In this section, we establish blow-up results of solutions with three levels of initial energy and estimate bounds of blow-up time. For this, we need ab auxiliary lemma.
Lemma 3.1
([12])
Let \(B(t)\) be a positive, twice differentiable function verifying
$$ B(t) B''(t) - ( 1+ \theta ) \bigl( B'(t) \bigr)^{2} \geq 0 $$
for \(t > 0 \), where θ is a positive constant. If \(B(0) >0\) and \(B'(0) >0\), then there exists a \(T_{1} \leq \frac{B(0)}{\theta B'(0)} \) with \({ \lim_{t \to T_{1}^{-} } B(t) = \infty } \).
Taking the scalar product (1.1) by \(w_{t}\) in \(L^{2}(\Omega )\) and using (1.3), we get
$$\begin{aligned}& \frac{d}{dt} \biggl( \frac{1}{2} \Vert w_{t} \Vert _{2}^{2} + \frac{1}{2} \Vert \Delta w \Vert _{2}^{2} - \int _{\Omega } \frac{ \vert w \vert ^{ q (x)} }{q(x)} \,dx \biggr) - \int ^{t}_{0} h (t-s) \bigl( \Delta w(s) , \Delta w_{t} (t) \bigr) \,ds \\& \quad = - \Vert \nabla w_{t} \Vert _{2}^{2} + \bigl( \bigl[w, \chi (w) \bigr] , w_{t} \bigr) . \end{aligned}$$
From Lemma 2.4, (1.2) and (1.3), we have
$$\begin{aligned} \bigl( \bigl[w, \chi (w) \bigr] , w_{t} \bigr) = \frac{1}{2} \biggl( \frac{d}{dt} [w, w ] , \chi (w) \biggr) = - \frac{1}{4} \frac{d}{dt} \bigl\Vert \Delta \chi (w) \bigr\Vert _{2}^{2} . \end{aligned}$$
Using this and the relation
$$\begin{aligned}& - \int ^{t}_{0} h (t-s) \bigl( \Delta w(s) , \Delta w_{t} (t) \bigr) \,ds \\& \quad = \frac{1}{2} \frac{d}{dt} \biggl\{ ( h \circ \Delta w ) - \biggl( \int ^{t}_{0} h (s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} \biggr\} + \frac{h(t)}{2} \Vert \Delta w \Vert _{2}^{2} - \frac{1}{2} \bigl( h' \circ \Delta w \bigr) , \end{aligned}$$
we get
$$\begin{aligned} E'(t) = - \Vert \nabla w_{t} \Vert _{2}^{2} - \frac{h(t)}{2} \Vert \Delta w \Vert _{2}^{2} + \frac{1}{2} \bigl(h' \circ \Delta w \bigr) \leq - \Vert \nabla w_{t} \Vert _{2}^{2} \leq 0, \quad t\in [0, T_{*}) \end{aligned}$$
(3.1)
and
$$\begin{aligned} E(t) + \int ^{t}_{0} \bigl\Vert \nabla w_{t} (s) \bigr\Vert _{2}^{2} \,ds \leq E(0), \quad t \in [0, T_{*}) , \end{aligned}$$
(3.2)
where
$$\begin{aligned} \begin{aligned}[b] E(t) = {}&\frac{1}{2} \Vert w_{t} \Vert _{2}^{2} +\frac{1}{2} \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + \frac{1}{2} (h \circ \Delta w) \\&{}+ \frac{1}{4} \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} - \int _{\Omega } \frac{ \vert w \vert ^{q(x)}}{q(x)} \,dx,\end{aligned} \end{aligned}$$
(3.3)
here
$$ (h \circ \Delta w) (t) = \int ^{t}_{0} h(t-s) \bigl\Vert \Delta w(t) - \Delta w(s) \bigr\Vert _{2}^{2} \,ds , $$
and \(T_{*} = \sup \{ T : [0,T] \text{ is the existence interval of the solution to (1.1)--(1.4)} \}\).
3.1 Case of non-positive initial energy
Theorem 3.1
Let \(q_{1} \geq 4\) and the kernel function h satisfy
$$ \int ^{\infty }_{0} h(s) \,ds \leq \frac{ q_{1} ( q_{1} -2 ) }{ q_{1} ( q_{1} -2 ) + 1 }. $$
(3.4)
Let one of the following hold.
-
(i)
\(E(0) < 0 \);
-
(ii)
\(E( 0) = 0 \) and \((w_{0} , w_{1} ) > \frac{ 2 \Vert \nabla w_{0} \Vert _{2}^{2} }{ q_{1} -2} \).
