# Blow-up solutions and global solutions to discrete p-Laplacian parabolic equations.

1. Introduction

In this paper, we discuss the blow-up property and global existence of solutions to the following discrete p-Laplacian parabolic equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where q > 0, p > 1 and [lambda] > 0.

The continuous case of this equation has been studied by many authors, assuming some conditions q, p, and [lambda], 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 [lambda] > 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 [lambda] > 0 in [4], respectively, and so on.

On the other hand, the long time behavior (extinction and positivity) of solutions to evolution p-Laplace equation with absorption on networks is studied in the paper [6, 7].

The goal of this paper is to give a condition on p, q, and [lambda] 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. Let u be a solution of (1). Then one has the following.

(i) If 0 < p - 1 < q and q > 1, then the solution blows up in a finite time, provided [[bar.u].sub.0] > ([[omega].sub.0]/[lambda])].sup.1/(q-p+1)], where [[omega].sub.0] := [max.sub.x[member of]S] [[summation].sub.y[member of][bar.S]][omega] (x, y) and [[bar.u].sub.0] := [max.sub.x[member of]S][u.sub.0](x).

(ii) If 0 < q [less than or equal to] 1, then the nonnegative solution is global.

(iii) If 1 < 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:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [[omega].sub.0] := [max.sub.x[member of]S] [[summation].sub.y[member of][bar.S]][omega](x, y), and as a consequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

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 graph G = G(V, E), we mean finite sets V of vertices (or nodes) with a set E of two-element subsets of V (whose elements are called edges). 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 [member of] V or x [member of] G the facts that x is a vertex in G.

A graph G is said to be simple if it has neither multiple edges nor loops, and G is said to be connected if, for every pair of vertices x and y, there exists a sequence (called a path) of vertices x = [x.sub.0], [x.sub.1], ..., [x.sub.n-1], [x.sub.n] = y, such that [x.sub.j-1] and [x.sub.j] are connected by an edge (called adjacent) for j = 1, ..., n.

A graph S = S(V', E') is said to be a subgraph of G(V, E), if V' [subset] V and E' [subset] E.

A weight on a graph G is a function [omega]: V x V [right arrow] [0,+[infinity]) satisfying

(i) [omega](x, x) = 0, x [member of] V,

(ii) [omega](x, y) = [omega](y, x) if x ~ y,

(iii) [omega](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 a weight graph or a network.

For a subgraph S of a graph G(V, E), the (vertex) boundary [[partial derivative]S of S is the set of all vertices z [member of] V\S but is adjacent to some vertex in S; that is,

[partial derivative]S := {z [member of] V\S | z ~ y for some y [member of] S}. (4)

By [bar.S], we denote a graph, whose vertices and edges are in both S and [partial derivative]S.

Throughout this paper, all subgraphs S and [bar.S] in our concern are assumed to be simple and connected.

For a function u : [bar.S] [right arrow] R, the discrete p-Laplacian [[DELTA].sub.p,[omega]] on S is defined by

[[DELTA].sub.p,[omega]]u(x) := [summation over (y[member of][bar.S])] [absolute value of u (y) - u (x)].sup.p-2] [u (y) -u(x)][omega] (x, y) (5)

for x [member of] S.

The rest of this section is devoted to prove the comparison principle for the discrete p-Laplacian parabolic equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where [lambda] > 0, q > 0, p > 1, and the initial data [u.sub.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. Let T > 0 (T may be +[infinity]), [lambda] > 0, q [greater than or equal to] 1, and p > 1. Suppose that real-valued functions u(x,.), v(x,.) [member of] C[0, T) are differentiable in (0, T) for each x [member of] [bar.S] and satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Then u(x, t) [greater than or equal to] v(x, t) for all (x, t) [member of] [bar.S] x [0, T).

