In this section, we prove our main result for \(p>q\). For this purpose, we give defined restrictions on \(\sigma (x,t)\) and the relaxation function h such that
$$\begin{aligned} E \int_{0}^{\infty } \int_{D} \sigma^{2}(x,t)\,dx\,dt < \infty , \qquad \int_{0}^{\infty }h(s)\,ds < \frac{{p(p-2)}}{{(p-1)^{2}}}. \end{aligned}$$
(3.1)
Now, we define an energy function
$$\begin{aligned} F(t)&=\frac{{1}}{{\rho +2}}\bigl\Vert u_{t}(t) \bigr\Vert _{\rho +2}^{\rho +2} + \frac{{1}}{{2}}\bigl\Vert \nabla u_{t}(t) \bigr\Vert ^{2} \\ &\quad {} +\frac{{1}}{{2}}\biggl(1- \int_{0}^{t} h(s)\,ds\biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{{1}}{{2}}(h \circ \nabla u) (t) - \frac{{1}}{{p}}\bigl\Vert u(t) \bigr\Vert _{p}^{p}, \end{aligned}$$
(3.2)
where
$$\begin{aligned} (h \circ \nabla u) (t)= \int_{0}^{t} h(t-s)\bigl\Vert \nabla u(t)-\nabla u(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$
For each N, stopping time \(\tau_{N}\) is given as
$$ \tau_{N}=\inf \bigl\{ t>0: \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \geq N \bigr\} , $$
where \(\tau_{N}\) is increasing in N, and \(\tau_{\infty }= \lim_{N \to \infty } \tau_{N}\). In order to prove our blow-up result, we rewrite (1.1) as an equivalent Itô’s system
$$\begin{aligned}& \begin{aligned} &du=v\,dt, \\ &d\biggl(\frac{{1}}{{\rho +1}} \vert v \vert ^{\rho }v-\Delta v\biggr)= \biggl(\Delta u -\int_{0} ^{t} h(t-s) \Delta u(s)\,ds -\vert v \vert ^{q-2}v +\vert u \vert ^{p-2}u\biggr)\,dt \\ &\hphantom{\biggl(\frac{{1}}{{\rho +1}} \vert v \vert ^{\rho }v-\Delta v\biggr)=\quad }{}+\epsilon \sigma (x,t)\,dW_{t}(x,t), \quad (x,t)\in D \times (0,T), \\ & u(x,t)=0, \quad (x,t) \in \partial D \times (0,T), \\ & u(x,0)=u_{0}(x), \qquad v(x,0)=v_{0}(x)=u_{1}(x), \quad x\in D, \end{aligned} \end{aligned}$$
(3.3)
where \((u_{0},u_{1})\in H_{0}^{1}(D)\times L^{2}(D)\). Then the energy function \(F(t)\) becomes
$$\begin{aligned} F(t)&=\frac{{1}}{{\rho +2}}\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} + \frac{{1}}{{2}}\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {} +\frac{{1}}{{2}}\biggl(1- \int_{0}^{t} h(s)\,ds\biggr) \bigl\Vert \nabla u(t) \bigr\Vert ^{2} + \frac{{1}}{{2}}(h \circ \nabla u) (t) - \frac{{1}}{{p}}\bigl\Vert u(t) \bigr\Vert _{p}^{p}. \end{aligned}$$
(3.4)
Lemma 3.1
Let
\((u,v)\)
be a solution of Eq. (3.3) with the initial data
\((u_{0},v_{0})\in H_{0}^{1}(D)\times L^{2}(D)\). Then we have
$$\begin{aligned} \frac{{d}}{{dt}}E F(t)&= -E \bigl\Vert v(t) \bigr\Vert _{q}^{q} + \frac{{\epsilon ^{2}}}{{2}} \sum_{j=1}^{\infty } E\int_{D} \lambda _{j} e_{j}^{2}(x) \sigma^{2}(x,t)\,dx \\ & \quad {}-E\bigl(-h'\circ \nabla u\bigr) (t)-\frac{{1}}{{2}}h(t)E \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & \leq -E \bigl\Vert v(t) \bigr\Vert _{q}^{q} + \frac{{\epsilon ^{2}}}{{2}} \sum_{j=1} ^{\infty } E \int_{D} \lambda_{j} e_{j}^{2}(x) \sigma^{2}(x,t)\,dx, \end{aligned}$$
(3.5)
and
