Critical Exponents in a Doubly Degenerate Nonlinear Parabolic System with Inner Absorptions

Abstract

This paper deals with critical exponents for a doubly degenerate nonlinear parabolic system coupled via local sources and with inner absorptions under null Dirichlet boundary conditions in a smooth bounded domain. The author first establishes the comparison principle and local existence theorem for the above problem. Then under appropriate hypotheses, the author proves that the solution either exists globally or blows up in finite time depends on the initial data and the relations of the parameters in the system. The critical exponent of the system is simply described via a characteristic matrix equation introduced.

1 Introduction and Main Results

In this paper, we consider the following nonlocal doubly degenerate nonlinear parabolic system with inner absorptions

$$\begin{aligned} \begin{array}{ll} u_t-\Delta _{m,p}u=u^{\alpha _1}v^{\beta _1}-a u^{r}\!, &{}\,(x,t)\in \Omega _T,\\ v_t-\Delta _{n,q}v=u^{\alpha _2}v^{\beta _2}-b v^{s}\!, &{}\, (x,t)\in \Omega _T,\\ \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{ll} u(x,t)=v(x,t)=0, &{}\quad (x,t)\in \partial \Omega \times (0,T],\\ u(x,0)=u_0(x), v(x,0)=v_0(x), &{}\quad x\in \Omega ,\\ \end{array} \end{aligned}$$

(1.1)

where for $k>0,\gamma >2$ and $N\ge 1$, $\Delta _{k,\gamma }\Theta =\nabla \cdot (|\nabla \Theta ^{k}|^{\gamma -2} \cdot \nabla \Theta ^{k}), \quad \nabla \Theta ^{k}=k\Theta ^{k-1}(\Theta _{x_1},\ldots ,\Theta _{x_N}),$ $\Omega \subset \mathbb {R}^N(N\ge 1)$ is a bounded domain with appropriately smooth boundary $\partial \Omega$; $m, n, r, s\ge 1$, $p,q>2$, $\alpha _i,\beta _i\ge 0$, $i=1,2$, $\Omega _T=\Omega \times (0,T]$ and $a,b$ are positive constants and $u_0,v_0$ satisfies compatibility and the following conditions:

$$\begin{aligned}&(H)\ u^m_0\in C(\overline{\Omega })\cap W^{1,p}_0(\Omega ),\quad v^n_0\in C(\overline{\Omega })\cap W^{1,q}_0(\Omega ) \hbox { and } \nabla u_0^m\cdot \nu <0, \\&\quad \nabla v_0^n\cdot \nu <0 \hbox { on }\partial \Omega ,\, \hbox {where}\, \nu \, \hbox {is unit outer normal vector on}\, \partial \Omega . \end{aligned}$$

Parabolic systems like (1.1) arise in many applications in the fields of mechanics, physics, and biology like, for instance, the description of turbulent filtration in porous media, the theory of non-Newtonian fluids perturbed by nonlinear terms and forced by rather irregular period in time excitations, the flow of a gas through a porous medium in a turbulent regime or the spread of biological (see [1, 6, 8, 15] and references therein); In the non-Newtonian fluids theory, the pair $(p,q)$ is a characteristic quantity of medium. When $(m,n)\ge (1,1)$ and $(p,q)>(2,2)$, the system models the non-stationary, polytropic flow of a fluid in a porous medium; it has been intensively studied (see [2, 10, 13, 16, 18] and references therein).

The problems with nonlinear reaction term, absorption term, and nonlinear diffusion include blow-up and global existence conditions of solutions, blow-up rates and blow-up sets, etc. This degenerate system exhibiting a doubly nonlinearity generalizes the porous medium system $(p =q= 2)$ and the parabolic p-Laplace system $(m =n= 1)$, which has been studied by many authors. For $p=q=2$, $m=n=1$, it is a classical reaction-diffusion system of Fujita type. Bedjaoui and Souplet [3] considered the critical blow-up exponents for the following system

$$\begin{aligned} u_t=\Delta u+v^p-b_1u^{r},\quad v_t=\Delta v+u^q-b_2v^s,&x\in \Omega , t>0. \end{aligned}$$

(1.2)

By constructing self-similar weak subsolutions with compact supports, they obtained the critical exponent: $pq = \max (r, 1) \max (s, 1)$. Moreover scalar absorption-diffusion equations of the style $u_t-\Delta u=-u^r$ have also been widely studied (see [7, 9, 11] and references therein).

Zheng and Su [22] considered the quasilinear reaction-diffusion system with nonlocal sources and inner absorptions of the form

$$\begin{aligned} u_t=\Delta u^{m}+\int _{\Omega }v^pdx-au^{r},\quad v_t=\Delta v^n+\int _{\Omega }u^qdx-bv^s,&x\in \Omega , t>0. \end{aligned}$$

(1.3)

They established the critical exponent and the blow-up rate for the system subject to homogeneous Dirichlet conditions and nonnegative initial data. It was found that the critical exponent is determined by the interaction among all the six nonlinear exponents from all the three kinds of the nonlinearities.

For p-Laplacian systems, Yang and Lu [19] studied the following equations

$$\begin{aligned}&u_t-\hbox {div}(|\nabla u|^{p-2}\nabla u)=u^{\alpha _1}v^{\beta _1}, \quad (x,t)\in \Omega \times (0,T],\nonumber \\&v_t-\hbox {div}(|\nabla v|^{q-2}\nabla v)=u^{\alpha _2}v^{\beta _2}, \quad (x,t)\in \Omega \times (0,T], \end{aligned}$$

(1.4)

with the homogeneous Dirichlet boundary value conditions, they derived some estimates near the blow-up point for positive solutions and non-existence of positive solutions of the relate elliptic systems.

Very recently, Zhang et al. [21] further studied the blow-up properties of positive solutions for system (1.1) with nonlocal sources

$$\begin{aligned}&u_t-\hbox {div}(|\nabla u|^{p-2}\nabla u)=\int _{\Omega }v^{m}dx-u^{r}\!, \quad (x,t)\in \Omega \times (0,T],\nonumber \\&v_t-\hbox {div}(|\nabla v|^{q-2}\nabla v)=\int _{\Omega }u^{n}dx-v^s\!,\quad (x,t)\in \Omega \times (0,T] \end{aligned}$$

(1.5)

in a smooth bounded domain $\Omega \subset \mathbb {R}^N$. Under appropriate hypotheses, they discussed the global existence and blow-up of positive weak solutions using a comparison principle. For $r=s=0$, the system (1.1) is reduced to a local non-Newton polytropic filtration system without inner absorptions. And the author [16, 17] dealt with it under local and nonlocal sources. Under appropriate hypotheses, they all establish local theory of the solutions and prove that the solution either exists globally or blows up in finite time. More results for the non-Newton polytropic filtration system with sources can be found in [12, 20, 23] and the references therein.

However, as far as we know, there is little literature on the blow-up properties for problems (1.1) with the concentrated source and inner absorptions. Motivated by the above works, in this paper, we investigate the blow-up properties of solutions of the problem (1.1) and extend the results of [3, 16, 20, 21, 23] to more generalized cases. Due to the nonlinear diffusion terms and doubly degeneration for $u=v=0$ and $|\nabla u|=|\nabla v|=0$, we have some new difficulties to be overcome. Noticing that the system (1.1) includes the Newtonian filtration system $(p=q=2)$ and the non-Newtonian filtration system $(m=n=1)$ formally, so the method for it should be synthetic. In fact, we can use the methods for the above two systems to deal with it. In order to apply monotonicity, we establish the comparison principle for system (1.1) by choosing suitable test function and Gronwall’s inequality. Then by the first eigenvalue and its corresponding eigenfunctions to the eigenvalue problem for the non-Newtonian filtration system, we construct a pair of well-ordered positive supersolution and subsolution. Using comparison principle, we achieve our purpose and obtain the global existence and blow-up of solutions to the problem. We will show that the critical exponent is determined by the interaction among all the nonlinear exponents from all the three nonlinearities. Correspondingly, two kinds of characteristic algebraic systems are introduced to get simple descriptions for the critical exponent and the blow-up considered.

In order to state our results, we introduce some useful symbols. Throughout this paper, we let $\zeta (x)$ and $\vartheta (x)$ be the unique solution of the following elliptic equation (see [4, 23]),

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{m,p}\zeta =1, &{}\quad x\in \Omega ,\\ \zeta =0, &{}\quad x\in \partial \Omega ,\\ \end{array}\right. \qquad \left\{ \begin{array}{ll} -\Delta _{n,q}\vartheta =1, &{}\quad x\in \Omega ,\\ \vartheta =0, &{}\quad x\in \partial \Omega .\\ \end{array}\right. \end{aligned}$$

(1.6)

Before starting the main results, we introduce a pair of parameters $(\mu _1,\mu _2)$ solving the following characteristic algebraic system

$$\begin{aligned} \left( \begin{array}{cc} -\mu _1 &{} \beta _1 \\ \alpha _2 &{} -\mu _2 \\ \end{array} \right) \left( \begin{array}{c} \tau \\ \theta \\ \end{array} \right) =\left( \begin{array}{c} 1 \\ 1 \\ \end{array} \right) , \end{aligned}$$

namely,

$$\begin{aligned} \tau =\dfrac{\beta _1+\mu _2}{\beta _1\alpha _2-\mu _1\mu _2},\ \theta =\dfrac{\alpha _2+\mu _1}{\beta _1\alpha _2-\mu _1\mu _2} \end{aligned}$$

with

$$\begin{aligned} \mu _1=\max \{m(p-1)-\alpha _1,r-\alpha _1\},\ \mu _2=\max \{n(q-1)-\beta _2,s-\beta _2\}. \end{aligned}$$

It is obvious that $1/\tau$ and $1/\theta$ share the same signs. We claim that the critical exponent of problem (1.1) should be $(1/\tau , 1/\theta )=(0, 0)$, described by the following theorems.

Theorem 1.1

Assume that $(1/\tau , 1/\theta )<(0, 0)$, then there exist solutions of (1.1) being globally bounded.

Theorem 1.2

Assume that $(1/\tau , 1/\theta )>(0, 0)$, then the nonnegative solution of (1.1) blows up in finite time for sufficiently large initial values and exists globally for sufficiently small initial values.

Theorem 1.3

Assume that $(1/\tau , 1/\theta )=(0, 0)$, $\zeta (x)$ and $\vartheta (x)$ are defined in (1.6), respectively.

• (i) Suppose that $r>m(p-1)$ and $s>n(q-1)$. If

  $$\begin{aligned} a^{\alpha _2}b^{r-\alpha _1}\ge 1, \end{aligned}$$

  then the solutions are globally bounded for small initial data; if

  $$\begin{aligned} \vartheta ^{\beta _1}>a\zeta ^{r-\alpha _1}, \zeta ^{\alpha _2}>b\vartheta ^{s-\beta _2}, \end{aligned}$$

  then the solutions blow-up in finite time for large data.

