In this section, we discuss the blow-up phenomenon.
Theorem 3.1
Assume the conditions (A1)–(A3) and \(2<\rho +2<\min \{m,r\}\), \(\max \{m,r\}<2(p+2)\) hold, and \(g_{i}\) satisfies the condition
$$ \max \biggl\{ \int ^{\infty }_{0}g_{1}(\tau )\,d\tau , \int ^{\infty }_{0}g_{2}( \tau )\,d\tau \biggr\} < \frac{p+1}{p+1+\frac{1}{4(p+2)}}, $$
(3.1)
for \(i=1,2\). Let \((u,v)\) be the solution of system (1.1), satisfying
$$ \int _{\Omega }u(0)\frac{u_{t}(0) \vert u_{t}(0) \vert ^{\rho }}{\rho +1}\,dx+ \int _{\Omega }v(0)\frac{v_{t}(0) \vert v_{t}(0) \vert ^{\rho }}{\rho +1}\,dx>ME(0)>0, $$
(3.2)
then \((u,v)\) blows up in finite time, where
$$ M=\biggl(\frac{1}{\varepsilon _{1}^{*}}\biggr)^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma } $$
\(\gamma =\max \{m,r\}\), \(\gamma ^{*}=\min \{m,r\}\), \(\varepsilon ^{*}_{1}>0\) is a constant such that \((\frac{1}{\varepsilon _{1}})^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }\geq \frac{2(p+2)(1-\varepsilon )}{\beta }\), \(\varepsilon \in (0,1)\) is a small enough constant such that
$$\begin{aligned} & \kappa _{1}(\varepsilon ) = \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{1} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{1})>0, \\ &\kappa _{2}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{2} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{2})>0, \\ &\alpha =\min \biggl\{ \kappa _{1}(\varepsilon )\lambda - \frac{\varepsilon ^{m}_{1}}{m},\kappa _{2}(\varepsilon )\lambda - \frac{\varepsilon ^{r}_{1}}{r},c_{0}\varepsilon - \frac{\varepsilon ^{m}_{1}}{m},c_{0} \varepsilon - \frac{\varepsilon ^{r}_{1}}{r}\biggr\} , \\ &\beta =\min \biggl\{ \frac{1}{\rho +1}+ \frac{2(p+2)(1-\varepsilon )}{\rho +2},\alpha \biggr\} , \end{aligned}$$
and λ is the first eigenvalue of −Δ.
Proof
Suppose that \((u,v)\) is a global solution of system (1.1). Multiplying the first two equations of system (1.1) by u and v, respectively, and integrating over Ω, we obtain
$$\begin{aligned}& \bigl( \vert u_{t} \vert ^{\rho }u_{tt},u \bigr)+ \Vert \nabla u \Vert ^{2}_{2}+ \int _{\Omega } \int ^{t}_{0}g_{1}(t- \tau )\Delta u(\tau )\,d\tau u(t)\,dx+ \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx \\& \quad = \int _{\Omega }uf_{1}(u,v)\,dx, \end{aligned}$$
(3.3)
$$\begin{aligned}& \bigl( \vert v_{t} \vert ^{\rho }v_{tt},v \bigr)+ \Vert \nabla v \Vert ^{2}_{2}+ \int _{\Omega } \int ^{t}_{0}g_{2}(t- \tau )\Delta v(\tau )\,d\tau v(t)\,dx+ \int _{\Omega }vv_{t} \vert v_{t} \vert ^{m-2}\,dx \\& \quad = \int _{\Omega }vf_{2}(u,v)\,dx. \end{aligned}$$
(3.4)
Taking the derivative of \((u,\frac{u_{t}|u_{t}|^{\rho }}{\rho +1})\) and \((v,\frac{v_{t}|v_{t}|^{\rho }}{\rho +1})\), respectively, and combining (3.3) and (3.4), we have
$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)= \frac{1}{\rho +1} \bigl\Vert u_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} - \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(\tau )\nabla u(t)\,dx\,d\tau \\ & \hphantom{\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)=}{}- \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx+ \int _{\Omega }uf_{1}(u,v)\,dx, \end{aligned}$$
(3.5)
$$\begin{aligned} &\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)= \frac{1}{\rho +1} \bigl\Vert v_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} - \Vert \nabla v \Vert ^{2}_{2}+ \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(\tau )\nabla v(t)\,dx\,d\tau \\ &\hphantom{\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)=}{} - \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx+ \int _{\Omega }vf_{2}(u,v)\,dx. \end{aligned}$$
