Blow-up solutions in a Cauchy problem of parabolic equations with spatial coefficients

Abstract

This paper deals with a Cauchy problem of the parabolic equations

$$\begin{aligned} u_t =\Delta u + a_{1}(x) u^{p_{1}} + b_{1}(x) v^{q_{1}},\ v_t =\Delta v + a_{2}(x) u^{p_{2}} + b_{2}(x) v^{q_{2}}, \end{aligned}$$

where the exponents $p_{i}$, $q_{i}$ $(i=1,2)$ are positive constants; the coefficients $a_{i}(x)\sim |x|^{\alpha _{i}}$ and $b_{i}(x)\sim |x|^{\beta _{i}}$ as $|x|\rightarrow +\infty$ with the parameters $\alpha _{i}$, $\beta _{i}\in R$. For $\alpha _{i}$, $\beta _{i}\ge 0$, we determine the exponent regions where all of the solutions blow up for any nonnegative nontrivial initial data. For at least one negative parameter, we find different conditions on global existence of solutions according to different classifications of the parameters.

1 Introduction

In this paper, we study a Cauchy problem of the parabolic equations with different spatially dependent coefficients

$$\begin{aligned} \left\{ \begin{array}{lll} u_t =\Delta u + a_{1}(x) u^{p_{1}} + b_{1}(x) v^{q_{1}}, &{}\quad (x,t)\in R^N\times [0,T),\\ v_t =\Delta v + a_{2}(x) u^{p_{2}} + b_{2}(x) v^{q_{2}}, &{}\quad (x,t)\in R^N\times [0,T),\\ u(x, 0)=u_{0}(x),\quad v(x, 0)=v_{0}(x), &{}\quad x\in R^N,\\ \end{array}\right. \end{aligned}$$

(1.1)

where the exponents $p_{i}$, $q_{i}$ $(i=1,2)$ are positive constants; the coefficients $a_{i}(x)$, $b_{i}(x)\gneqq 0$ are locally Hölder-continuous satisfying that $a_{i}(x)\sim |x|^{\alpha _{i}}$ and $b_{i}(x)\sim |x|^{\beta _{i}}$ as $|x|\rightarrow +\infty$ for $\alpha _{i}$, $\beta _{i}\in R$ $(i=1,2)$; Initial data $u_0$, $v_0\gneqq 0$ are nonnegative bounded continuous functions. The uniqueness and local existence of classical solutions can be obtained by the standard procedure in [14]. Nonlinear parabolic equations coupled via nonlinear sources just as (1.1) are widely used in chemical reactions, population dynamics, and heat transfer process, where the components of the solutions represent the thickness of two kinds of chemical reactants, the densities of two biological populations during a migration, and the temperature of two different materials during a propagation (see, for example, [4, 5, 12]).

It is well known that the Fujita blow-up exponent $p_c =1+2/N$ is introduced for the Cauchy problem of $u_t =\Delta u + u^{p}$. If $1<p\le p_c$, any nonnegative nontrivial solutions blow up in finite time. Pinsky studied the weighted equation $u_t =\Delta u +a (x) u^{p}$ with $a(x)\sim |x|^{\alpha }$ as $|x|\rightarrow +\infty$ and obtained the Fujita exponent $p_c =1+(2+\alpha )/N$ in [16]. Escobedo and Herrero studied the Fujita exponents of the problem (1.1) with $a_1 =b_2 =0$ and $a_2 =b_1 =1$ in [4].

Li, Sun and Zhang in [8] considered the Fujita exponent to the Cauchy problem of the following reaction–diffusion equations:

$$\begin{aligned} u_t =\Delta u + v^p,\quad v_t =\Delta v + a(x) u^q ,\quad (x,t)\in R^N\times [0,T), \end{aligned}$$

(1.2)

where $a(x)\sim |x|^m$ as $|x|\rightarrow +\infty$ and $m\in R$. They proved that (i) if $0<pq\le 1$ and $m\ge 0$, all of the solutions are global; (ii) if $pq>1$ and $m\ge 0$, there are no global solutions provided that $\max \left\{ \frac{\frac{m+2}{2}p+1}{pq-1},\frac{\frac{m+2}{2}+q}{pq-1} \right\} \ge N/2$; (iii) if $\max \left\{ \frac{\frac{m+2}{2}p+1}{pq-1},\frac{\frac{m+2}{2}+q}{pq-1} \right\} < N/2$ and $m\ge 0$, the problem (1.2) possesses both global and blow-up solutions. For $m<0$, some results on the global existence of solutions are proved under some additional conditions.

Souplet and Tayachi in [15] discussed the following Cauchy problem:

$$\begin{aligned} u_{t} =\Delta u +u^{p_1} +v^{q_2},\quad v_{t} =\Delta v +v^{p_2}+u^{q_1},\quad (x,t)\in {R}^N\times (0,T), \end{aligned}$$

(1.3)

with constants $p_{i}$, $q_{i}> 1$ $(i=1,2)$. If $p_{1}>q_{1}+1$ or $p_{2}>q_{2}+1$, then there exist initial data, such that non-simultaneous blow-up happens; If $p_{1}<q_{1}+1$ and $p_{2}<q_{2}+1$, then simultaneous blow-up occurs for every initial data. Rossi and Souplet in [13] studied the parabolic equations (1.3) in a bounded domain, subject to homogeneous Dirichlet boundary conditions. The coexistence of non-simultaneous and simultaneous blow-up was first observed in the exponent region $\{p_1>q_1+1,\ p_2>q_2+1\}$.

Liu and Lin discussed the Cauchy problem of the following parabolic equations in [10]:

$$\begin{aligned} u_t =\Delta u + b(x) u^\alpha v^p,\quad v_t =\Delta v + a(x) u^q v^\beta , \quad (x,t)\in R^N\times [0,T), \end{aligned}$$

(1.4)

where $a(x)\sim |x|^m$ and $b(x)\sim |x|^n$ as $|x|\rightarrow +\infty$ for $m,n\ge 0$. Problem (1.4) does not possess global solutions for nonnegative nontrivial initial data provided that $(1-\alpha )(1-\beta )< pq\le (pq)_c$ or $1<\alpha \le \alpha _{c}:=1+(2+n)/{N}$ or $1<\beta \le \beta _{c}:=1+(2+m)/{N}$, where

$$\begin{aligned} (pq)_c := & {} (1-\alpha )(1-\beta )+\frac{2}{N}\max \left\{ p\left( \frac{m}{2}+1\right) \right. \\&\left. +(1-\beta )\left( \frac{n}{2}+1\right) ,\, q\left( \frac{n}{2}+1\right) +(1-\alpha )\left( \frac{m}{2}+1\right) \right\} . \end{aligned}$$

If $pq>(pq)_c$, $\alpha > \alpha _{c}$, and $\beta > \beta _{c}$, both global solutions and blow-up solutions exist according to the choice of the initial data. Li and Sun in [7] studied a time-weighted parabolic system, subject to null Dirichlet boundary conditions. The critical Fujita exponents are prescribed by the weighted functions and the first eigenvalue of Laplacian operator with zero Dirichlet boundary. Some related results can also be found in the works [3, 9, 17,18,19,20] and [1, 2, 11] also.

It could be checked that the Fujita exponents in [10] are compatible with the ones in [4, 8, 16] when the exponents and the parameters in (1.4) were taken of the special values. Because of the different coupled relationship between (1.1) and (1.4), the blow-up and global criteria on solutions in [10] are not applicable to the ones of (1.1). The coupled parabolic equations in (1.1) are much more complicated than the ones in (1.2). There are also many classifications for the parameters of coefficients. It could be imagined that the singular phenomena of solutions are much more complicated than the ones in [8]. Inspired by the works [7, 8, 10], we want to determine the exponent regions where any nonnegative solutions blow up in finite time for any nontrivial nonnegative initial data. Moreover, we want to discuss the influence of the four parameters in the coefficients and show the quantitative conditions on the global existence of solution.

This paper is arranged as follows. In the next section, the main results are given with respect to different cases (Theorems 2.1–2.6). The proof of the theorems can be found in Sects. 3, 4, and 5, respectively. At the last section, we show the conclusion.

2 Main Results

If $p_{1}$, $q_{2}$, $p_{2}q_{1}\le 1$, the problem (1.1) turns into the subcritical one, where all of the nonnegative solutions are global for any nonnegative initial data. If $\max \{p_{1},q_{2},p_{2}q_{1}\}>1$, there exist blow-up solutions for large initial data (see, e.g., [6]). For convenience, we give three notations

$$\begin{aligned} (p_{1})_{c} :&= 1+\max \left\{ \frac{2+\alpha _{1}}{N},\,0\right\} ,\quad (q_{2})_{c} :=1+\max \left\{ \frac{2+\beta _{2}}{N},\,0\right\} ,\\ (p_{2}q_{1})_c :&= 1+\frac{2}{N}\max \left\{ p_{2}\left( \frac{\beta _{1}}{2}+1\right) +\frac{\alpha _{2}}{2}+1,\, q_{1}\left( \frac{\alpha _{2}}{2}+1\right) +\frac{\beta _{1}}{2}+1,\, 0\right\} . \end{aligned}$$

Let ${\tilde{c}}_1$, ${\tilde{c}}_2>0$ be two constants satisfying that

$$\begin{aligned} {\tilde{c}}_1|x|^{\alpha _{i}}\le a_{i}(x)\le {\tilde{c}}_2|x|^{\alpha _{i}},\quad {\tilde{c}}_1|x|^{\beta _{i}}\le b_{i}(x)\le {\tilde{c}}_2|x|^{\beta _{i}}\mathrm{\quad for\ large}\ |x|,\ i=1, 2. \end{aligned}$$

(2.1)

Theorem 2.1

Let $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}$, and $\beta _{2}$ be positive. There are no global solutions of (1.1) for any nonnegative nontrivial initial data provided that

$$\begin{aligned} 1< p_2 q_1 \le (p_2 q_1)_c,\quad { or}\quad 1<p_1\le (p_1)_c,\quad { or}\quad 1<q_2 \le (q_2)_c. \end{aligned}$$

The blow-up criteria in Theorem 2.1 are compatible with the ones in [4, 8, 16]. The results of Theorem 2.1 and the ones in [8, 10] show that the coefficients of the sources play an important role in distinguishing global solutions from blow-up solutions. Positive parameters of the coefficients are helpful for the existence of blow-up solutions when |x| is large enough, while negative parameters are good for global existence of solutions.

If the exponents satisfy that

$$\begin{aligned} p_{2}q_{1}>(p_{2}q_{1})_{c},\quad p_{1}>(p_{1})_{c},\quad \mathrm{and }\quad q_{2}>(q_{2})_{c}, \end{aligned}$$

(2.2)

there are blow-up solutions for large initial data (see [6]). We care about the different quantitative conditions on the global existence of solutions according to different classifications of $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}$, and $\beta _{2}$. The assumption (2.2) is necessary in the following theorems.

The first result is given for the positive parameters $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}$, and $\beta _{2}$.

