In this section, we seek the lower bounds for the blow-up time of the solution to problem (1)–(3) in an N-dimensional space \(\Omega \subset{\mathbf{R}}^{N}\) (\({N \ge1}\)).
\(N=1\) case
Suppose \(\Omega = ( {0,l} )\), \(N = 1\), then problem (1)–(3) can be rewritten as
$$\begin{aligned}& u_{t} = \bigl( {u^{m} } \bigr)_{xx} + a ( x )u^{p} \int_{0}^{l} {u^{q} \,dx - u^{s} } ,\quad x \in ( {0,l} ),t \in \bigl( {0,t^{*} } \bigr), \end{aligned}$$
(23)
$$\begin{aligned}& u ( {0,t} ) = u ( {l,t} ) = 0,\quad t \in \bigl( {0,t^{*} } \bigr), \end{aligned}$$
(24)
$$\begin{aligned}& u ( {x,0} ) = u_{0} ( x ) \ge0,\quad x \in ( {0,l} ). \end{aligned}$$
(25)
Theorem 4
Suppose that
\(m > 0\), \(0 \le p < 1\), \(q > 1\), \(\Omega = ( {0,l} )\)
and the weight function
\(a ( x )\)
satisfies (\({a_{1} }\)) or (\({a_{2}}\)). Define an auxiliary function
$$\psi_{1} ( t ) = \int_{0}^{l} {u^{k + 1} \,dx} , $$
where
\(k > \max \{ {2 - m,q - 1} \}\). If the solution
u
to problem (23)–(25) blows up in
\(L^{k+1}\)-norm at
\(t^{*} \), then
\(t^{*} \)
is bounded below by
$$\int_{\psi_{1} ( 0 )}^{\infty}{\frac{{d\eta}}{{H_{1} \eta ^{\frac{{k + p + q}}{{k + 1}}} }}} , $$
where
\(\psi_{1} ( 0 ) = \int_{0}^{l} u_{0}^{k + 1} ( x )\,dx \), positive constant
\(H_{1}\)
will be given in the proof.
Proof
Differentiating \(\psi_{1} ( t )\) and using Green’s formula and Hölder’s inequality, we have
$$\begin{aligned} \psi_{1} ^{\prime}( t ) =& ( {k + 1} ) \int_{0}^{l} {u^{k} u_{t} \,dx} \\ =& - mk ( {k + 1} ) \int_{0}^{l} {u^{k + m - 2} u_{x}^{2} \,dx} + ( {k + 1} ) \int_{0}^{l} {a ( x )} u^{k + p} \,dx \int_{0}^{l} {u^{q} \,dx} \\ &{} - ( {k + 1} ) \int_{0}^{l} {u^{k + s} \,dx} \\ \le& H_{1} \biggl( { \int_{0}^{l} {u^{k + 1} \,dx} } \biggr)^{\frac{{k + p + q}}{{k + 1}}} , \end{aligned}$$
(26)
where \(H_{1} = ( {k + 1} )l^{\frac{{k + 1 - q}}{{k + 1}}} ( {\int_{0}^{l} { ( {a ( x )} )} ^{\frac{{k + 1}}{{1 - p}}} \,dx} )^{\frac{{1 - p}}{{k + 1}}} \).