• (ii) Suppose that $r<m(p-1)$ and $s<n(q-1)$. If

  $$\begin{aligned} \zeta ^{\frac{\alpha _2}{n(q-1)-\beta _2}+\frac{\alpha _1}{\beta _1}} \vartheta ^{\frac{n(q-1)}{n(q-1)-\beta _2}} \le 1, \end{aligned}$$

  then the solutions are globally bounded for small initial data; if

  $$\begin{aligned} \zeta ^{\alpha _1}\vartheta ^{\beta _1}>1, \zeta ^{\alpha _2}\vartheta ^{\beta _2}>1, \end{aligned}$$

  then the solutions blow-up in finite time for large data.

• (iii) Suppose that $r<m(p-1)$ and $s>n(q-1)$. If

  $$\begin{aligned} \zeta ^{\alpha _2+\frac{\alpha _1(s-\beta _2)}{\beta _1}}\le b, \end{aligned}$$

  then the solutions are globally bounded for small initial data; if

  $$\begin{aligned} \zeta ^{\alpha _1}\vartheta ^{\beta _1}>1, \zeta ^{\alpha _2}>b\vartheta ^{s-\beta _2}, \end{aligned}$$

  then the solutions blow-up in finite time for large data.

• (iv) Suppose that $r>m(p-1)$ and $s<n(q-1)$. If

  $$\begin{aligned} \vartheta ^{\alpha _1+\frac{\alpha _2(r-\beta _1)}{\beta _2}}\le a, \end{aligned}$$

  then the solutions are globally bounded for small initial data; if

  $$\begin{aligned} \vartheta ^{\beta _1}>a\zeta ^{r-\alpha _1}, \zeta ^{\alpha _2}\vartheta ^{\beta _2}>1, \end{aligned}$$

  then the solutions blow-up in finite time for sufficiently large data.

The rest of this paper is organized as follows. In Sect. 2, we shall establish the comparison principle and local existence theorem for problem (1.1). Theorems 1.1 and 1.2 will be proved in Sects. 3 and 4, respectively. Finally, we will give the proof of Theorem 1.3 in Sect. 5.

2 Preliminaries

In order to study the globally existing and blowing-up solutions to problem (1.1), we need to firstly prove the comparison principle for the weak solution of the system (1.1). It worth to mention, this statement plays a crucial role in the investigation. Additions, the existence of local-in-time weak solutions of (1.1) under appropriate hypotheses is also studied in this section. From a physical point of view, we need only to consider the non-negative solutions. Moreover, if we assume that $u_0(x),v_0(x)\ge 0$ in $\Omega$, by Lemma 2.1 (see it below), we can obtain that $(u(x,t),v(x,t))\ge (0,0)$ a.e. in $(\Omega \times (0,T))\times (\Omega \times (0,T))$. Thus, we only consider the non-negative solutions in later sections.

As it is well known that doubly degenerate equations need not have classical solutions, we give a precise definition of a weak solution for problem (1.1). Let $\Omega _T=\Omega \times (0,T]$, $S_T=\partial \Omega \times [0,T]$, $T>0$.

Definition 2.1

A pair of functions $(u,v)$ is called a solution of the problem (1.1) on $\overline{\Omega }_T\times \overline{\Omega }_T$ if and only if $u^m(x,t)\in C(0,T;L^\infty (\Omega ))\cap L^p(0,T;W_0^{1,p}(\Omega ))$, $v^n(x,t)\in C(0,T;L^\infty (\Omega ))\cap L^q(0,T;W_0^{1,q}(\Omega ))$, $(u^m)_t\in L^2(0,T;L^2(\Omega ))$, $(v^n)_t\in L^2(0,T;L^2(\Omega ))$, $u(x,0)=u_0(x), v(x,0)=v_0(x)$ and the equalities

$$\begin{aligned}&\int _{\Omega }u(x,t_2)\psi _1(x,t_2)\mathrm{d}x-\int _{\Omega }u(x,t_1) \psi _1(x,t_1)\mathrm{d}x=\int ^{t_2}_{t_1}\!\!\!\int _{\Omega }u\psi _{1t}\mathrm{d}x\mathrm{d}t\nonumber \\&-\int ^{t_2}_{t_1}\!\!\!\!\int _{\Omega }|\nabla u^m|^{p-2} \nabla u^m\cdot \nabla \psi _1\mathrm{d}x\mathrm{d}t+a\int ^{t_2}_{t_1}\!\!\!\!\int _{\Omega } \psi _1(x,t)(u^{\alpha _1}v^{\beta _1}-au^{r})\mathrm{d}x\mathrm{d}t,\end{aligned}$$

(2.1)

$$\begin{aligned}&\int _{\Omega }v(x,t_2)\psi _2(x,t_2)\mathrm{d}x-\int _{\Omega }v(x,t_1) \psi _2(x,t_1)\mathrm{d}x=\int ^{t_2}_{t_1}\int _{\Omega }v\psi _{2t}\mathrm{d}x\mathrm{d}t\nonumber \\&-\int ^{t_2}_{t_1}\!\!\!\int _{\Omega }|\nabla v^n|^{q-2} \nabla v^n\cdot \nabla \psi _2\mathrm{d}x\mathrm{d}t +b\int ^{t_2}_{t_1}\!\!\!\int _{\Omega }\psi _2(x,t)(u^{\alpha _2}v^{ \beta _2}-bv^{s})\mathrm{d}x\mathrm{d}t \end{aligned}$$

(2.2)

hold for all $0 < t_1 < t_2 < T$, where $\psi _1(x,t), \psi _2(x,t)\in C^{1,1}(\overline{Q}_T)$ such that $\psi _1(x,T)=\psi _2(x,T)=0$ and $\psi _1(x,t)=\psi _2(x,t)=0$ on $S_T$.

Similarly, to define a subsolution $(\underline{u}(x,t),\underline{v}(x,t))$ we need only to require that $\psi _1(x,t)\ge 0, \psi _2(x,t)\ge 0$, $(\underline{u}(x,0),\underline{v}(x,0))\le (u_0(x),v_0(x))$ on $\Omega \times \Omega$, $(\underline{u}(x,t),\underline{v}(x,t))\le (0,0)$ on $S_T\times S_T$ and the equalities in (2.1) and (2.2) are replaced by $\le$. A supersolution can be defined similarly.

Definition 2.2

We say the solution $(u,v)$ of the problem (1.1) blows up in finite time if there exists a positive constant $T^\star <\infty$, such that

$$\begin{aligned} \lim \limits _{t\rightarrow T^{\star -}}(|u(\cdot ,t)|_{L^\infty (\Omega )} +|v(\cdot ,t)|_{L^\infty (\Omega )})=+\infty . \end{aligned}$$

We say the solution $(u,v)$ exists globally if

$$\begin{aligned} \sup \limits _{t\in (0,+\infty )}(|u(\cdot ,t)|_{L^\infty (\Omega )}+ |v(\cdot ,t)|_{L^\infty (\Omega )})<+\infty . \end{aligned}$$

By a modification of the method given in [16–18], we obtain the following results.

Theorem 2.1

Suppose that $(u_0,v_0)\ge (0,0)$ and satisfies the conditions (H), then there exists a constant $T_0> 0$ such that the problem (1.1) admits a unique solution $(u,v)\in Q_{T_0}\times Q_{T_0}$, $u^m\in C(0,T;L^\infty (\Omega ))\cap L^p(0,T;W_0^{1,p}(\Omega ))$, $v^n\in C(0,T;L^\infty (\Omega ))\cap L^q(0,T;W_0^{1,q}(\Omega ))$.

Proof of Theorem 2.1

Consider the following approximate problems for the problem (1.1):

$$\begin{aligned}&u_{it}-\hbox {div}((|\nabla u^m_i|^2+\varepsilon _i)^{\frac{p-2}{2}}\nabla u^m_i)= u_i^{\alpha _1}v_i^{\beta _1}-au_i^{r}, (x,t)\in \Omega _T,\nonumber \\&v_{it}-\hbox {div}((|\nabla v^n_i|^2+\sigma _i)^{\frac{q-2}{2}}\nabla v^n_i)= u_i^{\alpha _2}v_i^{\beta _2}-bv_i^{s}, (x,t)\in \Omega _T,\nonumber \\&u_{i}(x,t)=\varepsilon _i, v_{i}(x,t)=\sigma _i, (x,t)\in S_T,\nonumber \\&u_{i}(x,0)=u_{0\varepsilon _i}(x)+\varepsilon _i, v_{i}(x,0)=v_{0\sigma _i}(x)+\sigma _i, x\in \Omega . \end{aligned}$$

(2.3)

Here $\varepsilon _i, \sigma _i$ are strictly decreasing sequences, $0<\varepsilon _i, \sigma _i<1$, and $\varepsilon _i\rightarrow 0^+, \sigma _i\rightarrow 0^+$ as $i\rightarrow +\infty$. $u_{0\varepsilon _i}, v_{0\sigma _i}\in C_0^\infty (\Omega )$ are approximation functions for the initial data $u_0(x)$ and $v_0(x)$, respectively. $|u_{0\varepsilon _i}+\varepsilon _i|_{L^\infty (\Omega )}\le |u_0+1|_{L^\infty (\Omega )}$, $|\nabla u_{0\varepsilon _i}^m|_{L^\infty (\Omega )}\le |\nabla u_0^m|_{L^\infty (\Omega )}$, for all $\varepsilon _i$, and $(u_{0\varepsilon _i}+\varepsilon _i)^m\rightarrow u_0^m$ strongly in $W_0^{1,p}(\Omega )$; $|v_{0\sigma _i}+\sigma _i|_{L^\infty (\Omega )}\le |v_0+1|_{L^\infty (\Omega )}$, $|\nabla v^n_{0\sigma _i}|_{L^\infty (\Omega )}\le |\nabla v^n_0|_{L^\infty (\Omega )}$, for all $\sigma _i$, and $(v_{0\sigma _i}+\sigma _i)^n\rightarrow v_0^n$ strongly in $W_0^{1,q}(\Omega )$.

(2.3) is a non-degenerate problem for each fixed $\varepsilon _i$ and $\sigma _i$; it is easy to prove that it admits a unique classic solution $(u_i,v_i)$ using the Schauder’s fixed point theorem and $(u_i, v_i)\ge (\varepsilon _i,\sigma _i)>(0,0)$ by the classical theory for parabolic equations(see [10]). To find limit function $u(x,t)$ and $v(x,t)$ of the sequence $\{(u_i, v_i)\}$, we need some priori estimates for the nonnegative approximate solutions by carefully choosing special test functions and a scaling argument. The left arguments are as same as those of Theorem 1 in [16], so we omit them. We complete the existence part by a standard limiting process.

