Proof. It is easy to see that (2.3) can be derived through an integration of the first equation in (1.3) along with Cauchy–Schwarz's inequality, and (2.4) is a direct consequence of $(\alpha _u u+\alpha _w w)v\geq 0$ due to the nonnegativity of $u,w$ and $v$.

In addition, integrating the $w$-equation as well as $u$-equation respectively and adding the corresponding results, we then have

(2.7)\begin{equation} \frac{{\rm d}}{{\rm d}t}\left(\int_{\Omega}u+\int_{\Omega}w\right)+\int_{\Omega}w+\mu_u\int_{\Omega}u\leq |\Omega| \mu_u, \end{equation}

which readily leads to (2.5) upon an ODE comparison. At last, thanks to (2.5), (2.6) clearly results from the integration of $z$-equation in (1.3).

Proof. We use the $z$-equation and $w$-equation to compute

(2.9)\begin{equation} \frac{{\rm d}}{{\rm d}t}\left(\int_{\Omega}w+\int_{\Omega}z\right)+(1-\beta)\int_{\Omega}w+\delta_z\int_{\Omega}z=0. \end{equation}

Due to $0<\beta <1$, this readily implies that

\[ \left(\int_{\Omega}w({\cdot},t)+\int_{\Omega}z({\cdot},t)\right)\leq \left(\int_{\Omega}w_0+\int_{\Omega}z_0\right){\rm e}^{-\min\{1-\beta,\delta_z\}t} \]

and hence (2.8) is valid with $C=\int _{\Omega } w_0+\int _{\Omega }z_0$.

Proof. From the first equation in (1.3), it follows that

(3.2)\begin{equation} ({\rm e}^{\chi_u v}a)_t=D_u\nabla \cdot ({\rm e}^{\chi_u v}\nabla a)+\mu_u {\rm e}^{\chi_u v}a(1-{\rm e}^{\chi_u v}a)-\rho {\rm e}^{\chi_u v}az. \end{equation}

Hence, a testing procedure on the first equation in (2.2) leads to

\[\begin{aligned} \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_u v}a\log a & =\displaystyle\int_{\Omega}({\rm e}^{\chi_u v}a)_t\log a+\int_{\Omega}{\rm e}^{\chi_u v}a_t\\ & \leq \displaystyle-D_u\int_{\Omega}{\rm e}^{\chi_u v}\frac{|\nabla a|^{2}}{a}+\mu_u\int_{\Omega}{\rm e}^{\chi_u v}a\log a-\mu_u\int_{\Omega}{\rm e}^{2\chi_u v}a^{2}\log a\nonumber\\ & \quad\displaystyle-\rho\int_{\Omega}{\rm e}^{\chi_u v}za\log a +\mu_u\int_{\Omega}a{\rm e}^{\chi_u v}-\rho\int_{\Omega}{\rm e}^{\chi_u v}za\nonumber\\ & \quad\displaystyle+\chi_u \alpha_u\int_{\Omega}{\rm e}^{2\chi_u v}v a^{2} +\chi_u \alpha_w\int_{\Omega}{\rm e}^{(\chi_w+\chi_u)v}vab. \end{aligned} \]

Thanks to lemma 2.2 and the elementary inequality $a\log a\geq -\frac {1}{e}$ valid for all $a>0$, one can find $c_1>0$ such that

\begin{align*} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_u v}a\log a+D_u\int_{\Omega}{\rm e}^{\chi_u v}\frac{|\nabla a|^{2}}{a} +\mu_u\int_{\Omega}{\rm e}^{2\chi_u v}a^{2}\log a\\ & \quad\leq \displaystyle\alpha_u\chi_u{\rm e}^{2\chi_u m_v}m_v\int_{\Omega} a^{2}+\alpha_w\chi_u{\rm e}^{(\chi_u+\chi_w)m_v}m_v\int_{\Omega}ab +\mu_u\int_{\Omega}{\rm e}^{\chi_u v}a\log a +c_1, \end{align*}

which along with Young's inequality implies that for any $\varepsilon >0$

\begin{align*} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_u v}a\log a+D_u\int_{\Omega}{\rm e}^{\chi_u v}\frac{|\nabla a|^{2}}{a} +\mu_u\int_{\Omega}{\rm e}^{2\chi_u v}a^{2}\log a\\ & \quad\leq \left(\displaystyle\alpha_u\chi_u{\rm e}^{2\chi_u m_v}m_v+ \frac 1{\varepsilon} \alpha_w^{2}\chi_u^{2}{\rm e}^{2(\chi_u+\chi_w)m_v}m_v^{2} \right) \int_{\Omega} a^{2}\\ & \quad + \varepsilon \int_{\Omega}b^{2}+ \mu_u\int_{\Omega}{\rm e}^{\chi_u v}a\log a+ c_1. \end{align*}

Further invoking the inequality $a^{2}\leq \varepsilon _1 a^{2}\log a+{\rm e}^{{2}/{\varepsilon _1}}$ for any $\varepsilon _1>0$, we arrive at

(3.3)\begin{equation} \begin{aligned} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_u v}a\log a+D_u\int_{\Omega}{\rm e}^{\chi_u v}\frac{|\nabla a|^{2}}{a} +\frac{3\mu_u}{4}\int_{\Omega}{\rm e}^{2\chi_u v}a^{2}\log a\\ & \quad\leq \varepsilon \displaystyle\int_{\Omega}b^{2}+\mu_u\int_{\Omega}{\rm e}^{\chi_u v}a\log a+ c_2(\varepsilon) \end{aligned} \end{equation}

with some $c_2(\varepsilon )>0$. Accordingly, (3.1) is a consequence of (3.3) and the fact that $a\log a\leq \varepsilon _2 a^{2}\log a- \varepsilon _2^{-1} \ln \varepsilon _2$ with $\varepsilon _2={\mu _u}/{(4(\mu _u+1))}$.

Proof. From the second equation in (1.3), it follows that

\[ (be ^{\chi_w v})_t=D_w\nabla\cdot({\rm e}^{\chi_w v}\nabla b)- w+\rho uz. \]

Relying on $0\leq v\leq m_v$ in $\overline {\Omega }\times (0,\infty )$, a straightforward calculation along with the Young's inequality yields

(3.5)\begin{align} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_\Omega {\rm e}^{\chi_w v}b\log b+ \int_\Omega {\rm e}^{\chi_w v}b\log b+ D_w\displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b}\nonumber\\ & \quad=\int_\Omega ({\rm e}^{\chi_w v}b)_t\log b+ \int_\Omega {\rm e}^{\chi_w v}b_t+ \int_\Omega {\rm e}^{\chi_w v}b\log b+ D_w\displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b} \nonumber\\ & \quad=\int_\Omega \log b (D_w\nabla\cdot({\rm e}^{\chi_w v}\nabla b)- w+\rho uz) + \int_\Omega {\rm e}^{\chi_w v} g(a,b,v,c)\nonumber\\ & \quad + \int_\Omega {\rm e}^{\chi_w v}b\log b+ D_w\displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b} \nonumber\\ & \quad=\int_\Omega (1+\log b) (\rho uz- w) +\chi_w \displaystyle \int_\Omega w(\alpha_u u + \alpha_w w )v+ \int_\Omega w\log b \\ & \quad\leq\rho\int_\Omega (1+\log b) uz +\chi_w \alpha_u\displaystyle \int_\Omega w u v + \chi_w \alpha_w\displaystyle \int_\Omega w^{2}v\nonumber\\ & \quad\leq \rho \int_{\Omega}az{\rm e}^{\chi_u v}\log b + \rho {\rm e}^{\chi_u m_v}\int_{\Omega}az+ \alpha_u\chi_w {\rm e}^{(\chi_u+\chi_w)m_v}m_v\int_{\Omega}ab \nonumber\\ & \quad +\alpha_w\chi_w {\rm e}^{2\chi_w m_v}m_v\int_{\Omega}b^{2}\nonumber\\ & \quad\leq c_1\int_{\Omega}b^{2}+\int_{\Omega}a^{2} + c_1\int_{\Omega}z^{2}+ c_1\displaystyle\int_{\{x\in \Omega;b(x,t)\geq 1\}}z^{2} \log ^{2} b\nonumber\\ & \quad\leq c_1\int_{\Omega}b^{2}+\int_{\Omega}a^{2} + c_1\int_{\Omega}z^{2}+ c_1\displaystyle\left(\int_{\Omega} z^{4}\right)^{ 1/2} \left(\int_{\{x\in \Omega;b(x,t)\geq 1\}}|\log b|^{4}\right) ^{ 1/2} \nonumber\\ & \quad\leq c_1\int_{\Omega}b^{2}+\int_{\Omega}a^{2} + c_1\int_{\Omega}z^{2}+ c_1\displaystyle\left(\int_{\Omega} z^{4}\right)^{ 1/2} \left(\int_{\Omega} b+c_2|\Omega|\right)^{ 1/2} \nonumber \end{align}