Theorem 2.2

Let (2.2) be in force. Assume the initial data satisfy

$$\begin{aligned} u_{0}(y)\le \delta _{1}G(k,0,y),\quad v_{0}(y)\le \delta _{2}G(k,0,y), \end{aligned}$$

(2.3)

for any $k>0$ and some constants $\delta _{1},\delta _{2}>0$. There exist global solutions of (1.1) provided that positive parameters $\beta _{2}$, $\alpha _{1}$, $\beta _{1}$, and $\alpha _{2}$ satisfy that

$$\begin{aligned} \beta _{2}&>q_{1}N, \end{aligned}$$

(2.4)

$$\begin{aligned} \alpha _{1}&>p_{2}N, \end{aligned}$$

(2.5)

$$\begin{aligned} (q_{2}-1)E_{v}&<\min \left\{ 1,\, {N}(q_{2}-1)/{2}-1- {\beta _{2}}/{2}\right\} , \end{aligned}$$

(2.6)

$$\begin{aligned} (p_{1}-1)E_{u}&<\min \left\{ 1,\, {N}(p_{1}-1)/{2}-1- {\alpha _{1}}/{2}\right\} , \end{aligned}$$

(2.7)

where $E_{v}:=\frac{N}{2}-\frac{(\frac{\beta _{1}}{2}+1)p_{2}+\frac{\alpha _{2}}{2}+1}{p_{2}q_{1}-1}$ and $E_{u}:=\frac{N}{2}-\frac{(\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}$.

It can be checked that the conditions (2.4–2.7) in Theorem 2.2 are easy to meet. In fact, (2.2) deduces that $p_{1}>1$ and $q_{2}>1$; At first, we easily choose $\beta _{2}$ and $\alpha _{1}$, such that (2.4–2.5) hold for any $N\ge 1$. Then, one could choose suitable $\beta _{1}$ and $\alpha _{2}$, such that (2.6–2.7) hold.

The following four results of (1.1) are given, containing at least one negative parameter in the coefficients.

Theorem 2.3

Let only one of the parameters $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}$, and $\beta _{2}$ be negative and (2.2) be in force. Assume the initial data satisfy (2.3).

• (i) $\alpha _{1}<0$. If the positive parameters $\beta _{2}$, $\alpha _{2}$, and $\beta _{1}$ satisfy

  $$\begin{aligned} (p_{1}-1)E_{u}\le \frac{2+\alpha _{1}}{2}\quad { for}\ p_{1}-1>p_{2}, \end{aligned}$$

  (2.8)

  (2.4) and (2.6), then system (1.1) has global solutions.

• (ii) $\beta _{2}<0$. If the positive parameters $\alpha _{1}$, $\alpha _{2}$, and $\beta _{1}$ satisfy

  $$\begin{aligned} (q_{2}-1)E_{v}\le \frac{2+\beta _{2}}{2}\quad { for}\ q_{2}-1>q_{1}, \end{aligned}$$

  (2.9)

  (2.5) and (2.7), then system (1.1) has global solutions.

• (iii) $\alpha _{2}<0$ or $\beta _{1}<0$. If the positive parameters $\beta _{2}$, $\alpha _{1}$, and $\beta _{1}$, or $\beta _{2}$, $\alpha _{1}$, and $\alpha _{2}$, respectively, satisfy (2.4–2.7) for $q_{2}-1>q_{1}$, then system (1.1) has global solutions.

Theorem 2.4

Let two of the parameters $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}$, and $\beta _{2}$ be negative and (2.2) be in force. Assume the initial data satisfy (2.3).

• (i) $\beta _{1}$, $\beta _{2}<0$ or $\alpha _{2}$, $\beta _{2}<0$. If positive parameters $\alpha _{1}$ and $\alpha _{2}$, or $\alpha _{1}$ and $\beta _{1}$, respectively, satisfy (2.5), (2.7) and (2.9) for $q_{2}-1>q_{1}$, then system (1.1) has global solutions.

• (ii) $\alpha _{1}$, $\alpha _{2}<0$ or $\alpha _{1}$, $\beta _{1}<0$. If positive parameters $\beta _{2}$ and $\beta _{1}$, or $\beta _{2}$ and $\alpha _{2}$, respectively, satisfy (2.4), (2.6) and (2.8) for $p_{1}-1>p_{2}$, then system (1.1) has global solutions.

• (iii) $\alpha _{1}$, $\beta _{2}<0$ or $\alpha _{2}$, $\beta _{1}<0$. If (2.4–2.7) are true, then system (1.1) has global solutions.

Theorem 2.5

Let three of the parameters $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}$, and $\beta _{2}$ be negative and (2.2) be in force. Assume the initial data satisfy (2.3).

• (i) $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}<0$. If (2.4), (2.6) and (2.8) are true and $p_{1}-1>p_{2}$, then system (1.1) has global solutions.

• (ii) $\alpha _{2}$, $\beta _{1}$, $\beta _{2}<0$. If (2.5), (2.7) and (2.9) are true and $q_{2}-1>q_{1}$, then system (1.1) has global solutions.

• (iii) $\alpha _{1}$, $\alpha _{2}$, $\beta _{2}<0$ or $\alpha _{1}$, $\beta _{1}$, $\beta _{2}<0$. If the parameter $\beta _{1}$ or $\alpha _{2}$, respectively, satisfies (2.8–2.9) and $q_{2}-1>q_{1}$ and $p_{1}-1>p_{2}$, then system (1.1) has global solutions.

Theorem 2.6

Let all of the parameters $\alpha _{1}$, $\alpha _{2}$, $\beta _{1}$ and $\beta _{2}$ be negative and (2.2) be in force. Assume the initial data satisfy (2.3). If (2.8–2.9) are true and $q_{2}-1>q_{1}$ and $p_{1}-1>p_{2}$, then system (1.1) has global solutions.

3 Proof of Theorem 2.1

Before the proof of Theorem 2.1, we give five lemmas. Denote $u(t):=u(x,t)$ and $v(t):=v(x,t)$ for simplicity. Let $G(t,x,y)=\frac{1}{(4\pi t)^{N/2}}e^{-\frac{|x-y|^2}{4t}}$ be the fundamental solution of the heat equation in $R^N$ and

$$\begin{aligned} (S(t)w_0)(x):=\displaystyle \int _{R^N}G(t,x,y)w_0(y)\mathrm{d}y. \end{aligned}$$

The first three lemmas are just [8, Lemmas 4.1–4.3], respectively. For completeness, we introduce the three results here.

Lemma 3.1

For any $m\ge 0$ and $t>0$, the function $H(x)=\displaystyle \int _{R^N}G(t,x,y)(1+|y|)^m\mathrm{d}y$ attains its minimum at $x=0$. $\square$

Lemma 3.2

For any $\beta >1$, $m\ge 0$, and $k=1,2,...,$ there is a constant $C_{1}>0$ independent of k, such that

$$\begin{aligned} \left( \int _{R^{N}}G(t,0,y)(1+|y|)^{\frac{m}{\beta ^{k}}}\mathrm{d}y\right) ^{\beta ^{k}}\ge C_{1}t^{\frac{m}{2}} \quad { for}\ t\ge 0. \end{aligned}$$

Moreover, the same result holds for $m<0$ and $t\ge 1$. $\square$

Lemma 3.3

For any $k>0$ and $m\ge 0$, there is a constant $C_{2}>0$ independent of k, such that

$$\begin{aligned} \int _{0}^{1}r^{k}[r(1-r)]^{\frac{m}{2}}\mathrm{d}r\ge \frac{C_{2}}{(k+\frac{m}{2}+1)^{\frac{m}{2}+1}}.\quad \end{aligned}$$

$\square$

The following three estimates are inspired by the [8, Lemmas 4.4, 4.7, and 4.8], respectively. Since the coupled relationship in (1.4) is much complicated than the main system in [8], similarly to them, we show the proof of the following three lemmas for completeness.

Lemma 3.4

Let (u(t), v(t)) be a global solution of (1.1) with $p_2 \ge 1$, $q_1 \ge 1$, and $p_{2}q_{1}>1$. Then, there exists a positive constant $C=C(p_{2},q_{1},\alpha _{2},\beta _{1})$, such that for any $t>0$

$$\begin{aligned} t^{\frac{(\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}}\Vert S(t)u_{0}\Vert _{\infty }\le C,\qquad t^{\frac{(\frac{\beta _{1}}{2}+1)p_{2}+\frac{\alpha _{2}}{2}+1}{p_{2}q_{1}-1}}\Vert S(t)v_{0}\Vert _{\infty }\le C. \end{aligned}$$

Proof

Considering that

$$\begin{aligned} \left\{ \begin{array}{lll} u(t) =S(t)u_{0}+\displaystyle \int _{0}^{t}S(t-s)a_{1}u^{p_{1}}(s)\mathrm{d}s + \displaystyle \int _{0}^{t}S(t-s)b_{1}v^{q_{1}}(s)\mathrm{d}s ,\\ v(t) =S(t)v_{0}+\displaystyle \int _{0}^{t}S(t-s)a_{2}u^{p_{2}}(s)\mathrm{d}s + \displaystyle \int _{0}^{t}S(t-s)b_{2}v^{q_{2}}(s)\mathrm{d}s, \end{array}\right. \end{aligned}$$

(3.1)

and by (3.1), we have

$$\begin{aligned} u(t)&\ge S(t)u_{0} ,\quad v(t) \ge S(t)v_{0} , \end{aligned}$$

(3.2)

$$\begin{aligned} u(t)&\ge \displaystyle \int _{0}^{t}S(t-s)b_{1}v^{q_{1}}(s)\mathrm{d}s ,\quad v(t) \ge \displaystyle \int _{0}^{t}S(t-s)a_{2}u^{p_{2}}(s)\mathrm{d}s . \end{aligned}$$

(3.3)

Using (3.2) and (3.3), we obtain $u(x,t)\ge b_{1}(x)t(S(t)v_{0}(x))^{q_{1}}$ and

$$\begin{aligned} v(t)&\ge \int _{0}^{t}S(t-s)a_{2}u^{p_{2}}(s)\mathrm{d}s\\&\ge \int _{0}^{t}s^{p_{2}}\left( \int _{R^{N}}\int _{R^{N}}G\left( t-s,x,y\right) a_{2}^{\frac{1}{p_{2}q_{1}}}(y)G(s,y,z) b_{1}^{\frac{1}{q_{1}}}(y)v_{0}(z)\mathrm{d}z\mathrm{d}y\right) ^ {p_{2}q_{1}}\mathrm{d}s\\&\ge \int _{0}^{t}s^{p_{2}}\left\{ \int _{R^{N}}\int _{R^{N}}[4\pi (t-s)]^{-\frac{{N}}{2}}(4\pi s)^{-\frac{{N}}{2}}e^{-\frac{|y-x|^{2}}{4(t-s)}-\frac{|z-y|^{2}}{4s}}a_{2}^{\frac{1}{p_{2}q_{1}}}\right. \\&\left. \quad (y)b_{1}^{\frac{1}{q_{1}}}(y)v_{0}(z)\mathrm{d}z\mathrm{d}y\right\} ^ {p_{2}q_{1}}\mathrm{d}s. \end{aligned}$$

