Abstract
We discuss the conditions under which blow-up occurs for the solutions of discrete -Laplacian parabolic equations on networks with boundary as follows: , ; , ; , where , , , and the initial data is nontrivial on . The main theorem states that the solution to the above equation satisfies the following: (i) if and , then the solution blows up in a finite time, provided , where and ; (ii) if , then the nonnegative solution is global; (iii) if , 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.
1. Introduction
In this paper, we discuss the blow-up property and global existence of solutions to the following discrete -Laplacian parabolic equation: where , and .
The continuous case of this equation has been studied by many authors, assuming some conditions , , and , in order to get a blow-up solution or global solution (see [1–5]). For example, they consider the case , , and in [2], the case in [5], the case in [1, 3], the case , , and in [4], respectively, and so on.
On the other hand, the long time behavior (extinction and positivity) of solutions to evolution -Laplace equation with absorption on networks is studied in the paper [6, 7].
The goal of this paper is to give a condition on , , and 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 be a solution of (1). Then one has the following.(i)If and , then the solution blows up in a finite time, provided , where and .(ii)If , then the nonnegative solution is global.(iii)If , 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: where , and as a consequence 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 , we mean finite sets of vertices (or nodes) with a set of two-element subsets of (whose elements are called edges). The set of vertices and edges of a graph are sometimes denoted by and , or simply and , respectively. Conventionally, we denote by or the facts that is a vertex in .
A graph is said to be simple if it has neither multiple edges nor loops, and is said to be connected if, for every pair of vertices and , there exists a sequence (called a path) of vertices , , such that and are connected by an edge (called adjacent) for .
A graph is said to be a subgraph of , if and .
A weight on a graph is a function satisfying(i), ,(ii) if ,(iii) if and only if .Here means that two vertices and are connected (adjacent) by an edge in . A graph associated with a weight is said to be a weight graph or a network.
For a subgraph of a graph , the (vertex) boundary of is the set of all vertices but is adjacent to some vertex in ; that is, By , we denote a graph, whose vertices and edges are in both and .
Throughout this paper, all subgraphs and in our concern are assumed to be simple and connected.
For a function , the discrete p-Laplacian on is defined by for .
The rest of this section is devoted to prove the comparison principle for the discrete -Laplacian parabolic equation: where , , , and the initial data is nontrivial on , 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 ( may be ), , , and . Suppose that real-valued functions are differentiable in for each and satisfy Then for all .
Proof. Let be arbitrarily given with . Then, by the mean value theorem, for each and ,
for some lying between and . Then it follows from (7) that we have
for all . Let be the functions defined by
where and .
Then inequality (9) can be written as
for all . Since is compact, there exists such that
Then we have only to show that . Suppose that , on the contrary. Since on both and , we have . Then we have
Since , we have
Combining (13) and (14), we obtain
which contradicts (11). Therefore, for all so that we get for all , since is arbitrarily given.
When , we obtain a strict comparison principle as follows.
Corollary 3 (strict comparison principle). Let ( may be ), , , and . Suppose that real-valued functions are differentiable in for each and satisfy If for some , then for all .
Proof. First, note that on by Theorem 2. Let be arbitrarily given with and let be a function defined by
Then for all . Since and for all , we obtain from inequality (16) that
for all . Then, by the mean value theorem, for each and with , it follows that
and , where .
Then inequality (18) gives
where . This implies that
Now, suppose that there exists such that
Then
Hence, inequality (18) gives
Therefore,
that is,
which implies that for all with . Now, for any , there exists a path:
since is connected. By applying the same argument as above inductively, we see that for every . This gives a contradiction to (21).
For the case , 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 ( may be ), , , and . Suppose that real-valued functions are differentiable in for each and satisfy Then for all .
Proof. Let and be arbitrarily given with and , respectively, where (called a parabolic boundary).
Now, let a function be a function defined by
Then on . Now, we suppose that . Then there exists such that(i),(ii), ,(iii), .
Then
and
since
Hence, (28) gives
which leads to a contradiction. Hence, for all so that we have for all , since and are arbitrary.
Using the same method as in Corollary 3, we obtain a strict comparison principle as follows.