with some $c_1>0$, where we use the fact that there exits $c_2>0$ such that $\log ^{4} s\leq s+c_2$ for all $s\geq 1$ and $w^{2}={\rm e}^{2\chi _w v}b^{2}\leq {\rm e}^{2\chi _w m_v} b^{2}$.

Furthermore, in order to appropriately estimate the first summand on the right-hand side of (3.5), we apply the two-dimensional Gagliardo–Nirenberg inequalities

\[ \|\varphi\|^{4}_{L^{4}(\Omega)}\!\leq\! C_{g}\|\nabla \varphi\|_{L^{2}(\Omega)}^{2}\|\varphi\|^{2}_{L^{2}(\Omega)}\!+\!C_g\|\varphi\|^{4}_{L^{2}(\Omega)} \text{ for some } C_g\!>\!0 \text{ and all } \varphi\!\in\! W^{1,2}(\Omega) \]

to get

(3.6)\begin{equation} \begin{aligned} c_1\displaystyle \int_{\Omega}b^{2} & = c_1\|\sqrt{b}\|^{4}_{L^{4}(\Omega)}\\ & \leq \displaystyle c_1C_g \displaystyle\int_\Omega b \displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b}+ c_1C_g \left(\displaystyle\int_\Omega b\right)^{2}\\ & \leq\displaystyle c_1C_g \displaystyle\int_\Omega w\displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b}+ c_1C_g \left(\displaystyle\int_\Omega w\right)^{2}\\ \end{aligned} \end{equation}

Therefore combining (3.6) with (3.5), we arrive at

(3.7)\begin{align} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_\Omega {\rm e}^{\chi_w v}b\log b+ \int_\Omega {\rm e}^{\chi_w v}b\log b+ D_w\displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b} \\ & \quad\leq \displaystyle c_1C_g \displaystyle\int_\Omega w \displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b}+ c_1C_g \left(\displaystyle\int_\Omega w\right)^{2}+\int_{\Omega}a^{2}\nonumber\\ & \quad + c_1\int_{\Omega}z^{2}+ c_1\displaystyle\left(\int_{\Omega} z^{4}\right)^{1/2} \left(\int_{\Omega} w+c_2|\Omega|\right)^{1/2}. \nonumber \end{align}

By lemma 2.3, we can pick $t_0>0$ suitably large such that

\[ c_1C_g \displaystyle\int_\Omega w\leq \frac {D_w}2 \]

and thereby for $t\geq t_0$,

\begin{align*} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_\Omega {\rm e}^{\chi_w v}b\log b+ \int_\Omega {\rm e}^{\chi_w v}b\log b+ \frac {D_w}2 \displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b}\nonumber \\ & \quad\leq c_1C_g \left(\displaystyle\int_\Omega w\right)^{2}+\int_{\Omega}a^{2} + c_1\int_{\Omega}z^{2}+ c_1\displaystyle\left(\int_{\Omega} z^{4}\right)^{1/2} \left(\int_{\Omega} w+c_2|\Omega|\right)^{1/2}, \end{align*}

which along with lemma 2.2 and the Young's inequality completes the proof.

Proof. Testing the fourth equation in (2.2) by $z$, we can see that

\begin{align*} \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}z^{2}+\int_{\Omega}z^{2}+2D_z\int_{\Omega}|\nabla z|^{2}& \leq \displaystyle 2\beta\int_{\Omega}{\rm e}^{\chi_w v}bz +\int_{\Omega}z^{2}\\ & \leq\displaystyle \frac{D_w}{4m_wC_g}\int_{\Omega}b^{2}+\left(1+ \frac{4m_w C_g{\rm e}^{2\chi_w m_v}\beta^{2}}{D_w}\right)\int_{\Omega}z^{2}. \end{align*}

Hence lemmas 3.1 and 3.2 provide positive constants $c_i>0~ (i=1,2)$ such that

(3.9)\begin{equation} \begin{aligned} & \mathcal{F'}(t) +\mathcal{F}(t) +\displaystyle D_w\int_{\Omega}{\rm e}^{\chi_w v}\frac{|\nabla b|^{2}}{b}+D_z\int_{\Omega}|\nabla z|^{2}+\frac{\mu_u}{2}\int_{\Omega}{\rm e}^{2\chi_u v}a^{2}\log a+\int_{\Omega}z^{2}\\ & \quad\leq \displaystyle\int_{\Omega}a^{2} + c_1\left(\int_{\Omega} z^{4}\right)^{1/2}+ \frac{D_w}{2m_wC_g}\int_{\Omega}b^{2}+c_2 \end{aligned} \end{equation}

for all $t\geq t_0$ with $t_0$ given by lemma 3.2.

Now again since $a^{2}\leq \varepsilon _1 a^{2}\log a+{\rm e}^{{2}/{\varepsilon _1}}$ for any $\varepsilon _1>0$,

(3.10)\begin{equation} \displaystyle\int_{\Omega}a^{2} \leq \frac{\mu_u}{4}\int_{\Omega}{\rm e}^{2\chi_u v}a^{2}\log a+c_3 \end{equation}

with $c_3>0$, whereas according to the two-dimensional Gagliardo–Nirenberg inequalities,

(3.11)\begin{equation} \begin{aligned} \displaystyle \frac{D_w}{2m_wC_g}\displaystyle \int_{\Omega}b^{2} & = \displaystyle \frac{D_w}{2m_wC_g}\|\sqrt{b}\|^{4}_{L^{4}(\Omega)}\\ & \leq\displaystyle \frac{D_w}{2m_w} \displaystyle\int_\Omega b \displaystyle\int_\Omega {\rm e}^{\chi_w v}\displaystyle\frac{|\nabla b|^{2}}{b}+ \displaystyle \frac{D_w}{2m_w} \left(\displaystyle\int_\Omega b\right)^{2}\\ & \leq\displaystyle \frac{D_w}2\int_{\Omega}{\rm e}^{\chi_w v}\frac{|\nabla b|^{2}}{b}+\displaystyle \frac{D_w}{2m_w} m_w^{2}. \end{aligned} \end{equation}

In summary, (3.11), (3.10) and (3.9) show that for $t\geq t_0$

(3.12)\begin{equation} \mathcal{F'}(t) +\mathcal{F}(t) +D_z\int_{\Omega}|\nabla z|^{2}+\int_{\Omega}z^{2}\leq c_1\left(\int_{\Omega} z^{4}\right)^{1/2}+ c_4 \end{equation}

with some $c_4>0$.