Hence, applying (26), we can derive that the lower bound for \(t^{*}\) satisfies
$$t^{*} \ge \int_{\psi_{1} ( 0 )}^{\infty}{\frac{{d\eta}}{{H_{1} \eta^{\frac{{k + p + q}}{{k + 1}}} }}} . $$
□
\(N=2\) case
Theorem 5
Suppose that
\(m > 0\), \(0\leq p <1\), \(q > 1\), \(\Omega \subset{\mathbf{R}}^{2} \)
and the weight function satisfies (\({a_{2}}\)) and
- (\({a_{3}}\)):
-
there exists
\(A = ( {A_{1} ,A_{2} } )\)
such that
\(- a ( x )A \le\nabla a ( x ) \le a ( x )A\),
where
\(x \in\Omega\), \(A_{i} > 0\), \(i = 1,2\), while
A
satisfies
\(\vert A \vert ^{2} < \lambda_{1} \). Define a weight function
$$\psi_{2} ( t ) = \int_{\Omega}{a ( x )} u^{k + 1} \,dx, $$
where
\(\lambda_{1} \)
is the first eigenvalue of the fixed membrane problem (8)–(9) for a two-dimensional space, \(k > \max \{ {q - 1,\frac{{ \vert A \vert m}}{{2 ( {\sqrt{\lambda _{1} } - \vert A \vert } )}}} \}\). If the solution
u
to problem (1)–(3) blows up in the measure
\(\psi_{2} \)
at
\(t^{*} \), then
\(t^{*} \)
is bounded below by
$$\int_{\psi_{2} ( 0 )}^{\infty}{\frac{{d\eta}}{{H_{2} \eta ^{\frac{{k + p + q}}{{k + 1}}} }}} , $$
where
\(\psi_{2} ( 0 ) = \int_{\Omega}{a ( x )u_{0}^{k + 1} ( x )} \,dx\), positive constant
\(H_{2}\)
will be given in the proof.
In order to prove Theorem 5, we firstly need to give a lemma.
Lemma 1
Suppose that
\(a ( x )\)
satisfies (\({a_{2}}\)), (\({a_{3}}\)), and
\(\vert A \vert ^{2} < \lambda_{1} \). If
\(u \in C^{1} ( \Omega )\)
is nonnegative, then we have the differential inequality
$$\biggl( {\sqrt{\lambda_{1} } - \frac{{ \vert A \vert }}{2}} \biggr) \biggl( { \int_{\Omega}{a ( x )u^{2k} \,dx} } \biggr)^{\frac{1}{2}} \le \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} ,\quad k > 0,x \in \Omega. $$
Proof
By virtue of the Rayleigh principle, we know
$$\lambda_{1} \int_{\Omega}{\omega^{2} \,dx} \le \int_{\Omega}{ \vert {\nabla \omega} \vert ^{2} \,dx},\quad \mbox{where } \omega|_{\partial\Omega} = 0. $$
Now, choosing \(\omega = a^{\frac{1}{2}} ( x )u^{k}\), by (\({a_{3}}\)) and Hölder’s inequality, we have
$$\begin{aligned} &\lambda_{1} \int_{\Omega}{a ( x )u^{2k} \,dx} \\ &\quad \le \int_{\Omega}{ \bigl\vert {\nabla \bigl( {a^{\frac{1}{2}} ( x )u^{k} } \bigr)} \bigr\vert } ^{2} \,dx \\ &\quad = \frac{1}{4} \int_{\Omega}{\frac{{ \vert {\nabla a ( x )} \vert ^{2} }}{{a ( x )}}u^{2k} \,dx} + \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} + \int_{\Omega}{u^{k} \nabla a ( x )\nabla u^{k} \,dx} \\ &\quad \le\frac{{ \vert A \vert ^{2} }}{4} \int_{\Omega}{a ( x )u^{2k} \,dx} + \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} \\ &\qquad {} + \vert A \vert \biggl( { \int_{\Omega}{a ( x )u^{2k} \,dx} } \biggr)^{\frac{1}{2}} \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} \\ &\quad \le \biggl[ {\frac{{ \vert A \vert }}{2} \biggl( { \int_{\Omega}{a ( x )u^{2k} \,dx} } \biggr)^{\frac{1}{2}} + \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} } \biggr]^{2} . \end{aligned}$$
Hence, we derive
$$\biggl( {\sqrt{\lambda_{1} } - \frac{{ \vert A \vert }}{2}} \biggr) \biggl( { \int_{\Omega}{a ( x )u^{2k} \,dx} } \biggr)^{\frac{1}{2}} \le \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} . $$
□
The proof of Theorem 5 can be given as follows.
