In this section, we prove our main result.
Lemma 3.1
Suppose
\(\Omega\subset\mathbb{R}^{N}\) (\(N\ge3\)) is a bounded domain with smooth boundaries, \(\chi_{1}, \chi_{2}\in\mathbb{R}^{+}\). For any
\(k>1\), \(\eta>0\)
and
\(s_{0}>0\), there exists
\(\mu_{k, \eta}>0\)
and
\(C=C(k, \vert \Omega \vert , \mu_{1}, \mu_{2}, \chi_{1}, \chi_{2}, \eta, u_{0}, v_{0}, w_{0})>0\)
such that if
\(\min\{\mu_{1}, \mu_{2}\}>\mu_{k, \eta}\), then
$$\begin{aligned} \bigl\Vert u(\cdot, t) \bigr\Vert _{L^{k}(\Omega)}+\bigl\Vert v( \cdot, t)\bigr\Vert _{L^{k}(\Omega)}\le C \end{aligned}$$
(3.1)
for all
\(t\in(s_{0},\infty)\).
Proof
We fix \(s_{0}\in(0,T_{\max})\) such that \(s_{0}\le1\). For any constant \(k>1\), we take \(u^{k-1}\) as a test function for the first equation in (1.1) and integrate by parts. Then we have
$$\begin{aligned} \frac{1}{k}\frac{d}{dt} \int_{\Omega}u^{k}&=-(k-1) \int_{\Omega}u^{k-2}\phi(u) \vert \nabla u \vert ^{2}+\chi_{1}(k-1) \int_{\Omega}u^{k-1}\nabla u\cdot\nabla w \\ &\quad{}+\mu_{1} \int_{\Omega}u^{k}-\mu_{1} \int_{\Omega}u^{k+1}-\mu_{1}a_{1} \int_{\Omega}u^{k}v \\ &\le\chi_{1}\frac{k-1}{k} \int_{\Omega}\nabla u^{k}\cdot\nabla w+\mu_{1} \int_{\Omega}u^{k}-\mu_{1} \int_{\Omega}u^{k+1}-\mu_{1}a_{1} \int_{\Omega}u^{k}v \\ &=-\chi_{1}\frac{k-1}{k} \int_{\Omega}u^{k}\Delta w+\mu_{1} \int_{\Omega}u^{k}-\mu_{1} \int_{\Omega}u^{k+1}-\mu_{1}a_{1} \int_{\Omega}u^{k}v \\ &\le-\frac{\gamma(k+1)}{k} \int_{\Omega}u^{k}-\chi_{1}\frac{k-1}{k} \int_{\Omega}u^{k}\Delta w \\ &\quad {}+ \biggl(\mu_{1}+ \frac{\gamma(k+1)}{k} \biggr) \int_{\Omega}u^{k}-\mu_{1} \int_{\Omega}u^{k+1} \end{aligned}$$
(3.2)
for all \(t\in(s_{0},T_{\max})\). Then Young’s inequality implies the following two inequalities for any \(\varepsilon>0\) (to be determined) and some constants \(c_{1}\) and \(c_{2}\):
$$\begin{aligned} \biggl(\mu_{1}+\frac{\gamma(k+1)}{k} \biggr) \int_{\Omega}u^{k}\le \varepsilon \int_{\Omega}u^{k+1}+c_{1} \vert \Omega \vert \end{aligned}$$
(3.3)
and
$$\begin{aligned} -\chi_{1}\frac{k-1}{k} \int_{\Omega}u^{k}\Delta w\le\chi_{1} \int_{\Omega}u^{k} \vert \Delta w \vert \le\eta \int_{\Omega}u^{k+1}+c_{2}\eta^{-k} \chi_{1}^{k+1} \int_{\Omega} \vert \Delta w \vert ^{k+1}, \end{aligned}$$
(3.4)
where \(c_{1}=c_{1}(\mu_{1}, \varepsilon, k, \gamma)=\frac{1}{k} (1+\frac{1}{k} )^{-(k+1)}\varepsilon^{-k} (\mu_{1}+\frac{\gamma(k+1)}{k} )^{k+1}\) and \(c_{2}=\sup_{k>1}\frac{1}{k} (1+\frac{1}{k} )^{-(k+1)}<\infty\). By substituting (3.3) and (3.4) into (3.2), we find that