Proof. Let T' > 0 be arbitrarily given with T' < T. Then, by the mean value theorem, for each x [member of] S and 0 [less than or equal to] t [less than or equal to] T',

[absolute value of u (x, t)].sup.q-1] u(x, t) - [[absolute value of v (x, t)].sup.q-1] v (x, t) = q[[absolute value of [xi](x,t)].sup.q-1] [u(x,t)- v(x,t)] (8)

for some [xi](x,t) lying between u(x,t) and v(x,t). Then it follows from (7) that we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

for all (x,t) [member of] S x (0,T']. Let u,v : [bar.S] x [0,T'] [right arrow] R be the functions defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then inequality (9) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

for all (x, t) [member of] S x (0, T']. Since [bar.S] x [0, T'] is compact, there exists ([x.sub.0], [t.sub.0]) [member of] [bar.S] x [0, T'] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Then we have only to show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], on the contrary. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on both [partial derivative]S x [0, T'] and [bar.S] x {0}, we have ([x.sub.0], [t.sub.0]) [member of] S x (0, T']. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Combining (13) and (14), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

which contradicts (11). Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all (x,t) [member of] S x (0,T'] so that we get u(x,t) [greater than or equal to] v(x, t) for all (x,t) [member of] S x[0,T), since T' < T is arbitrarily given.

When p [greater than or equal to] 2, we obtain a strict comparison principle as follows.

Corollary 3 (strict comparison principle). Let T > 0 (T may be +[infinity]), [lambda] > 0, q [greater than or equal to] 1, and p [greater than or equal to] 2. Suppose that real- valued functions u(x, *), v(x, *) [member of] C[0, T) are differentiable in (0,T) for each x [member of] [bar.S] and satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

If [u.sub.0]([x.sup.*]) > [v.sub.0]([x.sup.*]) for some [x.sup.*] [member of] S, then u(x,t) > v(x,t) for all (x, t) [member of] S x (0, T).

Proof. First, note that u [greater than or equal to] v on [bar.S] x [0,T) by Theorem 2. Let T' > 0 be arbitrarily given with T' < T and let [tau] : [bar.S] x [0, T'] [right arrow] R be a function defined by

[tau](x,t):=u(x,t)- v(x,t), (x,t) [member of] [bar.S] x [0, T']. (17)

Then [tau](x, t) [greater than or equal to] 0 for all (x, t) [member of] [bar.S] x [0, T']. Since [tau]([x.sup.*], 0) > 0 and [absolute value of u([x.sup.*],t)].sup.q-1] u([x.sup.*], t) [greater than or equal to] [[absolute value of v([x.sup.*, t)].sup.q-1]v([x.sup.*], t) for all 0 < t [less than or equal to] T', we obtain from inequality (16) that

[[tau].sub.t] ([x.sup.*], t) - [[[DELTA].sub.p,[omega]]u([x.sup.*], t) - [[DELTA].sub.p,[[omega]]v([x.sup.*], t)] [greater than or equal to] 0, (18)

for all 0 < t [less than or equal to] T'. Then, by the mean value theorem, for each y [member of] [bar.S] and t with 0 [less than or equal to] t [less than or equal to] T', it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

and [absolute value of [eta]([x.sup.*], y, t)] < 2M, where M := [max.sub.0<t[less than or equal to]T']{[absolute value of u([x.sup.*], t)], [absolute value of v([x.sup.*], t)]}.

Then inequality (18) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where d = [[summation].sub.y[member of][bar.s]][omega]([x.sup.*], y). This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Now, suppose that there exists ([x.sub.0], [t.sub.0]) [member of] S x (0, T'] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

Hence, inequality (18) gives

0 [less than or equal to] [[tau].sub.t] ([x.sub.0], [t.sub.0]) - [[DELTA].sub.p,[omega]]u([x.sub.0], [t.sub.0]) - [[DELTA].sub.p,[omega]]v ([x.sub.0], [t.sub.0])] [less than or equal to] 0. (24)

Therefore,

[[DELTA].sub.p,[omega]]u ([x.sub.0], [t.sub.0]) = [[DELTA].sub.p,[omega]]v([x.sub.0], [t.sub.0]); (25)

that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

which implies that [tau](y, [t.sub.0]) = 0 for all y [member of] [bar.S] with y ~ [x.sub.0]. Now, for any x [member of] [bar.S], there exists a path:

[x.sub.0] ~ [x.sub.1] ~ ... ~ [x.sub.n-1] ~ [x.sub.n] = x, (27)

since [bar.S] is connected. By applying the same argument as above inductively, we see that [tau](x, [t.sub.0]) = 0 for every x [member of] [bar.S]. This gives a contradiction to (21).

