In order to suitably regularize the original problem (4) for \(\varepsilon \in (0,1)\), let us firstly consider a family of approximate systems:
$$ \textstyle\begin{cases} u_{\varepsilon t} = \Delta u_{\varepsilon }- \nabla \cdot ( \frac{u_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}}\nabla v_{\varepsilon })+ \kappa u_{\varepsilon }-\mu u_{\varepsilon }^{\alpha }, & x\in \Omega, t>0, \\ v_{\varepsilon t} = \Delta v_{\varepsilon }- v_{\varepsilon }\frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }}, & x\in \Omega, t>0, \\ \frac{\partial u_{\varepsilon }}{\partial \nu }= \frac{\partial v_{\varepsilon }}{\partial \nu }=0, & x\in \partial \Omega, t>0, \\ u_{\varepsilon }(x,0)=u_{0}(x), \qquad v_{\varepsilon }(x,0)=v_{0}(x), & x\in \Omega. \end{cases} $$
(9)
All the above approximate problems admit local-in-time smooth solutions.
Lemma 2.1
Let \(u_{0},v_{0}\) satisfy (5), let \(\kappa \in \mathbb{R}\), \(\mu >0\), and \(q>N\). Then, for each \(\varepsilon \in (0,1)\), there exist \(0< T_{\mathrm{max},\varepsilon }\leq +\infty \) and uniquely determined functions
$$ \textstyle\begin{cases} u_{\varepsilon }\in C^{0} ([\overline{\Omega }\times [0,T_{\mathrm{max},\varepsilon }) ) \cap C^{2,1} (\overline{\Omega }\times (0,T_{\mathrm{max}, \varepsilon }) ), \\ v_{\varepsilon }\in \bigcap_{q>N} C^{0} ([0,T_{\mathrm{max},\varepsilon });W^{1,q}( \Omega ) ) \cap C^{2,1} (\overline{\Omega }\times (0,T_{\mathrm{max}, \varepsilon }) ), \end{cases} $$
(10)
which are such that \(u_{\varepsilon }\ge 0\) and \(v_{\varepsilon }\ge 0\) in \(\overline{\Omega }\times (0,T_{\mathrm{max},\varepsilon })\), and the pair \((u_{\varepsilon },v_{\varepsilon })\) solves (9) classically in \(\Omega \times (0,T_{\mathrm{max},\varepsilon })\). Moreover, if \(T_{\mathrm{max},\varepsilon }<\infty \), then
$$ {\limsup_{t\nearrow T_{\mathrm{max},\varepsilon }} \bigl( \bigl\Vert u_{\varepsilon }(\cdot,t) \bigr\Vert _{L^{ \infty }(\Omega )}+ \bigl\Vert v_{\varepsilon }(\cdot,t) \bigr\Vert _{W^{1,q}(\Omega )} \bigr)=\infty.} $$
(11)
Proof
The proof follows the reasoning of [31], Lemma 3.1 (see also [26], Lemma 2.1). □
In contrast to the situation without source terms, we cannot hope for mass conservation in the first component. Nevertheless, the following result still holds (see also other works involving logistic source e.g. [32–34]).
Lemma 2.2
Let \(u_{0},v_{0}\) satisfy (5), let \(\kappa \in \mathbb{R}\), \(\mu >0\). Then, for any \(\varepsilon \in (0,1)\), the solution of (9) satisfies
$$\begin{aligned} & \int _{\Omega }u_{\varepsilon }(t)\le \max \biggl\{ \int _{\Omega }u_{0}(x), \biggl(\frac{\kappa _{+}}{\mu } \biggr)^{\frac{1}{\alpha -1}} \vert \Omega \vert \biggr\} \quad \textit{for all } t\in (0, T_{\mathrm{max},\varepsilon }) , \end{aligned}$$
(12)
$$\begin{aligned} & \bigl\Vert v_{\varepsilon }(t) \bigr\Vert _{L^{\infty }(\Omega )}\le \bigl\Vert v_{0}(x) \bigr\Vert _{L^{\infty }( \Omega )}=:v_{\infty } \quad\textit{for all } t\in (0, T_{\mathrm{max},\varepsilon }), \end{aligned}$$
(13)
where \(\kappa _{+}= \max \{0, \kappa \}\).
