A Condition for Blow-up solutions to Discrete $p$-Laplacian Parabolic Equations under the mixed boundary conditions on Networks

The purpose of this paper is to investigate a condition \begin{equation*} (C_{p}) \hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0 \end{equation*} for some $\alpha>2$, $\gamma>0$, and $0\leq\beta\leq\frac{\left(\alpha-p\right)\lambda_{p,0}}{p}$, where $p>1$ and $\lambda_{p,0}$ is the first eigenvalue of the discrete $p$-Laplacian $\Delta_{p,\omega}$. Using the above condition, we obtain blow-up solutions to discrete $p$-Laplacian parabolic equations \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=\Delta_{p,\omega}u\left(x,t\right)+f(u(x,t)),&\left(x,t\right)\in S\times\left(0,+\infty\right), \mu(z)\frac{\partial u}{\partial_{p} n}(x,t)+\sigma(z)|u(x,t)|^{p-2}u(x,t)=0,&\left(x,t\right)\in\partial S\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0(nontrivial),&x\in S, \end{cases} \end{equation*} on a discrete network $S$, where $\frac{\partial u}{\partial_{p}n}$ denotes the discrete $p$-normal derivative. Here, $\mu$ and $\sigma$ are nonnegative functions on the boundary $\partial S$ of $S$, with $\mu(z)+\sigma(z)>0$, $z\in \partial S$. In fact, it will be seen that the condition $(C_{p})$, the generalized version of the condition $(C)$, improves the conditions known so far.


Introduction
These days, the discrete version of differential equations has attracted many researcher's attention. In particular, p-Laplacian ∆ p,ω on networks(or weighted graphs) is used to observe various social and scientific phenomena(see [1]- [3] and references therein), which is modeled by discrete p-Laplacian parabolic equations u t (x, t) = ∆ p,ω + u q (x, t) , (x, t) ∈ S × (0, ∞) with some boundary and initial conditions where S is the set of chemicals and p > 1. Here, ∆ p,ω is the discrete p-Laplace operator on S, defined by From a similar point of view, we discuss, in this paper, the blow-up property of solutions to the following discrete p-Laplacian parabolic equations where p > 1, f is locally Lipschitz continuous on R, and B[u] = 0 on ∂S×[0, +∞) stands for the boundary condition µ(z) ∂u ∂ p n (z, t) + σ(z)|u(z, t)| p−2 u(z, t) = 0, (z, t) ∈ ∂S × [0, +∞) .
Here, µ, σ : ∂S → [0, +∞) are functions with µ(z) + σ(z) > 0, z ∈ ∂S and ∂u ∂pn denotes the discrete p-normal derivative (which is introduced in Section 1). It is easy to see that this boundary value problem includes the various boundary value problems such as the Dirichlet boundary, Neumann boundary, Robin boundary, and so on. We note here that one of the meaning of our result is an unified approach. The continuous case of this equation with some boundary conditions has been studied by many authors. For example, in 1973, Levine [19] considered the formally parabolic equations of the form P du dt = −A(t)u + f (u(t)), t ∈ [0, +∞), u(0) = u 0 , where P and A(t) are positive linear operators defined on a dense subdomain D of a real or complex Hilbert space H. Here, he first introduced "the concavity method" to obtained the blow-up solutions, under abstract conditions 2(α + 1)F (x) ≤ (x, f (x)), F (u 0 (x)) > 1 2 (u 0 (X), Au 0 (x)) for every x ∈ D, where F (x) = 1 0 (f (ρx), x)dρ. After this, Philippin and Proytcheva [25] have applied the above method to the equations      u t = ∆u + f (u), in Ω × (0, +∞), u(x, t) = 0, on ∂Ω × (0, +∞), u(x, 0) = u 0 (x) ≥ 0, and obtained a blow-up solution, under the condition (A) : (2 + ǫ)F (u) ≤ uf (u), u > 0 and the initial data u 0 satisfying Besides, in [23,24] Payne et al. obtained the blow-up solutions to the equations when the Neumann boundary data f satisfies the condition (A).
Recently, Ding and Hu [17] adopted the condition (A) to get blow-up solutions to the equation with the nonnegative initial value and the null Drichlet boundary condition.