For the case 0 < 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. Let T > 0 (T may be +[infinity]), [lambda] > 0, q > 0, and p > 1. Suppose that real-valued functions u(x,.), v(x,.) [member of] C[0, T) are differentiable in (0, T) for each x [member of] [bar.S] and satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

Then u(x, t) [greater than or equal to] v(x, t) for all (x, t) [member of] S x (0, T).

Proof. Let T' > 0 and [delta] > 0 be arbitrarily given with T' < T and 0 < [delta] < [min.sub.(x,t)[member of][GAMMA]][u(x, t) - v(x, T)], respectively, where [GAMMA] := {(x, t) [member of] [bar.S] x [0,T']] t = 0 or x [member of] [partial derivative]S} (called a parabolic boundary).

Now, let a function [tau]: [bar.S] x (0,T'] [right arrow] R be a function defined by

[tau](x, t) := [u (x, t) - v (x, t)] -[delta], (x, t) [member of] [bar.S] x(0,T']. (29)

Then [tau](x, t) > 0 on [GAMMA]. Now, we suppose that [min.SUB.x[member of]S,0<t[less than or equal to]T'][tau]t(x, t) < 0. Then there exists ([x.sub.0], [t.sub.0]) [member of] S x (0, T'] such that

(i) [[tau]([x.sub.0], [t.sub.0]) = 0,

(ii) [tau](y,[t.sub.0]) [greater than or equal to] [tau]([x.sub.0], [t.sub.0]) = 0, y [member of] S,

(iii) [tau](x, t) > 0, (x, t) [member of] S x (0, [t.sub.0]).

Then

[[tau].sub.t]([x.sub.0], [t.sub.0]) [less than or equal to] 0 (30)

and

[[DELTA].sub.p,[omega]]u([x.sub.0],[t.sub.0])[greater than or equal to][[DELTA].sub.p,[omega]]v ([x.sub.0],[t.sub.0]), (31)

since

u(y, [t.sub.0]) -u([x.sub.0], [t.sub.0]) [greater than or equal to] v (y, [t.sub.0]) - v ([x.sub.0], [t.sub.0]). (32)

Hence, (28) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

which leads to a contradiction. Hence, [tau](x, t) [greater than or equal to] 0 for all (x,t) [member of] S x (0, T'] so that we have u(x,t) [greater than or equal to] v(x,t) for all (x, t) [member of] S x (0, T), since [delta] and T' are arbitrary.

Using the same method as in Corollary 3, we obtain a strict comparison principle as follows.

Corollary 5 (strict comparison principle). Let T > 0 (T may be +[infinity]), [lambda] > 0, q > 0, and p [greater than or equal to] 2. Suppose that real-valued functions u(x, *), v(x,*) [member of] C[0, T) are differentiable in (0,T) for each x [member of] [bar.S] and satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

If [u.sub.0]([x.sup.*]) > [v.sub.0] ([x.sup.*]) for some [x.sup.*] [member of] S, then u(x,t) > v(x,t) for all (x, t) [member of] S x (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 solution u to an equation defined on a network [bar.S] blows up in finite time T, if there exists x [member of] S such that [absolute value of u(x, t)] [right arrow] +[infinity] as t [??] [T.sup.-].

According to the comparison principle in the previous section, we are guaranteed to get a solution to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

when p > 1, q > 0, [lambda] > 0, and the initial data [u.sub.0] is nontrivial on S.

We now state the main theorem of this paper as follows.