(3.6)
For the third term on the right side of (3.5), we get
$$\begin{aligned} \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(\tau )\nabla u(t)\,dx\,d \tau ={}& \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(t)\nabla \bigl( u(\tau )-u(t) \bigr)\,dx\,d\tau \\ &{} + \int _{0}^{t} g_{1}(t-\tau ) \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}\,d\tau \\ ={}& \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(t) \bigl(\nabla u( \tau )-\nabla u(t) \bigr)\,dx\,d\tau \\ &{} + \int _{0}^{t} g_{1}(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}. \end{aligned}$$
(3.7)
In the same way, we have
$$\begin{aligned} \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(\tau )\nabla v(t)\,dx\,d \tau={}& \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(t) \bigl( \nabla v(\tau )-\nabla v(t) \bigr)\,dx\,d\tau \\ &{} + \int _{0}^{t} g_{2}(\tau )\,d\tau \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2}. \end{aligned}$$
(3.8)
Then, inserting (3.7) and (3.8) into (3.5) and (3.6), respectively,
$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ &\quad = \frac{1}{\rho +1} \bigl\Vert u_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} - \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(t) \bigl(\nabla u(\tau )-\nabla u(t) \bigr)\,dx\,d\tau \\ &\qquad {} + \int _{0}^{t} g_{1}(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}- \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx+ \int _{\Omega }uf_{1}(u,v)\,dx, \end{aligned}$$
(3.9)
$$\begin{aligned} &\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ &\quad = \frac{1}{\rho +1} \bigl\Vert v_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} - \Vert \nabla v \Vert ^{2}_{2}+ \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(t) \bigl(\nabla v(\tau )-\nabla v(t) \bigr)\,dx\,d\tau \\ &\qquad {} + \int _{0}^{t} g_{2}(\tau )\,d\tau \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2}- \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx+ \int _{\Omega }vf_{2}(u,v)\,dx. \end{aligned}$$
(3.10)
For the third term on the right side of (3.9), applying the Cauchy inequality, we obtain
$$\begin{aligned} & \int ^{t}_{0} g_{1}(t-\tau ) \int _{\Omega }\nabla u(t) \bigl(\nabla u( \tau )-\nabla u(t) \bigr)\,dx\,d\tau \\ & \quad \geq -\frac{2(p+2)(1-\varepsilon )}{2}(g_{1}\circ \nabla u) (t)- \frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}, \end{aligned}$$
(3.11)
where \(\varepsilon \in (0,1)\), similarly
$$\begin{aligned} & \int ^{t}_{0} g_{2}(t-\tau ) \int _{\Omega }\nabla v(t) \bigl(\nabla v( \tau )-\nabla v(t) \bigr)\,dx\,d\tau \\ & \quad \geq -\frac{2(p+2)(1-\varepsilon )}{2}(g_{2}\circ \nabla v) (t)- \frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2}. \end{aligned}$$
(3.12)
Inserting (3.11) and (3.12) into (3.9) and (3.10), respectively, we can get
$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq \frac{1}{\rho +1} \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} - \Vert \nabla u \Vert ^{2}_{2} -\frac{2(p+2)(1-\varepsilon )}{2}(g_{1}\circ \nabla u) (t) \\ &\hphantom{\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq}{} + \int _{0}^{t} g_{1}(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}_{2}- \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx+ \int _{\Omega }uf_{1}(u,v)\,dx \\ &\hphantom{\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq}{} -\frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}, \end{aligned}$$