On the other hands, the condition (A) was relaxed by Bandle and Brunner [4] as follows: for some ǫ > 0 and γ > 0.
It is easy to see that the conditions (A) and (B) above are independent of the eigenvalue of Laplace operator which depends on the domain and the condition (C) is depend on the eigenvalue.
From this point of view, we generalized the condition (C) with respect to discrete p-Laplace operator ∆ p,ω , which is the main results of this paper, will be introduced as follows: for some α > 2, β ≥ 0, and γ > 0, , p > 1, and λ p,0 is the first eigenvalue of the discrete p-Laplacian ∆ p,ω . Here, we note that the term βu p is depending on the domain graph.
From this observation, we may understand the condition (A) and (B) with respect to the p-Laplace operator ∇(|∇u| p−2 ∇u) as follows: for p > 1, for some α > p with α > 2 and γ > 0. Above conditions (A p ), (B p ), and (C p ) are discussed in Section 3.
As far as the authors know, it seems that there have been no paper which deal with the blow-up solutions to the equation (1) for 1 < p < 2 in the discrete case, not even in the continuous case.
In fact, it is expected that, with the condition (C p ), more interesting results should be obtained even in the continuous case, which will be our forth-coming work.
We organize this paper as follows: in Section 1, we introduce briefly the preliminary concepts on networks and comparison principles. Section 2 is the main section, which is devoted to blow-up solutions using the concavity method with the condition (C p ). Finally in Section 3, we discuss the condition (C p ), comparing with the conditions (A p ) and (B p ), together with the condition B(0) > 0 for the initial data.

Preliminaries and Discrete Comparison Principles
In this section, we start with the theoretic graph notions frequently used throughout this paper. For more detailed information on notations, notions, and conventions, we refer the reader to [10].
set E of edges (two-element subsets of V ). Conventionally used, we denote by x ∈ V or x ∈ G the fact that x is a vertex in G.
(ii) A graph G is called simple if it has neither multiple edges nor loops (iii) G is called connected if for every pair of vertices x and y, there exists a sequence(called a path) of vertices x = x 0 , x 1 , · · · , x n−1 , x n = y such that x j−1 and x j are connected by an edge(called adjacent) for j = 1, · · · , n.
In this case, G is a host graph of G ′ . If E ′ consists of all the edges from E which connect the vertices of V ′ in its host graph G, then G ′ is called an induced subgraph.
We note that an induced subgraph of a connected host graph may not be connected.
Throughout this paper, all the subgraphs are assumed to be induced, simple and connected. Definition 1.2. For an induced subgraph S of a graph G = G (V, E), the (vertex) boundary ∂S of S is defined by Also, we denote by S a graph whose vertices and edges are in S ∪ ∂S. We note that by definition the set, S is an induced subgraph of G. Definition 1.3. A weight on a graph G is a symmetric function ω : V × V → [0, +∞) satisfying the following: and a graph G with a weight ω is called a network.
Definition 1.5. For p > 1 and a function u : S → R, the discrete p-Laplacian ∆ p,ω on S is defined by Definition 1.6. For p > 1 and a function u : S → R, the discrete p-normal derivative ∂u ∂pn on ∂S is defined by for z ∈ ∂S.
The following two lemmas are used throughout this paper.
In particular, in the case g = f , we have Here, Γ := {z ∈ ∂S | µ(z) > 0} and µ, σ : ∂S → [0, +∞) are functions with µ(z) + σ(z) > 0 for all z ∈ ∂S. Moreover, λ p,0 is given by In the above, the number λ p,0 is called the first eigenvalue of ∆ p,ω on a network S with corresponding eigenfunction φ 0 (see [5] and [16] for the spectral theory of the Laplacian operators). In fact, we note that if Γ is empty set, then z∈Γ σ(z) µ(z) |u(z)| p implies 0. Remark 1.9. It is clear that the first eigenvalue λ p,0 is nonnegative. Moreover, we note here that the first eigenvalue λ p,0 satisfies the following statements: We now discuss the local existence of a solution to the equation (1) which is where p > 1 and f is locally Lipschitz continuous on R. Here, B[u] on ∂S × [0, +∞) stands for the boundary condition (2) which is where µ, σ : ∂S → [0, +∞) are functions with µ(z) + σ(z) > 0 for all z ∈ ∂S.
