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Fujita type theorem for a class of coupled quasilinear convection–diffusion equations

Abstract

In this paper, we establish the Fujita type theorem for a homogeneous Neumann outer problem of the coupled quasilinear convection–diffusion equations and formulate the critical Fujita exponent. Besides, the influence of diffusion term, reaction term, and convection term on the global existence and the blow-up property of the problem is revealed. Finally, we discuss the large time behavior of the solution to the outer problem in the critical case and describe the asymptotic behavior of the solution.

Introduction

In this paper, we consider the critical Fujita exponent of the following coupled quasilinear convection–diffusion equations:

$$\begin{aligned} &\frac{\partial u}{\partial t}=\Delta u^{m}+\kappa \frac{x}{ \vert x \vert ^{2}} \cdot \nabla u^{m}+ \vert x \vert ^{\lambda }v^{p}, \quad x\in \mathbb{R}^{n} \backslash \overline{B}_{1}, t>0, \end{aligned}$$
(1)
$$\begin{aligned} &\frac{\partial v}{\partial t}=\Delta v^{m}+\kappa \frac{x}{ \vert x \vert ^{2}} \cdot \nabla v^{m}+ \vert x \vert ^{\mu }u^{q}, \quad x\in \mathbb{R}^{n} \backslash \overline{B}_{1}, t>0, \end{aligned}$$
(2)
$$\begin{aligned} &\frac{\partial u^{m}}{\partial {\boldsymbol{\nu }}}(x,t)= \frac{\partial v^{m}}{\partial {\boldsymbol{\nu }}}(x,t)=0, \quad x\in \partial B_{1}, t>0, \end{aligned}$$
(3)
$$\begin{aligned} &u(x,0)=u_{0}(x),\quad\quad v(x,0)=v_{0}(x), \quad x\in \mathbb{R}^{n} \backslash \overline{B}_{1}, \end{aligned}$$
(4)

where \(p, q>m>1\), \(\kappa \in \mathbb{R}\), \(\lambda \ge 0\), \(\mu = \frac{\lambda (q-m)+2(q-p)}{p-m}\ge 0\). In addition, \(B_{1}\) denotes the unit ball in \(\mathbb{R}^{n}\), ν denotes the unit inner normal vector to \(\partial B_{1}\), and \(0\le u_{0}\), \(v_{0}\in C_{0}(\mathbb{R}^{n})\) are nontrivial.

In 1966, the first result of the exponent of the quasilinear diffusion equation was introduced by Fujita [5]. Precisely, he investigated the Cauchy problem of the semilinear equation

$$ \frac{\partial u}{\partial t}=\Delta u+u^{p},\quad x\in \mathbb{R}^{n}, t>0, $$

and showed that the problem does not have any nontrivial global nonnegative solution if \(1< p< p_{c}=1+2/n\), whereas there exist both nontrivial global (with small initial data) and nonglobal nonnegative (with large initial data) solutions when \(p>p_{c}=1+2/n\). Among many other results, it proved \(p=p_{c}\) belonged to the blow-up case by Hayakawa [12], Kobayashi et al. [13], and Weissler [27]. This is what we know as the blow-up theorem of Fujita type, and \(p_{c}\) is called the critical Fujita exponent. Early results of the Fujita type theorem can be seen in the review articles of Deng [2], Levine [15], and relevant references. In recent years, there are a lot of Fujita’s results, such as [2, 3, 8, 10, 11, 14, 1621, 2426, 30, 31] and the references therein.

Among those works, Galaktionov et al. [6, 7] considered the critical Fujita exponent of the Cauchy problem

$$ \frac{\partial u}{\partial t}=\Delta u^{m}+u^{p}, \quad x\in \mathbb{R}^{n}, t>0 \ (p,m>1) $$

and proved that the critical Fujita exponent is \(p_{c}=m+2/n\). Aguirre and Escobedo [1] demonstrated the Fujita type theorem of the following convective–diffusion equation:

$$ \frac{\partial u}{\partial t}=\Delta u^{m}+\boldsymbol{b}_{0}\cdot \nabla u^{q}+u^{p}, \quad x\in \mathbb{R}^{n}, t>0, $$

where \(q\ge 1\), \(p>1\), \(\boldsymbol{b}_{0}\in \mathbb{R}^{n}\). They demonstrated that the critical Fujita exponent was

$$ p_{c}=\min \biggl\{ 1+\frac{2}{n}, 1+\frac{2q}{n+1} \biggr\} . $$

Zheng and Wang [29] studied more general nonlinear convection–diffusion systems

$$\begin{aligned} & \vert x \vert ^{\lambda _{1}}\frac{\partial u}{\partial t}=\Delta u^{m}+ \kappa \frac{x}{ \vert x \vert ^{2}}\cdot \nabla u^{m}+ \vert x \vert ^{\lambda _{2}}u^{p}, \quad x\in \mathbb{R}^{n}\backslash \overline{\Omega }, t>0, \\ &\frac{\partial u^{m}}{\partial \boldsymbol{\nu }}(x,t)=0, \quad x\in \partial \Omega , t>0, \\ &u(x,0)=u_{0}(x), \quad x\in \mathbb{R}^{n}\backslash \overline{\Omega }, \end{aligned}$$

where \(p>m\ge 1\), \(\kappa \in \mathbb{R}\), \(-2<\lambda _{1}\le \lambda _{2}\), Ω is the bounded area in \(\mathbb{R}^{n}\) with a smooth boundary Ω and \(B_{R_{1}}\subset \Omega \subset B_{R_{2}}\) for some \(0< R_{1}\le R_{2}\), and \(B_{R}\) denotes the ball in \(\mathbb{R}^{n}\) with radius R and center at the origin, and ν is a unit outer normal vector to \(\partial B_{1}\). It displayed that the critical Fujita exponent is

$$ p_{c}= \textstyle\begin{cases} m+\frac{2+\lambda _{2}}{n+\kappa +\lambda _{1}},&\kappa >-n- \lambda _{1}, \\ +\infty ,&\kappa \le -n-\lambda _{1}. \end{cases} $$

Following from a lot of results, it shows that critical Fujita exponent of a single equation is usually a constant, while the critical Fujita exponent of the coupled equations is usually a curve which is called the critical Fujita curve. In 1991, Escobedo and Herrero [4] investigated the following coupled systems:

$$ \frac{\partial u}{\partial t}=\Delta u+v^{p},\quad\quad \frac{\partial v}{\partial t}=\Delta v+u^{q}, \quad x\in \mathbb{R}^{n}, t>0, $$

where \(p,q>0\), and showed that the Fujita curve is

$$ (pq)_{c}=1+\frac{2}{n}\max \{p+1, q+1\}. $$

In [22], the authors studied the following Newtonian filtration system:

$$\begin{aligned} \frac{\partial u}{\partial t}=\Delta u^{m}+v^{p},\quad \quad \frac{\partial v}{\partial t}=\Delta v^{m}+u^{q}, \quad x\in \mathbb{R}^{n}, t>0, \end{aligned}$$
(5)

where \(0< m<1\), \(p,q\ge 1\), and \(pq>1\). It was proved that the critical Fujita curve is

$$ (pq)_{c}=m^{2}+\frac{2}{n}\max \{p+m,q+m\}. $$

In [9], the authors studied the Fujita type theorem for the outer problem of the following coupled nonlinear diffusion equations with convective terms:

$$\begin{aligned} &\frac{\partial u}{\partial t}=\Delta u+\kappa \frac{x}{ \vert x \vert ^{2}} \cdot \nabla u+ \vert x \vert ^{\lambda _{1}} v^{p}, \quad x\in \mathbb{R}^{n} \backslash \overline{B}_{1}, t>0, \\ &\frac{\partial v}{\partial t}=\Delta v+\kappa \frac{x}{ \vert x \vert ^{2}} \cdot \nabla v+ \vert x \vert ^{\lambda _{2}} u^{q}, \quad x\in \mathbb{R}^{n} \backslash \overline{B}_{1}, t>0, \end{aligned}$$

and obtained

$$\begin{aligned} (pq)_{c}= \textstyle\begin{cases} 1+ \frac{\max \{p(2+\lambda _{2})+(2+\lambda _{1}), q(2+\lambda _{1})+(2+\lambda _{2})\}}{ n+\kappa },& \kappa >-n, \\ +\infty ,&\kappa \le -n. \end{cases}\displaystyle \end{aligned}$$

In this paper, we prove that the critical Fujita exponent is

$$\begin{aligned} p_{c}= \textstyle\begin{cases} m+\frac{\lambda +2}{n+\kappa },&\kappa >-n, \\ +\infty ,&\kappa \le -n. \end{cases}\displaystyle \end{aligned}$$
(6)

The main attention of this paper is to prove the global existence and blow-up properties of solutions. For the global existence of the problem solution, we use the method of constructing the self-similar solution and the comparison principle to prove our conclusion. For the blow-up properties of solutions, we adopt the integral estimation method. It is noted that when discussing the global existence of solutions, we construct the self-similar upper solution to the system. In order to let the self-similar solutions have the same compact supported set, we introduce the perturbation term \(( \vert x \vert +1)^{\mu }\). But the disturbance term has a negative impact on our results, which is the problem we need to solve.