$$\begin{aligned} &E\biggl(u(t),\frac{{1}}{{\rho +1}}\bigl\vert v(t) \bigr\vert ^{\rho }v(t) - \Delta v(t)\biggr) \\ &\quad =\biggl(u_{0}, \frac{{1}}{{\rho +1}}\vert u_{1} \vert ^{\rho }u_{1}- \Delta u_{1}\biggr)- \int_{0}^{t} E\bigl\Vert \nabla u(s) \bigr\Vert ^{2}\,ds \\ & \quad \quad {} - \int_{0}^{t} E\bigl(u(s),\bigl\vert v(s) \bigr\vert ^{q-2}v(s)\bigr)\,ds + \int_{0}^{t} E \bigl\Vert u(s) \bigr\Vert _{p} ^{p}\,ds \\ &\quad \quad {} +E \int_{0}^{t} \int_{0}^{s} h(s-\tau ) \bigl(\nabla u(\tau ), \nabla u(s)\bigr)\,d\tau \,ds \\ &\quad \quad {} +\frac{{1}}{{\rho +1}}E \int_{0}^{t} \bigl\Vert v(s) \bigr\Vert _{\rho +2}^{\rho +2}\,ds +E \int_{0}^{t} \bigl\Vert \nabla v(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$
(3.6)
Proof
By multiplying Eq. (3.3) by \(v(t)\) and using Itô’s formula, we deduce (3.5). Also, multiplying Eq. (3.3) by \(u(t)\) and integrating by parts over \((0,T)\), we arrive at (3.6) (see [24]). □
Let
$$ G(t) =\frac{{\epsilon ^{2}}}{{2}}\sum_{j=1}^{\infty } E \int_{0}^{t} \int_{D} \lambda_{j} e_{j}^{2}(x) \sigma^{2}(x,s)\,dx \,ds. $$
(3.7)
Due to (3.1), we deduce
$$\begin{aligned} G(\infty )&= \frac{{\epsilon ^{2}}}{{2}}\sum_{j=1}^{\infty } E \int _{0}^{\infty } \int_{D} \lambda_{j} e_{j}^{2}(x) \sigma^{2}(x,s)\,dx \,ds \\ & \leq \frac{{\epsilon ^{2}}}{{2}} \operatorname{Tr}(Q) c_{0}^{2} E \int_{0}^{\infty } \int_{D} \sigma^{2}(x,s)\,dx \,ds=E_{1} < \infty . \end{aligned}$$
(3.8)
We set
$$ H(t)=G(t)-E\bigl[F(t)\bigr]. $$
Then (3.5) implies that
$$ H^{\prime }(t)=G^{\prime }(t) - \frac{{d}}{{dt}}E \bigl[F(t)\bigr] \geq E\bigl\Vert v(t) \bigr\Vert _{q} ^{q} \geq 0. $$
(3.9)
Lemma 3.2
Let
\((u,v)\)
be a solution of Eq. (3.3). Then there exists a positive constant
C
such that
$$\begin{aligned} E\bigl\Vert u(t) \bigr\Vert _{p}^{s} & \leq C\bigl[ G(t)-H(t) -E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {} -E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} +E\bigl\Vert u(t) \bigr\Vert _{p}^{p} - E(h \circ \nabla u) (t) \bigr], \quad 2\leq s\leq p. \end{aligned}$$
(3.10)
Proof
If \(\Vert u \Vert _{p} \leq 1\), then \(\Vert u \Vert _{p}^{s} \leq \Vert u \Vert _{p}^{2} \leq C\Vert \nabla u \Vert ^{2}\) by the Sobolev embedding theorem. If \(\Vert u \Vert _{p} \geq 1\), then \(\Vert u \Vert _{p}^{s} \leq \Vert u \Vert _{p}^{p}\). Thus there exists a constant \(C>0\) such that \(E\Vert u \Vert _{p}^{s} \leq C( E\Vert \nabla u \Vert ^{2} + E\Vert u \Vert _{p}^{p})\). Therefore, combining with the definition of energy function, we get (3.10). □
Theorem 3.1
Assume that (H1)-(H3) and (3.1) hold. Let
\((u,v)\)
be a solution of Eq. (3.3) with the initial data
\((u_{0},v _{0}) \in H_{0}^{1}(D)\times L^{2}(D)\)
satisfying
$$\begin{aligned} F(0) \leq -(1+\beta )E_{1}, \end{aligned}$$
(3.11)
where
\(\beta >0\)
is an arbitrary constant and
\(E_{1}\)
is defined in (3.8). If
\(p> \{ q, \rho +2 \}\), then the solution
\((u,v)\)
and the lifespan
\(\tau_{\infty }\)
defined above are either
-
(1)
\(P(\tau_{\infty }<\infty ) > 0\), that is, \(\Vert \nabla u(t) \Vert \)
blows up in finite time with positive probability, or
-
(2)
there exists a positive time
\(T^{*} \in [0,T_{0}]\)
such that
$$ \lim_{t \to T^{*} } E\bigl[F(t)\bigr]= +\infty , $$
(3.12)
where
$$\begin{aligned} \begin{aligned} & T_{0} = \frac{1-\alpha }{\alpha K L^{\alpha /(1-\alpha )}(0)} , \\ &L(0)=H^{1-\alpha }(0) + \delta E\biggl(u_{0}, \frac{1}{\rho +1} \vert u_{1} \vert ^{ \rho } u_{1} - \Delta u_{1} \biggr) >0, \end{aligned} \end{aligned}$$
(3.13)
and
α, K
are given later.
