Blow-Up Analysis for a Reaction–Diffusion System Coupled via $L^{\alpha }$-Norm-Type Sources under Positive Boundary Value Conditions

Abstract

This article mainly deals with the blow-up properties of nonnegative solutions for a reaction–diffusion system coupled with norm-type sources under positive boundary value conditions. The local existence of a nonnegative solution and the comparison principle are given. The criteria for the global existence or finite time blow-up of the solutions are obtained by constructing new functions and utilizing the super- and -sub-solution method. The results reveal a correlation between the blow-up profiles of the solutions and the size of the domain, as well as the positive boundary value.

1. Introduction

In this study, we consider the following reaction–diffusion system via $L^{\alpha }$-norm-type sources, for $i=1,2,\cdots ,k,$

$$u_i{}_t=\Delta u_i^{m_i}+a_i{\Vert u_i\Vert }_{{\alpha }_i}^{p_i}{\Vert u_{i+1}\Vert }_{{\beta }_i}^{q_i}, u_{k+1} =u_1 , x\in \mathrm{\Omega },t>0,$$

(1)

subject to positive boundary value conditions

$$u_i(x,t)={\epsilon }_0>0 , x\in \mathsf{\partial }\mathrm{\Omega },t>0,$$

(2)

and initial data

$$u_i(x,0)=u_{i,0}(x), x\in \mathrm{\Omega },$$

(3)

where $\mathrm{\Omega }\subset R^N$ is a bounded domain with smooth boundary $\mathsf{\partial }\mathrm{\Omega }$, $a_i>0$, $m_i>1,$ ${\alpha }_i\ge 1,$ ${\beta }_i\ge 1$, $p_i\ge 0, q_i>0$, the initial data $u_{i,0}(x)\ge {\epsilon }_0$ $(i=1,2,\cdots ,k)$ are bounded functions in $\mathrm{\Omega }$, and where ${\Vert ·\Vert }_{\alpha }^{\alpha }={\displaystyle {\int }_{\mathrm{\Omega }}{|·|}^{\alpha }dx}$.

The diffusion systems like (1)–(3) emerge in numerous applications across the fields of physics, chemistry, and biology; for instance, the study of the flow of a fluid through a homogeneous isotropic rigid porous medium or the investigation of combustion theory (see [1,2,3,4]), as well as the discussion of population models where communication occurs through chemical means (see [5,6,7,8]).

Over the last few decades, extensive research has been conducted on the blow-up properties of solutions to nonlinear parabolic equations with nonlocal sources under homogeneous Dirichlet boundary conditions. For example, in [9], Deng et al. investigated the following degenerate parabolic equation with the initial and boundary conditions

$$\begin{array}{l}{v}_t={(v^m)}_{xx}+a{\displaystyle {\int }_{-l}^l{v}^{q}dx}, x\in (-l,l),t>0,\\ v(\pm l,t)=0, t>0,\\ v(x,0)=v_0(x), x\in [-l,l],\end{array}$$

(4)

with $l>0, a>0$ and $q>m>1$. The authors proved that (4) has a global solution if $a$ is small enough, while the solution of (4) blows up in finite time if $a{\displaystyle {\int }_{-l}^l{v}_0^{q}dx}\ge 1$ and $\lambda <{(2al)}^{\frac{m}{q}-1}a{\displaystyle {\int }_{-l}^l\phi dx}$, where $\lambda$ and $\phi$ represent the first eigenvalue and the corresponding eigenfunction of the eigenvalue problem $-{\phi }_{xx}=\lambda \phi , x\in (-l,l)$, $\phi (\pm l)=0,$ respectively. Moreover, the blow-up set is the entire interval $[-l,l]$. Later, Duan et al. [4] extended problem (4) to a parabolic system with a nonlocal source and established the uniform blow-up profiles of solutions.

In [10], Deng et al. studied the following problem

$$u_t=\Delta u^m+a{\Vert v\Vert }_{\alpha }^p,v_t=\Delta v^n+b{\Vert u\Vert }_{\beta }^q , x\in \mathrm{\Omega },t>0 .$$

(5)

with zero homogeneous Dirichlet boundary data. They proved that if $pq<mn$, the non-negative solutions are global. For the case $pq\ge mn$, there are both global solutions and blow-up solutions, which depend on the initial data and the size of domain $\mathrm{\Omega }$.

Recently, Liu et al. [11] considered the following degenerate parabolic equations

$$u_t=\Delta u^m+a{\Vert v\Vert }_{{\alpha }_1}^{p_1}{\Vert u\Vert }_{{\alpha }_2}^{p_2}, v_t=\Delta v^n+b{\Vert u\Vert }_{\beta 1}^{q_1}{\Vert v\Vert }_{{\beta }_2}^{q_2}, x\in \mathrm{\Omega },t>0$$

(6)

subject to zero Dirichlet conditions. Based on the studies in [10,12], the authors discussed the criteria for determining whether the solutions of (6) would either exist globally or blow up in finite time. Furthermore, they established various forms of uniform blow-up behavior for simultaneous blow-up solutions.

For other related works on parabolic equations with nonlocal source and more interesting results, we refer the readers to [13,14,15,16,17,18,19,20] and the references therein.

However, there is also a lot of literature concerning the global existence and blow-up properties of solutions for parabolic equations under other boundary conditions, including nonlocal boundary conditions, Neumann conditions, Robin or like-Robin boundary conditions, and so on (see [21,22,23,24,25,26,27,28,29,30] and the references therein). Notably, Ling [21] focused on Equation (5) under positive Dirichlet boundary value conditions and pointed out that small diffusion exponents $m, n$ or large coupling exponents $p, q$ may lead to the blow-up of solutions. Simultaneously, the author demonstrated that the boundary values also play a significant role in determining the occurrence of blow-up.

Motivated by the works mentioned above, we would like to study the influence of the condition (3) on determining the behavior of both the global and blow-up solutions. Denoting $D={\displaystyle \prod _{i=1}^k(m_i-p_i)}-{\displaystyle \prod _{i=1}^k{q}_i}$, we have the following results.

Theorem 1.

If the solution of (1)–(3) is global, then $m_i\ge p_i$, for every $i\in \{1,2,\cdots ,k\}$.

Theorem 2.

Assuming that for every $i\in \{1,2,\cdots ,k\}$,  $m_i>p_i$,  $D>0$, then the solution of (1)–(3) exists globally.

Theorem 3.

Assuming that for every $i\in \{1,2,\cdots ,k\}$,  $m_i>p_i$,  $D=0$.

 (1)

If $a_i (i=1,2,\cdots ,k)$ are sufficiently small, then the solution of (1)–(3) exists globally;

 (2)

If the domain $\mathrm{\Omega }$ is sufficiently small, then the nonnegative solution of (1)–(3) is global;

 (3)

If the domain $\mathrm{\Omega }$ contains a sufficiently large ball, then the solution of (1)–(3) blows up in finite time provided that $u_{i,0}(x)$ are positive and continuous in $\mathrm{\Omega }$.

Theorem 4.

Assuming that for every $i\in \{1,2,\cdots ,k\}$,  $m_i\ge p_i$,  $D<0$, then the solution of (1)–(3) blows up in finite time provided that initial data $u_{i,0}(x)$ are large enough.

