In this section, we prove the blow-up result of solution of problem (1.1).
First, we define the functional
$$\begin{aligned} \mathbb{H}(t)=-E(t) =&-\frac{1}{\eta +2} \bigl[ \Vert u_{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2} \bigr]- \frac{1}{2} \bigl[ \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} \bigr] \\ &{}-\frac{1}{2(\gamma +1)} \bigl[ \Vert \nabla u \Vert _{2}^{2(\gamma +1)}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)} \bigr] \\ &{}-\frac{1}{2} \biggl[ \biggl(1- \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \biggl(1- \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert _{2}^{2} \biggr] \\ &{}-\frac{1}{2} \bigl[(h_{1}o\nabla u) (t)+(h_{2}o\nabla v) (t) \bigr] \\ &{}+\frac{1}{2(p+2)}\bigl[ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2}\bigr]. \end{aligned}$$
(3.1)
Theorem 3.1
Assume that (2.1)–(2.2) and (2.4) hold, and suppose that \(E(0)<0\) and
$$ 2(p+2)>\max \bigl\{ k+j+1;l+j+1;\theta +s+1;\varrho +s+1;2(\gamma +1) \bigr\} . $$
(3.2)
Then the solution of problem (1.1) blows up in finite time.
Proof
From (2.5), we have
$$ E(t)\leq E(0)\leq 0. $$
(3.3)
Therefore
$$\begin{aligned} \mathbb{H}'(t)=-E'(t) \geq & \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j+1}\,dx+ \int _{\Omega }\bigl( \vert v \vert ^{ \theta }+ \vert u \vert ^{\varrho }\bigr) \vert v_{t} \vert ^{s+1}\,dx. \end{aligned}$$
(3.4)
Hence
$$\begin{aligned}& \begin{gathered} \mathbb{H}'(t)\geq \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j+1}\,dx\geq 0, \\ \mathbb{H}'(t)\geq \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v_{t} \vert ^{s+1}\,dx\geq 0 . \end{gathered} \end{aligned}$$
(3.5)
By (3.1) and (2.3), we have
$$\begin{aligned} 0 \leq& \mathbb{H}(0)\leq \mathbb{H}(t)\leq \frac{1}{2(p+2)} \bigl( \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2} \bigr) \\ \leq & \frac{c_{1}}{2(p+2)} \bigl( \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)} \bigr) . \end{aligned}$$
(3.6)
We set
$$\begin{aligned} \mathcal{K}(t) =&\mathbb{H}^{1-\alpha }+\frac{\varepsilon }{\eta +1} \int _{\Omega } \bigl[u \vert u_{t} \vert ^{\eta }u_{t}+v \vert v_{t} \vert ^{ \eta }v_{t} \bigr]\,dx \\ &{}+\varepsilon \int _{\Omega } [\nabla u_{t}\nabla u+\nabla v_{t} \nabla v ]\,dx , \end{aligned}$$
(3.7)
where \(\varepsilon >0\) is to be assigned later, and we assume that
$$\begin{aligned} 0< \alpha < &\min \biggl\{ \biggl(1-\frac{1}{2(p+2)}- \frac{1}{\eta +2} \biggr),\frac{1+2\gamma }{4(\gamma +1)}, \frac{2p+3-(k+j)}{2j(p+2)}, \\ & \frac{2p+3-(l+j)}{2j(p+2)},\frac{2p+3-(\theta +s)}{2s(p+2)}, \frac{2p+3-(\varrho +s)}{2s(p+2)} \biggr\} < 1. \end{aligned}$$
(3.8)
By multiplying (1.1)1, (1.1)2 by u, v and with a derivative of (3.7), we get