Proof
Differentiating \(\psi_{2} ( t )\) and using Green’s formula, Hölder’s inequality, and Lemma 1, we have
$$\begin{aligned} \psi_{2} ^{\prime}( t ) =& ( {k + 1} ) \int_{\Omega}{a ( x )u^{k} u_{t} \,dx} \\ =& - ( {k + 1} ) \int_{\Omega}{ \bigl( {\nabla a ( x )u^{k} + ka ( x )u^{k - 1} \nabla u} \bigr)mu^{m - 1} \nabla u\,dx} \\ &{} + ( {k + 1} ) \int_{\Omega}{a^{2} ( x )u^{k + p} \,dx} \int_{\Omega}{u^{q} \,dx} - ( {k + 1} ) \int_{\Omega}{a ( x )u^{k + s} \,dx} \\ \le&\frac{{2m ( {k + 1} ) \vert A \vert }}{{k + m}} \int _{\Omega}{a ( x )u^{\frac{{k + m}}{2}} \bigl\vert {\nabla u^{\frac {{k + m}}{2}} } \bigr\vert \,dx} \\ &{} -\frac{{4mk ( {k + 1} )}}{{ ( {k + m} )^{2} }} \int _{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{k + m}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + ( {k + 1} ) \int_{\Omega}{a^{2} ( x )u^{k + p} \,dx} \int_{\Omega}{u^{q} \,dx} - ( {k + 1} ) \int_{\Omega}{a ( x )u^{k + s} \,dx} \\ \le&\frac{{2m ( {k + 1} ) \vert A \vert }}{{k + m}} \biggl( { \int_{\Omega}{a ( x )u^{k + m} \,dx} } \biggr)^{\frac{1}{2}} \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{k + m}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} \\ &{} - \frac{{4mk ( {k + 1} )}}{{ ( {k + m} )^{2} }} \int _{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{k + m}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + ( {k + 1} ) \int_{\Omega}{a^{2} ( x )u^{k + p} \,dx} \int_{\Omega}{u^{q} \,dx} \\ \le& \biggl[ {\frac{{4m ( {k + 1} ) \vert A \vert }}{{ ( {2\sqrt{\lambda_{1} } - \vert A \vert } ) ( {k + m} )}} - \frac{{4mk ( {k + 1} )}}{{ ( {k + m} )^{2} }}} \biggr] \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{k + m}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + ( {k + 1} ) \int_{\Omega}{a^{2} ( x )u^{k + p} \,dx} \int_{\Omega}{u^{q} \,dx} . \end{aligned}$$
(27)
By the value of k, we can easily get that the coefficient of \(\int_{\Omega}a ( x ) \vert {\nabla u^{\frac{{k + m}}{2}} } \vert ^{2} \,dx\) in (28) is negative. Then, applying Hölder’s inequality, we have
$$ \psi_{2} ^{\prime}( t ) \le H_{2} \psi_{2}^{\frac{{k + p + q}}{{k + 1}}} , $$
(28)
where \(H_{2} = ( {k + 1} ) ( {\int_{\Omega}( {a ( x )} )^{\frac{{k + 2 - p}}{{1-p}}} \,dx } )^{\frac{{1 - p}}{{k + 1}}} (\int_{\Omega}{ ( {a ( x )} )^{ - \frac{q}{{k + 1 - q}}} \,dx} )^{\frac{{k + 1 - q}}{{k + 1}}} \).
Hence, using (27), we can obtain that the lower bound for \(t^{*}\) satisfies
$$t^{*} \ge \int_{\psi_{2} ( 0 )}^{\infty}\frac{{d\eta }}{{H_{2} \eta^{\frac{{k + p + q}}{{k + 1}}} }} . $$
□
\(N \geq3\) case
Theorem 6
Suppose that
\(m > 0\), \(p + q > \max \{ {m,s} \}\), \(\Omega \subset{\mathbf{R}}^{N}\) (\({N \ge3}\)) is a bounded convex region with smooth boundary, the weight function
\(a ( x )\)
satisfies (\(a_{1}\)) or (\(a_{2}\)) and
-
\((a_{3})^{\prime}\)
:
-
there exists
\(A = ( {A_{1} , \ldots,A_{N} } )\)
such that
\(- a ( x )A \le\nabla a ( x ) \le a ( x )A\),
where
\(x \in\Omega\), \(A_{i} > 0\), \(i = 1, \ldots,N\), \(N \geq3\).