Proof
By Hölder’s inequality, \((\int _{\Omega }u_{\varepsilon })^{\alpha }\le (\int _{\Omega }u_{\varepsilon }^{\alpha })| \Omega |^{\alpha -1}\). An integration of the first equation in (9) yields
$$\begin{aligned} \biggl( \int _{\Omega }u_{\varepsilon }\biggr)_{t} &\le \kappa _{+} \int _{ \Omega }u_{\varepsilon }-\mu \int _{\Omega }u_{\varepsilon }^{\alpha } \\ &\le \kappa _{+} \int _{\Omega }u_{\varepsilon }-\mu \vert \Omega \vert ^{1-\alpha } \biggl( \int _{\Omega }u_{\varepsilon }\biggr)^{\alpha }. \end{aligned}$$
(14)
We can obtain (12) by an ODE comparison argument. Estimate (13) is a consequence of the parabolic comparison principle. □
Lemma 2.3
Let \(u_{0},v_{0}\) satisfy (5), let \(\kappa \in \mathbb{R}\), \(\mu >0\). There is \(C(\tau )>0\) such that, for any \(\varepsilon \in (0,1)\),
$$ \int ^{t+\tau }_{t} \int _{\Omega }u_{\varepsilon }^{\alpha }\le C(\tau ) \quad\textit{for all } t\in (0, T_{\mathrm{max},\varepsilon }-\tau ), $$
(15)
where \(\tau:=\min \{1,\frac{1}{2}T_{\mathrm{max},\varepsilon }\}\).
Proof
Estimate (15) results from (14) after time-integration. □
Next we want to derive a (quasi-)energy inequality for the functional
$$\begin{aligned} \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }+ \frac{1}{2} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon } \vert ^{2}}{v_{\varepsilon }} \end{aligned}$$
to get some essential estimates of \((u_{\varepsilon },v_{\varepsilon })\). The method used here is from [35].
Lemma 2.4
Let \(u_{0},v_{0}\) satisfy (5), let \(\kappa \in \mathbb{R}\), \(\mu >0\). Then there exist K, \(C>0\) such that
$$\begin{aligned} &\frac{d}{dt} \biggl( \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }+ \frac{1}{2} \int _{\Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }} \biggr)+ \int _{ \Omega }\frac{ \vert \nabla u_{\varepsilon }\vert ^{2}}{u_{\varepsilon }}+K \int _{\Omega }v_{\varepsilon }\bigl\vert D^{2} \ln v_{\varepsilon }\bigr\vert ^{2} \\ &\quad{}+\frac{\mu }{2} \int _{\Omega }u_{\varepsilon }^{\alpha }\ln u_{\varepsilon } +K \int _{\Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{4}}{v_{\varepsilon }^{3}} \le C \end{aligned}$$
(16)
on \((0,T_{\mathrm{max},\varepsilon })\) for all \(\varepsilon \in (0,1)\).
Proof
From integration by parts, we obtain that on \((0, T_{\mathrm{max},\varepsilon })\)
$$\begin{aligned} \frac{d}{dt} \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }={}& \int _{\Omega }u_{ \varepsilon t}+ \int _{\Omega }u_{\varepsilon t}\ln u_{\varepsilon } \\ ={}& \kappa \int _{\Omega }u_{\varepsilon }-\mu \int _{\Omega }u_{\varepsilon }^{\alpha }- \int _{\Omega }\frac{ \vert \nabla u_{\varepsilon }\vert ^{2}}{u_{\varepsilon }}+ \int _{\Omega } \frac{\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}} \\ & {} +\kappa \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }-\mu \int _{\Omega }u_{\varepsilon }^{\alpha }\ln u_{\varepsilon }. \end{aligned}$$
(17)
As \(\alpha >1\), we can see that \(s \mapsto \kappa s-\mu s^{\alpha }\), \(s\in [0, \infty )\) and \(s\mapsto (\kappa s-\frac{\mu }{2} s^{\alpha })\ln s\), \(s\in [0, \infty )\) are bounded from above by some constant \(C_{1}\). We thus can estimate
$$\begin{aligned} & \frac{d}{dt} \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }+ \frac{\mu }{2} \int _{ \Omega }u_{\varepsilon }^{\alpha }\ln u_{\varepsilon }+ \int _{\Omega } \frac{ \vert \nabla u_{\varepsilon }\vert ^{2}}{u_{\varepsilon }} \le \int _{\Omega } \frac{\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}}+2C_{1} \\ &\quad \text{for all } t\in (0, T_{\mathrm{max},\varepsilon }). \end{aligned}$$
(18)