The rest of the article is organized as follows. In Section 2, we establish the local existence and the comparison principle which will be used for problems (1)–(3). Then, we give the proof of Theorems 1–4 concerning the global existence and blow-up in finite time in Section 3. Finally, some conclusions are summarized in Section 4.

2. Local Existence and Comparison Principle

Since equations of (1) are degenerate, there are usually no classical solutions. Therefore, we may give a definition of a weak solution for problem (1)–(3). For convenience, we denote $Q_T=\mathrm{\Omega }\times (0,T)$, $S_T=\mathsf{\partial }\mathrm{\Omega }\times (0,T)$ and define a class of test functions as

$$\mathrm{\Phi }\equiv \{\psi \in C(\overline{Q_T});{\psi }_t,\Delta \psi \in C(Q_T)\cap L^2(Q_T);\psi \ge 0,\psi (x,t)|{}_{S_T}=0\}$$

Definition 1.

A vector function $(u_1,u_2,\cdots ,u_k)$ is called a sub- (or super-) solution of (1)–(3) in $Q_T$, if the following conditions hold for every $i\in \{1,2,\cdots ,k\}$,

 (1)

$u_i\in L^{\infty }(Q_T)$;

 (2)

$u_i\le (\ge ){\epsilon }_0$, $(x,t)\in S_T$, and  $u_i(x,0)\le (\ge )u_{i,0}(x)$, $x\in \mathrm{\Omega }$;

 (3)

For every $t\in (0,T)$  and  ${\psi }_i\in \mathrm{\Phi }$,

$$\begin{array}{l} {\displaystyle {\int }_{\mathrm{\Omega }}(u_i(x,t){\psi }_i(x,t)-u_{i,0}(x){\psi }_i(x,0))dx}\le (\ge )\\ {\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathrm{\Omega }}(u_i{\psi }_{is}+u_i^{m_i}\Delta {\psi }_i+a_i{\Vert u_i\Vert }_{{\alpha }_i}^{p_i}{\Vert u_{i+1}\Vert }_{{\beta }_i}^{q_i}{\psi }_i)dx}}ds -{\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathsf{\partial }\mathrm{\Omega }}{u}_i^{m_i}\frac{\mathsf{\partial }{\psi }_i}{\mathsf{\partial }\nu }d\sigma ds}}, u_{k+1} =u_1 . \end{array}$$

(7)

Accordingly, we can say that the vector function $(u_1,u_2,\cdots ,u_k)$ is a weak solution of problem (1)–(3) if it is both a sub-solution and a super-solution. Moreover, we say $(u_1,u_2,\cdots ,u_k)$ is global, if $(u_1,u_2,\cdots ,u_k)$ is a solution of (1)–(3) in $Q_T$ for some $T<\infty$.

Next, we give a maximum principle, which is important for proving the local existence of a solution to (1)–(3).

Lemma 1.

Assuming that $w_i(x,t)\in C^{2,1}(Q_T)\cap C(\overline{Q_T})$  $(i=1,2,\cdots ,k)$ and satisfies

$$\{\begin{array}{l}{w}_{it}-d_i\Delta w_i\ge c_{i1}{w}_i+c_{i2}{w}_{i+1}+c_{i3}{\displaystyle {\int }_{\mathrm{\Omega }}{c}_{i4}{w}_{i+1}dx}+c_{i5}{\displaystyle {\int }_{\mathrm{\Omega }}{c}_{i6}{w}_{i}dx}, w_{k+1}=w_1 , (x,t)\in Q_T\\ w_i(x,t)\ge {\epsilon }_0, (x,t)\in S_T\\ w_i(x,0)\ge 0,x\in \mathrm{\Omega }\end{array}$$

where $c_{ij}(i=1,2,\cdots ,k;j=1,2,3,4,5,6)$ are bounded functions,

$$c_{ij}(x,y), d_i(x,y)\ge 0 (i=1,2,\cdots ,k;j=1,2,\cdots ,6)$$

in $Q_T$.

Then, $w_i(x,t)\ge 0$ on $\overline{Q_T}$.

Proof. 

It can be proven by the similar method in [25,26], so we omit it here. □

The local existence of solutions may be proven by the regularization procedure, so we adopt a similar method (see [15]) and consider the following regularized system, for $i=1,2,\cdots ,k$, and $n\in Z^{+},$

$$\{\begin{array}{l}{u}_{i,nt}=\Delta f_{i,n}(u_{i,n})+a_i{\Vert g_{i,n}(u_{i,n})\Vert }_{{\alpha }_i}^{p_i}·{\Vert g_{i+1,n}(u_{i+1,n})\Vert }_{{\beta }_i}^{q_i}, u_{k+1,n} =u_{1,n},(x,t)\in Q_T,\\ u_{i,n}(x,t)={\epsilon }_0+1/n, \hspace{1em}(x,t)\in S_T,\\ u_{i,n}(x,0)=u_{i,0}^j+1/n, x\in \mathrm{\Omega }.\end{array}$$

(8)

where $u_{i,0}^j$ is a smooth approximation of $u_{i,0}(x)$ with $\mathrm{supp}{u}_{i,0}^j\subset \mathrm{\Omega }$, and

$$f_{i,n}(u_{i,n})=\{\begin{array}{l}{u}_{i,n}^{m_i}, u_{i,n}\ge 1/n,\\ (1/n{)}^{m_i}, u_{i,n}<1/n,\end{array} g_{i,n}(u_{i,n})=\{\begin{array}{l}{u}_{i,n}, u_{i,n}\ge 1/n,\\ 1/n, u_{i,n}<1/n.\end{array}$$

By using a discussion similar to that of Theorems A.1–A.4 in [13], it is known that the problem of (8) has a unique classical solution $u_{i,n}^j$$\in C(\overline{\mathrm{\Omega }}\times [0,T_j(n)))$$\cap C^{2,1}(\mathrm{\Omega } \times$$(0,T_j(n)))$ for $0<T_j(n)<\infty$, where $T_j(n)$ is the maximal existence time. A direct calculation and the classical maximum principle lead to $u_{i,n}^j\ge {\epsilon }_0+1/n$. Therefore, $u_{i,n}^j(i=1,2,\cdots ,k)$ satisfies

$$u_{i,n}^j{}_t=\Delta {(u_{i,n}^j)}^{m_i}+a_i{\Vert u_{i,n}^j\Vert }_{{\alpha }_i}^{p_i}·{\Vert u_{i+1,n}^j\Vert }_{{\beta }_i}^{q_i}, u_{k+1,n}^j=u_{1,n}^j, (x,t)\in Q_{T_j(n)},$$

(9)

with the corresponding initial and boundary conditions. Obviously, passing to the limit $j\to \infty$, it follows that

$$u_{i,n}\equiv \underset{j\to \infty }{\lim }{u}_{{}_{i,n}}^j,$$

and $(u_{1,n},u_{2,n},\cdots ,u_{k,n})$ is a weak solution of

$$u_{i,n}{}_t=\Delta u_{i,n}^{m_i}+a_i{\Vert u_{i,n}\Vert }_{{\alpha }_i}^{p_i}·{\Vert u_{i+1,n}\Vert }_{{\beta }_i}^{q_i}, u_{k+1,n}=u_{1,n}, (x,t)\in Q_{T(n)}$$