The uniqueness of the solution is obvious. In fact, assume that $(u_1,v_1), (u_2,v_2)$ are two non-negative solutions of (1.1), using Lemma 2.1 repeatedly, we can get $u_1=u_2, v_1=v_2$ a.e. in $\overline{\Omega }\times [0,T_0]$.$\square$

We first give a comparison lemma for the non-degenerate parabolic system, which plays a crucial role in the proof of our results.

Proposition 2.1

(Comparison Principle) Suppose that $(\underline{u}(x,t),\underline{v}(x,t))$ and $(\overline{u}(x,t),\overline{v}(x,t))$ are the lower and upper solution of problem (1.1) on $\overline{\Omega }_T\times \overline{\Omega }_T$, respectively. Then $(\underline{u}(x,t),\underline{v}(x,t))\le (\overline{u}(x,t),\overline{v}(x,t))$ a.e. on $\overline{\Omega }_T\times \overline{\Omega }_T$.

Proof of Proposition 2.1

For small $\sigma >0$, set $\psi _\sigma (\xi )=\min \{1,\max \{\xi /\sigma ,0\}\}$, $\xi \in \mathbb {R}$. Then $\psi _\sigma (\xi )$ is a piecewise differentiable function. Let $\psi _1=\psi _\sigma (\underline{u}^m-u^m)$, $\psi _2=\psi _\sigma (\underline{v}^n-v^n)$, it is easy to verify that $\psi _1$ and $\psi _2$ are admissible test functions in (2.1) and (2.2).

Since $(\underline{u}, \underline{v})$ and $(\overline{u}, \overline{v})$ are subsolution and supersolution of (1.1), let $t_1=\tau$, $t_2=\tau +h$, $\tau ,h>0$, $\tau +h<T$ and $w=\underline{u}-\overline{u}$, $z=\underline{v}-\overline{v}$, $w_1=\underline{u}^m-\overline{u}^m$, $z_1=\underline{v}^n-\overline{v}^n$, then we obtain

$$\begin{aligned}&\int _{\Omega }w(x,\tau +h)\psi _1(x,\tau +h)\mathrm{d}x-\int _{\Omega }w(x,\tau ) \psi _1(x,\tau )\mathrm{d}x\nonumber \\&=\int ^{\tau +h}_{\tau }\int _{\Omega }w\psi _{1t}\mathrm{d}x\mathrm{d}s -\int ^{\tau +h}_{\tau }\int _{\Omega }(|\nabla \underline{u}^m|^{p-2} \nabla \underline{u}^m-|\nabla \overline{u}^m|^{p-2}\nabla \overline{u}^m)\cdot \nabla \psi _1\mathrm{d}x\mathrm{d}s\nonumber \\&\quad +\int ^{\tau +h}_{\tau }\int _{\Omega }\psi _1(x,t) \left[ (\underline{u}^{\alpha _1}\underline{v}^{\beta _1}-\overline{u}^{ \alpha _1}\overline{v}^{\beta _1})-a(\underline{u}^{r}-\overline{u}^{r})\right] \mathrm{d}x\mathrm{d}s, \end{aligned}$$

(2.4)

$$\begin{aligned}&\int _{\Omega }z(x,\tau +h)\psi _2(x,\tau +h)\mathrm{d}x-\int _{\Omega } z(x,\tau )\psi _2(x,\tau )\mathrm{d}x\nonumber \\&=\int ^{\tau +h}_{\tau }\int _{\Omega }z\psi _{2t}\mathrm{d}x\mathrm{d}s -\int ^{\tau +h}_{\tau }\int _{\Omega }(|\nabla \underline{v}^n|^{ q-2}\nabla \underline{v}^n-|\nabla \overline{v}^m|^{q-2}\nabla \overline{v}^n)\cdot \nabla \psi _2\mathrm{d}x\mathrm{d}s\nonumber \\&\quad +\int ^{\tau +h}_{\tau }\int _{\Omega }\psi _2(x,t) \left[ (\underline{u}^{\alpha _2}\underline{v}^{\beta _2}-\overline{u}^ {\alpha _2}\overline{v}^{\beta _2})-b(\underline{v}^{s}-\overline{v}^{s})\right] \mathrm{d}x\mathrm{d}s, \end{aligned}$$

(2.5)

Dividing (2.4) and (2.5) by $h$ and integrating $\tau$ over $(0, t)$ gives

$$\begin{aligned}&\int _0^t\frac{1}{h}\int _{\Omega }(w(x,\tau +h)\psi _1(x,\tau +h) -w(x,\tau )\psi _1(x,\tau ))\mathrm{d}x\mathrm{d}\tau \nonumber \\&=\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }w\psi _{1t}\mathrm{d}x\mathrm{d}s\mathrm{d}\tau \nonumber \\&\quad -\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega } (|\nabla \underline{u}^m|^{p-2}\nabla \underline{u}^m-|\nabla \overline{u}^m|^{p-2}\nabla \overline{u}^m)\cdot \nabla \psi _1\mathrm{d}x\mathrm{d}s\mathrm{d}\tau \nonumber \\&\quad +\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }\psi _1(x,t) [(\underline{u}^{\alpha _1}\underline{v}^{\beta _1}-\overline{u} ^{\alpha _1}\overline{v}^{\beta _1}) -a(\underline{u}^{r}-\overline{u}^{r})]\mathrm{d}x\mathrm{d}s\mathrm{d}\tau ,\end{aligned}$$

(2.6)

$$\begin{aligned}&\int _0^t\frac{1}{h}\int _{\Omega }z(x,\tau +h)\psi _2(x,\tau +h)dx- \int _{\Omega }z(x,\tau )\psi _2(x,\tau )\mathrm{d}x\mathrm{d}\tau \nonumber \\&=\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }z\psi _{2t}\mathrm{d}x\mathrm{d}s\mathrm{d}\tau \nonumber \\&\quad -\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }\left( |\nabla \underline{v}^n|^{q-2}\nabla \underline{v}^n-|\nabla \overline{v}^m|^{q-2}\nabla \overline{v}^n\right) \cdot \nabla \psi _2\mathrm{d}x\mathrm{d}s\mathrm{d}\tau \nonumber \\&\quad +\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }\psi _2(x,t) \left[ (\underline{u}^{\alpha _2}\underline{v}^{\beta _2}-\overline{u}^{ \alpha _2}\overline{v}^{\beta _2}) -b(\underline{v}^{s}-\overline{v}^{s})\right] \mathrm{d}x\mathrm{d}s\mathrm{d}\tau . \end{aligned}$$

(2.7)

By the properties of Steklov’s averages ([5], Lemma 1.3.2), we get

$$\begin{aligned}&\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }w\psi _{1t} \mathrm{d}x\mathrm{d}s\mathrm{d}\tau \rightarrow \int _0^t\int _{\Omega }w\psi _{1t}\mathrm{d}x\mathrm{d}s \hbox { as } h\rightarrow 0^+,\end{aligned}$$

(2.8)

$$\begin{aligned}&\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }z\psi _{2t}\mathrm{d}x\mathrm{d}s\mathrm{d} \tau \rightarrow \int _0^t\int _{\Omega }z\psi _{2t}\mathrm{d}x\mathrm{d}s \hbox { as } h\rightarrow 0^+,\end{aligned}$$

(2.9)

$$\begin{aligned}&\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }(|\nabla \underline{u}^m|^{p-2}\nabla \underline{u}^m-|\nabla \overline{u}^m|^{p-2}\nabla \overline{u}^m)\cdot \nabla \psi _1\mathrm{d}x\mathrm{d}s\mathrm{d}\tau \nonumber \\&\quad \rightarrow \int _0^t\int _{\Omega }(|\nabla \underline{u}^m|^{p-2}\nabla \underline{u}^m-|\nabla \overline{u}^m|^{p-2}\nabla \overline{u}^m)\cdot \nabla \psi _1\mathrm{d}x\mathrm{d}s \hbox { as } h\rightarrow 0^+,\end{aligned}$$

(2.10)

$$\begin{aligned}&\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }(|\nabla \underline{v}^n|^{q-2}\nabla \underline{v}^n-|\nabla \overline{v}^m|^{q-2}\nabla \overline{v}^n)\cdot \nabla \psi _2\mathrm{d}x\mathrm{d}s\mathrm{d}\tau \nonumber \\&\quad \rightarrow \int _0^t\int _{\Omega }(|\nabla \underline{v}^n|^{q-2}\nabla \underline{v}^n-|\nabla \overline{v}^m|^{q-2}\nabla \overline{v}^n)\cdot \nabla \psi _2\mathrm{d}x\mathrm{d}s \hbox { as } h\rightarrow 0^+,\end{aligned}$$

(2.11)

$$\begin{aligned}&\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }\psi _1(x,t) [(\underline{u}^{\alpha _1}\underline{v}^{\beta _1}-\overline{u}^ {\alpha _1}\overline{v}^{\beta _1})-a(\underline{u}^{r}-\overline{u}^{r})]\mathrm{d}x\mathrm{d}s\mathrm{d}\tau \nonumber \\&\quad \rightarrow \int _0^t\int _{\Omega }\psi _1(x,t) [(\underline{u}^{\alpha _1}\underline{v}^{\beta _1}-\overline{u}^ {\alpha _1}\overline{v}^{\beta _1})-a(\underline{u}^{r}-\overline{u}^{r})]\mathrm{d}x\mathrm{d}s \hbox { as } h\rightarrow 0^+, \nonumber \\&\int _0^t\frac{1}{h}\int ^{\tau +h}_{\tau }\int _{\Omega }\psi _2(x,t) [(\underline{u}^{\alpha _2}\underline{v}^{\beta _2}-\overline{u}^{\alpha _2} \overline{v}^{\beta _2})-b(\underline{v}^{s}-\overline{v}^{s})]\mathrm{d}x\mathrm{d}s\mathrm{d}\tau , \nonumber \\&\quad \rightarrow \int _0^t\int _{\Omega }\psi _2(x,t) [(\underline{u}^{\alpha _2}\underline{v}^{\beta _2}-\overline{u}^{\alpha _2} \overline{v}^{\beta _2})-b(\underline{v}^{s}-\overline{v}^{s})]\mathrm{d}x\mathrm{d}s \hbox { as } h\rightarrow 0^+. \end{aligned}$$

(2.12)

Now, we claim that

$$\begin{aligned}&\int _0^t\frac{1}{h}\int _{\Omega }(w(x,\tau +h)\psi _1(x,\tau +h) -w(x,\tau )\psi _1(x,\tau ))\mathrm{d}x\mathrm{d}\tau \nonumber \\&\quad \rightarrow \int _{\Omega }(w(x,t)\psi _1(x,t)-w(x,0)\psi _1(x,0))\mathrm{d}x,\end{aligned}$$