Then the solution w of problem (1.1)–(1.4) blows up in a finite time \(T_{*}\), that is,
$$ \lim_{t \to T_{*}^{-}} \bigl\Vert \Delta w(t) \bigr\Vert _{2} = \infty . $$
(3.5)
In addition, \(T_{*}\) satisfies
$$ T_{*} \leq \frac{ 2 \Vert w_{0} \Vert _{2}^{2} + 2 \alpha \beta ^{2} }{ ( q_{1} -2) \{ (w_{0}, w_{1}) + \alpha \beta \} -2 \Vert \nabla w_{0} \Vert _{2}^{2} } , $$
(3.6)
where
$$\begin{aligned} \textstyle\begin{cases} 0 < \alpha \leq -2E(0) \quad \textit{and} \quad \beta > \max \{ 0 , - \frac{( w_{0}, w_{1} )}{ \alpha } , \frac{ 2 \Vert \nabla w_{0} \Vert _{2}^{2} - ( q_{1} -2 ) ( w_{0} , w_{1}) }{ q_{1} -2 } \} \quad \textit{if } E( 0) < 0 ; \\ \alpha =0 \quad \textit{if } E( 0) = 0 \quad \textit{and}\quad (w_{0} , w_{1} ) > \frac{ 2 \Vert \nabla w_{0} \Vert _{2}^{2} }{ q_{1} -2} . \end{cases}\displaystyle \end{aligned}$$
(3.7)
Proof
Suppose that w is global. For \(0 < T < T_{*} \), we define a function F on \([0,T]\) by
$$ F(t) = \Vert w \Vert _{2}^{2} + \int ^{t}_{0} \bigl\Vert \nabla w(s) \bigr\Vert _{2}^{2} \,ds + (T-t) \Vert \nabla w_{0} \Vert _{2}^{2} + \alpha (t+ \beta )^{2}, $$
(3.8)
where \(\alpha \geq 0 \) and \(\beta >0 \) are the constants satisfying (3.7), then
$$\begin{aligned}& F(t) > 0 \quad \text{for } t \in [0,T], \end{aligned}$$
(3.9)
$$\begin{aligned}& F'(t) = 2(w, w_{t}) + 2 \int ^{t}_{0} \bigl(\nabla w(s), \nabla w_{t}(s) \bigr) \,ds + 2 \alpha (t+ \beta ), \end{aligned}$$
(3.10)
and
$$\begin{aligned} F''(t) = & 2 \Vert w_{t} \Vert _{2}^{2} - 2 \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + 2 \bigl( w, \bigl[ w, \chi ( w ) \bigr] \bigr) + 2 \int _{ \Omega } \vert w \vert ^{q(x)} \,dx \\ & {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds + 2 \alpha . \end{aligned}$$
From Lemma 2.4, (1.2), and (1.3), we have
$$\begin{aligned} 2 \bigl( w, \bigl[ w, \chi ( w ) \bigr] \bigr) = 2 \bigl( [ w, w ] , \chi ( w ) \bigr) = - 2 \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} . \end{aligned}$$
Thus, we get
$$\begin{aligned} F''(t) = & 2 \Vert w_{t} \Vert _{2}^{2} - 2 \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} - 2 \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} + 2 \int _{\Omega } \vert w \vert ^{q(x)} \,dx \\ & {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds + 2 \alpha . \end{aligned}$$
(3.11)
Using the inequality \(( a \xi _{1} + b \xi _{2} + c \xi _{3} )^{2} \leq ( a^{2} + b^{2} + c^{2} ) ( \xi _{1}^{2} + \xi _{2}^{2} + \xi _{3}^{2} ) \) and (3.8), we get
$$\begin{aligned} \bigl(F'(t) \bigr)^{2} \leq 4 F(t) \biggl( \Vert w_{t} \Vert _{2}^{2} + \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert _{2}^{2} \,ds + \alpha \biggr) . \end{aligned}$$
(3.12)
Using (3.11), (3.12), (3.3), (3.2), and Young’s inequality, one finds
$$\begin{aligned}& F(t) F''(t) - \frac{q_{1} + 2}{4} \bigl(F'(t) \bigr)^{2} \\& \quad \geq F(t) \biggl\{ - q_{1} \Vert w_{t} \Vert _{2}^{2} - 2 \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} - 2 \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} + 2 \int _{\Omega } \vert w \vert ^{q(x)} \,dx \\& \quad \quad {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds - ( q_{1} +2 ) \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert _{2}^{2} \,ds - q_{1} \alpha \biggr\} \\& \quad = F(t) \biggl\{ - 2 q_{1} E(t)+ ( q_{1} - 2 ) \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + q_{1} (h \circ \Delta w) \\& \quad \quad {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds + \biggl( \frac{q_{1}}{2} -2 \biggr) \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} \\& \quad \quad {} - ( q_{1} +2 ) \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert _{2}^{2} \,ds - 2 q_{1} \int _{\Omega } \frac{ \vert w \vert ^{q(x)} }{q(x)} \,dx + 2 \int _{ \Omega } \vert w \vert ^{q(x)} \,dx - q_{1} \alpha \biggr\} \\& \quad \geq F(t) \biggl\{ - 2 q_{1} E(0)+ ( q_{1} - 2 ) \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + q_{1} (h \circ \Delta w) \\& \quad \quad {} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w(t), \Delta w(t) - \Delta w(s) \bigr) \,ds + \biggl( \frac{q_{1}}{2} -2 \biggr) \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} \\& \quad \quad {} + ( q_{1} - 2 ) \int ^{t}_{0} \bigl\Vert \nabla w_{t}(s) \bigr\Vert _{2}^{2} \,ds - 2 q_{1} \int _{\Omega } \frac{ \vert u \vert ^{q(x)} }{q(x)} \,dx + 2 \int _{ \Omega } \vert w \vert ^{q(x)} \,dx - q_{1} \alpha \biggr\} . \end{aligned}$$
(3.13)
Using the relations
$$\begin{aligned} - 2 q_{1} \int _{\Omega } \frac{ \vert w \vert ^{q(x)} }{q(x)} \,dx \geq - 2 \int _{\Omega } \vert w \vert ^{q(x)} \,dx \end{aligned}$$
and
$$\begin{aligned} - 2 \int ^{t}_{0} h(t-s) \bigl(\Delta w (t), \Delta w(t) - \Delta w(s) \bigr) \,ds \geq - \epsilon (h \circ \Delta w ) - \frac{1}{\epsilon } \biggl( \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} \end{aligned}$$
for \(\epsilon >0 \), we infer
$$\begin{aligned} F(t) F''(t) - \frac{q_{1} + 2}{4} \bigl(F'(t) \bigr)^{2} \geq F(t)G(t) , \end{aligned}$$
(3.14)
where
$$\begin{aligned} G(t) = & {-} 2 q_{1} E(0) - q_{1} \alpha + ( q_{1} - \epsilon ) (h \circ \Delta w) \\ & {} + \biggl\{ ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{\epsilon } \biggr) \int ^{t}_{0} h(s) \,ds \biggr\} \Vert \Delta w \Vert _{2}^{2} . \end{aligned}$$
(3.15)
Taking \(\epsilon = q_{1}\) in (3.15), and using (3.4) and (3.7), we find
$$\begin{aligned} G(t) \geq - 2 q_{1} E(0) + \biggl\{ ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{ q_{1} } \biggr) \int ^{t}_{0} h(s) \,ds \biggr\} \Vert \Delta w \Vert ^{2} - q_{1} \alpha \geq 0. \end{aligned}$$
From the condition (3.7), it is clear that
$$ F'(0) = 2 (w_{0}, w_{1}) + 2 \alpha \beta >0 . $$
Thus, applying Lemma 3.1, we get the existence of \(T_{*}\) satisfying
$$ T_{*} \leq \frac{ 4 F(0)}{(q_{1} -2) F '(0)} = \frac{ 2 ( \Vert w_{0} \Vert _{2}^{2} + T \Vert \nabla w_{0} \Vert _{2}^{2} + \alpha \beta ^{2} )}{ (q_{1} -2) \{ ( w_{0}, w_{1}) + \alpha \beta \} } $$
(3.16)
and
$$ \lim_{t \to T_{*}^{-} } F(t) = \infty , $$
which gives
$$ \lim_{t \to T_{*}^{-}} \bigl\Vert \Delta w(t) \bigr\Vert _{2} = \infty . $$
Moreover, using (3.16) and the relation \(0 < T < T_{*} \), we see
$$ T_{*} \leq \frac{ 2 ( \Vert w_{0} \Vert _{2}^{2} + T_{*} \Vert \nabla w_{0} \Vert _{2}^{2} + \alpha \beta ^{2} )}{ (q_{1} -2) \{ ( w_{0}, w_{1}) + \alpha \beta \} } . $$
This gives (3.6) under the condition β given in (3.7). □
3.2 Case of certain positive initial energy
We set
$$ {\overline{B}} = \max \biggl\{ 1, \frac{B}{\sqrt{l}} \biggr\} , \quad\quad \eta _{1} = \biggl( \frac{1}{ {\overline{B}} } \biggr)^{\frac{ q_{1} }{ q_{1} -2}}, \quad\quad E_{1} = \frac{( q_{1} -2 ) \eta _{1}^{2}}{2 q_{1} }, $$
(3.17)
where B is the embedding constant given in (2.2), and define a function g by
$$ g(\eta ) = \frac{1}{2} \eta ^{2} - \frac{ {\overline{B}}^{ q_{1} }}{ q_{1} } \eta ^{ q_{1} } . $$
(3.18)
Then one knows
-
(i)
\(g(0)=0 \) and \({ \lim_{\eta \to + \infty } g(\eta ) = -\infty }\),
-
(ii)
g is increasing on \((0, \eta _{1})\) and decreasing on \((\eta _{1}, \infty ) \),
-
(iii)
g has the maximum value \(g(\eta _{1}) = E_{1} \).
