Blow-up in a $p$-Laplacian mutualistic model based on graphs

1.   Introduction

In recent years, evolution problems on complex networks have been studied extensively, for example, in the field of epidemic processes or population ecology ^[1,2,3,4,5]. A network is mathematically described as a undirected graph $T = (\Omega, E)$, which contains a set $\Omega$ of vertices and a set $E$ of edges $(x, y)$ connecting vertex $x$ and vertex $y$. If vertices $x$ and $y$ are connected by an edge (i.e., they are adjacent), we write $x\sim y$. $T$ is called a finite-dimensional graph if it has a finite number of edges and vertices. A graph is weighted if each adjacent $x$ and $y$ is assigned a weight function $\omega(x, y)$. Here $\omega:\Omega\times \Omega\rightarrow [0, +\infty)$ satisfies that $\omega(x, y) = \omega(y, x)$ and $\omega(x, y) > 0$ if and only if $x\sim y$. Throughout this paper, $T = (\Omega, E)$ is assumed to be a weighted finite-dimensional graph with $\Omega = \{1, 2, \ldots, n\}$.

In order to describe our problem more conveniently, we first introduce the following discrete $p\,$-Laplacian operators defined on a network.

Definition 1.1. For a function $v:\Omega\rightarrow \mathbb{R}$ and $p\in(2, +\infty)$, the discrete $p$-Laplacian $\Delta^{p}_{\omega}$ on $\Omega$ is defined by

When $p = 2$, it is called the discrete Laplacian $\Delta_{\omega}: = \Delta^2_{\omega}$ on $\Omega$, which is defined by

Recently, the classical Laplacian $\Delta$ was substituted by the discrete Laplacian $\Delta_{\omega}$ in graph Laplacian problems, and various methods and techniques to study the existence and qualitative properties of solutions have been developed ^{[2,5,6,7,8,9]}. Here we should emphasize that the discrete $p\,$-Laplacian operator $\Delta^{p}_{\omega}\, (p > 2)$ is actually nonlinear, which is different from the classical Laplacian $\Delta$ or the discrete Laplacian $\Delta_{\omega}$.

We are mainly interested in studying the blow-up properties of the solution of the following mutualistic model with a $p\,$-Laplacian ($p > 2$) defined on the networks

Here $v_i$ represents the spatial density of the $i^{th}$ species at time $t$ and $d_i$ represents its respective diffusion rate. The nonnegative constant $a_i$ is the birth rate, $b_i$ is its respective intraspecific competition and the parameter $c_i$ denotes the interspecific cooperation of the $i^{th}$ species.

Under the condition that $\Delta^{p}_{\omega}$ is replaced by the classical Laplacian in (1.3), the strong mutualistic ($b_1/c_1 < c_2/b_2$) population-based dynamical system experiences blow-up if the intrinsic growth rates of the population are large or the initial data size is sufficiently large ^[10]. In the case that $p = 2$ in (1.3), Liu et al. ^[1] proved that the solution blows up for all $x\in \Omega$, $\min\{v_1^0(x), v_2^0(x)\}\not\equiv 0$, under the strong mutualistic condition $b_1/c_1 < c_2/b_2$ and given $\min\{a_1/d_1, \, a_2/d_2\}\geq 1$.

In this paper, when $p > 2$, we can overcome the difficulties caused by the nonlinear operator $p\,$-Laplacian $\Delta^{p}_{\omega}$ and study the blow-up properties for the solution of system (1.3). First, we prove the Green formula for the nonlinear operator $\Delta^{p}_{\omega}$ and consider the eigenvalue problem $\Delta^{p}_{\omega}$. Second, with the help of the following important inequality (see Lemma 2.4)

the comparison principle of system (1.3) is constructed (see Theorem 2.5). Finally, we propose a new strong mutualistic condition

When condition (1.4) holds, it is proved that the solution of (1.3) blows up for all $x\in \Omega$, $\min\{v_1^0(x), v_2^0(x)\}\not\equiv 0$ (see Theorem 3.2).

2.   Preliminaries

Lemma 2.1. (Green formula for $\Delta^{p}_{\omega}$) For any functions $u, v:\Omega\rightarrow \mathbb{R}$, the $p\,$-Laplacian $\Delta^{p}_{\omega}$ satisfies that

Moreover, if $u = v$, we have

Proof. Using (1.1), we get

Meanwhile, we also deduce that

Hence, using (2.3) and (2.4), we obtain

which completes the proof. □

Lemma 2.2. Consider the following eigenvalue problem:

There exists

and $\Phi_1(x) > 0$ in $\Omega$ satisfying the conditions of the above system (2.5), and they are called the first eigenvalue and eigenfunction of (2.5), respectively. Furthermore, we have that $\lambda_1 = 0$.