The paper is organized as follows. In Sect. 2, we state some definitions and some theorems. Then, several useful auxiliary lemmas are given. In Sect. 4, we derive a Fujita type theorem for problem (1)–(4). At last, we study the asymptotic behavior of the solution to problem (1)–(4) in the critical case.

Preliminaries

In this section, we introduce the definition of the solutions to problem (1)–(4) that will be useful for the rest of the paper.

Definition 2.1

Let \(0< T\le +\infty \). A pair of nonnegative functions \((u,v)\) is called a super (sub) solution to problem (1)–(4) in \((0,T)\) if

$$ u, v\in C \bigl([0,T ),L_{\mathrm{loc}}^{m} \bigl( \mathbb{R}^{n} \bigr)\bigr)\cap L^{\infty }_{ \mathrm{loc}} \bigl(0,T;L^{\infty } \bigl(\mathbb{R}^{n} \bigr) \bigr), $$

and the following integral inequalities

$$\begin{aligned} & \int _{0}^{T} \int _{\mathbb{R}^{n}\backslash B_{1}}u(x,t) \frac{\partial \varphi }{\partial t}(x,t)\,{\mathrm{d}}x\,{ \mathrm{d}}t+ \int _{0}^{T} \int _{\mathbb{R}^{n}\backslash B_{1}}u^{m}(x,t)\Delta \varphi (x,t)\,{ \mathrm{d}}x\,{\mathrm{d}}t \\ &\quad{} -\kappa \int _{0}^{T} \int _{\mathbb{R}^{n}\backslash B_{1}}u^{m}(x,t) \operatorname{div} \biggl( \frac{1}{ \vert x \vert ^{2}}\varphi (x,t)x \biggr)\,{\mathrm{d}}x\,{\mathrm{d}}t \\ &\quad{}+ \int _{0}^{T} \int _{\mathbb{R}^{n}\backslash B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \varphi (x,t)\,{\mathrm{d}}x\,{\mathrm{d}}t \\ &\quad{}- \int _{0}^{T} \int _{\partial B_{1}}u^{m}(x,t) \biggl( \frac{\partial \varphi }{\partial {\boldsymbol{\nu }}}(x,t)- \frac{\kappa }{ \vert x \vert ^{2}} \varphi (x,t)x\cdot {\boldsymbol{\nu }} \biggr)\,{\mathrm{d}} \sigma\,{\mathrm{d}}t \\ &\quad{}+ \int _{\mathbb{R}^{n}\backslash B_{1}}u_{0}(x)\varphi (x,0)\,{\mathrm{d}}x \le ( \ge )\, 0, \\ & \int _{0}^{T} \int _{\mathbb{R}^{n}\backslash B_{1}}v(x,t) \frac{\partial \psi }{\partial t}(x,t)\,{\mathrm{d}}x\,{ \mathrm{d}}t+ \int _{0}^{T} \int _{\mathbb{R}^{n}\backslash B_{1}}v^{m}(x,t)\Delta \psi (x,t)\,{\mathrm{d}}x \,{\mathrm{d}}t \\ &\quad{}-\kappa \int _{0}^{T} \int _{\mathbb{R}^{n}\backslash B_{1}}v^{m}(x,t) \operatorname{div} \biggl( \frac{1}{ \vert x \vert ^{2}}\psi (x,t)x \biggr)\,{\mathrm{d}}x\,{\mathrm{d}}t \\ &\quad{}+ \int _{0}^{T} \int _{\mathbb{R}^{n}\backslash B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi (x,t)\,{\mathrm{d}}x\,{\mathrm{d}}t \\ &\quad{}- \int _{0}^{T} \int _{\partial B_{1}}v^{m}(x,t) \biggl( \frac{\partial \psi }{\partial {\boldsymbol{\nu }}}(x,t) -\frac{\kappa }{ \vert x \vert ^{2}} \psi (x,t)x\cdot {\boldsymbol{\nu }} \biggr)\,{\mathrm{d}} \sigma\,{\mathrm{d}}t \\ &\quad{}+ \int _{\mathbb{R}^{n}\backslash B_{1}}v_{0}(x)\psi (x,0)\,{\mathrm{d}}x\le ( \ge )\,0 \end{aligned}$$

are fulfilled for any \(0\le \varphi \), \(\psi \in C^{2,1}(\mathbb{R}^{n}\times [0,T))\) vanishing when t is near T or \(\vert x \vert \) is sufficiently large. \((u,v)\) is called a solution to problem (1)–(4) in \((0,T)\) if it is both a supersolution and a subsolution.

Definition 2.2

A solution \((u,v)\) to problem (1)–(4) is said to blow up in a finite time \(0< T<+\infty \) if

$$ \bigl\Vert u(\cdot ,t) \bigr\Vert _{L^{\infty }(\mathbb{R}^{n}\backslash B_{1})} + \bigl\Vert v( \cdot ,t) \bigr\Vert _{L^{\infty }(\mathbb{R}^{n}\backslash B_{1})} \to +\infty \quad \text{as }t\to T^{-}, $$

which T is called the blow-up time. Otherwise, \((u,v)\) is said to be global.

The following existence theorem and the comparison principle to problem (1)–(4) play an important role in proving our main results.

Theorem 2.1

(Local existence)

When \(0\le u_{0}\), \(v_{0}\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})\cap L^{\infty }(\mathbb{R}^{n})\), the Cauchy problem (1)(4) admits at least one solution locally in time.

Theorem 2.2

(Comparison principle)

For \(0< T\leq +\infty \), assume that \((u^{*},v^{*})\) and \((u^{**},v^{**})\) are two solutions to system (1) and (2) with nonnegative initial data \(u_{0}^{*}(x)\), \(v_{0}^{*}(x)\) and \(u_{0}^{**}(x)\), \(v_{0}^{**}(x)\) in \((0, T)\), respectively. If \((u_{0}^{*}(x), v_{0}^{*}(x))\le (u_{0}^{**}(x), v_{0}^{**}(x))\) a.e. in \(\mathbb{R}^{n}\), then \((u^{*},v^{*})\le (u^{**},v^{**})\) a.e. in \(\mathbb{R}^{n}\times (0,T)\).

The proofs of Theorem 2.1 and Theorem 2.2 are the same as the one in [23, 25, 28] and are omitted here.

Auxiliary lemmas

In order to research the blow-up property of solutions to problem (1)–(4), we need the following auxiliary lemmas.

Lemma 3.1

Assume that \((u,v)\) is a solution to problem (1)(4). Then there exists \(R_{0}>0\) depending only on n and κ such that, for any \(l>R_{0}\),

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t} \int _{\mathbb{R}^{n}\backslash B_{1}}u(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \ge-C_{0}l^{-2} \int _{B_{\delta l}\setminus B_{l}} u^{m}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x + \int _{\mathbb{R}^{n}\backslash B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x, \end{aligned}$$
(7)
$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t} \int _{\mathbb{R}^{n}\backslash B_{1}}v(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \ge-C_{0}l^{-2} \int _{B_{\delta l}\setminus B_{l}} v^{m}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x + \int _{\mathbb{R}^{n}\backslash B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x, \end{aligned}$$
(8)

where

$$ \delta = \textstyle\begin{cases} 2,&n+\kappa -1\le 0, \\ \frac{\pi }{n+\kappa -1}+1, &n+\kappa -1>0, \end{cases}\displaystyle \qquad C_{0}= \frac{\pi ^{2}}{(\delta -1)^{2}}, $$

and

$$\begin{aligned} \psi _{l}(r)= \textstyle\begin{cases} r^{\kappa },&1\le r\le l, \\ \frac{1}{2}r^{\kappa } (1+\cos \frac{(r-l)\pi }{(\delta -1)l} ), &l< r< \delta l, \\ 0,&r\ge \delta l. \end{cases}\displaystyle \end{aligned}$$