Lemma 3.2
Let w be the solution of problem (1.1)–(1.4). Assume that
$$ E(0) < E_{1} \quad \textit{and}\quad \eta _{1} < \sqrt{l} \Vert \Delta w_{0} \Vert _{2} \leq \frac{1}{ {\overline{B}} }. $$
(3.19)
Then there exists a constant \(\eta _{*} > \eta _{1}\) such that
$$ l \Vert \Delta w \Vert _{2}^{2} \geq \eta _{*}^{2} \quad \textit{for } 0 \leq t < T_{*} . $$
(3.20)
Proof
From (3.3), Lemma 2.1, (2.2) and (3.17), we have
$$\begin{aligned} E(t) \geq & \frac{1}{2} \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} - \int _{\Omega } \frac{ \vert w \vert ^{q(x)} }{q(x)} \,dx \\ \geq & \frac{l}{2} \Vert \Delta w \Vert _{2}^{2} - \frac{1}{ q_{1} } \max \bigl\{ \Vert w \Vert ^{ q_{1} }_{q(\cdot )}, \Vert w \Vert ^{ q_{2} }_{q(\cdot )} \bigr\} \\ \geq & \frac{l}{2} \Vert \Delta w \Vert _{2}^{2} - \frac{1}{ q_{1} } \max \bigl\{ B^{ q_{1} } \Vert \Delta w \Vert _{2}^{ q_{1} }, B^{ q_{2} } \Vert \Delta w \Vert _{2}^{ q_{2} } \bigr\} \} \\ \geq & \frac{1}{2} \bigl( \sqrt{l} \Vert \Delta w \Vert _{2} \bigr) ^{2} - \frac{1}{ q_{1} } \max \bigl\{ { \overline{B}}^{ q_{1} } \bigl( \sqrt{l} \Vert \Delta w \Vert _{2} \bigr)^{q_{1} }, {\overline{B}}^{ q_{2} } \bigl( \sqrt{l} \Vert \Delta w \Vert _{2} \bigr)^{ q_{2} } \bigr\} \\ = & f \bigl( \sqrt{l} \Vert \Delta w \Vert _{2} \bigr), \end{aligned}$$
(3.21)
where
$$ f(\eta ) = \frac{1}{2} \eta ^{2} - \frac{1}{ q_{1} } \max \bigl\{ { \overline{B}}^{ q_{1} } \eta ^{ q_{1} }, {\overline{B}}^{ q_{2} } \eta ^{q_{2} } \bigr\} . $$
It is easily seen that
$$ f(\eta ) = g(\eta ) \quad \text{for } 0 \leq \eta \leq \frac{1}{ {\overline{B}} }. $$
(3.22)
Since \(E(0) < E_{1}\), there exists \(\eta _{*} > \eta _{1}\) such that
$$ E(0) = g(\eta _{*}). $$
(3.23)
From (3.23), (3.21), and (3.19), we observe
$$ g(\eta _{*}) = E(0) \geq f \bigl( \sqrt{l} \Vert \Delta w_{0} \Vert _{2} \bigr) = g \bigl( \sqrt{l} \Vert \Delta w_{0} \Vert _{2} \bigr) . $$
Since g is decreasing on \((\eta _{1}, \infty )\), we see that
$$ \eta _{*} \leq \sqrt{l} \Vert \Delta w_{0} \Vert _{2}. $$
From (3.19), we also know
$$ \eta _{1} < \eta _{*} \leq \frac{1}{{\overline{B}}} . $$
(3.24)
We will show (3.20) by contradiction. Suppose that there exists \(t_{0} \in [0, T_{*} ) \) such that
$$ \sqrt{l} \bigl\Vert \Delta w ( t_{0}) \bigr\Vert _{2} < \eta _{*} . $$
Because the solution w is continuous in t, there exists \(t_{1} >0\) such that
$$ \eta _{1} < \sqrt{l} \bigl\Vert \Delta w(t_{1}) \bigr\Vert _{2} < \eta _{*} . $$
(3.25)
Noting that g is decreasing on \((\eta _{1}, \infty )\) and using (3.23), (3.25), (3.22), (3.24), (3.21), (3.1), we get
$$ E(0) = g( \eta _{*} ) < g \bigl( \sqrt{l} \bigl\Vert \Delta w (t_{1}) \bigr\Vert _{2} \bigr) = f \bigl( \sqrt{l} \bigl\Vert \Delta w (t_{1}) \bigr\Vert _{2} \bigr) \leq E(t_{1}) \leq E(0). $$
This is a contradiction. □
Theorem 3.2