Proof. Multiplying the first equation of (2.5) by $\varphi$ and integrating with respect to $\Omega$, we get

By (2.2), we deduce that

Hence we obtain

where the minimum can be attained by taking $\Phi_1 = \frac{1}{n}$, where $n$ is the number of vertices in $\Omega$ and $\Phi_1$ satisfies that $\sum_{x\in\Omega} \Phi_1(x) = 1$. Therefore, by taking $\Phi_1 = \frac{1}{n}$, we can get that $\lambda_1 = 0$; the proof is completed. □

Definition 2.3. For any $T > 0$, assume that for each $x\in \Omega$, $\hat{v}_1(x, \cdot), \hat{v}_2(x, \cdot)\in C([0, T])$ are differentiable in the range of $(0, T]$. If $(\hat{v}_1, \hat{v}_2)$ satisfies the following:

$(\hat{v}_1, \hat{v}_2)$ is called a lower solution (an upper solution) of (1.3) on $\Omega\times [0, T]$.

It is worth noting that the existence of the nonlinear operator $\Delta^{p}_{\omega}\, (p > 2)$ introduces difficulties when we construct the comparison principle of system (1.3). We introduce the following classical inequalities which will be used in the proof of the comparison principle. For the proofs the readers can refer to ^[11] (Section 10).

Lemma 2.4. (Lemma B.4 in ^[12]) For $p > 2$, $J_p(t): = |t|^{p-2}t$, we have

Moreover, if $b\geq a$, we have

With the help of inequality (2.8), we propose the following important comparison principle.

Theorem 2.5. (Comparison principle) Suppose that $(v_1, v_2)$ is a solution of system (1.3). If $(\hat{v}_1, \hat{v}_2)$ is a lower solution of (1.3) on $\Omega\times [0, T]$, then $(v_1, v_2)\geq(\hat{v}_1, \hat{v}_2)$ for $\Omega\times [0, T]$.

Proof. Denote $z_1: = (v_1-\hat{v}_1)e^{-Kt}$ and $z_2: = (v_2-\hat{v}_2)e^{-Kt}$, where $K > 0$ will be determined later. Notice that $\Omega$ is finite and $z_i(x, t)\, (i = 1, 2)$ is continuous in the range of $[0, T]$ for each $x\in \Omega$; there exists $(x_0, t_0)\in \Omega\times[0, T]$ such that

which immediately implies that

This is equivalent to

and

Recalling the definition of $\Delta^{p}_{\omega}$, we have

At the same time, due to the differentiability of $z_1(x, t)$ in the range of $(0, T]$, we obtain

Note that

we have

Denote

In view of (2.11), we have that $b_y\geq a_y$ for any $y\sim x_0$ and $y\in\Omega$. Combining this with (2.8) in Lemma 2.4, we deduce that

which implies that

Combining (2.16) with (2.14), we have

Note that $(v_1, v_2)$ is a solution and $(\hat{v}_1, \hat{v}_2)$ is a lower solution to system (1.3). That is, $(v_1, v_2)$ and $(\hat{v}_1, \hat{v}_2)$ respectively satisfy

and

Recall that $z_1: = (v_1-\hat{v}_1)e^{-Kt}$; we have

Combining (2.18)–(2.20), we obtain

Combining (2.17) with (2.21), we deduce that

where

Substituting (2.12) and (2.13) into (2.22), we deduce that

Next, we will prove that $z_1(x_0, t_0)\geq 0$ by contradiction. Alternatively, suppose that $z_1(x_0, t_0) = -\delta < 0$. Choosing

we obtain that $\big((-K+b_{11})z_1+b_{12}z_2\big)(x_0, t_0) > 0$, which contradicts (2.24). Hence we have that $z_1(x_0, t_0)\geq 0$. In view of (2.9), it follows that $z_1(x, t)\geq 0$ for $x\in \Omega, \, t\in[0, T]$. By a similar argument to that for $z_2$, we can also obtain that $z_2(x, t)\geq 0$ for $x\in \Omega, \, t\in[0, T]$. Thus, we obtain that $v_i\geq \hat{v}_i$ $(i = 1, 2)$ for $x\in \Omega, \, t\in[0, T]$. □

3.   Blow up result

Theorem 3.1. Let $U(x, t)$ be a solution of the following problem:

where $d, \alpha$ and $\beta$ are constants satisfying that $d > 0$ and $\beta > 0$. We have the following blow-up properties:

${\bf{(i)}}$ When $\alpha\geq 0$, $U(x, t)$ blows up for all nontrivial initial data.