Furthermore to estimate $\left (\int _{\Omega }z^{4}\right )^{{1}/{2}}$ on the right-hand side of (3.12), we employ (2.8), the Gagliardo–Nirenberg inequalities and Young's inequality once more to conclude that there exists $t_1>t_0$ such that for all $t\geq t_1$

(3.13)\begin{equation} \begin{aligned} c_1\|z({\cdot},t)\|^{2}_{L^{4}(\Omega)} & \leq c_1 C_g \|\nabla z({\cdot},t)\|^{ 3/2}_{L^{2}(\Omega)}\|z({\cdot},t)\|^{1/2}_{L^{1}(\Omega)} +c_1 C_g\|z({\cdot},t)\|^{2}_{L^{1}(\Omega)}\\ & \leq \displaystyle \frac{D_z}2\int_{\Omega}|\nabla z|^{2}+c_5 \end{aligned} \end{equation}

with $c_5>0$, which together with (3.12) readily establishes (3.8).

Proof. As in lemma 3.4 of [Reference Li and Wang15], by the inequality $a\log a>-{\rm e}^{-1}$ in $\Omega \times (0,\infty )$, we have

\begin{align*} \displaystyle\int_{\Omega}a({\cdot},t)|\log a({\cdot},t)|& \leq \displaystyle\int_{\Omega}{\rm e}^{\chi_u v}a({\cdot},t)\log a({\cdot},t)-2\int_{a<1}{\rm e}^{\chi_{u} v}a({\cdot},t)\log a({\cdot},t) \\ & \leq \displaystyle\int_{\Omega}{\rm e}^{\chi_u v}a({\cdot},t)\log a({\cdot},t)+\frac{2|\Omega|{\rm e}^{\chi_u m_v}}{e}. \end{align*}

Likewise, we can also obtain

\[ \int_{\Omega}b({\cdot},t)|\log b({\cdot},t)|\leq \int_{\Omega}{\rm e}^{\chi_w v}b({\cdot},t)\log b({\cdot},t)+\frac{2|\Omega|{\rm e}^{\chi_w m_v}}{e}. \]

According to (3.8), there exits $c_1>0$ such that

(3.16)\begin{equation} \mathcal{F}(t)\leq c_1. \end{equation}

Therefore by the definition of $\mathcal {F}(t)$, we can see that

\[ \int_{\Omega}a({\cdot},t)|\log a({\cdot},t)|+\int_{\Omega}b({\cdot},t)|\log b({\cdot},t)|\leq \mathcal{F}(t)+ \frac{2|\Omega|{\rm e}^{\chi_u m_v}}{e}+\frac{2|\Omega|{\rm e}^{\chi_w m_v}}{e} \leq c_2 \]

with $c_2= ({2|\Omega |}/ e)({\rm e}^{\chi _w m_v} +{\rm e}^{\chi _u m_v} )+ c_1$ for all $t>t_0$, and thus complete the proof.

Proof. Testing the first equation in (2.2) by ${\rm e}^{\chi _u v}a^{r-1}$ with $r\geq 2$, we obtain $c_1(r)>0$

\begin{align*} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_u v}a^{r}+\frac{4(r-1)D_u}{r}\int_{\Omega}{\rm e}^{\chi_u v}|\nabla a^{{r}/{2}}|^{2}+ r\mu_u\int_{\Omega}{\rm e}^{2\chi_v}a^{r+1}\\ & \quad\leq\displaystyle\mu_u r\int_{\Omega}{\rm e}^{\chi_u v}a^{r}+r \chi_u \int_{\Omega}{\rm e}^{\chi_u v}a^{r}(\alpha_u u +\alpha_w w)v\\ & \quad\leq\displaystyle\mu_u r\int_{\Omega}{\rm e}^{\chi_u v}a^{r}+c_1(r) \int_{\Omega}a^{r}(a +b), \end{align*}

which together with the Young's inequality, leads to

(3.18)\begin{equation} \begin{aligned} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_u v}a^{r}+2 D_u \int_{\Omega}|\nabla a^{{r}/{2}}|^{2}+ \int_{\Omega}{\rm e}^{\chi_u v}a^{r}\\ & \quad\leq \displaystyle c_2(r)\int_{\Omega}a^{r+1}+c_2(r)\int_{\Omega}b^{r+1}+c_2(r) \end{aligned} \end{equation}

with constant $c_2(r)>0$. Likewise, there exist $c_i(r)>0~ (i=3,4)$ such that

(3.19)\begin{equation} \begin{aligned} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_w v}b^{r}+2 D_w \int_{\Omega}|\nabla b^{{r}/{2}}|^{2}+ r\int_{\Omega}{\rm e}^{\chi_v}b^{r}\\ & \quad\leq\displaystyle c_3(r)\int_{\Omega}b^{r-1}az +c_3(r)\int_{\Omega}a^{r+1} +c_3(r)\int_{\Omega}b^{r+1}\\ & \quad\leq\displaystyle c_4(r)\int_{\Omega} a^{r+1}+c_4(r)\int_{\Omega}b^{r+1} +c_4(r)\int_{\Omega}z^{r+1} \end{aligned} \end{equation}

as well as

(3.20)\begin{equation} \frac{{\rm d}}{{\rm d}t}\int_{\Omega}z^{r}+2 D_z \int_{\Omega}|\nabla z^{{r}/{2}}|^{2}+\frac{r\delta_z}{2}\int_{\Omega}z^{r}\leq \frac{2r\beta^{r}}{\delta_z}\int_{\Omega}w^{r}. \end{equation}

Collecting (3.18)–(3.20), we then arrive at

(3.21)\begin{equation} \begin{aligned} & \displaystyle \frac{{\rm d}}{{\rm d}t} \int_{\Omega}({\rm e}^{\chi_u v}a^{r}+{\rm e}^{\chi_w v} b^{r}+ z^{r})\nonumber\\ & \quad +c_5 \int_{\Omega}({\rm e}^{\chi_u v}a^{r}+{\rm e}^{\chi_w v} b^{r}+ z^{r}) +c_5 \int_{\Omega}(|\nabla a^{{r}/{2}}|^{2}+|\nabla b^{{r}/{2}}|^{2} +|\nabla z^{{r}/{2}}|^{2}) \\ & \quad\leq \displaystyle c_6(r)\left(\int_{\Omega}a^{r+1}+\int_{\Omega}b^{r+1}+\int_{\Omega}z^{r+1}\right)+c_6(r) \end{aligned} \end{equation}

with $c_5>0$ and $c_6(r)>0$.

