# Blow-Up Solutions and Global Solutions to Discretep-Laplacian Parabolic Equations

Authors: Soon-Yeong Chung; Min-Jun Choi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We discuss the conditions under which blow-up occurs for the solutions of discretep-Laplacian parabolic equations on networks S with boundary ∂
S as follows: u
t
(
x
,
t
)
=
Δ
p
,
ω
u
(
x
,
t
)
+
λ
|
u
(
x
,
t
)
|
q
-
1
u
(
x
,
t
), (
x
,
t
)
∈
S
×
(
0
,
+
∞
); u
(
x
,
t
)
=
0, (
x
,
t
)
∈
∂
S
×
(
0
,
+
∞
); u
(
x
,
0
)
=
u
0
≥
0
,
x
∈
S
¯, where p
>
1, q
>
0, λ
>
0, and the initial data u
0 is nontrivial on S. The main theorem states that the solution u  to the above equation satisfies the following: (i)  if  0
<
p
-
1
<
q and q
>
1, then  the  solution  blows  up  in  a  finite  time, provided u
¯
0
>
ω
0
/
λ
1
/
q
-
p
+
1, where ω
0
:
=
max
x
∈
S
⁡
∑
y
∈
S
¯
‍
ω
(
x
,
y
) and u
¯
0
:
=
max
x
∈
S
u
0
(
x
); (ii) if 0
<
q
≤
1, then the nonnegative solution  is  global; (iii) if 1
<
q
<
p
-
1, then the solution is global. In order to prove the main theorem, we first derive the comparison principles for the solution of the equation above, which play an important role throughout this paper. Moreover, when the solution blows up, we give an estimate for the blow-up time and also provide the blow-up rate. Finally, we give some numerical illustrations which exploit the main results.


---

## Body

## 1. Introduction

In this paper, we discuss the blow-up property and global existence of solutions to the following discretep-Laplacian parabolic equation:
(1)
u
t
x
,
t
=
Δ
p
,
w
u
x
,
t
+
λ
u
x
,
t
q
-
1
u
x
,
t
,
x
,
t
∈
S
×
0
,
+
∞
,
u
x
,
t
=
0
,
x
,
t
∈
∂
S
×
0
,
+
∞
,
u
x
,
0
=
u
0
≥
0
,
x
∈
S
¯
,
where q
>
0, p
>
1 and λ
>
0.The continuous case of this equation has been studied by many authors, assuming some conditionsq, p, and λ, in order to get a blow-up solution or global solution (see [1–5]). For example, they consider the case 1
<
p
<
2, q
>
0, and λ
>
0 in [2], the case q
>
p
-
1
>
1 in [5], the case p
=
q
>
2 in [1, 3], the case 1
<
p
<
2, q
>
0, and λ
>
0 in [4], respectively, and so on.On the other hand, the long time behavior (extinction and positivity) of solutions to evolutionp-Laplace equation with absorption on networks is studied in the paper [6, 7].The goal of this paper is to give a condition onp, q, and λ for the solution to (1) to be blow-up or global. In fact, we prove the following as one of the main theorems.Theorem 1.
Letu be a solution of (1). Then one has the following.(i)
If0
<
p
-
1
<
q and q
>
1, then the solution blows up in a finite time, provided u
¯
0
>
ω
0
/
λ
1
/
(
q
-
p
+
1
), where ω
0
:
=
max
⁡
x
∈
S
⁡
∑
y
∈
S
¯
ω
(
x
,
y
) and u
¯
0
:
=
max
⁡
x
∈
S
u
0
(
x
).
(ii)
If0
<
q
≤
1, then the nonnegative solution is global.
(iii)
If1
<
q
<
p
-
1, then the solution is global.In order to prove the above theorem, we give comparison principles for the solutions of (1) in Section 2. Moreover, when the solutions to (1) blow up, we derive the blow-up rate as follows:
(2)
λ
q
-
1
T
-
t
-
1
/
q
-
1
≤
max
⁡
x
∈
S
u
x
,
t
≤
q
-
1
T
-
t
2
q
-
p
/
q
-
1
λ
q
-
1
T
-
t
-
ω
0
λ
q
-
p
+
1
/
q
-
1
q
-
1
T
-
t
2
q
-
p
/
q
-
1
×
q
-
1
T
-
t
2
q
-
p
/
q
-
1
-
1
/
q
-
1
,
where ω
0
:
=
max
⁡
x
∈
S
∑
y
∈
S
¯
‍
ω
x
,
y, and as a consequence
(3)
lim
⁡
t
→
T
-
T
-
t
1
/
q
-
1
max
⁡
x
∈
S
¯
u
x
,
t
=
1
λ
q
-
1
1
/
q
-
1
.
We organized this paper as follows. In Section 2, we discuss the preliminary concepts on networks and the discrete version of comparison principles on networks. In Section 3, we are devoted to find out blow-up conditions of the solution and the blow-up rate with the blow-up time. Finally, in Section 4, we give some numerical illustrations to 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 [8–10], 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 are 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
=
x
0
,
x
1
,
…
,
x
n
-
1, x
n
=
y, such that x
j
-
1 and x
j 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 is adjacent to some vertex in S; that is,
(4)
∂
S
:
=
z
∈
V
∖
S
∣
z
~
y
for
some
y
∈
S
.