Next we compute \(\frac{1}{2}\frac{d}{dt}\int _{\Omega } \frac{|\nabla v_{\varepsilon }|^{2}}{v_{\varepsilon }}\). From the second equation of (9) we know that on \((0, T_{\mathrm{max},\varepsilon })\)
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }}={}& \int _{\Omega } \frac{\nabla v_{\varepsilon }\cdot \nabla v_{\varepsilon t}}{v_{\varepsilon }}- \frac{1}{2} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{2}\cdot v_{\varepsilon t}}{v_{\varepsilon }^{2}} \\ ={}& \int _{\Omega }\frac{\nabla v_{\varepsilon }}{v_{\varepsilon }}\nabla \cdot \biggl( \Delta v_{\varepsilon }-{v_{\varepsilon }}\frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }} \biggr) \\ &{} -\frac{1}{2} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }^{2}} \biggl(\Delta v_{\varepsilon }-{v_{\varepsilon }}\frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }} \biggr) \\ ={}& \int _{\Omega } \frac{\nabla v_{\varepsilon }\cdot \nabla \Delta v_{\varepsilon }}{v_{\varepsilon }}- \int _{ \Omega } \frac{\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}}- \int _{\Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{ v_{\varepsilon }} \frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }} \\ & {} -\frac{1}{2} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }^{2}}\Delta v_{\varepsilon }+ \frac{1}{2} \int _{\Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }} \frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }} \\ \le{} & {-} \int _{\Omega }\frac{ \vert \Delta v_{\varepsilon }\vert ^{2} }{v_{\varepsilon }}+ \frac{1}{2} \int _{\Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }^{2}} \Delta v_{\varepsilon }- \int _{\Omega } \frac{\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}}. \end{aligned}$$
(19)
We know from Lemma 2.7 of [35] that there exist ε-independent positive constants \(K>0\), \(K_{1}>0\) such that
$$ - \int _{\Omega }\frac{ \vert \Delta v_{\varepsilon }\vert ^{2} }{v_{\varepsilon }}+\frac{1}{2} \int _{ \Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }^{2}}\Delta v_{\varepsilon }\le -K \int _{\Omega }v_{\varepsilon }\bigl\vert D^{2} \ln v_{\varepsilon }\bigr\vert ^{2}-K \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{4}}{v_{\varepsilon }^{3}}+K_{1} \int _{\Omega }v_{\varepsilon }$$
on \((0, T_{\mathrm{max},\varepsilon })\). Thereupon, we derive that
$$ \frac{1}{2}\frac{d}{dt} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }} \le K \int _{\Omega }v_{\varepsilon }\bigl\vert D^{2} \ln v_{\varepsilon }\bigr\vert ^{2}-K \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{4}}{v_{\varepsilon }^{3}}+K_{1} \int _{\Omega }v_{\varepsilon }- \int _{\Omega } \frac{\nabla u_{\varepsilon }\cdot \nabla v_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}} $$
(20)
on \((0, T_{\mathrm{max},\varepsilon })\). Combining (18) and (20), we obtain that, for any \(\varepsilon \in (0,1)\),
$$\begin{aligned} & \frac{d}{dt} \biggl( \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }+ \frac{1}{2} \int _{\Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }} \biggr)+ \int _{ \Omega }\frac{ \vert \nabla u_{\varepsilon }\vert ^{2}}{u_{\varepsilon }}+K \int _{\Omega }v_{\varepsilon }\bigl\vert D^{2} \ln v_{\varepsilon }\bigr\vert ^{2} \\ &\qquad{}+\frac{\mu }{2} \int _{\Omega }u_{\varepsilon }^{\alpha }\ln u_{\varepsilon }+K \int _{ \Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{4}}{v_{\varepsilon }^{3}} \\ &\quad \le K_{1} \int _{\Omega }v_{\varepsilon }+2C_{1} \le C\quad \text{on } t \in (0, T_{\mathrm{max},\varepsilon }) \end{aligned}$$
with \(C:=K_{1} v_{\infty }+2C_{1}\). □
Lemma 2.4 immediately entails the following boundedness estimates.