Proof

It follows from Definition 2.1 that

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t} \int _{\mathbb{R}^{n}\backslash B_{1}}u(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad = \int _{B_{\delta l}\backslash B_{1}}u^{m}(x,t) \biggl(\Delta \psi _{l} \bigl( \vert x \vert \bigr)- \kappa \operatorname{div} \biggl(\frac{1}{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)x \biggr) \biggr)\,{\mathrm{d}}x \\ &\quad\quad{} - \int _{\partial B_{1}}v^{m}(x,t) \biggl( \frac{\partial \psi _{l}( \vert x \vert )}{\partial {\boldsymbol{\nu }}} - \frac{\kappa }{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)x\cdot { \boldsymbol{\nu }} \biggr)\,{\mathrm{d}}\sigma + \int _{\mathbb{R}^{n}\backslash B_{1}}v^{p}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad = \int _{B_{\delta l}\backslash B_{1}}u^{m}(x,t) \biggl(\Delta \psi _{l} \bigl( \vert x \vert \bigr)- \kappa \operatorname{div} \biggl(\frac{1}{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)x \biggr) \biggr)\,{\mathrm{d}}x \\ &\quad\quad{} + \int _{\mathbb{R}^{n}\backslash B_{1}}v^{p}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x, \quad t>0, \end{aligned}$$
(9)

where \(\psi _{l}(r)\in C^{1}([0,+\infty ))\) satisfies \(\psi _{l}'(0)=0\) and

$$ \frac{\partial \psi _{l}( \vert x \vert )}{\partial {\boldsymbol{\nu }}}- \frac{\kappa }{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)x\cdot {\boldsymbol{\nu }}=0, \quad x\in \partial B_{1}. $$

For \(0\le \vert x \vert \le l\), it is easily verified that

$$\begin{aligned} &\Delta \psi _{l} \bigl( \vert x \vert \bigr)- \kappa \operatorname{div} \biggl(\frac{1}{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)x \biggr) \\ &\quad =\psi _{l}'' \bigl( \vert x \vert \bigr)+\frac{n-\kappa -1}{ \vert x \vert }\psi _{l}' \bigl( \vert x \vert \bigr)-\kappa \frac{n-2}{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)=0. \end{aligned}$$
(10)

While for \(l\le \vert x \vert \le \delta l\), a direct calculation gives

$$\begin{aligned} &\Delta \psi _{l} \bigl( \vert x \vert \bigr)-\kappa \operatorname{div} \biggl(\frac{1}{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)x \biggr) \\ &\quad =-\frac{1}{2}(\delta -1)^{-1}\pi (n+\kappa -1)l^{-1} \vert x \vert ^{\kappa -1} \sin \frac{( \vert x \vert -l)\pi }{(\delta -1)l} \\ &\quad\quad{} -\frac{1}{2}(\delta -1)^{-2}\pi ^{2}l^{-2} \vert x \vert ^{\kappa }\cos \frac{( \vert x \vert -l)\pi }{(\delta -1)l}. \end{aligned}$$

If \(n+\kappa -1\le 0\), one gets

$$\begin{aligned} \Delta \psi _{l} \bigl( \vert x \vert \bigr)- \kappa \operatorname{div} \biggl(\frac{1}{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)x \biggr) &\ge-\frac{1}{2}(\delta -1)^{-2}\pi ^{2}l^{-2} \vert x \vert ^{\kappa } \cos \frac{( \vert x \vert -l)\pi }{(\delta -1)l} \\ &\ge -\frac{1}{2}(\delta -1)^{-2}\pi ^{2}l^{-2} \psi _{l} \bigl( \vert x \vert \bigr). \end{aligned}$$
(11)

If \(n+\kappa -1>0\), we have

$$\begin{aligned} &\Delta \psi _{l} \bigl( \vert x \vert \bigr)- \kappa \operatorname{div} \biggl(\frac{1}{ \vert x \vert ^{2}}\psi _{l} \bigl( \vert x \vert \bigr)x \biggr) \\ &\quad \ge-\frac{1}{2}(\delta -1)^{-2}\pi ^{2}l^{-2} \vert x \vert ^{\kappa }\sin \frac{( \vert x \vert -l)\pi }{(\delta -1)l} -\frac{1}{2}( \delta -1)^{-2}\pi ^{2}l^{-2} \vert x \vert ^{ \kappa }\cos \frac{( \vert x \vert -l)\pi }{(\delta -1)l} \\ &\quad \geq-\frac{1}{2}(\delta -1)^{-2}\pi ^{2}l^{-2} \psi _{l} \bigl( \vert x \vert \bigr). \end{aligned}$$
(12)

By (9)–(12), we obtain (7). Similarly, one can prove that (8) holds. □

To prove the existence of a nontrivial global solution to problem (1)–(4), we introduce the following form of self-similar supersolutions to system (1) and (2):

$$\begin{aligned} &u(x,t)=(t+1)^{-\alpha }U \bigl((t+1)^{-\beta } \vert x \vert \bigr),\quad x\in \mathbb{R}^{n} \backslash B_{1}, t \ge 0, \end{aligned}$$
(13)
$$\begin{aligned} &v(x,t)=(t+1)^{-\alpha }V \bigl((t+1)^{-\beta } \vert x \vert \bigr),\quad x\in \mathbb{R}^{n} \backslash B_{1}, t \ge 0, \end{aligned}$$
(14)

where

$$ \alpha =\frac{\lambda +2}{\lambda (m-1)+2(p-1)},\quad\quad \beta = \frac{(p-m)\alpha }{\lambda +2}. $$

By a simple calculation, we show that

$$\begin{aligned} & \bigl(U^{m} \bigr)''(r)+ \frac{n+\kappa -1}{r} \bigl(U^{m} \bigr)'(r)+\beta rU'(r)+\alpha U(r)+r^{\lambda }V^{p}(r)\le 0, \end{aligned}$$
(15)
$$\begin{aligned} & \bigl(V^{m} \bigr)''(r)+ \frac{n+\kappa -1}{r} \bigl(V^{m} \bigr)'(r)+\beta rV'(r)+\alpha V(r)+r^{\mu }U^{q}(r)\le 0, \end{aligned}$$
(16)

for any \(r>0\). Then the self-similar function \((u,v)\) with the structure (13)–(14) is a supersolution to (1) and (2).

Lemma 3.2

Assume that \(m>1\), \(\kappa >-n\), \(p>p_{c}\) and set

$$ U(r)=V(r)= \bigl(\eta -A r^{2} \bigr)_{+}^{1/(m-1)}, \quad r\ge 0, $$
(17)

where \(s_{+}= \max \{0,s\}\), \(\eta >0\), and

$$ A=\frac{(m-1)(p-m)\alpha }{m(n+\kappa )(p+p_{c}-2m)}. $$

Then there exists sufficiently small \(\eta >0\) such that \((u,v)\) given by (13), (14), and (17) is a supersolution to system (1) and (2).

Proof

It is clear that \(U^{m}\) and \(V^{m}\) satisfy (15) and (16) when \(r\ge (\eta /A)^{1/2}\). For \(0< r<(\eta /A)^{1/2}\), a simple computation can obtain

$$\begin{aligned} & \bigl(U^{m} \bigr)''(r)+\frac{n+\kappa -1}{r} \bigl(U^{m} \bigr)'(r)+\beta rU'(r)+\alpha U(r) \\ &\quad = \biggl(\frac{2A}{m-1} \biggl(\frac{2Am}{m-1}-\beta \biggr)U^{1-m}(r) + \biggl(\alpha -\frac{2Am(n+\kappa )}{m-1} \biggr) \biggr)U(r) \end{aligned}$$

and

$$\begin{aligned} & \bigl(V^{m} \bigr)''(r)+\frac{n+\kappa -1}{r} \bigl(V^{m} \bigr)'(r)+\beta rV'(r)+\alpha V(r) \\ &\quad = \biggl(\frac{2A}{m-1} \biggl(\frac{2Am}{m-1}-\beta \biggr)V^{1-m}(r) + \biggl(\alpha -\frac{2Am(n+\kappa )}{m-1} \biggr) \biggr)V(r). \end{aligned}$$

Due to \(\frac{2Am}{m-1}<\beta \), there exists sufficiently small \(\eta _{1}>0\) such that, for \(0<\eta <\eta _{1}\),

$$\begin{aligned} &\bigl(U^{m} \bigr)''(r)+ \frac{n+\kappa -1}{r} \bigl(U^{m} \bigr)'(r)+\beta rU'(r)+\alpha U(r)< - \frac{(p-p_{c})\alpha U(r)}{2(p+p_{c}-2m)}, \end{aligned}$$
(18)
$$\begin{aligned} &\bigl(V^{m} \bigr)''(r)+ \frac{n+\kappa -1}{r} \bigl(V^{m} \bigr)'(r)+\beta rV'(r)+\alpha V(r)< - \frac{(p-p_{c})\alpha V(r)}{2(p+p_{c}-2m)}. \end{aligned}$$
(19)