According to lemma 3.4, there exists $c_7(r)>0$ such that

(3.22)\begin{equation} \int_{\Omega}a\cdot|\log |a^{{r}/{2}}||+ \int_{\Omega}b\cdot|\log |b^{{r}/{2}}||+ \int_{\Omega}z\cdot|\log |z^{{r}/{2}}||\leq c_7(r). \end{equation}

Now we invoke the logarithm-type Gagliardo–Nirenberg inequality (we refer to lemma A.5 in [Reference Tao and Winkler28] for details) to obtain that there exists $c_8(r)>0$ such that

(3.23)\begin{equation} c_6(r)\int_{\Omega}a^{r+1}\!\leq\! \frac{c_5}{2c_7(r)}\|\nabla a^{{r}/{2}}\|^{2}_{L^{2}(\Omega)}\cdot \int_{\Omega}a\cdot|\log |a^{{r}/{2}}||\!+\!c_8(r)\left(\|a^{{r}/{2}}\|^{({2(r+1)}/{r})}_{L^{{2}/{r}}(\Omega)}\!+\!1\right), \end{equation}

and

(3.24)\begin{equation} c_6(r)\int_{\Omega}b^{r+1}\leq \frac{c_5}{2c_7(r)}\|\nabla b^{{r}/{2}}\|^{2}_{L^{2}(\Omega)}\cdot\int_{\Omega}b\cdot|\log |b^{{r}/{2}}||+c_8(r)\left(\|b^{{r}/{2}}\|^{({2(r+1)}/{r})}_{L^{{2}/{r}}(\Omega)}+1\right) \end{equation}

as well as

(3.25)\begin{equation} c_6(r)\int_{\Omega}z^{r+1}\!\leq\! \frac{c_5}{2c_7(r)}\|\nabla z^{{r}/{2}}\|^{2}_{L^{2}(\Omega)}\cdot\int_{\Omega}z\cdot|\log |z^{{r}/{2}}||\!+\!c_8(r)\left(\|z^{{r}/{2}}\|^{({2(r+1)}/{r})}_{L^{{2}/{r}}(\Omega)}\!+\!1\right). \end{equation}

Therefore combining (3.22)–(3.25) and by lemma 2.2, we arrive at

\[ \displaystyle \frac{{\rm d}}{{\rm d}t} \int_{\Omega}({\rm e}^{\chi_u v}a^{r}+{\rm e}^{\chi_w v} b^{r}+ z^{r}) +c_5 \int_{\Omega}({\rm e}^{\chi_u v}a^{r}+{\rm e}^{\chi_w v} b^{r}+ z^{r})\leq c_9(r) \]

and thereby

\[ \int_{\Omega}a^{r}({\cdot},t)+\int_{\Omega}b^{r}({\cdot},t)+\int_{\Omega}z^{r}({\cdot},t)\leq c_{10}(r) \]

with $c_{9}(r)> 0, c_{10}(r)>0$ by a standard ODE comparison argument. At this position, one can derive a bound for $a,b,z$ with respect to the norm in $L^{\infty }(\Omega )$ by means of a Moser-type iteration argument in quite a standard manner. We omit the proof thereof, and would like refer to [Reference Li and Wang15, Reference Tao and Winkler26, Reference Tao and Winkler28] for details in a closely related setting.

Proof. We invoke lemma 2.3 along with (3.17) to see that there exist $c_1>0$ and $c_2>0$ such that

(4.2)\begin{equation} \|w({\cdot},t)\|_{L^{2}(\Omega)}^{2}\leq c_1\|w({\cdot},t)\|_{L^{\infty}(\Omega)} {\rm e}^{-\delta t}\leq c_2 {\rm e}^{-\delta t} \end{equation}

with $\delta =\min \{1-\beta,\delta _z\}>0$. According to known smoothing properties of the Neumann heat semigroup $( {\rm e}^{\sigma \Delta } )_{\sigma > 0}$ on the domain $\Omega \subset \mathbb {R}^{2}$ [Reference Winkler31], there exists $c_3 >0$ such that for all $\sigma >0$ and $\varphi \in C^{0}(\Omega )$,

(4.3)\begin{equation} \left \| {\rm e}^{\sigma D_z\Delta } \varphi\right \|_{L^{\infty}(\Omega)} \leq c_3( 1+\sigma^{-(1/2)}) \| \varphi \|_{L^{2}(\Omega)}. \end{equation}

Due to the nonnegativity of $z$ and the comparison principle, we may use (4.2) and (4.3) to infer that

(4.4)\begin{equation} \begin{aligned} z({\cdot},t) & = {\rm e}^{t(D_z\Delta-\delta_z)}z_0+\displaystyle\int^{t}_0{\rm e}^{(t-s)(D_z\Delta-\delta_z)}(\beta w-\rho uz)({\cdot},s){\rm d}s \\ & \leq {\rm e}^{t(D_z\Delta-\delta_z)}z_0+\beta \displaystyle\int^{t}_0{\rm e}^{(t-s)(D_z\Delta-\delta_z)}w({\cdot},s){\rm d}s \\ & \leq \displaystyle {\rm e}^{-\delta_z t}\|z_0\|_{L^{\infty}(\Omega)}+\beta c_3 \int^{t}_0(1+(t-s)^{-(1/2)}){\rm e}^{-\delta_z(t-s)}\|w({\cdot},s)\|_{L^{2}(\Omega)}{\rm d}s\\ & \leq \displaystyle {\rm e}^{-\delta_z t}\|z_0\|_{L^{\infty}(\Omega)}+\beta c_2 c_3\int^{t}_0(1+(t-s)^{-(1/2)}){\rm e}^{-\delta_z(t-s)} {\rm e}^{-({\delta s}/2)}{\rm d}s\\ & \leq \displaystyle {\rm e}^{-\delta_z t}\|z_0\|_{L^{\infty}(\Omega)}+\beta c_2 c_3 c_4 {\rm e}^{-{\rm min}\{\delta_z, {\delta }/2)t } \end{aligned} \end{equation}

with some $c_4>0$, which along with the nonnegativity of $z$ entails that (4.1) holds with $C=\|z_0\|_{L^{\infty }(\Omega )} +\beta c_2 c_3 c_4.$

Proof. According to lemma 4.1, one can pick $t_1>0$ sufficiently large such that for all $t>t_1$

(4.6)\begin{equation} \|z({\cdot},t)\|_{L^{\infty}(\Omega)}\leq \frac{\mu_u}{2\rho}. \end{equation}

Hence by means of a straightforward computation based on (2.2), one can see that

\begin{align*} a_t& \geq D_u{\rm e}^{-\chi_u v}\nabla\cdot({\rm e}^{\chi_u v}\nabla a)+\mu_u a(1-a{\rm e}^{\chi_u v})-\displaystyle\rho az\\ & \geq D_u{\rm e}^{-\chi_u v}\nabla \cdot({\rm e}^{\chi_u v}\nabla a)+a\left(\frac{\mu_u}{2}-{\rm e}^{\chi_u \|v_0\|_{L^{\infty}(\Omega)}}a \right) \end{align*}

for all $t>t_1$.

Now let $\underline {a}(t)$ be the smooth solution to the initial value problem:

(4.7)\begin{equation} \left\{ \begin{array}{@{}l} \underline{a}_t=\underline{a}\left(\displaystyle\dfrac{\mu_u}{2}-{\rm e}^{\chi_u \|v_0\|_{L^{\infty}(\Omega)}}\underline{a}\right),\\ \underline{a}(t_1)=\inf\limits_{x\in{\Omega}} \{ u(x,t_1) {\rm e}^{-\chi_u \|v_0\|_{L^{\infty}(\Omega)}}\}, \end{array} \right. \end{equation}

then through the explicit solution of above Bernoulli-type ODE, we have

(4.8)\begin{equation} \underline{a}(t)\geq c_1:= \min\left\{\underline{a}(t_1),\frac{\mu_u}{2}{\rm e}^{-\chi_u \|v_0\|_{L^{\infty}(\Omega)}}\right\}>0 \end{equation}

for all $t>t_1$. It is observed that

(4.9)\begin{equation} \underline{a}_t=D_u {\rm e}^{-\chi_u v}\nabla \cdot({\rm e}^{\chi_u v}\nabla \underline{a})+ \underline{a}\left(\frac{\mu_u}{2}-{\rm e}^{\chi_u \|v_0\|_{L^{\infty}(\Omega)}}\underline{a}\right)\end{equation}

and $a(x,t_1)\geq \underline {a}(t_1)$. Hence from the comparison principle of the parabolic equation, one can conclude that

(4.10)\begin{equation} a(x,t)\geq \underline{a} (t)\geq c_1\text{ for all }(x,t)\in \Omega\times(t_1,\infty). \end{equation}

and thereby (4.5) is valid with $\gamma = c_1$.

