Journal of Applied Mathematics

Volume 2011, Article ID 937079, 12 pages

http://dx.doi.org/10.1155/2011/937079

## Extinction and Positivity of the Solutions for a -Laplacian Equation with Absorption on Graphs

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received 1 April 2011; Accepted 4 July 2011

Academic Editor: Alain Miranville

Copyright © 2011 Qiao Xin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [8], the existence, uniqueness, regularity, and behavior of solutions to the initial-boundary value problem for (1.1) has been studied. Moreover, in [9], 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 [10] and Hongjun et al. [11] 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 [16], 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 .

#### 2. Preliminary

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 [17], Lemma 2.1). *For any and , one has
**
where C _{1} 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 [16].

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 [16], 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 [16], but the method is very different from that previously used in [16].

#### 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.

#### Acknowledgments

This paper is supported in part by NSF of China 11071266 and in part by Natural Science Foundation Project of CQ CSTC (2010BB9218).

#### References

- S.-Y. Chung and C. A. Berenstein, “$\omega $-harmonic functions and inverse conductivity problems on networks,”
*SIAM Journal on Applied Mathematics*, vol. 65, no. 4, pp. 1200–1226, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. B. Curtis and J. A. Morrow, “Determining the resistors in a network,”
*SIAM Journal on Applied Mathematics*, vol. 50, no. 3, pp. 918–930, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. B. Curtis and J. A. Morrow, “The Dirichlet to Neumann map for a resistor network,”
*SIAM Journal on Applied Mathematics*, vol. 51, no. 4, pp. 1011–1029, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Yamasaki, “Discrete potentials on an infinite network,”
*Memoirs of the Faculty of Literature and Science. Shimane University*, vol. 13, pp. 31–44, 1979. View at Google Scholar · View at Zentralblatt MATH - A. Elmoataz, O. Lezoray, and S. Bougleux, “Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing,”
*IEEE Transactions on Image Processing*, vol. 17, no. 7, pp. 1047–1060, 2008. View at Publisher · View at Google Scholar - S.-Y. Ha and D. Levy, “Particle, kinetic and fluid models for phototaxis,”
*Discrete and Continuous Dynamical Systems. Series B*, vol. 12, no. 1, pp. 77–108, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Kindermann, S. Osher, and P. W. Jones, “Deblurring and denoising of images by nonlocal functionals,”
*Multiscale Modeling & Simulation*, vol. 4, no. 4, pp. 1091–1115, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Tsutsumi, “On solutions of some doubly nonlinear degenerate parabolic equations with absorption,”
*Journal of Mathematical Analysis and Applications*, vol. 132, no. 1, pp. 187–212, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. G. Gu, “Necessary and sufficient conditions for extinction of solutions to parabolic equations,”
*Acta Mathematica Sinica*, vol. 37, no. 1, pp. 73–79, 1994 (Chinese). View at Google Scholar - E. DiBenedetto,
*Degenerate Parabolic Equations*, Universitext, Springer, New York, NY, USA, 1993. - Y. Hongjun, L. Songzhe, G. Wenjie, X. Xiaojing, and C. Chunling, “Extinction and positivity for the evolution
*p*-Laplacian equation in ${R}^{N}$,”*Nonlinear Analysis. Theory, Methods & Applications*, vol. 60, no. 6, pp. 1085–1091, 2005. View at Publisher · View at Google Scholar - N. Biggs,
*Algebraic Graph Theory*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, Mass, USA, 2nd edition, 1993. View at Zentralblatt MATH - S.-Y. Chung, Y.-S. Chung, and J.-H. Kim, “Diffusion and elastic equations on networks,”
*Publications of the Research Institute for Mathematical Sciences*, vol. 43, no. 3, pp. 699–726, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Chung and S.-T. Yau, “Discrete Green's functions,”
*Journal of Combinatorial Theory. Series A*, vol. 91, no. 1-2, pp. 191–214, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. K. Wojciechowski, “Heat kernel and essential spectrum of infinite graphs,”
*Indiana University Mathematics Journal*, vol. 58, no. 3, pp. 1419–1441, 2009. View at Publisher · View at Google Scholar - Y.-S. Chung, Y.-S. Lee, and S.-Y. Chung, “Extinction and positivity of the solutions of the heat equations with absorption on networks,”
*Journal of Mathematical Analysis and Applications*, vol. 380, no. 2, pp. 642–652, 2011. View at Publisher · View at Google Scholar - J. W. Barrett and W. B. Liu, “Finite element approximation of the parabolic
*p*-Laplacian,”*SIAM Journal on Numerical Analysis*, vol. 31, no. 2, pp. 413–428, 1994. View at Publisher · View at Google Scholar