By S
¯, we denote a graph, whose vertices and edges are in both S and ∂
S.Throughout this paper, all subgraphsS and S
¯ in our concern are assumed to be simple and connected.For a functionu
:
S
¯
→
R, thediscrete p-Laplacian
Δ
p
,
ω on S is defined by
(5)
Δ
p
,
ω
u
(
x
)
:
=
∑
y
∈
S
¯
‍
u
y
-
u
x
p
-
2
u
y
-
u
x
ω
(
x
,
y
)
for x
∈
S.The rest of this section is devoted to prove the comparison principle for the discretep-Laplacian parabolic equation:
(6)
u
t
x
,
t
=
Δ
p
,
ω
u
x
,
t
+
λ
u
x
,
t
q
-
1
u
x
,
t
,
x
,
t
∈
S
×
0
,
+
∞
,
u
x
,
t
=
0
,
x
,
t
∈
∂
S
×
0
,
+
∞
,
u
x
,
0
=
u
0
≥
0
,
x
∈
S
¯
,
where λ
>
0, q
>
0, p
>
1, and the initial data u
0 is nontrivial on S, 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 2.
LetT
>
0 (T may be +
∞), λ
>
0, q
≥
1, and p
>
1. Suppose that real-valued functions u
(
x
,
·
)
,
v
(
x
,
·
)
∈
C
[
0
,
T
) are differentiable in (
0
,
T
) for each x
∈
S
¯ and satisfy
(7)
u
t
x
,
t
-
Δ
p
,
ω
u
x
,
t
-
λ
u
x
,
t
q
-
1
u
(
x
,
t
)
≥
v
t
x
,
t
-
Δ
p
,
ω
v
x
,
t
-
λ
v
x
,
t
q
-
1
v
(
x
,
t
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
(
x
,
t
)
∈
S
×
0
,
T
,
u
x
,
t
≥
v
x
,
t
,
(
x
,
t
)
∈
∂
S
×
0
,
T
,
u
x
,
0
≥
v
x
,
0
,
x
∈
S
¯
.
Then u
(
x
,
t
)
≥
v
(
x
,
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
′,
(8)
u
x
,
t
q
-
1
u
x
,
t
-
v
x
,
t
q
-
1
v
x
,
t
=
q
ξ
x
,
t
q
-
1
u
x
,
t
-
v
x
,
t
for some ξ
(
x
,
t
) lying between u
(
x
,
t
) and v
(
x
,
t
). Then it follows from (7) that we have
(9)
u
t
x
,
t
-
Δ
p
,
ω
u
x
,
t
-
λ
q
ξ
x
,
t
q
-
1
u
x
,
t
≥
v
t
x
,
t
-
Δ
p
,
ω
v
x
,
t
-
λ
q
ξ
x
,
t
q
-
1
v
x
,
t
for all (
x
,
t
)
∈
S
×
(
0
,
T
′
]. Let u
~
,
v
~
:
S
¯
×
[
0
,
T
′
]
→
R be the functions defined by
(10)
u
~
x
,
t
:
=
e
-
2
λ
q
L
t
u
x
,
t
,
v
~
x
,
t
:
=
e
-
2
λ
q
L
t
v
x
,
t
,
where L
:
=
max
⁡
r
≤
M
r
q
-
1 and M
:
=
max
⁡
x
∈
S
¯
,
t
∈
[
0
,
T
′
]
u
x
,
t
,
v
x
,
t.
Then inequality (9) can be written as
(11)
u
~
t
x
,
t
-
v
~
t
x
,
t
-
e
2
q
λ
L
p
-
2
t
Δ
p
,
ω
u
~
x
,
t
-
Δ
p
,
ω
v
~
x
,
t
+
λ
q
2
L
-
ξ
x
,
t
q
-
1
u
~
x
,
t
-
v
~
x
,
t
≥
0
for all (
x
,
t
)
∈
S
×
0
,
T
′. Since S
¯
×
[
0
,
T
′
] is compact, there exists (
x
0
,
t
0
)
∈
S
¯
×
[
0
,
T
′
] such that
(12)
(
u
~
-
v
~
)
(
x
0
,
t
0
)
=
min
⁡
x
∈
S
¯
min
⁡
0
≤
t
≤
T
′
(
u
~
-
v
~
)
x
,
t
.
Then we have only to show that (
u
~
-
v
~
)
(
x
0
,
t
0
)
≥
0. Suppose that (
u
~
-
v
~
)
(
x
0
,
t
0
)
<
0, on the contrary. Since (
u
~
-
v
~
)
(
x
,
t
)
≥
0 on both ∂
S
×
[
0
,
T
′
] and S
¯
×
0, we have (
x
0
,
t
0
)
∈
S
×
0
,
T
′. Then we have
(13)
u
~
t
x
0
,
t
0
≤
v
~
t
x
0
,
t
0
,
Δ
p
,
ω
u
~
x
0
,
t
0
≥
Δ
p
,
ω
v
~
x
0
,
t
0
.
