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

We deal with the extinction of the solutions of the initial-boundary value problem of the discrete pLaplacian equation with absorption ut Δp,ωu−u 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 p ≥ 2, q ≥ 1 and q ≥ p − 1. Finally, a numerical experiment on a simple graph with standard weight is given.


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 p-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 p-Laplacian equation with absorption: where Ω is a bounded domain in R n .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 λ > 0, if p ∈ 1, 2 or q ∈ 0, 1 the solutions of the problem vanish in finite time, but if p ≥ 2 and q ≥ 1, there is nonextinction 9, Example 2 and Theorem 3.

1.2
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 0 < q < 1, a nontrivial solution of 1.2 becomes extinction in finite time, but if q ≥ 1, 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 p-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 p-Laplacian equation: where G is a finite simple graph, p > 1, q > 0, and u 0 x is a nonnegative function on graph G.The main work of this paper is to show, for 0 < q < 1, the nontrivial solution becomes extinction in finite time while it remains strict positive in the case p ≥ 2, q ≥ 1 and q ≥ p − 1.

Preliminary
In this section, we will begin with some definitions of graph theoretic notions, which are frequently used throughout this paper.
Let G be a finite simple, connected, and undirected graph, V G and E G denote its vertex set and its edge set, respectively.Two vertices x, y are adjacent if they are connected by an edge, in this case, we write x ∼ y or x, y ∈ E.Moreover, we also omit the subscript G in V G , E G , and so forth, if G is clear from context.In general, we can split the set of vertexes V into two disjoint subsets S and ∂S such that V S ∂S, which are called the interior and the boundary of Since the set of edge is uniquely determined by the weight, thus the simple weighted graph G V, E; ω can be simply denoted by V ; ω .Throughout this paper, we consider the space of functions on vertex sets of the graph.The integration of function f : V → R on a simple weighted graph G V ; ω is defined as As usual, the set where T is a fixed positive real number or ∞.
Finally, in the case of p > 1, for a function f : V → R, the graph p-directional derivative of f to the direction y for x ∈ V is defined by and the graph p-Laplacian Δ p,ω of a function on G V ; ω is defined as follows: where Note that, the p-Laplacian operator is nonlinear, with the exception of p 2. For p 2, it becomes the standard graph Laplacian.

p-Laplacian Equations with Absorption on Graphs
Before studying our problem, we will give a lemma, which will be frequently used in our later proofs.

3.1
where C 1 and C 2 are positive constants depending only on p. Theorem 3.2 uniqueness for BVP .Let Ψ be a continuous and increasing function, then the initialboundary value problem Proof.Suppose both u and u are two solutions of 3.2 and let v : u − u.Next, we introduce an energy functional as the form where 0 ≤ t < T. Taking the derivative of E t with respect to t, and applying Lemma 3.1 and Fubini's theorem, we get which means E t ≤ E 0 0 for all 0 ≤ t < T. Furthermore, we can conclude that v ≡ 0 in V T .
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

3.6
then, u ≤ u for all x, t ∈ S T .
Proof.Letting v : u − u, 3.7 then we have

3.8
Putting v : max{v, 0}, 3.9 then it is obvious that v 0 on Γ T .Now, multiplying both sides of 3.8 by v , and integrating over 0, t , we obtain

3.11
We will assume that J t / ∅ for each t ∈ 0, T , and establish a contradiction.Integrating 3.8 over J t and applying Fubini's theorem, we get 3.17 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 .

3.19
Then, u ≥ 0 in S T .
Remark 3.6.It follows from the above corollary that 3.19 is equivalent to the equation 3.20

Extinction and Positivity of the Solution
In this section, we investigate the extinction phenomenon and the positivity property of the solutions of the discrete p-Laplacian with absorption on graphs with boundary.

4.1
there exists a finite time T > 0 such that u x, t ≡ 0 for all x, t ∈ V × T, ∞ .
Proof.The proof is similar to Theorem 4.3 in 16 , we omit the details here.

4.2
Then u x, t > 0 for all x, t ∈ S × 0, ∞ .Moreover, one has Proof.In order to obtain the positivity of the solutions, we first introduce a transformation, v x, s ρe s u x, ρ q−1 s , s ρ − q−1 t,

4.4
then after a simple computation, it is easy to verify that v is the solution of the following problem:

4.5
Multiplying both sides of 4.5 1 by v, and integrating on S, we have 1 2

4.6
Meanwhile, multiplying 4.5 1 by v s , and integrating on S, we get x∈S y∈S ω x, y v y, s − v x, s p q − 1 q 1 e 1−q s S v q 1 0.

