In this section, we define the auxiliary function \(\varphi=\varphi(t)\) as follows (see [1]):
$$ \varphi(t)= \int_{\Omega}u^{2(n-1)(q+1)+2}\,dx= \int_{\Omega}u^{\sigma }\,dx \quad\mbox{with } \sigma=2(n-1) (q+1)+2. $$
(2.1)
We establish the following theorem.
Theorem 1
Assume that
\(u=u(x,t)\)
is the classical nonnegative solution of the mixed problem (1.1)-(1.3) in a bounded domain
\(\Omega\in\mathbb{R}^{N} \) (\(N\geq3\)). Then the quantity
\(\varphi(t)\)
defined in (2.1) satisfies the differential inequality
$$ \varphi'(t) \leq\sigma a_{1}| \Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma }}+k_{1} \bigl[ \phi(t) \bigr]^{\frac{(N-2)\alpha}{N\alpha-2}}+ k_{2} \bigl[\phi(t) \bigr]^{\frac{(N-2)\alpha'}{N\alpha'-2}}, $$
(2.2)
which yields that the blow-up time
\(t^{*}\)
is bounded from below. We have
$$ t ^{*}\geq \int_{\phi(0)}^{+\infty}\frac{d\xi}{\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}[\xi]^{\frac{\sigma-1}{\sigma}}+k_{1} [\xi ]^{\frac{(N-2)\alpha}{N\alpha-2}} +k_{2} [\xi ]^{\frac{(N-2)\alpha'}{N\alpha'-2}}}, $$
(2.3)
where
\(|\Omega|\)
is the volume of the domain Ω, and
\(k_{1}\), \(k_{2}\)
are positive constants that will be defined later.
Proof
First, we compute
$$\begin{aligned} \varphi'(t)={}&\sigma \int_{\Omega}u^{\sigma-1} \bigl[\bigl(\rho\bigl(|\nabla u|^{2}\bigr)u_{,i}\bigr)_{,i}+f(u) \bigr]\,dx \\ ={}&{-} \sigma(\sigma-1) \int_{\Omega}u^{\sigma-2}\rho\bigl(|\nabla u|^{2} \bigr)|\nabla u|^{2}\,dx+\sigma \int_{\Omega}u^{\sigma-1}f(u)\,dx \\ \leq{}& {-} \sigma(\sigma-1) \int_{\Omega}u^{2(n-1)(q+1)}|\nabla u|^{2} \bigl(b_{1}+b_{2}|\nabla u|^{2q}\bigr)\,dx \\ &{}+\sigma \int_{\Omega}u^{\sigma-1}\bigl(a_{1}+a_{2}u^{p} \bigr)\,dx. \end{aligned}$$
(2.4)
Using the equality
$$ \bigl|\nabla u^{n}\bigr|^{2(q+1)}=\bigl|nu^{n-1}\nabla u\bigr|^{2(q+1)}=n^{2(q+1)}u^{2(n-1)(q+1)}|\nabla u|^{2(q+1)} $$
and the Hölder inequality, we get
$$ \varphi'(t) \leq - \frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}} \int_{\Omega}\bigl|\nabla u^{n}\bigr|^{2(q+1)}\,dx+\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma}}+\sigma a_{2} \int_{\Omega}u^{\sigma+p-1}\,dx. $$
(2.5)
If we set \(v=u^{n}\), then we obtain
$$ \varphi'(t) \leq - \frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}} \int_{\Omega}|\nabla v|^{2(q+1)}\,dx+\sigma a_{1}| \Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma}}+\sigma a_{2} \int_{\Omega}v^{2(q+1)+\frac{\gamma}{n}}\,dx, $$
(2.6)
where \(\gamma=p-1-2q>0\). After application of the Hölder and Schwarz inequalities, it follows
$$\begin{aligned} \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx& \leq(q+1)^{2} \biggl( \int_{\Omega}|\nabla v|^{2(q+1)}\,dx \biggr)^{\frac{1}{q+1}} \biggl( \int_{\Omega}| v|^{2(q+1)}\,dx \biggr)^{\frac{q}{q+1}} \\ &\leq(q+1) \int_{\Omega}|\nabla v|^{2(q+1)}\,dx+(q+1)q \int_{\Omega}| v|^{2(q+1)}\,dx. \end{aligned}$$
(2.7)
Combining (2.6) and (2.7), we easily obtain
$$\begin{aligned} \varphi'(t) \leq{}& {-} \frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}(q+1)} \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx+ \frac{q\sigma(\sigma-1)b_{2}}{n^{2(q+1)}} \int_{\Omega}v^{2(q+1)}\,dx+\sigma a_{1}| \Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma}} \\ &{}+\sigma a_{2} \int_{\Omega}v^{2(q+1)+\frac{\gamma}{n}}\,dx. \end{aligned}$$