Theorem 7. Let u be a solution of (35). Then one has the following.

(i) If 0 < p - 1 < q and q > 1, then the solution blows up in a finite time, provided [[bar.u].sub.0] > [([[omega].sub.0]/[lambda]).sup.1/q-p+1)], where [[omega].sub.0] := [max.sub.x[member of]S] [[summation].sub.y[member of][bar.S]][omega](x,y) and [[bar.u].sub.0] := [max.sub.x[member of]S][u.sub.0](x).

(ii) If 0 < q [less than or equal to] 1, then the nonnegative solution is global.

(iii) If 1 < q < p - 1, then the solution is global.

Proof. First, we prove (i). We note that u(x,t) [greater than or equal to] 0, for all (x,t) [member of] [bar.S] x [0,+[infinity]), by Theorem 2. Assume that 0 < p - 1 < q, q > 1, and [[bar.u].sub.0] > ([[omega].sub.0]/[lambda]).sup.1/(q-p+1)], where [[bar.u].sub.0] := [max.sub.x[member of]S][u.sub.0](x). For each t > 0, let [x.sub.t] [member of] S be a node such that u([x.sub.t], t) := [max.sub.x[member of]S]u(x, t). In fact, we note that [max.sub.x[member of]S]u(x, t) is differentiable, for almost all t > 0. Then (35) can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

for almost all s > 0. We need to show that [max.sub.x[member of]S]u(x, t) > [[bar.u].sub.0], for all t > 0. Since u(x, t) [greater than or equal to] 0 on S x (0, [infinity]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

u([x.sub.s],s) is increasing in some interval (0, [s.sub.1]). Suppose that there exists s > 0 somewhere at which u([x.sub.s],s) [less than or equal to] [[bar.u].sub.0]. Then now take the interval (0, [s.sub.1]) to be maximal on which u([x.sub.s], s) > [[bar.u].sub.0], s [member of] (0, [s.sub.1]), and u([x.sub.s],s) = [[bar.u].sub.0]. Then there exists [s.sup.*] [member of] (0, [s.sub.1]) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] but

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38)

which leads to a contradiction. Thus it follows that u([x.sub.s], s) > [[bar.u].sub.0], s [member of] (0, +[infinity]).

Let F : [[[bar.u].sub.0], +[infinity]) [right arrow] (0, F([[bar.u].sub.0])] be a function defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

We note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then F is a decreasing continuous function from [[bar.u].sub.0], +[infinity]) onto (0, F([[bar.u].sub.0])] with its inverse function G. Integrating (36) from 0 to t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

This can be written as

F(u([x.sub.t], t))[less than or equal to]F([[bar.u].sub.0])-t (41)

and, equivalently,

u ([x.sub.t], t) [greater than or equal to]G[F ([[bar.u].sub.0]) - t], (42)

which implies that u([x.sub.t],t) blows up, as t [right arrow] F([[bar.u].sub.0]).

Secondly, we prove (ii). Consider the following ODE problem:

d/dt z (t) = [lambda][z.sup.q] (t), t > 0, z (0) = [[bar.u].sub.0] + 1. (43)

Then, we have

z(t) = [[(1 - q) [lambda]t + [z.sup.1-q] (0)].sup.1/(1-q)], q [not member of] 1, z(t) = z (0) [e.sup.[lambda]t], q = 1, (44)

for every t [greater than or equal to] 0.

Take v(x, t) := z(t), for all x [member of] [bar.S] and t [greater than or equal to] 0. Then it is easy to see that v(x, t) > u(x, t), (x, t) [member of] [partial derivative]S x (0, +[infinity]), v(x, 0) = z(0) >[[bar.u].sub.0], x [member of] [bar.S], and

[v.sub.t](x, t) - [[DELTA].sub.p,[omega]]v(x, t) - [lambda][v.sup.q] (x, t) = d/dt z(t) - [lambda][z.sup.q](t) = 0. (45)

Thus, 0 [less than or equal to] u(x, t) [less than or equal to] v(x, t) = z(t) for every (x, t) [member of] [bar.S] x(0, +[infinity]) by Theorem 4. This implies that u must be global.