Since ξ
q
-
1
x
,
t
≤
max
⁡
r
≤
M
r
q
-
1
=
L, we have
(14)
2
L
-
ξ
x
0
,
t
0
q
-
1
u
~
-
v
~
x
0
,
t
0
≤
L
u
~
-
v
~
x
0
,
t
0
<
0
.
Combining (13) and (14), we obtain
(15)
u
~
t
x
0
,
t
0
-
v
~
t
x
0
,
t
0
-
e
2
q
λ
L
p
-
2
t
0
[
Δ
p
,
ω
u
~
x
0
,
t
0
-
Δ
p
,
ω
v
~
x
0
,
t
0
]
+
λ
q
2
L
-
ξ
x
0
,
t
0
q
-
1
[
u
~
x
0
,
t
0
-
v
~
x
0
,
t
0
]
<
0
which contradicts (11). Therefore, u
~
x
,
t
-
v
~
x
,
t
≥
0 for all (
x
,
t
)
∈
S
×
(
0
,
T
′
] so that we get u
x
,
t
≥
v
x
,
t for all (
x
,
t
)
∈
S
¯
×
0
,
T, since T
′
<
T is arbitrarily given.Whenp
≥
2, we obtain a strict comparison principle as follows.Corollary 3 (strict comparison principle).
LetT
>
0 (T may be +
∞), λ
>
0, q
≥
1, 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
(16)
u
t
x
,
t
-
Δ
p
,
ω
u
x
,
t
-
λ
u
x
,
t
q
-
1
u
(
x
,
t
)
≥
v
t
x
,
t
-
Δ
p
,
ω
v
x
,
t
-
λ
v
x
,
t
q
-
1
v
(
x
,
t
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
(
x
,
t
)
∈
S
×
0
,
T
,
u
x
,
t
≥
v
x
,
t
,
(
x
,
t
)
∈
∂
S
×
0
,
T
,
u
x
,
0
≥
v
x
,
0
,
x
∈
S
¯
.
If u
0
(
x
*
)
>
v
0
(
x
*
) for some x
*
∈
S, then u
x
,
t
>
v
x
,
t for all (
x
,
t
)
∈
S
×
(
0
,
T
).Proof.
First, note thatu
≥
v on S
¯
×
0
,
T by Theorem 2. Let T
′
>
0 be arbitrarily given with T
′
<
T and let τ
:
S
¯
×
[
0
,
T
′
]
→
R be a function defined by
(17)
τ
x
,
t
:
=
u
x
,
t
-
v
x
,
t
,
x
,
t
∈
S
¯
×
0
,
T
′
.
Then τ
(
x
,
t
)
≥
0 for all (
x
,
t
)
∈
S
¯
×
[
0
,
T
′
]. Since τ
(
x
*
,
0
)
>
0 and u
x
*
,
t
q
-
1
u
(
x
*
,
t
)
≥
v
x
*
,
t
q
-
1
v
(
x
*
,
t
) for all 0
<
t
≤
T
′, we obtain from inequality (16) that
(18)
τ
t
x
*
,
t
-
Δ
p
,
ω
u
x
*
,
t
-
Δ
p
,
ω
v
x
*
,
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
(19)
u
y
,
t
-
u
x
*
,
t
p
-
2
u
y
,
t
-
u
x
*
,
t
-
v
y
,
t
-
v
x
*
,
t
p
-
2
v
y
,
t
-
v
x
*
,
t
=
p
-
1
η
x
*
,
y
,
t
p
-
2
τ
y
,
t
-
τ
x
*
,
t
,
and η
x
*
,
y
,
t
<
2
M, where M
:
=
max
⁡
0
<
t
≤
T
′
u
x
*
,
t
,
v
x
*
,
t.
Then inequality (18) gives
(20)
τ
t
x
*
,
t
≥
∑
y
∈
S
¯
‍
p
-
1
η
x
*
,
y
,
t
p
-
2
τ
y
,
t
-
τ
x
*
,
t
ω
x
*
,
y
≥
-
d
p
-
1
2
M
p
-
2
τ
x
*
,
t
,
where d
=
∑
y
∈
S
¯
‍
ω
x
*
,
y. This implies that
(21)
τ
x
*
,
t
≥
τ
x
*
,
0
e
-
d
p
-
1
2
M
p
-
2
t
>
0
,
t
∈
(
0
,
T
′
]
.
Now, suppose that there exists (
x
0
,
t
0
)
∈
S
×
0
,
T
′ such that
(22)
τ
x
0
,
t
0
=
min
⁡
x
∈
S
,
0
<
t
≤
T
′
⁡
τ
x
,
t
=
0
.
Then
(23)
τ
t
x
0
,
t
0
≤
0
,
Δ
p
,
ω
u
x
0
,
t
0
≥
Δ
p
,
ω
v
x
0
,
t
0
.
Hence, inequality (18) gives
(24)
0
≤
τ
t
x
0
,
t
0
-
Δ
p
,
ω
u
x
0
,
t
0
-
Δ
p
,
ω
v
x
0
,
t
0
≤
0
.