$$\begin{aligned} \mathcal{K}'(t) = &(1-\alpha )\mathbb{H}^{-\alpha } \mathbb{H}'(t)+ \frac{\varepsilon }{\eta +1}\bigl( \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2}\bigr) \\ &{}+ \underbrace{\varepsilon \int _{\Omega }\nabla u \int ^{t}_{0}g(t-s) \nabla u(s) \,dsdx}_{J_{1}} + \underbrace{\varepsilon \int _{\Omega }\nabla v \int ^{t}_{0}h(t-s) \nabla v(s) \,dsdx}_{J_{2}} \\ &{}- \underbrace{\varepsilon \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j-1}u_{t}.udx}_{J_{3}}- \underbrace{\varepsilon \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v_{t} \vert ^{s-1}v_{t}.vdx}_{J_{4}} \\ &{}-\varepsilon \bigl( \Vert \nabla u \Vert _{2}^{2}+ \Vert \nabla v \Vert _{2}^{2}\bigr)- \varepsilon \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+ \underbrace{\varepsilon \bigl[ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2} \bigr]}_{J_{5}}. \end{aligned}$$
(3.9)
We have
$$\begin{aligned} J_{1} =&\varepsilon \int _{0}^{t}h_{1}(t-s)\,ds \int _{\Omega }\nabla u.\bigl( \nabla u(s)-\nabla u(t)\bigr) \,dx\,ds+\varepsilon \int _{0}^{t}h_{1}(s)\,ds \Vert \nabla u \Vert _{2}^{2} \\ \geq & \frac{\varepsilon }{2} \int _{0}^{t}h_{1}(s)\,ds \Vert \nabla u \Vert _{2}^{2}-\frac{\varepsilon }{2}(h_{1}o \nabla u). \end{aligned}$$
(3.10)
$$\begin{aligned} J_{2} =&\varepsilon \int _{0}^{t}h_{2}(t-s)\,ds \int _{\Omega }\nabla v.\bigl( \nabla v(s)-\nabla v(t)\bigr) \,dx\,ds+\varepsilon \int _{0}^{t}h_{2}(s)\,ds \Vert \nabla v \Vert _{2}^{2} \\ \geq & \frac{\varepsilon }{2} \int _{0}^{t}h_{2}(s)\,ds \Vert \nabla v \Vert _{2}^{2}-\frac{\varepsilon }{2}(h_{2}o \nabla v). \end{aligned}$$
(3.11)
From (3.9), we find
$$\begin{aligned} \mathcal{K}'(t) \geq &(1-\alpha )\mathbb{H}^{-\alpha } \mathbb{H}'(t)+ \frac{\varepsilon }{\eta +1}\bigl( \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}\bigr)+\varepsilon \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2}\bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \biggl(1- \frac{1}{2} \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert _{2}^{2} \biggr] \\ &{}-\frac{\varepsilon }{2}(h_{1}o\nabla u)-\frac{\varepsilon }{2}(h_{2}o \nabla v)-\varepsilon \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}-J_{3}-J_{4}+J_{5} . \end{aligned}$$
(3.12)
At this point, we use Young’s inequality for \(\delta >0\)
$$ XY\leq \frac{\delta ^{\alpha }X^{\alpha }}{\alpha }+ \frac{\delta ^{-\beta }X^{\beta }}{\beta }, \quad \alpha ,\beta >0, \frac{1}{\alpha }+\frac{1}{\beta }=1, $$
(3.13)
we get, for \(\delta _{1},\delta _{2}>0\),
$$\begin{aligned} \bigl\vert u \vert u_{t} \vert ^{j-1}u_{t} \bigr\vert \leq &\frac{\delta _{1}^{j+1}}{j+1} \vert u \vert ^{j+1}+ \frac{j}{j+1}\delta _{1}^{-(\frac{j+1}{j})} \vert u_{t} \vert ^{j+1}, \\ \bigl\vert v \vert v_{t} \vert ^{s-1}v_{t} \bigr\vert \leq &\frac{\delta _{2}^{s+1}}{s+1} \vert v \vert ^{s+1}+ \frac{s}{s+1}\delta _{2}^{-(\frac{s+1}{s})} \vert v_{t} \vert ^{s+1}. \end{aligned}$$
(3.14)
Hence, we have