(10)

with the corresponding initial and boundary conditions on $(0,T(n))$, where $T(n)\equiv$ $\underset{j\to \infty }{\lim }{T}_j(n)$ is the maximal existence time. Here, a weak solution of (10) is defined similarly to that for problem (1)–(3), only the equalities for $u_i$, (7) may be replaced with

$$\begin{array}{l}{\displaystyle {\int }_{\mathrm{\Omega }}[u_{i,n}(x,t){\psi }_i(x,t)-(u_{i,0}(x)+\frac{1}{n}){\psi }_i(x,0)]dx}\\ ={\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathrm{\Omega }}[u_{i,n}{\psi }_{is}+u_{i,n}^{m_i}\Delta {\psi }_i+a_i{\Vert u_{i,n}\Vert }_{{\alpha }_i}^{p_i}{\Vert u_{i+1,n}\Vert }_{{\beta }_i}^{q_i}{\psi }_i]dx}}ds-{({\epsilon }_0+\frac{1}{n})}^{m_i}{\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathsf{\partial }\mathrm{\Omega }}\frac{\mathsf{\partial }{\psi }_i}{\mathsf{\partial }\nu }d\sigma }}ds.\end{array}$$

(11)

Due to $u_{i,n}\ge {\epsilon }_0+\frac{1}{n}$ and Lemma 1, we have Lemmas 2 and 3 which will be used to prove the local existence of solutions. The proof is standard (see [15]).

Lemma 2.

Assuming that $w_i(x,t)$$\in C(\overline{\mathrm{\Omega }}\times [0,T_j(n)))\cap C^{2,1}(\mathrm{\Omega }\times (0,T_j(n)))$ for $0<$$T_j(n)<\infty$, and $(w_1,w_2,\cdots ,w_n)$ is a sub- (or super-) solution of 9). Then, $(w_1,w_2,\cdots ,$$w_n)\le (\ge )$$(u_{1,n}^j,u_{2,n}^j,$$\cdots ,u_{k,n}^j)$ on $\overline{\mathrm{\Omega }}\times [0,T_j(n))$.

Lemma 3.

If $n_1>n_2$, then $u_{i,n_1}^j\le u_{i,n_2}^j, (i=1,2,\cdots ,k)$ on $\overline{\mathrm{\Omega }}\times [0,T_j(n_2))$ and $T_j(n_1)>T_j(n_2)$.

Hence, the limit $T^{*}\equiv \underset{n\to \infty }{\lim }T(n)$ exists and there are the pointwise limits

$$u_i(x,t)\equiv \underset{n\to \infty }{\lim }{u}_{i,n}(x,t), (i=1,2,\cdots ,k),$$

for any $(x,t)\in \overline{\mathrm{\Omega }}\times [0,T^{*})$. In addition, as the convergence of the sequence $u_{i,n}$ is monotone, passage to the limit $n\to \infty$ in identities (11) for ${\psi }_i\in \mathrm{\Phi }$ and $t\in [0,T^{*})$, the following theorem can be established by monotone and dominated convergence theorems.

Theorem 5.

(Local existence). Assuming that for every $i\in \{1,2,\cdots ,k\}$, nonnegative functions $u_{i,0}\in L^{\infty }(\mathrm{\Omega })$, there is some $T^{*}=T^{*}(u_{i,0})>0$ such that there exists a nonnegative weak solution $(u_1,u_2,\cdots ,u_k)$ of (1)–(3) for each $T<T^{*}$. Furthermore, either $T^{*}=\infty$ or $\underset{t\to T^{*}}{\lim }\underset{x\in \overline{\mathrm{\Omega }}}{\max }{\displaystyle \sum _{i=1}^k\left|u_i(x,t)\right|}=+\infty$.

Proposition 1.

(Comparison principle). Let $({\underset{\_}{u}}_1,{\underset{\_}{u}}_2,\cdots ,{\underset{\_}{u}}_k)$ and $({\overline{u}}_1,{\overline{u}}_2,\cdots ,{\overline{u}}_k)$ be a nonnegative sub-solution and a super-solution of (1)–(3), respectively. If $({\underset{\_}{u}}_{1,0},{\underset{\_}{u}}_{2,0},\cdots ,$${\underset{\_}{u}}_{k,0})\le ({\overline{u}}_{1,0},{\overline{u}}_{2,0},\dots ,{\overline{u}}_{k,0})$ and either

$${\displaystyle {\int }_{\mathrm{\Omega }}{\underset{\_}{u}}_i^{{\alpha }_i}}dx\ge \delta >0,{\displaystyle {\int }_{\mathrm{\Omega }}{\underset{\_}{u}}_{i+1}^{{\beta }_i}}dx\ge \delta >0, {\underset{\_}{u}}_{i+1}={\underset{\_}{u}}_1, i=1,2,\cdots ,k,$$

(12)

or

$${\displaystyle {\int }_{\mathrm{\Omega }}{\overline{u}}_i^{{\alpha }_i}}dx\ge \delta >0,{\displaystyle {\int }_{\mathrm{\Omega }}{\overline{u}}_{i+1}^{{\beta }_i}}dx\ge \delta >0, {\overline{u}}_{i+1}={\overline{u}}_1, i=1,2,\cdots ,k,$$

(13)

 hold. Then, $({\underset{\_}{u}}_1,{\underset{\_}{u}}_2,\cdots ,{\underset{\_}{u}}_k)\le ({\overline{u}}_1,{\overline{u}}_2,\cdots ,{\overline{u}}_k)$ on $Q_T$.

Although the proof is quite standard and similar to that in [8,15], the comparison principle is very important in proving the existence or blow-up of the solution of (1)–(3). Therefore, we sketch the outline for the reader’s convenience.

Proof. 

Subtracting the first inequalities of (7) for ${\underset{\_}{u}}_i$ and ${\overline{u}}_i$ yields

$$\begin{array}{l}{\displaystyle {\int }_{\mathrm{\Omega }}[{\underset{\_}{u}}_i(x,t)-{\overline{u}}_i(x,t)]{\psi }_i(x,t)dx}\le {\displaystyle {\int }_{\mathrm{\Omega }}({\underset{\_}{u}}_{i,0}-{\overline{u}}_{i,0}){\psi }_i(x,0)dx}\\ +{\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathrm{\Omega }}({\underset{\_}{u}}_i-{\overline{u}}_i)[{\psi }_{is}+{\mathrm{\Phi }}_i(x,s)\Delta {\psi }_i]dx}}ds\\ +a_i{\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathrm{\Omega }}{\Vert {\underset{\_}{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i}({\displaystyle {\int }_{\mathrm{\Omega }}{\psi }_i}dx)}{D}_i(s)G_i(x,s)({\underset{\_}{u}}_i-{\overline{u}}_i)}dxds\\ +a_i{\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathrm{\Omega }}{\Vert {\overline{u}}_i\Vert }_{{\alpha }_i}^{p_i}({\displaystyle {\int }_{\mathrm{\Omega }}{\psi }_i}dx)}{E}_i(s)H_i(x,s)({\underset{\_}{u}}_{i+1}-{\overline{u}}_{i+1})}dxds,\end{array}$$

where

$${\mathrm{\Phi }}_i(x,s)={\displaystyle {\int }_0^1{m}_i{[\theta {\overline{u}}_i+(1-\theta ){\underset{\_}{u}}_i]}^{m_i-1}}d\theta$$