(2.13)

$$\begin{aligned}&\int _0^t\frac{1}{h}\int _{\Omega }z(x,\tau +h)\psi _2(x,\tau +h)\mathrm{d}x -\int _{\Omega }z(x,\tau )\psi _2(x,\tau )dxd\tau \nonumber \\&\quad \rightarrow \int _{\Omega }z(x,t)\psi _2(x,t)\mathrm{d}x -\int _{\Omega }z(x,0)\psi _2(x,0)\mathrm{d}x. \end{aligned}$$

(2.14)

By (2.4)–(2.14), we obtain

$$\begin{aligned}&\int _{\Omega }\!w(x,t)\psi _\sigma (w_1(x,t)) \mathrm{d}x\le \int _{\Omega }\! w(x,0)\psi _\sigma (w_1(x,0)) \mathrm{d}x \!+\!\int _0^t\int _{\Omega }w\psi _\sigma ^\prime (w_1)w_{1s}\mathrm{d}x\mathrm{d}s\nonumber \\&\quad -\int _0^t\int _{\Omega }(|\nabla \underline{u}^m|^{p-2}\nabla \underline{u}^m-|\nabla \overline{u}^m|^{p-2}\nabla \overline{u}^m)\cdot \nabla \psi _\sigma (\underline{u}^m-\overline{u}^m)\mathrm{d}x\mathrm{d}s \nonumber \\&\quad +\int _0^t\int _{\Omega }\psi _\sigma (w_1(x,t)) [(\underline{u}^{\alpha _1}\underline{v}^{\beta _1}-\overline{u}^{\alpha _1} \overline{v}^{\beta _1}) -a(\underline{u}^{r}-\overline{u}^{r})]\mathrm{d}x\mathrm{d}s,\end{aligned}$$

(2.15)

$$\begin{aligned}&\int _{\Omega }z(x,t)\psi _\sigma (z_1(x,t)) dx\le \int _{\Omega } z(x,0)\psi _\sigma (z_1(x,0)) \mathrm{d}x +\int _0^t\int _{\Omega }z\psi _\sigma ^\prime (z_1)z_{1s}\mathrm{d}x\mathrm{d}s\nonumber \\&\quad -\int _0^t\int _{\Omega }(|\nabla \underline{v}^n|^{q-2}\nabla \underline{v}^n-|\nabla \overline{v}^n|^{q-2}\nabla \overline{v}^n)\cdot \nabla \psi _\sigma (\underline{v}^n-\overline{v}^n)\mathrm{d}x\mathrm{d}s \nonumber \\&\quad +\int _0^t\int _{\Omega }\psi _\sigma (z_1(x,t)) [(\underline{u}^{\alpha _2}\underline{v}^{\beta _2}- \overline{u}^{\alpha _2}\overline{v}^{\beta _2}) -b(\underline{v}^{s}-\overline{v}^{s})]\mathrm{d}x\mathrm{d}s. \end{aligned}$$

(2.16)

Now, we deal with the terms in (2.15) and (2.16). First, we have

$$\begin{aligned}&\int _0^t\int _{\Omega }\psi _\sigma (w_1(x,t)) [(\underline{u}^{\alpha _1}\underline{v}^{\beta _1} -\overline{u}^{\alpha _1}\overline{v}^{\beta _1}) -a(\underline{u}^{r}-\overline{u}^{r})]\mathrm{d}x\mathrm{d}s\\&\quad \le \beta _1M_1^{\alpha _1}M_2^{\beta _1-1}\int _0^t\int _{\Omega } (\underline{v}-\overline{v})_+\mathrm{d}x +\alpha _1M_1^{\alpha _1-1}M_2^{\beta _1}\int _0^t\int _{\Omega } (\underline{u}-\overline{u})_+\mathrm{d}x\mathrm{d}s\\&\quad +arM_1^{r-1}\int _0^t\int _{\Omega }(\underline{u}-\overline{u})_+\mathrm{d}x\mathrm{d}s,\\&\int _0^t\int _{\Omega }\psi _\sigma (z_1(x,t)) [(\underline{u}^{\alpha _2}\underline{v}^{\beta _2}-\overline{u}^{\alpha _2} \overline{v}^{\beta _2}) -b(\underline{v}^{s}-\overline{v}^{s})]\mathrm{d}x\mathrm{d}s\\&\quad \le \alpha _2M_1^{\alpha _2-1}M_2^{\beta _2}\int _0^t\int _{\Omega } (\underline{u}-\overline{u})_+\mathrm{d}x +\quad \beta _2M_1^{\alpha _2}M_2^{\beta _2-1}\int _0^t\int _{\Omega } (\underline{v}-\overline{v})_+\mathrm{d}x\mathrm{d}s\\&+bsM_2^{s-1}\int _0^t\int _{\Omega }(\underline{v}-\overline{v})_+\mathrm{d}x\mathrm{d}s \end{aligned}$$

for some positive constants $M_1,M_2$, and as $\sigma \rightarrow 0^+$,

$$\begin{aligned}&\left| \int _0^t\int _{\Omega }w\psi _\sigma ^\prime (w_1)w_{1s}\mathrm{d}x\mathrm{d}s\right| \le \int _0^t\int _{\Omega }w_+|\psi _\sigma ^\prime (w_1)||w_{1s}|\mathrm{d}x\mathrm{d}s\\&\quad =\frac{1}{\sigma }\int _0^{\sigma }\int _{\Omega }w_+|w_{1s}|\mathrm{d}x\mathrm{d}s\rightarrow 0,\\&\left| \int _0^t\int _{\Omega }z\psi _\sigma ^\prime (z_1)z_{1s}\mathrm{d}x\mathrm{d}s\right| \le \int _0^t\int _{\Omega }z_+|\psi _\sigma ^\prime (z_1)||z_{1s}|\mathrm{d}x\mathrm{d}s\\&\quad =\frac{1}{\sigma }\int _0^{\sigma }\int _{\Omega }z_+|z_{1s}|\mathrm{d}x\mathrm{d}s\rightarrow 0. \end{aligned}$$

Second, by Lemma 1.4.4 in [5], we get

$$\begin{aligned} (|\nabla \underline{u}^m|^{p-2}\nabla \underline{u}^m-|\nabla \overline{u}^m|^{p-2}\nabla \overline{u}^m)\cdot \nabla \psi _\sigma (\underline{u}^m\!-\!\overline{u}^m)&\ge \min \left\{ 0,\gamma _1|\nabla (\underline{u}^m-\overline{u}^m)_+|^p\right\} ,\\ (|\nabla \underline{v}^n|^{q-2}\nabla \underline{v}^n-|\nabla \overline{v}^n|^{q-2}\nabla \overline{v}^n)\cdot \nabla \psi _\sigma (\underline{v}^n-\overline{v}^n)&\ge \min \left\{ 0,\gamma _2|\nabla (\underline{v}^n-\overline{v}^n)_+|^q\right\} \end{aligned}$$

for some $\gamma _1,\gamma _2>0$.

Finally, we have $\int _\Omega w(x,0)\psi _\sigma (w_1(x,0))\mathrm{d}x\equiv 0$, $\int _\Omega z(x,0)\psi _\sigma (z_1(x,0))\mathrm{d}x\equiv 0$ and $\psi _\sigma ^\prime \ge 0$ a.e. in $\mathrm {R}$, $w\psi _\sigma ^\prime (w_1)w_{1s}$, $z\psi _\sigma ^\prime (z_1)z_{1s}$ increase and tend to $w_+$, $z_+$ as $\sigma \rightarrow 0^+$, respectively. Hence, we may let $\sigma \rightarrow 0^+$ in (2.15) and (2.16) to yield

$$\begin{aligned} \int _{\Omega }w_+(x,t)\mathrm{d}x&\le C_1\int _0^t\int _{\Omega }w_+ (x,s)dxds+C_2\int _0^t\int _{\Omega }z_+(x,s)\mathrm{d}x\mathrm{d}s,\\ \int _{\Omega }z_+(x,t)\mathrm{d}x&\le C_3\int _0^t\int _{\Omega }w_+ (x,s)\mathrm{d}x\mathrm{d}s+C_4\int _0^t\int _{\Omega }z_+(x,s)\mathrm{d}x\mathrm{d}s. \end{aligned}$$

Hence,

$$\begin{aligned} \int _{\Omega }(w_+(x,t)+z_+(x,t))\mathrm{d}x\le C\int _0^t\int _{\Omega }(w_+(x,s)+z_+(x,s))\mathrm{d}x\mathrm{d}s. \end{aligned}$$

By the Gronwall’s inequality, we obtain $\int _{\Omega }(w_+(x,t)+z_+(x,t))\mathrm{d}x=0$, i.e. $\underline{u}\le \overline{u}$, $\underline{v}\le \overline{v}$, a.e. on $\overline{\Omega }_T$. This completes the proof.$\square$

3 Proof of Theorem 1.1

In this section, we investigate the global existence property of the solutions to Problem (1.1) and prove Theorem 1.1. The main method is constructing a globally upper solution and using comparison principle to achieve our purpose.

In order to study the globally existing solutions to Problem (1.1), we need to study the following elliptic system

$$\begin{aligned} -\Delta _{k,\gamma }\Theta =1,\ x\in \Omega ,\quad \Theta =1,\ x\in \partial \Omega , \end{aligned}$$

(3.1)

where $\Delta _{k,\gamma }\Theta$ is defined in (1.1), and we obtain the following lemma.

Lemma 3.1

problem (3.1) has a unique solution $\Theta (x)$, and satisfies the following relations,

$$\begin{aligned} \Theta (x)>1 \hbox { in } \Omega ,\ \nabla \Theta \cdot \nu <0 \hbox { on }\partial \Omega ,\ \sup \limits _{x\in \Omega }\Theta =M<+\infty , \end{aligned}$$

where $M$ is a positive constant.

Proof of this lemma is similar to that given in [23], we omit it here.

Proof of Theorem 1.1

Let $\varphi (x)$ and $\psi (x)$ be the unique solution of the following elliptic problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{m,p}\varphi =1, &{}\quad x\in \Omega ,\\ \varphi =1, &{}\quad x\in \partial \Omega ,\\ \end{array}\right. \qquad \left\{ \begin{array}{ll} -\Delta _{n,q}\psi =1, &{}\quad x\in \Omega ,\\ \psi =1, &{}\quad x\in \partial \Omega .\\ \end{array}\right. \end{aligned}$$

(3.2)

Then from Lemma 3.1, we obtain the following relations

$$\begin{aligned}&\varphi (x), \psi (x)>1 \hbox { in } \Omega ,\quad \nabla \varphi \cdot \nu , \nabla \psi \cdot \nu <0 \hbox { on }\partial \Omega ,\end{aligned}$$

(3.3)

$$\begin{aligned}&M_1=\min \{\inf \limits _{x\in \Omega }\varphi ,\ \inf \limits _{x\in \Omega }\psi \}<+\infty , M_2=\max \{\sup \limits _{x\in \Omega }\varphi ,\ \sup \limits _{x\in \Omega }\psi \}<+\infty , \end{aligned}$$

(3.4)

where $M_1,M_2>0$ is a positive constant.

