Research Article | Open Access
Qiao Xin, Chunlai Mu, Dengming Liu, "Extinction and Positivity of the Solutions for a -Laplacian Equation with Absorption on Graphs", Journal of Applied Mathematics, vol. 2011, Article ID 937079, 12 pages, 2011. https://doi.org/10.1155/2011/937079
Extinction and Positivity of the Solutions for a -Laplacian Equation with Absorption on Graphs
We deal with the extinction of the solutions of the initial-boundary value problem of the discrete p-Laplacian equation with absorption with p > 1, q > 0, which is said to be the discrete p-Laplacian equation on weighted graphs. For 0 < q < 1, we show that the nontrivial solution becomes extinction in finite time while it remains strictly positive for , and . Finally, a numerical experiment on a simple graph with standard weight is given.
The discrete analogue of the Laplacian on networks, the so-called discrete Laplacian, can be used in various areas, for example, modeling energy flows through a network or modeling vibration of molecules [1–4]. However, many phenomena on some cases cannot be expressed well by the discrete Laplacian. In view of this, a nonlinear operator, called the discrete -Laplacian, has recently been studied by many researchers in various fields, such as dynamical systems and image processing [5–7].
Our interest in this work can be considered as a discrete analogue of the following initial boundary value problem for the -Laplacian equation with absorption: where is a bounded domain in . In fact, all the continuous regularization methods (local or nonlocal) with a given discretization scheme can be considered as particular cases of our proposed discrete regularization. Since the proposed framework is directly expressed in a discrete setting, no partial difference equations resolution is needed. Equation (1.1) has been extensively studied. In , the existence, uniqueness, regularity, and behavior of solutions to the initial-boundary value problem for (1.1) has been studied. Moreover, in , Gu proved that, for , if or the solutions of the problem vanish in finite time, but if and , there is nonextinction [9, Example 2 and Theorem 3.3]. In the absence of absorption (i.e., ), DiBenedetto  and Hongjun et al.  proved that the necessary and sufficient conditions for the extinction to occur is .
The -heat equation , which can be interpreted as a heat (or energy) diffusion equation on electric networks, has been studied by a number of authors such as [1, 3, 12–15] and so on, for example, the solvability of direct problems such as the Dirichlet and Neumann boundary value problems of the -Laplace equation, the global uniqueness of the inverse problem of the equation under the monotonicity condition, moreover, finding solutions to their initial and boundary problems, and representing them by means of their kernels have also been studied. Recently, in , Chung et al. considered the homogeneous Dirichlet boundary value problem for the -heat equation with absorption on a network: The absorption term denotes that the heat flows through networks are influenced by the reactive forces proportional to the power of their potentials. The authors proved that if , a nontrivial solution of (1.2) becomes extinction in finite time, but if , it remains strictly positive. However, a lot of material usually have complicated interconnection governed by their intrinsic characteristics and to express such a feature, it needs to be a more complex systems than simple linear equations on networks. So many authors have adapted nonlinear operators which is useful to describe natures on networks and one of those operators is a discrete -Laplacian which is a generalized nonlinear operator of the discrete Laplacian. And then, the present paper is devoted to the discrete analogues of (1.1) on networks, that is, we consider the following discrete -Laplacian equation: where is a finite simple graph, , , and is a nonnegative function on graph . The main work of this paper is to show, for , the nontrivial solution becomes extinction in finite time while it remains strict positive in the case , and .
In this section, we will begin with some definitions of graph theoretic notions, which are frequently used throughout this paper.
Let be a finite simple, connected, and undirected graph, and denote its vertex set and its edge set, respectively. Two vertices are adjacent if they are connected by an edge, in this case, we write or . Moreover, we also omit the subscript in , , and so forth, if is clear from context. In general, we can split the set of vertexes into two disjoint subsets and such that , which are called the interior and the boundary of . A weight on a graph is a function satisfying Since the set of edge is uniquely determined by the weight, thus the simple weighted graph can be simply denoted by .
Throughout this paper, we consider the space of functions on vertex sets of the graph. The integration of function on a simple weighted graph is defined as
As usual, the set consists of all functions defined on which is satisfy for each . Further, for convenience, we denote where is a fixed positive real number or .
