therefore, we can conclude that yt=0, t>T, for some T>0, since y′(t)≤0 in inequality (79).Remark 15.
(i) Whenp=2 in the above, we do not need the assumption that u is nonnegative, which follows automatically from the comparison principle (Theorem 4).
(ii) In fact, inequality (80) gives us the extinction time T, estimated by
(81)0<T≤∑x∈Su02x1-p/22-pλ0-λ.


## 4. Examples and Numerical Illustrations

In this section, we show numerical illustrations to exploit our results in the previous section.Now, consider a graphS={x1,x2,x3} with the boundary ∂S={x4,x5} and the weight
(82)ω(x1,x2)=ω(x2,x3)=0.05,ω(x1,x3)=0.1ω(x1,x4)=ω(x2,x5)=ω(x3,x5)=1
as in Figure 1.Figure 1
GraphS¯.Example 1.
For (36) on (Figure 1) graph S¯ with p=3, λ=1.1, consider initial data given by u0(x1)=20, u0(x2)=30, and u0(x3)=10. Then by easy calculation, we get the first eigenvalue λ0=1<λ=1.1 and the corresponding eigenfunction ϕ0(x1)=ϕ0(x2)=ϕ0(x3)≒0.6934. Figure 2 shows that the solution to (36) blows up and the computed blow-up time T is estimated as T≒0.4433079.
On the other hand, consider the same equation (36) with p=3, λ=0.9 and the same initial data. Then λ=0.9<λ0=1 and Figure 3 shows that the solution to (36) is global.Figure 2
Behavior of each node forp=3 and λ=1.1.Figure 3
Behavior of each node forp=3 and λ=0.9.Example 2.
For (36) on (Figure 1) graph S¯ with p=1.5, λ=0.5, and the same initial data u0 given by u0(x1)=20, u0(x2)=30, and u0(x3)=10, then Figure 4 shows that the solution to (36) is global and extinctive.Figure 4
Behavior of each node forp=1.5.



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*Source: 716327-2014-12-14.xml*