Therefore,
(25)
Δ
p
,
ω
u
x
0
,
t
0
=
Δ
p
,
ω
v
x
0
,
t
0
;
that is,
(26)
∑
y
∈
S
¯
‍
u
y
,
t
0
-
u
x
0
,
t
0
p
-
2
u
y
,
t
0
-
u
x
0
,
t
0
iiiiiiiiiii
-
v
y
,
t
0
-
v
x
0
,
t
0
p
-
2
i
i
i
i
i
i
i
i
i
i
×
u
y
,
t
0
-
u
x
0
,
t
0
p
-
2
v
y
,
t
0
-
v
x
0
,
t
0
ω
x
0
,
y
=
0
,
which implies that τ
y
,
t
0
=
0 for all y
∈
S
¯ with y
~
x
0. Now, for any x
∈
S
¯, there exists a path:
(27)
x
0
~
x
1
~
⋯
~
x
n
-
1

~
x
n
=
x
,
since S
¯ is connected. By applying the same argument as above inductively, we see that τ
(
x
,
t
0
)
=
0 for every x
∈
S
¯. This gives a contradiction to (21).For the case0
<
q
<
1, it is well known that (6) may not have unique solution, in general, and the comparison principle in usual form as in Theorem 2 may not hold. Instead, with a strict condition on the parabolic boundary, we obtain a similar comparison principle as follows.Theorem 4.
LetT
>
0 (T may be +
∞), λ
>
0, q
>
0, and p
>
1. Suppose that real-valued functions u
x
,
·
,
v
x
,
·
∈
C
0
,
T are differentiable in 0
,
T for each x
∈
S
¯ and satisfy
(28)
u
t
x
,
t
-
Δ
p
,
ω
u
x
,
t
-
λ
u
x
,
t
q
-
1
u
(
x
,
t
)
≥
v
t
x
,
t
-
Δ
p
,
ω
v
x
,
t
-
λ
v
x
,
t
q
-
1
v
(
x
,
t
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
(
x
,
t
)
∈
S
×
0
,
T
,
u
x
,
t
>
v
x
,
t
,
(
x
,
t
)
∈
∂
S
×
0
,
T
,
u
x
,
0
>
v
x
,
0
,
x
∈
S
¯
.
Then u
x
,
t
≥
v
x
,
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
=
0
or
x
∈
∂
S
} (called a parabolic boundary).
Now, let a functionτ
:
S
¯
×
(
0
,
T
′
]
→
R be a function defined by
(29)
τ
x
,
t
:
=
u
x
,
t
-
v
x
,
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 (
x
0
,
t
0
)
∈
S
×
0
,
T
′ such that(i)
τ
(
x
0
,
t
0
)
=
0,
(ii)
τ
y
,
t
0
≥
τ
x
0
,
t
0
=
0, y
∈
S,
(iii)
τ
(
x
,
t
)
>
0, (
x
,
t
)
∈
S
×
(
0
,
t
0
).
Then(30)
τ
t
(
x
0
,
t
0
)
≤
0
and
(31)
Δ
p
,
ω
u
x
0
,
t
0
≥
Δ
p
,
ω
v
x
0
,
t
0
,
since
(32)
u
y
,
t
0
-
u
x
0
,
t
0
≥
v
y
,
t
0
-
v
x
0
,
t
0
.
Hence, (28) gives
(33)
0
≥
τ
t
x
0
,
t
0
≥
λ
u
x
0
,
t
0
q
-
1
u
x
0
,
t
0
-
v
x
0
,
t
0
q
-
1
v
x
0
,
t
0
=
λ
v
x
0
,
t
0
+
δ
q
-
1
v
x
0
,
t
0
+
δ
I
I
I
I
I
I
I
-
v
x
0
,
t
0
q
-
1
v
x
0
,
t
0
>
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.Using the same method as in Corollary3, we obtain a strict comparison principle as follows.Corollary 5 (strict comparison principle).
LetT
>
0 (T may be +
∞), λ
>
0, q
>
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
(34)
u
t
x
,
t
-
Δ
p
,
ω
u
x
,
t
-
λ
u
x
,
t
q
-
1
u
(
x
,
t
)
≥
v
t
x
,
t
-
Δ
p
,
ω
v
x
,
t
-
λ
v
x
,
t
q
-
1
v
(
x
,
t
)
,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
(
x
,
t
)
∈
S
×
0
,
T
,
u
x
,
t
>
v
x
,
t
,
x
,
t
∈
∂
S
×
0
,
T
,
u
x
,
0
>
v
x
,
0
,
x
∈
S
¯
.
If u
0
(
x
*
)
>
v
0
(
x
*
) for some x
*
∈
S, then u
x
,
t
>
v
x
,
t for all (
x
,
t
)
∈
S
×
0
,
T.


## 3. Blow-Up and Blow-Up Estimates

In this section, we discuss the blow-up phenomena of the solutions to discrete reaction-diffusion equation defined on networks, which is a main part of this paper.We first introduce the concept of the blow-up as follows.Definition 6 (blow-up).