Notice that $(1/\tau , 1/\theta )<(0, 0)$ implies

$$\begin{aligned} \beta _1\alpha _2<\mu _1\mu _2=\max \{m(p-1)-\alpha _1,r-\alpha _1\} \max \{n(q-1)-\beta _2,s-\beta _2\}. \end{aligned}$$

We will prove Theorem 1.1 in four subcases.

• (a) For $\mu _1=r-\alpha _1$, $\mu _2=s-\beta _2$, we then have $\beta _1\alpha _2<(r-\alpha _1)(s-\beta _2)$. Let $(\overline{u},\overline{v})=(\Lambda _1,\Lambda _2)$, where $\Lambda _1\ge \max \limits _{x\in \overline{\Omega }}u_0(x)$, $\Lambda _2\ge \max \limits _{x\in \overline{\Omega }}v_0(x)$ will be determined later. After a simple computation, we have

  $$\begin{aligned} \overline{u}_t-\Delta _{m,p}\overline{u}-\overline{u}^{\alpha _1} \overline{v}^{\beta _1}+a\overline{u}^r =a\Lambda _1^r-\Lambda _1^{\alpha _1}\Lambda _2^{\beta _1},\\ \overline{v}_t-\Delta _{n,q}\overline{v}-\overline{u}^{\alpha _2} \overline{v}^{\beta _2}+b\overline{v}^s =b\Lambda _2^s-\Lambda _1^{\alpha _2}\Lambda _2^{\beta _2}. \end{aligned}$$

  So, $(\overline{u},\overline{v})=(\Lambda _1,\Lambda _2)$ is a time-independent supersolution of problem (1.1) if

  $$\begin{aligned} a\Lambda _1^{r-\alpha _1}\ge \Lambda _2^{\beta _1}\hbox { and }b\Lambda _2^{s-\beta _2}\ge \Lambda _1^{\alpha _2}, \end{aligned}$$

  i.e.

  $$\begin{aligned} \Lambda _2^{\frac{\beta _1}{r-\alpha _1}}(\frac{1}{a})^{\frac{1}{r-\alpha _1}}\le \Lambda _1 \le \Lambda _2^{\frac{s-\beta _2}{\alpha _2}}(b)^{\frac{1}{\alpha _2}}. \end{aligned}$$

• (b) For $\mu _1=m(p-1)-\alpha _1$, $\mu _2=n(q-1)\!-\!\beta _2$, we then have $\beta _1\alpha _2<mn(p-1)(q-1)$. Let $(\overline{u},\overline{v})=(\Lambda _1\varphi (x),\Lambda _2\psi (x))$, where $\Lambda _1, \Lambda _2>0$ will be determined later. Then with a direct computation we obtain

  $$\begin{aligned} \overline{u}_t-\Delta _{m,p}\overline{u}-\overline{u}^{\alpha _1} \overline{v}^{\beta _1}+a\overline{u}^r \ge \Lambda _1^{m(p-1)}-\Lambda _1^{\alpha _1}\Lambda _2^{\beta _1} M_2^{\alpha _1+\beta _1},\\ \overline{v}_t-\Delta _{n,q}\overline{v}-\overline{u}^ {\alpha _2}\overline{v}^{\beta _2}+b\overline{v}^s \ge \Lambda _2^{n(q-1)}-\Lambda _1^{\alpha _2}\Lambda _2^{\beta _2}M_2^{\alpha _2+\beta _2}, \end{aligned}$$

  So, $(\overline{u}(x,t),\overline{v}(x,t))$ is an upper solution of problem (1.1), if

  $$\begin{aligned}&\Lambda _1^{m(p-1)}\ge \Lambda _1^{\alpha _1}\Lambda _2^{\beta _1}M_2^{\alpha _1+\beta _1},\ \Lambda _2^{n(q-1)}\ge \Lambda _1^{\alpha _2}\Lambda _2^{\beta _2}M_2^{\alpha _2+\beta _2},\nonumber \\&\overline{u}(x,t)\mid _{\partial \Omega }\ge 0,\ \overline{v}(x,t)\mid _{\partial \Omega }\ge 0,\ \overline{u}(x,0) =u_0(x),\ \overline{v}(x,0)=v_0(x). \end{aligned}$$

  (3.5)

  Then (3.5) holds if we choose $\Lambda _1$, $\Lambda _2$ large enough such that

  $$\begin{aligned}&\Lambda _1>\max \left\{ \max \limits _{x\in \overline{\Omega }}u_0(x), \left( M_2^{\alpha _1+\beta _1+\frac{(\alpha _2+\beta _2)\beta _1}{n(q-1)-\beta _2}} \right) ^{\frac{1}{m(p-1)-\alpha _1-\frac{\alpha _2\beta _1}{n(q-1)-\beta _2}}}\right\} ,\\&\Lambda _2>\max \left\{ \max \limits _{x\in \overline{\Omega }}v_0(x), \left( M_2^{\alpha _2+\beta _2+\frac{(\alpha _1+\beta _1)\alpha _2}{m(p-1)-\alpha _1}} \right) ^{\frac{1}{n(q-1)-\beta _2-\frac{\alpha _2\beta _1}{m(p-1)-\alpha _1}}}\right\} . \end{aligned}$$

• (c) For $\mu _1=r-\alpha _1$, $\mu _2=n(q-1)-\beta _2$, we then have $\beta _1\alpha _2<(r-\alpha _1)[n(q-1)-\beta _2]$. Choose $\Lambda _1\ge \max \limits _{x\in \overline{\Omega }}u_0(x)$ and $\Lambda _2\ge \max \limits _{x\in \overline{\Omega }}v_0(x)$ satisfy

  $$\begin{aligned} (\Lambda _1^{\alpha _2}M_2^{\beta _2})^{\frac{1}{n(q-1)-\beta _2}}\le \Lambda _2\le (a\Lambda _1^{r-\alpha _1}M_2^{-\beta _1})^{\frac{1}{\beta _1}}. \end{aligned}$$

  Let $(\overline{u},\overline{v})=(\Lambda _1,\Lambda _2\psi (x))$ with $\psi (x)$ defined by (3.2). By direct computation, we arrive at

  $$\begin{aligned} \overline{u}_t-\Delta _{m,p}\overline{u}-\overline{u}^{\alpha _1} \overline{v}^{\beta _1}+a\overline{u}^r&\ge a\Lambda _1^r-\Lambda _1^{\alpha _1}\Lambda _2^{\beta _1}M_2^{\beta _1} \ge 0,\nonumber \\ \overline{v}_t-\Delta _{n,q}\overline{v}-\overline{u}^{\alpha _2} \overline{v}^{\beta _2}+b\overline{v}^s&\ge \Lambda _2^{n(q-1)}-\Lambda _1^{\alpha _2}\Lambda _2^{\beta _2}M_2^{\beta _2} \ge 0. \end{aligned}$$

  (3.6)

• (d) For $\mu _1=m(p-1)-\alpha _1$, $\mu _2=s-\beta _2$, we then have $\beta _1\alpha _2<[m(p-1)-\alpha _1](s-\beta _2)$. Let $(\overline{u},\overline{v})=(\Lambda _1\varphi (x),\Lambda _2)$ with $\varphi (x)$ defined by (3.2), where $\Lambda _1\ge \max \limits _{x\in \overline{\Omega }}u_0(x)$ and $\Lambda _2\ge \max \limits _{x\in \overline{\Omega }}v_0(x)$. Then, (3.6) hold if

  $$\begin{aligned} (\Lambda _2^{\alpha _1}M_2^{\beta _1})^{\frac{1}{m(p-1)-\beta _1}}\le \Lambda _1\le (b\Lambda _2^{s-\alpha _2}M_2^{-\beta _2})^{\frac{1}{\beta _2}}. \end{aligned}$$

The proof of Theorem 1.1 is complete.$\square$

4 Proof of Theorem 1.2

In this section, we investigate the blow-up property of the solutions to problem (1.1) and prove Theorem 1.2. The main method is constructing a blowing-up lower solution and using the comparison principle to achieve our purpose.

Proof of Theorem 1.2

Observe that $(1/\tau , 1/\theta )>(0, 0)$ implies

$$\begin{aligned} \beta _1\alpha _2>\mu _1\mu _2=\max \{m(p-1)-\alpha _1,r-\alpha _1\} \max \{n(q-1)-\beta _2,s-\beta _2\}. \end{aligned}$$

For $\mu _1=r-\alpha _1$, $\mu _2=s-\beta _2$. Choosing

$$\begin{aligned} \Lambda _1=\frac{1}{2}\left[ (\frac{1}{a})^{\frac{1}{r-\alpha _1}} \Lambda _2^{\frac{\beta _1}{r-\alpha _1}} +b^{\frac{1}{\alpha _2}}\Lambda _2^{\frac{s-\beta _2}{\alpha _2}}\right] , \Lambda _2=\left( a^{\alpha _2}b^{r-\alpha _1}\right) ^{\frac{1}{\beta _1\alpha _2-(r-\alpha _1)(s-\beta _2)}}, \end{aligned}$$

then $(\overline{u},\overline{v})=(\Lambda _1,\Lambda _2)$ is a global supersolution for problem (1.1) provided that $\Lambda _1\ge \max \limits _{x\in \overline{\Omega }}u_0(x)$ and $\Lambda _2\ge \max \limits _{x\in \overline{\Omega }}v_0(x)$.

For $\mu _1=m(p-1)-\alpha _1$, $\mu _2=n(q-1)-\beta _2$. Let $(\overline{u},\overline{v})=(\Lambda _1\varphi (x),\Lambda _2\psi (x))$, where $\varphi (x)$ and $\psi (x)$ satisfying (3.2), respectively. Choosing

$$\begin{aligned}&\Lambda _1=\frac{1}{2}\left( M_2^{\frac{\alpha _1+\beta _1}{m(p-1)-\alpha _1}} \Lambda _2^{\frac{\beta _1}{m(p-1)-\alpha _1}} +M_2^{-\frac{\alpha _2+\beta _2}{\alpha _2}}\Lambda _2^{\frac{n(q-1) -\beta _2}{\alpha _2}}\right) ,\\&\Lambda _2=\left( M_2^{\alpha _2+\beta _2+\frac{(\alpha _1+\beta _1)\alpha _2}{m(p-1)-\alpha _1}} \right) ^{\frac{1}{n(q-1)-\beta _2-\frac{\alpha _2\beta _1}{m(p-1)-\alpha _1}}}, \end{aligned}$$

therefore, $(\overline{u},\overline{v})$ is a global supersolution for system (1.1) if $\Lambda _1\ge \max \limits _{x\in \overline{\Omega }}u_0(x)$ and $\Lambda _2\ge \max \limits _{x\in \overline{\Omega }}v_0(x)$.