Lemma 2.5
Let \(u_{0},v_{0}\) satisfy (5), let \(\kappa \in \mathbb{R}\), \(\mu >0\). Then there exists \(C>0\) such that, for any \(\varepsilon \in (0,1)\),
$$\begin{aligned} \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }+ \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }}\le C \quad\textit{for all } t \in (0,T_{\mathrm{max},\varepsilon }) \end{aligned}$$
(21)
and such that
$$\begin{aligned} & \int ^{t+\tau }_{t} \int _{\Omega } \frac{ \vert \nabla u_{\varepsilon }\vert ^{2}}{u_{\varepsilon }}+ \int ^{t+\tau }_{t} \int _{\Omega }u_{\varepsilon }^{\alpha }\ln u_{\varepsilon }\le C, \end{aligned}$$
(22)
$$\begin{aligned} & \int ^{t+\tau }_{t} \int _{\Omega }v_{\varepsilon }\bigl\vert D^{2}\ln v_{\varepsilon }\bigr\vert ^{2}+ \int ^{t+\tau }_{t} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{4}}{v_{\varepsilon }^{3}}\le C, \end{aligned}$$
(23)
$$\begin{aligned} & \int _{\Omega } \vert \nabla v_{\varepsilon }\vert ^{2}\leq C \end{aligned}$$
(24)
for all \(\varepsilon \in (0,1)\) and any \(t\in [0,T_{\mathrm{max},\varepsilon }-\tau )\), where \(\tau:=\min \{1,\frac{1}{2}T_{\mathrm{max},\varepsilon }\}\).
Proof
Fix \(p\in (1,1+\frac{2}{N}) \) and observe that
$$\begin{aligned} \xi \ln \xi \leq \frac{1}{p(p-1)}\xi ^{p} \quad \text{for all }\xi >0. \end{aligned}$$
An application of the Gagliardo–Nirenberg inequality yields ε-independent positive constant \(C_{1} \) such that
$$\begin{aligned} \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }&\leq \frac{1}{p(p-1)} \int _{\Omega }u_{\varepsilon }^{p} \\ &=\frac{1}{p(p-1)} \bigl\Vert u_{\varepsilon }^{\frac{1}{2}} \bigr\Vert ^{2p}_{L^{2p}(\Omega )} \\ &\leq C_{1} \bigl( \bigl\Vert \nabla u_{\varepsilon }^{\frac{1}{2}} \bigr\Vert ^{2p\cdot \theta }_{L^{2}( \Omega )}\cdot \bigl\Vert u_{\varepsilon }^{\frac{1}{2}} \bigr\Vert ^{2p(1-\theta )}_{L^{2}( \Omega )}+ \bigl\Vert u_{\varepsilon }^{\frac{1}{2}} \bigr\Vert ^{2p}_{L^{2}(\Omega )} \bigr) \quad\text{for all }t\in (0,T_{\mathrm{max},\varepsilon }), \end{aligned}$$
where \(\theta:=\frac{(p-1)N}{2p} \in (0,1) \) and \(2p\cdot \theta <2 \) due to \(p\in (1,1+\frac{2}{N}) \). Thereupon, we can find some ε-independent positive constants \(C_{2}, C_{3} \) such that
$$\begin{aligned} \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }\leq C_{2} \bigl( \bigl\Vert \nabla u_{\varepsilon }^{ \frac{1}{2}} \bigr\Vert ^{2}_{L^{2}(\Omega )}+1 \bigr)\leq C_{3} \int _{\Omega } \frac{ \vert \nabla u_{\varepsilon }\vert ^{2}}{u_{\varepsilon }}+ C_{2} \quad\text{for all }t\in (0,T_{\mathrm{max},\varepsilon }) \end{aligned}$$
(25)
by making use of (12). On the other hand, making use of the boundedness of \(\Vert v_{\varepsilon }\Vert _{L^{\infty }(\Omega )} \) and the Young inequality, we know there exist ε-independent positive constants \(C_{4}, C_{5} \) fulfilling
$$\begin{aligned} \int _{\Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }} &\leq C_{4} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{4}}{v_{\varepsilon }^{3}}+ \frac{1}{4C_{4}} \int _{\Omega }v_{\varepsilon } \\ &\leq C_{4} \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{4}}{v_{\varepsilon }^{3}}+C_{5}\quad \text{for all }t\in (0,T_{\mathrm{max},\varepsilon }). \end{aligned}$$