Then, due to \(\lambda , \mu >0\) and the definition of U, V, there exists \(\eta _{2}>0\) such that, for any \(0<\eta <\eta _{2}\),

$$\begin{aligned} \begin{aligned} &r^{\mu } U^{q-1}(r)\le A^{-\mu /2}\eta ^{(q-1)/(m-1)+\mu /2}< \frac{(p-p_{c})\alpha }{2(p+p_{c}-2m)},\quad 0< r< \biggl(\frac{\eta }{A} \biggr)^{1/2}, \\ &r^{\lambda }V^{p-1}(r)\le A^{-\lambda /2}\eta ^{(p-1)/(m-1)+\lambda /2}< \frac{(p-p_{c})\alpha }{2(p+p_{c}-2m)},\quad 0< r< \biggl(\frac{\eta }{A} \biggr)^{1/2}. \end{aligned} \end{aligned}$$

Combining the above inequations with (18) and (19), we can see that for sufficiently small \(0<\eta _{2}<\eta _{1}\) and \(0<\eta <\eta _{2}<\eta _{1}\), one gets (15) and (16). Thus, \((u,v)\) given by (13), (14), and (17) is a supersolution of system (1) and (2). □

Blow-up theorems of Fujita type

In this section, we establish the blow-up theorems of Fujita type for problem (1)–(4). First, we consider the case \(\kappa \le -n\).

Theorem 4.1

Assume that \(p, q>m\), \(\lambda , \mu \ge 0\), \(\kappa \le -n\), and \(0\le u_{0}\), \(v_{0}\in C_{0}(\mathbb{R}^{n}\setminus B_{1})\) are nontrivial, the solution to problem (1)(4) blows up in a finite time.

Proof

Let \((u,v)\) be the solution to problem (1)–(4). Denote

$$\begin{aligned} w_{l}(t)= \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t)+v(x,t) \bigr) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x,\quad t\ge 0. \end{aligned}$$
(20)

For any \(l>R_{0}\), Lemma 3.1 shows that

$$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}}t}w_{l}(t) &\ge -\frac{C_{0}}{l^{2}} \int _{{B_{ \delta l}\setminus B_{l}}}u^{m}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x + \int _{ \mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad{} -\frac{C_{0}}{l^{2}} \int _{B_{\delta l}\setminus B_{l}}v^{m}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x + \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x. \end{aligned}$$
(21)

The Hölder inequality leads to

$$\begin{aligned} & \int _{B_{\delta l}\setminus B_{l}}u^{m}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le C_{1}l^{n+\kappa -m(n+\kappa +\mu )/q} \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/q}, \end{aligned}$$
(22)
$$\begin{aligned} & \int _{B_{\delta l}\setminus B_{l}}v^{m}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le C_{1}l^{n+\kappa -m(n+\kappa +\lambda )/p} \biggl( \int _{ \mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/p}, \end{aligned}$$
(23)

where \(C_{1}>0\) is a positive constant independent of l. Substituting (22) and (23) into (21) shows that

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t}w_{l}(t) \\ &\quad \ge \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/q} \biggl( \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(q-m)/q} \\ &\qquad{} -C_{0}C_{1}l^{-2+n+\kappa -m(n+\kappa +\mu )/q} \biggr) \\ &\quad\quad{} + \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/p} \biggl( \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(p-m)/p} \\ &\qquad {}-C_{0}C_{1}l^{-2+n+\kappa -m(n+\kappa +\lambda )/p} \biggr). \end{aligned}$$
(24)

Owing to the Hölder inequality, for any \(t>0\), we have

$$\begin{aligned} & \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le \biggl( \int _{B_{\delta l}\setminus B_{1}} \vert x \vert ^{-\mu /(q-1)}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(q-1)/q} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{1/ q}, \\ & \int _{\mathbb{R}^{n}\setminus B_{1}}v(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le \biggl( \int _{B_{\delta l}\setminus B_{1}} \vert x \vert ^{-\lambda /(p-1)} \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(p-1)/p} \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{1/p}, \end{aligned}$$

which imply

$$\begin{aligned} & \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \ge \textstyle\begin{cases} C_{2} (\int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t)\psi _{l}( \vert x \vert )\,{\mathrm{d}}x )^{q} l^{n+\kappa +\mu -q(n+\kappa )}, &\text{if } A(q,\mu )< 0, \\ C_{2} (\int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t)\psi _{l}( \vert x \vert )\,{\mathrm{d}}x )^{q}(\ln l)^{-(q-1)}, &\text{if }A(q,\mu )=0, \\ C_{2} (\int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t)\psi _{l}( \vert x \vert )\,{\mathrm{d}}x )^{q}, &\text{if }A(q,\mu )>0, \end{cases}\displaystyle \end{aligned}$$
(25)
$$\begin{aligned} & \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \ge \textstyle\begin{cases} C_{2} (\int _{\mathbb{R}^{n}\setminus B_{1}}v(x,t)\psi _{l}( \vert x \vert )\,{\mathrm{d}}x )^{p} l^{n+\kappa +\lambda -p(n+\kappa )}, &\text{if } A(p,\lambda )< 0, \\ C_{2} (\int _{\mathbb{R}^{n}\setminus B_{1}}v(x,t)\psi _{l}( \vert x \vert )\,{\mathrm{d}}x )^{p}(\ln l)^{-(p-1)}, &\text{if }A(p,\lambda )=0, \\ C_{2} (\int _{\mathbb{R}^{n}\setminus B_{1}}v(x,t)\psi _{l}( \vert x \vert )\,{\mathrm{d}}x )^{p}, &\text{if }A(p,\lambda )>0, \end{cases}\displaystyle \end{aligned}$$
(26)

where \(C_{2}>0\) is a positive constant independent of l and \(A(q,\mu )=n+\kappa +\mu -q(n+\kappa )\), \(A(p,\lambda )=n+\kappa + \lambda -p(n+\kappa )\). Here, it should be pointed out that the above discussion only requires \(p,q >m\).

Due to \(\kappa \le -n\), it is easy to verify that \(A(q,\mu )>0\), \(A(p,\lambda )>0\). From (24)–(26),

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t}w_{l}(t) \\ &\quad \ge C_{2}^{m/q} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m} \biggl(C_{2}^{(q-m)/q} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q-m} \\ &\quad\quad{} -C_{0}C_{1}l^{-2+n+\kappa -m(n+\kappa +\mu )/q} \biggr) \\ &\quad\quad{} +C_{2}^{m/p} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}v(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m} \biggl(C_{2}^{(p-m)/p} \biggl( \int _{ \mathbb{R}^{n}\setminus B_{1}}v(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p-m} \\ &\quad \quad {}-C_{0}C_{1}l^{-2+n+\kappa -m(n+\kappa +\lambda )/p} \biggr). \end{aligned}$$
(27)

For sufficiently large \(l_{1}>1\), and note that \(-2+n+\kappa -m(n+\kappa +\mu )/q<0\), \(-2+n+\kappa -m(n+\kappa +\lambda )/p<0\), one can get

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t}w_{l_{1}}(t) \\ &\quad \ge C_{2}^{m/q} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t) \psi _{l_{1}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m} \\ &\qquad {}\times \frac{1}{2} C_{2}^{(q-m)/q} \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}}u(x,t)\psi _{l_{1}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q-m} \\ &\quad\quad{} +C_{2}^{m/p} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}v(x,t)\psi _{l_{1}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m} \\ &\qquad {}\times \frac{1}{2} C_{2}^{(p-m)/p} \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}}v(x,t)\psi _{l_{1}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p-m} \\ &\quad \ge C_{3} \biggl( \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t) \psi _{l_{1}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q}+ \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}}v(x,t)\psi _{l_{1}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p} \biggr) \\ &\quad \ge 2^{p+q}C_{3}\cdot \min \bigl\{ w_{l_{1}}^{p}(t), w_{l_{1}}^{q}(t) \bigr\} , \end{aligned}$$

where \(C_{3}>0\) is a constant depending on \(l_{1}\). Since \(p, q>m>1\), there exists \(0< T<+\infty \) such that

$$ w_{l_{1}}(t)= \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t)+v(x,t) \bigr)\psi _{l_{1}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x\rightarrow +\infty ,\quad t \rightarrow T^{-}. $$

Obviously, \(\operatorname{supp}\psi _{l_{1}}(x)=B_{2{l_{1}}}\). Then one gets

$$ \bigl\Vert u(\cdot ,t) \bigr\Vert _{L^{\infty }(\mathbb{R}^{n}\setminus B_{1})}+ \bigl\Vert v(\cdot ,t) \bigr\Vert _{L^{\infty }(\mathbb{R}^{n}\setminus B_{1})}\rightarrow +\infty , \quad t\rightarrow T^{-}. $$

That is to say, \((u,v)\) blows up in a finite time. □

Next, we discuss the case \(\kappa >-n\).