$$\begin{aligned} \frac{d}{dt} \biggl(\frac{1}{k} \int_{\Omega}u^{k} \biggr)&\le -\gamma(k+1) \biggl( \frac{1}{k} \int_{\Omega}u^{k} \biggr)-(\mu_{1}- \varepsilon-\eta) \int_{\Omega}u^{k+1} \\ &\quad {}+c_{2}\eta^{-k}\chi_{1}^{k+1} \int_{\Omega}\vert \Delta w \vert ^{k+1}+c_{1} \vert \Omega\vert . \end{aligned}$$
(3.5)
Similarly, for some constants \(c_{3}\) and \(c_{4}\), we have
$$\begin{aligned} \frac{d}{dt} \biggl(\frac{1}{k} \int_{\Omega}v^{k} \biggr)&\le-\gamma(k+1) \biggl( \frac{1}{k} \int_{\Omega}v^{k} \biggr)-(\mu_{2}- \varepsilon-\eta) \int_{\Omega}v^{k+1} \\ &\quad {}+c_{4}\eta^{-k}\chi_{2}^{k+1} \int_{\Omega}\vert \Delta w \vert ^{k+1}+c_{3} \vert \Omega\vert . \end{aligned}$$
(3.6)
Applying the variation-of-constants formula to the above inequalities shows that
$$\begin{aligned} \frac{1}{k} \int_{\Omega}u^{k}(\cdot,t)&\le e^{-\gamma(k+1)(t-s_{0})} \frac{1}{k} \int_{\Omega}u^{k}(\cdot,s_{0})-( \mu_{1}-\varepsilon-\eta) \int_{s_{0}}^{t}e^{-\gamma (k+1)(t-s)} \int_{\Omega}u^{k+1} \\ &\quad{}+c_{2}\eta^{-k}\chi_{1}^{k+1} \int_{s_{0}}^{t}e^{-\gamma(k+1)(t-s)} \int _{\Omega}\vert \Delta w \vert ^{k+1}+c_{1} \vert \Omega\vert \int_{s_{0}}^{t}e^{-\gamma(k+1)(t-s)} \\ &\le-(\mu_{1}-\varepsilon-\eta)e^{-\gamma(k+1)t} \int_{s_{0}}^{t} \int_{\Omega}e^{\gamma(k+1)s} u^{k+1} \\ &\quad{}+c_{2}\eta^{-k}\chi_{1}^{k+1}e^{-\gamma(k+1)t} \int_{s_{0}}^{t} \int _{\Omega}e^{\gamma(k+1)s} \vert \Delta w \vert ^{k+1}+c_{5} \end{aligned}$$
(3.7)
and
$$\begin{aligned} \frac{1}{k} \int_{\Omega}v^{k}(\cdot,t) &\le-(\mu_{2}- \varepsilon-\eta)e^{-\gamma(k+1)t} \int_{s_{0}}^{t} \int_{\Omega}e^{\gamma(k+1)s} v^{k+1} \\ &\quad{}+c_{4}\eta^{-k}\chi_{2}^{k+1}e^{-\gamma(k+1)t} \int_{s_{0}}^{t} \int _{\Omega}e^{\gamma(k+1)s} \vert \Delta w \vert ^{k+1}+c_{6} \end{aligned}$$
(3.8)
for all \(t\in(s_{0},T_{\max})\), where
$$\begin{aligned} c_{5}=c_{1} \vert \Omega \vert \int_{s_{0}}^{t}e^{-\gamma(k+1)(t-s)}+\frac{1}{k} \int_{\Omega}u^{k}(\cdot,s_{0}) \end{aligned}$$
and
$$\begin{aligned} c_{6}=c_{3} \vert \Omega \vert \int_{s_{0}}^{t}e^{-\gamma(k+1)(t-s)}+\frac{1}{k} \int_{\Omega}v^{k}(\cdot,s_{0}) \end{aligned}$$
are independent of t. Now, we apply Lemma 2.2 to see that there is \(C_{k}>0\) such that
$$\begin{aligned} \int_{s_{0}}^{t} \int_{\Omega}e^{\gamma(k+1)s}\vert \Delta w\vert ^{k+1} &\le C_{k} \int_{s_{0}}^{t} \int_{\Omega}e^{\gamma (k+1)s}u^{k+1}+C_{k} \int_{s_{0}}^{t} \int_{\Omega}e^{\gamma (k+1)s}v^{k+1} \\ &\quad {}+C_{k} \bigl\Vert w(\cdot, s_{0}) \bigr\Vert ^{k+1}_{W^{2,k+1}(\Omega)}. \end{aligned}$$