For other cases, the solutions of (1.1) should be global due to the above discussion.

Next, we begin to prove our blow-up conclusion under large enough initial data. Due to the requirement of the comparison principle, we will construct blow-up subsolutions in some subdomain of $\Omega$ in which $u, v>0$. We use an idea from Souplet [14] and apply it to degenerate equations. Since problem (1.1) does not make sense for negative values of $(u, v)$, we actually consider the following problem

$$\begin{aligned} \begin{array}{ll} \displaystyle Pu(x,t)\equiv u_t-\Delta _{m,p}u-u_{+}^{\alpha _1}v_{+}^{\beta _1}+a u_{+}^{r},&{}\quad \displaystyle x\in \Omega ,t>0,\\ \displaystyle Qv(x,t)\equiv v_t-\Delta _{n,q}v-u_{+}^{\alpha _2}v_{+}^{\beta _2}+b v_{+}^{s},&{}\quad \displaystyle x\in \Omega ,t>0,\\ \displaystyle u(x,t)=v(x,t)=0, &{}\quad \displaystyle x\in \partial \Omega ,t>0,\\ \displaystyle u(x,0)=u_0(x), \quad v(x,0)=v_0(x), &{}\quad \displaystyle x\in \overline{\Omega }, \end{array} \end{aligned}$$

where $u_+=\max \{0, u\}$, $v_+=\max \{0, v\}$. Let $\varpi (x)$ be a nontrivial nonnegative continuous function and vanish on $\partial \Omega$. Without loss of generality, we may assume that $0\in \Omega$ and $\varpi (0)>0$. We shall construct a self-similar blow-up subsolution to complete our proof.

Set

$$\begin{aligned}&\underline{u}(x,t)=(\tau -t)^{-\gamma _1}V_1(\xi ), \xi =|x|(\tau -t)^{-\sigma _1}, V_1(\xi )=\left( {1+\frac{A}{2}-\frac{\xi ^2}{2A}}\right) _+^{1/m},\\&\underline{v}(x,t)=(\tau -t)^{-\gamma _2}V_2(\eta ), \eta =|x|(\tau -t)^{-\sigma _2}, V_2(\eta )=\left( {1+\frac{A}{2}-\frac{\eta ^2}{2A}}\right) _+^{1/n}, \end{aligned}$$

where $\gamma _i,\sigma _i>0(i=1,2)$, $A>1$ and $0<\tau <1$ are parameters to be determined. It is easy to see that $\underline{u}(x,t),\underline{v}(x,t)$ blow-up at time $\tau$, so it is enough to prove that $(\underline{u}(x,t),\underline{v}(x,t))$ is a lower solution of problem (1.1). If we choose $\tau$ small enough such that

$$\begin{aligned} \hbox {supp}\underline{u}(\cdot ,t)&=\overline{B(0,R(\tau -t)^{\sigma _1})} \subset \overline{B(0,R\tau ^{\sigma _1})}\subset \Omega ,\\ \hbox {supp}\underline{v}(\cdot ,t)&=\overline{B(0,R(\tau -t)^{\sigma _2})} \subset \overline{B(0,R\tau ^{\sigma _2})}\subset \Omega , \end{aligned}$$

where $R=(A(2+A))^{1/2}$, then $\underline{u}(x,t)\mid _{\partial \Omega }=0, \underline{v}(x,t)\mid _{\partial \Omega }=0$. Next if we choose the initial data large enough such that

$$\begin{aligned} u_0(x)\ge \dfrac{1}{\tau ^{\gamma _1}}V_1(\dfrac{|x|}{\tau ^{\sigma _1}}),\ v_0(x)\ge \dfrac{1}{\tau ^{\gamma _2}}V_2(\dfrac{|x|}{\tau ^{\sigma _2}}), \end{aligned}$$

then $(\underline{u}(x,t),\underline{v}(x,t))$ is a lower solution of problem (1.1) if for any $(x,t)\in \Omega \times (0,\tau ]$,

$$\begin{aligned}&\underline{u}_t-\Delta _{m,p}\underline{u}\le a\underline{u}^{\alpha _1}\underline{v}^{\beta _1},\end{aligned}$$

(4.1)

$$\begin{aligned}&\underline{v}_t-\Delta _{n,q}\underline{v}\le b\underline{u}^{\alpha _2}\underline{v}^{\beta _2}. \end{aligned}$$

(4.2)

After a direct computation, we obtain

$$\begin{aligned}&\underline{u}_t\!=\!\frac{\gamma _1V_1(\xi )+\sigma _1\xi V_1^\prime (\xi )}{(\tau -t)^{\gamma _1+1}},\quad \nabla \underline{u}^m\!=\!\frac{x}{A(\tau -t)^{m\gamma _1+2\sigma _1}}, -\Delta \underline{u}^m=\frac{N}{A(\tau -t)^{m\gamma _1+2\sigma _1}}, \nonumber \\&\underline{v}_t=\frac{\gamma _2V_2(\eta )+\sigma _2\eta V_2^\prime (\eta )}{(\tau -t)^{\gamma _2+1}}, \nabla \underline{v}^n=\frac{x}{A(\tau -t)^{n\gamma _2+2\sigma _2}}, -\Delta \underline{v}^n=\frac{N}{A(\tau -t)^{n\gamma _2+2\sigma _2}}, \end{aligned}$$

(4.3)

$$\begin{aligned}&\qquad \quad -\Delta _{m,p}\underline{u} =|\nabla \underline{u}^m|^{p-2}\Delta \underline{u}^m +(p-2)|\nabla \underline{u}^m|^{p-4}(\nabla \underline{u}^m)^{\tau } \cdot (H_{x}(\underline{u}^m))\cdot \nabla \underline{u}^m\nonumber \\&\quad =|\nabla \underline{u}^m|^{p-2}\Delta \underline{u}^m +(p-2)|\nabla \underline{u}^m|^{p-4}\sum \limits _{j=1}^{N}\sum \limits _{i=1}^{N} \frac{\partial \underline{u}^m}{\partial x_i}\frac{\partial ^2\underline{u}^m}{\partial x_i\partial x_j} \frac{\partial \underline{u}^m}{\partial x_j},\end{aligned}$$

(4.4)

$$\begin{aligned}&\qquad \quad -\Delta _{n,q}\underline{v} =|\nabla \underline{v}^n|^{q-2}\Delta \underline{v}^n +(q-2)|\nabla \underline{v}^n|^{q-4}(\nabla \underline{v}^n)^{\tau } \cdot (H_{x}(\underline{v}^n))\cdot \nabla \underline{v}^n\nonumber \\&\quad =|\nabla \underline{v}^n|^{q-2}\Delta \underline{v}^n +(q-2)|\nabla \underline{v}^n|^{q-4}\sum \limits _{j=1}^{N}\sum \limits _{i=1}^{N} \frac{\partial \underline{v}^n}{\partial x_i}\frac{\partial ^2\underline{v}^n}{\partial x_i\partial x_j} \frac{\partial \underline{v}^n}{\partial x_j}, \end{aligned}$$

(4.5)

where $H_x(\underline{u}^m)$, $H_x(\underline{v}^n)$ denote the Hessian matrix of $\underline{u}^m(x,t)$, $\underline{v}^n(x,t)$, respectively.

Use the notation $d(\Omega )=\hbox {diam}(\Omega )$, then from (4.4) and (4.5), we obtain

$$\begin{aligned} |\Delta _{m,p}\underline{u}|\le&\frac{N}{A(\tau -t)^{m\gamma _1+2\sigma _1}} \left( \frac{d(\Omega )}{(\tau -t)^{m\gamma _1+2\sigma _1}}\right) ^{p-2}\nonumber \\&+(p-2)\left( \frac{d(\Omega )}{(\tau -t)^{m\gamma _1+2\sigma _1}}\right) ^{p-4} \left( \frac{d(\Omega )}{(\tau -t)^{m\gamma _1+2\sigma _1}}\right) ^{2} \frac{N}{A(\tau -t)^{m\gamma _1+2\sigma _1}}\nonumber \\ =&\frac{N(p-1)(d(\Omega ))^{p-2}}{A(\tau -t)^{(m\gamma _1+2\sigma _1)(p-1)}}. \end{aligned}$$

(4.6)

Similarly, from (4.4) and (4.5) we obtain

$$\begin{aligned} |\Delta _{n,q}\underline{v}| \le&\frac{N}{A(\tau -t)^{n\gamma _2+2\sigma _2}} \left( \frac{d(\Omega )}{(\tau -t)^{n\gamma _2+2\sigma _2}}\right) ^{q-2}\nonumber \\&+(q-2)\left( \frac{d(\Omega )}{(\tau -t)^{n\gamma _2+2\sigma _2}}\right) ^{q-4} \left( \frac{d(\Omega )}{(\tau -t)^{n\gamma _2+2\sigma _2}}\right) ^{2} \frac{N}{A(\tau -t)^{n\gamma _2+2\sigma _2}}\nonumber \\&=\frac{N(q-1)(d(\Omega ))^{q-2}}{A(\tau -t)^{(n\gamma _2+2\sigma _2)(q-1)}}. \end{aligned}$$

(4.7)

Next, we compute the local term of (4.1)

$$\begin{aligned} \underline{u}^{\alpha _1}\underline{v}^{\beta _1}&= \frac{1}{(\tau -t)^{\gamma _1\alpha _1+\gamma _2\beta _1}} V_1^{\alpha _1}\left( \frac{|x|}{(\tau -t)^{\sigma _1}}\right) V_2^{\beta _1}\left( \frac{|x|}{(\tau -t)^{\sigma _2}}\right) ,\nonumber \\ \underline{u}^{\alpha _2}\underline{v}^{\beta _2}&=\frac{1}{(\tau -t)^{\gamma _1\alpha _2+\gamma _2\beta _2}} V_1^{\alpha _2}\left( \frac{|x|}{(\tau -t)^{\sigma _1}}\right) V_2^{\beta _2}\left( \frac{|x|}{(\tau -t)^{\sigma _2}}\right) . \end{aligned}$$

(4.8)