Theorem 4.2

Assume that \(p, q>m>1\), \(\lambda , \mu >0\), \(\kappa >-n\), and \(0\le u_{0}\), \(v_{0}\in C_{0}(\mathbb{R}^{n}\setminus B_{1})\) are nontrivial. Then, for \(p< p_{c}\), any nontrivial solution to problem (1)(4) blows up in a finite time.

Proof

Let \((u,v)\) be a nontrivial solution to problem (1)–(4). Set

$$\begin{aligned} w_{l}(t)= \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t)+l^{\theta }v(x,t) \bigr)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x,\quad t\ge 0, \end{aligned}$$
(28)

where θ is a constant determined below. According to Lemma 3.1, for any \(l>R_{0}\),

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t}w_{l}(t) \\ &\quad \ge -C_{0}l^{-2} \int _{{B_{\delta l}\setminus B_{l}}}u^{m}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x +l^{\theta } \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad\quad{} -C_{0}l^{-2+\theta } \int _{B_{\delta l}\setminus B_{l}}v^{m}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x + \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x. \end{aligned}$$
(29)

Substituting (22) and (23) into (29) shows that

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t}w_{l}(t) \\ &\quad \ge \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/q} \\ &\quad\quad{} \cdot \biggl(l^{\theta } \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(q-m)/q}-C_{0}C_{1}l^{-2+n+ \kappa -m(n+\kappa +\mu )/q} \biggr) \\ &\quad\quad{} + \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/p} \biggl( \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(p-m)/p} \\ &\qquad{} -C_{0}C_{1}l^{-2+\theta +n+\kappa -m(n+\kappa +\lambda )/p} \biggr). \end{aligned}$$
(30)

Let us discuss the classification of symbols of \(A(q,\mu )\) and \(A(p,\lambda )\) in (25), (26).

If \(A(q,\mu )<0\), \(A(p,\lambda )<0\), we substitute (25) and (26) into (30), and this yields that

$$\begin{aligned} &\frac{{\mathrm{d}}w_{l}(t)}{{\mathrm{d}}t} \\ &\quad \ge -C_{0}C_{4}l^{m(\theta )}w_{l}^{m}(t)+C_{2}l^{-q(n+\kappa )+n+ \kappa +\mu +\theta } \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q} \\ &\quad\quad{} +C_{2}l^{-p(n+\kappa )+n+\kappa +\lambda -p\theta } \biggl( \int _{ \mathbb{R}^{n}\setminus B_{1}}l^{\theta }v(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p}, \end{aligned}$$
(31)

where \(C_{4}=\max \{C_{2}^{m/p}, C_{2}^{m/q}\}>0\), and

$$ m(\theta )=\max \bigl\{ (1-m) (n+\kappa )-2, (1-m) (n+\kappa )-2-(m-1) \theta \bigr\} . $$

Set

$$ \theta =\frac{q-p}{p+1} \biggl(n+\kappa -\frac{\lambda +2}{p-m} \biggr), $$

which implies that

$$ -p(n+\kappa )+n+\kappa +\lambda -p\theta =-q(n+\kappa )+n+\kappa + \mu +\theta = \Theta , $$

namely,

$$ \Theta = \frac{(-p^{2}q+pqm+p-m)(n+\kappa )+(\lambda +2)(pq-p^{2})}{(p+1)(p-m)}+ \lambda . $$

By a simple calculation,

$$\begin{aligned} &\frac{{\mathrm{d}}w_{l}(t)}{{\mathrm{d}}t} \\ &\quad \ge -C_{0}C_{4}l^{m(\theta )}w_{l}^{m}(t) \\ &\quad\quad{} +C_{2}l^{\Theta } \biggl( \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q}+ \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}l^{\theta }v(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p} \biggr) \\ &\quad \ge w_{l}^{m}(t) \bigl(-C_{0}C_{4}l^{m(\theta )}+2^{-(p+q)}C_{2}l^{ \Theta } \cdot \min \bigl\{ w_{l}^{p-m}(t), w_{l}^{q-m}(t) \bigr\} \bigr). \end{aligned}$$
(32)

Note that if \(p< p_{c}\), then \(m(\theta )<\Theta \). Further, \(w_{l}(0)\) is nondecreasing with respect to \(l\in (0,+\infty )\) and

$$ \sup \bigl\{ w_{l}(0): l\in (0,+\infty ) \bigr\} >0. $$

Then there exists sufficiently large \(l_{2}>1\) such that

$$\begin{aligned} C_{0}C_{4}l_{2}^{m(\theta )} \le 2^{-(p+q+1)}C_{2}l_{2}^{\Theta }\cdot \min \bigl\{ w_{l_{2}}^{p-m}(0), w_{l_{2}}^{q-m}(0) \bigr\} . \end{aligned}$$
(33)

Combining (32) with (33), we get

$$ \frac{{\mathrm{d}}w_{l_{2}}(t)}{{\mathrm{d}}t}\ge 2^{-(p+q+1)}C_{2}l_{2}^{ \Theta } \cdot \min \bigl\{ w_{l_{2}}^{p}(t), w_{l_{2}}^{q}(t) \bigr\} . $$

Just like the proof of Theorem 4.1, we can obtain that \((u,v)\) blows up in a finite time.

For \(A(q,\mu )=0\), \(A(p,\lambda )<0\), we set \(\theta =0\). It follows from (25), (26), and (30) that

$$\begin{aligned} &\frac{{\mathrm{d}}w_{l}(t)}{{\mathrm{d}}t} \\ &\quad \ge \biggl(C_{2}(\ln l)^{-(q-1)} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q} \biggr)^{m/q} \\ &\qquad{} \times \biggl(C_{2}^{(q-m)/q}(\ln l)^{-(q-1)(q-m)/q} \biggl( \int _{ \mathbb{R}^{n}\setminus B_{1}}u(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q-m} \\ &\quad\quad{} -C_{0}C_{1}l^{n+\kappa -2-m(n+\kappa +\mu )/q} \biggr) \\ &\quad\quad{} + \biggl(C_{2}l^{-p(n+\kappa )+n+\kappa +\lambda } \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}}v(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p} \biggr)^{m/p} \\ &\qquad{} \times \biggl(C_{2}^{(p-m)/p}l^{(-p(n+\kappa )+n+\kappa + \lambda )(p-m)/p} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}v(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p-m} \\ &\quad\quad{} -C_{0}C_{1}l^{n+\kappa -2-m(n+\kappa +\lambda )/p} \biggr). \end{aligned}$$
(34)

Here

$$\begin{aligned} &n+\kappa -2-m(n+\kappa +\mu )/q< 0, \\ &n+\kappa -2-m(n+\kappa +\lambda )/p< \bigl(-p(n+\kappa )+n+\kappa + \lambda \bigr) (p-m)/p. \end{aligned}$$

Then there exists sufficiently large \(l_{3}\) such that

$$\begin{aligned} &\frac{{\mathrm{d}}w_{l_{3}}(t)}{{\mathrm{d}}t} \\ &\quad \ge C_{2}^{m/q}(\ln l_{3})^{m(1-q)/q} \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}}u(x,t)\psi _{l_{3}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m} \\ &\qquad{} \times \frac{1}{2}C_{2}^{(q-m)/q}(\ln l_{3})^{(1-q)(q-m)/q} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t)\psi _{l_{3}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q-m} \\ &\quad\quad{} +C_{2}^{m/p}l_{3}^{-m(n+\kappa )+m(n+\kappa +\lambda )/p} \biggl( \int _{ \mathbb{R}^{n}\setminus B_{1}}v(x,t)\psi _{l_{3}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m} \\ &\qquad{} \times \frac{1}{2}C_{2}^{(p-m)/p}l_{3}^{(-p(n+\kappa )+n+ \kappa +\lambda )(p-m)/p} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}v(x,t) \psi _{l_{3}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p-m} \\ &\quad \ge C_{5} \biggl( \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}}u(x,t) \psi _{l_{3}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{q} + \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}}v(x,t)\psi _{l_{3}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{p} \biggr) \\ &\quad \ge 2^{-(p+q)}C_{5}\cdot \min \bigl\{ w_{l_{3}}^{p}(t),w_{l_{3}}^{q}(t) \bigr\} , \end{aligned}$$

where \(C_{5}>0\) is a positive constant depending only on \(l_{3}\). Therefore, we can obtain that \((u,v)\) blows up in a finite time by a similar proof process of Theorem 4.1.