$$ \begin{gathered} J_{3}\leq \varepsilon \frac{\delta _{1}^{j+1}}{j+1} \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx+ \varepsilon \frac{j\delta _{1}^{-(\frac{j+1}{j})}}{j+1} \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u_{t} \vert ^{j+1}\,dx, \\ J_{4}\leq \varepsilon \frac{\delta _{2}^{s+1}}{s+1} \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx+ \varepsilon \frac{s\delta _{2}^{-(\frac{s+1}{s})}}{s+1} \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v_{t} \vert ^{s+1}\,dx. \end{gathered} $$
(3.15)
Therefore, using (3.5) and by setting \(\delta _{1}\), \(\delta _{1}\) so that
$$ \frac{j\delta _{1}^{-(\frac{j+1}{j})}}{j+1}= \frac{\kappa \mathbb{H}^{-\alpha }(t)}{2}, \qquad \frac{s\delta _{2}^{-(\frac{s+1}{s})}}{s+1}= \frac{\kappa \mathbb{H}^{-\alpha }(t)}{2}, $$
substituting in (3.12), we get
$$\begin{aligned} \mathcal{K}'(t) \geq &\bigl[(1-\alpha )-\varepsilon \kappa \bigr] \mathbb{H}^{- \alpha }\mathbb{H}'(t)+\frac{\varepsilon }{\eta +1} \bigl( \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2} \bigr) \\ &{}-\varepsilon \biggl[ \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \biggl(1- \frac{1}{2} \int _{0}^{t}h_{2}(s)\,ds \biggr) \Vert \nabla v \Vert _{2}^{2} \biggr] \\ &{}+\varepsilon \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} \bigr)-\frac{\varepsilon }{2}(h_{1}o \nabla u)- \frac{\varepsilon }{2}(h_{2}o\nabla v) \\ &{}-\varepsilon C_{1}(\kappa )\mathbb{H}^{\alpha j}(t) \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx \\ &{}-\varepsilon C_{2}(\kappa )\mathbb{H}^{\alpha s}(t) \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx \\ &{}-\varepsilon \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr)+J_{5} , \end{aligned}$$
(3.16)
where
$$ C_{1}(\kappa ):= \biggl(\frac{2j}{\kappa (j+1)} \biggr)^{j+1} \frac{1}{j+1}, \qquad C_{2}(\kappa ):= \biggl( \frac{2s}{\kappa (s+1)} \biggr)^{s+1} \frac{1}{s+1}, $$
(3.17)
we have
$$\begin{aligned} \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx =& \Vert u \Vert ^{k+j+1}_{k+j+1}+ \int _{\Omega } \vert v \vert ^{l} \vert u \vert ^{j+1}\,dx, \\ \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx =& \Vert v \Vert ^{\theta +s+1}_{\theta +s+1}+ \int _{ \Omega } \vert u \vert ^{\varrho } \vert v \vert ^{s+1}\,dx. \end{aligned}$$
(3.18)
By Young’s inequality, we find for \(\delta _{3},\delta _{4}>0\)
$$ \begin{gathered} \int _{\Omega } \vert v \vert ^{l} \vert u \vert ^{j+1}\,dx\leq \frac{l}{l+j+1} \delta _{3}^{(\frac{l+j+1}{l})} \Vert v \Vert ^{l+j+1}_{l+j+1}+ \frac{j+1}{l+j+1}\delta _{3}^{-(\frac{l+j+1}{l})} \Vert u \Vert ^{l+j+1}_{l+j+1}, \\ \int _{\Omega } \vert u \vert ^{\varrho } \vert v \vert ^{s+1}\,dx\leq \frac{\varrho }{\varrho +s+1} \delta _{4}^{(\frac{\varrho +s+1}{\varrho })} \Vert u \Vert ^{\varrho +s+1}_{ \varrho +s+1}+ \frac{s+1}{\varrho +s+1}\delta _{4}^{-( \frac{\varrho +s+1}{\varrho })} \Vert v \Vert ^{\varrho +s+1}_{\varrho +s+1}. \end{gathered} $$
(3.19)
Hence