Define a weight function
$$\psi_{3} ( t ) = \int_{\Omega}{a ( x )} u^{Nk} \,dx, $$
where
\(\lambda_{1} \)
is the first eigenvalue of the fixed membrane problem (8)–(9) for the
N-dimensional space, \(k > \max \{ {\frac{1}{N},\frac{{2 ( {N - 2} ) ( {p + q - 1} )}}{N},\frac{{2 - 2m}}{3}} \}\). If the solution
u
to problem (1)–(3) blows up in the measure
\(\psi_{3} \)
at
\(t^{*} \), then
\(t^{*} \)
is bounded below by
$$\int_{\psi_{3} ( 0 )}^{\infty}{\frac{{d\eta}}{{H_{3} \eta ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} + H_{4} \eta^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} + H_{5} }}} , $$
where
\(\psi_{3} ( 0 ) = \int_{\Omega}{a ( x )u_{0}^{Nk} ( x )} \,dx\)
and positive constants
\(H_{3} \), \(H_{4} \), \(H_{5} \), γ, θ
will be given in the proof.
Proof
Differentiating \(\psi_{3} ( t )\) and using Green’s formula, Hölder’s, and Young’s inequalities, and condition \((a_{3})^{\prime}\), we have
$$\begin{aligned} \psi_{3} ^{\prime}( t ) =& Nk \int_{\Omega}{a ( x )u^{Nk - 1} u_{t} \,dx} \\ =& Nk \int_{\Omega}{a ( x )u^{Nk - 1} \Delta u^{m} } \,dx + Nk \int _{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx} \\ &{} - Nk \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx \\ =& - mNk ( {Nk - 1} ) \int_{\Omega}{a ( x )u^{Nk + m - 3} \vert {\nabla u} \vert ^{2} \,dx} \\ &{} - mNk \int_{\Omega}{\nabla a ( x )u^{Nk + m - 2} \nabla u\,dx} \\ &{} + Nk \int_{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx - Nk} \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx \\ \le& - \frac{{4mNk ( {Nk - 1} )}}{{ ( {Nk + m - 1} )^{2} }} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + mNk \vert A \vert \biggl( { \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} } \biggr)^{\frac{1}{2}} \\ &{} \times \biggl( {\frac{4}{{ ( {Nk + m - 1} )^{2} }} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} \\ &{} + Nk \int_{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx - Nk} \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx \\ \le& \biggl[ { - \frac{{4mNk ( {Nk - 1} )}}{{ ( {Nk + m - 1} )^{2} }} + \frac{{2mNk \vert A \vert \varepsilon_{1} }}{{ ( {Nk + m - 1} )^{2} }}} \biggr] \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + \frac{{mNk \vert A \vert }}{{2\varepsilon_{1} }} \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} + Nk \int_{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx} \\ &{} - Nk \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx, \end{aligned}$$
(29)
where \(\varepsilon_{1} \) is a positive constant to be determined later.
To begin with, applying Hölder’s inequality, we get the inequality
$$\begin{aligned} & \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{ \frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{ \bigl( {a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr)} ^{\frac{{2N}}{{N - 2}}} \,dx} \biggr)^{1 - \theta} , \end{aligned}$$
(30)
where \(\theta = \frac{{3k + 2m - 2}}{{4k + 2m - 2}}\). Using Hölder’s and Young’s inequalities to estimate the second term on the right-hand side of (29), we have
$$\begin{aligned} \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} &\le \biggl( { \int _{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac {{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{\gamma}\biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{1} } \,dx} } \biggr)^{1 - \gamma} \\ & \le\gamma \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} + ( {1 - \gamma} ) \int_{\Omega}\bigl( {a ( x )} \bigr)^{\sigma_{1} } \,dx , \end{aligned}$$
(31)
where \(\gamma = \frac{{2 ( {N - 2} ) ( {Nk + m - 1} )}}{{Nk ( {2N - 3} )}}\), \(\sigma_{1}\) such that \(1 = \frac{{N - 2\theta}}{{N - 2}}\gamma + \sigma_{1} ( {1 - \gamma} )\).