Proof. By recalling the outcomes of lemma 4.2, we have

\[ v_t={-}(\alpha_u u+\alpha_w w)v\leq{-}\alpha_u\gamma v \]

for all $t>t_1$ and hence $v(x,t)\leq v_0(x) {\rm e}^{-\alpha _u\gamma (t-t_1)}\leq \|v_0\|_{L^{\infty }(\Omega )} {\rm e}^{\alpha _u\gamma t_1}{\rm e}^{-\alpha _u\gamma t}$. On the other hand, $v(x,t)\leq \|v_0\|_{L^{\infty }(\Omega )}\leq \|v_0\|_{L^{\infty }(\Omega )} {\rm e}^{\alpha _u\gamma t_1}{\rm e}^{-\alpha _u\gamma t}$ for $t\in (0, t_1)$. Hence (4.11) is valid with $C=\|v_0\|_{L^{\infty }(\Omega )} {\rm e}^{\alpha _u\gamma t_1}$.

Proof. In view of $s-1-\log s>0$ for all $s>0$ and $v_t<0$, we can conclude that

(4.14)\begin{equation} \begin{aligned} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_u v}(a-1-\log a)\\ & \quad= \displaystyle\int_{\Omega}{\rm e}^{\chi_u v}(a-1-\log a)v_t+\int_{\Omega}{\rm e}^{\chi_u v}\left(\frac{a-1}{a}\right)a_t\\ & \quad\leq \displaystyle-D_u\int_{\Omega}{\rm e}^{\chi_u v}\frac{|\nabla a|^{2}}{a^{2}}+\mu_u\int_{\Omega} {\rm e}^{\chi_u v}(a-1)(1-u)\\ & \qquad+\displaystyle\chi_u\int_{\Omega}{\rm e}^{\chi_u v}(a-1)(\alpha_uu+\alpha_w w)v- \displaystyle\rho\int_{\Omega}{\rm e}^{\chi_u v}z(a-1). \end{aligned} \end{equation}

Here by Young's inequality,

(4.15)\begin{equation} \begin{aligned} (1-a)(1-u) & =(1-u)^{2}+(u-a)(1-u)\\ & \geq \displaystyle\frac 12 (1-u)^{2}- a^{2}({\rm e}^{\chi_u v}-1)^{2}. \end{aligned} \end{equation}

Due to the fact that ${\rm e}^{s}\leq 1+2s$ for all $s\in [0,\log 2]$, (4.11) allows us to fix a $t_1>1$ suitably large such that for all $t\geq t_1$,

\[ ({\rm e}^{\chi_u v({\cdot},t)}-1)^{2}\leq 4\chi_u ^{2} v^{2}({\cdot},t), \]

which together with (4.15) entails that for $t\geq t_1$

\[ (1-a)(1-u)\geq \displaystyle\frac 12 (1-u)^{2}-4 a^{2}\chi_u ^{2} v^{2}. \]

Therefore we infer from (3.17) and (4.14) that for all $t\geq t_1$

\begin{align*} & \displaystyle \frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_u v}(a-1-\log a)+D_u\int_{\Omega}{\rm e}^{\chi_u v}\frac{|\nabla a|^{2}}{a^{2}}+ \mu_u\int_{\Omega}(u-1) ^{2}\\ & \quad\leq \displaystyle c_1\int_{\Omega}z+c_1\int_{\Omega}v \end{align*}

with some $c_1>0$. After a time integration this leads to

\begin{align*} & \displaystyle D_u\int^{t}_{t_1}\int_{\Omega}{\rm e}^{\chi_u v}\frac{|\nabla a|^{2}}{a^{2}}+ \mu_u\int^{t}_{t_1} \int_{\Omega}(u-1)^{2}\\ & \quad\leq c_1 \displaystyle\int^{t}_{t_1}\int_{\Omega}z+c_1\int^{t}_{t_1}\int_{\Omega}v+ \int_{\Omega}{\rm e}^{\chi_u m_v}(a({\cdot},t_1)-1-\log a ({\cdot},t_1)) \end{align*}

and thereby implies that both (4.12) and (4.13) is valid thanks to (4.1) and (4.11).

Proof. Multiplying the second equation in (2.2) by ${\rm e}^{\chi _w v}b$ and integrating by parts, one can conclude that

\begin{align*} & \frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_w v}b^{2}+2D_w\int_{\Omega}{\rm e}^{\chi_w v}|\nabla b|^{2}+2 \int_{\Omega}{\rm e}^{\chi_w v}b^{2}\\ & \quad= 2\rho\int_{\Omega}abz {\rm e}^{\chi_w v}+2\chi_w\int_{\Omega}{\rm e}^{\chi_w v}b^{2}(\alpha_u a {\rm e}^{\chi_u v}+\alpha_w b {\rm e}^{\chi_w v})v, \end{align*}

which together with the global-in-time boundedness property of $a$ and $b$, implies that

\[ \frac{{\rm d}}{{\rm d}t}\int_{\Omega}{\rm e}^{\chi_w v}b^{2}+2D_w\int_{\Omega}{\rm e}^{\chi_w v}|\nabla b|^{2}+2\int_{\Omega}{\rm e}^{\chi_w v}b^{2} \leq c_1\left(\int_{\Omega}v+\int_{\Omega}z\right) \]

for some $c_1>0$. Hence according to lemmas 4.3 and 2.3, we can get

(4.17)\begin{equation} \int^{\infty}_0\int_{\Omega}|\nabla b|^{2}<\infty. \end{equation}

Now since

\[ \nabla v_t={-}(\alpha_u \nabla u + \alpha_w \nabla w )v-(\alpha_u u + \alpha_w w ) \nabla v, \]

a direct computation shows that

\begin{align*} & \displaystyle \frac{1}{2}\frac{{\rm d}}{{\rm d}t}\int_{\Omega}|\nabla v|^{2}+\int_{\Omega}(u+w)|\nabla v|^{2}\\ & \quad={-}\alpha_w \chi_w\displaystyle \int_{\Omega}v {\rm e}^{\chi_w v} b|\nabla v|^{2}- \alpha_w\displaystyle\int_{\Omega}v {\rm e}^{\chi_w v}\nabla v\cdot \nabla b- \alpha_u \displaystyle\int_{\Omega}v\nabla v\cdot \nabla u\\ & \quad\leq{-}\alpha_w\displaystyle\int_{\Omega}v {\rm e}^{\chi_w v}\nabla v\cdot \nabla b -\alpha_u \displaystyle\int_{\Omega}v {\rm e}^{\chi_u v}\nabla v\cdot \nabla a. \end{align*}

Therefore, recalling the pointwise lower bound in (4.5), (3.17) and by the Young's inequality, we can find a constant $c_2>0$ such that

(4.18)\begin{equation} \frac{{\rm d}}{{\rm d}t}\int_{\Omega}|\nabla v|^{2}+\gamma \int_{\Omega}|\nabla v|^{2}\leq c_2 \left(\int_{\Omega}|\nabla b|^{2}+\int_{\Omega}\frac{|\nabla a|^{2}}{a^{2}}\right) \end{equation}

and thus for any $t>0$,

\[ \gamma \int^{t}_0 \int_{\Omega}|\nabla v|^{2}\leq \int_{\Omega}|\nabla v_0|^{2}+c_2 \left( \int^{\infty}_0 \int_{\Omega}|\nabla b|^{2}+\int^{\infty}_0 \int_{\Omega}\frac{|\nabla a|^{2}}{a^{2}} \right), \]

which along with (4.17) and (4.12) makes sure that (4.16) is actually valid.