For other cases, select \(\theta =0\). By the similar argument as \(A(q,\mu )=0\), \(A(p,\lambda )<0\), we can also prove that any nontrivial solution blows up in a finite time. □

Theorem 4.3

Assume that \(p, q>m>1\), \(\lambda , \mu >0\), \(\kappa >-n\), and \(0\le u_{0}\), \(v_{0}\in C_{0}(\mathbb{R}^{n}\setminus B_{1})\) are nontrivial. Then, if \(p>p_{c}\), there exist both nontrivial global and blow-up solutions to problem (1)(4).

Proof

The comparison principle and Lemma 3.1 can prove the existence of the nontrivial global solution to problem (1)–(4) with sufficiently small initial value. Next, we study the blow-up solution to problem (1)–(4) when the initial value is sufficiently large.

For \(l>1\) and \((u,v)\) is the solution to problem (1)–(4), set

$$ \tilde{w}_{l}(t)= \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t)+v(x,t) \bigr)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x,\quad t\ge 0. $$

According to the Hölder inequality and (30), we have

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{w}_{l}(t) \\ &\quad \ge \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/q} \biggl( \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(q-m)/q} \\ &\qquad{} -C_{0}C_{1}l^{-2+n+\kappa -m(n+\kappa +\mu )/q} \biggr) \\ &\quad\quad{} + \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/p} \biggl( \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(p-m)/p} \\ &\qquad {}-C_{0}C_{1}l^{-2+n+\kappa -m(n+\kappa +\lambda )/p} \biggr) \\ &\quad \ge \tilde{w}_{l}^{m}(t) \bigl(-C_{0}C_{1}C_{6}+2^{-(p+q)}C_{7} \cdot \min \bigl\{ \tilde{w}_{l}^{p-m}(t), \tilde{w}_{l}^{q-m}(t) \bigr\} \bigr), \end{aligned}$$
(35)

where

$$\begin{aligned}& C_{6}=\max \biggl\{ l^{-2+n+\kappa -m(n+\kappa +\mu )/q} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{-\mu /(q-1)}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(1-q)m/q}, \\& \hphantom{C_{6}}\quad {}l^{-2+n+\kappa -m(n+\kappa +\lambda )/p} \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{-\lambda /(p-1)}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(1-p)m/p} \biggr\} , \\& C_{7}=\min \biggl\{ \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{ \mu /(1-q)}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{1-q}, \biggl( \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\lambda /(1-p)}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{1-p} \biggr\} . \end{aligned}$$

If \((u_{0},v_{0})\) is so large that

$$ C_{0}C_{1}C_{6}\le 2^{-(p+q+1)}C_{7} \cdot \min \bigl\{ \tilde{w}_{l}^{p-m}(0), \tilde{w}_{l}^{q-m}(0) \bigr\} , $$

then (35) leads to

$$ \frac{{\mathrm{d}}\tilde{w}_{l}(t)}{{\mathrm{d}}t}\ge 2^{-(p+q+1)}C_{7}\cdot \min \bigl\{ \tilde{w}_{l}^{p}(t), \tilde{w}_{l}^{q}(t) \bigr\} ,\quad t>0. $$

By a similar argument in the proof of Theorem 4.1, one can show that \((u,v)\) blows up in a finite time. □

The critical case

In this section, we consider the critical case

$$\begin{aligned} p=p_{c}=m+\frac{2+\lambda }{n+\kappa }. \end{aligned}$$
(36)

Obviously, we can prove that (29), (32) still hold, and

$$\begin{aligned} n+\kappa +\mu -q(n+\kappa )=n+\kappa +\lambda -p_{c}(n+\kappa )=(1-m) (n+ \kappa )-2. \end{aligned}$$
(37)

The result of the critical case is based on the following three lemmas.

Lemma 5.1

Assume that \((u,v)\) is a nontrivial global solution to problem (1)(4) with \(p=p_{c}\), then there exists \(M_{0}>0\) independent of t such that

$$\begin{aligned} \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t)+v(x,t) \bigr) \vert x \vert ^{ \kappa }\,{\mathrm{d}}x\le M_{0},\quad t>0. \end{aligned}$$
(38)

Proof

For any sufficiently large \(l>1\), it follows from (32) that

$$\begin{aligned} &\frac{{\mathrm{d}}w_{l}(t)}{{\mathrm{d}}t} \\ &\quad \ge w_{l}^{m}(t)l^{-(m-1)(n+\kappa )-2} \bigl(-C_{0}C_{4}+2^{-(p_{c}+q)}C_{2} \cdot \min \bigl\{ w_{l}^{p_{c}-m}(t), w_{l}^{q-m}(t) \bigr\} \bigr), \end{aligned}$$

where \(w_{l}\) is defined by (28) with \(\theta =0\). Similar to the end of the proof of Theorem 4.1, there exists some \(l_{3}>1\) such that, for any \(l>l_{3}\),

$$ 2^{-(p_{c}+q+1)}C_{2}\cdot \min \bigl\{ w_{l}^{p_{c}-m}(t), w_{l}^{q-m}(t) \bigr\} \le C_{0}C_{4}, $$

which implies

$$\begin{aligned} w_{l}(t)\le \max \bigl\{ \bigl(C_{0}C_{4}C_{2}^{-1}2^{p_{c}+q+1} \bigr)^{1/(p_{c}-m)}, \bigl(C_{0}C_{4}C_{2}^{-1}2^{p_{c}+q+1} \bigr)^{1/(q-m)} \bigr\} . \end{aligned}$$

Let \(l\to +\infty \) in the above inequality, then we can obtain (38). □

Lemma 5.2

Under the assumption of Lemma 5.1, there exist three positive constants \(M_{1}, M_{2}, M_{3}>0\) independent of l and t such that, for any sufficiently large \(l>1\),

$$\begin{aligned} \frac{{\mathrm{d}}w_{l}(t)}{{\mathrm{d}}t} &\ge M_{1}^{m-\tau }l^{(1-m)(n+ \kappa )-2}w_{l}^{m-\tau }(t) \\ &\quad{} \times \biggl(-M_{2} \biggl( \int _{B_{\delta l}\setminus B_{l}} \bigl(u(x,t)+v(x,t) \bigr)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{\tau } \\ &\quad{} +M_{1}^{-(m-\tau )}M_{3}\cdot \min \bigl\{ w_{l}^{p_{c}-m+ \tau }(t),w_{l}^{q-m+\tau }(t) \bigr\} \biggr), \end{aligned}$$
(39)

where

$$ 0< \tau < \min \biggl\{ \frac{p_{c}-m}{p_{c}-1}, \frac{q-m}{q-1} \biggr\} . $$

Proof

It is easy to verify that

$$\begin{aligned} &n+\kappa -2-m(n+\kappa +\mu )/q+\tau \bigl(\mu -(q-1) (n+\kappa ) \bigr)/q \\ &\quad = \bigl((1-q) (n+\kappa )+\mu \bigr) (q-m+\tau )/q, \end{aligned}$$
(40)
$$\begin{aligned} &n+\kappa -2-m(n+\kappa +\lambda )/p_{c}+\tau \bigl( \lambda -(p_{c}-1) (n+ \kappa ) \bigr)/p_{c} \\ &\quad = \bigl((1-p_{c}) (n+\kappa )+\lambda \bigr) (p_{c}-m+\tau )/p_{c}. \end{aligned}$$
(41)