$$\begin{aligned} \mathbb{H}^{\alpha j}(t) \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx \leq & \mathbb{H}^{\alpha j}(t) \Vert u \Vert ^{k+j+1}_{k+j+1}+ \frac{l\mathbb{H}^{\alpha j}(t)}{l+j+1}\delta _{3}^{( \frac{l+j+1}{l})} \Vert v \Vert ^{l+j+1}_{l+j+1} \\ &{}+\frac{(j+1)\mathbb{H}^{\alpha j}(t)}{l+j+1}\delta _{3}^{-( \frac{l+j+1}{l})} \Vert u \Vert ^{l+j+1}_{l+j+1}, \\ \mathbb{H}^{\alpha s}(t) \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx \leq & \mathbb{H}^{\alpha s}(t) \Vert v \Vert ^{\theta +s+1}_{\theta +s+1}+ \frac{\varrho \mathbb{H}^{\alpha s}(t)}{\varrho +s+1}\delta _{4}^{( \frac{\varrho +s+1}{\varrho })} \Vert u \Vert ^{\varrho +s+1}_{\varrho +s+1} \\ &{}+\frac{(s+1)\mathbb{H}^{\alpha s}(t)}{\varrho +s+1}\delta _{4}^{-( \frac{\varrho +s+1}{\varrho })} \Vert v \Vert ^{\varrho +s+1}_{\varrho +s+1}. \end{aligned}$$
(3.20)
Since (2.4) holds, we obtain by using (3.6) and (3.8)
$$\begin{aligned}& \mathbb{H}^{\alpha j}(t) \Vert u \Vert _{k+j+1}^{k+j+1} \leq c_{1}\bigl( \Vert u \Vert ^{2\alpha j(p+2)+k+j+1}_{2(p+2)}+ \Vert v \Vert ^{2\alpha j(p+2)}_{2(p+2)} \Vert u \Vert _{k+j+1}^{k+j+1}\bigr), \\& \mathbb{H}^{\alpha j}(t) \Vert v \Vert _{k+j+1}^{k+j+1} \leq c_{2}\bigl( \Vert v \Vert ^{2\alpha j(p+2)+k+j+1}_{2(p+2)}+ \Vert u \Vert ^{2\alpha j(p+2)}_{2(p+2)} \Vert v \Vert _{k+j+1}^{k+j+1}\bigr), \\& \mathbb{H}^{\alpha s}(t) \Vert v \Vert _{\theta +s+1}^{\theta +s+1} \leq c_{3}\bigl( \Vert v \Vert ^{2\alpha s(p+2)+\theta +s+1}_{2(p+2)}+ \Vert u \Vert ^{2\alpha s(p+2)}_{2(p+2)} \Vert v \Vert _{\theta +s+1}^{ \theta +s+1}\bigr), \\& \mathbb{H}^{\alpha s}(t) \Vert u \Vert _{\theta +s+1}^{\theta +s+1} \leq c_{4}\bigl( \Vert u \Vert ^{2\alpha s(p+2)+\theta +s+1}_{2(p+2)}+ \Vert v \Vert ^{2\alpha s(p+2)}_{2(p+2)} \Vert u \Vert _{\theta +s+1}^{ \theta +s+1}\bigr) \end{aligned}$$
(3.21)
for some positive constants \(c_{i},i=1,\ldots,4\). By using (3.8) and the algebraic inequality
$$ B^{\varsigma }\leq (B+1)\leq \biggl(1+\frac{1}{b}\biggr) (B+b), \quad \forall B>0, 0< \varsigma < 1, b>0, $$
(3.22)
we have \(\forall t>0\)
$$\begin{aligned}& \begin{gathered} \Vert u \Vert ^{2\alpha j(p+2)+k+j+1}_{2(p+2)}\leq d\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(0)\bigr)\leq d\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(t)\bigr) , \\ \Vert v \Vert ^{2\alpha j(p+2)+k+j+1}_{2(p+2)}\leq d\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(0)\bigr)\leq d\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(t)\bigr), \\ \Vert v \Vert ^{2\alpha s(p+2)+\theta +s+1}_{2(p+2)}\leq d\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(0)\bigr)\leq d\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \\ \Vert u \Vert ^{2\alpha s(p+2)+\theta +s+1}_{2(p+2)}\leq d\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(0)\bigr)\leq d\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \end{gathered} \end{aligned}$$
(3.23)
where \(d=1+\frac{1}{\mathbb{H}(0)}\). Also, since
$$ (X+Y)^{\gamma } \leq C\bigl(X^{\gamma }+Y^{\gamma }\bigr), \quad X,Y>0, \gamma >0, $$
(3.24)
we conclude