$$D_i(s)={{\displaystyle {\int }_0^1\frac{p_i}{{\alpha }_i}[\theta {\displaystyle {\int }_{\mathrm{\Omega }}{\overline{u}}_i^{{\alpha }_i}}dx+(1-\theta ){\displaystyle {\int }_{\mathrm{\Omega }}{\underset{\_}{u}}_i^{{\alpha }_i}}dx]}}^{\frac{p_i}{{\alpha }_i}-1}d\theta$$

$$G_i(x,s)={\displaystyle {\int }_0^1{\alpha }_i{[\theta {\overline{u}}_i+(1-\theta ){\underset{\_}{u}}_i]}^{{\alpha }_i-1}}d\theta$$

$$E_i(s)={{\displaystyle {\int }_0^1\frac{q_i}{{\beta }_i}[\theta {\displaystyle {\int }_{\mathrm{\Omega }}{\overline{u}}_{i+1}^{{\beta }_i}}dx+(1-\theta ){\displaystyle {\int }_{\mathrm{\Omega }}{\underset{\_}{u}}_{i+1}^{{\beta }_i}}dx]}}^{\frac{q_i}{{\beta }_i}-1}d\theta$$

$$H_i(x,s)={\displaystyle {\int }_0^1{\beta }_i{[\theta {\overline{u}}_{i+1}+(1-\theta ){\underset{\_}{u}}_{i+1}]}^{{\beta }_i-1}}d\theta$$

Since ${\underset{\_}{u}}_i$ and ${\overline{u}}_i$ are bounded $Q_T$, it follows from $m_i>1, {\alpha }_i, {\beta }_i\ge 1$ that ${\mathrm{\Phi }}_i(x,s),$ $H_i(x,s)$ are bounded. Clearly, if $p_i/{\alpha }_i,$ $q_i/{\beta }_i\ge 1$, then $D_i(s)$ and $E_i(s)$ are also bounded. On the other hand, if $p_i/{\alpha }_i,$ $q_i/{\beta }_i<1$, we have $D_i(s)\le {\delta }^{p_i/{\alpha }_i-1},$ $E_i(s)\le {\delta }^{q_i/{\beta }_i-1}$ according to the assumptions (12) or (13). Therefore, we may choose some suitable test function ${\psi }_i\in \mathrm{\Phi }$ as in [15] (pp. 118–123) to obtain

$$\begin{array}{l}{\displaystyle {\int }_{\mathrm{\Omega }}{[{\underset{\_}{u}}_i(x,t)-{\overline{u}}_{i+1}(x,t)]}_{+}dx}\\ \le {\Vert {\psi }_i\Vert }_{\infty }{\displaystyle {\int }_{\mathrm{\Omega }}{({\underset{\_}{u}}_{i,0}-{\overline{u}}_{i,0})}_{+}dx} +C_i{\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathrm{\Omega }}[{({\underset{\_}{u}}_i-{\overline{u}}_i)}_{+}}}+{({\underset{\_}{u}}_{i+1}-{\overline{u}}_{i+1})}_{+}]dxds,\end{array}$$

(14)

where $U_{+}\equiv \max \{U(x,t),0\}$ and $C_i$ are bounded constants. Thus,

$$\begin{array}{l}{\displaystyle {\int }_{\mathrm{\Omega }}{\displaystyle \sum _{i=1}^k{[{\underset{\_}{u}}_i(x,t)-{\overline{u}}_i(x,t)]}_{+}}dx}\\ \le \underset{1\le i\le k}{\max }\{{\Vert {\psi }_i\Vert }_{\infty }\}{\displaystyle {\int }_{\mathrm{\Omega }}{\displaystyle \sum _{i=1}^k{({\underset{\_}{u}}_{i,0}-{\overline{u}}_{i,0})}_{+}}dx}+C{\displaystyle {\int }_0^t{\displaystyle {\int }_{\mathrm{\Omega }}{\displaystyle \sum _{i=1}^k{({\underset{\_}{u}}_i-{\overline{u}}_i)}_{+}}}}dxds,\end{array}$$

(15)

where $C=\underset{1\le i\le k}{\max }\{C_i+C_{i+1}\}, C_{k+1}=C_1$. By (15), it follows from the Gronwall lemma that ${\underset{\_}{u}}_i\le {\overline{u}}_i$ since ${\underset{\_}{u}}_{i,0}\le {\overline{u}}_{i,0}$. □

Based on the above argument, it can be observed that the solutions of (1)–(3) are unique provided that $p_i\ge {\alpha }_i, q_i\ge {\beta }_i$. As a result, we have the following corollary.

Corollary 1.

Assume that $p_i\ge {\alpha }_i, q_i\ge {\beta }_i$$(i=1,2,\cdots ,k)$. Let $({\underset{\_}{u}}_1,{\underset{\_}{u}}_2,\cdots ,{\underset{\_}{u}}_k)$ and $({\overline{u}}_1,{\overline{u}}_2,$$\cdots ,{\overline{u}}_k)$ be a sub-solution and a super-solution of (1)–(3), respectively. Then, $({\underset{\_}{u}}_1,{\underset{\_}{u}}_2,\cdots ,{\underset{\_}{u}}_k)\le ({\overline{u}}_1,{\overline{u}}_2,\cdots ,{\overline{u}}_k)$ on $Q_T$ if ${\underset{\_}{u}}_{i,0}\le {\overline{u}}_{i,0}$.

Denote

$$A=\left(\begin{array}{ccccc}{m}_1-p_1& -q_1& \cdots & 0& 0\\ 0& m_2-p_2& \cdots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& \cdots & m_{k-1}-p_{k-1}& -q_{k-1}\\ -q_k& 0& \cdots & 0& m_k-p_k\end{array}\right),l=\left(\begin{array}{c}{l}_1\\ l_2\\ ⋮\\ l_{k-1}\\ l_k\end{array}\right)$$

By the theory of linear system, we give the following lemma, which will be used later.

Lemma 4.

Assuming that $m_i>p_i$ and $D=0$, then there exist positive constants $l_1,l_2,\cdots ,l_k$, such that $Al={(0,0,\cdots ,0)}^T$.

Proof. 

Due to $D=0$, then $rank(A)<k$, where $rank(A)$ represents the rank of matrix $A$. It implies that there must exist nonzero solutions to the linear system $Al={(0,0,\cdots ,0)}^T$. Without loss of generality, by taking $l_1=\theta >0$, and $l_j={\displaystyle \prod _{i=1}^{j-1}\frac{(m_i-p_i)\theta }{q_i}}>0$, $(j=2,3,\cdots ,k)$, the lemma has been proven directly. □

3. Proof of Global Existence and Blow-Up

In this section, we will employ the super- and sub-solution method to prove the Theorems 1–4. By the comparison principle, we only need to construct appropriate super-solutions or sub-solutions for problem (1)–(3). Firstly, we give the proof of Theorem 1, which is a necessary condition for global solutions.

Proof of Theorem 1.