It can be checked that

$$\begin{aligned} -\frac{|y-x|^{2}}{4(t-s)}-\frac{|z-y|^{2}}{4s}=-\frac{t}{4s(t-s)}\left| y-\frac{sx+(t-s)z}{t}\right| ^{2}-\frac{|x-z|^{2}}{4t}. \end{aligned}$$

Then, by (2.1) and Lemma 3.2, one can obtain that $a_{2}(x)\ge c_{1}(1+|x|)^{\alpha _{2}}$ and $b_{1}(x)\ge c_{1}(1+|x|)^{\beta _{1}}$ for large |x|, and hence

$$\begin{aligned} v(t)&\ge \int _{0}^{t}s^{p_{2}}\left\{ \int _{R^{N}}\int _{R^{N}} [4\pi (t-s)]^{-\frac{{N}}{2}}(4\pi s)^{-\frac{{N}}{2}}e^{-\frac{t|y|^{2}}{4s(t-s)}-\frac{|x-z|^{2}}{4t}}\right. \\&\left. \quad c_{1}(1+|y|)^{\frac{\alpha _{2}+p_{2}\beta _{1}}{p_{2}q_{1}}}v_{0}(z)\mathrm{d}z\mathrm{d}y\right\} ^{p_{2}q_{1}}\mathrm{d}s\\&\ge \int _{0}^{t}s^{p_{2}}\left[ \int _{R^{N}}\int _{R^{N}}G\left( \frac{(t-s)s}{t},0,y\right) c_{1} (1+|y|)^{\frac{\alpha _{2}+p_{2}\beta _{1}}{p_{2}q_{1}}}\mathrm{d}y {G(t,x,z)}v_{0}(z)\mathrm{d}z\right] ^{p_{2}q_{1}}\mathrm{d}s\\&\ge \int _{0}^{t}s^{p_{2}}c_{1}C_{1}\left[ \frac{(t-s)s}{t}\right] ^{\frac{\alpha _{2}+p_{2}\beta _{1}}{2}}\mathrm{d}s(S(t)v_{0})^{p_{2}q_{1}}, \end{aligned}$$

where $C_{1}$ is given in Lemma 3.2. Using Lemma 3.3, we let $s=rt$ and obtain

$$\begin{aligned} \int _{0}^{t}s^{p_{2}}\left[ \frac{s(t-s)}{t}\right] ^{\frac{\alpha _{2}+p_{2}\beta _{1}}{2}}\mathrm{d}s {\ge }\frac{C_{2}t^{\frac{\alpha _{2}+p_{2}\beta _{1}}{2}+p_{2}+1}}{\left( \frac{\alpha _{2}+p_{2}\beta _{1}}{2}+p_{2}+1\right) ^{\frac{\alpha _{2}+p_{2}\beta _{1}}{2}+1}}. \end{aligned}$$

In fact, in Lemma 3.3, $\displaystyle \int _0^1r^{k}[r(1-r)]^{\frac{m}{2}}\mathrm{d}r$ is bounded if $k\le 0$ or $m<0$. Here, we solve the case for $p_{2}>0$. Then, we have

$$\begin{aligned}&v(t)\ge \frac{c_{1}C_{1}C_{2}t^{\left( \frac{\beta _{1}}{2}+1\right) p_{2}+\frac{\alpha _{2}}{2}+1}(S(t)v_{0})^{p_{2}q_{1}}}{\left[ \left( \frac{\beta _{1}}{2}+1\right) p_{2} +\frac{\alpha _{2}}{2}+1\right] ^{\frac{\alpha _{2}+p_{2}\beta _{1}}{2}+1}},\nonumber \\&\quad \mathrm{similarly},\ u(t)\ge \frac{c_{1}C_{1}C_{2}t^{\left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1}(S(t)u_{0})^{p_{2}q_{1}}}{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1} +\frac{\beta _{1}}{2}+1\right] ^{\frac{\alpha _{2}+p_{2}\beta _{1}}{2}+1}}, \end{aligned}$$

(3.4)

where $C_1$ and $c_1$ are two positive constants.

Substituting (3.4) into (3.1), we have

$$\begin{aligned} v(t)&\ge \int _{0}^{t}S(t-s)a_{2}u^{p_{2}}(s)\mathrm{d}s\\&\ge \int _{0}^{t}S(t-s)a_{2}\left\{ \frac{c_{1}C_{1}C_{2}s^{\left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1}(S(s)u_{0})^{p_{2}q_{1}}}{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] ^{\frac{\alpha _{2}+p_{2}\beta _{1}}{2}+1}}\right\} ^{p_{2}}\mathrm{d}s\\&=\frac{(c_{1}C_{1}C_{2})^{p_{2}}}{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1} +\frac{\beta _{1}}{2}+1\right] ^{\left( \frac{\alpha _{2}+p_{2}\beta _{1}}{2}+1\right) p_{2}}} \int _{0}^{t}S(t-s)a_{2}s^{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] p_{2}}\\&\quad (S(s)u_{0})^{p_{2}^{2}q_{1}}\mathrm{d}s. \end{aligned}$$

In fact, by Lemma 3.2, we have

$$\begin{aligned}&\ \int _{0}^{t}S(t-s)a_{2}(S(s)u_{0})^{p_{2}^{2}q_{1}}\mathrm{d}s\\&\quad \ge \int _{0}^{t}\left( \int _{R^{N}}\int _{R^{N}}G\left( t-s,x,y\right) a_{2}^{\frac{1}{p_{2}^{2}q_{1}}}(y) G(s,y,z)u_{0}(z)\mathrm{d}z\mathrm{d}y\right) ^{p_{2}^{2}q_{1}}\mathrm{d}s\\&\quad \ge \int _{0}^{t}\left\{ \int _{R^{N}}\int _{R^{N}}[4\pi (t{-}s)]^{{-}\frac{{N}}{2}}(4\pi s)^{{-}\frac{{N}}{2}}e^{{-}\frac{t|y|^{2}}{4s(t{-}s)}{-}\frac{|x-z|^{2}}{4t}}a_{2}^{\frac{1}{p_{2}^{2}q_{1}}}(y)u_{0}(z)\mathrm{d}z\mathrm{d}y\right\} ^{p_{2}^{2}q_{1}}\mathrm{d}s\\&\quad \ge \int _{0}^{t}\left[ \int _{R^{N}}G\left( \frac{(t-s)s}{t},0,y\right) (1+|y|)^{\frac{\alpha _{2}}{p_{2}^{2}q_{1}}}S(t)u_{0}\right] ^{p_{2}^{2}q_{1}}\mathrm{d}s\\&\quad \ge \int _{0}^{t}C_{1}\left[ \frac{(t-s)s}{t}\right] ^{\frac{\alpha _{2}}{2}}(S(t)u_{0})^{p_{2}^{2}q_{1}}\mathrm{d}s. \end{aligned}$$

By Lemma 3.3, we have

$$\begin{aligned}&\int _{0}^{t}S(t-s)a_{2}s^{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] p_{2}}(S(s)u_{0})^{p_{2}^{2}q_{1}}\mathrm{d}s\\&\quad \ge \int _{0}^{t}S^{p_{2}^{2}q_{1}}(t-s)a_{2}s^{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] p_{2}}(S(s)u_{0})^{p_{2}^{2}q_{1}}\mathrm{d}s\\&\quad \ge \int _{0}^{t}C_{1}\left[ \frac{(t-s)s}{t}\right] ^{\frac{\alpha _{2}}{2}}(S(s)u_{0})^{p_{2}^{2}q_{1}}s^{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] p_{2}}\mathrm{d}s\\&\quad \ge C_{1}t^{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] p_{2}+\frac{\alpha _{2}}{2}+1} \int _{0}^{1}r^{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] p_{2}} [r(1-r)]^{\frac{\alpha _{2}}{2}}\mathrm{d}r(S(t)u_{0})^{p_{2}^{2}q_{1}}\\&\quad \ge \frac{C_{1}C_{2}t^{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] p_{2}+\frac{\alpha _{2}}{2}+1}(S(t)u_{0})^{p_{2}^{2}q_{1}}}{\left\{ \left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] p_{2}+\frac{\alpha _{2}}{2}+1\right\} ^{\frac{\alpha _{2}}{2}+1}}. \end{aligned}$$

For convenience, we denote $E:= (\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1$, $F:= (\frac{\beta _{1}}{2}+1)p_{2}+\frac{\alpha _{2}}{2}+1.$ Therefore, we have

$$\begin{aligned}&v(t)\ge \frac{(c_{1}C_{1}C_{2})^{p_{2}+1}t^{Ep_{2}+\frac{\alpha _{2}}{2}+1}(S(t)u_{0})^{p_{2}^{2}q_{1}}}{E^{\left( \frac{\alpha _{2}+p_{2}\beta _{1}}{2}+1\right) p_{2}} (Ep_{2}+\frac{\alpha _{2}}{2}+1)^{\frac{\alpha _{2}}{2}+1}},\\&\quad \mathrm{similarly},\ u(t)\ge \frac{(c_{1}C_{1}C_{2})^{q_{1}+1}t^{Fq_{1}+\frac{\beta _{1}}{2}+1}(S(t)v_{0})^{p_{2}q_{1}^{2}}}{F^{\left( \frac{\beta _{1}+q_{1}\alpha _{2}}{2}+1\right) q_{1}} (Fq_{1}+\frac{\beta _{1}}{2}+1)^{\frac{\beta _{1}}{2}+1}}. \end{aligned}$$

Using Lemmas 3.2 and 3.3, we obtain

$$\begin{aligned}&\int _{0}^{t}S(t-s)a_{2}s^{(Fq_{1}+\frac{\beta _{1}}{2}+1)p_{2}}\mathrm{d}s\\&\quad \ge c_{1}C_{1}\int _{0}^{t}s^{(Fq_{1}+\frac{\beta _{1}}{2}+1)p_{2}}\left[ \frac{s(t-s)}{t}\right] ^{\frac{\alpha _{2}}{2}}\mathrm{d}s\\&\quad =c_{1}C_{1}t^{(Fq_{1}+\frac{\beta _{1}}{2}+1)p_{2}+\frac{{\alpha _{2}}}{2}+1}\int _{0}^{1}r^{(Fq_{1}+\frac{\beta _{1}}{2}+1)p_{2}}[r(1-r)]^{\frac{\alpha _{2}}{2}}\mathrm{d}r\\&\quad \ge \frac{c_{1}C_{1}C_{2}t^{(Fq_{1}+\frac{\beta _{1}}{2}+1)p_{2}+\frac{{\alpha _{2}}}{2}+1}}{[(Fq_{1}+\frac{\beta _{1}}{2}+1)p_{2} +\frac{{\alpha _{2}}}{2}+1] ^{\frac{\alpha _{2}}{2}+1}} \ge \frac{c_{1}C_{1}C_{2}t^{Fp_{2}q_{1}+F}}{(Fp_{2}q_{1}+F)^{\frac{{\alpha _{2}}}{2}+1}}. \end{aligned}$$