(26)
Substituting (25), (26) into (16), we conclude that there exist positive constants \(C_{6} \) and \(C_{7} \) such that, for any \(\varepsilon >0 \),
$$\begin{aligned} &\frac{d}{dt} \biggl( \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon }+ \frac{1}{2} \int _{ \Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }} \biggr)+ C_{6} \biggl( \int _{\Omega }u_{\varepsilon }\ln u_{\varepsilon } + \frac{1}{2} \int _{\Omega }\frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }} \biggr) \\ &\qquad{}+K \int _{\Omega }v_{\varepsilon }\bigl\vert D^{2}\ln v_{\varepsilon }\bigr\vert ^{2}+\frac{\mu }{2} \int _{ \Omega }u_{\varepsilon }^{\alpha }\ln u_{\varepsilon } \\ &\quad\le C_{7} \quad\text{for all }t\in (0,T_{\mathrm{max},\varepsilon }) \end{aligned}$$
(27)
with K as given by Lemma 2.4. We can conclude the validity of (21). The result of (22) and (23) can be obtained by an integration of (16) and the fact \(\int _{\Omega }u_{\varepsilon } \mathrm{ln} u_{\varepsilon }\geq - \frac{|\Omega |}{e} \). Furthermore, for (24), by Lemma 2.2, for any \(\varepsilon >0 \),
$$\begin{aligned} \int _{\Omega } \bigl\vert \nabla v_{\varepsilon }(t) \bigr\vert ^{2} \leq v_{\infty } \int _{\Omega } \frac{ \vert \nabla v_{\varepsilon }\vert ^{2}}{v_{\varepsilon }}\quad \text{for all }t \in (0,T_{\mathrm{max},\varepsilon }), \end{aligned}$$
which is bounded due to (21). □
By using interpolation inequalities, we can derive some further estimates from Lemma 2.5.
Lemma 2.6
Let \(u_{0},v_{0}\) satisfy (5), let \(\kappa \in \mathbb{R}\), \(\mu >0\). Then there exists \(C>0\) such that, for any \(\varepsilon \in (0,1)\),
$$\begin{aligned} &\int ^{t+\tau }_{t} \int _{\Omega }u_{\varepsilon }^{\frac{N+2}{N}} \le C, \end{aligned}$$
(28)
$$\begin{aligned} &\int ^{t+\tau }_{t} \int _{\Omega } \vert \nabla u_{\varepsilon }\vert ^{\frac{N+2}{N+1}} \le C \end{aligned}$$
(29)
for any \(t\in [0,T_{\mathrm{max},\varepsilon }-\tau )\), where \(\tau:=\min \{1,\frac{1}{2}T_{\mathrm{max},\varepsilon }\}\).
Proof
From (22), we know that \(\int ^{t+\tau }_{t}\int _{\Omega }|\nabla u_{\varepsilon }^{\frac{1}{2}}|^{2} \le C_{1}\) with some ε-independent constant \(C_{1}>0\). Then with the aid of the Gagliardo–Nirenberg inequality, we obtain
$$\begin{aligned} \int ^{t+\tau }_{t} \int _{\Omega }u_{\varepsilon }^{\frac{N+2}{N}}&= \int ^{t+ \tau }_{t} \int _{\Omega } \bigl\Vert u_{\varepsilon }^{\frac{1}{2}} \bigr\Vert ^{\frac{2(N+2)}{N}}_{L^{ \frac{2(N+2)}{N}}} \\ &\le \int ^{t+\tau }_{t} C_{2} \bigl( \bigl\Vert \nabla u_{\varepsilon }^{\frac{1}{2}} \bigr\Vert _{L^{2}( \Omega )}^{2} \cdot \bigl\Vert u_{\varepsilon }^{\frac{1}{2}} \bigr\Vert _{L^{2}(\Omega )}^{ \frac{4}{N}}+ \bigl\Vert u_{\varepsilon }^{\frac{1}{2}} \bigr\Vert _{L^{2}(\Omega )}^{ \frac{2(N+2)}{N}} \bigr) \\ &\le C_{3} \quad\text{for all }t\in (0,T_{\mathrm{max}, \varepsilon }-\tau ) \end{aligned}$$
with some ε-independent positive constants \(C_{2}\), \(C_{3}\).