If the inference is not true, we may assume without loss of generality that $p_1>m_1$ and consider the following problem

$$\{\begin{array}{l}{w}_t=\Delta w^{m_1}+a_1{\left|\mathrm{\Omega }\right|}^{q_1/{\beta }_1}{\epsilon }_0^{q_1}{\Vert w\Vert }_{{\alpha }_1}^{p_1}, (x,t)\in Q_T,\\ w(x,t)={\epsilon }_0>0, (x,t)\in S_T,\\ w(x,0)=u_{1,0}(x), x\in \mathrm{\Omega }.\end{array}$$

(16)

By [17], the solution $w$ of (16) blows up in finite time. Since $u_i\ge {\epsilon }_0$$(i=2,3,$$\cdots ,k)$, ${\epsilon }_0^{q_1}{\left|\mathrm{\Omega }\right|}^{q_1/{\beta }_1}{\Vert u_1\Vert }_{{\alpha }_1}^{p_1}\le {\Vert u_1\Vert }_{{\alpha }_1}^{p_1}{\Vert u_2\Vert }_{{\beta }_1}^{q_1}$, it implies that $(w,{\epsilon }_0,\cdots ,{\epsilon }_0)$ is the sub-solution of (1)–(3). Thus, by the comparison principle, the solution $(u_1,u_2,\cdots ,u_k)$ of (1)–(3) blows up in finite time. □

Proof of Theorem 2.

Now, let $\phi (x)$ be the unique positive solution of the following elliptic problem:

$$-\Delta \phi (x)=1, x\in \mathrm{\Omega }; \phi (x)=1, x\in \mathsf{\partial }\mathrm{\Omega }.$$

(17)

Taking $C_0=\underset{x\in \mathrm{\Omega }}{\max }\phi (x)$, then $1\le \phi (x)\le C_0, x\in \mathrm{\Omega } .$

Define ${\overline{u}}_1,\cdots ,{\overline{u}}_k$ as the following:

$${\overline{u}}_1=A^{l_1}{\phi }^{1/m_1}(x), {\overline{u}}_2=A^{l_2}{\phi }^{1/m_2}(x),\cdots ,{\overline{u}}_k=A^{l_k}{\phi }^{1/m_k}(x).$$

(18)

where $l_i$$(i=1,2,\cdots ,k)$ and $A>0$ are positive constants to be fixed later. Clearly, ${\overline{u}}_i\ge A^{l_i}$. For $i=1,2,\cdots ,k$, a series of computations yields

$$\begin{array}{l}\Delta {\overline{u}}_i^{m_i}=-A^{m_i{l}_i},\\ a_i{\Vert {\overline{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\overline{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i}=a_i{A}^{p_i{l}_i+q_i{l}_{i+1}}{\Vert {\phi }^{1/m_i}\Vert }_{{\alpha }_i}^{p_i}{\Vert {\phi }^{1/m_{i+1}}\Vert }_{{\beta }_i}^{q_i}\le a_i{A}^{p_i{l}_i+q_i{l}_{i+1}}{C}_0{}^{\frac{p_i}{m_i}+\frac{q_i}{m_{i+1}}}{\left|\mathrm{\Omega }\right|}^{\frac{p_i}{{\alpha }_i}+\frac{q_i}{{\beta }_i}},\end{array}$$

(19)

where $l_{k+1}=l_1, m_{k+1}=m_1.$ Denote

$$A_i={(a_i{C}_0{}^{\frac{p_i}{m_i}+\frac{q_i}{m_{i+1}}}{\left|\mathrm{\Omega }\right|}^{\frac{p_i}{{\alpha }_i}+\frac{q_i}{{\beta }_i}})}^{1/(m_i{l}_i-p_i{l}_i-q_i{l}_{i+1})}, i=1,2,\cdots ,k.$$

(20)

Due to $m_i>p_i$, $D>0$, one may find two positive constants $l_1$ and $l_k$ such that

$$\frac{m_k-p_k}{q_k}>\frac{l_1}{l_k}>\frac{{\displaystyle \prod _{i=1}^{k-1}{q}_i}}{{\displaystyle \prod _{i=1}^{k-1}(m_i-p_i)}}$$

(21)

Furthermore, there exist constants $l_i>0 (i=2,3,$$\cdots ,k-1)$ that satisfy

$$\frac{l_1}{l_2}>\frac{q_1}{m_1-p_1}, \frac{l_2}{l_3}>\frac{q_2}{m_2-p_2},\cdots ,\frac{l_{k-1}}{l_k}>\frac{q_{k-1}}{m_{k-1}-p_{k-1}}.$$

Thus,

$$p_1{l}_1+q_1{l}_2<m_1{l}_1, p_2{l}_2+q_2{l}_3<m_2{l}_2,\cdots ,p_k{l}_k+q_k{l}_1<m_k{l}_k.$$

(22)

By (20) and (22), we choose $A$ sufficiently large to satisfy

$$A\ge \underset{1\le i\le k}{\max }\{A_i,{\epsilon }_0^{1/l_i},{\left({\Vert u_{i,0}(x)\Vert }_{\infty }\right)}^{1/l_i}\}$$

(23)

Due to ${\overline{u}}_i{}_t=0$ ($i=1,2,\cdots ,k$), we have

$${\overline{u}}_i{}_t-\Delta {\overline{u}}_i^{m_i}-a_i{\Vert {\overline{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\overline{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i}\ge 0, (x,t)\in Q_T.$$

(24)

On the other hand, for $(x,t)\in S_T$,

$${\overline{u}}_i=A^{l_i}{\phi }^{1/m_i}(x)\ge {\epsilon }_0$$

(25)

and

$${\overline{u}}_i(x,0)\ge u_{i,0}(x),x\in \mathrm{\Omega }.$$

(26)

It follows from (24)–(26) that $({\overline{u}}_1,\cdots ,{\overline{u}}_k)$ is a positive super-solution of (1)–(3). Hence, $(u_1,u_2,\cdots ,u_k)\le ({\overline{u}}_1,\cdots ,{\overline{u}}_k)$ by comparison principle, and it implies that $(u_1,$ $u_2,\cdots ,u_k)$ exists globally. □

Proof of Theorem 3.

Case (1). According to Lemma 4, there exist positive constants $l_i (1\le i\le k)$ satisfying

$$m{l}_1=p_1{l}_1+q_1{l}_2, \cdots ,m_{k-1}{l}_{k-1}=p_{k-1}{l}_{k-1}+q_{k-1}{l}_k,m_k{l}_k=p_k{l}_k+q_k{l}_1.$$

(27)

Clearly, if $a_i$$(i=1,2,\cdots ,k)$ are sufficiently small such that

$$a_i\le (C_0{}^{\frac{p_i}{m_i}+\frac{q_i}{m_{i+1}}}{\left|\mathrm{\Omega }\right|}^{\frac{p_i}{{\alpha }_i}+\frac{q_i}{{\beta }_i}}{)}^{-1} ; m_{k+1}=m_1 .$$

(28)

Choosing $A\ge \underset{1\le i\le k}{\max }\{{\epsilon }_0^{1/l_i},{\left({\Vert u_{i,0}(x)\Vert }_{\infty }\right)}^{1/l_i}\}$, it follows from (24)–(26) that $({\overline{u}}_1,\cdots ,{\overline{u}}_k)$ defined by (18) is a positive super-solution of (1)–(3). This means that the solution $(u_1,\cdots ,u_k)$ of (1)–(3) exists globally.

Case (2). Next, we consider the case where the domain $\mathrm{\Omega }$ is small enough.