Therefore, there is

$$\begin{aligned} v(t)&\ge \int _{0}^{t}S(t-s)a_{2}\left[ \frac{(c_{1}C_{1}C_{2})^{q_{1}+1}s^{Fq_{1}+\frac{\beta _{1}}{2}+1} (S(s)v_{0})^{p_{2}q_{1}^{2}}}{F^{\left( \frac{\beta _{1}+q_{1}\alpha _{2}}{2}+1\right) q_{1}} \left( Fq_{1}+\frac{\beta _{1}}{2}+1\right) ^{\frac{{\beta _{1}}}{2}+1}}\right] ^{p_{2}}\mathrm{d}s\\&=\frac{(c_{1}C_{1}C_{2})^{(q_{1}+1)p_{2}}}{F^{\left( \frac{\beta _{1}+q_{1}\alpha _{2}}{2}+1\right) p_{2}q_{1}} \left( Fq_{1}+\frac{\beta _{1}}{2}+1\right) ^{\left( \frac{{\beta _{1}}}{2}+1\right) p_{2}}}\\&\quad \times \int _{0}^{t}S(t-s)a_{2}s^{(Fq_{1}+\frac{\beta _{1}}{2}+1)p_{2}}(S(s)v_{0})^{(p_{2}q_{1})^{2}}\mathrm{d}s\\&\ge \frac{(c_{1}C_{1}C_{2})^{(p_{2}q_{1}+1)}t^{Fp_{2}q_{1}+F}(S(t)v_{0})^{(p_{2}q_{1})^{2}}}{F^{\left( \frac{\beta _{1}+q_{1}\alpha _{2}}{2}+1\right) p_{2}q_{1}} \left( Fq_{1}+\frac{\beta _{1}}{2}+1\right) ^{\left( \frac{{\beta _{1}}}{2}+1\right) p_{2}}(Fp_{2}q_{1}+F)^{\frac{\alpha _{2}}{2}+1}}. \end{aligned}$$

Using the induction process, we have

$$\begin{aligned} v(t)\ge (c_{1}C_{1}C_{2})^{\frac{(p_{2}q_{1})^{k}-1}{p_{2}q_{1}}}A_{k}B_{k} t^{\frac{(p_{2}q_{1})^{k}-1}{p_{2}q_{1}}}(S(t)v_{0})^{(p_{2}q_{1})^{k}}, \end{aligned}$$

(3.5)

where constants

$$\begin{aligned} A_{k}:&= F^{-\left( \frac{\beta _{1}+q_{1}\alpha _{2}}{2}+1\right) (p_{2}q_{1})^{k-1}}\prod _{i=1}^{k-1}\left[ Fq_{1}\frac{(p_{2}q_{1})^{i}-1}{p_{2}q_{1}-1}+\frac{\beta _{1}}{2}+1\right] ^ {-\left( \frac{\beta _{1}}{2}+1\right) (p_{2}q_{1})^{k-i}p_{2}},\\ B_{k}:&= (Fp_{2}q_{1}+F)^{-(\frac{\alpha _{2}}{2}+1)(p_{2}q_{1})^{k-2}}\prod _{i=1}^{k-1}\left[ \frac{(p_{2}q_{1})^{i}-1}{p_{2}q_{1}-1}\right] ^ {-\left( \frac{\alpha _{2}}{2}+1\right) (p_{2}q_{1})^{k-i}}. \end{aligned}$$

Then by (3.5), we have

$$\begin{aligned}&t^{\left[ \left( \frac{\beta _{1}}{2}+1\right) p_{2}+\frac{\alpha _{2}}{2}+1\right] \frac{(p_{2}q_{1})^{k}-1}{p_{2}q_{1}-1}\frac{1}{(p_{2}q_{1})^{k}}}(S(t)v_{0})\\&\quad \le (c_{1}C_{1}C_{2})^{-\frac{(p_{2}q_{1})^{k}-1}{p_{2}q_{1}-1}\frac{1}{(p_{2}q_{1})^{k}}}A_{k}^{-\frac{1}{(p_{2}q_{1})^{k}}}B_{k}^{-\frac{1}{(p_{2}q_{1})^{k}}} \Vert v(t)\Vert _{\infty }^{\frac{1}{(p_{2}q_{1})^{k}}}. \end{aligned}$$

It can be found out that $(A_kB_k)^{-\frac{1}{(p_{2}q_{1})^{k}}}$ has a finite limit as $k\rightarrow \infty$. Thus, for some constant $C>0$ and any $t\in [0,T)$

$$\begin{aligned} t^{\frac{\left( \frac{\beta _{1}}{2}+1\right) p_{2}+\frac{\alpha _{2}}{2}+1}{p_{2}q_{1}-1}}\Vert S(t)v_{0}\Vert _{\infty }\le C<+\infty . \end{aligned}$$

Then, we treat $(u(t+\tau ),v(t+\tau ))$ for t, $\tau \ge 0$ as the solution of (1.1) with the initial value $(u(\tau ),v(\tau ))$. Replace $(u_0,v_0)$ by $(u(\tau ),v(\tau ))$ and those estimates hold also. Setting $t=\tau$, one can obtain the conclusion. The proof for u(t) is similar. $\square$

Lemma 3.5

Assume the global solution (u, v) of (1.1) satisfies that

$$\begin{aligned} u(x,t)\ge c_{0}t^{l_{1}}e^{-\frac{|x|^{2}}{t}},\quad v(x,t)\ge c_{0}t^{l_{2}}e^{-\frac{|x|^{2}}{t}},\quad t\ge t_{0}>0,\ x\in R^{N}, \end{aligned}$$

(3.6)

where $t_{0}$, $c_0>0$, and $l_1\in [- {N}/{2},\infty ),$ $l_2\in [- {N}/{2},\infty ).$ Then, for $a_{i}(x)\sim |x|^{\alpha _{i}}$, $b_{i}(x)\sim |x|^{\beta _{i}}$, $i=1,2$, as $|x|\rightarrow \infty$, there exist positive constants c, $t_1$, such that for $t\ge t_1$

$$\begin{aligned}&\int _{0}^{t}S(t-s)b_{1}v^{q_{1}}(s)\mathrm{d}s\ge \left\{ \begin{array}{llll} ct^{1+q_{1}l_{2}+\frac{\beta _{1}}{2}}e^{-\frac{|x|^{2}}{t}}, &{}{ if }1+q_{1}l_{2}+\frac{\beta _{1}}{2}>-\frac{N}{2},\\ ct^{-\frac{N}{2}}\log (1+t)e^{-\frac{|x|^{2}}{t}}, &{}{ if }1+q_{1}l_{2}+\frac{\beta _{1}}{2}=-\frac{N}{2}; \end{array} \right. \end{aligned}$$

(3.7)

$$\begin{aligned}&\int _{0}^{t}S(t-s)a_{2}u^{p_{2}}(s)\mathrm{d}s\ge \left\{ \begin{array}{llll} ct^{1+p_{2}l_{1}+\frac{\alpha _{2}}{2}}e^{-\frac{|x|^{2}}{t}}, &{}{ if\ }1+p_{2}l_{1}+\frac{\alpha _{2}}{2}>-\frac{N}{2},\\ ct^{-\frac{N}{2}}\log (1+t)e^{-\frac{|x|^{2}}{t}}, &{}{ if\ }1+p_{2}l_{1}+\frac{\alpha _{2}}{2}=-\frac{N}{2}. \end{array} \right. \end{aligned}$$

(3.8)

Proof

It follows from (3.3) and (3.6) that:

$$\begin{aligned} u(t)\ge c_{0}^{q_{1}}\int _{t_{0}}^{t}\int _{R^{N}}[4\pi (t-s)]^{-\frac{{N}}{2}}e^{-\frac{|y-x|^{2}}{4(t-s)}}b_{1}(y)s^{q_{1}l_{2}}e^{-\frac{q_{1}|y|^{2}}{s}}\mathrm{d}y\mathrm{d}s, \quad t\ge t_{0}. \end{aligned}$$

Here

$$\begin{aligned} e^{-\frac{|y-x|^2}{4(t-s)}-\frac{q_{1}|y|^2}{s}}=e^{-\frac{|y-r(s,t)x|^2}{4r(s,t)(t-s)}}e^{-\frac{q_{1}r(s,t)|x|^2}{s}}, \end{aligned}$$

and for $s\in [0, \frac{t}{2}]$

$$\begin{aligned} -\frac{q_{1}r(s,t)}{s}|x|^{2}-\frac{r(s,t)}{2(t-s)}|x|^{2}>-\frac{|x|^{2}}{t}, \end{aligned}$$

where $r(s,t)=\frac{s}{s+4q_{1}(t-s)}$. Then

$$\begin{aligned} u(t)&=c_{0}^{q_{1}}\int _{t_{0}}^{t}\int _{R^{N}}G(r(s,t)(t-s),r(s,t)x,y)e^{-\frac{q_{1}r(s,t)|x|^2}{s}}b_{1}(y) s^{q_{1}l_{2}}r^{\frac{N}{2}}(s,t)\mathrm{d}y\mathrm{d}s\\&\ge c_{0}^{q_{1}}\int _{t_{0}}^{t}\int _{R^{N}}2^{-\frac{N}{2}} G\left( \frac{r(s,t)(t-s)}{2},0,y\right) e^{-\frac{q_{1}r(s,t)}{s}|x|^{2}-\frac{r(s,t)}{2(t-s)}|x|^{2}}\\&\quad b_{1}(y)s^{q_{1}l_{2}}r^{\frac{N}{2}}(s,t)\mathrm{d}y\mathrm{d}s\\&\ge c_{0}^{q_{1}}2^{-\frac{N}{2}}\int _{t_{0}}^{t}\int _{R^{N}}G\left( \frac{r(s,t)(t-s)}{2},0,y\right) e^{-\frac{|x|^{2}}{t}}b_{1}(y) s^{q_{1}l_{2}}r^{\frac{N}{2}}(s,t)\mathrm{d}y\mathrm{d}s. \end{aligned}$$

For $t\ge 2t_{0}$, we obtain

$$\begin{aligned}&\int _{0}^{t}S(t-s)b_{1}v^{q_{1}}(s)\mathrm{d}s\\&\quad \ge c_{0}^{q_{1}}2^{-\frac{N}{2}}e^{-\frac{|x|^{2}}{t}}\int _{t_{0}}^{\frac{t}{2}}\int _{R^{N}}G\left( \frac{r(s,t)(t-s)}{2},0,y\right) b_{1}(y) s^{q_{1}l_{2}}r^{\frac{N}{2}}(s,t)\mathrm{d}y\mathrm{d}s. \end{aligned}$$