Proof
For the lifespan \(\tau_{\infty }\) of the solution \(\{ u(t): t>0 \}\) of Eq. (3.3) with \(H_{0}^{1}(D)\) norm, we treat the case when \(P( \tau_{\infty }= +\infty ) <1\). Then, for sufficiently large \(T>0\), by (3.9) and (3.11), we obtain
$$\begin{aligned} 0< (1+\beta )E_{1} \leq -F(0)=H(0)\leq H(t)\leq G(t) +\frac{1}{p}E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \leq E_{1} + \frac{1}{p}E\bigl\Vert u(t) \bigr\Vert _{p} ^{p}. \end{aligned}$$
(3.14)
Define
$$ L(t)=H^{1-\alpha }(t) + \delta E\biggl(u(t), \frac{1}{\rho +1} \bigl\vert v(t) \bigr\vert ^{ \rho }v(t) - \Delta v(t) \biggr), $$
where
$$ 0< \alpha < \min \biggl\{ \frac{1}{2}, \frac{p-2}{2p}, \frac{p-q}{pq}, \frac{1}{ \rho +2}-\frac{1}{p}\biggr\} $$
(3.15)
and δ is a very small constant to be determined later.
Using (3.6) and (3.9), we deduce
$$\begin{aligned} L^{\prime }(t)& =(1-\alpha ) H^{-\alpha }(t)H^{\prime }(t) +\delta \biggl[ -E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} -E\bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2} v(t)\bigr) \\ & \quad {}+ E \bigl\Vert u(t) \bigr\Vert _{p}^{p} +E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u( \tau ),\nabla u(t)\bigr)\,d\tau \\ & \quad {}+ \frac{{1} }{{\rho +1}}E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \biggr] \\ &\geq (1-\alpha ) H^{-\alpha }(t) E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \delta p \bigl[H(t) -G(t) + E F(t)\bigr] \\ & \quad {}-\delta E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} -\delta E \bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2}v(t)\bigr) + \delta E \bigl\Vert u(t) \bigr\Vert _{p}^{p} \\ & \quad {} +\delta E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ),\nabla u(t)\bigr)\,d\tau + \frac{{\delta } }{{\rho +1}}E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} + \delta E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ & \geq (1-\alpha ) H^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \delta p H(t) \\ & \quad {}+ \delta \biggl(\frac{{p} }{{\rho +2}}+ \frac{{1} }{{\rho +1}}\biggr) E\bigl\Vert v(t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ & \quad {}+ \delta \biggl( \frac{p}{2} -1\biggr) E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ & \quad {}+\delta \biggl( \frac{{p} }{{2}}+1\biggr)E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\delta E\bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2}v(t)\bigr) \\ & \quad {}+\delta E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ),\nabla u(t)\bigr)\,d\tau \\ & \quad {}+\frac{{\delta p}}{{2}} E(h \circ \nabla u) (t)- \frac{{\delta p}}{{2}}E \int_{0}^{t} h(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2} -\delta p G(t). \end{aligned}$$
(3.16)
On the other hand, we have
$$\begin{aligned} & \delta E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ),\nabla u(t)\bigr)\,d\tau \\ & \quad =\delta E \int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ) - \nabla u(t) ,\nabla u(t)\bigr)\,d\tau \\ & \qquad {}+\delta E \int_{0}^{t} h(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}, \end{aligned}$$
(3.17)
and by Hölder’s inequality, we get