(3.13)
$$\begin{aligned} & \frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq \frac{1}{\rho +1} \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} - \Vert \nabla v \Vert ^{2}_{2} -\frac{2(p+2)(1-\varepsilon )}{2}(g_{2}\circ \nabla v) (t) \\ &\hphantom{\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq}{} + \int _{0}^{t} g_{2}(\tau )\,d\tau \bigl\Vert \nabla v(t) \bigr\Vert ^{2}_{2}- \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx+ \int _{\Omega }vf_{2}(u,v)\,dx \\ &\hphantom{\frac{d}{dt}\biggl(v,\frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)\geq}{} -\frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2}. \end{aligned}$$
(3.14)
Adding (3.13) and (3.14), we derive that
$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \frac{d}{dt}\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ &\quad \geq \frac{1}{\rho +1} \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} \bigr)+2(p+2) \int _{\Omega }F(u,v)\,dx \\ &\qquad {} -\biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2}-\biggl(1- \int _{0}^{t}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {} -\frac{2(p+2)(1-\varepsilon )}{2} \bigl((g_{1}\circ \nabla u) (t)+(g_{2} \circ \nabla v) (t) \bigr) \\ &\qquad {} -\frac{1}{4(p+2)(1-\varepsilon )} \biggl( \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0}g_{2}(\tau ) \,d\tau \Vert \nabla v \Vert ^{2}_{2} \biggr) \\ &\qquad {} - \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx- \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx. \end{aligned}$$
(3.15)
Adding \(2(p+2)(1-\varepsilon )E(t)\) on the right side of (3.15), we can get
$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \frac{d}{dt}\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ &\quad \geq \biggl( \frac{1}{\rho +1}+ \frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} \bigr) \\ &\qquad {} +2(p+2)\varepsilon \int _{\Omega }F(u,v) -2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2} \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int _{0}^{t}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {} -\frac{1}{4(p+2)(1-\varepsilon )} \biggl( \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2} \biggr) \\ &\qquad {} - \int _{\Omega }uu_{t} \vert u_{t} \vert ^{m-2}\,dx- \int _{\Omega }vv_{t} \vert v_{t} \vert ^{r-2}\,dx. \end{aligned}$$
(3.16)
For the last two terms on the right side of (3.16), applying the Hölder inequality and the Young inequality, we arrive at
$$\begin{aligned} \biggl\vert \int _{\Omega } \vert u_{t} \vert ^{m-2}u_{t}u\,dx \biggr\vert &\leq \frac{\varepsilon ^{m}_{1} \Vert u \Vert ^{m}_{m}}{m}+\biggl( \frac{1}{\varepsilon _{1}} \biggr)^{\frac{m}{m-1}} \frac{(m-1) \Vert u_{t} \Vert ^{m}_{m}}{m}, \end{aligned}$$
(3.17)
where \(\varepsilon _{1}>0\). By Lemma 2.4 and the conditions of the theorem, one can deduce that \(\|u\|^{m}_{m}\leq \|u\|^{2}_{2}+\|u\|^{2(p+2)}_{2(p+2)}\), then the inequality (3.17) can be rewritten as
$$\begin{aligned} \biggl\vert \int _{\Omega } \vert u_{t} \vert ^{m-2}u_{t}u\,dx \biggr\vert &\leq \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr)+\biggl(\frac{1}{\varepsilon _{1}}\biggr)^{\frac{m}{m-1}} \frac{(m-1) \Vert u_{t} \Vert ^{m}_{m}}{m}, \end{aligned}$$