(2.8)
We choose \(x_{1}\), \(x_{2}\), and α such that
$$ x_{1}+x_{2}=2(q+1), \qquad x_{1}\cdot \frac{1}{\alpha}=\frac{\sigma}{n},\qquad x_{2}\cdot \frac{1}{1-\alpha}=(q+1)\frac{2N}{N-2}, $$
so that
$$ \begin{aligned} &x_{1}=\frac{\sigma}{n}\frac{2(q+1)\frac{2}{N-2}}{2(q+1)\frac {N}{N-2}-\frac{\sigma}{n}}, \qquad x_{2}=2(q+1)- \frac{\sigma}{n}\frac{2(q+1)\frac{2}{N-2}}{2(q+1)\frac{N}{N-2}-\frac {\sigma}{n}}, \\ &\alpha=\frac{2(q+1)\frac{2}{N-2}}{2(q+1)\frac{N}{N-2}-\frac{\sigma}{n}}. \end{aligned} $$
Then the Hölder inequality (1.7) yields
$$ \int_{\Omega}v^{2(q+1)}\,dx\leq \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\alpha}\biggl( \int_{\Omega}v^{(q+1)\frac{2N}{N-2}}\,dx \biggr)^{1-\alpha}. $$
(2.9)
We follow the same procedure for \(x_{1}'\), \(x_{2}'\), and \(\alpha'\), that is, we choose them such that
$$ x_{1}'+x_{2}'=2(q+1)+ \frac{\gamma}{n}, \qquad x_{1}\cdot \frac{1}{\alpha'}= \frac{\sigma}{n}, \qquad x_{2}'\cdot\frac{1}{1-\alpha'}=(q+1)\frac{2N}{N-2}, $$
so that
$$ \begin{aligned} &x_{1}'=\frac{\sigma}{n} \frac{2(q+1)\frac{2}{N-2}-\frac{\gamma }{n}}{2(q+1)\frac{N}{N-2}-\frac{\sigma}{n}}, \\ &x_{2}'=2(q+1)+\frac{\gamma}{n}- \frac{\sigma}{n} \frac{2(q+1)\frac{2}{N-2}-\frac{\gamma}{n}}{2(q+1)\frac {N}{N-2}-\frac{\sigma}{n}}, \\ &\alpha'=\frac{2(q+1)\frac{2}{N-2}-\frac{\gamma}{n}}{2(q+1)\frac {N}{N-2}-\frac{\sigma}{n}}, \end{aligned} $$
and obtain
$$ \int_{\Omega}v^{2(q+1)+\frac{\gamma}{n}}\,dx\leq \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\alpha'} \biggl( \int_{\Omega}v^{(q+1)\frac{2N}{N-2}}\,dx \biggr)^{1-\alpha'}. $$
(2.10)
Stressing the Sobolev inequality gives \(W_{0}^{1,2}\hookrightarrow L^{\frac{2N}{N-2}}\) for \(N\geq3\). Consequently, we get
$$ \bigl\| v^{q+1}\bigr\| _{L^{\frac{2N}{N-2}}}^{\frac{2N}{N-2}(1-\alpha)}\leq c_{1}^{\frac{2N}{N-2}(1-\alpha)}\bigl\| \nabla v^{q+1}\bigr\| _{L^{2}}^{\frac{2N}{N-2}(1-\alpha)} $$
(2.11)
and
$$ \bigl\| v^{q+1}\bigr\| _{L^{\frac{2N}{N-2}}}^{\frac{2N}{N-2}(1-\alpha')}\leq c_{1}^{\frac{2N}{N-2}(1-\alpha')}\bigl\| \nabla v^{q+1}\bigr\| _{L^{2}}^{\frac{2N}{N-2}(1-\alpha')}, $$
(2.12)
where \(c_{1}\) is the best embedding constant (see [18]).
A combination of (2.9) and (2.11) leads to
$$ \int_{\Omega}v^{2(q+1)}\,dx\leq c_{1}^{\frac{2N(1-\alpha)}{N-2}} \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\alpha}\biggl( \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx \biggr)^{\frac{N(1-\alpha)}{N-2}}. $$
(2.13)
An application of the Young inequality yields
$$\begin{aligned} \int_{\Omega}v^{2(q+1)}\,dx\leq{}& \frac{N\alpha-2}{N-2}c_{1}^{\frac{2N(1-\alpha)}{N\alpha-2}} \varepsilon _{1}^{-{\frac{N(1-\alpha)}{N\alpha-2}}} \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\frac{(N-2)\alpha}{N\alpha-2}} \\ &{}+ \frac {N(1-\alpha)}{N-2}\varepsilon_{1} \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx, \end{aligned}$$
(2.14)
where \(\varepsilon_{1}\) is a positive constant to be determined later.