One says 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
-.According to the comparison principle in the previous section, we are guaranteed to get a solution to(35)
u
t
x
,
t
=
Δ
p
,
ω
u
x
,
t
+
λ
u
q
x
,
t
,
x
,
t
∈
S
×
0
,
∞
,
u
x
,
t
=
0
,
x
,
t
∈
∂
S
×
0
,
∞
,
u
x
,
0
=
u
0
≥
0
,
x
∈
S
¯
,
when p
>
1, q
>
0, λ
>
0, and the initial data u
0 is nontrivial on S.We now state the main theorem of this paper as follows.Theorem 7.
Letu be a solution of (35). Then one has the following.(i)
If0
<
p
-
1
<
q and q
>
1, then the solution blows up in a finite time, provided u
¯
0
>
ω
0
/
λ
1
/
(
q
-
p
+
1
), where ω
0
:
=
max
⁡
x
∈
S
⁡
∑
y
∈
S
¯
ω
(
x
,
y
) and u
¯
0
:
=
max
⁡
x
∈
S
u
0
(
x
).
(ii)
If0
<
q
≤
1, then the nonnegative solution is global.
(iii)
If1
<
q
<
p
-
1, then the solution is global.Proof.
First, we prove (i). We note thatu
(
x
,
t
)
≥
0, for all x
,
t
∈
S
¯
×
0
,
+
∞, by Theorem 2. Assume that 0
<
p
-
1
<
q, q
>
1, and u
¯
0
>
ω
0
/
λ
1
/
(
q
-
p
+
1
), where u
¯
0
:
=
max
⁡
x
∈
S
u
0
(
x
). For each t
>
0, let x
t
∈
S be a node such that u
x
t
,
t
:
=
max
⁡
x
∈
S
u
x
,
t. In fact, we note that max
⁡
x
∈
S
u
x
,
t is differentiable, for almost all t
>
0. Then (35) can be written as follows:
(36)
u
t
x
s
,
s
=
Δ
p
,
ω
u
x
s
,
s
+
λ
u
q
x
s
,
s
=
∑
y
∈
S
¯
‍
u
y
,
s
-
u
x
,
s
p
-
2
×
u
y
,
s
-
u
x
s
,
s
ω
x
s
,
y
+
λ
u
q
x
s
,
s
≥
-
ω
0
u
p
-
1
x
s
,
s
+
λ
u
q
x
s
,
s
for almost all s
>
0. We need to show that max
⁡
x
∈
S
u
x
,
t
>
u
¯
0, for all t
>
0. Since u
(
x
,
t
)
≥
0 on S
×
(
0
,
∞
) and
(37)
lim
⁡
s
→
0
+
u
t
x
,
t
=
lim
⁡
s
→
0
+
∑
y
∈
S
¯
‍
u
y
,
s
-
u
x
s
,
s
p
-
2
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
×
u
y
,
s
-
u
x
s
,
s
ω
x
s
,
y
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
∑
y
∈
S
¯
‍
+
λ
u
q
x
s
,
s
≥
lim
⁡
s
→
0
+
-
ω
0
u
p
-
1
x
s
,
s
+
λ
u
q
x
s
,
s
=
-
ω
0
u
¯
0
p
-
1
+
λ
u
¯
0
q
>
0
,
u
(
x
s
,
s
) is increasing in some interval (
0
,
s
1
). Suppose that there exists s
>
0 somewhere at which u
(
x
s
,
s
)
≤
u
¯
0. Then now take the interval (
0
,
s
1
) to be maximal on which u
(
x
s
,
s
)
>
u
¯
0, s
∈
(
0
,
s
1
), and u
(
x
s
,
s
)
=
u
¯
0. Then there exists s
*
∈
(
0
,
s
1
) such that u
t
(
x
s
*
,
s
*
)
<
0 but
(38)
0
>
u
t
x
s
*
,
s
*
=
∑
y
∈
S
¯
‍
u
y
,
s
*
-
u
x
s
*
,
s
*
p
-
2
×
u
y
,
s
*
-
u
x
s
*
,
s
*
ω
x
s
*
,
s
*
+
λ
u
q
x
s
*
,
s
*
≥
-
ω
0
u
p
-
1
x
s
*
,
s
*
+
λ
u
q
x
s
*
,
s
*
>
0
,
which leads to a contradiction. Thus it follows that u
(
x
s
,
s
)
>
u
¯
0, s
∈
(
0
,
+
∞
).
LetF
:
[
u
¯
0
,
+
∞
)
→
(
0
,
F
u
¯
0
] be a function defined by
(39)
F
y
:
=
∫
y
+
∞
‍
d
s
-
ω
0
s
p
-
1
+
λ
s
q
<
+
∞
,
y
≥
u
¯
0
.
We note that-
ω
0
s
p
-
1
+
λ
s
q
>
0, for s
≥
u
¯
0, since u
¯
0
>
ω
0
/
λ
1
/
(
q
-
p
+
1
).
ThenF is a decreasing continuous function from u
¯
0
,
+
∞ onto 0
,
F
u
¯
0 with its inverse function G. Integrating (36) from 0 to t, we have
(40)
t
≤
∫
0
t
‍
u
t
x
s
,
s
-
ω
0
u
p
-
1
x
s
,
s
+
λ
u
q
x
s
,
s
d
u
=
∫
u
¯
0
u
x
t
,
t
d
s
-
ω
0
s
p
-
1
+
λ
s
q
.