A simple calculation reveals, for sufficiently large $k>0$, if $t\ge 2k$ and $k\le s\le t-k$, then $r(s,t)(t-s)/2\ge 1$. Without loss of generality, we take $k\ge t_0$ and $r_{1}= {s}/{t}$. By [8, Lemma 2.2], we have

$$\begin{aligned} u(t)\ge & {} c_{1}c_{0}^{q_{1}}2^{-\frac{N+\beta _{1}}{2}}e^{-\frac{|x|^{2}}{t}}\int _{k}^{\frac{t}{2}}(t-s)^{\frac{\beta _{1}}{2}}s^{q_{1}l_{2}} r^{\frac{N+\beta _{1}}{2}}\mathrm{d}s\nonumber \\\ge & {} c_{1}c_{0}^{q_{1}}2^{-\frac{N+\beta _{1}}{2}}e^{-\frac{|x|^{2}}{t}}\int _{\frac{k}{t}}^{\frac{1}{2}} (t-tr_{1})^{\frac{\beta _{1}}{2}}(tr_{1})^{q_{1}l_{2}} r^{\frac{N+\beta _{1}}{2}}t\mathrm{d}r_{1}\nonumber \\= & {} c_{1}c_{0}^{q_{1}}2^{-\frac{N+\beta _{1}}{2}}e^{-\frac{|x|^{2}}{t}}\int _{\frac{k}{t}}^{\frac{1}{2}}(t-tr_{1})^{\frac{\beta _{1}}{2}}(tr_{1})^{q_{1}l_{2}} \left[ \frac{r_{1}t}{r_{1}t+4q_{1}(t-tr_{1})}\right] ^{\frac{N+\beta _{1}}{2}}t\mathrm{d}r_{1}\nonumber \\= & {} c_{1}c_{0}^{q_{1}}2^{-\frac{N+\beta _{1}}{2}}e^{-\frac{|x|^{2}}{t}}t^{1+q_{1}l_{2}+\frac{\beta _{1}}{2}}\int _{\frac{k}{t}}^{\frac{1}{2}} \frac{r_{1}^{q_{1}l_{2}+\frac{\beta _{1}+N}{2}}(1-r_{1})^{\frac{\beta _{1}}{2}}}{[r_{1}+4q_{1}(1-r_{1})]^{\frac{N+\beta _{1}}{2}}}\mathrm{d}r_{1},\quad t\ge 2k.\nonumber \\ \end{aligned}$$

(3.9)

By (3.9), one could find positive constants $c_{2}$ and $t_1$, such that

$$\begin{aligned} \displaystyle \int _{0}^{t}S(t-s)b_{1}v^{q_{1}}(s)\mathrm{d}s\ge \ c_2 t^{1+q_{1}l_{2}+\frac{\beta _{1}}{2}}e^{-\frac{|x|^{2}}{t}},\quad t\ge t_1 . \end{aligned}$$

This shows the first inequality in (3.7). If $1+q_{1}l_{2}+\frac{\beta _{1}}{2}=-\frac{N}{2}$, then $q_{1}l_{2}+\frac{\beta _{1}}{2}+\frac{N}{2}=-1$, and the integral of (3.7) is on $\log t$ for large t. Thus

$$\begin{aligned} \displaystyle \int _{0}^{t}S(t-s)b_{1}v^{q_{1}}(s)\mathrm{d}s\ge \ ct^{-\frac{N}{2}}\log (1+t)e^{-\frac{|x|^{2}}{t}},\quad t\ge t_1. \end{aligned}$$

This finishes the proof of (3.7). Similarly, we have (3.8). $\square$

Lemma 3.6

Assume that $p_{2}q_{1}>1$ and (u(t), v(t)) be a global solution of (1.1).

• (i) If $\frac{(\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}>\frac{N}{2}$, then there exist positive constants $r_1$, $t_{r_1}$, and C, such that $u(x,t)\ge Ct^{r_1}e^{-\frac{|x|^{2}}{t}}$ for $t\ge t_{r_1}$, $x\in R^N$; If $\frac{(\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}=\frac{N}{2}$, then there exist positive constants $t_1$, C, such that $u(x,t)\ge Ct^{-\frac{N}{2}}\log (1+t)e^{-\frac{|x|^{2}}{t}}$ for $t\ge t_1$, $x\in R^N$.

• (ii) If $\frac{(\frac{\beta _{1}}{2}+1)p_{2}+\frac{\alpha _{2}}{2}+1}{p_{2}q_{1}-1}>\frac{N}{2}$, then there exist positive constants $r_2$, $t_{r_2}$, and C, such that $v(x,t)\ge Ct^{r_2}e^{-\frac{|x|^{2}}{t}}$ for $t\ge t_{r_2}$, $x\in R^N$; If $\frac{(\frac{\beta _{1}}{2}+1)p_{2}+\frac{\alpha _{2}}{2}+1}{p_{2}q_{1}-1}=\frac{N}{2}$, then there exist positive constants $t_2$, C, such that $v(x,t)\ge Ct^{-\frac{N}{2}}\log (1+t)e^{-\frac{|x|^{2}}{t}}$ for $t\ge t_2$, $x\in R^N$.

Proof

By (3.1), we have

$$\begin{aligned} u(t)&\ge S(t)u_{0}+\int _{0}^{t}S(t-s)b_{1}v^{q_{1}}(s)\mathrm{d}s, \end{aligned}$$

(3.10)

$$\begin{aligned} v(t)&\ge S(t)v_{0}+\int _{0}^{t}S(t-s)a_{2}u^{p_{2}}(s)\mathrm{d}s. \end{aligned}$$

(3.11)

Using (3.11) and (3.10), we obtain

$$\begin{aligned} u(t)\ge S(t)u_{0}+\int _{0}^{t}S(t-s)b_{1}\left[ \int _{0}^{s_{1}}S(s_{1}-s_{2})a_{2}u^{p_{2}}(s_{2})\mathrm{d}s_{2}\right] ^{q_{1}}\mathrm{d}s_{1}. \end{aligned}$$

We have $u(t)\ge u_{0}(x,t)+u_{1}(x,t)$. Define

$$\begin{aligned} u_{1}(t):= \int _{0}^{t}S(t-s_{1})b_{1}\left[ \int _{0}^{s_{1}}S(s_{1}-s_{2})a_{2}u^{p_{2}}(s_{2})\mathrm{d}s_{2}\right] ^{q_{1}}\mathrm{d}s_{1}. \end{aligned}$$

(3.12)

Let $l_{1}=l_{2}=-\frac{N}{2}$. By [8, Lemma 2.2], substituting (3.8) into (3.12), we obtain

$$\begin{aligned} u_{1}(t)&\ge \int _{0}^{t}\int _{R^{N}}G(t-s_{1},x,y)b_{1}(y)(cs_{2}^{1-\frac{N}{2}p_{2}+\frac{\alpha _{2}}{2}}e^{-\frac{|x|^{2}}{s_{2}}})^{q_{1}}\mathrm{d}y\mathrm{d}s_{1}\\&\ge C_{1}t^{\left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1-\frac{N}{2}p_{2}q_{1}}e^{-\frac{|x|^{2}}{t}},\\&\quad \mathrm{similarly}, v_{1}(t)= C_{1}t^{\left( \frac{\beta _{1}}{2}+1\right) p_{2}+\frac{\alpha _{2}}{2}+1-\frac{N}{2}p_{2}q_{1}}e^{-\frac{|x|^{2}}{t}}, \end{aligned}$$

where constant $C_{1}>0$. Let

$$\begin{aligned} u_{2}(t):= \int _{0}^{t}S(t-s_{1})b_{1}\left[ \int _{0}^{s_{1}}S(s_{1}-s_{2})a_{2}u_{1}^{p_{2}}(s_{2})\mathrm{d}s_{2}\right] ^{q_{1}}\mathrm{d}s_{1}. \end{aligned}$$

We have

$$\begin{aligned} u_{2}(t)&\ge \int _{0}^{t}\int _{R^{N}}G(t-s_{1},x,y)b_{1}(y)\left\{ cs_{2}^{\left[ \left( 1+\frac{\alpha _{2}}{2}\right) q_{1}+1+\frac{\beta _{1}}{2}-\frac{N}{2}p_{2}q_{1}\right] p_{2}+1 +\frac{\alpha _{2}}{2}}e^{-\frac{|x|^{2}}{s_{2}}}\right\} ^{q_{1}}\mathrm{d}y\mathrm{d}s_{1}\\&\ge C_{2}t^{\left\{ \left[ \left( 1+\frac{\alpha _{2}}{2}\right) q_{1}+1+\frac{\beta _{1}}{2}-\frac{N}{2}p_{2}q_{1}\right] p_{2}+1 +\frac{\alpha _{2}}{2}\right\} q_{1}+1+\frac{\beta _{1}}{2}}e^{-\frac{|x|^{2}}{t}}\\&= C_{2}t^{\left[ \left( 1+\frac{\alpha _{2}}{2}\right) q_{1}+1+\frac{\beta _{1}}{2}\right] (p_{2}q_{1}+1)-\frac{N}{2}(p_{2}q_{1})^{2}}e^{-\frac{|x|^{2}}{t}}. \end{aligned}$$

By induction, we have

$$\begin{aligned} u_{k}(t)&= \int _{0}^{t}S(t-s_{1})b_{1}\left( \int _{0}^{s_{1}}S(s_{1}-s_{2})a_{2}u_{k-1}^{p_{2}}(s_{2})\mathrm{d}s_{2}\right) ^{q_{1}}\mathrm{d}s_{1}\\&= C_{k}t^{\left[ \left( 1+\frac{\alpha _{2}}{2}\right) q_{1}+1+\frac{\beta _{1}}{2}\right] \frac{\left[ \left( p_{2}q_{1}\right) ^{k}-1\right] }{p_{2}q_{1}-1}-\frac{N}{2}(p_{2}q_{1})^{k}} e^{-\frac{|x|^{2}}{t}}. \end{aligned}$$

Then, there exist positive constants $C_k$, $t_k$, such that

$$\begin{aligned}&u(x,t)\ge C_ku_k(x,t)\ge c_kC_kt^{\left[ \left( 1+\frac{\alpha _{2}}{2}\right) q_{1}+1+\frac{\beta _{1}}{2}\right] \frac{\left[ (p_{2}q_{1})^{k}-1\right] }{p_{2}q_{1}-1} -\frac{N}{2}(p_{2}q_{1})^{k}}e^{-\frac{|x|^{2}}{t}},\\&\quad t\ge t_k,\ x\in R^N. \end{aligned}$$

Since $p_{2}q_{1}>1$, we obtain that, if $\frac{(\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}>\frac{N}{2}$, there exists a constant $K>0$, such that

$$\begin{aligned} r_1=\frac{\left[ \left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1\right] \left[ (p_{2}q_{1})^{K}-1\right] }{p_{2}q_{1}-1}-\frac{N}{2}(p_{2}q_{1})^{K}>0, \end{aligned}$$

and $u(x,t)\ge Ct^{r_1}e^{-\frac{|x|^{2}}{t}}.$ If $\frac{(\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}=\frac{N}{2}$, by Lemma 3.5, we have $u(x,t)\ge Ct^{-\frac{N}{2}}\log (1+t)e^{-\frac{|x|^{2}}{t}}.$ Similarly, the case (ii) of Lemma 3.6 is proved. $\square$

Inspired by the proof of [10, Theorem 1.1(i)], we show the proof of Theorem 2.1 of the present paper.