$$\begin{aligned} & \delta E\int_{0}^{t} h(t-\tau ) \bigl(\nabla u(\tau ) - \nabla u(t) , \nabla u(t)\bigr)\,d\tau \\ & \quad \geq -\frac{{\delta p}}{{2}} E(h \circ \nabla u) (t) - \frac{{\delta }}{{2p}} E \int_{0}^{t} h(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.18)
Inserting (3.17) and (3.18) into (3.16), we obtain
$$\begin{aligned} L^{\prime }(t) &\geq (1-\alpha )H^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \delta p H(t) \\ &\quad {} + \delta \biggl( \frac{{p}}{{\rho +2}}+ \frac{{1}}{{\rho +1}}\biggr) E\bigl\Vert v(t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &\quad {} + \delta \biggl( \frac{p}{2} -1\biggr)E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {} + \delta \biggl(\frac{{p}}{{2}}+1\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\delta E\bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2}v(t)\bigr) \\ &\quad {}-\delta p G(t) +\delta \biggl(1-\frac{{p^{2} +1}}{{2p}}\biggr) E \int_{0}^{t} h( \tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.19)
For \(q< p\), by \(E\Vert u(t) \Vert _{q}^{q} \leq cE\Vert u(t) \Vert _{p}^{q}\) and Hölder’s inequality, we deduce the following estimate (see [23]):
$$\begin{aligned} E\bigl(u(t), \bigl\vert v(t) \bigr\vert ^{q-2}v(t) \bigr)& \leq \bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{ \frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{1}}{{q}}} \\ & \leq C\bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{q} \bigr)^{ \frac{{1}}{{q}}} \\ & \leq C \bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{ \frac{{1}}{{p}}} \\ & \leq C\bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{ \frac{{1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{\frac{{1}}{{p}} -\frac{{1}}{{q}}} \end{aligned}$$
(3.20)
and Young’s inequality
$$\begin{aligned} & \bigl(E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \bigr)^{\frac{{q-1}}{{q}}} \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{ \frac{{1}}{{q}}} \leq \frac{{q-1}}{{q}}\mu E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \frac{{\mu ^{1-q}}}{{q}}E\bigl\Vert u(t) \bigr\Vert _{p}^{p}, \end{aligned}$$
(3.21)
where μ is a constant to be determined later. In view of (3.14), we get
$$ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \geq p\bigl(H(t)-G(t)\bigr)\geq \widetilde{\rho } H(t), $$
(3.22)
where \(\widetilde{\rho } = p\beta /(1+\beta )\). With the assumption of \(H(0)>1\), (3.21), (3.22), and (3.15) imply that
$$ \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \bigr)^{\frac{{1}}{{p}} -\frac{{1}}{{q}}} \leq \widetilde{\rho }^{\frac{{1}}{{p}} -\frac{{1}}{{q}}}H(t)^{ \frac{{1}}{{p}} -\frac{{1}}{{q}}} \leq \widetilde{\rho }^{ \frac{{1}}{{p}} -\frac{{1}}{{q}}}H^{-\alpha }(t) \leq \widetilde{\rho } ^{\frac{{1}}{{p}} -\frac{{1}}{{q}}}H^{-\alpha }(0). $$
(3.23)
Combining with (3.20), (3.21), and (3.23), we arrive at
$$ \bigl\vert E\bigl(u(t), \bigr\vert v(t)\bigl\vert ^{q-2}v(t)\bigr) \bigr\vert \leq a_{1} \frac{{q-1}}{{q}}\mu E\bigl\Vert v(t) \bigr\Vert _{q} ^{q} H^{-\alpha }(t) + a_{1} \frac{{\mu ^{1-q}}}{{q}}E\bigl\Vert u(t) \bigr\Vert _{p}^{p} H^{-\alpha }(t), $$
(3.24)
where \(a_{1}=C\widetilde{\rho }^{\frac{{1}}{{p}} -\frac{{1}}{{q}}}\). Hence, substituting (3.24) for (3.19), we get