Denote by ${\lambda }_1$ the first eigenvalue of the following eigenvalue problem,

$$-\Delta \varphi (x)=\lambda \varphi (x), x\in \mathrm{\Omega }; \varphi (x)=0, x\in \mathsf{\partial }\mathrm{\Omega }$$

(29)

with the corresponding eigenfunction ${\varphi }_1(x)$. Then, ${\lambda }_1>0$ and ${\varphi }_1(x)>0$ in $\mathrm{\Omega }$. It is well known that ${\lambda }_1$ continuously depends on the domain $\mathrm{\Omega }$. Then, for an arbitrary constant ${\lambda }_2\in (0,{\lambda }_1)$, we can find a bounded domain ${\mathrm{\Omega }}_1\supset \mathrm{\Omega }$ such that ${\lambda }_2$ is the first eigenvalue of the following eigenvalue problem

$$\Delta \varphi (x)+\lambda \varphi (x)=0, x\in {\mathrm{\Omega }}_1; \varphi (x)=0, x\in \mathsf{\partial }{\mathrm{\Omega }}_1$$

(30)

Let ${\varphi }_2(x)$ be the corresponding eigenfunction of (30) with ${\lambda }_2$. Here, ${\varphi }_2(x)$ can be normalized as ${\Vert {\varphi }_2(x)\Vert }_{\infty }=1$ and ${\varphi }_2(x)>0$ in ${\mathrm{\Omega }}_1$. So, there exists some constant ${\delta }_0>0$ such that ${\varphi }_2(x)>{\delta }_0$ on $\mathrm{\Omega }$.

Now, we define the functions ${\overline{u}}_i(x,t),(i=1,2,\cdots ,k)$ as follows:

$${\overline{u}}_i(x,t)=M^{l_i}{\varphi }_2{}^{1/m_i}(x), x\in \mathrm{\Omega },t>0,$$

where $l_i$ satisfies (27), and $M>0$ will be determined later.

Note ${\overline{u}}_{it}=0$. After a direct calculation, we get

$$\Delta {\overline{u}}_i^{m_i}+a_i{\Vert {\overline{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\overline{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i}=M^{m_i{l}_i}{\varphi }_2(-{\lambda }_2+c_i{\varphi }_2{}^{-1})\le M^{m_i{l}_i}{\varphi }_2(-{\lambda }_2+c_i{\delta }_0{}^{-1})$$

(31)

where

$$c_i=a_i{\Vert {\varphi }_2{}^{1/m_i}(x)\Vert }_{{\alpha }_i}^{p_i}{\Vert {\varphi }_2{}^{1/m_{i+1}}(x)\Vert }_{{\beta }_i}^{q_i}>0, m_{k+1}=m_1$$

(32)

By the relationship between the domain and the corresponding first eigenvalues of problem (30) (see [31]), if the domain $\mathrm{\Omega }$ is sufficiently small such that ${\lambda }_1>$${\lambda }_0=\underset{1\le i\le k}{\max }\{c_i{\delta }_0{}^{-1}\}$, then we can choose ${\lambda }_2$, satisfying ${\lambda }_0<{\lambda }_2<{\lambda }_1$. Thus, (31) implies that

$${\overline{u}}_i{}_t\ge \Delta {\overline{u}}_i^{m_i}-a_i{\Vert {\overline{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\overline{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i}, x\in \mathrm{\Omega },t>0.$$

(33)

By taking $M$ appropriately large to satisfy $M\ge \underset{1\le i\le k}{\max }\{{({\delta }_0{}^{-1/m_i}{\Vert u_{i,0}\Vert }_{\infty })}^{1/l_i}\}$, we have

$${\overline{u}}_i(x,t)\ge {\epsilon }_0 , x\in \mathsf{\partial }\mathrm{\Omega },t>0.$$

(34)

and

$${\overline{u}}_i(x,0)\ge u_{i,0}(x),$$

(35)

Thus, we prove that $({\overline{u}}_1,{\overline{u}}_2,$$\cdots ,{\overline{u}}_k)$ is a super-solution of (1)–(3). By the comparison principle, the nonnegative solution $(u_1,\cdots ,u_k)$ of (1)–(3) is global.

Case (3). Thanks to ${\displaystyle \prod _{i=1}^k(m_i-p_i)}={\displaystyle \prod _{i=1}^k{q}_i}$, there exist positive constants $l_i^{\prime }$$(i=1,2,\cdots ,$ $k)$ that are appropriately large such that

$$\frac{m_i-p_i}{q_i}=\frac{l_{i+1}^{\prime }}{l_i^{\prime }}, \mathrm{and} (m_i-1)l_i^{\prime }>1, l_{i+1}^{\prime }=l_1^{\prime }.$$

(36)

Thus,

$$l_i^{\prime }+1<m_i{l}_i^{\prime }=p_i{l}_i^{\prime }+q_i{l}_{i+1}^{\prime }$$

(37)

To complete the proof, without loss of generality, we can assume that $0\in \mathrm{\Omega }$, and $B_R\subset \subset \mathrm{\Omega }$ is an open ball with center at 0 and radius $R$.

Denoted by ${\lambda }_{B_R}>0$ the first eigenvalue of the following eigenvalue problem

$$-{\varphi }^{″}(r)-\frac{N-1}{r}{\varphi }^{\prime }(r)=\lambda \varphi (r), r\in (0,R); {\varphi }^{\prime }(0)=0, (R)=0$$

(38)

with the corresponding eigenfunction ${\varphi }_R(r)$. Normalized as ${\varphi }_R(r)>0$ in $B_R$ and ${\varphi }_R(0)=\underset{B_R}{\max }{\varphi }_R(r)=1$. Utilizing the scaling property (let $\tau =\frac{r}{R}$) of eigenvalues and eigenfunctions, we obtain that ${\lambda }_{B_R}={\lambda }_{B_1}{R}^{-2}$, ${\varphi }_R(r)=$${\varphi }_1(\frac{r}{R})$$={\varphi }_1(\tau )$, where ${\lambda }_{B_1}$ and ${\varphi }_1(\tau )$ are the first eigenvalue and the corresponding normalized eigenfunction of the eigenvalue problem in the unit ball $B_1$. Thus, $\underset{B_1}{\max }{\varphi }_1={\varphi }_1(0)={\varphi }_R(0)=\underset{B_R}{\max }{\varphi }_R=1$.

Constructing functions ${\underset{\_}{u}}_i(x,t) , (i=1,2,\cdots ,k)$ as follows:

$${\underset{\_}{u}}_i(x,t)={\epsilon }_0{(1+d(1-ct))}^{-l_i^{\prime }}{\varphi }_R{}^{l_i^{\prime }}(\left|x\right|), (x,t)\in B_R\times (0,\frac{1+d}{cd})$$

(39)

where $c,d>0$ are determined later. Clearly,$({\underset{\_}{u}}_1,{\underset{\_}{u}}_2,\cdots ,{\underset{\_}{u}}_k)$ blows up in finite time $T\le \frac{1+d}{cd}$.