(3.9)
Put the inequalities (3.7) and (3.8) together and apply (3.9); then we arrive at
$$\begin{aligned} &\frac{1}{k} \biggl( \int_{\Omega}u^{k}(\cdot,t)+ \int_{\Omega}v^{k}(\cdot,t) \biggr) \\ &\quad\le -\bigl(\mu_{1}-\varepsilon-\eta-c_{2} \eta^{-k}\chi_{1}^{k+1}C_{k}\bigr) \int_{s_{0}}^{t} \int _{\Omega}e^{\gamma(k+1)s} u^{k+1} \\ &\qquad{}-\bigl(\mu_{2}-\varepsilon-\eta-c_{4} \eta^{-k}\chi_{2}^{k+1}C_{k}\bigr) \int _{s_{0}}^{t} \int_{\Omega}e^{\gamma(k+1)s} v^{k+1}+c_{7} \end{aligned}$$
(3.10)
for all \(t\in(s_{0},T_{\max})\), with the constant \(c_{7}>0\) being independent of t.
Let \(\mu_{k,\eta}=\max\{\eta+c_{2}\eta^{-k}\chi_{1}^{k+1}C_{k}, \eta+c_{4}\eta^{-k}\chi_{2}^{k+1}C_{k}\}\), which is independent of ε. We can choose \(\varepsilon\in(0,\min\{\mu_{1},\mu_{2}\}-\mu_{k,\eta})\) such that
$$\begin{aligned} \mu_{1}-\varepsilon-\eta-c_{2}\eta^{-k} \chi_{1}^{k+1}C_{k}>0,\qquad\mu _{2}- \varepsilon-\eta-c_{4}\eta^{-k}\chi_{2}^{k+1}C_{k}>0. \end{aligned}$$
It entails
$$\begin{aligned} \frac{1}{k} \biggl( \int_{\Omega}u^{k}(\cdot,t)+ \int_{\Omega}v^{k}(\cdot,t) \biggr)\le c_{8} \end{aligned}$$
(3.11)
for all \(t\in(s_{0},T_{\max})\), with the constant \(c_{8}=c_{8}(\mu_{1}, \varepsilon, \eta, k, \gamma, w(s_{0}))\) being independent of t. This completes the proof. □
In order to prove Theorem 1.1, we should give an estimation for \((u,v.w)\) when \(t\in(0,s_{0})\). We know by Lemma 2.1 that \(u(\cdot,s_{0}), v(\cdot,s_{0}), w(\cdot,s_{0})\in C^{2}(\bar{\Omega})\) with \(\frac{\partial w(\cdot,s_{0})}{\partial n}=0\) on ∂Ω, so that we can pick \(M>0\) such that
$$\begin{aligned} \textstyle\begin{cases} \sup_{0\le t\le s_{0}} \Vert u(\cdot,t) \Vert _{L^{\infty}(\Omega)}\le M,\qquad \sup_{0\le t\le s_{0}} \Vert v(\cdot,t) \Vert _{L^{\infty}(\Omega)}\le M,\\ \sup_{0\le t\le s_{0}} \Vert w(\cdot,t) \Vert _{L^{\infty}(\Omega)}\le M,\qquad \sup_{0\le t\le s_{0}}\Vert \Delta w(\cdot,t)\Vert _{L^{\infty}(\Omega)}\le M. \end{cases}\displaystyle \end{aligned}$$
(3.12)
Combining Lemma 3.1 with the estimates (3.12), we readily arrive at our main result.
Proof of Theorem 1.1
Let \(\mu_{0}=\inf_{\eta>0}\mu_{k_{0},\eta}\). We know by Lemma 3.1 and (3.12) that (2.6) holds when \(\min\{\mu_{1},\mu_{2}\}>\mu_{0}\), and hence (2.7) is true. Lemma (2.1) shows that \((u,v)\) is global. □