Finally, in the case of , for a function , the graph -directional derivative of to the direction for is defined by and the graph -Laplacian of a function on is defined as follows: where Note that, the -Laplacian operator is nonlinear, with the exception of . For , it becomes the standard graph Laplacian.
3. -Laplacian Equations with Absorption on Graphs
Before studying our problem, we will give a lemma, which will be frequently used in our later proofs.
Lemma 3.1 (see , Lemma 2.1). For any and , one has where C1 and are positive constants depending only on .
Theorem 3.2 (uniqueness for BVP). Let be a continuous and increasing function, then the initial-boundary value problem admits a unique solution in .
Proof. Suppose both and are two solutions of (3.2) and let . Next, we introduce an energy functional as the form where . Taking the derivative of with respect to , and applying Lemma 3.1 and Fubini's theorem, we get which means for all . Furthermore, we can conclude that in .
Remark 3.3. Similar to the process of the proof of Theorem 3.2, it is easy to prove that the uniqueness of the following initial value problem
holds in .
Now, we give a comparison principle.
Theorem 3.4. Let be a continuous and increasing function, and suppose satisfy then, for all .
Proof. Letting then we have Putting then it is obvious that on . Now, multiplying both sides of (3.8) by , and integrating over , we obtain Setting We will assume that for each , and establish a contradiction. Integrating (3.8) over and applying Fubini's theorem, we get Defining and using Lemma 3.1, then for any and , we arrive at On the other hand, we have Next, our goal is to estimate the two terms in the right side of the above equality. First, applying Fubini' theorem and (3.12), we have On the other hand, if and , we then have and . Furthermore, we get . Thus, by (3.14), we get In addition, noticing that therefore, the right side of (3.12) is negative, which is a contradiction. The proof of Theorem 3.4 is complete.
From the above theorem, we can obtain the following result which is similar to .
Corollary 3.5. Assume that satisfies Then, in .
Remark 3.6. It follows from the above corollary that (3.19) is equivalent to the equation
4. Extinction and Positivity of the Solution
In this section, we investigate the extinction phenomenon and the positivity property of the solutions of the discrete -Laplacian with absorption on graphs with boundary.
Theorem 4.1 (extinction). Let . Suppose satisfies there exists a finite time such that for all .
Proof. The proof is similar to Theorem 4.3 in , we omit the details here.
Theorem 4.2 (positivity). Let , . Suppose and satisfies Then for all . Moreover, one has for , where is independent on .
Proof. In order to obtain the positivity of the solutions, we first introduce a transformation, then after a simple computation, it is easy to verify that is the solution of the following problem: Multiplying both sides of (4.5)1 by , and integrating on , we have Meanwhile, multiplying (4.5)1 by , and integrating on , we get Define Noticing that and , we have Therefore, we can choose small enough such that . Moreover, from (4.8) and (4.9), it follows that On the other hand, taking in (4.6), and combining (4.6) with (4.10), we get Setting , since , we have Due to (4.12), there exists , , such that for . By transformation (4.4), we get for , where is independent on .
Remark 4.3. By Theorem 4.2, when and , the solution also remain strictly positive. In this case, our result is consistent with that in , but the method is very different from that previously used in .
5. Numerical Experiment
In this section, we consider a graph whose vertices and are linked as the above figure (see Figure 1) with the standard weight (i.e., ).
Let be a solution of with the following initial condition and boundary condition . Then, we obtain the following system of first order ordinary differential equations in terms of interior nodes: Since the above ordinary differential equations is nonlinear, we choose the following explicit difference scheme to compute the numerical solution: where , for and . We should point out that the time step must be set small enough, if not, the images of the function , will appear oscillation phenomena near zero. Set , take and , respectively. By Theorem 4.1, we have functions , will vanish in finite time.
In the numerical experiment, the time step is chosen as , the numerical experiment result is shown in Figure 2, the solutions extinct after 1800 iterations, that is, . The green curve is the image of function , the image of function is the red curve, and the image of function is expressed by the blue one for and , respectively.
This paper is supported in part by NSF of China 11071266 and in part by Natural Science Foundation Project of CQ CSTC (2010BB9218).
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