${\bf{(ii)}}$ When $\alpha < 0$, $U(x, t)$ blows up for sufficiently large initial data.

Proof. Denote $M(t): = \sum_{x\in\Omega} \Phi_1(x)U(x, t)$, where $\Phi_1(x)$ is defined in Lemma 2.2. Deriving $M(t)$ with respect to $t$ and using (3.1), we have

Note that $\Phi_1(x)$ is a constant; then, due to (2.1) in Lemma 2.1, we have

Hence, combining the above equation with $\sum_{x\in\Omega} \Phi_1(x) = 1$ in Lemma 2.2, we deduce that

where Hölder's inequality is used.

When $\alpha\geq 0$, using (3.2), we immediately obtain the blow-up result.

When $\alpha < 0$, we choose a sufficiently large initial function $U(x, 0)$ which satisfies

It follows from (3.2) that $U(x, t)$ blows up. □

Theorem 3.2. If the strong mutualistic condition

is satisfied, the solution $(v_1, v_2)$ of (1.3) blows up for all $x\in \Omega$, $\min\{v_1^0(x), v_2^0(x)\}\not\equiv 0$.

Proof. We define $(\hat{v}_1(x, t), \hat{v}_2(x, t)): = (\delta_1 U(x, t), \delta_2U(x, t))$, where the constants $\delta_1, \delta_2$ and function $U(x, t)$ will be determined later. In order to ensure that $(\hat{v}_1(x, t), \hat{v}_2(x, t))$ is a lower solution of (1.3), we need to prove that $(\hat{v}_1(x, t), \hat{v}_2(x, t))\leq (v_{1}^0(x), v_{2}^0(x))$ and

Since the parameters satisfy that $\frac{b_1}{c_1} < \big(\frac{d_1}{d_2}\big)^{\frac{1}{p-2}} < \frac{c_2}{b_2}$, by calculation, we get

Thus, for a sufficiently small positive constant $\varepsilon$, we choose $\delta_1: = \varepsilon, \, \delta_2: = \big(\frac{d_1}{d_2}\big)^{\frac{1}{p-2}}\varepsilon$ such that $d_1\delta_1^{p-2} = d_2\delta_2^{p-2}$,

Denote

To prove (3.3), it suffices to show the following:

Hence $(\hat{v}_1(x, t), \hat{v}_2(x, t))$ is a lower solution of (1.3) provided that $(\hat{v}_1(x, 0), \hat{v}_2(x, 0))\leq (v_{1}^0(x), v_{2}^0(x))$. Let $w(x, 0): = \min\{v_{1}^0(x), \, v_{2}^0(x)\}$ and $\varepsilon$ be small enough such that $\delta_1, \delta_2 < 1$. Thus, $(\hat{v}_1(x, 0), \hat{v}_2(x, 0))\leq (v_{1}^0(x), v_{2}^0(x))$ holds.

Let $U$ be a solution of the following problem:

By applying Theorem 3.1 to $U$, we have that $U(x, t)$ blows up, which implies that $(\hat{v}_1(x, t), \hat{v}_2(x, t))$ blows up. With the help of Lemma 2.2, the blow-up properties of the solution of system (1.3) are then obtained. □

4.   Conclusions

This research contributes to the blow-up properties of a $p\,$-Laplacian ($p > 2$) reaction-diffusion system based on weighted graphs. The discrete $p\,$-Laplacian operator $\Delta^{p}_{\omega}\, (p > 2)$ is actually nonlinear, which is different from the classical Laplacian $\Delta$ or the discrete Laplacian $\Delta_{\omega}$. To overcome the difficulties caused by the nonlinearity, we establish Green Formula and comparison principle for the $p\,$-Laplacian operator. Hence, we develop a new strong mutualistic condition and prove the blow-up properties of the solution for any nontrivial initial data. In this sense, we extend the blow-up results of models with a graph Laplacian ($p = 2$) in ^[1] to a general graph $p\,$-Laplacian ($p > 2$).

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence tools in the creation of this article.

Acknowledgments

The authors would like to express their sincere thanks to the anonymous reviewers for their helpful comments. The work was partially supported by the National Natural Science Foundation of China (11771380) and Natural Science Foundation of Jiangsu Province (BK20191436).

Conflict of interest

The authors declare that there are no conflicts of interest.