Proof. On the basis of the $v$-equation in (1.3), we have

\[ \nabla v({\cdot},t)\!=\! \nabla v({\cdot},0){\rm e}^{-\int^{t}_0(u+w)({\cdot},s){\rm d}s} \!-\!v({\cdot},0){\rm e}^{-\int^{t}_0(u+w)({\cdot},s){\rm d}s} \int^{t}_0(\nabla u({\cdot},s)\!+\!\nabla w({\cdot},s)){\rm d}s \]

which along with (4.5) and the Young's inequality entails that

(4.20)\begin{align} & \int_{\Omega}|\nabla v|^{2}\leq 2{\rm e}^{{-}2\gamma t}\|\nabla v_0\|^{2}_{L^{2}(\Omega)}\nonumber\\ & \quad +4 t{\rm e}^{{-}2\gamma t}\|v_0\|^{2}_{L^{\infty}(\Omega)} \left(\int^{t}_0\int_{\Omega}|\nabla u|^{2}{\rm d}s+\int^{t}_0\int_{\Omega}|\nabla w|^{2}{\rm d}s\right) . \end{align}

Furthermore observing that

\[ |\nabla w|\leq \chi_w {\rm e}^{\chi_w v}|\nabla v|b+{\rm e}^{\chi_w v}|\nabla b| \]

as well as

\[ |\nabla u|\leq \chi_u {\rm e}^{\chi_u v}|\nabla v|a+{\rm e}^{\chi_u v}|\nabla a|, \]

we conclude from (4.20) that there exists $c_1>0$ such that

(4.21)\begin{equation} \begin{aligned} \displaystyle\int_{\Omega}|\nabla v({\cdot},t)|^{2} & \leq c_1{\rm e}^{{-}2\gamma t}+c_1 t{\rm e}^{{-}2\gamma t} \displaystyle\int^{t}_0\int_{\Omega}(|\nabla b|^{2}+|\nabla a|^{2} +|\nabla v|^{2}){\rm d}s\\ & \leq c_1{\rm e}^{{-}2\gamma t}+c_1 t{\rm e}^{{-}2\gamma t}\displaystyle\int^{\infty}_0\int_{\Omega}(|\nabla b|^{2}+|\nabla a|^{2} +|\nabla v|^{2}){\rm d}s \end{aligned} \end{equation}

and thus

\[ \displaystyle\int_{\Omega}|\nabla v({\cdot},t)|^{2}\leq c_2(t+1){\rm e}^{{-}2\gamma t} \]

with some $c_2>0$, thanks to (4.17), (4.12) and (4.16).

Proof. Testing the first equation in (1.3) by $u-1$ and integrating by parts, we have

\begin{align*} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}(u-1)^{2}+2D_u\int_{\Omega}|\nabla u|^{2}+2\mu_u\int_{\Omega}u(u-1)^{2}\\ & \quad=2\xi_u\displaystyle\int_{\Omega}u\nabla v\cdot\nabla u-2\rho\int_{\Omega}(u-1)uz. \end{align*}

We thereupon make use of lemmas 4.2, 3.5 along with the Young's inequality to get

(4.23)\begin{equation} \begin{aligned} & \displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}(u-1)^{2}+D_u\int_{\Omega}|\nabla u|^{2}+2\mu_u\gamma \int_{\Omega}(u-1)^{2}\\ & \quad\leq\displaystyle \frac{\xi^{2}_u}{D_u}\int_{\Omega}u^{2}|\nabla v|^{2}+2\rho\int_{\Omega}uz\\ & \quad\leq c_1\displaystyle \int_{\Omega}|\nabla v|^{2}+c_1\int_{\Omega}z \end{aligned} \end{equation}

with some $c_1>0$.

According to lemmas 4.6 and 2.3, (4.23) implies that

(4.24)\begin{equation} \int_{\Omega}(u-1)^{2}\leq c_2 {\rm e}^{-\eta_1 t} \end{equation}

with $\eta _1:=\min \{2\mu _u\gamma,2\gamma,\delta \}$ and $c_2>0$ for all $t>0$.

Recalling known smoothing estimates for the Neumann heat semigroup on $\Omega \subset \mathbb {R}^{2}$ [Reference Winkler31], there exist $c_3=c_3(p,q)>0$, $c_4=c_4(p,q)>0$ fulfilling

(4.25)\begin{equation} \left \| {\rm e}^{\sigma D_u\Delta } \varphi\right \|_{L^{p}(\Omega)} \leq c_3 \sigma^{-( 1/q- 1/p)} \| \varphi \|_{L^{q}(\Omega)} \end{equation}

for each $\varphi \in C^{0}(\Omega )$, and for all $\varphi \in ( L^{q}( \Omega ) )^{2}$,

(4.26)\begin{equation} \left \| {\rm e}^{\sigma D_u\Delta }\nabla\cdot \varphi\right \|_{L^{p}\left ( \Omega \right )}\leq c_{4} ( 1+\sigma^{-({1}/{2})-( {1}/{q}-{1}/{p}) } ){\rm e}^{-\lambda _{1}\sigma}\left \| \varphi \right \|_{L^{q}\left ( \Omega \right )} \end{equation}

with $\lambda _{1}> 0$ the first nonzero eigenvalue of $-\Delta$ in $\Omega$ under the Neumann boundary condition.

Relying on a variation-of-constants representation of $u$ related to the first equation in (1.3), we utilize (4.25) and (4.26) to infer that

(4.27)\begin{equation} \begin{aligned} & \|(u-1)({\cdot},t)\|_{L^{p}(\Omega)}\\ & \quad\leq \|{\rm e}^{t(D_u\Delta-\delta )}(u_0-1)\|_{L^{p}(\Omega)}+ \xi_u\displaystyle \int^{t}_0\|{\rm e}^{(t-s)(D_u\Delta-\delta)} \nabla\cdot(u\nabla v) \|_{L^{p}(\Omega)}{\rm d}s\\ & \qquad+\displaystyle\int^{t}_0\|{\rm e}^{(t-s)(D_u\Delta-\delta)}((\mu_u u-\delta)(1-u)-\rho uz)\|_{L^{p}(\Omega)}{\rm d}s \\ & \quad\leq \displaystyle {\rm e}^{-\delta t}\|u_0\!-1\|_{L^{p}(\Omega)}\!+ c_5(p) \int^{t}_0(1\!+(t-\!s)^{{-}1+1/p}){\rm e}^{-(\delta+\lambda_1)(t-s)}\|\nabla v({\cdot},s)\|_{L^{2}(\Omega)}{\rm d}s\\ & \qquad +c_5(p)\displaystyle\int^{t}_0(1+(t-s)^{-(1/2)+1/p}){\rm e}^{-\delta(t-s)} \|(u-1)({\cdot},s)\|_{L^{2}(\Omega)}{\rm d}s\\ & \qquad+c_5(p)\displaystyle\int^{t}_0(1+(t-s)^{{-}1+1/p}){\rm e}^{-\delta(t-s)} \|z({\cdot},s)\|_{L^{1}(\Omega)}{\rm d}s \end{aligned} \end{equation}

for some $c_5(p)>0$. Therefore by (4.24), (4.19), (2.8) and thanks to the fact that for $\alpha \in (0,1)$ $\gamma _1$ and $\delta _1$ positive constants with $\gamma _1 \neq \delta _1$, there exists $c_6> 0$ such that

\[ \int_{0}^{t} ( 1+( t-s ) ^{-\alpha}){\rm e}^{-\gamma_1 s}{\rm e}^{-\delta_1 ( t-s )}{\rm d}s \leq c_{6} {\rm e}^{-\min\left \{ \gamma_1 ,\delta_1 \right \}t}, \]

(4.22) readily results from (4.27) with $\eta _1=\frac 12 \min \{ \gamma, \mu _u \gamma, \frac {1-\beta }2, \frac {\delta _z}2\}$ and some $C(p)>0$.