This can be written as
(41)
F
u
x
t
,
t
≤
F
u
¯
0
-
t
and, equivalently,
(42)
u
x
t
,
t
≥
G
F
u
¯
0
-
t
,
which implies that u
x
t
,
t blows up, as t
→
F
u
¯
0.Secondly, we prove (ii). Consider the following ODE problem:(43)
d
d
t
z
t
=
λ
z
q
t
,
t
>
0
,
z
0
=
u
¯
0
+
1
.
Then, we have
(44)
z
t
=
1
-
q
λ
t
+
z
1
-
q
0
1
/
1
-
q
,
q
≠
1
,
z
t
=
z
0
e
λ
t
,
q
=
1
,
for every t
≥
0.Takev
x
,
t
:
=
z
t, for all x
∈
S
¯ and t
≥
0. Then it is easy to see that v
(
x
,
t
)
>
u
(
x
,
t
), (
x
,
t
)
∈
∂
S
×
(
0
,
+
∞
), v
(
x
,
0
)
=
z
(
0
)
>
u
¯
0, x
∈
S
¯, and
(45)
v
t
x
,
t
-
Δ
p
,
ω
v
x
,
t
-
λ
v
q
x
,
t
=
d
d
t
z
t
-
λ
z
q
t
=
0
.
Thus, 0
≤
u
(
x
,
t
)
≤
v
(
x
,
t
)
=
z
(
t
) for every (
x
,
t
)
∈
S
¯
×
(
0
,
+
∞
) by Theorem 4. This implies that u must be global.Finally, we prove (iii). Consider the following eigenvalue problem:(46)
-
Δ
p
,
ω
ϕ
x
=
λ
1
ϕ
x
p
-
2
ϕ
x
,
x
∈
S
,
ϕ
x
=
0
,
x
∈
∂
S
.
Note that it is well known that λ
1
>
0 and ϕ
x
>
0, for all x
∈
S (see [11, 12]).Now, takev
(
x
,
t
)
:
=
k
ϕ
(
x
), x
∈
S
¯, t
≥
0. Choosing k
>
0 so large that k
ϕ
(
x
)
>
u
¯
0 and k
ϕ
(
x
)
>
λ
/
λ
1
1
/
(
p
-
1
-
q
), then we see that v
(
x
,
0
)
=
k
ϕ
(
x
)
≥
u
0
(
x
)
=
u
(
x
,
0
), x
∈
S, and
(47)
v
t
x
,
t
-
Δ
p
,
ω
v
x
,
t
-
λ
v
q
x
,
t
=
λ
1
k
ϕ
x
p
-
1
-
λ
k
ϕ
x
q
≥
0
.
Therefore, 0
≤
u
(
x
,
t
)
≤
v
(
x
,
t
)
=
k
ϕ
(
x
) for every (
x
,
t
)
∈
S
¯
×
(
0
,
+
∞
) by Theorems 2 and 4, which is required.Remark 8.
(i) When the solution blows up in the above, the blow-up timeT can be estimated as
(48)
u
¯
0
1
-
q
λ
(
q
-
1
)
≤
T
≤
F
(
u
¯
0
)
:
=
∫
u
¯
0
+
∞
‍
d
s
-
ω
0
s
p
-
1
+
λ
s
q
.
In fact, the first inequality is derived as follows. By the definition of maximum function u
(
x
t
,
t
), (35) gives
(49)
u
t
x
s
,
s
≤
λ
u
q
x
s
,
s
,
for almost all s
>
0. Then integrating both sides, we have
(50)
t
≥
∫
0
t
‍
u
t
x
s
,
s
λ
u
q
x
s
,
s
d
s
=
∫
u
¯
0
u
x
t
,
t
‍
d
s
λ
s
q
,
so that we obtain T
≥
∫
u
¯
0
+
∞
‍
d
s
/
λ
s
q
=
u
¯
0
1
-
q
/
λ
(
q
-
1
), by taking the limit as t
→
T
-.
(ii) In the above, ifu
¯
0
:
=
max
⁡
x
∈
S
u
0
(
x
) is not sufficiently large, then the solution may be global. This can be seen in the numerical examples in Section 4.
(iii) In the above, the case where1
<
p
-
1
=
q was not discussed. As a matter of fact, the solution to (35) in this case may blow up or not, depending on the magnitude of the parameter λ. Each case is illustrated in Section 4. A full argument will be discussed in a forthcoming paper.We now derive the lower bound, the upper bound, and the blow-up rate for the maximum function of blow-up solutions.Theorem 9.
Letu be a solution of (35), which blows up at a finite time T, q
>
p
-
1
>
0, and q
>
1. Then one has the following.(i)
The lower bound is(51)
max
⁡
x
∈
S
u
(
x
,
t
)
≥
λ
(
q
-
1
)
(
T
-
t
)
-
1
/
(
q
-
1
)
,
0
<
t
<
T
.