Proof of Theorem 2.1

We only prove $1<p_{2}q_{1}\le (p_{2}q_{1})_c$. Assume for some $(u_0,v_0)\ne (0,0)$, system (1.1) has a solution which is bounded in any $S_T=[0,T)\times R^N$. By Lemma 3.6, if $\frac{(\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}=\frac{N}{2}$ and $p_{2}q_{1}>1$, then there exist constants $t_1,C>0$ satisfying that

$$\begin{aligned} S(t)u(t)&\ge \int _{R^N}G(t,x,y)Ct^{-\frac{N}{2}}\log (1+t)e^{-\frac{|y|^{2}}{t}}\mathrm{d}y\\&\ge Ct^{-\frac{N}{2}}\log (1+t)e^{-\frac{|x|^{2}}{t}}, \quad t\ge t_1,\ x\in R^N. \end{aligned}$$

Thus, for some $\tau >0$

$$\begin{aligned} t^{\frac{\left( \frac{\alpha _{2}}{2}+1\right) q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}}S(t)u(t)\ge C\log (1+t)e^{-\frac{|x|^{2}}{t}},\quad t\ge \tau ,\ x\in R^N, \end{aligned}$$

and these imply the left-hand side comes to infinite at point $x=0$ as $t\rightarrow +\infty$, a contradiction to Lemma 3.4. If $\frac{(\frac{\alpha _{2}}{2}+1)q_{1}+\frac{\beta _{1}}{2}+1}{p_{2}q_{1}-1}>\frac{N}{2}$ and $p_{2}q_{1}>1$, there exist constants $t_1,C>0$, such that

$$\begin{aligned} S(t)u(t)\ge \displaystyle \int _{R^N}G(t,x,y)Ct^{r_1}e^{-\frac{|y|^{2}}{t}}\mathrm{d}y\ge Ct^{r_1}e^{-\frac{|x|^{2}}{t}},\quad t\ge t_{r_1},\ x\in R^N . \end{aligned}$$

Similarly, one could obtain a contradiction also at point $x=0$ as $t\rightarrow +\infty$. The proof for v(t) is similar. $\square$

4 Proof of Theorem 2.2

The proof is inspired and similar to the part (I) in the proof of [8, Theorem 1.2(b) and Theorem 1.3].

Proof of Theorem 2.2

Define $u_{0}(x,t) =S(t)u_{0}$, $v_{0}(x,t)=S(t)v_{0}$, and

$$\begin{aligned} u_{n+1}(x,t)&=u_{0}(x,t)+\int _{0}^{t}S(t-s)a_{1}u_{n}^{p_{1}}(s)\mathrm{d}s + \int _{0}^{t}S(t-s)b_{1}v_{n}^{q_{1}}(s)\mathrm{d}s, \end{aligned}$$

(4.1)

$$\begin{aligned} v_{n+1}(x,t)&=v_{0}(x,t)+\int _{0}^{t}S(t-s)a_{2}u_{n}^{p_{2}}(s)\mathrm{d}s + \int _{0}^{t}S(t-s)b_{2}v_{n}^{q_{2}}(s)\mathrm{d}s. \end{aligned}$$

(4.2)

By induction, $u_{n+1}(x,t)\ge u_{n}(x,t)$ and $v_{n+1}(x,t)\ge v_{n}(x,t)$. If $u(x,t)=\lim \limits _{n \rightarrow \infty }u_{n}(x,t)<\infty$ and $v(x,t)=\lim \limits _{n \rightarrow \infty }v_{n}(x,t)<\infty$ for $x\in R^{N}$, $t\in [0,\infty )$, then by(4.1–4.2), (u, v) satisfies (3.1). Hence, (u, v) is global. Thus, it suffices to prove that if

$$\begin{aligned} u_{0}(y)\le \delta _{1}G(k,0,y),\quad v_{0}(y)\le \delta _{2}G(k,0,y), \end{aligned}$$

for any $k>0$ and some $\delta _{1}$, $\delta _{2}>0$, then

$$\begin{aligned} \sup _{n} u_{n}(x,t)<\infty \quad \sup _{n} v_{n}(x,t)<\infty \quad \mathrm{for} \,\ x\in R_{N},\ t\ge 0. \end{aligned}$$

Consider

$$\begin{aligned} \left\{ \begin{array}{lll} u_{n}(x,t)\le c_{1}(k+t)^{E_{u}}G(m_{n}(t+k),0,x) &{} \mathrm{for} \,\ x\in R_{N}, \ t\ge 0,\\ v_{n}(x,t)\le c_{2}(k+t)^{E_{v}}G(l_{n}(t+k),0,x) &{}\mathrm{for} \,\ x\in R_{N}, \ t\ge 0, \end{array}\right. \end{aligned}$$

(4.3)

where $c_{1}, c_{2}>0$ and

$$\begin{aligned} w(\varepsilon ,N)=\left\{ \begin{array}{llll} \dfrac{\varepsilon }{2}, &{}{ if }(\varepsilon ,N)\in \{(-1,+\infty )\times [1,+\infty )\}\cup \{(-2,-1]\times [2,+\infty )\},\\ -\dfrac{1}{2}, &{}{ if }(\varepsilon ,N)\in [-2,-1)\times \{1\},\\ \dfrac{\varepsilon }{2}+\delta , &{}{ if }(\varepsilon ,N)=\{\{-1\}\}\times \{1\}\}\cup \{\{-2\}\times [2,+\infty \}\}, \end{array}\right. \end{aligned}$$

with $\varepsilon \in \{\alpha _{i}, \beta _{i},\ i=1,2\}$ and small $\delta >0$. For $n=0,1,2,...$

$$\begin{aligned} m_n&=1,\ \frac{1+p_{2}}{p_{2}},\ \frac{1+q_{1}+p_{2}q_{1}}{p_{2}q_{1}},\ \frac{1+p_{2}+p_{2}q_{1}+p_{2}^{2}q_{1}}{p_{2}^{2}q_{1}},...,\\ l_n&=1,\ \frac{1+q_{1}}{q_{1}},\ \frac{1+p_{2}+p_{2}q_{1}}{p_{2}q_{1}},\ \frac{1+q_{1}+p_{2}q_{1}+p_{2}q_{1}^{2}}{p_{2}q_{1}^{2}},..., \end{aligned}$$

that is, for $k=0,1,2,...$

$$\begin{aligned} m_{n}= & {} \left\{ \begin{array}{llll} \dfrac{1+q_{1}-(p_{2}q_{1})^{k+1}-q_{1}(p_{2}q_{1})^{k}}{(1-p_{2}q_{1})(p_{2}q_{1})^{k}}, &{}{ \ }n=2k,\\ \dfrac{(1+p_{2})\left[ 1-(p_{2}q_{1})^{k+1}\right] }{(1-p_{2}q_{1})p_{2}^{k+1}q_{1}^{k}}, &{}{ \ }n=2k+1; \end{array} \right. \\ l_{n}= & {} \left\{ \begin{array}{llll} \dfrac{1+p_{2}-(p_{2}q_{1})^{k+1}-p_{2}(p_{2}q_{1})^{k}}{(1-p_{2}q_{1})(p_{2}q_{1})^{k}}, &{}{ \ }n=2k,\\ \dfrac{(1+q_{1})\left[ 1-(p_{2}q_{1})^{k+1}\right] }{(1-p_{2}q_{1})q_{1}^{k+1}p_{2}^{k}}, &{}{ \ }n=2k+1. \end{array}\right. \end{aligned}$$

One could see $\frac{m_{n}+q_{1}}{q_{1}}=l_{n+1}$, $\frac{l_{n}+p_{2}}{p_{2}}=m_{n+1}$, and $\{m_{n}\}$, $\{l_{n}\}$ are increasing with

$$\begin{aligned} \lim \limits _{n \rightarrow \infty }m_{n}=\frac{p_{2}q_{1}+q_{1}}{p_{2}q_{1}-1},\quad \lim \limits _{n \rightarrow \infty }l_{n}=\frac{p_{2}q_{1}+p_{2}}{p_{2}q_{1}-1} . \end{aligned}$$

Here, we only prove (4.3). First

$$\begin{aligned} u_{0}(x,t)&=S(t)u_{0}\le \delta _{1}\int _{R^{N}}G(t,x,y)G(k,0,y)\mathrm{d}y\le \delta _{1}(k+t)^{E_{u}}G(m_{n}(t+k),0,x),\\ v_{0}(x,t)&=S(t)v_{0}\le \delta _{2}\int _{R^{N}}G(t,x,y)G(k,0,y)\mathrm{d}y\le \delta _{2}(k+t)^{E_{v}}G(l_{n}(t+k),0,x). \end{aligned}$$

Assume that

$$\begin{aligned} u_{n}(x,t)&\le c_{1}(k+t)^{E_{u}}G(m_{n}(t+k),0,x) \quad \mathrm{for} \,\ x\in R^{N},\ t\ge 0, \end{aligned}$$

(4.4)

$$\begin{aligned} v_{n}(x,t)&\le c_{2}(k+t)^{E_{v}}G(l_{n}(t+k),0,x) \quad \mathrm{for} \,\ x\in R^{N},\ t\ge 0. \end{aligned}$$

(4.5)

By (4.4) and (4.2), we see by [8, (2.1)] that

$$\begin{aligned} v_{n+1}(x,t)\le & {} v_{0}(x,t)+\int _{0}^{t}S(t-s)a_{2}c_{1}^{p_{2}}(k+s)^{p_{2}{E_{u}}}G^{p_2}(m_{n}(s+k),0,x)\mathrm{d}s\nonumber \\&+ \int _{0}^{t}S(t-s)b_{2}c_{2}^{q_{2}}(k+s)^{q_{2}{E_{v}}}G^{q_2}(l_{n}(s+k),0,x)\mathrm{d}s\nonumber \\\le & {} v_{0}(x,t)+C_{1}\int _{0}^{t}c_{1}^{p_{2}}(k+s)^{p_{2}{E_{u}}+\frac{N}{2}(1-p_{2})}\nonumber \\&G\left( \frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}},0,x\right) \nonumber \\&\times \int _{R^{N}}a_{2}(y) G\left( \frac{(t-s)m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)},y,\frac{m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}x\right) \mathrm{d}y\mathrm{d}s\nonumber \\&+C_{2}\int _{0}^{t}c_{2}^{q_{2}}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}G\left( \frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}},0,x\right) \nonumber \\&\times \int _{R^{N}}b_{2}(y) G\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)},y,\frac{l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}x\right) \mathrm{d}y\mathrm{d}s.\qquad \end{aligned}$$

(4.6)

We study the case $\alpha _{2}$, $\beta _{2}>0$. By [8, Lemma 5.1], we have

$$\begin{aligned}&\int _{R^{N}}a_{2}(y)G\left( \frac{(t-s)m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)},y,\frac{m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}x\right) \mathrm{d}y\\&\quad \le C^{'}\left\{ 1+\left[ \frac{(t-s)m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{\alpha _{2}}{2}} +\left[ \frac{m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\alpha _{2}}|x|^{\alpha _{2}}\right\} ,\\&\quad \int _{R^{N}}b_{2}(y)G\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)},y,\frac{l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}x\right) \mathrm{d}y\\&\quad \le C^{''}\left\{ 1+\left[ \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{\beta _{2}}{2}} +\left[ \frac{l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\beta _{2}}|x|^{\beta _{2}}\right\} . \end{aligned}$$