Then it is easy to see that ψ is a continuous function which is strictly increasing and bijective on R. Therefore, there exists ρ ∈ R uniquely such that ψ(ρ) = 0. It means that for all z ∈ ∂S, we can define the value of u(z, 0) uniquely according to the boundary condition B[u] = 0 and initial data u 0 which are given. i.e. for every z ∈ ∂S, u(z, 0) is determined such that Remark 1.11. Considering the initial data with the boundary condition B We will use the Schauder fixed point theorem to prove local existence of the equation (1). For this reason, we need the modified version of the Arzelá-Ascoli theorem as follows.
Lemma 1.12 (Modified version of the Arzelá-Ascoli theorem). Let K be a compact subset of R and S be a network. Consider a Banach space C S × K with the maximum norm u S,K := max x∈S max t∈K |u (x, t)|. Then a subset A of C S × K is relatively compact if A is uniformly bounded on S × K and A is equicontinuous on K for each x ∈ S.
Proof. The proof of this version is similar to the original one (see [20]). Thus we only state the idea of the proof. Let ǫ > 0 be arbitrarily given. Since K is compact on R and A is equicontinuous on K, there is a finite open cover

is a function and define
Hence, A is totally bounded and the proof is complete.
Proof. We first start with the following Banach space: with the maximum norm u S,t0 := max x∈S max 0≤t≤t0 |u(x, t)|, where t 0 ∈ R is a positive constant which will be defined later. Now, consider a subspace Then it is clear that B t0 is convex. In order to apply the Schauder fixed point theorem, we have to show that B t0 is closed. Let g n be a sequence in B t0 which converges to g. Since the convergence is uniform, g is continuous. Moreover, | g n S,t0 − g S,t0 | ≤ g n − g S,t0 implies that g ∈ B t0 . Hence, B t0 is closed. On the other hand, for every u ∈ B t0 , we can define the value of u(z, t) uniquely according to the boundary condition B[u] = 0 by the similar way to Remark 1.10. i.e. for every u ∈ B t0 , u(z, t) satisfies for all (z, t) ∈ ∂S × [0, t 0 ], where µ, σ : ∂S → [0, +∞) are given functions with µ(z) + σ(z) > 0 for all z ∈ ∂S. Then by the boundary condition, it is clear that where u 0 : S → R is a given function.
Since f is locally Lipschitz continuous on R, there exists L > 0 such that where m = 2 u 0 S,t0 . Now, put where ω 0 := max x∈S y∈S ω(x, y). Then, it is easy to see that the operator D is well-defined. Now, we will show that D is continuous. The verification of the continuity is divided into 2 cases as follows: (i) 1 < p < 2 For u and v in B t0 , it follows that Consequently, for each p > 1, we obtain where C 1 and C 2 are constant depending only on u 0 , t 0 , p, L and ω 0 . Therefore, we obtain the continuity of D.
When p ≥ 2, we obtain a strong comparison principle as follows: Proof. First, note that u ≥ v on S × [0, T ) by above theorem. Let T ′ > 0 be arbitrarily given with T ′ < T . Since f is locally Lipschitz continuous on R, there exists L > 0 such that where m = max Then τ (x, t) ≥ 0 for all (x, t) ∈ S × [0, T ′ ]. From the inequality (13), we have for all 0 < t ≤ T ′ . Then by the mean value theorem, for each y ∈ S and 0 ≤ t ≤ T ′ , it follows that (14) and (16), the inequality (15) becomes This implies since τ (x * , 0) > 0. Now, suppose there exists (x 0 , t 0 ) ∈ S × (0, T ′ ] such that Hence, from the inequality (13), we obtain Therefore, we have which implies that τ (y, t 0 ) = 0 for all y ∈ S with y ∼ x 0 . Now, for any x ∈ S, there exists a path since S is connected. By applying the same argument as above inductively we see that τ (x, t 0 ) = 0 for every x ∈ S, which is a contradiction to (17).
By the boundary condition in (13), we have It means that there exists x 1 ∈ S with x 0 ∼ x 1 such that τ (x 1 , t 0 ) = 0, which contradicts to Case 1. Hence, we finally obtain that u (x, t) > v (x, t) for all (x, t) ∈ S × (0, T ), since T ′ < T is arbitrarily given.