If $0\le \xi ,\eta \le A$, then $1\le V_1(\xi )\le (1+A/2)^{1/m}$, $1\le V_2(\eta )\le (1+A/2)^{1/n}$ and $V_1^\prime (\xi )\le 0$, $V_2^\prime (\eta )\le 0$. Combining the above inequalities, we obtain

$$\begin{aligned} P\underline{u}(x,t)&\le \frac{\gamma _1(1+\frac{A}{2})^{1/m}}{(\tau -t)^{\gamma _1+1}} +\frac{N(p-1)(d(\Omega ))^{p-2}}{A(\tau -t)^{(m\gamma _1+2\sigma _1)(p-1)}} +\frac{a\left( 1+\frac{A}{2}\right) ^{r/m}}{(\tau -t)^{r\gamma _1}}\nonumber \\&\quad -\frac{1}{(\tau -t)^{\gamma _1\alpha _1+\gamma _2\beta _1}},\end{aligned}$$

(4.9)

$$\begin{aligned} Q\underline{v}(x,t)&\le \frac{\gamma _2(1+\frac{A}{2})^{1/n}}{(\tau -t)^{\gamma _2+1}} +\frac{N(q-1)(d(\Omega ))^{q-2}}{A(\tau -t)^{(n\gamma _2+2\sigma _2)(q-1)}} +\frac{b\left( 1+\frac{A}{2}\right) ^{r/n}}{(\tau -t)^{s\gamma _2}}\nonumber \\&\quad -\frac{1}{(\tau -t)^{\gamma _1\alpha _2+\gamma _2\beta _2}}. \end{aligned}$$

(4.10)

If $\xi ,\eta \ge A$, since $m, n\ge 1$, we obtain $V_1(\xi )\le 1$, $V_2(\eta )\le 1$ and $V_1^\prime (\xi )\le -1/m$, $V_2^\prime (\eta )\le -1/n$. Combining the above inequalities (4.3)–(4.8), we obtain

$$\begin{aligned} P\underline{u}(x,t)&\le \frac{\gamma _1-\frac{1}{m}\sigma _1A}{(\tau -t)^{\gamma _1+1}} +\frac{N(p-1)(d(\Omega ))^{p-2}}{A(\tau -t)^{(m\gamma _1+2\sigma _1)(p-1)}} +\frac{a}{(\tau -t)^{r\gamma _1}},\end{aligned}$$

(4.11)

$$\begin{aligned} Q\underline{v}(x,t)&\le \frac{\gamma _2-\frac{1}{n}\sigma _2A}{(\tau -t)^{\gamma _2+1}} +\frac{N(q-1)(d(\Omega ))^{q-2}}{A(\tau -t)^{(n\gamma _2+2\sigma _2)(q-1)}}+ \frac{b}{(\tau -t)^{s\gamma _2}}. \end{aligned}$$

(4.12)

If $0\le \xi \le A$ and $\eta \ge A$, we have that (4.9) and (4.12) hold. If $\xi \ge A$ and $0\le \eta \le A$, we have that (4.10) and (4.11) hold.

So, from the above discussions, (4.1) hold if the right-hand sides of (4.9)–(4.12) are nonpositive.

Since $1/\tau ,1/\theta <0$, we see that $\beta _1\alpha _2>\mu _1\mu _2$. In addition, it is clear that

$$\begin{aligned} \frac{\mu _1}{\beta _1}<\frac{\alpha _2+1}{\beta _1+1}\hbox { or }\frac{\mu _2}{\alpha _2}<\frac{\beta _1+1}{\alpha _2+1}. \end{aligned}$$

(4.13)

For $\mu _1/\beta _1<(\alpha _2+1)/(\beta _1+1)$, we choose $\gamma _1$ and $\gamma _2$ such that

$$\begin{aligned} \frac{\mu _1}{\beta _1}&<\frac{\gamma _2}{\gamma _1}<\min \{\frac{ \alpha _2+1}{\beta _1+1},\frac{\alpha _2}{\mu _2}\},\nonumber \\ \alpha _1+\mu _1&<\frac{1+\gamma _1}{\gamma _1}<\min \{\frac{r}{\gamma _1 (r-1)},\frac{\beta _1\gamma _2+\alpha _1\gamma _1}{\gamma _1}\}. \end{aligned}$$

(4.14)

Recall that $\mu _1=\max \{m(p-1)-\alpha _1,r-\alpha _1\}$ and $\mu _2=\max \{n(q-1)-\beta _2,s-\beta _2\}$, then (4.14) implies

$$\begin{aligned} \beta _1\gamma _2+\alpha _1\gamma _1&>r\gamma _1,\beta _1\gamma _2+ \alpha _1\gamma _1>m(p-1)\gamma _1, \beta _1\gamma _2+\alpha _1\gamma _1>\gamma _1+1>r\gamma _1,\\ \beta _2\gamma _2+\alpha _2\gamma _1&>s\gamma _2,\beta _2\gamma _2+ \alpha _2\gamma _1>n(q-1)\gamma _2, \beta _2\gamma _2+\alpha _2\gamma _1>\gamma _2+1>s\gamma _2. \end{aligned}$$

Next, we can choose positive constants $\sigma _1$, $\sigma _2$ sufficiently small such that

$$\begin{aligned}&\sigma _1=\sigma _2<\min \Bigg \{\frac{\beta _1\gamma _2+\alpha _1\gamma _1-\gamma _1-1}{2N},\frac{\beta _1\gamma _2+\alpha _1\gamma _1-m(p-1)\gamma _1}{2(N+p-1)},\\&\quad \frac{\beta _1\gamma _2+\alpha _1\gamma _1-r\gamma _1}{2N}, \frac{\gamma _1+1+m(p-1)\gamma _1}{2(p-1)}, \frac{\beta _2\gamma _2+\alpha _2\gamma _1-\gamma _2-1}{2N},\\&\qquad \frac{\beta _2\gamma _2+\alpha _2\gamma _1-n(q-1)\gamma _2}{2(N+q-1)}, \frac{\beta _2\gamma _2+\alpha _2\gamma _1-s\gamma _2}{2N}, \frac{\gamma _2+1+n(q-1)\gamma _2}{2(q-1)}\Bigg \}, \end{aligned}$$

consequently, we have

$$\begin{aligned} \beta _1\gamma _2+\alpha _1\gamma _1&>\max \Big \{\gamma _1+1, (m\gamma _1+2\sigma _1)(p-1),r\gamma _1\Big \},\nonumber \\ \beta _2\gamma _2+\alpha _2\gamma _1&>\max \Big \{\gamma _2+1, (n\gamma _2+2\sigma _2)(q-1),s\gamma _2\Big \},\nonumber \\ \gamma _1+1&>\max \Big \{r\gamma _1,(m\gamma _1+2\sigma _1)(p-1)\Big \},\nonumber \\ \gamma _2+1&>\max \Big \{s\gamma _2,(m\gamma _2+2\sigma _2)(q-1)\Big \}. \end{aligned}$$

(4.15)

For $\mu _2/\alpha _2<(\beta _1+1)/(\alpha _2+1)$, we fix $\gamma _1$ and $\gamma _2$ to satisfy

$$\begin{aligned} \frac{\mu _2}{\alpha _2}&<\frac{\gamma _1}{\gamma _2}<\min \{\frac{\beta _1+1}{\alpha _2+1},\frac{\beta _1}{\mu _1}\},\nonumber \\ \beta _2+\mu _2&<\frac{1+\gamma _2}{\gamma _2}<\min \{\frac{s}{\gamma _2(s-1)}, \frac{\beta _2\gamma _2+\alpha _2\gamma _1}{\gamma _2}\}, \end{aligned}$$

(4.16)

then we can also select $\sigma _1$, $\sigma _2$ small enough such that (4.15) holds.

Furthermore, if we choose $A>\max \{1, m\gamma _1/\sigma _1, n\gamma _2/\sigma _2\}$, then for $\tau >0$ sufficiently small, the right-hand sides of (4.9)–(4.12) are nonpositive, so (4.1) and (4.2) holds, and we obtain Theorem 1.2.$\square$

5 Proof of Theorem 1.3

Proof of Theorem 1.3

In the critical case of $(1/\tau , 1/\theta )=(0, 0)$, we have

$$\begin{aligned} \beta _1\alpha _2=\mu _1\mu _2=\max \{m(p-1)-\alpha _1,r-\alpha _1\} \max \{n(q-1)-\beta _2,s-\beta _2\}. \end{aligned}$$

(i) For $r>m(p-1)$, $s>n(q-1)$, we know $\beta _1\alpha _2=(r-\alpha _1)(s-\beta _2)$. Thanks to $a^{\alpha _2}b^{r-\alpha _1}\ge 1$, we can choose $\Lambda _1$ and $\Lambda _2$ sufficiently large such that $\Lambda _1\ge \max \limits _{x\in \overline{\Omega }}u_0(x)$, $\Lambda _2\ge \max \limits _{x\in \overline{\Omega }}v_0(x)$ and

$$\begin{aligned} \Lambda _2^{\frac{\beta _1}{r-\alpha _1}}(\frac{1}{a})^{\frac{1}{r-\alpha _1}}\le \Lambda _1 \le \Lambda _2^{\frac{s-\beta _2}{\alpha _2}}b^{\frac{1}{\alpha _2}}. \end{aligned}$$

Clearly, $(\overline{u}, \overline{v})=(\Lambda _1, \Lambda _2)$ is a supersolution of problem (1.1), then by comparison principle, the solution of (1.1) should be global.

Next, we begin to prove our blow-up conclusion.