(3.18)
similarly, we have
$$\begin{aligned} \biggl\vert \int _{\Omega } \vert v_{t} \vert ^{r-2}v_{t}v\,dx \biggr\vert &\leq \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr)+\biggl(\frac{1}{\varepsilon _{1}}\biggr)^{\frac{r}{r-1}} \frac{(r-1) \Vert v_{t} \Vert ^{r}_{r}}{r}. \end{aligned}$$
(3.19)
Substituting (3.18) and (3.19) into (3.16), then we have
$$\begin{aligned} &\frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \frac{d}{dt}\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) +\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{m}{m-1}}\frac{(m-1) \Vert u_{t} \Vert ^{m}_{m}}{m}+ \biggl( \frac{1}{\varepsilon _{1}} \biggr)^{\frac{r}{r-1}} \frac{(r-1) \Vert v_{t} \Vert ^{r}_{r}}{r} \\ &\quad \geq \biggl(\frac{1}{\rho +1}+ \frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2} \bigr) \\ & \qquad {}+2(p+2)\varepsilon \int _{\Omega }F(u,v) -2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2} \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int _{0}^{t}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {} -\frac{1}{4(p+2)(1-\varepsilon )} \biggl( \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2} \biggr) \\ &\qquad {} - \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) - \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr). \end{aligned}$$
(3.20)
Take \(\gamma =\max \{m,r\}\), \(\gamma ^{*}=\min \{m,r\}\). Combining with (2.5), we know \(E'(t)\leq -\|u_{t}\|^{m}_{m}-\|v_{t}\|^{r}_{r}\), that is, \(-E'(t)\geq \|u_{t}\|^{m}_{m}+\|v_{t}\|^{r}_{r}\), then (3.20) can be rewritten as
$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) \biggr)+\biggl(\frac{1}{\varepsilon _{1}}\biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }\bigl( \Vert u_{t} \Vert ^{m}_{m}+ \Vert v_{t} \Vert ^{r}_{r}\bigr) \\ &\quad \geq \frac{d}{dt}\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\frac{d}{dt} \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) +\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{m}{m-1}}\frac{(m-1) \Vert u_{t} \Vert ^{m}_{m}}{m} \\ &\qquad {}+ \biggl( \frac{1}{\varepsilon _{1}} \biggr)^{\frac{r}{r-1}} \frac{(r-1) \Vert v_{t} \Vert ^{r}_{r}}{r} \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} +2(p+2)\varepsilon \int _{\Omega }F(u,v)\,dx-2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int ^{t}_{0}g_{1}(\tau )\,d\tau \biggr) \Vert \nabla u \Vert ^{2}_{2} \\ &\qquad {} + \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int _{0}^{t}g_{2}(\tau )\,d\tau \biggr) \Vert \nabla v \Vert ^{2}_{2} \\ & \qquad {}-\frac{1}{4(p+2)(1-\varepsilon )} \biggl( \int ^{t}_{0}g_{1}(\tau )\,d\tau \Vert \nabla u \Vert ^{2}_{2}+ \int ^{t}_{0}g_{2}(\tau )\,d\tau \Vert \nabla v \Vert ^{2}_{2} \biggr) \\ &\qquad {} - \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) - \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr). \end{aligned}$$
(3.21)
Combining Lemma 2.3 and the Poincaré inequality, we can deduce