$$\begin{aligned} \begin{gathered} \Vert v \Vert ^{2\alpha j(p+2)}_{2(p+2)} \Vert u \Vert _{k+j+1}^{k+j+1} \leq c_{5}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr), \\ \Vert u \Vert ^{2\alpha j(p+2)}_{2(p+2)} \Vert v \Vert _{k+j+1}^{k+j+1} \leq c_{6}\bigl( \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \Vert v \Vert ^{2(p+2)}_{2(p+2)} \bigr), \\ \Vert u \Vert ^{2\alpha s(p+2)}_{2(p+2)} \Vert v \Vert _{\theta +s+1}^{ \theta +s+1}\leq c_{7}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr), \\ \Vert v \Vert ^{2\alpha s(p+2)}_{2(p+2)} \Vert u \Vert _{\theta +s+1}^{ \theta +s+1}\leq c_{8}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)} \bigr). \end{gathered} \end{aligned}$$
(3.25)
Substituting (3.23) and (3.25) in (3.21), we get
$$\begin{aligned} \mathbb{H}^{\alpha j}(t) \Vert u \Vert _{k+j+1}^{k+j+1} \leq &c_{9}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \\ \mathbb{H}^{\alpha j}(t) \Vert v \Vert _{k+j+1}^{k+j+1} \leq &c_{10}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \\ \mathbb{H}^{\alpha s}(t) \Vert v \Vert _{\theta +s+1}^{\theta +s+1} \leq &c_{11}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr), \\ \mathbb{H}^{\alpha s}(t) \Vert u \Vert _{\theta +s+1}^{\theta +s+1} \leq &c_{12}\bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t)\bigr). \end{aligned}$$
(3.26)
Hence, by fixed \(\delta _{3},\delta _{4}>0\), we get
$$\begin{aligned}& \begin{gathered} \mathbb{H}^{\alpha j}(t) \int _{\Omega }\bigl( \vert u \vert ^{k}+ \vert v \vert ^{l}\bigr) \vert u \vert ^{j+1}\,dx \\ \quad \leq M_{1} \biggl(1+\frac{l\delta _{3}^{(\frac{l+j+1}{l})}}{l+j+1}+ \frac{(j+1)\delta _{3}^{-(\frac{l+j+1}{l})}}{l+j+1} \biggr) \bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+\mathbb{H}(t) \bigr), \\ \mathbb{H}^{\alpha s}(t) \int _{\Omega }\bigl( \vert v \vert ^{\theta }+ \vert u \vert ^{\varrho }\bigr) \vert v \vert ^{s+1}\,dx \\ \quad \leq M_{2} \biggl(1+ \frac{\varrho \delta _{4}^{(\frac{\varrho +s+1}{\varrho })}}{\varrho +s+1}+ \frac{(s+1)\delta _{4}^{-(\frac{\varrho +s+1}{\varrho })}}{\varrho +s+1} \biggr) \bigl( \Vert v \Vert ^{2(p+2)}_{2(p+2)}+ \Vert u \Vert ^{2(p+2)}_{2(p+2)}+ \mathbb{H}(t) \bigr) \end{gathered} \end{aligned}$$
(3.27)
for some constants \(M_{1},M_{2}>0\).
Now, for \(0< a<1\), from (3.1)
$$\begin{aligned} J_{5} =&\varepsilon \bigl[ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2}\bigr] \\ =&\varepsilon a \bigl[ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2} \bigr] \\ &{}+\frac{2\varepsilon (p+2)(1-a)}{\eta +2}\bigl( \Vert u_{t} \Vert _{\eta +2}^{ \eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}\bigr) \\ &{}+\varepsilon (p+2) (1-a) \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2}\bigr) \\ &{}+\varepsilon (p+2) (1-a) \biggl(1- \int _{0}^{t}g(s)\,ds\biggr) \Vert \nabla u \Vert _{2}^{2} \\ &{}+\varepsilon (p+2) (1-a) \biggl(1- \int _{0}^{t}h(s)\,ds\biggr) \Vert \nabla v \Vert _{2}^{2} \\ &{}+\varepsilon (p+2) (1-a) \bigl((h_{1}o\nabla u)+(h_{2}o \nabla v)\bigr) \\ &{}+\frac{\varepsilon (p+2)(1-a)}{\gamma +1}\bigl( \Vert \nabla u \Vert ^{2( \gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+\varepsilon 2(p+2) (1-a)\mathbb{H}(t). \end{aligned}$$