Let the conditions of Lemma 3.2are valid. If \(E(0) = \gamma E_{1} \), where \(0< \gamma <1 \), and
$$ \int ^{\infty }_{0} h(s) \,ds \leq \frac{ q_{1} -2}{ q_{1} -2 + \frac{1}{(1- \gamma )^{2} q_{1} + 2 \gamma (1- \gamma ) } }, $$
(3.26)
the solution w to problem (1.1)–(1.4) blows up in a finite time \(T_{*} \). Moreover, \(T_{*}\) satisfies
$$ T_{*} \leq \frac{ 2 \Vert w_{0} \Vert _{2}^{2} + 2 \alpha \beta ^{2} }{ ( q_{1} -2) \{ (w_{0}, w_{1}) + \alpha \beta \} -2 \Vert \nabla w_{0} \Vert _{2}^{2} } , $$
where
$$ \begin{gathered} 0 < \alpha \leq \frac{ \gamma \lambda ( q_{1} -2 ) }{ q_{1} }, \quad \textit{and} \\ \beta > \max \biggl\{ 0 , - \frac{( w_{0}, w_{1} )}{ \alpha } , \frac{ 2 \Vert w_{0} \Vert _{2}^{2} - ( q_{1} -2 ) ( w_{0} , w_{1}) }{ q_{1} -2 } \biggr\} , \end{gathered} $$
(3.27)
here \(0 < \lambda < \eta _{*}^{2} - \eta _{1}^{2} \).
Proof
Let F be the function given in (3.8) with (3.27). Then (3.9), (3.10), (3.11), (3.14), and (3.15) are valid. Taking \(\epsilon = (1- \gamma ) q_{1} + 2 \gamma \) in (3.15), we have
$$\begin{aligned} G(t) \geq & {-} 2 q_{1} E(0) - q_{1} \alpha + \gamma ( q_{1} -2) (h \circ \Delta w ) \\ & {} + \biggl\{ ( q_{1} -2) - \biggl( q_{1} - 2 + \frac{1}{(1- \gamma ) q_{1} + 2 \gamma } \biggr) \int ^{t}_{0} h(s)\,ds \biggr\} \Vert \Delta w \Vert _{2}^{2} . \end{aligned}$$
(3.28)
The condition (3.26) implies
$$\begin{aligned} ( q_{1} -2) - \biggl( q_{1} -2 + \frac{1}{ (1- \gamma ) q_{1} + 2 \gamma } \biggr) \int ^{t}_{0} h(s)\,ds \geq \gamma ( q_{1} -2) \biggl( 1 - \int ^{\infty }_{0} h(s)\,ds \biggr). \end{aligned}$$
Since w is continuous in t, Lemma 3.2 guarantees the existence of \(\lambda >0\) with
$$\begin{aligned} \eta _{1}^{2} + \lambda < \eta _{*}^{2} \leq l \Vert \Delta w \Vert _{2}^{2} \quad \text{for all } t\in [0,T] . \end{aligned}$$
Adapting these too, noting the definition of \(E_{1}\) given in (3.17), and using (3.27), we have
$$\begin{aligned} G(t) \geq & {-} 2 q_{1} E(0) - q_{1} \alpha + \gamma ( q_{1} -2) l \Vert \Delta w \Vert _{2}^{2} \\ \geq & {-} 2 q_{1} \gamma E_{1} - q_{1} \alpha + \gamma ( q_{1} -2 ) \bigl( \eta _{1}^{2} + \lambda \bigr) \\ = & \gamma ( q_{1} - 2 ) \lambda - q_{1} \alpha \geq 0. \end{aligned}$$
By the same argument of Theorem 3.1, we complete the proof. □
3.3 Case of high initial energy
Lemma 3.3
If \(q_{1} \geq 4\) and h satisfies (3.4), it fulfills
$$\begin{aligned} \biggl( (w, w_{t}) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(t) \biggr) \geq e^{rt} \biggl( (w_{0} , w_{1}) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(0) \biggr) \quad \textit{for } t\in [0,T_{*}), \end{aligned}$$
(3.29)
here
$$ \begin{gathered} Q = ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{q_{1} } \biggr) (1- l ) , \\ k = \min \biggl\{ \frac{ q_{1} +2}{ q_{1} -2}, \frac{1}{B_{1}^{2} B_{2}^{2}} \biggr\} , \quad\quad r = \frac{2 k q_{1} Q }{2 q_{1} + k B_{2}^{2}} .\end{gathered} $$