(ii)
The upper bound is(52)
max
⁡
x
∈
S
u
x
,
t
≤
λ
(
q
-
1
)
(
T
-
t
)
-
α
(
T
-
t
)
(
2
q
-
p
)
/
(
q
-
1
)
-
1
/
(
q
-
1
)
,
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
0
<
t
<
T
,
where α
:
=
ω
0
λ
(
q
-
p
+
1
)
/
(
q
-
1
)
(
q
-
1
)
(
2
q
-
p
)
/
(
q
-
1
) and ω
0
=
max
⁡
x
∈
S
∑
y
∈
S
¯
‍
ω
x
,
y.
(iii)
The blow-up rate is(53)
lim
⁡
t
→
T
-
T
-
t
1
/
q
-
1
max
⁡
x
∈
S
¯
u
x
,
t
=
1
λ
q
-
1
1
/
q
-
1
,
0
<
t
<
T
.Proof.
First, we prove (i). As in the previous theorem, letx
t
∈
S be a node such that u
x
t
,
t
:
=
max
⁡
x
∈
S
u
x
,
t, for each t
>
0. Then it follows from (35) that
(54)
u
t
x
s
,
s
≤
λ
u
q
x
s
,
s
,
for almost alls
>
0. Then integrating from t to T, we get
(55)
λ
T
-
t
≥
∫
t
T
‍
u
t
x
s
,
s
u
q
x
s
,
s
d
s
=
∫
u
x
t
,
t
+
∞
‍
d
s
s
q
=
1
q
-
1
u
1
-
q
(
x
t
,
t
)
.
Hence, we obtain
(56)
u
(
x
t
,
t
)
≥
λ
(
q
-
1
)
(
T
-
t
)
-
1
/
(
q
-
1
)
,
0
<
t
<
T
.
Next, we prove (ii). Since the solution u is positive, we get
(57)
u
t
x
s
,
s
≥
-
∑
y
∈
S
¯
‍
u
p
-
1
x
s
,
s
ω
x
s
,
y
+
λ
u
q
x
s
,
s
≥
-
ω
0
u
p
-
1
x
s
,
s
+
λ
u
q
x
s
,
s
=
u
q
x
s
,
s
λ
-
ω
0
u
p
-
1
-
q
x
s
,
s
,
for almost all s
>
0 and ω
0
=
max
⁡
x
∈
S
∑
y
∈
S
¯
‍
ω
x
,
y. Then, it follows from (i) (lower bound) that we have
(58)
u
t
(
x
s
,
s
)
≥
u
q
x
s
,
s
λ
-
ω
0
λ
(
q
-
1
)
(
T
-
t
)
(
q
-
p
+
1
)
/
(
q
-
1
)
.
Integrating from t to T, we get
(59)
u
x
t
,
t
≤
λ
(
q
-
1
)
(
T
-
t
)
-
α
T
-
t
(
2
q
-
p
)
/
(
q
-
1
)
-
1
/
(
q
-
1
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
0
<
t
<
T
,
where α
:
=
ω
0
λ
(
q
-
p
+
1
)
/
(
q
-
1
)
q
-
1
(
2
q
-
p
)
/
(
q
-
1
).
Finally, (iii) can be easily obtained by (i) and (ii).


## 4. Examples and Numerical Illustrations

In this section, we show numerical illustrations to exploit our results in the previous section.Now, consider a graphS
=
{
x
1
,
…
,
x
29
} with the boundary ∂
S
=
{
x
30
,
x
31
} and the weight
(60)
ω
x
i
,
x
j
=
0.05
,
j
=
i
+
1
,
0.1
,
j
=
i
+
2
,
0.05
,
i
=
1
,
j
=
30
,
i
=
29
,
j
=
31
,
0
,
otherwise
,
where i
=
1
,
…
,
28 (see Figure 1). Then, we note that ω
0
:
=
max
⁡
x
∈
S
∑
y
∈
S
¯
‍
ω
(
x
,
y
)
=
0.3.Figure 1
GraphS
¯.Example 1 (1
<
p
-
1
<
q).
For the graphS
¯ (see Figure 1), consider q
=
3, p
=
2.5, λ
=
0.5, and the initial data u
0 given by Table 1.Table 1
Initial data ofu.
Nodei
u
0
(
x
i
)
1
0.7
2
0.9
3
1.1
4
1.3
5
1.5
6
1.3
7
1.1
8
0.9
9
0.7
10
0
11
0.7
12
0.9
13
1.1
14
1.3
15
1.5
16
1.3
17
1.1
18
0.9
19
0.7
20
0
21
0.7
22
0.9
23
1.1
24
1.3
25
1.5
26
1.3
27
1.1
28
0.9
29
0.7
30
0
31
0Then1
<
p
-
1
=
1.5
<
q
=
3 and max
⁡
x
∈
S
u
0
(
x
)
=
1.5
>
ω
0
/
λ
1
/
q
-
p
+
1
≒
0.71
1. Then Figure 2 shows that the solution to (35) blows up and the computed blow-up time T is estimated as T
≒
0.4617817 and
(61)
0.444
≒
∫
1.5
+
∞
‍
d
s
0.5
s
3
≤
T
≤
∫
1.5
+
∞
‍
d
s
-
0.3
s
1.5
+
0.5
s
3
≒
0.553
.Behavior of each node forq
=
3 and p
=
2.5.