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} +\delta p H(t) \\ &\quad {}+ \delta \biggl( \frac{{p}}{{\rho +2}}+ \frac{{1}}{{\rho +1}}\biggr) E\bigl\Vert v(t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &\quad {}+ \delta \biggl( \frac{p}{2} -1\biggr)E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {}+ \delta \biggl(\frac{{p}}{{2}}+1\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\delta p G(t) \\ &\quad {}+ \delta \biggl(1-\frac{{p^{2} +1}}{{2p}}\biggr) \int_{0}^{t} h(\tau )\,d\tau E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {}-\delta a_{1} \frac{ \mu^{1-q}}{q} E\bigl\Vert u(t) \bigr\Vert _{p}^{p} H^{- \alpha } (0). \end{aligned}$$
(3.25)
Using Lemma 3.2 with \(s=p\) and (3.25), we have
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} +\delta p H(t) \\ &\quad {} + \delta \biggl( \frac{{p}}{{\rho +2}}+ \frac{{1}}{{\rho +1}}\biggr) E\bigl\Vert v(t) \bigr\Vert _{ \rho +2}^{\rho +2} \\ &\quad {} + \delta \biggl( \frac{p}{2} -1\biggr)E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {} + \delta \biggl(\frac{{p}}{{2}}+1\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\delta p G(t) \\ &\quad {} +\delta \biggl(1-\frac{{p^{2} +1}}{{2p}}\biggr) E \int_{0}^{t} h(\tau )\,d\tau \bigl\Vert \nabla u(t) \bigr\Vert ^{2} \\ &\quad {}-\delta a_{2} \mu^{1-q} \bigl[G(t) -H(t) - E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {} -E\bigl\Vert \nabla v(t) \bigr\Vert ^{2}+E\bigl\Vert u(t) \bigr\Vert _{p}^{p} -E(h\circ \nabla u) (t)\bigr] \\ &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q} ^{q} \\ &\quad {}+\delta \bigl( p +a_{2} \mu^{1-q}\bigr) H(t) -\delta \bigl( p+ a_{2} \mu^{1-q}\bigr)G(t) \\ &\quad {} + \delta \biggl( \frac{{p}}{{\rho +2}}+ \frac{{1}}{{\rho +1}}+ a_{2} \mu ^{1-q}\biggr) E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {} + \delta \biggl(\frac{{p}}{{2}}+1 +a_{2} \mu^{1-q} \biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} - \delta a_{2} \mu^{1-q}E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \\ &\quad {} +\delta a_{2} \mu^{1-q}E(h\circ \nabla u) (t) \\ &\quad {}+\delta \biggl[\frac{{p}}{{2}}-1 +\biggl(1-\frac{{p^{2} +1}}{{2p}}\biggr) \int_{0}^{t} h( \tau )\,d\tau \biggr]E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}, \end{aligned}$$
(3.26)
where \(a_{2}=Ca_{1} H^{-\alpha }(0)/q \).
Note that
$$\begin{aligned} H(t) &\geq G(t) +\frac{{1}}{{p}} E\bigl\Vert u(t) \bigr\Vert _{p}^{p} - \frac{{1}}{{\rho +2}}E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {}-\frac{{1}}{{2}}E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} -\frac{1}{2} \bigl\Vert \nabla u(t) \bigr\Vert ^{2} - \frac{{1}}{{2}}E(h\circ \nabla u) (t) \end{aligned}$$
(3.27)
with
$$ a_{3}= \frac{{p}}{{2}}-1 +\biggl(1- \frac{{p^{2} +1}}{{2p}}\biggr) \int_{0}^{t} h( \tau )\,d\tau >0, $$
(3.28)
we write \(p=2a_{4}+(p-2a_{4})\) with \(a_{4}=\min \{ a_{1}, a_{3}\}\), then estimate (3.26) yields
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \\ & \quad {}+ \delta \bigl( p-2a_{4} +a_{2} \mu^{1-q}\bigr) H(t) - \delta \bigl( p-2a_{4} +a _{2} \mu^{1-q} \bigr) G(t) \\ & \quad {}+ \delta \biggl( \frac{{p}}{{\rho +2}}-\frac{{2a_{4}}}{{\rho +2}}+ \frac{{1}}{{\rho +1}} +a_{2} \mu^{1-q}\biggr) E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ & \quad {}+ \delta \biggl(\frac{{p}}{{2}}-a_{4}+1+a_{2}\mu^{1-q}\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {}+ \delta \biggl(- a_{2} \mu^{1-q}+\frac{{2a_{4}}}{{p}}\biggr) E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \\ & \quad {}+ \delta \bigl( a_{2} \mu^{1-q}-a_{4}\bigr)E(h\circ \nabla u) (t) + \delta (a _{3}-a_{4})E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.29)