(3.30)
Proof
Using (1.1)–(1.4) and Young’s inequality, we get
$$\begin{aligned} \frac{d}{dt}(w, w_{t} ) = & \Vert w_{t} \Vert _{2}^{2} - \biggl( 1- \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} - (\nabla w, \nabla w_{t} ) - \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} \\ & {} + \int ^{t}_{0} h(t-s) \bigl( \Delta w(s) - \Delta w(t), \Delta w(t) \bigr) \,ds + \int _{\Omega } \vert w \vert ^{q(x)} \,dx \\ \geq & \Vert w_{t} \Vert _{2}^{2} - \biggl\{ \biggl( 1- \int ^{t}_{0} h(s) \,ds \biggr) + \frac{1}{2 q_{1}} \int ^{t}_{0} h(s) \,ds \biggr\} \Vert \Delta w \Vert _{2}^{2} - \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} \\ & {} - \frac{ \delta }{2} \Vert \nabla w \Vert _{2}^{2} - \frac{ 1 }{2 \delta } \Vert \nabla w_{t} \Vert _{2}^{2} - \frac{q_{1}}{2} ( h \circ \Delta w) + \int _{\Omega } \vert w \vert ^{q(x)} \,dx \end{aligned}$$
(3.31)
for \(\delta >0 \). From (3.3), we observe
$$\begin{aligned} \int _{\Omega } \vert w \vert ^{q(x)} \,dx \geq& - q_{1} E(t) + \frac{ q_{1} }{2} \Vert w_{t} \Vert _{2}^{2} + \frac{ q_{1} }{2} \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} \\ &{}+ \frac{ q_{1} }{2} (h \circ \Delta w) + \frac{ q_{1} }{4} \bigl\Vert \Delta \chi ( w ) \bigr\Vert _{2}^{2} . \end{aligned}$$
Applying this to (3.31) and using (3.1), we have
$$\begin{aligned} \frac{d}{dt}(w, w_{t} ) \geq & \frac{ q_{1} +2}{2} \Vert w_{t} \Vert _{2}^{2} + \frac{1}{2 \delta } E'(t) - q_{1} E(t) \\ & {} + \biggl[ \biggl\{ ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{q_{1}} \biggr) \int ^{t}_{0} h(s) \,ds \biggr\} B_{2}^{-2} - \delta \biggr] \frac{ \Vert \nabla w \Vert _{2}^{2} }{2} \\ \geq & \frac{ q_{1} +2}{2} \Vert w_{t} \Vert _{2}^{2} + \frac{1}{2 \delta } E'(t) - q_{1} E(t) \\ & {} + \frac{1}{2 B_{1}^{2} B_{2}^{2} } \biggl[ \biggl\{ ( q_{1} - 2 ) - \biggl( q_{1} - 2 + \frac{1}{q_{1}} \biggr) \int ^{t}_{0} h(s) \,ds \biggr\} - B_{2}^{2} \delta \biggr] \Vert w \Vert _{2}^{2} . \end{aligned}$$
Recalling (3.30), we have
$$\begin{aligned}& \frac{d}{dt} \biggl( (w, w_{t} ) - \frac{1}{2 \delta } E(t) \biggr) \\& \quad \geq \frac{ q_{1} +2}{2} \Vert w_{t} \Vert _{2}^{2} + \frac{ Q - B_{2}^{2} \delta }{2 B_{1}^{2} B_{2}^{2}} \Vert w \Vert _{2}^{2} - q_{1} E(t) \\& \quad \geq \frac{ k ( q_{1} - 2 ) }{2} \Vert w_{t} \Vert _{2}^{2} + \frac{ k ( Q - B_{2}^{2} \delta ) }{2} \Vert w \Vert _{2}^{2} - q_{1} E(t) \\& \quad \geq \frac{ k ( Q - B_{2}^{2} \delta )}{2} \bigl( \Vert w_{t} \Vert _{2}^{2} + \Vert w \Vert _{2}^{2} \bigr) - q_{1} E(t) \\& \quad \geq k \bigl( Q - B_{2}^{2} \delta \bigr) \biggl( ( w, w_{t} ) - \frac{ q_{1} }{ k ( Q - B_{2}^{2} \delta ) } E(t) \biggr) \quad \text{for } 0 < \delta < \frac{ Q }{B_{2}^{2}} . \end{aligned}$$
Taking
$$ \delta = \frac{ k Q }{ 2 q_{1} + k B_{2}^{2} } , $$
we find