We note that by the comparison principle, if f (0) = 0 then solutions u to the equation (1) are nonnegative. On the other hand, it is natural that f is assumed to be positive on (0, +∞) when we deal with the blow-up theory. Hence, we always assume that f is a locally Lipschitz continuous function on R which is positive in (0, +∞) and, f (0) = 0. Moreover, we assume that the initial data u 0 is nontrivial and nonnegative.

Blow-Up: the Concavity Method
In this section, we discuss the blow-up phenomena of the solutions to the equation (1) by using concavity method, which is the main part of this paper. This method, introduced by Levine [19], uses the concavity of an auxiliary function. In fact, the concavity method is an elegant tool for deriving estimates and giving criteria for blow-up. In order to state and prove our result, we introduce the following condition: for some α > 2, β ≥ 0, and γ > 0 with 0 ≤ β ≤ (α−p)λp,0 p . Remark 2.2. Observing that λ p,0 = 0 if and only if σ ≡ 0, we can easily obtain that the condition of α in (C p ) is difference in each p > 1 and boundary conditions as follows: (i) For all p > 1, if σ ≡ 0, then α > 2.
We now state the main theorem of this paper: x,y∈S x,y∈S Then we have from the equation (1) and Lemma 1.7 that Applying the condition (C p ) and Lemma 1.8, we can see that (19) implies |u(x, t) − u(y, t)| p ω(x, y) Here, it is easy to see that if λ p,0 = 0 or α = p, then β = 0. Therefore, even though λ p,0 = 0 or α = p, (20) is true.
On the other hand, we have from the equation (1) and Lemma 1.7 that (21) Now, we will show that for all t > 0. Using the Schwarz inequality, we obtain from (20) and (21) that for all t > 0. Therefore, the inequality (22) is true, which implies that Solving the differential inequality (23), we obtain Hence, A(t) blows up in finite time T with 0 < T ≤ Remark 2.4. The above blow-up time can be estimated roughly as . Remark 2.5. Chung and Choi [11] obtained the blow-up results for the equation (1) under the Dirichlet boundary condition in the continuous setting, where p ≥ 2 by using the (C p ) condition. In fact, their condition had assumption α > p, which is one of main difference to us.
3. Discussion on the Condition (C p ) with the initial data conditions  (x, t)).
From the above equality, we can obtain that the time-behavior of x∈S u(x, t) is determined by x∈S f (u(x, t)). Therefore, by the definition of the blow-up, we can expect that the blow-up condition for the solution u depends only on f , not on p. On the other hand, for all p > 1, the (C p ) condition is represented by (2 + ǫ)F (u) ≤ uf (u) + γ for some ǫ > 0 and γ > 0, which also doesn't depend on p.
By the similar way, assuming ǫ = 0 we have d du Hence, (24) and (25)  for some constants ǫ ≥ 0, a ≥ 0, and b > 0 with 0 ≤ a ≤ λp,0 p , where h 1 , h 2 , and h 3 are nondecreasing function on (0, +∞). Here also, the constants ǫ, a, and b may be different in each case. We note here that the nondecreasing function h 1 is nonnegative on (0, +∞), but h 2 , and h 3 may not be nonnegative, in general.
Case 2: 1 < p ≤ 2. We obtain that (C p ) is equivalent to d du which implies that for every u > 0 and 1 < p ≤ 2, Remark 3.2. Chung and Choi studied the case f (u) = λu q in the Dirichlet boundary condition with respect to blow-up property (see [11,12]). In their results, the solution u blows up in finite time if (i) 0 < p − 1 < q, q > 1, and the initial data u 0 is sufficiently large.
Considering the case (i) and (ii), we obtain that the solution u doesn't blow up in finite time whenever q ≤ 1. From this observation, we can easily obtain that ǫ in the (C p ) condition cannot be 0 when 1 < p ≤ 2, since F (u) = u 2+ǫ h 3 (u) + au p + b. Theorem 3.3. For p > 1, let f be a real-valued function satisfying the condition (C p ). Suppose that f (u) ≥ λu p−1 , u > 0 for some λ > λ p,0 . Then the following statements are true.