For $i=1,2,\cdots ,k$, by directly calculating

$$\begin{array}{l}{\underset{\_}{u}}_i{}_t=l_i^{\prime }cd{\epsilon }_0{(1+d(1-ct))}^{-l_i^{\prime }-1}{\varphi }_R{}^{l_i^{\prime }},\\ \Delta {\underset{\_}{u}}_i^{m_i}={(1+d(1-ct))}^{-m_i{l}_i^{\prime }}[-{\lambda }_{B_R}{m}_i{l}_i^{\prime }{\epsilon }_0^{m_i}{\varphi }_R{}^{m_i{l}_i^{\prime }}+m_i{l}_i^{\prime }(m_i{l}_i^{\prime }-1){\epsilon }_0^{m_i}{\varphi }_R{}^{m_i{l}_i^{\prime }-2}{{\varphi }^{\prime }}_R{}^2],\\ {\Vert {\underset{\_}{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\underset{\_}{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i} =M_{i,i+1}{\epsilon }_0^{p_i+q_i}{(1+d(1-ct))}^{-p_i{l}_i^{\prime }-q_i{l}_{i+1}^{\prime }},M_{i,i+1}={\Vert {\varphi }_R{}^{l_i^{\prime }}\Vert }_{{\alpha }_i}^{p_i}{\Vert {\varphi }_R{}^{l_{i+1}^{\prime }}\Vert }_{{\beta }_i}^{q_i}, l_{k+1}^{\prime }=l_1^{\prime }.\end{array}$$

Then, for $(x,t)\in \mathrm{\Omega }\times (0,\frac{1+d}{cd})$, we have

$$\begin{array}{l}{\underset{\_}{u}}_i{}_t-\Delta {\underset{\_}{u}}_i^{m_i}-a_i{\Vert {\underset{\_}{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\underset{\_}{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i} \\ \le \frac{d{\epsilon }_0{\varphi }_R{}^{l_i^{\prime }}}{{[1+d(1-ct)]}^{l_i^{\prime }+1}}\left[c{l}_i^{\prime }-\frac{{\epsilon }_0^{m_i-1}}{d{[1+d(1-ct)]}^{m_i{l}_i^{\prime }-l_i^{\prime }-1}}\left(-{\lambda }_{B_R}{m}_i{l}_i^{\prime }+a_i{M}_{i,i+1}{\epsilon }_0^{p_i+q_i-m_i}\right)\right].\end{array}$$

(40)

Letting

$${\lambda }_{i,i+1}=\frac{a_i{M}_{i,i+1}{\epsilon }_0^{p_i+q_i-m_i}}{m_i{l}_i^{\prime }}$$

In view of ${\lambda }_{B_R}={\lambda }_{B_1}{R}^{-2}$, we may assume that the ball $B_R$ is sufficiently large, i.e., the radius $R$ is large enough such that

$${\lambda }_{B_R}<\underset{1\le i\le k}{\min }\left\{{\lambda }_{i,i+1}\right\}$$

then

$${\delta }_{i,i+1}=a_i{M}_{i,i+1}{\epsilon }_0^{p_i+q_i-m_i}-{\lambda }_{B_R}{m}_i{l}_i^{\prime }>0,(i=1,2,\cdots ,k).$$

(41)

Taking $c\le \underset{1\le i\le k}{\min }\left\{\frac{{\epsilon }_0^{m_i-1}{\delta }_{i,i+1}}{l_i^{\prime }d{(1+d)}^{m_i{l}_i-l_i-1}}\right\}$.

Since $u_{i,0}$ are a positive and continuous in $\mathrm{\Omega }$, we choose $d>0$ large enough to satisfy

$${\underset{\_}{u}}_i(x,0)\le u_{i,0}(x),x\in B_R$$

(42)

On the other hand,

$${\underset{\_}{u}}_i(x,t)=0<{\epsilon }_0, x\in \mathsf{\partial }{B}_R\times (0,\frac{1+d}{cd}).$$

(43)

By (40)–(43), $({\underset{\_}{u}}_1,{\underset{\_}{u}}_2,\cdots ,{\underset{\_}{u}}_k)$ is a positive sub-solution of (1)–(3), which blows up in finite time in the ball $B_R$. It implies that the solution of (1)–(3) blows up in the larger domain $\mathrm{\Omega }$. □

Proof of Theorem 4.

Due to $m_i\ge p_i$, ${\displaystyle \prod _{i=1}^k(m_i-p_i)}<{\displaystyle \prod _{i=1}^k{q}_i}$ for $i\in \{1,2,\cdots ,k\}$, there exist positive constants $l_i$$(i=1,2,\cdots ,k)$ such that

$$(m_1-p_1)l_1<q_1{l}_2, (m_2-p_2)l_2 <q_2{l}_3,\cdots , (m_k-p_k)l_k<q_k{l}_1, \mathrm{and} (m_i-1)l_i>1.$$

(44)

For these positive constants $l_i$, we construct functions ${\underset{\_}{u}}_i(x,t)$ as follows:

$${\underset{\_}{u}}_i(x,t)={\epsilon }_0{(1+d(1-ct))}^{-l_i}{e}^{l_i{\varphi }_1(x)}, (x,t)\in \mathrm{\Omega }\times (0,\frac{1+d}{cd}),$$

(45)

where $c,d>0$ can be determined later, and ${\varphi }_1(x)$ is the first eigenfunction of the problem (29) with the corresponding first eigenvalue ${\lambda }_1>0$. Here, ${\varphi }_1(x)$ can be normalized as ${\Vert {\varphi }_1\Vert }_{\infty }=1$ and ${\varphi }_1(x)>0$ in $\mathrm{\Omega }$.

Clearly, for $(x,t)\in \mathsf{\partial }\mathrm{\Omega }\times (0,\frac{1+d}{cd})$, and $i\in \{1,2,\cdots ,k\}$,

$${\underset{\_}{u}}_i(x,t)\le {\epsilon }_0$$

(46)

Moreover, for $(x,t)\in \mathrm{\Omega }\times (0,\frac{1+d}{cd})$, we have

$$\begin{array}{l} {\underset{\_}{u}}_i{}_t={\epsilon }_0{l}_{i}cd{(1+d(1-ct))}^{-l_i-1}{e}^{l_i{\varphi }_1(x)},\\ \Delta {\underset{\_}{u}}_i^{m_i}={(1+d(1-ct))}^{-m_i{l}_i}[-{\lambda }_1{m}_i{l}_i{\epsilon }_0^{m_i}{e}^{m_i{l}_i{\varphi }_1(x)}{\varphi }_1+{(m_i{l}_i)}^2{\epsilon }_0^{m_i}{e}^{m_i{l}_i{\varphi }_1(x)}{\left|\nabla {\varphi }_1\right|}^2],\\ {\Vert {\underset{\_}{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\underset{\_}{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i}=b_{i,i+1}{\epsilon }_0^{p_i+q_i}{(1+d(1-ct))}^{-p_i{l}_i-q_i{l}_{i+1}}, b_{i,i+1}={\Vert e^{l_i{\varphi }_1}\Vert }_{{\alpha }_i}^{p_i}{\Vert e^{l_{i+1}{\varphi }_1}\Vert }_{{\beta }_i}^{q_i}, l_{k+1}=l_1\end{array}$$