$$\begin{aligned} \frac{d}{dt} \biggl( (w, w_{t} ) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(t) \biggr) \geq \frac{2 k q_{1} Q }{2 q_{1} + k B_{2}^{2}} \biggl( ( w, w_{t} ) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(t) \biggr) . \end{aligned}$$
This completes the proof. □
Theorem 3.3
Let \(q_{1} \geq 4 \) and h satisfy (3.4). If \(0 < E(0) < \frac{2k Q }{2 q_{1} + k B_{2}^{2}} (w_{0}, w_{1}) \), then the solution w blows up in a finite time \(T_{*}\). Moreover, if \(E(0) < \frac{ Q \Vert u_{0} \Vert _{2}^{2} }{ 2 q_{1} B_{1}^{2} B_{2}^{2}} \), then \(T_{*}\) satisfies
$$ T_{*} \leq \frac{ 2 \Vert w_{0} \Vert _{2}^{2} + 2 \alpha \beta ^{2} }{ ( q_{1} -2) \{ (w_{0}, w_{1}) + \alpha \beta \} -2 \Vert \nabla w_{0} \Vert _{2}^{2} }, $$
where
$$\begin{aligned} \begin{gathered} 0 < \alpha \leq \frac{ - 2 q_{1} B_{1}^{2} B_{2}^{2} E(0) + Q \Vert w_{0} \Vert _{2}^{2} }{ q_{1} B_{1}^{2} B_{2}^{2} } \quad \textit{and} \\ \beta > \max \biggl\{ 0, \frac{ - ( q_{1} -2)(w_{0}, w_{1}) + 2 \Vert \nabla w_{0} \Vert _{2}^{2} }{ ( q_{1} -2) \alpha } \biggr\} . \end{gathered} \end{aligned}$$
(3.32)
Proof
Suppose that w is global. Then, using (3.2), we get
$$\begin{aligned} \Vert w \Vert _{2} \leq & \Vert w_{0} \Vert _{2} + \int ^{t}_{0} \bigl\Vert w_{t} (s) \bigr\Vert _{2} \,ds \leq \Vert w_{0} \Vert _{2} + B_{1} t^{\frac{1}{2}} \biggl( \int ^{t}_{0} \bigl\Vert \nabla w_{t} (s) \bigr\Vert _{2}^{2} \,ds \biggr)^{\frac{1}{2}} \\ \leq & \Vert w_{0} \Vert _{2} + B_{2} \bigl( t \bigl( E(0) -E(t) \bigr) \bigr)^{\frac{1}{2}}, \quad t \geq 0 . \end{aligned}$$
(3.33)
In the case \(E(t) \geq 0\) for all \(t\geq 0 \), from (3.33), we see
$$\begin{aligned} \Vert w \Vert _{2}^{2} \leq 2 \Vert w_{0} \Vert _{2}^{2} + 2 B_{2}^{2} E(0)t, \quad t\geq 0. \end{aligned}$$
(3.34)
Applying Lemma 3.3, we also have
$$\begin{aligned} \Vert w \Vert _{2}^{2} = & \Vert w_{0} \Vert _{2}^{2} + 2 \int ^{t}_{0} \bigl(w(s), w_{t} (s) \bigr) \,ds \\ \geq & \Vert w_{0} \Vert _{2}^{2} + 2 \biggl\{ \int ^{t}_{0} e^{rs} \biggl( (w_{0} , w_{1}) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(0) \biggr) \,ds + \int ^{t}_{0} \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(s) \,ds \biggr\} \\ \geq & \Vert w_{0} \Vert _{2}^{2} + 2 \biggl( (w_{0} , w_{1}) - \frac{2 q_{1} + k B_{2}^{2}}{ 2k Q } E(0) \biggr) \frac{ e^{rt} -1 }{r} \quad \text{for all } t\geq 0. \end{aligned}$$
(3.35)
But this contradicts (3.34) for t appropriately large. In the case \(E( t_{1} ) < 0\) for some \(t_{1} > 0 \), there exists the first \(t_{2} >0\) with \(0 < t_{2} < t_{1} \) satisfying \(E(t_{2}) =0 \), \(E(t) >0\) for \(0 \leq t < t_{2}\), and \(E(t_{0}) < 0\) for some \(t_{0} > t_{2}\). Taking \(w(t_{0})\) as a new initial datum, by Theorem 3.1, the solution w blows up after the time \(t_{0}\). This also is a contradiction. Consequently, \(T_{*} < \infty \).