$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} +c_{0}\varepsilon \bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr)-2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \biggl[ \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int ^{t}_{0}g_{1}( \tau )\,d\tau \biggr) - \frac{1}{4(p+2)(1-\varepsilon )} \int _{0}^{t} g_{1}( \tau )\,d\tau \biggr] \lambda \Vert u \Vert ^{2}_{2} \\ &\qquad {} + \biggl[ \bigl((p+2) (1-\varepsilon )-1 \bigr) \biggl(1- \int _{0}^{t}g_{2}( \tau )\,d\tau \biggr)- \frac{1}{4(p+2)(1-\varepsilon )} \int ^{t}_{0}g_{2}( \tau )\,d\tau \biggr] \lambda \Vert v \Vert ^{2}_{2} \\ &\qquad {} - \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) - \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr) \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} +c_{0}\varepsilon \bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr)-2(p+2) (1-\varepsilon )E(t) \\ &\qquad {} + \biggl[ \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{1} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{1}) \biggr]\lambda \Vert u \Vert ^{2}_{2} \\ &\qquad {} + \biggl[ \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{2}- \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{2}) \biggr]\lambda \Vert v \Vert ^{2}_{2} \\ &\qquad {} - \frac{\varepsilon ^{r}_{1}}{r} \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) - \frac{\varepsilon ^{m}_{1}}{m} \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr), \end{aligned}$$
(3.22)
where λ is the first eigenvalue of −Δ, now we take
$$\begin{aligned} &\kappa _{1}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{1} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{1}), \\ &\kappa _{2}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{2} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{2}). \end{aligned}$$
Then (3.22) can be rewritten as
$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} + \biggl(\kappa _{1}(\varepsilon )\lambda - \frac{\varepsilon ^{m}_{1}}{m} \biggr) \Vert u \Vert ^{2}_{2}+ \biggl(\kappa _{2}( \varepsilon )\lambda - \frac{\varepsilon ^{r}_{1}}{r} \biggr) \Vert v \Vert ^{2}_{2} \\ &\qquad {} + \biggl(c_{0}\varepsilon - \frac{\varepsilon ^{r}_{1}}{r} \biggr) \Vert v \Vert ^{2(p+2)}_{2(p+2)} + \biggl(c_{0}\varepsilon - \frac{\varepsilon ^{m}_{1}}{m} \biggr) \Vert u \Vert ^{2(p+2)}_{2(p+2)} \\ & \qquad {}-2(p+2) (1-\varepsilon )E(t). \end{aligned}$$
(3.23)
By the condition
$$ \max \biggl\{ \int ^{\infty }_{0}g_{1}(\tau )\,d\tau , \int ^{\infty }_{0}g_{2}( \tau )\,d\tau \biggr\} < \frac{p+1}{p+1+\frac{1}{4(p+2)}}, $$
we can obtain
$$\begin{aligned} & \bigl((p+2)-1 \bigr)l_{1} -\frac{1}{4(p+2)}(1-l_{1})>0, \\ & \bigl((p+2)-1 \bigr)l_{2} -\frac{1}{4(p+2)}(1-l_{2})>0. \end{aligned}$$
Then we choose ε small enough such that
$$\begin{aligned} &\kappa _{1}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{1} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{1})>0, \\ &\kappa _{2}(\varepsilon )= \bigl((p+2) (1-\varepsilon )-1 \bigr)l_{2} - \frac{1}{4(p+2)(1-\varepsilon )}(1-l_{2})>0. \end{aligned}$$
And we pick \(\varepsilon _{1}\) small enough such that
$$\begin{aligned} &\min \biggl\{ \kappa _{1}(\varepsilon )\lambda - \frac{\varepsilon ^{m}_{1}}{m},\kappa _{2}(\varepsilon )\lambda - \frac{\varepsilon ^{r}_{1}}{r}\biggr\} >0, \\ & \min \biggl\{ c_{0}\varepsilon - \frac{\varepsilon ^{m}_{1}}{m},c_{0} \varepsilon - \frac{\varepsilon ^{r}_{1}}{r}\biggr\} >0. \end{aligned}$$
(3.24)
Then we choose
$$\begin{aligned} &\alpha=\min \biggl\{ \kappa _{1}(\varepsilon )\lambda - \frac{\varepsilon ^{m}_{1}}{m},\kappa _{2}(\varepsilon )\lambda - \frac{\varepsilon ^{r}_{1}}{r},c_{0}\varepsilon - \frac{\varepsilon ^{m}_{1}}{m},c_{0} \varepsilon - \frac{\varepsilon ^{r}_{1}}{r}\biggr\} , \\ &\beta=\min \biggl\{ \frac{1}{\rho +1}+ \frac{2(p+2)(1-\varepsilon )}{\rho +2},\alpha \biggr\} . \end{aligned}$$