(3.28)
Substituting in (3.16), and by using (2.3), we get
$$\begin{aligned} \mathcal{K}'(t) \geq & \bigl\{ (1-\alpha )-\varepsilon \kappa \bigr\} \mathbb{H}^{-\alpha }\mathbb{H}'(t)+\varepsilon \bigl\{ (p+2) (1-a)+1 \bigr\} \bigl( \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2}\bigr) \\ &{}+\varepsilon \biggl\{ \frac{2\varepsilon (p+2)(1-a)}{\eta +2}+ \frac{1}{\eta +1} \biggr\} \bigl( \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}\bigr) \\ &{}+\varepsilon \biggl\{ (p+2) (1-a) \biggl(1- \int _{0}^{t}h_{1}(s)\,ds \biggr)- \biggl(1-\frac{1}{2} \int _{0}^{t}h_{1}(s)\,ds \biggr) \biggr\} \Vert \nabla u \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ (p+2) (1-a) \biggl(1- \int _{0}^{t}h_{2}(s)\,ds \biggr)- \biggl(1-\frac{1}{2} \int _{0}^{t}h_{2}(s)\,ds \biggr) \biggr\} \Vert \nabla v \Vert _{2}^{2} \\ &{}+\varepsilon \biggl\{ (p+2) (1-a)-\frac{1}{2} \biggr\} (h_{1}o\nabla u+h_{2}o \nabla v) \\ &{}+\varepsilon \biggl\{ \frac{(p+2)(1-a)}{\gamma +1}-1 \biggr\} \bigl( \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}\bigr) \\ &{}+\varepsilon \bigl\{ c_{0}a- \bigl(M_{3}C_{1}( \kappa )+M_{4}C_{2}( \kappa ) \bigr) \bigr\} \bigl( \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)} \bigr) \\ &{}+\varepsilon \bigl\{ ( 2(p+2) (1-a)- \bigl(M_{3}C_{1}( \kappa )+M_{4}C_{2}( \kappa ) \bigr) \bigr\} \mathbb{H}(t), \end{aligned}$$
(3.29)
where
$$\begin{aligned}& M_{3}:=M_{1} \biggl(1+\frac{l\delta _{3}^{(\frac{l+j+1}{l})}}{l+j+1}+ \frac{(j+1)\delta _{3}^{-(\frac{l+j+1}{l})}}{l+j+1} \biggr)>0 . \\& M_{4}:=M_{2} \biggl(1+ \frac{\varrho \delta _{4}^{(\frac{\varrho +s+1}{\varrho })}}{\varrho +s+1}+ \frac{(s+1)\delta _{4}^{-(\frac{\varrho +s+1}{\varrho })}}{\varrho +s+1} \biggr)>0. \end{aligned}$$
At this stage, we take \(a>0\) small enough so that
$$ (p+2) (1-a)>1+\gamma , $$
we have
$$\begin{aligned}& \lambda _{1}:=(p+2) (1-a)-1>0 \\& \lambda _{2}:=(p+2) (1-a)-\frac{1}{2}>0 \\& \lambda _{3}:=\frac{(p+2)(1-a)}{\gamma +1}-1>0 \end{aligned}$$
and we assume that
$$ \max \biggl\{ \int _{0}^{\infty }h_{1}(s)\,ds, \int _{0}^{\infty }h_{2}(s)\,ds \biggr\} < \frac{(p+2)(1-a)-1}{((p+2)(1-a)-\frac{1}{2})}= \frac{2\lambda _{1}}{2\lambda _{1}+1} $$
(3.30)
gives
$$\begin{aligned} \lambda _{4} =& \biggl\{ \bigl((p+2) (1-a)-1 \bigr)- \int _{0}^{t}h_{1}(s)\,ds \biggl((p+2) (1-a)-\frac{1}{2} \biggr) \biggr\} >0, \\ \lambda _{5} =& \biggl\{ \bigl((p+2) (1-a)-1 \bigr)- \int _{0}^{t}h_{2}(s)\,ds \biggl((p+2) (1-a)-\frac{1}{2} \biggr) \biggr\} >0, \end{aligned}$$
then we choose κ so large that
$$\begin{aligned} \lambda _{6} =&ac_{0}- \bigl(M_{3}C_{1}( \kappa )+M_{4}C_{2}(\kappa ) \bigr)>0, \\ \lambda _{7} =&2(p+2) (1-a)- \bigl(M_{3}C_{1}( \kappa )+M_{4}C_{2}( \kappa ) \bigr)>0. \end{aligned}$$
Finally, we fix κ, a, and we appoint ε small enough so that