Next, applying Hölder’s inequality to the third term on the right-hand side of (29), we obtain the inequality
$$\begin{aligned} & \int_{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx} \\ &\quad \le \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma _{2} } \,dx} } \biggr)^{\frac{q}{{Nk + p + q - 1}}} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{3} } \,dx} } \biggr)^{\frac {{Nk + p - 1}}{{Nk + p + q - 1}}} \\ &\qquad {}\times \int_{\Omega}{a^{m_{1} + \frac{{N - 2\theta}}{{N - 2}}m_{2} } ( x )} u^{Nk + p + q - 1} \,dx, \end{aligned}$$
(32)
where \(m_{1} = \frac{{Nk - 2N ( {p + q} ) + 2N + 4 ( {p + q} ) - 4}}{{Nk - 2Ns + 2N + 4s - 4}}\), \(m_{2} = \frac{{2 ( {p + q - s} ) ( {N - 2} )}}{{Nk - 2Ns + 2N + 4s - 4}}\), \(\sigma_{2} ,\sigma_{3} \) such that
$$\begin{aligned}& 2 = \frac{{Nk + p - 1}}{{Nk + p + q - 1}} \biggl( {m_{1} + \frac{{N - 2\theta}}{{N - 2}}m_{2} } \biggr) + \frac{q}{{Nk + p + q - 1}}\sigma_{2}, \\& 0 = \frac{q}{{Nk + p + q - 1}} \biggl( {m_{1} + \frac{{N - 2\theta }}{{N - 2}}m_{2} } \biggr) + \frac{{Nk + p - 1}}{{Nk + p + q - 1}}\sigma_{3} . \end{aligned}$$
Thus, applying Hölder’s and Young’s inequalities to the third integral term on the right-hand side of (32), we can derive the inequality
$$\begin{aligned} & \int_{\Omega}{a^{m_{1} + \frac{{N - 2\theta}}{{N - 2}}m_{2} } ( x )} u^{Nk + p + q - 1} \,dx \\ &\quad \le \biggl( { \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx} \biggr)^{m_{1} } \biggl( { \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{m_{2} } \\ &\quad \le m_{1} \varepsilon_{2} \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx + m_{2} \varepsilon_{2} ^{ - \frac{{m_{1} }}{{m_{2} }}} \int_{\Omega}a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx , \end{aligned}$$
(33)
where \(\varepsilon_{2} > 0\) is a constant to be determined later.
Now, we substitute (31)–(33) into (29) and choose suitable \(\varepsilon_{2} > 0\) to make the coefficient of \(\int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx\) in (29) vanish, that is,
$$m_{1} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma _{2} } } \,dx} \biggr)^{\frac{p}{{Nk + p + q - 1}}} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{3} } } \,dx} \biggr)^{\frac{{Nk + q - 1}}{{Nk + p + q - 1}}} \varepsilon_{2} = 1. $$
We can obtain the inequality
$$ \psi_{3} ^{\prime}( t ) \le C_{1} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} + C_{2} \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} + C_{3} , $$
(34)
where
$$\begin{aligned} &C_{1} = - \frac{{4mNk ( {Nk - 1} )}}{{ ( {Nk + m - 1} )^{2} }} + \frac{{2mNk \vert A \vert \varepsilon_{1} }}{{ ( {Nk + m - 1} )^{2} }}, \\ &C_{2} = Nkm_{2} \varepsilon_{2} ^{ - \frac{{m_{1} }}{{m_{2} }}} \biggl( { \int _{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{2} } \,dx} } \biggr)^{\frac{p}{{Nk + p + q - 1}}} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{3} } \,dx} } \biggr)^{\frac{{Nk + q - 1}}{{Nk + p + q - 1}}} + \frac{{mNk \vert A \vert \gamma}}{{2\varepsilon_{1} }}, \\ &C_{3} = \frac{{mNk \vert A \vert ( {1 - \gamma} )}}{{2\varepsilon_{1} }} \int_{\Omega}\bigl( {a ( x )} \bigr)^{\sigma_{1} } \,dx. \end{aligned}$$
In order to deal with the gradient term in (34), we will use Sobolev’s inequality