4.8
Noticing that p ≥ 2 and q ≥ 1, we have Therefore, we can choose ρ ρ 0 small enough such that E 0 < 0.Moreover, from 4.8 and 4.9 , it follows that

4.10
On the other hand, taking ρ ρ 0 in 4.6 , and combining 4.6 with 4.10 , we get

4.11
Setting y s S v 2 , since q 1 − p ≥ 0, we have d ds y s q − 1 y s > 0.

4.12
Due to 4.12 , there exists s 0 > 0, c > 0, such that y s ≥ c for s > s 0 .By transformation 4.4 , we get for t > T 1 , where c, ρ 0 , T 1 is independent on u x, t .Remark 4.3.By Theorem 4.2, when p 2 and q ≥ 1, 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 .

Numerical Experiment
In this section, we consider a graph G whose vertices S {x 1 , x 2 , x 3 } and ∂S {x 4 , x 5 } are linked as the above figure see Figure 1 with the standard weight i.e., w x, y 1 .Let u be a solution of u t x, t Δ p,ω u x, t − u q , 5.1 with the following initial condition u 0 x 1 10, u 0 x 2 5, u 0 x 3 0.5 and boundary condition u x 4 , t 0, u x 5 , t 0.Then, we obtain the following system of first order ordinary differential equations in terms of interior nodes:

5.2
Since the above ordinary differential equations is nonlinear, we choose the following explicit difference scheme to compute the numerical solution: |u n x i − u n x 2 | p−2 u n x i − u n x 2 − u n x 2 q , u n 1 x 3 − u n x 3 Δt i 1,2,5 |u n x i − u n x 3 | p−2 u n x i − u n x 3 − u n x 3 q , 5.3 where u n x i u x i , nΔt , for i 1, 2, . . ., 5 and n 0, 1, 2, . ... We should point out that the time step must be set small enough, if not, the images of the function u x i , t , i 1, 2, 3 will u(x 1 , t), for p = 2, q = 0.3 u(x 2 , t), for p = 2, q = 0.3 u(x 2 , t), for p = 2, q = 0.3 u(x 1 , t), for p = 3.1, q = 0.3 u(x 2 , t), for p = 3.1, q = 0.3 u(x 3 , t), for p = 3.1, q = 0.3 appear oscillation phenomena near zero.Set q 0.3, take p 3.1 and p 2, respectively.By Theorem 4.1, we have functions u i x, t , i 1, 2, 3 will vanish in finite time.
In the numerical experiment, the time step is chosen as 0.005, the numerical experiment result is shown in Figure 2, the solutions extinct after 1800 iterations, that is, t 9.The green curve is the image of function u x 1 , t , the image of function u x 2 , t is the red curve, and the image of function u x 3 , t is expressed by the blue one for q 3.1 and q 2, respectively.

3 . 5 holds in C 1 Theorem 3 . 4 .
V T .Now, we give a comparison principle.Let Ψ be a continuous and increasing function, and suppose u, u ∈ C 1 G T satisfy

Figure 2 :
Figure 2: Extinction of the solutions.
3 .In the absence of absorption i.e., λ 0 , DiBenedetto 10 and Hongjun et al. 11 proved that the necessary and sufficient conditions for the extinction to occur is p ∈ 1, 2 .The ω-heat equation u t x, t Δ ω u x, t , 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 ω-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 ω -heat equation with absorption on a network: On the other hand, if x ∈ J t and y ∈ V \ J t , we then have v x, t > 0 and v y, t < 0. Furthermore, we get v y, t − v x, t < 0. Thus, by 3.14 , we get dt. 3.12 Defining F ω,p u, u x, y, t ω x, y u y, t − u x, t p−2 u y, t − u x, t − u y, t − u x, t p−2 u y, t − u x, t , 3.13 and using Lemma 3.1, then for any x, t and y, t ∈ V T , we arrive at F ω,p u, u x, y, t v y, t − v x, t ≥ 0. v x, t F ω,p u, u x, y, t − 1 2 x∈J t y∈J t v y, t − v x, t F ω,p u, u x, y, t ≤ 0. 3.16