Thus, for every $i\in \{1,2,\cdots ,k\}$,

$$\begin{array}{l}{\underset{\_}{u}}_i{}_t-\Delta {\underset{\_}{u}}_i^{m_i}-a_i{\Vert {\underset{\_}{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\underset{\_}{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i}\\ \le \frac{l_{i}cd{\epsilon }_0}{{[1+d(1-ct)]}^{l_i+1}}{e}^{l_i{\varphi }_1(x)}+\frac{{\lambda }_1{m}_i{l}_i{\epsilon }_0^{m_i}{e}^{m_i{l}_i{\varphi }_1(x)}{\varphi }_1}{{[1+d(1-ct)]}^{m_i{l}_i}}-\frac{a_i{b}_{i,i+1}{\epsilon }_0^{p_i+q_i}}{{[1+d(1-ct)]}^{p_i{l}_i+q_i{l}_{i+1}}}\end{array}$$

(47)

$$\begin{array}{l}=\frac{d{\epsilon }_0{e}^{l_i{\varphi }_1(x)}}{{[1+d(1-ct)]}^{l_i+1}}\left[c{l}_i+\frac{{\lambda }_1{m}_i{l}_i{\epsilon }_0^{m_i-1}{e}^{(m_i-1)l_i{\varphi }_1(x)}{\varphi }_1}{d{[1+d(1-ct)]}^{m_i{l}_i-l_i-1}}-\frac{a_i{b}_{i,i+1}{\epsilon }_0^{p_i+q_i-1}{e}^{-l_i{\varphi }_1(x)}}{d{[1+d(1-ct)]}^{p_i{l}_i+q_i{l}_{i+1}-l_i-1}}\right]\\ \le \frac{d{\epsilon }_0{e}^{l_i{\varphi }_1(x)}}{{[1+d(1-ct)]}^{l_i+1}}\left[c{l}_i-\frac{{\epsilon }_0^{m_i-1}}{d{e}^{l_i}{[1+d(1-ct)]}^{m_i{l}_i-l_i-1}}\left(-{\lambda }_1{m}_i{l}_i{e}^{m_i{l}_i}+\frac{a_i{b}_{i,i+1}{\epsilon }_0^{p_i+q_i-m_i}}{{(1+d)}^{p_i{l}_i+q_i{l}_{i+1}-m_i{l}_i}}\right)\right].\end{array}$$

Letting

$$d<\underset{1\le i\le k}{\min }\left\{\frac{1-d_{i,i+1}}{d_{i,i+1}}\right\}$$

(48)

where $d_{i,i+1}={\left(\frac{{\lambda }_1{m}_i{l}_i{e}^{m_i{l}_i}}{a_i{b}_{i,i+1}{\epsilon }_0^{p_i+q_i-m_i}}\right)}^{1/(p_i{l}_i+q_i{l}_{i+1}-m_i{l}_i)}$.

Thus,

$${\delta }_{i,i+1}=-{\lambda }_1{m}_i{l}_i{e}^{m_i{l}_i}+\frac{a_i{b}_{i,i+1}{\epsilon }_0^{p_i+q_i-m_i}}{{(1+d)}^{p_i{l}_i+q_i{l}_{i+1}-m_i{l}_i}}>0.$$

(49)

Taking

$$c\le \underset{1\le i\le k}{\min }\left\{\frac{{\epsilon }_0^{m_i-1}{\delta }_{i,i+1}}{l_{i}d{e}^{l_i}{(1+d)}^{m_i{l}_i-l_i-1}}\right\}$$

By (47)–(49), we have

$${\underset{\_}{u}}_i{}_t-\Delta {\underset{\_}{u}}_i^{m_i}-a_i{\Vert {\underset{\_}{u}}_i\Vert }_{{\alpha }_i}^{p_i}{\Vert {\underset{\_}{u}}_{i+1}\Vert }_{{\beta }_i}^{q_i} \le 0, (x,t)\in \mathrm{\Omega }\times (0,\frac{1+d}{cd}).$$

(50)

Assuming that $u_{i,0}(x)$ are properly large such that

$${\underset{\_}{u}}_i(x,0)\le u_{i,0}(x),x\in \mathrm{\Omega }.$$

(51)

Then, it follows from (47), (50), and (51) that $({\underset{\_}{u}}_1,{\underset{\_}{u}}_2,\cdots ,{\underset{\_}{u}}_k)$ is a positive sub-solution of (1)–(3). Due to $({\underset{\_}{u}}_1,{\underset{\_}{u}}_2,\cdots ,{\underset{\_}{u}}_k)$, it blows up in time $T\le \frac{1+d}{cd}$, and so does the solution $(u_1,\cdots ,u_k)$ of (1)–(3). □

4. Conclusions

In this article, we discuss the existence and blow-up properties of a diffusion system with positive boundary value conditions. As the above arguments, the positive boundary value conditions guarantee the local existence of solutions of (1)–(3), which also influence the occurrence of the blow-up phenomenon. In the meantime, the blow-up profiles are also affected by the interactions among the multi-nonlinearities $m_i, p_i, q_i$. We derive a critical case ${\displaystyle \prod _{i=1}^k(m_i-p_i)}$$={\displaystyle \prod _{i=1}^k{q}_i}$, which belongs to the scenario of global existence (or blow-up) under other assumptions, such as small $a_i$ or small (or large) size of the domain Ω. Comparing the results of this article with those under homogeneous Dirichlet boundary conditions, as in [11] (with $k=2$ in Equation (1)), we observe differences in the blow-up profiles of the solutions when considering positive value boundary conditions. In this case, the application of specialized processing skills and the construction of new auxiliary functions become necessary to overcome the difficulties caused by boundary values. It is worth noting that the global existence of the two situations appears to be similar, as stated in Theorems 2 and 3(1). The global solutions of (1)–(3) may be independent of the boundary value ${\epsilon }_0$. However, Theorems 3(3) and 4 show that the blow-up situation is slightly different, where ${\epsilon }_0$ plays a significant role.

In fact, it follows from the proof of Theorem 3(3) that the global nonexistence of solutions mainly depends on the relation between the boundary value ${\epsilon }_0$ and ${\lambda }_{B_R}$, where ${\lambda }_{B_R}$ is the characteristic of the size of $B_R$. That is to say, if we fix the boundary value ${\epsilon }_0$, that ${\lambda }_{B_R}$ should be properly small (i.e., $B_R$ is large enough) such that

$${\lambda }_{B_R}<\underset{1\le i\le k}{\min }\left\{a_i{M}_{i,i+1}{\epsilon }_0^{p_i+q_i-m_i}{(m_i{l}_i^{\prime })}^{-1}\right\}$$

Correspondingly, for fixed $\mathrm{\Omega }$ and $p_i+q_i>m_i$, ${\epsilon }_0$ should be suitably large to satisfy

$${\epsilon }_0\ge \underset{1\le i\le k}{\max }\{{[{\lambda }_{B_R}{m}_i{l}_i^{\prime }{(a_i{M}_{i,i+1})}^{-1}]}^{1/(p_i+q_i-m_i)}\}$$

Similarly, we can observe the relation according to the inequality (49).

As a special case, when the domain $\mathrm{\Omega }$ is symmetric, particularly when it takes the form of a ball (or an interval), the results of this paper also hold. However, the following problem is still unsolved: how to construct blow-up solutions in an explicit form supporting the theoretical results obtained above for system (1)–(3). Symmetry-based methods (see, e.g., [32,33]) can be useful for its solving.