From (3.8) and (3.14), we deduce
$$\begin{aligned} \bigl( p-2a_{4} +a_{2} \mu^{1-q} \bigr) G(t) &\leq \bigl( p-2a_{4} +a_{2} \mu^{1-q} \bigr)E _{1} \\ & \leq \frac{{ p-2a_{4} +a_{2} \mu ^{1-q}}}{{1+\beta }} H(t) . \end{aligned}$$
(3.30)
Substituting (3.30) with (3.29), we get
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} \\ &\quad {} + \delta \bigl( p-2a_{4} +a_{2} \mu^{1-q}\bigr) \frac{{\beta }}{{1+\beta }} H(t) \\ &\quad {} + \delta \biggl( \frac{{p}}{{\rho +2}}-\frac{{2a_{4}}}{{\rho +2}}+ \frac{{1}}{{\rho +1}} +a_{2} \mu^{1-q}\biggr) E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {} + \delta \biggl(\frac{{p}}{{2}}-a_{4}+1 +a_{2} \mu^{1-q}\biggr) E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {} + \delta \biggl(- a_{2} \mu^{1-q}+\frac{{2a_{4}}}{{p}} \biggr) E\bigl\Vert u(t) \bigr\Vert _{p}^{p} \\ &\quad {} + \delta \bigl( a_{2} \mu^{1-q}-a_{4}\bigr)E(h \circ \nabla u) (t) + \delta (a _{3}-a_{4})E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}. \end{aligned}$$
(3.31)
Next, we can choose μ large enough so that (3.31) becomes
$$\begin{aligned} L^{\prime }(t) &\geq \biggl(1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta \biggr)H ^{-\alpha }(t)E\bigl\Vert v(t) \bigr\Vert _{q}^{q} + \delta \gamma \bigl(H(t) +E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ & \quad {}+ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} + E(h \circ \nabla u) (t)+E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \bigr), \end{aligned}$$
(3.32)
where
$$\begin{aligned} \gamma &= \min \biggl\{ \bigl( p-2a_{4} +a_{2} \mu^{1-q}\bigr) \frac{{\beta }}{{1+\beta }}, \frac{{p}}{{\rho +2}}- \frac{{2a_{4}}}{{\rho +2}}+ \frac{{1}}{{\rho +1}} +a_{2} \mu^{1-q}, \\ & \quad {} \frac{{p}}{{2}}-a_{4}+1+a_{2} \mu^{1-q}, - a_{2} \mu^{1-q}+ \frac{{2a_{4}}}{{p}}, a_{2} \mu^{1-q}-a_{4}, a_{3}-a_{4} \biggr\} >0. \end{aligned}$$
Once μ is fixed, we pick δ small enough so that
$$ 1-\alpha -a_{1} \frac{{q-1}}{{q}}\mu \delta >0. $$
Using this, (3.32) takes the form
$$\begin{aligned} L^{\prime }(t) &\geq \delta \gamma \bigl(H(t) +E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{ \rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {}+ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} + E(h \circ \nabla u) (t)+E\bigl\Vert \nabla u(t) \bigr\Vert ^{2} \bigr). \end{aligned}$$
(3.33)
Thus, we see that
$$\begin{aligned} L(t) \geq L(0)=H^{1-\alpha }(0) + \delta E \biggl(u_{0}, \frac{{1}}{{\rho +1}}\vert u_{1} \vert ^{\rho }u_{1} - \Delta u_{1}\biggr)>0, \quad \forall t \geq0. \end{aligned}$$
(3.34)
Consequently, we get
$$ L(t) \geq L(0)>0, \quad \forall t \geq 0. $$
(3.35)
Since
$$\begin{aligned} \biggl\vert E\int_{D} \bigl\vert v(t)\bigr\vert ^{\rho }v(t)u(t) \,dx \biggr\vert &\leq E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{ \rho +1} E\bigl\Vert u(t) \bigr\Vert _{\rho +2} \\ & \leq CE\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +1} E\bigl\Vert u(t) \bigr\Vert _{p}, \end{aligned}$$