(a)
(b)On the other hand, consider a small initial datau
0 given by Table 2.Table 2
Initial data ofu.
Nodei
u
0
(
x
i
)
1
0.002
2
0.004
3
0.006
4
0.008
5
0.01
6
0.008
7
0.006
8
0.004
9
0.002
10
0
11
0.002
12
0.004
13
0.006
14
0.008
15
0.01
16
0.008
17
0.006
18
0.004
19
0.002
20
0
21
0.002
22
0.004
23
0.006
24
0.008
25
0.01
26
0.008
27
0.006
28
0.004
29
0.002
30
0
31
0Thenmax
⁡
x
∈
S
u
0
(
x
)
=
0.01
≯
ω
0
/
λ
1
/
(
q
-
p
+
1
)
≒
0.711 and Figure 3 shows that the solution to (35) is global.Behavior of each node forq
=
3 and p
=
2.5.
(a)
(b)Example 2 (0
<
p
-
1
<
1
<
q).
For the graphS
¯ (see Figure 1), consider q
=
3, p
=
1.5, λ
=
0.1, and the initial data u
0 given by Table 3.Table 3
Initial data ofu.
Nodei
u
0
(
x
i
)
1
1
2
1.5
3
2
4
2.5
5
3
6
2.5
7
2
8
1.5
9
1
10
0
11
1
12
1.5
13
2
14
2.5
15
3
16
2.5
17
2
18
1.5
19
1
20
0
21
1
22
1.5
23
2
24
2.5
25
3
26
2.5
27
2
28
1.5
29
1
30
0
31
0Then0
<
p
-
1
=
0.5
<
q
=
3 and max
⁡
x
∈
S
u
0
(
x
)
=
3
>
ω
0
/
λ
1
/
(
q
-
p
+
1
)
≒
1.55. Then Figure 4 shows that the solution to (35) blows up and the computed blow-up time T is estimated as T
≒
0.5864884 and
(62)
0.555
≒
∫
3
+
∞
‍
d
s
0.1
s
3
≤
T
≤
∫
3
+
∞
‍
d
s
-
0.3
s
p
-
1
+
0.1
s
q
≒
0.610
.Behavior of each node forq
=
3 and p
=
1.5.
(a)
(b)Example 3 (1
<
q
<
p
-
1).
For the graphS
¯ (see Figure 1), consider q
=
1.5, p
=
3, λ
=
0.1, and the initial data u
0 given by Table 3 in Example 2. Then 1
<
q
=
1.5
<
p
-
1
=
2 and Figure 5 shows that the solution to (35) is global.Behavior of each node forq
=
1.5 and p
=
3.
(a)
(b)Example 4 (0
<
q
≤
1).
For the graphS
¯ (see Figure 1), consider q
=
0.5, p
=
3, λ
=
0.1, and the initial data u
0 given by Table 3 in Example 2. Then 0
<
q
=
0.5
≤
1 and Figure 6 shows that the solution to (35) is global.Behavior of each node forq
=
0.5 and p
=
3.
(a)
(b)Example 5 (1
<
p
-
1
=
q).
For the graphS
¯ (see Figure 1), consider q
=
2, p
=
3, λ
=
2, and the initial data u
0 given by Table 3 in Example 2. Then 1
<
q
=
p
-
1
=
2 and Figure 7 shows that the solution to (35) blows up.
On the contrary, whenλ
=
0.00001, the solution to (35) is global, as seen in Figure 8.Behavior of each node forq
=
2 and p
=
3.
(a)
(b)Behavior of each node forq
=
2 and p
=
3.
(a)
(b)


## 5. Conclusion

We discuss the conditions under which blow-up occurs for the solutions of discretep-Laplacian parabolic equations on networks S with boundary ∂
S:
(63)
u
t
(
x
,
t
)
=
Δ
p
,
ω
u
(
x
,
t
)
+
λ
u
x
,
t
q
-
1
u
(
x
,
t
)
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
x
,
t
∈
S
×
0
,
+
∞
,
u
x
,
t
=
0
,
(
x
,
t
)
∈
∂
S
×
(
0
,
+
∞
)
,
u
(
x
,
0
)
=
u
0
≥
0
,
x
∈
S
¯
,
where p
>
1, q
>
0, λ
>
0, and the initial data u
0 is nontrivial on S.The main theorem states that the solutionu to the above equation satisfies the following:(i)
if0
<
p
-
1
<
q and q
>
1, then the solution blows up in a finite time, provided u
¯
0
>
ω
0
/
λ
1
/
(
q
-
p
+
1
), where ω
0
:
=
max
⁡
x
∈
S
⁡
∑
y
∈
S
¯
ω
(
x
,
y
) and u
¯
0
:
=
max
⁡
x
∈
S
⁡
u
0
(
x
);
(ii)
if0
<
q
≤
1, then the nonnegative solution is global;
(iii)
if1
<
q
<
p
-
1, then the solution is global. In addition, we give an estimate for the blow-up time and the blow-up rate for the blow-up solution. Finally, we give some numerical illustrations which exploit the main results.



---

*Source: 351675-2014-11-24.xml*