Since $\beta _1\alpha _2=\mu _1\mu _2$, we can choose constants $l_1, l_2>1$ such that

$$\begin{aligned} \frac{n(q-1)-\beta _2-1}{r-\alpha _1-1}<\frac{s-\beta _2}{\alpha _2}= \frac{l_1}{l_2}=\frac{\beta _1}{r-\alpha _1}<\frac{s-\beta _2-1}{m(p-1)-\alpha _1-1}. \end{aligned}$$

(5.1)

According to Proposition 2.1, we only need to construct a suitable blow-up subsolution of problem (1.1) on $\overline{\Omega }_T\times \overline{\Omega }_T$. Let $\gamma (t)$ be the solution of the following ordinary differential equation

$$\begin{aligned} \gamma ^{\prime }(t)=c_1\gamma ^{\delta _1}-c_2\gamma ^{\delta _2},\ \gamma (0)=\gamma _0>0,\quad t>0, \end{aligned}$$

where

$$\begin{aligned} c_1&= \min \left\{ \dfrac{\zeta ^{\alpha _1}\vartheta ^{\beta _1}-a\zeta ^{r}}{l_1\zeta }, \dfrac{\zeta ^{\alpha _2}\vartheta ^{\beta _2}-b\vartheta ^{s}}{l_2\vartheta }\right\} , c_2=\max \left\{ \dfrac{1}{l_1\zeta },\dfrac{1}{l_2\vartheta }\right\} ,\\ \delta _1&= \min \left\{ l_1(r-1)+1,(s-1)l_2+1\right\} ,\\ \delta _2&= \max \left\{ [m(p-1)-1]l_1+1,[n(q-1)-1]l_2+1\right\} . \end{aligned}$$

Since $\vartheta ^{\beta _1}>a\zeta ^{r-\alpha _1}$ and $\zeta ^{\alpha _2}>b\vartheta ^{s-\beta _2}$, we have $c_1>0$. On the other hand, by virtue of (5.1), it is easy to see that $\delta _1>\delta _2$. Then, it is obvious that there exists a constant $0 <T^{\star }<+\infty$ such that

$$\begin{aligned} \lim \limits _{t\rightarrow T^{\star }}\gamma (t)=+\infty . \end{aligned}$$

Construct

$$\begin{aligned} (\underline{u}(x,t),\underline{v}(x,t))=(\gamma ^{l_1}(t)\zeta (x), \gamma ^{l_2}(t)\vartheta (x)), \end{aligned}$$

where $\zeta (x), \vartheta (x)$ satisfying (1.6). Moreover, by the assumptions on initial data, we can take small enough constant $\gamma _0$ such that

$$\begin{aligned} u_0(x)\ge \gamma _0^{l_1}M_1\hbox { and } v_0(x)\ge \gamma _0^{l_2}M_2\quad \hbox { for all } x\in \Omega , \end{aligned}$$

(5.2)

where $M_1=\max \limits _{x\in \Omega }\zeta (x)$, $M_2=\max \limits _{x\in \Omega }\vartheta (x)$.

Now, we begin to verify that $(\overline{u}(x,t),\overline{v}(x,t))$ is a blow-up subsolution of the problem (1.1) on $\overline{\Omega }_T\times \overline{\Omega }_T$, $T <T^{\star }$. In fact, $\forall (x,t)\in \Omega _T\times (0, T)$, a series of computations show

$$\begin{aligned} P\underline{u}(x,t)&\equiv \underline{u}_t-\Delta _{m,p}\underline{u}-\underline{u}^{\alpha _1} \underline{v}^{\beta _1}+a \underline{u}^{r}\nonumber \\&=l_1\zeta \gamma ^{l_1-1}\gamma ^{\prime }(t)+\gamma ^{m(p-1)l_1} -\gamma ^{l_1\alpha _1+l_2\beta _1}\zeta ^{\alpha _1}\vartheta ^{\beta _1} +a\gamma ^{rl_1}\zeta ^r\\&=l_1\zeta \gamma ^{l_1-1}\left( \gamma ^{\prime }(t)+ \frac{1}{l_1\zeta }\gamma ^{m(p-1)l_1-l_1+1} -\frac{\zeta ^{\alpha _1}\vartheta ^{\beta _1}-a\zeta ^r}{l_1\zeta } \gamma ^{l_1(r-1)+1}\right) \nonumber \\&\le 0.\nonumber \end{aligned}$$

(5.3)

Similarly, we also have

$$\begin{aligned} Q\underline{v}(x,t)\equiv \underline{v}_t-\Delta _{n,q}\underline{v}-\underline{u}^{\alpha _2} \underline{v}^{\beta _2}+b \underline{v}^{s} \le 0. \end{aligned}$$

(5.4)

On the other hand, $\forall t\in [0, T]$, we have

$$\begin{aligned} \underline{u}(x,t)|_{x\in \partial \Omega }=\gamma ^{l_1}(t)\zeta (x) |_{x\in \partial \Omega }=0,\quad \underline{v}(x,t)|_{x\in \partial \Omega }=\gamma ^{l_2}(t)\vartheta (x) |_{x\in \partial \Omega }=0. \end{aligned}$$

(5.5)

Combining now (5.2)-(5.5), we see that $(\underline{u}, \underline{v})$ is a subsolution of (1.1) and $(\underline{u}, \underline{v})< (u, v)$ on $\overline{\Omega }_T\times \overline{\Omega }_T$ by comparison principle, thus $(u, v)$ must blow-up in finite time since $(\underline{u}, \underline{v})$ does.

(ii) For $r<m(p-1)$, $s<n(q-1)$, we know $\beta _1\alpha _2=[m(p-1)-\alpha _1][n(q-1)-\beta _2]$. Under the assumption $(\zeta ^{\alpha _2}\vartheta ^{\beta _2})^{1/[n(q-1)-\beta _2]} (\zeta ^{\alpha _1}\vartheta ^{\beta _1})^{1/\beta _1}\le 1$, we can choose $\Lambda _1, \Lambda _2$ such that

$$\begin{aligned} \Lambda _1^{\frac{\alpha _2}{n(q-1)-\beta _2}}(\zeta ^{\alpha _2} \vartheta ^{\beta _2})^{\frac{1}{n(q-1)-\beta _2}} \le \Lambda _2\le \Lambda _1^{\frac{m(p-1)-\alpha _1}{\beta _1}}(\zeta ^{\alpha _1} \vartheta ^{\beta _1})^{-\frac{1}{\beta _1}}. \end{aligned}$$

Then $(\overline{u}, \overline{v})=(\Lambda _1, \Lambda _2)$ is a global supersolution of (1.1).

Since $\beta _1\alpha _2=[m(p-1)-\alpha _1][n(q-1)-\beta _2]$, we can choose constants $l_1, l_2 > 1$ such that

$$\begin{aligned} \frac{s-1}{m(p-1)-1}<\frac{n(q-1)-\beta _2}{\alpha _2}=\frac{l_1}{l_2} =\frac{\beta _1}{m(p-1)-\alpha _1}<\frac{n(q-1)-1}{r-1}. \end{aligned}$$

(5.6)

Next, we consider the following ordinary differential equation

$$\begin{aligned} \gamma ^{\prime }(t)=c_1\gamma ^{\delta _1}-c_2\gamma ^{\delta _2},\ \gamma (0)=\gamma _0>0,\ t>0, \end{aligned}$$

where

$$\begin{aligned} c_1&= \min \{\zeta ^{\alpha _1}\vartheta ^{\beta _1}-1, \zeta ^{\alpha _2}\vartheta ^{\beta _2}-1\},\quad c_2=\max \left\{ \dfrac{a\zeta ^{r-1}}{l_1},\dfrac{b\vartheta ^{s-1}}{l_2}\right\} ,\\ \delta _1&= \min \left\{ [m(p-1)-1]l_1+1,[n(q-1)-1]l_2+1\right\} ,\\ \delta _2&= \max \{l_1(r-1)+1,(s-1)l_2+1\}. \end{aligned}$$

Since $\zeta ^{\alpha _1}\vartheta ^{\beta _1}>1$, $\zeta ^{\alpha _2}\vartheta ^{\beta _2}>1$, we have $c_1>0$. On the other hand, in light of (5.6), it is easy to show that $\delta _1>\delta _2$. Then, it is clear that $\gamma (t)$ will become infinite in a finite time $T^{\star }<+\infty$.

Let

$$\begin{aligned} (\underline{u}(x,t),\underline{v}(x,t))=(\gamma ^{l_1}(t)\zeta (x), \gamma ^{l_2}(t)\vartheta (x)), \end{aligned}$$

where $\zeta (x), \vartheta (x)$ satisfying (1.6). Similar to the arguments for the case $r>m(p-1)$, $s>n(q-1)$, we can prove that $(\underline{u}(x, t), \underline{v}(x, t))$ is a blow-up subsolution of the problem (1.1) on $\overline{\Omega }_T\times \overline{\Omega }_T$, $T <T^{\star }$. Then, the solution $(u, v)$ of (1.1) blows up in finite time.

(iii) For $r<m(p-1)$, $s>n(q-1)$, we know $\beta _1\alpha _2=[m(p-1)-\alpha _1][s-\beta _2]$. Since $(\zeta ^{\alpha _2})(\zeta ^{\alpha _1})^{(s-\beta _2)/\beta _1}\le b$, we can choose $\Lambda _1,\Lambda _2$, such that

$$\begin{aligned} b^{-\frac{1}{s-\beta _2}}\Lambda _1^{\frac{\alpha _2}{s-\beta _2}} (\zeta ^{\alpha _2})^{\frac{1}{s-\beta _2}} \le \Lambda _2\le \Lambda _1^{\frac{m(p-1)-\alpha _1}{\beta _1}} (\zeta ^{\alpha _1})^{-\frac{1}{\beta _1}}. \end{aligned}$$

We can check $(\overline{u}, \overline{v}) = (\Lambda _1\zeta , \Lambda _2)$ is a global supersolution of (1.1).

Thanks to $\beta _1\alpha _2=[m(p-1)-\alpha _1][s-\beta _2]$, we can choose constants $l_1, l_2>1$ such that

$$\begin{aligned} \dfrac{n(q-1)-\beta _2}{\alpha _2}<\dfrac{s-\beta _2}{\alpha _2}=\dfrac{l_1}{l_2} =\dfrac{\beta _1}{m(p-1)-\alpha _1}<\dfrac{\beta _1}{r-\alpha _1}. \end{aligned}$$

Let

$$\begin{aligned} (\underline{u}(x,t),\underline{v}(x,t))=(\gamma ^{l_1}(t)\zeta (x), \gamma ^{l_2}(t)\vartheta (x)), \end{aligned}$$

where $\zeta (x), \vartheta (x)$ are defined in (1.6), and $\Gamma (t)$ satisfies the following Cauchy problem

$$\begin{aligned} \gamma ^{\prime }(t)=c_1\gamma ^{\delta _1}-c_2\gamma ^{\delta _2},\ \gamma (0)=\gamma _0>0,\ t>0, \end{aligned}$$

where

$$\begin{aligned} c_1&= \min \left\{ \zeta ^{\alpha _1}\vartheta ^{\beta _1}-1, \dfrac{\zeta ^{\alpha _2}\vartheta ^{\beta _2}-b\vartheta ^{s}}{l_2\vartheta }\right\} , c_2=\max \left\{ \dfrac{a\zeta ^{r-1}}{l_1},\dfrac{1}{l_2\vartheta }\right\} ,\\ \delta _1&= \min \left\{ [m(p-1)-1]l_1+1,(s-1)l_2+1\right\} ,\\ \delta _2&= \max \left\{ l_1(r-1)+1,[n(q-1)-1]l_2+1\right\} . \end{aligned}$$

Then, the left arguments are the same as those for the case $r>m(p-1)$, $s>n(q-1)$, so we omit them.

(iv) The proof of this case is parallel to (iii). The proof of Theorem 1.3 is complete. $\square$