we have
$$\begin{aligned} \biggl\vert E\int_{D} \bigl\vert v(t)\bigr\vert ^{\rho }v(t)u(t)\,dx \biggr\vert ^{\frac{{1}}{{1-\alpha }}} &\leq E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\frac{{\rho +1}}{{1-\alpha }}} E\bigl\Vert u(t) \bigr\Vert _{ \rho +2}^{\frac{{1}}{{1-\alpha }}} \\ &\leq C \bigl[\bigl(E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2}\bigr)^{ \frac{ {(\rho +1)}}{{(\rho +2)(1-\alpha )}}\zeta } + \bigl(E\bigl\Vert u(t) \bigr\Vert _{p} ^{2}\bigr)^{\frac{{\theta }}{{2(1-\alpha )}}}\bigr], \end{aligned}$$
(3.36)
where \(\frac{{1}}{{\zeta }} +\frac{{1}}{{\theta }}=1\). By choosing \(\zeta = \frac{{(1-\alpha )(\rho +2)}}{{\rho +1}} (>1)\), we have \(\frac{{\theta }}{{2(1-\alpha )}}= \frac{{\rho +2}}{{2[(1-\alpha )(\rho +2)-(\rho +1)]}} <\frac{{p}}{{2}}\). And with (3.15), (3.36) becomes
$$\begin{aligned} \biggl\vert E \int_{D} \bigl\vert v(t)\bigr\vert ^{\rho }v(t)u(t) \,dx \biggr\vert ^{\frac{{1}}{{1-\alpha }}} \leq C \bigl\{ E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl(E\bigl\Vert u(t) \bigr\Vert _{p}^{2}\bigr)^{ \frac{{\rho +2}}{{2[(1-\alpha )(\rho +2)-(\rho +1)]}}}\bigr\} . \end{aligned}$$
Using Lemma 3.2 with \(s= \frac{{\rho +2}}{{2[(1-\alpha )(\rho +2)-(\rho +1)]}}\), we obtain
$$\begin{aligned} \biggl\vert E\int_{D} \bigl\vert v(t)\bigr\vert ^{\rho }v(t)u(t)\,dx \biggr\vert ^{\frac{{1}}{{1-\alpha }}} & \leq C \bigl[ H(t) +E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {}+ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} + E(h \circ \nabla u) (t)+E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}\bigr] . \end{aligned}$$
Therefore, we deduce, for all \(t\geq 0\),
$$\begin{aligned} L^{\frac{{1}}{{1-\alpha }}}(t)&=\biggl(H^{1-\alpha }(t) + \frac{{\delta }}{{\rho +1}} E \int_{D} \bigl\vert v(t) \bigr\vert ^{\rho }v(t)u(t) \,dx +\delta E \int_{D} \nabla v(t) \cdot \nabla u(t)\,dx \biggr)^{ \frac{{1}}{{1-\alpha }}} \\ & \leq C \bigl[ H(t) +E\bigl\Vert v(t) \bigr\Vert _{\rho +2}^{\rho +2} +E\bigl\Vert \nabla v(t) \bigr\Vert ^{2} \\ &\quad {}+ E\bigl\Vert u(t) \bigr\Vert _{p}^{p} + E(h \circ \nabla u) (t)+E\bigl\Vert \nabla u(t) \bigr\Vert ^{2}\bigr] . \end{aligned}$$
(3.37)
Combining (3.33) and (3.37)
$$\begin{aligned} L^{\prime }(t) \geq KL^{\frac{{1}}{{1-\alpha }}}(t),\quad \forall t\geq 0 \end{aligned}$$
with a positive constant K depending on C and δγ, it follows that
$$\begin{aligned} L^{\frac{{\alpha }}{{1-\alpha }}}(t) \geq \frac{{1-\alpha }}{{(1-\alpha )L^{-{{\alpha }\over {1-\alpha }}}(0)-\alpha K t }}. \end{aligned}$$
Let
$$\begin{aligned} T_{0} = \frac{{1-\alpha }}{{\alpha K L^{{{\alpha }\over {1-\alpha }}}(0)}}. \end{aligned}$$
Then \(L(t) \to \infty \) as \(t \to T_{0}\). This means that there exists a positive time \(T^{*} \in (0,T_{0}]\) such that
$$\begin{aligned} \lim_{t \to T^{*}} E\bigl[F(t)\bigr] = + \infty . \end{aligned}$$
As for the case when \(P(\tau_{\infty }=+\infty ) < 1\) (i.e., \(P(\tau_{\infty } < + \infty )>0\)), then \(\Vert \nabla u(t) \Vert \) blows up in finite time \(T^{*} \in (0, \tau_{\infty })\) with positive probability. Thus, the proof of Theorem 3.1 is completed. □