Then

$$\begin{aligned} v_{n+1}(x,t)&=v_{0}(x,t) +C_{1}\int _{0}^{t}c_{1}^{p_{2}}(k+s)^{p_{2}{E_{u}}+\frac{N}{2}(1-p_{2})}G\left( \frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}},0,x\right) \\&\quad \times \left\{ 1+\left[ \frac{(t-s)m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{\alpha _{2}}{2}} +\left[ \frac{m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\alpha _{2}}|x|^{\alpha _{2}}\right\} \mathrm{d}s\\&\quad +C_{2}\int _{0}^{t}c_{2}^{q_{2}}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}G\left( \frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}},0,x\right) \\&\quad \times \left\{ 1+\left[ \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{\beta _{2}}{2}} +\left[ \frac{l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\beta _{2}}|x|^{\beta _{2}}\right\} \mathrm{d}s. \end{aligned}$$

Let

$$\begin{aligned} J_{1}&=\int _{0}^{t}(k+s)^{p_{2}{E_{u}}+\frac{N}{2}(1-p_{2})}G\left( \frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}},0,x\right) \mathrm{d}s,\\ J_{2}&=\int _{0}^{t}(k+s)^{p_{2}{E_{u}}+\frac{N}{2}(1-p_{2})}G\left( \frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}},0,x\right) \\&\quad \left[ \frac{(t-s)m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{\alpha _{2}}{2}}\mathrm{d}s,\\ J_{3}&=\int _{0}^{t}(k+s)^{p_{2}{E_{u}}+\frac{N}{2}(1-p_{2})}G\left( \frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}},0,x\right) \\&\quad \left[ \frac{m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\alpha _{2}}|x|^{\alpha _{2}}\mathrm{d}s,\\ J_{4}&=\int _{0}^{t}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}G\left( \frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}},0,x\right) \mathrm{d}s,\\ J_{5}&=\int _{0}^{t}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}G\left( \frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}},0,x\right) \\&\quad \left[ \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{\beta _{2}}{2}}\mathrm{d}s,\\ J_{6}&=\int _{0}^{t}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}G\left( \frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}},0,x\right) \\&\quad \left[ \frac{l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\beta _{2}}|x|^{\beta _{2}}\mathrm{d}s. \end{aligned}$$

Then

$$\begin{aligned} v_{n+1}(x,t)\le v_{0}(x,t)+C(J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}) \end{aligned}$$

with $C:=\max \{c_{1}^{p_{2}}C_{1},c_{2}^{q_{2}}C_{2}\}$. Since $\frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}}\le \frac{m_{n}+p_{2}}{p_{2}}(t+k)=l_{n+1}(t+k)$, one has

$$\begin{aligned} G\left( \frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}},0,x\right)&\le \left[ 4\pi \frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}}\right] ^{-\frac{N}{2}} e^{-\frac{|x|^{2}}{4\frac{(m_{n}+p_{2})(t+s)}{p_{2}}}}\\&\le C\left[ \frac{t+k}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{N}{2}}G(l_{n+1}(t+k),0,x). \end{aligned}$$

As for $J_{1}$, we have

$$\begin{aligned} J_{1}&\le C\int _{0}^{t}(k+s)^{p_{2}{E_{u}}+\frac{N}{2}(1-p_{2})}\left[ \frac{t+k}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{N}{2}} G(l_{n+1}(t+k),0,x)\mathrm{d}s\\&=CG(l_{n+1}(t+k),0,x)\int _{0}^{t}(k+s)^{E_{v}-\frac{\alpha _{2}}{2}-1} \left[ \frac{t+k}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{N}{2}}\mathrm{d}s. \end{aligned}$$

In view of $E_{v}>0$, we obtain

$$\begin{aligned} J_{1}=C(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x)\int _{0}^{t}\frac{1}{(k+s)^{\frac{\alpha _{2}}{2}+1}} \left[ \frac{t+k}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{N}{2}}\mathrm{d}s. \end{aligned}$$

Let $r= {s}/{t}$. We have

$$\begin{aligned}&\int _{0}^{t}\frac{1}{(k+s)^{\frac{\alpha _{2}}{2}+1}}\left[ \frac{t+k}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{N}{2}}\mathrm{d}s\\&\quad =\int _{0}^{1}\frac{1}{(k+rt)^{\frac{\alpha _{2}}{2}+1}}\left[ \frac{t+k}{m_{n}(rt+k)+p_{2}(t-rt)}\right] ^{\frac{N}{2}}t\mathrm{d}r. \end{aligned}$$

Moreover

$$\begin{aligned} \frac{t+k}{m_{n}(rt+k)+p_{2}(t-rt)}&\rightarrow \frac{1}{m_{n}}\quad \mathrm{as}\,\ t\rightarrow 0,\\ \frac{t+k}{m_{n}(rt+k)+p_{2}(t-rt)}&\rightarrow \frac{1}{p_{2}-(p_{2}-m_{n})r}\quad \mathrm{as}\,\ t\rightarrow \infty . \end{aligned}$$

Since $1\le m_{n}\le \frac{p_{2}q_{1}+q_{1}}{p_{2}q_{1}-1}$ and $r=\frac{p_{2}}{p_{2}-m_{n}}>1$, $\frac{t+k}{m_{n}(rt+k)+p_{2}(t-rt)}$ is bounded. By $\frac{\alpha _{2}}{2}>0$, we have $\frac{1}{(k+rt)^{\frac{\alpha _{2}}{2}+1}}t$ is integrable in [0, 1] and $\displaystyle \int _{0}^{1}\frac{1}{(k+rt)^{\frac{\alpha _{2}}{2}+1}}t\mathrm{d}r$ is bounded in $[0,\infty )$. Hence, $J_{1}\le C(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x)$.

The estimate for $J_{2}$ is similar to the one for $J_1$ as $J_{2}\le C(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x)$. We omit the detail here.

As for $J_{3}$, we have

$$\begin{aligned} J_{3}&\le CG(l_{n+1}(t+k),0,x)\int _{0}^{t}(k+s)^{p_{2}{E_{u}}+\frac{N}{2}(1-p_{2})}\left[ \frac{m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\alpha _{2}} \\&\quad \times \left[ \frac{t+k}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{N}{2}}e^{-\frac{p_{2}|x|^{2}}{4}R} |x|^{\alpha _{2}}\mathrm{d}s. \end{aligned}$$

Here

$$\begin{aligned} R=\frac{m_{n}t+p_{2}k-(p_{2}-m_{n})s}{[m_{n}(s+k)+p_{2}(t-s)][(m_{n}+p_{2})(t+s)]}. \end{aligned}$$

As for $e^{-\frac{p_{2}|x|^{2}}{4}R} |x|^{\alpha _{2}}$, let $z=|x|^{2}$ and

$$\begin{aligned} f(z)=e^{-\frac{p_{2}|x|^{2}}{4}R}|x|^{\alpha _{2}}=e^{-\frac{p_{2}Rz}{4}}|z|^{\frac{\alpha _{2}}{2}}. \end{aligned}$$

f(z) shows its maximum at $z=\frac{2\alpha _{2}}{p_{2}R}$ and its maximum is $e^{-\frac{\alpha _{2}}{2}}(\frac{2\alpha _{2}}{p_{2}R})^{\frac{\alpha _{2}}{2}}$. Then

$$\begin{aligned} J_{3}&\le CG(l_{n+1}(t+k),0,x)\int _{0}^{t}(k+s)^{E_{v}-1}\\&\quad \times \left[ \frac{s+k}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{\alpha _{2}}{2}} \left[ \frac{t+k}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{N}{2}+\frac{\alpha _{2}}{2}}\mathrm{d}s. \end{aligned}$$

Similarly to the proof of $J_{1}$ and $J_{2}$, one could find a constant $C>0$, such that

$$\begin{aligned} J_{3}\le C(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x). \end{aligned}$$

Since the proofs for $J_{4}$, $J_{5}$, and $J_{6}$ are all very similar, we only prove $J_{5}$ here. In the case of $q_{2}>(q_{2})_{c}$, if $\beta _{2}>q_{1}N$ and $E_{v}(q_{2}-1)<\min \Big \{1,\, \frac{N}{2}(q_{2}-1)-1-\frac{\beta _{2}}{2}\Big \}$, we obtain $q_{2}E_{v}+\frac{N}{2}(1-q_{2})+\theta \le E_{v}$ and $\theta >\frac{\beta _{2}}{2}+1$, where $\theta$ is a constant. Therefore, we have

$$\begin{aligned} J_{5}&\le C(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x)\int _{0}^{t}\frac{1}{(k+s)^{\theta -\frac{\beta _{2}}{2}}}[4\pi l_{n+1}(t+k)]^{\frac{N}{2}}\\&\quad \times \left[ 4\pi \frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}}\right] ^{-\frac{N}{2}}e^{-\frac{|x|^{2}}{4}\left[ \frac{q_{2}}{l_{n}(s+k)+q_{2}(t-s)} -\frac{1}{l_{n+1}(t+k)}\right] }\\&\quad \left[ \frac{t-s}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{\beta _{2}}{2}}\mathrm{d}s. \end{aligned}$$

Since $\frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}}\le \frac{l_{n}+q_{2}}{q_{2}}(t+k)$, we have $e^{-\frac{|x|^{2}q_{2}}{4[l_{n}(s+k)+q_{2}(t-s)]}}\le e^{-\frac{|x|^{2}}{4(l_{n}+q_{2})(t+s)}}$. Let

$$\begin{aligned} R=\frac{q_{2}}{(l_{n}+q_{2})(t+s)}-\frac{1}{l_{n+1}(t+k)}. \end{aligned}$$

In virtue of $q_{2}>(q_{2})_{c}$ and $\frac{2+\beta _{2}}{N}>q_{1}$, one could find that $l_{n}>l_{2}=1+\frac{1}{q_{1}}>1+\frac{1}{q_{2}-1}$. Hence, $R>0$ and

$$\begin{aligned} J_{5}&=C(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x)\int _{0}^{t}\frac{1}{(k+s)^{\theta -\frac{\beta _{2}}{2}}}\\&\quad \times \left[ \frac{l_{n+1}(t+k)q_{2}}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{N}{2}}\left[ \frac{t-s}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{\beta _{2}}{2}}\mathrm{d}s. \end{aligned}$$

Let $r= {s}/{t}$. We have

$$\begin{aligned}&\int _{0}^{t}\frac{1}{(k+s)^{\theta -\frac{\beta _{2}}{2}}}\left[ \frac{l_{n+1}(t+k)q_{2}}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{N}{2}} \left[ \frac{t-s}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{\beta _{2}}{2}}\mathrm{d}s\\&\quad =\int _{0}^{1}\frac{1}{(k+rt)^{\theta -\frac{\beta _{2}}{2}}}\left[ \frac{l_{n+1}(t+k)q_{2}}{l_{n}(rt+k)+q_{2}(t-rt)}\right] ^{\frac{N}{2}} \left[ \frac{t-rt}{l_{n}(rt+k)+q_{2}(t-rt)}\right] ^{\frac{\beta _{2}}{2}}t\mathrm{d}r. \end{aligned}$$