Proof. Testing the identity

\[ a_t=D_u \triangle a + D_u \nabla v \cdot \nabla a + f(x,t), \quad x\in\Omega, \ t > 0 \]

with $f(x,t)=\mu _u a(1-u)-\displaystyle \rho a z+\chi _u a(\alpha _u u + \alpha _w w)v$ by $-\triangle a$, and using Young's inequality, we get

(4.30)\begin{equation} \begin{aligned} & \displaystyle \frac {\rm d} {{\rm d}t}\int_\Omega |\nabla a|^{2} +2D_u\displaystyle \int_\Omega|\triangle a|^{2} \\ & \quad={-}2D_u \chi_u \displaystyle \int_\Omega (\nabla a\cdot \nabla v)\triangle a- 2\int_\Omega f\triangle a\\ & \quad\leq D_u\displaystyle \int_\Omega|\triangle a|^{2} +2D_u\chi_u^{2} \int_\Omega |\nabla a|^{2} |\nabla v|^{2}+\frac 2{D_u}\int_\Omega |f|^{2}. \end{aligned} \end{equation}

Note that by the Gagliardo–Nirenberg type interpolation with standard elliptic regularity theory and Poincaré's inequality, one can find constants $c_1> 0$ and $c_2> 0$ such that for all $\varphi \in W^{2,2}(\Omega )$ with ${\partial \varphi }/{\partial \nu }=0$ on $\partial \Omega$,

\[ \|\nabla \varphi\|^{4}_ {L^{4}(\Omega)}\leq c_1 \| \Delta \varphi\|^{2}_{L^{2}(\Omega)}\| \varphi\|^{2}_{L^{\infty}(\Omega)} \]

and

\[ \|\nabla \varphi\|^{2}_ {L^{2}(\Omega)}\leq c_2 \| \Delta \varphi\|^{2}_{L^{2}(\Omega)} \]

(see lemmas A.1 and A.3 in [Reference Fuest7]). Hence thanks to lemma 3.5, we can pick $c_3>0$ such that

\[ a(x,t)\leq c_3, \quad b(x,t)\leq c_3 \]

for all $x \in \Omega$ and $t > 0$, and thereby have

(4.31)\begin{equation} \|\nabla a\|^{4}_ {L^{4}(\Omega)}\leq c_1c_3^{2} \| \Delta a\|^{2}_{L^{2}(\Omega)},\quad \|\nabla a\|^{2}_ {L^{2}(\Omega)}\leq c_2 \| \Delta a\|^{2}_{L^{2}(\Omega)} \end{equation}

as well as

(4.32)\begin{equation} \|\nabla b\|^{4}_ {L^{4}(\Omega)}\leq c_1c_3^{2} \| \Delta b\|^{2}_{L^{2}(\Omega)},\quad \|\nabla b\|^{2}_ {L^{2}(\Omega)}\leq c_2 \| \Delta b\|^{2}_{L^{2}(\Omega)}. \end{equation}

Combining (4.31) with (4.30), the Young's inequality shows that

(4.33)\begin{equation} \begin{aligned} & \displaystyle \frac {\rm d} {{\rm d}t}\|\nabla a\|^{2}_ {L^{2}(\Omega)} + \frac{D_u}{2c_2}\|\nabla a\|^{2}_ {L^{2}(\Omega)}+ D_u\displaystyle \|\triangle a\|^{2}_ {L^{2}(\Omega)} \\ & \quad\leq \displaystyle\frac{D_u}{4c_1c_3^{2}} \|\nabla a\|^{4}_ {L^{4}(\Omega)} + \displaystyle\frac{D_u}{2c_2}\|\nabla a\|^{2}_ {L^{2}(\Omega)} + 4D_u\chi_u^{4} c_1c_3^{2} \|\nabla v\|^{4}_ {L^{4}(\Omega)} +\frac 2{D_u}\|f\|_{L^{2}(\Omega)}^{2}\\ & \quad\leq \displaystyle \frac {3D_u}4\|\triangle a\|^{2}_ {L^{2}(\Omega)} +4D_u\chi_u^{4} c_1 c_3^{2} \|\nabla v\|^{4}_ {L^{4}(\Omega)} +\frac 2{D_u}\|f\|_{L^{2}(\Omega)}^{2}, \end{aligned} \end{equation}

which readily implies that

(4.34)\begin{equation} \begin{aligned} & \displaystyle \frac {\rm d} {{\rm d}t}\|\nabla a\|^{2}_ {L^{2}(\Omega)} + \frac{D_u}{2c_2}\|\nabla a\|^{2}_ {L^{2}(\Omega)}+ \displaystyle\frac {D_u}4 \|\triangle a\|^{2}_ {L^{2}(\Omega)} \\ & \quad\leq \displaystyle 4D_u\chi_u^{4} c_1 c_3^{2} \|\nabla v\|^{4}_ {L^{4}(\Omega)} +\frac 2{D_u}\|f\|_{L^{2}(\Omega)}^{2}. \end{aligned} \end{equation}

Likely, we can get

(4.35)\begin{equation} \begin{aligned} & \displaystyle \frac d {dt}\|\nabla b\|^{2}_ {L^{2}(\Omega)} + \frac{D_w}{2c_2}\|\nabla b\|^{2}_ {L^{2}(\Omega)}+ \displaystyle\frac {D_w}4 \|\triangle b\|^{2}_ {L^{2}(\Omega)} \\ & \quad\leq \displaystyle 4D_w\chi_w^{4} c_1 c_3^{2} \|\nabla v\|^{4}_ {L^{4}(\Omega)} +\frac 2{D_w}\|g\|_{L^{2}(\Omega)}^{2} \end{aligned} \end{equation}

with $g(x,t)=- b+\displaystyle \rho u{\rm e}^{-\chi _w v}z + \chi _w b(\alpha _u u+ \alpha _w w)v$.

Now in order to appropriately compensate the first summand on right-hand side of (4.34) and (4.35), we use the third equation in (2.2) to see that

(4.36)\begin{equation} \begin{aligned} & \displaystyle \frac 14 \frac {\rm d} {{\rm d}t}\int_\Omega |\nabla v|^{4}\\ & \quad={-}\displaystyle \int_\Omega|\nabla v|^{2} \nabla v\cdot\nabla v_t\\ & \quad={-}\alpha_u \displaystyle \int_\Omega a(v+\chi_u){\rm e}^{\chi_u v}|\nabla v|^{4}- \alpha_u \displaystyle \int_\Omega v{\rm e}^{\chi_u v}|\nabla v|^{2} \nabla v\cdot\nabla a\\ & \qquad-\alpha_w \displaystyle \int_\Omega b(v+\chi_w){\rm e}^{\chi_w v}|\nabla v|^{4}- \alpha_w \displaystyle \int_\Omega v{\rm e}^{\chi_w v}|\nabla v|^{2} \nabla v\cdot\nabla b. \end{aligned} \end{equation}

Here, recalling the uniform positivity of $a$ stated in lemma 4.2, we can pick $c_4>0$ fulfilling

\[ \alpha_u \displaystyle \int_\Omega a(v+\chi_u){\rm e}^{\chi_u v}|\nabla v|^{4}\geq \alpha_u \chi_uc_4 \displaystyle \int_\Omega |\nabla v|^{4} \]

and thus infer by the Young's inequality and lemma 2.2 that for all $t>t_0$

(4.37)\begin{equation} \begin{aligned} & \displaystyle \frac {\rm d} {{\rm d}t}\int_\Omega |\nabla v|^{4} +c_5\int_\Omega |\nabla v|^{4} \\ & \quad\leq c_6\|v({\cdot},t_0)\|_{L^{\infty}(\Omega)}\displaystyle \int_\Omega (|\nabla a|^{4}+ |\nabla b|^{4}) \end{aligned} \end{equation}

with constants $c_5>0$, $c_6>0$.