Furthermore, we can make use of the Young inequality to obtain that
$$\begin{aligned} \int ^{t+\tau }_{t} \int _{\Omega } \vert \nabla u_{\varepsilon }\vert ^{\frac{N+2}{N+1}}&= \int ^{t+ \tau }_{t} \int _{\Omega } \biggl(\frac{ \vert \nabla u_{\varepsilon }\vert ^{2}}{u_{\varepsilon }} \biggr)^{\frac{N+2}{2N+2}} \cdot u_{\varepsilon }^{\frac{N+2}{2N+2}} \\ &\le C_{4} \int ^{t+\tau }_{t} \int _{\Omega } \frac{ \vert \nabla u_{\varepsilon }\vert ^{2}}{u_{\varepsilon }}+C_{5} \int ^{t+\tau }_{t} \int _{ \Omega } u_{\varepsilon }^{\frac{N+2}{N}} \\ &\le C_{6} \quad \text{for all }t\in (0,T_{\mathrm{max}, \varepsilon }-\tau ) \end{aligned}$$
with some ε-independent positive constants \(C_{i}\ (i=4,5,6)\). □
Lemma 2.7
Let \(u_{0},v_{0}\) satisfy (5), let \(\kappa \in \mathbb{R}\), \(\mu >0\). There exists \(C>0\) such that, for any \(\varepsilon \in (0,1)\),
$$ \int ^{t+\tau }_{t} \int _{\Omega } \vert u_{\varepsilon }\nabla v_{\varepsilon }\vert ^{ \frac{4(N+2)}{5N+2}} \le C $$
(30)
for any \(t\in [0,T_{\mathrm{max},\varepsilon }-\tau )\), where \(\tau:=\min \{1,\frac{1}{2}T_{\mathrm{max},\varepsilon }\}\).
Proof
As \(\|v_{\varepsilon }\|_{L^{\infty }(\Omega )}\le v_{\infty }\), the time-spatial estimate (23) implies that there exists ε-independent constant \(C_{1}>0\) such that
$$ \int ^{t+\tau }_{t} \int _{\Omega } \vert \nabla v_{\varepsilon }\vert ^{4} \leq C_{1}\quad \text{for all }t\in (0,T_{\mathrm{max},\varepsilon }-\tau ). $$
(31)
Noticing (28), we can use the Young inequality to estimate
$$\begin{aligned} \int ^{t+\tau }_{t} \int _{\Omega } \vert u_{\varepsilon }\nabla v_{\varepsilon }\vert ^{ \frac{4(N+2)}{5N+2}}&\le C_{2} \int ^{t+\tau }_{t} \int _{\Omega } \vert u_{\varepsilon }\vert ^{ \frac{N+2}{N}}+C_{3} \int ^{t+\tau }_{t} \int _{\Omega } \vert \nabla v_{\varepsilon }\vert ^{4} \\ &\le C_{4} \quad \text{for all }t\in (0,T_{\mathrm{max},\varepsilon }-\tau ), \end{aligned}$$
where \(C_{i}>0\ (i=2,3,4)\) are all independent of ε. □
Lemma 2.8
Let \(u_{0},v_{0}\) satisfy (5), let \(\kappa \in \mathbb{R}\), \(\mu >0\). Then there exists \(C>0\) such that, for any \(\varepsilon \in (0,1)\),
$$ \int ^{t+\tau }_{t} \int _{\Omega } \vert u_{\varepsilon }\nabla v_{\varepsilon }\vert ^{ \frac{4\alpha }{4+\alpha }}\le C $$
(32)
for any \(t\in [0,T_{\mathrm{max},\varepsilon }-\tau )\), where \(\tau:=\min \{1,\frac{1}{2}T_{\mathrm{max},\varepsilon }\}\).
Proof
We can use the Young inequality, (31), and (15) to estimate
$$\begin{aligned} & \int ^{t+\tau }_{t} \int _{\Omega } \vert u_{\varepsilon }\nabla v_{\varepsilon }\vert ^{ \frac{4\alpha }{4+\alpha }} \\ &\quad\le C_{1} \int ^{t+\tau }_{t} \int _{\Omega } \vert u_{\varepsilon }\vert ^{\alpha }+C_{2} \int ^{t+\tau }_{t} \int _{\Omega } \vert \nabla v_{\varepsilon }\vert ^{4} \le C_{3}\quad \text{for all }t\in (0,T_{\mathrm{max},\varepsilon }-\tau ), \end{aligned}$$
with some ε-independent positive constants \(C_{i}\ (i=1,2,3)\). □
We are now in the position to prove that the classical solution \((u_{\varepsilon },v_{\varepsilon })\) to the approximate systems (9) is global for each \(\varepsilon \in (0,1)\).