For any sufficiently large \(l>1\), it follows from the Hölder inequality that

$$\begin{aligned} & \int _{B_{\delta l}\setminus B_{l}} u^{m}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le \biggl( \int _{B_{\delta l}\setminus B_{l}} \vert x \vert ^{-(m-\tau )\mu /(q-m-(q-1) \tau )}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(q-m-(q-1)\tau )/q} \\ &\quad\quad{} \times \biggl( \int _{B_{\delta l}\setminus B_{l}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(m-\tau )/q} \biggl( \int _{B_{\delta l} \setminus B_{l}}u(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{\tau } \\ &\quad \le C_{8}l^{n+\kappa -(n+\kappa +\mu )m/q+\tau (\mu -(q-1)(n+ \kappa ))/q} \\ &\quad\quad{} \times \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(m-\tau )/q} \biggl( \int _{B_{\delta l} \setminus B_{l}}u(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{\tau }, \\ & \int _{B_{\delta l}\setminus B_{l}}v^{m}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le \biggl( \int _{B_{\delta l}\setminus B_{l}} \vert x \vert ^{-(m-\tau ) \lambda /(p_{c}-m-(p_{c}-1)\tau )}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(p_{c}-m-(p_{c}-1) \tau )/p_{c}} \\ &\quad\quad{} \times \biggl( \int _{B_{\delta l}\setminus B_{l}} \vert x \vert ^{\lambda }v^{p_{c}}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(m-\tau )/p_{c}} \biggl( \int _{B_{\delta l} \setminus B_{l}}v(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{\tau } \\ &\quad \le C_{8}l^{n+\kappa -(n+\kappa +\lambda )m/p_{c}+\tau (\lambda -(p_{c}-1)(n+ \kappa ))/p_{c}} \\ &\quad\quad{} \times \biggl( \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p_{c}}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(m-\tau )/p_{c}} \biggl( \int _{B_{\delta l} \setminus B_{l}}v(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{\tau }, \end{aligned}$$

where \(C_{8}>0\) is a constant independent of l. Substituting the above two inequalities into (29) with \(\theta =0\), it follows from (25), (26), (37), (40), and (41) that

$$\begin{aligned} &\frac{{\mathrm{d}}}{{\mathrm{d}}t}w_{l}(t) \\ &\quad \ge -C_{0}C_{8}l^{-(m-1)(n+\kappa )-2} \bigl(M_{1}w_{l}(t) \bigr)^{m-\tau } \biggl( \int _{B_{2l}\setminus B_{l}} \bigl(u(x,t)+v(x,t) \bigr)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{\tau } \\ &\qquad{} +2^{-(p_{c}+q)}C_{2}l^{-(m-1)(n+\kappa )-2}\cdot \min \bigl\{ w_{l}^{p_{c}}(t), w_{l}^{q}(t) \bigr\} , \end{aligned}$$

which yields (39) by choosing

$$ M_{1}=\max \bigl\{ C_{2}^{1/p_{c}},C_{2}^{1/q} \bigr\} ,\quad\quad M_{2}=C_{0}C_{8}, \quad\quad M_{3}=2^{-(p_{c}+q)}C_{2}. $$

 □

Lemma 5.3

Under the assumption of Lemma 5.1, there exists a constant \(M_{4}>0\) independent of l and t such that, for any sufficiently large \(l>1\),

$$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}}t} \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t)+v(x,t) \bigr)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x\ge -M_{4}l^{(p_{c}(n+\kappa -2)-m(n+ \kappa +\lambda ))/(p_{c}-m)}. \end{aligned}$$
(42)

Proof

Owing to the Hölder inequality, one obtains

$$\begin{aligned} & \int _{B_{2l}\setminus B_{l}}u^{m}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le \biggl( \int _{B_{2l}\setminus B_{l}} \vert x \vert ^{-\frac{m\mu }{q-m}}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(q-m)/q} \biggl( \int _{B_{2l}\setminus B_{l}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/q} \\ &\quad \le C_{9}l^{n+\kappa -m(n+\kappa +\mu )/q} \biggl( \int _{B_{2l} \setminus B_{l}} \vert x \vert ^{\mu }u^{q}(x,t) \,{\mathrm{d}}x \biggr)^{m/q}, \\ & \int _{B_{2l}\setminus B_{l}}v^{m}(x,t)\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le \biggl( \int _{B_{2l}\setminus B_{l}} \vert x \vert ^{- \frac{m\lambda }{p_{c}-m}}\psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{(p_{c}-m)/p_{c}} \biggl( \int _{B_{2l}\setminus B_{l}} \vert x \vert ^{\mu }v^{p_{c}}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \biggr)^{m/p_{c}} \\ &\quad \le C_{9}l^{n+\kappa -m(n+\kappa +\lambda )/p_{c}} \biggl( \int _{B_{2l} \setminus B_{l}} \vert x \vert ^{\lambda }v^{p_{c}}(x,t) \,{\mathrm{d}}x \biggr)^{m/p_{c}}, \end{aligned}$$

where \(C_{9}>0\), independent of l. Substitute the above results into (29) and

$$ \frac{q(n+\kappa -2)-m(n+\kappa +\mu )}{q-m} = \frac{p_{c}(n+\kappa -2)-m(n+\kappa +\lambda )}{p_{c}-m}, $$

then it follows from the Young inequality that

$$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}}t}w_{l}(t) &\ge -C_{0}C_{9}l^{n+\kappa -2-m(n+ \kappa +\mu )/q} \biggl( \int _{B_{2l}\setminus B_{l}} \vert x \vert ^{\mu }u^{q}(x,t) \,{\mathrm{d}}x \biggr)^{m/q} \\ &\quad{} + \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad {} -C_{0}C_{9}l^{n+\kappa -2-m(n+\kappa +\lambda )/p_{c}} \biggl( \int _{B_{2l} \setminus B_{l}} \vert x \vert ^{\lambda }v^{p_{c}}(x,t) \,{\mathrm{d}}x \biggr)^{m/p_{c}} \\ &\quad{} + \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\ge -\frac{m}{q} \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x + \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\mu }u^{q}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad{} -\frac{q-m}{q}(C_{0}C_{9})^{q/(q-m)}l^{(q(n+\kappa -2)-m(n+ \kappa +\mu ))/(q-m)} \\ &\quad{} -\frac{m}{p_{c}} \int _{\mathbb{R}^{n}\setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x + \int _{\mathbb{R}^{n} \setminus B_{1}} \vert x \vert ^{\lambda }v^{p}(x,t) \psi _{l} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad{} -\frac{p_{c}-m}{p_{c}}(C_{0}C_{9})^{p_{c}/(p_{c}-m)}l^{(p_{c}(n+ \kappa -2)-m(n+\kappa +\lambda ))/(p_{c}-m)} \\ &\ge -M_{4}l^{(p_{c}(n+\kappa -2)-m(n+\kappa +\lambda ))/(p_{c}-m)}, \end{aligned}$$

where

$$ M_{4}=\max \biggl\{ \frac{q-m}{q}(C_{0}C_{9})^{q/(q-m)}, \frac{p_{c}-m}{p_{c}}(C_{0}C_{9})^{p_{c}/(p_{c}-m)} \biggr\} . $$

 □

Now we prove the following theorem.

Theorem 5.1

Assume that \(\kappa >-n\). Then any nontrivial solution to problem (1)(4) with \(p=p_{c}\) blows up in a finite time.

Proof

We prove the theorem by contradiction. Assume that \((u,v)\) is a nontrivial global solution to problem (1)–(4) with \(p=p_{c}\). Set

$$\begin{aligned} \Lambda =\sup_{l>0, t>0}w_{l}(t)=\sup _{t>0} \int _{\mathbb{R}^{n} \setminus B_{1}} \bigl(u(x,t)+v(x,t) \bigr) \vert x \vert ^{\kappa }\,{\mathrm{d}}x. \end{aligned}$$
(43)

It follows from (38) and the nontriviality of \((u,v)\) that \(0<\Lambda <+\infty \). Owing to (43) and the monotonicity of \(w_{l}(t)\) with respect to \(l\in (0, +\infty )\), there exist \(l_{0}>1\) and \(t_{0}>0\) such that, for any \(0<\varepsilon <\Lambda \),

$$ w_{l_{0}/\delta }(t_{0})\ge \Lambda -\varepsilon . $$

From Lemma 5.3, for \(s\ge t_{0}\), we obtain

$$\begin{aligned} & \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,s)+v(x,s) \bigr)\psi _{l_{0}/ \delta } \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \ge \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t_{0})+v(x,t_{0}) \bigr)\psi _{l_{0}/\delta } \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\qquad{} -M_{4}(l_{0}/\delta )^{(p_{c}(n+\kappa -2)-m(n+\kappa + \lambda ))/(p_{c}-m)}(s-t_{0}) \\ &\quad \ge \Lambda -\varepsilon -M_{4}(l_{0}/\delta )^{(p_{c}(n+\kappa -2)-m(n+ \kappa +\lambda ))/(p_{c}-m)}(s-t_{0}), \end{aligned}$$

which yields that

$$\begin{aligned} & \int _{B_{\delta l_{0}}\setminus B_{l_{0}}} \bigl(u(x,s)+v(x,s) \bigr) \psi _{l_{0}} \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t)+v(x,t) \bigr) \vert x \vert ^{ \kappa }\,{\mathrm{d}}x - \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,s)+v(x,s) \bigr)\psi _{l_{0}/\delta } \bigl( \vert x \vert \bigr)\,{\mathrm{d}}x \\ &\quad \le \varepsilon +M_{4}(l_{0}/\delta )^{(p_{c}(n+\kappa -2)-m(n+ \kappa +\lambda ))/(p_{c}-m)}(s-t_{0}),\quad s\ge t_{0}. \end{aligned}$$