Using Lemma 2.4, (3.23) can be deduced as
$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \biggl(\frac{1}{\rho +1}+\frac{2(p+2)(1-\varepsilon )}{\rho +2} \biggr) \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{\rho +2}_{ \rho +2} \bigr) \\ &\qquad {} +\alpha \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr)+ \alpha \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) \\ &\qquad {} -2(p+2) (1-\varepsilon )E(t) \\ &\quad \geq \beta \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} \bigr) \\ &\qquad {} +\beta \bigl( \Vert u \Vert ^{2}_{2}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr)+\beta \bigl( \Vert v \Vert ^{2}_{2}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr) \\ &\qquad {} -2(p+2) (1-\varepsilon )E(t) \\ &\quad \geq \beta \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{\rho +2}_{\rho +2}+ \bigl\Vert v_{t}(t) \bigr\Vert ^{ \rho +2}_{\rho +2} \bigr) +\beta \Vert u \Vert ^{\rho +2}_{\rho +2}+\beta \Vert v \Vert ^{ \rho +2}_{\rho +2} -2(p+2) (1-\varepsilon )E(t). \end{aligned}$$
(3.25)
By applying the Hölder and Young inequalities, we can deduce that
$$ \biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)\leq \Vert u \Vert ^{\rho +2}_{\rho +2}+ \Vert u_{t} \Vert ^{ \rho +2}_{\rho +2}, $$
then we have
$$\begin{aligned} &\frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}} \biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \beta \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)- \frac{2(p+2)(1-\varepsilon )}{\beta }E(t) \biggr). \end{aligned}$$
(3.26)
It is easy to see that
$$ \biggl(\frac{1}{\varepsilon _{1}}\biggr)^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }\rightarrow + \infty ,\quad \varepsilon _{1} \rightarrow 0^{+}, $$
and \(\frac{2(p+2)(1-\varepsilon )}{\beta }\) is a positive constant, hence there exists a constant \(\varepsilon _{1}^{*}\) such that
$$ \biggl(\frac{1}{\varepsilon _{1}}\biggr)^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }\geq \frac{2(p+2)(1-\varepsilon )}{\beta }. $$
Therefore, we have
$$\begin{aligned} & \frac{d}{dt} \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+ \biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr) -\biggl(\frac{1}{\varepsilon _{1}^{*}}\biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma }E(t) \biggr) \\ &\quad \geq \beta \biggl(\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)-\biggl(\frac{1}{\varepsilon _{1}^{*}}\biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma } E(t) \biggr). \end{aligned}$$
(3.27)
Take
$$ H(t)=\biggl(u,\frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v, \frac{v_{t} \vert v_{t} \vert ^{\rho }}{\rho +1}\biggr)- \biggl(\frac{1}{\varepsilon _{1}^{*}}\biggr)^{ \frac{\gamma ^{*}}{\gamma ^{*}-1}}\frac{\gamma -1}{\gamma } E(t), $$
from (3.2), we know
$$ H(0)=\biggl(u(0),\frac{u_{t}(0) \vert u_{t}(0) \vert ^{\rho }}{\rho +1}\biggr)+\biggl(v(0), \frac{v_{t}(0) \vert v_{t}(0) \vert ^{\rho }}{\rho +1} \biggr)-\biggl( \frac{1}{\varepsilon _{1}^{*}}\biggr)^{\frac{\gamma ^{*}}{\gamma ^{*}-1}} \frac{\gamma -1}{\gamma } E(0)>0. $$
By calculating \(H'(t)\geq \beta H(t)\), we can get
$$ H(t)\geq e^{\beta t}H(0), \quad \forall t\geq 0. $$
(3.28)