Moreover

$$\begin{aligned} \frac{l_{n+1}(t+k)q_{2}}{l_{n}(rt+k)+q_{2}(t-rt)}&\rightarrow \frac{l_{n+1}q_{2}}{l_{n}}\quad \mathrm{as}\,\ t\rightarrow 0,\\ \frac{l_{n+1}(t+k)q_{2}}{l_{n}(rt+k)+q_{2}(t-rt)}&\rightarrow \frac{l_{n+1}q_{2}}{q_{2}-(q_{2}-l_{n})r}\quad \mathrm{as}\,\ t\rightarrow \infty . \end{aligned}$$

Since $1\le l_{n}\le \frac{p_{2}q_{1}+p_{2}}{p_{2}q_{1}-1}$ and $r=\frac{q_{2}}{q_{2}-l_{n}}>1$, there is a contradiction here with premise $r\in (0,1)$, and hence, the function $\frac{l_{n+1}(t+k)q_{2}}{l_{n}(rt+k)+q_{2}(t-rt)}$ is bounded. Similarly, $\frac{t-tr}{m_{n}(rt+k)+p_{2}(t-rt)}$ is also bounded. In virtue of $\theta -\frac{\beta _{2}}{2}>1$, we know that $\frac{1}{(k+rt)^{\theta -\frac{\beta _{2}}{2}}}t$ is integrable in [0, 1] and $\displaystyle \int _{0}^{1}\frac{1}{(k+rt)^{\theta -\frac{\beta _{2}}{2}}}t\mathrm{d}r$ is bounded in $[0,\infty )$. Then, there exists some positive constant C, such that $J_{5}\le C(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x)$. Consequently, we obtain

$$\begin{aligned} v_{n+1}(x,t)&\le v_{0}(x,t)+{{\tilde{C}}}(J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6})\\&\le (\delta _{1}+{{\tilde{C}}})(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x), \end{aligned}$$

where ${{\tilde{C}}}=\max \{c_{1}^{p_{1}}C_{1},\,c_{2}^{q_{1}}C_{2}\}$. The calculation steps for $u_{n+1}$ and $v_{n+1}$ are very similar and the result is

$$\begin{aligned} u_{n+1}(x,t)\le (\delta _{2}+{{\hat{C}}})(k+t)^{E_{u}}G(m_{n+1}(t+k),0,x), \end{aligned}$$

where ${{\hat{C}}}=\max \{c_{1}^{q_{2}}C_{1},\,c_{2}^{p_{2}}C_{2}\}$. We can do former procedure again and obtain that

$$\begin{aligned} \left\{ \begin{array}{lll} u_{n+2}(x,t)\le u_{0}(x,t)+C(k+t)^{E_{u}}G(m_{n+1}(t+k),0,x),\\ v_{n+2}(x,t)\le v_{0}(x,t)+C(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x). \end{array}\right. \end{aligned}$$

That is, if (4.4–4.5) hold for n, the estimates hold for $n+2$. This completes the proof. $\square$

5 Proof of Theorems 2.3–2.6

In this section, we prove Theorems 2.3–2.6. We only give the proof of Theorem 2.4 (iii). The other cases can be proved similarly. The proof is inspired and similar to the part (II) in the proof of [8, Theorem 1.2(b) and Theorem 1.3].

Proof of Theorem 2.4(iii). We study the case for $\alpha _{1}, \beta _{2}<0$ and the other cases can be obtained similarly. There exists some positive constant C, such that $a_{2}(x)\le C(1+|x|)^{\alpha _{2}}$, $b_{2}(x)\le C(1+|x|)^{\beta _{2}}$. By [8, Lemma 5.2], we obtain from (4.6) that

$$\begin{aligned}&\int _{R^{N}}b_{2}(y)G\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)},y,\frac{l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}x\right) \mathrm{d}y\\&\quad \le \int _{R^{N}}b_{2}(y)G\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)},0,y\right) \mathrm{d}y. \end{aligned}$$

For $t\in [0, 1]$ and $\beta _{2}<0$, $\displaystyle \int _{R^{N}}G(t,0,y)(1+|y|)^{\beta _{2}}\mathrm{d}y\le 1$. Using [8, Lemma 2.2]

$$\begin{aligned} \int _{R^{N}}G(t,0,y)(1+|y|)^{\beta _{2}}\mathrm{d}y\le Cf(t),\quad t\ge 0,\ \beta _{2}\in [-2,0], \end{aligned}$$

where

$$\begin{aligned} f(t)=\left\{ \begin{array}{llll} 1, &{}{ if\ }t\le 1,\\ t^{w(\beta _{2},N)}, &{}{ if\ }t>1. \end{array} \right. \end{aligned}$$

By (4.6), we see

$$\begin{aligned} v_{n+1}(x,t)&\le v_{0}(x,t) +C_{1}\int _{0}^{t}c_{1}^{p_{2}}(k+s)^{p_{2}{E_{u}}+\frac{N}{2}(1-p_{2})}G\left( \frac{m_{n}(s+k)+p_{2}(t-s)}{p_{2}},0,x\right) \\&\quad \times \left\{ 1+\left[ \frac{(t-s)m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\frac{\alpha _{2}}{2}} +\left[ \frac{m_{n}(s+k)}{m_{n}(s+k)+p_{2}(t-s)}\right] ^{\alpha _{2}}|x|^{\alpha _{2}}\right\} \mathrm{d}s\\&\quad +C_{2}\int _{0}^{t}c_{2}^{q_{2}}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}G\left( \frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}},0,x\right) \\&\quad \times f\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right) \mathrm{d}s. \end{aligned}$$

Define

$$\begin{aligned} K_{1}:= & {} \displaystyle \int _{0}^{t}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}G\left( \frac{l_{n}(s+k)+q_{2}(t-s)}{q_{2}},0,x\right) \\&f\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right) \mathrm{d}s. \end{aligned}$$

Therefore, $v_{n+1}(x,t)\le v_{0}(x,t)+C(J_{1}+J_{2}+J_{3}+K_{1})$ with $C=\max \{c_{1}^{p_{2}}C_{1},c_{2}^{q_{2}}C_{2}\}$. The discussion on $J_{1}$, $J_{2}$, and $J_{3}$ could be obtained by the former ones. We only deal with $K_{1}$. If $p_{1}-1>p_{2}$,

$$\begin{aligned} K_{1}&\le G(l_{n+1}(t+k),0,x)\int _{0}^{t}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}\\&\quad \times \left[ \frac{l_{n+1}(t+k)q_{2}}{l_{n}(s+k)+q_{2}(t-s)}\right] ^{\frac{N}{2}} f\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right) \mathrm{d}s. \end{aligned}$$

From the analysis on $J_{5}$, we know that $\frac{l_{n+1}(t+k)q_{2}}{l_{n}(s+k)+q_{2}(t-s)}$ is bounded. Then

$$\begin{aligned} K_{1}\le G(l_{n+1}(t+k),0,x)\int _{0}^{t}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}f\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right) \mathrm{d}s. \end{aligned}$$

Let $r= {s}/{t}$. We obtain that

$$\begin{aligned}&\int _{0}^{t}(k+s)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}f\left( \frac{(t-s)l_{n}(s+k)}{l_{n}(s+k)+q_{2}(t-s)}\right) \mathrm{d}s\\&\quad = \int _{0}^{1}(k+rt)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}f\left( \frac{(t-rt)l_{n}(rt+k)}{l_{n}(rt+k)+q_{2}(t-rt)}\right) t\mathrm{d}r. \end{aligned}$$

If $\frac{(t-rt)l_{n}(rt+k)}{l_{n}(rt+k)+q_{2}(t-rt)}\le 1$, one can see that t is bounded. Then, if $(q_{2}-1)E_{v}\le \frac{2+\beta _{2}}{2}$, it is obvious that there exists a constant $C>0$, such that

$$\begin{aligned} \int _{0}^{1}(k+rt)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}f\left( \frac{(t-rt)l_{n}(rt+k)}{l_{n}(rt+k)+q_{2}(t-rt)}\right) t\mathrm{d}r\le C(k+t)^{E_{v}}. \end{aligned}$$

If $\frac{(t-rt)l_{n}(rt+k)}{l_{n}(rt+k)+q_{2}(t-rt)}>1$, there exists a constant $C>0$, such that $\frac{t-rt}{l_{n}(rt+k)+q_{2}(t-rt)}>C$. Then

$$\begin{aligned}&\int _{0}^{1}(k+rt)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})}f\left( \frac{(t-rt)l_{n}(rt+k)}{l_{n}(rt+k)+q_{2}(t-rt)}\right) t\mathrm{d}r\\&\quad \le \int _{0}^{1}(k+rt)^{q_{2}{E_{v}}+\frac{N}{2}(1-q_{2})+w(\beta _{2},N)}t\mathrm{d}r \le C(k+t)^{E_{v}}. \end{aligned}$$

Hence

$$\begin{aligned} v_{n+1}(x,t)&\le v_{0}(x,t)+c_{i}^{p}C_{i}(J_{1}+J_{2}+J_{3}+K_{1})\\&\le (\delta _{2}+c^{p}C)(k+t)^{E_{v}}G(l_{n+1}(t+k),0,x). \end{aligned}$$

Similarly, $u_{n+1}(x,t)\le c(k+t)^{E_{u}}G(m_{n+1}(t+k),0,x)$. $\square$

6 Conclusion

As noted above, for $\alpha _i$, $\beta _i>0$, $i = 1,2$, we prove that there are no global solutions of (1.1) for any nonnegative nontrivial initial data provided that

$$\begin{aligned} 1< p_2 q_1 \le (p_2 q_1)_c,\quad { or}\quad 1<p_1\le (p_1)_c,\quad { or}\quad 1<q_2 \le (q_2)_c. \end{aligned}$$

Besides the case for $\alpha _i$, $\beta _i>0$, $i = 1,2$, at least one out of $\alpha _i$, $\beta _i$, $i = 1,2$ is negative, we show the global existence of solution provided that

$$\begin{aligned} p_{2}q_{1}>(p_{2}q_{1})_{c},\quad p_{1}>(p_{1})_{c},\quad \mathrm{and }\quad q_{2}>(q_{2})_{c}, \end{aligned}$$

where some other conditions on $p_i$, $q_i$, $\alpha _i$, $\beta _i$, $i = 1,2$, are needed. Therefore, it was a pity that we have not obtain the precise Fujita exponents of (1.4). We thought about that the exponents $(p_2 q_1)_c$, $(p_1)_c$, $(q_2)_c$ were correct, because they are compatible with the ones in [4, 8, 16]. In fact, the more complicated coupled relation and the unbounded variable coefficients bring much more difficulty in the discussion of the global existence of solutions. And the semigroup method used in [10] would not be used anymore. We need to overcome more difficulty in dealing with the interactive terms, such as $J_i$, $i=1,2,\cdots ,6$, in the proof of Theorems 2.2–2.6, respectively.