A combination of (2.9) and (2.11) also leads to
$$\begin{aligned} \int_{\Omega}v^{2(q+1)+\frac{\gamma}{n}}\,dx\leq{}& \frac{N\alpha'-2}{N-2}c_{1}^{\frac{2N(1-\alpha')}{N\alpha'-2}} \varepsilon _{2}^{-{\frac{N(1-\alpha')}{N\alpha'-2}}} \biggl( \int_{\Omega}v^{\frac{\sigma}{n}}\,dx \biggr)^{\frac{(N-2)\alpha'}{N\alpha'-2}} \\ &{}+\frac {N(1-\alpha')}{N-2}\varepsilon_{2} \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx, \end{aligned}$$
(2.15)
where \(\varepsilon_{2}\) is a positive constant to be determined later.
Combining (2.8), (2.14), and (2.15), we obtain
$$\begin{aligned} \varphi'(t) \leq{}& {-} \biggl[\frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}(q+1)}- \frac{q\sigma(\sigma -1)b_{2}}{n^{2(q+1)}}\frac{N(1-\alpha)}{N-2}\varepsilon_{1}-\sigma a_{2}\frac{N(1-\alpha')}{N-2}\varepsilon_{2} \biggr] \int_{\Omega}\bigl|\nabla v^{q+1}\bigr|^{2}\,dx \\ &{}+\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}\bigl[\phi(t) \bigr]^{\frac{\sigma-1}{\sigma}}+\frac {N\alpha-2}{N-2}c_{1}^{\frac{2N(1-\alpha)}{N\alpha-2}} \varepsilon_{1}^{-{\frac{N(1-\alpha)}{N\alpha-2}}}\frac{q\sigma (\sigma-1)b_{2}}{n^{2(q+1)}} \bigl[\phi(t) \bigr]^{\frac{(N-2)\alpha }{N\alpha-2}} \\ &{}+\frac{N\alpha'-2}{N-2}c_{1}^{\frac{2N(1-\alpha')}{N\alpha'-2}} \varepsilon_{2}^{-{\frac{N(1-\alpha')}{N\alpha'-2}}} \bigl[\phi(t) \bigr]^{\frac {(N-2)\alpha'}{N\alpha'-2}}. \end{aligned}$$
(2.16)
By choosing \(\varepsilon_{1}\) and \(\varepsilon_{2}\) small enough such that
$$ \frac{\sigma(\sigma-1)b_{2}}{n^{2(q+1)}(q+1)}-\frac{q\sigma(\sigma -1)b_{2}}{n^{2(q+1)}}\frac{N(1-\alpha)}{N-2} \varepsilon_{1}-\sigma a_{2}\frac{N(1-\alpha')}{N-2} \varepsilon_{2}\geq0 $$
(2.17)
we get the differential inequality
$$ \varphi'(t) \leq\sigma a_{1}| \Omega|^{\frac{1}{\sigma}}\bigl[\phi(t)\bigr]^{\frac{\sigma-1}{\sigma }}+k_{1} \bigl[ \phi(t) \bigr]^{\frac{(N-2)\alpha}{N\alpha-2}}+ k_{2} \bigl[\phi(t) \bigr]^{\frac{(N-2)\alpha'}{N\alpha'-2}} $$
(2.18)
with \(k_{1}=\frac{N\alpha-2}{N-2}c_{1}^{\frac{2N(1-\alpha)}{N\alpha -2}}\varepsilon_{1}^{-{\frac{N(1-\alpha)}{N\alpha-2}}}\) and \(k_{2}=\frac{N\alpha'-2}{N-2}c_{1}^{\frac{2N(1-\alpha')}{N\alpha'-2}} \varepsilon_{2}^{-{\frac{N(1-\alpha')}{N\alpha'-2}}}\).
Inequality (2.18) can be rewritten as
$$ \frac{d\phi}{\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}[\phi(t)]^{\frac{\sigma-1}{\sigma }}+k_{1} [\phi(t) ]^{\frac{(N-2)\alpha}{N\alpha-2}} +k_{2} [\phi(t) ]^{\frac{(N-2)\alpha'}{N\alpha'-2}}} \leq \,dt. $$
(2.19)
An integration of (2.19) from 0 to t leads to
$$ \int_{\phi(0)}^{\phi(t)}\frac{d\xi}{\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}[\xi]^{\frac{\sigma-1}{\sigma}}+k_{1} [\xi ]^{\frac{(N-2)\alpha}{N\alpha-2}} +k_{2} [\xi ]^{\frac{(N-2)\alpha'}{N\alpha'-2}}} \leq t. $$
(2.20)
Taking the limit as \(t\longrightarrow t^{*}\), we obtain
$$ \int_{\phi(0)}^{+\infty}\frac{d\xi}{\sigma a_{1}|\Omega|^{\frac{1}{\sigma}}[\xi]^{\frac{\sigma-1}{\sigma}}+k_{1} [\xi ]^{\frac{(N-2)\alpha}{N\alpha-2}} +k_{2} [\xi ]^{\frac{(N-2)\alpha'}{N\alpha'-2}}} \leq t^{*}, $$
(2.21)
and the proof is complete. □