Since \((u,v)\) shows global existence, by Lemma 2.2 and Lemma 2.6, we have \(0< E(t)\leq E(0)\), \(t\in [0,+\infty )\), then
$$\begin{aligned} \Vert u \Vert ^{\rho +2}_{\rho +2}+ \Vert u_{t} \Vert ^{\rho +2}_{\rho +2}+ \Vert v \Vert ^{\rho +2}_{ \rho +2}+ \Vert v_{t} \Vert ^{\rho +2}_{\rho +2} \geq& \biggl(u, \frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr)+\biggl(u, \frac{u_{t} \vert u_{t} \vert ^{\rho }}{\rho +1}\biggr) \\ \geq& e^{\beta t}H(0). \end{aligned}$$
(3.29)
Using the Hölder inequality, Lemma 2.2 and Lemma 2.6, we have
$$\begin{aligned} & \bigl\Vert u(t) \bigr\Vert _{\rho +2}+ \Vert v \Vert _{\rho +2} \\ &\quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{ \rho +2}+ \int ^{t}_{0} \bigl\Vert u_{t}(\tau ) \bigr\Vert _{\rho +2}\,d\tau + \int ^{t}_{0} \bigl\Vert v_{t}( \tau ) \bigr\Vert _{\rho +2}\,d\tau \\ &\quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{\rho +2}+C_{1} \int ^{t}_{0} \bigl\Vert u_{t}( \tau ) \bigr\Vert _{m}\,d\tau +C_{2} \int ^{t}_{0} \bigl\Vert v_{t}(\tau ) \bigr\Vert _{r}\,d\tau \\ & \quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{\rho +2}+C_{1}t^{ \frac{m-1}{m}}\biggl( \int ^{t}_{0} \bigl\Vert u_{t}(\tau ) \bigr\Vert _{m}\,d\tau \biggr)^{\frac{1}{m}} \\ &\qquad {}+C_{2}t^{ \frac{r-1}{r}} \biggl( \int ^{t}_{0} \bigl\Vert v_{t}(\tau ) \bigr\Vert _{r}\,d\tau \biggr)^{\frac{1}{r}} \\ &\quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{\rho +2}+C_{1}t^{ \frac{m-1}{m}} \bigl(E(0)-E(t) \bigr)^{\frac{1}{m}}+C_{2}t^{\frac{r-1}{r}} \bigl(E(0)-E(t) \bigr)^{\frac{1}{r}} \\ &\quad \leq \bigl\Vert u(0) \bigr\Vert _{\rho +2}+ \bigl\Vert v(0) \bigr\Vert _{\rho +2}+C_{1}t^{ \frac{m-1}{m}} \bigl(E(0) \bigr)^{\frac{1}{m}}+C_{2}t^{\frac{r-1}{r}} \bigl(E(0) \bigr)^{\frac{1}{r}}, \end{aligned}$$
(3.30)
where \(C_{1}\) and \(C_{2}\) are positive constants. By combining (3.29) and (3.30), we know \(\|u\|^{\rho +2}_{\rho +2}+\|v\|^{\rho +2}_{\rho +2}\) shows polynomial growth and \(\|u_{t}\|^{\rho +2}_{\rho +2}+\|v_{t}\|^{\rho +2}_{\rho +2}\) shows exponential growth. By (2.5) and \(E(t)\) being nonnegative, we can deduce
$$\begin{aligned} \int ^{t}_{0} \bigl\Vert u_{t}(\tau ) \bigr\Vert ^{m}_{m}\,d\tau + \int ^{t}_{0} \bigl\Vert v_{t}( \tau ) \bigr\Vert ^{r}_{r}\,d\tau \leq E(0), \end{aligned}$$
(3.31)
and thanks to Lemma 2.5 and the assumption \(2<\rho +2<\min \{m,r\}\), we have
$$\begin{aligned} & \bigl\Vert u_{t}(\tau ) \bigr\Vert ^{\rho +2}_{m}< \bigl\Vert u_{t}(\tau ) \bigr\Vert ^{m}_{m}+1, \\ &\bigl\Vert v_{t}(\tau ) \bigr\Vert ^{\rho +2}_{r}< \bigl\Vert v_{t}(\tau ) \bigr\Vert ^{r}_{r}+1. \end{aligned}$$
(3.32)
By using the Sobolev embedding theorem and combining (3.31) and (3.32), we can get
$$\begin{aligned} &\int ^{t}_{0} \bigl\Vert u_{t}(\tau ) \bigr\Vert ^{\rho +2}_{\rho +2}\,d\tau + \int ^{t}_{0} \bigl\Vert v_{t}( \tau ) \bigr\Vert ^{\rho +2}_{\rho +2}\,d\tau \\ &\quad \leq C \biggl( \int ^{t}_{0} \bigl\Vert u_{t}( \tau ) \bigr\Vert ^{\rho +2}_{m}\,d\tau + \int ^{t}_{0} \bigl\Vert v_{t}(\tau ) \bigr\Vert ^{\rho +2}_{r}\,d\tau \biggr) \\ &\quad \leq C \biggl( \int ^{t}_{0}\bigl( \bigl\Vert u_{t}( \tau ) \bigr\Vert ^{m}_{m}+1\bigr)\,d\tau + \int ^{t}_{0}\bigl( \bigl\Vert v_{t}( \tau ) \bigr\Vert ^{r}_{r}+1\bigr)\,d\tau \biggr) \\ &\quad \leq CE(0)+2Ct, \end{aligned}$$
which contradicts with \(\|u_{t}\|^{\rho +2}_{\rho +2}+\|v_{t}\|^{\rho +2}_{\rho +2}\) showing exponential growth. Hence the theorem is proved. □