$$ \biggl( { \int_{\Omega}{ \bigl( {a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr)} ^{\frac{{2N}}{{N - 2}}} \,dx} \biggr)^{\frac{{N - 2}}{{2N}}} \le C_{s} \biggl( { \int_{\Omega}{ \bigl\vert {\nabla \bigl( {a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr)} \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} , $$
(35)
where \(C_{s} \) is the optimal Sobolev constant. Applying condition \((a_{3})^{\prime}\) and Hölder’s inequality to the right-hand side of (35), we have
$$\begin{aligned} & \int_{\Omega}{ \bigl\vert {\nabla \bigl( {a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr)} \bigr\vert ^{2} \,dx} \\ &\quad \le\frac{{ \vert A \vert ^{2} }}{4} \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} + \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} \\ &\qquad {} + \vert A \vert \int_{\Omega}{a ( x )u^{\frac{{Nk + m - 1}}{2}} \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert \,dx} \\ &\quad \le \biggl[ {\frac{{ \vert A \vert }}{2} \biggl( { \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} } \biggr)^{\frac{1}{2}} + \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} } \biggr]^{2} . \end{aligned}$$
(36)
Now, using (30), (35), and (36) to estimate the second term on the right-hand side of (34), we obtain
$$\begin{aligned} & \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le C_{s}^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \biggl( { \int _{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int _{\Omega}{ \bigl\vert {\nabla a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )}}{{N - 2}}} \\ &\quad \le C_{s} ^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta} \\ & \qquad {}\times \biggl[ {\frac{{ \vert A \vert }}{2} \biggl( { \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} } \biggr)^{\frac{1}{2}} + \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} } \biggr]^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} . \end{aligned}$$
(37)
Next, applying the inequality
$$( {x + y} )^{k} \le \bigl( {2\max \{ {x,y} \}} \bigr)^{k} \le2^{k} \max \bigl\{ {x^{k} ,y^{k} } \bigr\} \le2^{k} \bigl( {x^{k} + y^{k} } \bigr), \quad \forall x,y,k \ge0, $$
and (31), we derive an estimate for the summation of the bracket in (37) as follows:
$$\begin{aligned} & \biggl[ {\frac{{ \vert A \vert }}{2} \biggl( { \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} } \biggr)^{\frac{1}{2}} + \biggl( { \int _{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} } \biggr]^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \\ &\quad \le \vert A \vert ^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma _{1} } \,dx} } \biggr)^{\frac{{N ( {1 - \theta} ) ( {1 - \gamma} )}}{{N - 2}}} \biggl( { \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )\gamma}}{{N - 2}}} \\ &\qquad {} + 2^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )}}{{N - 2}}} . \end{aligned}$$
(38)
Substituting (38) into (37), we have
$$\begin{aligned} & \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le C_{4} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )\gamma}}{{N - 2}}} \\ &\qquad {} + C_{5} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )}}{{N - 2}}} , \end{aligned}$$
(39)
where \(C_{4} = ( {C_{s} \vert A \vert } )^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} ( {\int_{\Omega}{ ( {a ( x )} )^{\sigma_{1} } \,dx} } )^{\frac{{N ( {1 - \theta} ) ( {1 - \gamma} )}}{{N - 2}}} \), \(C_{5} = ( {2C_{s} } )^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \).