Let \(l=l_{0}\) in (39), from the above inequality, one gets that

$$\begin{aligned} \frac{{\mathrm{d}}w_{l_{0}}(t)}{{\mathrm{d}}t} &\ge M_{1}^{m-\tau }l_{0}^{(1-m)(n+ \kappa )-2}w_{l_{0}}^{m-\tau }(t) \\ &\quad{} \times \biggl(-M_{2} \biggl( \int _{B_{\delta l_{0}}\setminus B_{l_{0}}} \bigl(u(x,t)+v(x,t) \bigr) \psi _{l_{0}}\,{ \mathrm{d}}x \biggr)^{\tau } \\ &\quad{} +M_{1}^{-(m-\tau )}M_{3}\cdot \min \bigl\{ w_{l_{0}}^{p_{c}-m+ \tau }(t),w_{l_{0}}^{q-m+\tau }(t) \bigr\} \biggr) \\ &\ge M_{1}^{m-\tau }l_{0}^{(1-m)(n+\kappa )-2}w_{l_{0}}^{m-\tau }(t) \\ &\quad{} \times \bigl(-M_{2} \bigl(\varepsilon +M_{4}(l_{0}/ \delta )^{(p_{c}(n+ \kappa -2)-m(n+\kappa +\lambda ))/(p_{c}-m)}(s-t_{0}) \bigr)^{\tau } \\ &\quad{} +M_{1}^{-(m-\tau )}M_{3}\cdot \min \bigl\{ w_{l_{0}}^{p_{c}-m+ \tau }(t),w_{l_{0}}^{q-m+\tau }(t) \bigr\} \bigr). \end{aligned}$$

Take \(\varepsilon _{0}\in (0,\Lambda )\) and \(M_{5}>0\) to get

$$\begin{aligned} &M_{2}(\varepsilon _{0}+M_{5})^{\tau } \le \frac{1}{2}M_{1}^{-(m-\tau )}M_{3} \cdot \min \bigl\{ (\Lambda -\varepsilon )^{p_{c}-m+\tau }(t),(\Lambda - \varepsilon )^{q-m+\tau }(t) \bigr\} , \end{aligned}$$

where \(\varepsilon _{0}\) and \(M_{5}\) are independent of \(l_{0}\), \(0<\tau <\min \{\frac{p_{c}-m}{p_{c}-1}, \frac{q-m}{q-1} \}\). Then we obtain

$$\begin{aligned} \frac{{\mathrm{d}}w_{l_{0}}(t)}{{\mathrm{d}}t}\ge \frac{1}{2}M_{3}l_{0}^{(1-m)(n+ \kappa )-2} \cdot \min \bigl\{ w_{l_{0}}^{p_{c}}(t), w_{l_{0}}^{q}(t) \bigr\} , \quad t_{0}< t< t_{1}, \end{aligned}$$
(44)

where

$$ t_{1}=t_{0}+\frac{M_{5}}{M_{4}}(l_{0}/\delta )^{(-p_{c}(n+\kappa -2)+m(n+ \kappa +\lambda ))/(p_{c}-m)}. $$

Integrating (44) over \((t_{0},t_{1})\) with respect to t and using

$$ \bigl(p_{c}(n+\kappa -2)-m(n+\kappa +\lambda ) \bigr)/(p_{c}-m)=(1-m) (n+\kappa )-2, $$

one gets that

$$\begin{aligned} &w_{l_{0}}(t_{1}) \\ &\quad \ge w_{l_{0}}(t_{0})+\frac{1}{2}M_{3}l_{0}^{(1-m)(n+\kappa )-2} \cdot \min \bigl\{ (\Lambda -\varepsilon _{0})^{p_{c}}, ( \Lambda - \varepsilon _{0})^{q} \bigr\} (t_{1}-t_{0}) \\ &\quad \ge w_{l_{0}/\delta }(t_{0})+\frac{1}{2}M_{3}l_{0}^{(1-m)(n+\kappa )-2} \cdot \min \bigl\{ (\Lambda -\varepsilon _{0})^{p_{c}}, ( \Lambda - \varepsilon _{0})^{q} \bigr\} \\ &\qquad{} \times \frac{M_{5}}{M_{4}}(l_{0}/\delta )^{(-p_{c}(n+\kappa -2)+m(n+ \kappa +\lambda ))/(p_{c}-m)} \\ &\quad =w_{l_{0}/\delta }(t_{0}) \\ &\quad\quad{} +\frac{M_{3} M_{5}}{2 M_{4}}\delta ^{(p_{c}(n+\kappa -2)-m(n+ \kappa +\lambda ))/(p_{c}-m)} \cdot \min \bigl\{ ( \Lambda -\varepsilon _{0})^{p_{c}}, (\Lambda -\varepsilon _{0})^{q} \bigr\} . \end{aligned}$$

That is to say,

$$ \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t_{1})+v(x,t_{1}) \bigr) \vert x \vert ^{ \kappa }\,{\mathrm{d}}x \ge w_{l_{0}}(t_{1}) \ge w_{l_{0}/\delta }(t_{0})+ \gamma _{0}\ge \Lambda - \varepsilon _{0}+\gamma _{0}, $$

where

$$\begin{aligned} \gamma _{0}=\frac{M_{3}M_{5}}{2 M_{4}}\delta ^{(p_{c}(n+\kappa -2)-m(n+ \kappa +\lambda ))/(p_{c}-m)} \cdot \min \bigl\{ (\Lambda -\varepsilon _{0})^{p_{c}}, (\Lambda - \varepsilon _{0})^{q} \bigr\} \end{aligned}$$

is a positive constant independent of \(l_{0}\). It is obviously verified that

$$ w_{(\delta l_{0})/\delta }(t_{1})=w_{l_{0}}(t_{1})\ge \Lambda - \varepsilon _{0}+\gamma _{0}\ge \Lambda - \varepsilon _{0}. $$

Using the same method, one gets

$$ \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t_{2})+v(x,t_{2}) \bigr) \vert x \vert ^{ \kappa }\,{\mathrm{d}}x\ge w_{\delta l_{0}}(t_{2}) \ge w_{l_{0}}(t_{1})+ \gamma _{0}\ge \Lambda - \varepsilon _{0}+2\gamma _{0}, $$

where

$$ t_{2}=t_{1}+\frac{M_{5}}{M_{4}}l_{0}^{(-p_{c}(n+\kappa -2)+m(n+ \kappa +\lambda ))/(p_{c}-m)}. $$

Similarly, for any positive integer i, we obtain

$$\begin{aligned} & \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t_{i})+v(x,t_{i}) \bigr) \vert x \vert ^{ \kappa }\,{\mathrm{d}}x \\ &\quad \ge w_{\delta ^{i-1}l_{0}}(t_{i})\ge w_{\delta ^{i-2}l_{0}}(t_{i-1})+ \delta _{0}\ge \Lambda -\varepsilon _{0}+i\gamma _{0}, \end{aligned}$$
(45)

where

$$ t_{i}=t_{i-1}+\frac{M_{5}}{M_{4}} \bigl(\delta ^{i-2}l_{0} \bigr)^{-(p_{c}(n+ \kappa -2)+m(n+\kappa +\lambda ))/(p_{c}-m)}. $$

Letting \(i\to +\infty \) in (45) implies

$$ \sup_{t>0} \int _{\mathbb{R}^{n}\setminus B_{1}} \bigl(u(x,t)+v(x,t) \bigr) \vert x \vert ^{\kappa }\,{\mathrm{d}}x=+\infty , $$

which contradicts (38). □

Availability of data and materials

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Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which improved the original manuscript.

Funding

This work is supported by the National Natural Science Foundation of China (No. 11871133), by the Department of Science and Technology of Jilin Province (YDZJ202101ZYTS044).

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All the authors contributed to each part of this study equally and approved the final version of the manuscript.

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Correspondence to Yuanyuan Nie.

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Zhou, Y., Leng, Y. & Nie, Y. Fujita type theorem for a class of coupled quasilinear convection–diffusion equations. Bound Value Probl 2021, 36 (2021). https://doi.org/10.1186/s13661-021-01513-w

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MSC

  • 35B33
  • 35K20
  • 35K59

Keywords

  • Critical Fujita exponent
  • Fujita type theorem
  • Convection–diffusion equations
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