Let F be the function given in (3.8) with (3.32). Then (3.9), (3.10), (3.11), (3.14), and (3.15) are also valid. Taking \(\epsilon = q_{1}\) in (3.15) and using (3.35), we have
$$\begin{aligned} F(t) F''(t) - \frac{q_{1} + 2}{4} \bigl(F'(t) \bigr)^{2} \geq & F(t) \bigl( - 2 q_{1} E(0) - q_{1} \alpha + Q \Vert \Delta w \Vert _{2}^{2} \bigr) \\ \geq & F(t) \biggl( - 2 q_{1} E(0) - q_{1} \alpha + \frac{ Q }{ B_{1}^{2} B_{2}^{2}} \Vert w_{0} \Vert _{2}^{2} \biggr). \end{aligned}$$
From (3.32), we observe
$$ F(t) F''(t) - \frac{ q_{1} +2}{4} \bigl( F'(t) \bigr)^{2} \geq 0 . $$
By the same argument of Theorem 3.1, we complete the proof. □
Theorem 3.4
Let the conditions of one of Theorem 3.1–Theorem 3.3are satisfied. Then the blow-up time \(T_{*}\) verifies
$$ \int ^{\infty }_{D(0)} \frac{1}{2 y + d_{1} y^{3} + d_{2} y^{ q_{1} -1 } + d_{3} y^{ q_{2} -1 } } \,dy \leq T_{*}, $$
(3.36)
where \(D(0) = \Vert w_{1} \Vert _{2}^{2} + \Vert \Delta w_{0} \Vert _{2}^{2} \) and \(d_{i} >0\) (\(i=1,2,3\)) are certain constants.
Proof
We let
$$ D(t) = \Vert w_{t} \Vert _{2}^{2} + \biggl( 1 - \int ^{t}_{0} h(s) \,ds \biggr) \Vert \Delta w \Vert _{2}^{2} + ( h \circ \Delta w ). $$
(3.37)
From (3.5), it is observed
$$ \lim_{t \to T_{*}^{-}} D(t) \geq \lim_{t \to T_{*}^{-}} \bigl( \Vert w_{t} \Vert _{2}^{2} + l \Vert \Delta w \Vert _{2}^{2} \bigr) = \infty . $$
(3.38)
Using (1.1)–(1.4), one gets
$$\begin{aligned} D'(t) = & {-} 2 \Vert \nabla w_{t} \Vert _{2}^{2} - h(t) \Vert \Delta w \Vert _{2}^{2} + \bigl(h' \circ \Delta w \bigr) + 2 \bigl( w_{t}, \bigl[w, \chi ( w ) \bigr] \bigr) + 2 \int _{ \Omega } w_{t} \vert w \vert ^{ q(x) -2} w \,dx \\ \leq & 2 \Vert w_{t} \Vert _{2}^{2} + \bigl\Vert \bigl[w, \chi ( w ) \bigr] \bigr\Vert _{2}^{2} + \int _{ \Omega } \vert w \vert ^{2(q(x) -1 )} \,dx. \end{aligned}$$
(3.39)
From Lemma 2.2 and Lemma 2.3, we see
$$\begin{aligned} \bigl\Vert \bigl[w, \chi ( w ) \bigr] \bigr\Vert _{2}^{2} \leq & \bigl( a_{2} \Vert w \Vert _{H^{2}(\Omega )} \bigl\Vert \chi ( w ) \bigr\Vert _{W^{2, \infty }(\Omega )} \bigr)^{2} \\ \leq & \bigl( a_{1} a_{2} \Vert w \Vert _{H^{2}(\Omega )}^{3} \bigr)^{2} \\ \leq & b_{1} \Vert \Delta w \Vert _{2}^{6} \end{aligned}$$
(3.40)
for some \(b_{1} >0 \). The last term of (3.39) is estimated as
$$\begin{aligned} \int _{\Omega } \vert w \vert ^{2(q(x) -1 )} \,dx \leq & \int _{ \vert w \vert < 1} \vert w \vert ^{2( q_{1} - 1 )} \,dx + \int _{ \vert w \vert \geq 1 } \vert w \vert ^{2( q_{2} - 1 )} \,dx \\ \leq & \Vert w \Vert _{2 ( q_{1} -1 )}^{2 ( q_{1} -1 )} + \Vert w \Vert _{2 ( q_{2} -1 )}^{2 ( q_{2} -1 )} \\ \leq & b_{2} \Vert \Delta w \Vert _{2}^{2 ( q_{1} -1 )} + b_{3} \Vert \Delta w \Vert _{2}^{2 ( q_{2} -1 )} \end{aligned}$$
(3.41)
for some \(b_{2}, b_{3} >0 \). From (3.39), (3.40), (3.41), and (3.37), we arrive at
$$\begin{aligned} D'(t) \leq & 2 \Vert w_{t} \Vert _{2}^{2} + b_{1} \Vert \Delta w \Vert _{2}^{6} + b_{2} \Vert \Delta w \Vert _{2}^{2 ( q_{1} -1 )} + b_{3} \Vert \Delta w \Vert _{2}^{2 ( q_{2} -1 )} \\ \leq & 2 D(t) + d_{1} \bigl(D(t) \bigr)^{3} + d_{2} \bigl(D(t) \bigr)^{ q_{1} -1 } + d_{3} \bigl(D(t) \bigr)^{ q_{2} -1 } \end{aligned}$$
for some \(d_{1}, d_{2}, d_{3} >0\). Using the integration of substitution and (3.38), we get (3.36). □