Finally, we prove (iii). Consider the following eigenvalue problem:

-[[DELTA].sub.p,[omega]][phi](x) = [[lambda].sub.1][[absolute value of [phi](x)].sup.p-1][phi](x), x [member of] S, [phi](x) = 0, x [member of [partial derivative]S. (46)

Note that it is well known that [[lambda].sub.1] > 0 and [phi](x) > 0, for all x [member of] S (see [11, 12]).

Now, take v(x,t) := k[phi](x), x [member of] [bar.S], t [greater than or equal to] 0. Choosing k > 0 so large that k[phi](x) > [[bar.u].sub.0] and k[phi](x) > ([lambda]/[[lambda].sub.1]).sup.1/(p-1-q)], then we see that v(x, 0) = k[phi](x) [greater than or equal to] [u.sub.0](x) = u(x, 0), x [member of] S, and

[v.sub.t](x, t) - [[DELTA]sub.p,[omega]]v(x, t) - [lambda][v.sup.q] (x, t) = [[lambda].sub.1] [(k[phi] (x)).sup.p - 1] - [lambda][(k[phi] (x)).sup.q] [greater than or equal to] 0. (47)

Therefore, 0 [less than or equal to] u(x, t) [less than or equal to] v(x, t) = k[phi](x) for every (x, t) [member of] [bar.S] x (0, +[infinity]) by Theorems 2 and 4, which is required.

Remark 8. (i) When the solution blows up in the above, the blow-up time T can be estimated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (48)

In fact, the first inequality is derived as follows. By the definition of maximum function u([x.sub.t], t), (35) gives

[u.sub.t] ([x.sub.s], s) [less than or equal to] [lambda][u.sup.q] ([x.sub.s], s), (49)

for almost all s > 0. Then integrating both sides, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (50)

so that we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by taking the limit as t [right arrow] [T.sup.-].

(ii) In the above, if [[bar.u].sub.0] := [max.sub.x[member of]S][u.sub.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 where 1 < 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 [lambda]. 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. Let u 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (51)

(ii) The upper bound is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (52)

where [alpha] := [[omega].sub.0][[lambda].sup.(q-p+1)/(q-1)][(q - 1).sup.(2q-p)/(q-1)] and [[omega].sub.0] = [max.sub.x[member of]S] [[summation].sub.y[member of][bar.S]][omega](x,y)].

(iii) The blow-up rate is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (53)

Proof. First, we prove (i). As in the previous theorem, let [x.sub.t] [member of] S be a node such that u([x.sub.t],t) := [max.sub.x[member of]S]u(x, t),for each t > 0. Then it follows from (35) that

[u.sub.t] ([x.sub.s], s) [less than or equal to] [lambda][u.sup.q] ([x.sub.s], s), (54)

for almost all s > 0. Then integrating from t to T, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (55)

Hence, we obtain

u([x.sub.t], t) [greater than or equal to] [[[lambda](q - 1)(T - t)].sup.-1/(q-1)], 0 < t < T. (56)

Next, we prove (ii). Since the solution u is positive, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (57)

for almost all s > 0 and [[omega].sub.0] = [max.sub.x[member of]S] [[summation].sub.y[member of][bar.S]][[summation].sub.y[member of][bar.S]][omega](x, y). Then, it follows from (i) (lower bound) that we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (58)

Integrating from t to T, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (59)

where [alpha] := [[omega].sub.0][[lambda].sup.(q-P+1)/(q-1)](q - 1).sup.(2q-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 graph S = [[x.sub.1], [x.sub.29]} with the boundary [partial derivative] = {[x.sub.30], [x.sub.31]} and the weight

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (60)

where i = 1, ..., 28 (see Figure 1). Then, we note that [[omega].sub.0] := [max.sub.x[member of]S] [[summation].sub.y[member of][bar.S]][omega](x,y) = 0.3.