Lemma 2.9
Let \(\kappa \in \mathbb{R}\), \(\mu >0\) and assume that \(u_{0},v_{0}\) satisfy (5). For any \(\varepsilon \in (0,1)\), \(T_{\mathrm{max},\varepsilon }=\infty \).
Proof
Assume that \(T_{\mathrm{max},\varepsilon }\) is finite for some \(\varepsilon \in (0,1)\). To deduce a contradiction from this, we fix a suitably large \(q\geq N+1\) and use the standard estimate for the Neumann heat semigroup (see e.g. [36]) together with Lemma 2.2 and the fact that \(\frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }}\leq \frac{1}{\varepsilon }\) to obtain \(C_{i}>0\ (i=1,2,3,4) \) such that
$$\begin{aligned} \Vert \nabla v_{\varepsilon } \Vert _{L^{{q}}}&= \biggl\Vert \nabla e^{t \Delta }v_{\varepsilon }(\cdot,0) - \int _{0}^{t} \nabla \biggl(e^{(t-s) \Delta } \biggl(v_{\varepsilon }\cdot \frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }} \biggr) \biggr)\,ds \biggr\Vert _{L^{q}} \\ &\le \bigl\Vert \nabla e^{t\Delta }v_{0} \bigr\Vert _{L^{q}}+C_{1} \int _{0}^{t} \bigl(1+(t-s)^{- \frac{1}{2}} \bigr)e^{-\lambda t} \biggl\Vert v_{\varepsilon }\cdot \frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }} \biggr\Vert _{L^{q}}\,ds \\ &\leq C_{2} \Vert \nabla v_{0} \Vert _{L^{q}}+ \frac{C_{3}}{\varepsilon } \int _{0}^{t} \bigl(1+(t-s)^{-\frac{1}{2}} \bigr) \,ds \\ &\leq C_{4}(\varepsilon,T)\quad \text{for all }t \in (0,T_{\mathrm{max},\varepsilon }). \end{aligned}$$
(33)
Similarly, according to the fact \(\frac{u_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}}\leq \frac{u_{\varepsilon }}{1+\varepsilon u_{\varepsilon }}\leq \frac{1}{\varepsilon }\) and \(\kappa u_{\varepsilon }-\mu u_{\varepsilon }^{\alpha }\) is bounded, there exist \(C_{i}>0\ (i=5,6,7,8) \) such that
$$\begin{aligned} \Vert u_{\varepsilon } \Vert _{L^{{\infty }}}={}& \biggl\Vert e^{t \Delta }u_{\varepsilon }(\cdot,0) + \int _{0}^{t}e^{(t-s)\Delta } \biggl(- \nabla \biggl( \frac{u_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}} \cdot \nabla v_{\varepsilon } \biggr)+\kappa u_{\varepsilon }- \mu u_{ \varepsilon }^{\alpha } \biggr)\,ds \biggr\Vert _{L^{\infty }} \\ \leq {}& \bigl\Vert \nabla e^{t\Delta }u_{0} \bigr\Vert _{L^{\infty }} +C_{5} \int _{0}^{t} \bigl(1+(t-s)^{- \frac{1}{2}-\frac{N}{2N+2}} \bigr)e^{-\lambda t} \biggl\Vert \frac{u_{\varepsilon }}{(1+\varepsilon u_{\varepsilon })^{2}}\cdot \nabla v_{\varepsilon } \biggr\Vert _{L^{N+1}}\,ds \\ & {}+ \int _{0}^{t} \bigl\Vert e^{(t-s)\Delta }\kappa u_{\varepsilon }-\mu u_{ \varepsilon }^{\alpha } \bigr\Vert _{L^{\infty }} \,ds \\ \leq {}& \bigl\Vert \nabla e^{t\Delta }u_{0} \bigr\Vert _{L^{\infty }}+ \frac{C_{6}}{\varepsilon } \int _{0}^{t} \bigl(1+(t-s)^{-\frac{1}{2}- \frac{N}{2N+2}} \bigr) \Vert \nabla v_{\varepsilon } \Vert _{L^{N+1}}\,ds+C_{7} \\ \leq{} & C_{8}(\varepsilon,T)\quad \text{for all }t \in (0,T_{\mathrm{max},\varepsilon }). \end{aligned}$$
Together with (33), this contradicts criterion (11) in Lemma 2.1 and thereby entails that actually \(T_{\mathrm{max},\varepsilon }=\infty \), as claimed. □