Now if we write $d_1:=({8c_1c_3^{2}( D_u\chi _u^{4}+ D_w\chi _w^{4})})/{c_5}$, combining (4.37), (4.34) with (4.35) yields

(4.38)\begin{equation} \begin{aligned} & \displaystyle \frac {\rm d} {{\rm d}t}\left(\|\nabla a\|^{2}_ {L^{2}(\Omega)} + \|\nabla b\|^{2}_ {L^{2}(\Omega)}+d_1\|\nabla v\|_ {L^{4}(\Omega)}^{4}\right)\\ & \qquad+\displaystyle \frac{D_u}{2c_2}\|\nabla a\|^{2}_ {L^{2}(\Omega)}+ \displaystyle\frac {D_u}4 \|\triangle a\|^{2}_ {L^{2}(\Omega)} + \frac{D_w}{2c_2}\|\nabla b\|^{2}_ {L^{2}(\Omega)}\\ & \quad\quad+ \displaystyle\frac {D_w}4 \|\triangle b\|^{2}_ {L^{2}(\Omega)} +\frac{c_5 d_1}2\|\nabla v\|_ {L^{4}(\Omega)}^{4} \\ & \quad\leq c_6d_1\|v({\cdot},t_0)\|_{L^{\infty}(\Omega)} (\|\nabla a\|_ {L^{4}(\Omega)}^{4} +\|\nabla b\|_ {L^{4}(\Omega)}^{4}) +\displaystyle\frac 2{D_u}\|f\|_{L^{2}(\Omega)}^{2}+\frac 2{D_w}\|g\|_{L^{2}(\Omega)}^{2}, \end{aligned} \end{equation}

which together with (4.31), (4.32) and lemma 4.3 entails that there exists $t_1>1$ suitably large such that for all $t>t_1$,

(4.39)\begin{equation} \begin{aligned} & \displaystyle\frac {\rm d} {{\rm d}t}\left(\|\nabla a\|^{2}_ {L^{2}(\Omega)} + \|\nabla b\|^{2}_ {L^{2}(\Omega)}+d_1\|\nabla v\|_ {L^{4}(\Omega)}^{4}\right)\\ & \qquad+\displaystyle\frac{D_u}{2c_2}\|\nabla a\|^{2}_ {L^{2}(\Omega)}+ \frac{D_w}{2c_2}\|\nabla b\|^{2}_ {L^{2}(\Omega)}+\frac{c_5 d_1}2\|\nabla v\|_ {L^{4}(\Omega)}^{4} \\ & \quad\leq \displaystyle\frac 2{D_u}\|f\|_{L^{2}(\Omega)}^{2}+\frac 2{D_w}\|g\|_{L^{2}(\Omega)}^{2}. \end{aligned} \end{equation}

Due to lemma 3.5, we see that

\[ |f(x,t)|^{2}+|g(x,t)|^{2}\leq c_7(|u(x,t)-1|^{2}+|w(x,t)|^{2}+ |z(x,t)|^{2}+|v(x,t)|^{2}) \]

with some $c_7>0$, and thus there exist $\eta _3>0$ and $c_8>0$ such that

(4.40)\begin{equation} \int_{\Omega}|f(x,t)|^{2}+ |g(x,t)|^{2} \leq c_8 {\rm e}^{-\eta_3 t}\text{ for all }t>t_1, \end{equation}

thanks to lemmas 4.7, 4.8, 4.1 and 4.3. Therefore (4.29) readily results from (4.40) and (4.39).

Proof. Since the proof is similar to that of lemma 4.7, we only give a short proof of (4.41). In view to known smoothing estimates for the Neumann heat semigroup on $\Omega \subset \mathbb {R}^{2}$ [Reference Winkler31], there exist $c_1>0$, $c_2>0$ fulfilling

(4.43)\begin{equation} \left \| {\rm e}^{\sigma D_u\Delta } \varphi\right \|_{L^{\infty}(\Omega)} \leq c_1 \sigma^{-(1/2)} \| \varphi \|_{L^{2}(\Omega)} \end{equation}

for each $\varphi \in C^{0}(\Omega )$, and for all $\varphi \in ( L^{4}( \Omega ) )^{2}$,

(4.44)\begin{equation} \left \| {\rm e}^{\sigma D_u\Delta }\nabla\cdot \varphi\right \|_{L^{\infty}\left ( \Omega \right )}\leq c_2 ( 1+\sigma^{-(3/4)} ){\rm e}^{-\lambda_{1}\sigma}\left \| \varphi \right \|_{L^{4}\left ( \Omega \right )} \end{equation}

with $\lambda _{1}> 0$ the first nonzero eigenvalue of $-\Delta$ in $\Omega$ under the Neumann boundary condition.

According to the variation-of-constants representation of $u$ related to the first equation in (1.3), we utilize (4.43) and (4.44) to infer that

(4.45)\begin{equation} \begin{aligned} & \|(u-1)({\cdot},t)\|_{L^{\infty}(\Omega)}\\ & \quad\leq \|{\rm e}^{t(D_u\Delta-1 )}(u_0-1)\|_{L^{\infty}(\Omega)}+ \xi_u\displaystyle \int^{t}_0\|{\rm e}^{(t-s)(D_u\Delta-1)} \nabla\cdot(u\nabla v) \|_{L^{\infty}(\Omega)}{\rm d}s\\ & \qquad+\displaystyle\int^{t}_0\|{\rm e}^{(t-s)(D_u\Delta-1)}((\mu_u u-1)(1-u)-\rho uz)\|_{L^{\infty}(\Omega)}{\rm d}s \\ & \quad\leq\displaystyle {\rm e}^{{-}t}\|u_0-1\|_{L^{\infty}(\Omega)}+ c_3 \int^{t}_0(1+(t-s)^{-(3/4)}){\rm e}^{-(1+\lambda_1)(t-s)}\|\nabla v({\cdot},s)\|_{L^{4}(\Omega)}ds\\ & \qquad+c_3\displaystyle\int^{t}_0(1+(t-s)^{-\frac 12}){\rm e}^{-(t-s)} (\|(u-1)({\cdot},s)\|_{L^{2}(\Omega)} + \|z({\cdot},s)\|_{L^{2}(\Omega)})ds \end{aligned} \end{equation}

with some $c_3>0$. This readily establishes (4.41) with appropriate $\eta >0$ in view of (4.29), (4.22) and (4.1).

Proof of theorem 1.1. The statement on global boundedness of classical solutions has been asserted by lemma 3.5. The convergence properties in (1.6)–(1.9) are precisely established by lemmas 4.10, 4.1 and 4.3, respectively.