Example 1 (1 < p - 1 < q). For the graph [bar.S] (see Figure 1), consider q = 3, p = 2.5, [lambda] = 0.5, and the initial data [u.sub.0] given by Table 1.

Then 1 < p-1 = 1.5 < = 3 and [max.sub.x[member of]S][u.sub.0](x) = 1.5 > ([[omega].sub.0]/[lambda]).sup.1/(q-p+1)] =[??] 0.711. 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (61)

On the other hand, consider a small initial data [u.sub.0] given by Table 2.

Then [max.sub.x[member of]S][u.sub.0](x) = 0.01 [??] [([[omega].sub.0]/[lambda]).sup.1/(q-p+1)] [??] 0.711 and Figure 3 shows that the solution to (35) is global.

Example 2 (0 < p-1 < 1 < q). For the graph [bar.S] (see Figure 1), consider q = 3, p = 1.5, [lambda] = 0.1, and the initial data [u.sub.0] given by Table 3.

Then 0 < p - 1 = 0.5 < q = 3 and [max.sub.x[member of]S][u.sub.0](x) = 3 > [([[omega].sub.0]/[lambda]).sup.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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (62)

Example 3 (1 < q < p - 1). For the graph [bar.S] (see Figure 1), consider q = 1.5, p = 3, [lambda] = 0.1, and the initial data [u.sub.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.

Example 4 (0 < q [less than or equal to] 1). For the graph [bar.S] (see Figure 1), consider q = 0.5, p = 3, [lambda] = 0.1, and the initial data [u.sub.0] given by Table 3 in Example 2. Then 0 < q = 0.5 [less than or equal to] 1 and Figure 6 shows that the solution to (35) is global.

Example 5 (1 < p - 1 = q). For the graph [bar.S] (see Figure 1), consider q = 2, p = 3, [lambda] = 2, and the initial data [u.sub.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, when [lambda] = 0.00001, the solution to (35) is global, as seen in Figure 8.

5. Conclusion

We discuss the conditions under which blow-up occurs for the solutions of discrete p-Laplacian parabolic equations on networks S with boundary [partial derivative]S:

[u.sub.t](x, t) = [[DELTA].sub.p,[omega]]u (x, t) + [lambda] [absolute value of u (x, t)].sup.q-1]] u(x, t), (x, t) [member of] S x (0, +[infinity]),

u(x, t) = 0, (x, t) [member of] [partial derivative]S x (0, +[infinity]), u (x, 0) = [u.sub.0] [greater than or equal to] 0, x [member of] [bar.S], (63)

where p > 1, q > 0, [lambda] > 0, and the initial data [u.sub.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 [[bar.u].sub.0] > [([[omega].sub.0]/[lambda]).sup.1/(q-p+1)], where [[omega].sub.0] : = [max.sub.x[member of S] [[summation].sub.y[member of]S][omega](x,y) and [[bar.u].sub.0] := [max.sub.x[member of]S][u.sub.0](x);

(ii) if 0 < q [less than or equal to] 1, then the nonnegative solution is global;

(iii) if 1 < 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.

http://dx.doi.org/10.1155/2014/351675

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MOE) (no. 2012R1A1A2004689) and Sogang University Research Grant of 2014 (no. 201410044).

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Soon-Yeong Chung (1) and Min-Jun Choi (2)

(1) Department of Mathematics and Program of Integrated Biotechnology, Sogang University, Seoul 121-742, Republic of Korea

(2) Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Correspondence should be addressed to Soon-Yeong Chung; sychung@sogang.ac.kr

Received 21 August 2014; Accepted 16 October 2014; Published 24 November 2014

```
TABLE 1: Initial data of n.

Node i   [u.sub.0] ([x.sub.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                 0

TABLE 2: Initial data of n.

Node i   [u.sub.0] ([x.sub.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                 0

TABLE 3: Initial data of u.

Node i   [u.sub.0] ([x.sub.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                 0
```