Abstract
The goal of this paper is to investigate the blow-up and the global existence of the solutions to the discrete p-Laplacian parabolic equation , , , , , depending on the parameters and . 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.
1. Introduction
A discrete p-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 with some boundary and initial conditions where is the set of chemicals and , . Here, is the discrete -Laplace operator on a network , defined by 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 , , and (see [1–7] and see the book [2] for more information about the various blow-up phenomena). In particular, the case where has been known as the critical case at which solutions may blow up or exist globally, depending on the parameter .
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 discrete -Laplacian parabolic equation on a network 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 was studied in [3, 6] and the case where was studied in [3, 4].
For the discrete case of (1), the authors studied the general theory of the equation for the case in [8] and Xin et al. studied the blow-up solutions and global solutions for the case where and in [9]. On the other hand, for the case in (1), Chung et al. [10, 11] investigate the positive solutions and extinctive solutions, depending on the parameters and . But, the case where and 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) For and , the solution to
blows up in a finite time , for every nonnegative and nontrivial initial data .
Here, is the first eigenvalue of the p-Laplace operator on . Moreover, it is shown that the blow-up rate of the solution is estimated by
(ii) For and , the solution to (4) is global, for every nonnegative initial data .
(iii) For , the nonnegative solution to (4) is global for every nonnegative initial data . In particular, when and , there exists (extinction time) such that for all .
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 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 is 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 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 following lemmas are useful throughout this paper.
Lemma 2 (see [14, 15]). For functions , one has the following: (i); (ii).
Lemma 3 (see [14, 15]). For , there exist and , , such that Moreover, where .
In the above, the number is called the first eigenvalue of on a network with corresponding eigenfunction (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 discrete -Laplacian parabolic equation where and , 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. 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 (11) that we have
for all . Let be the functions defined by
where and .
Then inequality (13) 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 (17) and (18), we obtain
which contradicts (15). Therefore, for all so that we get for all , since is arbitrarily given.
Now, we obtain a strict comparison principle as a corollary.
Corollary 5 (strict comparison principle). In Theorem 4, if for some , then for all .
Proof. First, note that on by Theorem 4. Let be arbitrarily given with and let be a function defined by
Then for all . Since for all , and , we obtain from inequality (15) that
for all . Then by the mean value theorem, for each and with , it follows that
and where .
Then inequality (21) gives
where . This implies
Now, suppose there exists such that
Then
Hence, inequality (21) 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 (24).
For the case , 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. 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
since . Hence, inequality (31) gives
which leads to a contradiction. Hence, for all so that we have for all , since and are arbitrary.
3. Blow-Up and Global Existence
In this section, we discuss the blow-up and global existence of the equation where , , and the initial data is nontrivial on . According to the comparison principle in the previous section, the nonhomogeneous term can be written as a simpler form as , for the case where .
Definition 7 (blow-up). We say that a solution to an equation defined on a network blows up in finite time , if there exists such that as .
Theorem 8. For and , the solution to blows up at some , provided that
Proof. First, we note that on , by the strict comparison principle (Corollary 5). Now, we define a functional by
Then by (37),
Multiplying (36) by and summing up over , we obtain from Lemma 2
Multiplying (36) by and summing up over , we obtain from Lemma 2
Then it follows that
Moreover, it follows from (42) that
Now, we introduce a new function
where is a constant to be determined later. Then by (40) we have
Moreover,
Using the Schwarz inequality, we obtain
where is arbitrary. Combining the above estimates (46), (47), and (48), we obtain that, for ,
Since by assumption, we can choose to be large enough so that
Thus inequality (50) implies that, for ,
Therefore, it follows that cannot remain finite for all . In other words, the solutions blow up at some time .
Remark 9. (i) Condition (37) implies that
where is the first eigenvalue of .
(ii) The initial data with always exists. In fact, consider an eigenvalue and eigenfunction in Lemma 3. Taking , then we have
where .
(iii) When the solution to (37) is global, then we must have
for all .
(iv) The blow-up time in the above can be estimated roughly. Taking
we see that
which implies
where . Then the blow-up time satisfies
Now, we state the main theorem as follows.
Theorem 10. The solution to (36) with and blows up in finite time, for every nonnegative and nontrivial initial data .
Proof. First, we note that (36) has a unique solution such that Take , arbitrarily and consider an equation where is the eigenfunction corresponding to the first eigenvalue . Then the initial data satisfies Then, by Theorem 8, the solution is positive and blows up in a finite time. Since for a small , the comparison principle enables us to see which completes the proof.
Remark 11. According to Remark 9 (iv) to Theorem 8, the blow-up time for the solution to (36) is estimated by
We now derive the lower bound for the maximum function of blow-up solutions.
Theorem 12. Let be the solutions to (36) blowing up at finite time . Then it follows that for all .
Proof. For each , let be the node such that
In fact, we note that is continuous on and differentiable for almost all . Then (36) can be written as
for almost all .
Now define a function by
Then , , if . Multiplying (67) by and summing over , we obtain, for almost all ,
or, equivalently,
Integrating over , we have
which is desired.
Now, we state the global existence of the solutions.
Theorem 13. For and , the solution to (36) is global for every nonnegative initial data .
Proof. Consider an eigenvalue and eigenfunction in Lemma 3 and , , . Then by taking large enough to be , , we have for all and . Then by the comparison principle, we see that , , , which is desired.
We have so far discussed the blow-up or the global existence (35) for the case . Now, we discuss the case . Here, we note that when , the solution to the equation may not have unique positive solution.
Theorem 14. For , the nonnegative solution to is global for every . In particular, when and , then there exists T (extinction time) such that for .
Proof. First, consider ODE
where . Then
Then by the comparison principle (Theorem 6), we see that
which implies that must be global.
Now, assume and . When is a trivial solution, then we are done. So now we assume that is nontrivial. Multiplying (73) by , as done in (40), we have
where the last inequality follows from the elementary inequality
for . Now suppose for , on the contrary. Since , , , we obtain
Integrating (79) over [], we obtain
which is absurd for large . Hence, for some and, therefore, we can conclude that , , for some , since in inequality (79).
Remark 15. (i) When in the above, we do not need the assumption that is nonnegative, which follows automatically from the comparison principle (Theorem 4).
(ii) In fact, inequality (80) gives us the extinction time , estimated by
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 as in Figure 1.
Example 1. For (36) on (Figure 1) graph with , , consider initial data given by , , and . Then by easy calculation, we get the first eigenvalue and the corresponding eigenfunction . Figure 2 shows that the solution to (36) blows up and the computed blow-up time is estimated as .
On the other hand, consider the same equation (36) with , and the same initial data. Then and Figure 3 shows that the solution to (36) is global.
Example 2. For (36) on (Figure 1) graph with , , and the same initial data given by , , and , then Figure 4 shows that the solution to (36) is global and extinctive.
Conflict of Interests
The author declares 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).