$$ \lambda _{8}=(1-\alpha )-\varepsilon \kappa >0. $$
Thus, for some \(\beta >0\), estimate (3.29) becomes
$$\begin{aligned} \mathcal{K}'(t) \geq &\beta \bigl\{ \mathbb{H}(t)+ \Vert u_{t} \Vert _{ \eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} + \Vert \nabla u \Vert _{2}^{2}+ \Vert \nabla v \Vert _{2}^{2} \\ &{} + \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2( \gamma +1)} +(h_{1}o \nabla u)+(h_{2}o\nabla v)+ \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert u \Vert _{2(p+2)}^{2(p+2)} \bigr\} . \end{aligned}$$
(3.31)
By (2.3), for some \(\beta _{1}>0\), we obtain
$$\begin{aligned} \mathcal{K}'(t) \geq &\beta _{1} \bigl\{ \mathbb{H}(t)+ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} + \Vert \nabla u \Vert _{2}^{2} \\ &{}+ \Vert \nabla v \Vert _{2}^{2}+ \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert _{2}^{2(\gamma +1)}+(h_{1}o\nabla u)+(h_{2}o \nabla v) \\ &{}+ \Vert u+v \Vert _{2(p+2)}^{2(p+2)}+2 \Vert uv \Vert _{p+2}^{p+2} \bigr\} \end{aligned}$$
(3.32)
and
$$ \mathcal{K}(t)\geq \mathcal{K}(0)>0, \quad t>0. $$
(3.33)
Next, using Holder’s and Young’s inequalities, we have
$$\begin{aligned} \biggl\vert \int _{\Omega }\bigl(u \vert u_{t} \vert ^{\eta } u_{t}+v \vert v_{t} \vert ^{\eta }v_{t}\bigr)\,dx \biggr\vert ^{\frac{1}{1-\alpha }} \leq &C \bigl[ \Vert u \Vert _{2(p+2)}^{\frac{\theta }{1-\alpha }}+ \Vert u_{t} \Vert _{ \eta +2}^{\frac{\mu }{1-\alpha }} \\ & + \Vert v \Vert _{2(p+2)}^{\frac{\theta }{1-\alpha }}+ \Vert v_{t} \Vert _{ \eta +2}^{\frac{\mu }{1-\alpha }} \bigr] , \end{aligned}$$
(3.34)
where \(\frac{1}{\mu }+\frac{1}{\theta }=1\).
We take \(\mu =(\eta +2)(1-\alpha )\) to get
$$ \frac{\theta }{1-\alpha }=\frac{\eta +2}{(1-\alpha )(\eta +2)-1}\leq 2(p+2). $$
Subsequently, by using (3.8), (3.6), and (3.22), we obtain
$$\begin{aligned} \Vert u \Vert _{2(p+2)}^{\frac{\eta +2}{(1-\alpha )(\eta +2)-1}} \leq &d\bigl( \Vert u \Vert _{2(p+2)}^{2(p+2)}+\mathbb{H}(t)\bigr) , \\ \Vert v \Vert _{2(p+2)}^{\frac{\eta +2}{(1-\alpha )(\eta +2)-1}} \leq &d\bigl( \Vert v \Vert _{2(p+2)}^{2(p+2)}+\mathbb{H}(t)\bigr), \quad \forall t \geq 0. \end{aligned}$$
Therefore,
$$\begin{aligned}& \biggl\vert \int _{\Omega }\bigl(u \vert u_{t} \vert ^{\eta } u_{t}+v \vert v_{t} \vert ^{\eta }v_{t}\bigr)\,dx \biggr\vert ^{\frac{1}{1-\alpha }} \\& \quad \leq c_{13} \bigl\{ \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)}+ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{ \eta +2}+\mathbb{H}(t) \bigr\} . \end{aligned}$$
(3.35)
Similarly, we have
$$\begin{aligned} \biggl\vert \int _{\Omega }(\nabla u\nabla u_{t}+\nabla v \nabla v_{t})\,dx \biggr\vert ^{\frac{1}{1-\alpha }} \leq &C \bigl[ \Vert \nabla u \Vert _{2}^{ \frac{\theta }{1-\alpha }}+ \Vert \nabla u_{t} \Vert _{2}^{ \frac{\mu }{1-\alpha }} + \Vert \nabla v \Vert _{2}^{\frac{\theta }{1-\alpha }}+ \Vert \nabla v_{t} \Vert _{2}^{\frac{\mu }{1-\alpha }} \bigr] , \end{aligned}$$
where \(\frac{1}{\mu }+\frac{1}{\theta }=1\).