Now, applying Young’s inequality to the two terms on the right-hand side of (39), we can get
$$\begin{aligned} & \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )\gamma}}{{N - 2}}} \\ &\quad \le\frac{{N - 2 - N ( {1 - \theta} )\gamma}}{{N - 2}}\varepsilon_{3} ^{ - \frac{{N ( {1 - \theta} )\gamma }}{{N - 2 - N ( {1 - \theta} )\gamma}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac {{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} \\ &\qquad {} + \frac{{N ( {1 - \theta} )\gamma\varepsilon_{3} }}{{N - 2}} \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} , \end{aligned}$$
(40)
and
$$\begin{aligned} & \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )}}{{N - 2}}} \\ &\quad \le\frac{{N - 2 - N ( {1 - \theta} )}}{{N - 2}}\varepsilon_{4} ^{ - \frac{{N ( {1 - \theta} )}}{{N - 2 - N ( {1 - \theta} )}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac {{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} \\ &\qquad {} + \frac{{N ( {1 - \theta} )\varepsilon_{4} }}{{N - 2}} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} , \end{aligned}$$
(41)
where \(\varepsilon_{3} ,\varepsilon_{4} > 0\) are constants to be determined later. Substituting (40), (41) into (39) leads to
$$\begin{aligned} & \biggl( {1 - \frac{{C_{4} N ( {1 - \theta} )\gamma \varepsilon_{3} }}{{N - 2}}} \biggr) \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le\frac{{N - 2 - N ( {1 - \theta} )\gamma}}{{N - 2}}C_{4} \varepsilon_{3}^{ - \frac{{N ( {1 - \theta} )\gamma }}{{N - 2 - N ( {1 - \theta} )\gamma}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac {{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} \\ &\qquad {} + \frac{{N - 2 - N ( {1 - \theta} )}}{{N - 2}}C_{5} \varepsilon_{4} ^{ - \frac{{N ( {1 - \theta} )}}{{N - 2 - N ( {1 - \theta} )}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta } )}}} \\ &\qquad {} + \frac{{C_{5} N ( {1 - \theta} )\varepsilon_{4} }}{{N - 2}} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} . \end{aligned}$$
Choose \(\varepsilon_{3} > 0\) small enough such that \(\rho: = 1 - \frac{{C_{4} N ( {1 - \theta} )\gamma \varepsilon_{3} }}{{N - 2}} > 0\).
It follows that the second term on the right-hand side of (34) satisfies
$$\begin{aligned} & \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le C_{6} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} + C_{7} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} \\ &\qquad {} + C_{8} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} , \end{aligned}$$
(42)
where
$$\begin{aligned} &C_{6} = \frac{{N - 2 - N ( {1 - \theta} )\gamma }}{{ ( {N - 2} )\rho}}C_{4} \varepsilon_{3} ^{ - \frac{{\gamma N ( {1 - \theta} )}}{{N - 2 - \gamma N ( {1 - \theta} )}}} , \\ &C_{7} = C_{5} \frac{{N - 2 - N ( {1 - \theta} )}}{{ ( {N - 2} )\rho}} \varepsilon_{4} ^{\frac{{N ( {\theta - 1} )}}{{N - 2 - N ( {1 - \theta} )}}} , \\ &C_{8} = \frac{{C_{5} N ( {1 - \theta} )\varepsilon_{4} }}{{ ( {N - 2} )\rho}}. \end{aligned}$$
Then, substituting (42) into (44), we can derive
$$\begin{aligned} \psi_{3} ^{\prime}( t ) &\le ( {C_{1} + C_{2} C_{8} } ) \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} \\ & \quad {}+ C_{2} C_{6} \psi_{3} ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} + C_{2} C_{7} \psi_{3} ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} + C_{3} . \end{aligned}$$
(43)
Choose \(\varepsilon_{1} \) small enough such that \(C_{1} < 0\) and \(\varepsilon_{4} \) such that \(C_{1} + C_{2} C_{8} = 0\). Therefore, (43) can be rewritten as
$$ \psi_{3} ^{\prime}( t ) \le H_{3} \psi_{3} ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma }}} + H_{4} \psi_{3} ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} + H_{5} , $$
(44)
where \(H_{3} = C_{2} C_{6} \), \(H_{4} = C_{2} C_{7} \), \(H_{5} = C_{3} \).
Note that \(\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}} > 1\), then integrating (44) from 0 to \(t^{*} \), we derive
$$t^{*} \ge \int_{\psi_{3} ( 0 )}^{\infty}{\frac{{d\eta }}{{H_{3} \eta^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} + H_{4} \eta^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} + H_{5} }}} . $$
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Remark 3
If the null Dirichlet boundary condition (2) is replaced by the null Neumann boundary condition
$$ \frac{{\partial u}}{{\partial\nu}} ( {x,t} ) = 0,\quad (x,t)\in \partial\Omega\times \bigl(0,t^{\ast}\bigr), $$
(2′)
where ν is the unit outward normal vector on ∂Ω, then Theorem 1 is valid for the case \(m \ge1\), and Theorems 4–6 are also valid.