Corollary 5 (strict comparison principle). Let ( may be ), , , and . Suppose that real-valued functions are differentiable in for each and satisfy If for some , then for all .
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 to an equation defined on a network blows up in finite time , if there exists such that as .
According to the comparison principle in the previous section, we are guaranteed to get a solution to when , , , and the initial data is nontrivial on .
We now state the main theorem of this paper as follows.
Theorem 7. Let be a solution of (35). Then one has the following.(i)If and , then the solution blows up in a finite time, provided , where and .(ii)If , then the nonnegative solution is global.(iii)If , then the solution is global.
Proof. First, we prove (i). We note that , for all , by Theorem 2. Assume that , , and , where . For each , let be a node such that . In fact, we note that is differentiable, for almost all . Then (35) can be written as follows:
for almost all . We need to show that , for all . Since on and
is increasing in some interval . Suppose that there exists somewhere at which . Then now take the interval to be maximal on which , , and . Then there exists such that but
which leads to a contradiction. Thus it follows that , .
Let be a function defined by
We note that , for , since .
Then is a decreasing continuous function from onto with its inverse function . Integrating (36) from to , we have
This can be written as
and, equivalently,
which implies that blows up, as .
Secondly, we prove (ii). Consider the following ODE problem: Then, we have for every .
Take , for all and . Then it is easy to see that , , , , and Thus, for every by Theorem 4. This implies that must be global.
Finally, we prove (iii). Consider the following eigenvalue problem: Note that it is well known that and , for all (see [11, 12]).
Now, take , , . Choosing so large that and , then we see that , , and Therefore, for every by Theorems 2 and 4, which is required.
Remark 8. (i) When the solution blows up in the above, the blow-up time can be estimated as
In fact, the first inequality is derived as follows. By the definition of maximum function , (35) gives
for almost all . Then integrating both sides, we have
so that we obtain , by taking the limit as .
(ii) In the above, if 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 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 . 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 be a solution of (35), which blows up at a finite time , , and . Then one has the following.(i)The lower bound is (ii)The upper bound is where and .(iii)The blow-up rate is
Proof. First, we prove (i). As in the previous theorem, let be a node such that , for each . Then it follows from (35) that
for almost all . Then integrating from to , we get
Hence, we obtain
Next, we prove (ii). Since the solution is positive, we get
for almost all and . Then, it follows from (i) (lower bound) that we have
Integrating from to , we get
where .
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 with the boundary and the weight where (see Figure 1). Then, we note that .
Example 1 (). For the graph (see Figure 1), consider , , , and the initial data given by Table 1.
Then and . Then Figure 2 shows that the solution to (35) blows up and the computed blow-up time is estimated as and
(a)
(b)
On the other hand, consider a small initial data given by Table 2.
Then and Figure 3 shows that the solution to (35) is global.
(a)
(b)
Example 2 (). For the graph (see Figure 1), consider , , , and the initial data given by Table 3.
Then and . Then Figure 4 shows that the solution to (35) blows up and the computed blow-up time is estimated as and
(a)
(b)
Example 3 (). For the graph (see Figure 1), consider , , , and the initial data given by Table 3 in Example 2. Then and Figure 5 shows that the solution to (35) is global.
(a)
(b)
Example 4 (). For the graph (see Figure 1), consider , , , and the initial data given by Table 3 in Example 2. Then and Figure 6 shows that the solution to (35) is global.
(a)
(b)
Example 5 ().
For the graph (see Figure 1), consider , , , and the initial data given by Table 3 in Example 2. Then and Figure 7 shows that the solution to (35) blows up.
On the contrary, when , the solution to (35) is global, as seen in Figure 8.
(a)
(b)
(a)
(b)
5. Conclusion
We discuss the conditions under which blow-up occurs for the solutions of discrete -Laplacian parabolic equations on networks with boundary : where , , , and the initial data is nontrivial on .
The main theorem states that the solution to the above equation satisfies the following:(i)if and , then the solution blows up in a finite time, provided , where and ;(ii)if , then the nonnegative solution is global;(iii)if , 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.
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).