We take \(\theta =2(\gamma +1)(1-\alpha )\) to get
$$\begin{aligned}& \frac{\mu }{1-\alpha }=\frac{2(\gamma +1)}{2(1-\alpha )(\gamma +1)-1} \leq 2, \\& \begin{aligned}[b] \biggl\vert \int _{\Omega }(\nabla u\nabla u_{t}+\nabla v \nabla v_{t})\,dx \biggr\vert ^{\frac{1}{1-\alpha }}\leq{} &c_{14} \bigl\{ \Vert \nabla u \Vert ^{2(\gamma +1)}_{2}+ \Vert \nabla v \Vert ^{2(\gamma +1)}_{2} \\ & {} + \Vert \nabla u_{t} \Vert _{2}^{2}+ \Vert \nabla v_{t} \Vert _{2}^{2} \bigr\} . \end{aligned} \end{aligned}$$
(3.36)
Hence, by (3.35) and (3.36),
$$\begin{aligned} \mathcal{K}^{\frac{1}{1-\alpha }}(t) =& \biggl(\mathbb{H}^{1-\alpha }+ \frac{\varepsilon }{\eta +1} \int _{\Omega }\bigl(u \vert u_{t} \vert ^{\eta }u_{t}+v \vert v_{t} \vert ^{\eta }v_{t}\bigr)\,dx \\ &{}+\varepsilon \int _{\Omega }(\nabla u_{t}\nabla u+\nabla v_{t} \nabla v)\,dx \biggr)^{\frac{1}{1-\alpha }} \\ \leq &c \biggl(\mathbb{H}(t)+ \biggl\vert \int _{\Omega }\bigl(u \vert u_{t} \vert ^{ \eta } u_{t}+v \vert v_{t} \vert ^{\eta }v_{t}\bigr) \,dx \biggr\vert ^{ \frac{1}{1-\alpha }}+ \Vert \nabla u \Vert _{2}^{\frac{2}{1-\alpha }}+ \Vert \nabla v \Vert _{2}^{\frac{2}{1-\alpha }} \\ & + \Vert \nabla u_{t} \Vert _{2}^{\frac{2}{1-\alpha }}+ \Vert \nabla v_{t} \Vert _{2}^{\frac{2}{1-\alpha }} \biggr) \\ \leq &c \bigl(\mathbb{H}(t)+ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla u \Vert ^{\gamma +1}_{2}+ \Vert \nabla v \Vert _{2}^{\gamma +1}+ \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}+ \Vert \nabla v_{t} \Vert _{2}^{2}+(h_{1}o \nabla u)+(h_{2}o\nabla v)+ \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)} \bigr) \\ \leq &c \bigl(\mathbb{H}(t)+ \Vert u_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert v_{t} \Vert _{\eta +2}^{\eta +2}+ \Vert \nabla u \Vert ^{\gamma +1}_{2}+ \Vert \nabla v \Vert _{2}^{\gamma +1}+ \Vert \nabla u_{t} \Vert _{2}^{2} \\ &{}+ \Vert \nabla v_{t} \Vert _{2}^{2}+ \Vert \nabla u \Vert _{2}^{2}+ \Vert \nabla v \Vert _{2}^{2}+(h_{1}o\nabla u)+(h_{2}o\nabla v) \\ &{}+ \Vert u \Vert _{2(p+2)}^{2(p+2)}+ \Vert v \Vert _{2(p+2)}^{2(p+2)} \bigr) . \end{aligned}$$
(3.37)
From (3.31) and (3.37), it gives
$$ \mathcal{K}'(t)\geq \lambda \mathcal{K}^{\frac{1}{1-\alpha }}(t), $$
(3.38)
where \(\lambda > 0 \), this depends only on β and c.
By integration of (3.38), we obtain
$$ \mathcal{K}^{\frac{\alpha }{1-\alpha }}(t)\geq \frac{1}{\mathcal{K}^{\frac{-\alpha }{1-\alpha }}(0)-\lambda \frac{\alpha }{(1-\alpha )} t}. $$
Hence, \(\mathcal{K}(t)\) blows up in time
$$ T\leq T^{*}= \frac{1-\alpha }{\lambda \alpha \mathcal{K}^{\alpha /(1-\alpha )}(0)}. $$
Then the proof is completed. □