Proof. We first multiply equation (1.9) $_1$ by $n_{\epsilon}^{2\alpha-1}$ , integrate by parts over $\Omega$ and use the solenoidality of $u_\epsilon$ and the Young inequality to deduce that

(3.2) \begin{align} &\quad -\frac{1}{2\alpha}\frac{d}{dt}\int_{\Omega}n_{\epsilon}^{2\alpha}+(1-2\alpha)\int_{\Omega}n_{\epsilon}^{2\alpha-2}|\nabla n_{\epsilon}|^2 \nonumber\\[3pt] & =-\frac{1}{2\alpha}\int_{\Omega} u_\epsilon\cdot\nabla n_{\epsilon}^{2\alpha}+(1-2\alpha)\int_{\Omega}n_{\epsilon}^{2\alpha-1}\nabla n_{\epsilon}\cdot \big(S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_{\epsilon}\big) \nonumber\\[3pt] & =(1-2\alpha)\int_{\Omega}n_{\epsilon}^{2\alpha-1}\nabla n_{\epsilon}\cdot \big(S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_{\epsilon}\big) \nonumber\\[3pt] & \le \frac{1-2\alpha}{2} \int_{\Omega}n_{\epsilon}^{2\alpha-2} |\nabla n_{\epsilon}|^2+ \frac{1-2\alpha}{2}C_{S}^2\int_{\Omega}n_{\epsilon}^{2\alpha}(1+n_{\epsilon})^{-2\alpha}|\nabla c_{\epsilon}|^2 \nonumber\\[3pt] &\le \frac{1-2\alpha}{2} \int_{\Omega}n_{\epsilon}^{2\alpha-2} |\nabla n_{\epsilon}|^2+ C_1\int_{\Omega}|\nabla c_{\epsilon}|^2\end{align}

for all $t>0$ , where we also used the upper estimate (1.11) for S. To compensate the rightmost summand herein properly, we multiply equation (1.9) $_2$ by $c_{\epsilon}$ and utilise the solenoidality of $u_\epsilon$ again to find that

(3.3) \begin{equation}\frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega}c_{\epsilon}^{2}+\int_{\Omega}|\nabla c_{\epsilon}|^2+\int_{\Omega}c_{\epsilon}^{2}= \int_{\Omega}n_{\epsilon}c_{\epsilon} \qquad \textrm{for all}\quad t>0.\end{equation}

For any fixed $\theta\in\left(1,\frac{1}{1-\alpha}\right)$ , we use the Hölder inequality to obtain

\begin{equation*}\int_{\Omega}n_{\epsilon}c_{\epsilon}\leq \|n_{\epsilon}\|_{L^{\theta}(\Omega)}\|c_{\epsilon}\|_{L^{\frac{\theta}{\theta-1}}(\Omega)}=\|n_{\epsilon}^\alpha\|_{L^{\frac{\theta}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}\|c_{\epsilon}\|_{L^{\frac{\theta}{\theta-1}}(\Omega)} \qquad \textrm{for all}\quad t>0.\end{equation*}

Since the Sobolev embedding $W^{1,2}(\Omega)\hookrightarrow L^{\frac{\theta}{\theta-1}}(\Omega)$ and (2.2) imply that

\begin{equation*}\|c_{\epsilon}\|_{L^{\frac{\theta}{\theta-1}}(\Omega)}^2\leq C_2\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^2+ C_2\|c_{\epsilon}\|_{L^{1}(\Omega)}^2\leq C_2\int_{\Omega}|\nabla c_{\epsilon}|^2+C_3 \qquad \textrm{for all}\quad t>0,\end{equation*}

we make use of the Young inequality to deduce that

(3.4) \begin{equation}\int_{\Omega}n_{\epsilon}c_{\epsilon}\leq \frac{1}{2C_2}\|c_{\epsilon}\|_{L^{\frac{\theta}{\theta-1}}(\Omega)}^2+\frac{C_2}{2}\|n_{\epsilon}^\alpha\|_{L^{\frac{\theta}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}\le \frac12\int_{\Omega}|\nabla c_{\epsilon}|^2+\frac{C_3}{2C_2}+ \frac{C_2}{2}\|n_{\epsilon}^\alpha\|_{L^{\frac{\theta}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}\end{equation}

for all $t>0$ . In order to guarantee that the last summand here can be absorbed by the dissipated quantity in (3.2), we next apply the Gagliardo–Nirenberg inequality and the mass conservation (2.1) to see that

(3.5) \begin{equation}\frac{C_2}{2}\|n_{\epsilon}^\alpha\|_{L^{\frac{\theta}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}\leq C_4\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}\|n_{\epsilon}^\alpha\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{\alpha\theta}}+C_4\|n_{\epsilon}^\alpha\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}\leq C_5\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}+C_5\end{equation}

for all $t>0$ , whereupon substituting (3.4) and (3.5) into (3.3), we have

\begin{equation*}\epsilon\frac{d}{dt}\int_{\Omega}c_{\epsilon}^{2}+\int_{\Omega}|\nabla c_{\epsilon}|^2+2\int_{\Omega}c_{\epsilon}^{2}\leq 2C_5\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}+C_6 \qquad \textrm{for all}\quad t>0\end{equation*}

with $C_6:=2C_5+\frac{C_3}{C_2}$ . This together with (3.2) entails that

\begin{align*}&\frac{d}{dt}\left\{\!-\frac{1}{2\alpha}\int_{\Omega}n_{\epsilon}^{2\alpha}+2C_1\epsilon\int_{\Omega}c_{\epsilon}^{2}\right\}+\frac{1-2\alpha}{2\alpha^2}\int_{\Omega}|\nabla n_{\epsilon}^\alpha|^2+C_1\int_{\Omega}|\nabla c_{\epsilon}|^{2}+4C_1\int_{\Omega}c_{\epsilon}^{2}\\[3pt] & \le 4C_1C_5\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}+2C_1C_6\, \le \, \frac{1-2\alpha}{4\alpha^2}\int_{\Omega}|\nabla n_{\epsilon}^\alpha|^2+C_7 \qquad \textrm{for all}\quad t>0\end{align*}

and thus that

(3.6) \begin{equation}\frac{d}{dt}\left\{\!-\frac{1}{2\alpha}\int_{\Omega}n_{\epsilon}^{2\alpha}+2C_1\epsilon\int_{\Omega}c_{\epsilon}^{2}\right\}+\frac{1-2\alpha}{4\alpha^2}\int_{\Omega}|\nabla n_{\epsilon}^\alpha|^2+C_1\int_{\Omega}|\nabla c_{\epsilon}|^{2}+4C_1\int_{\Omega}c_{\epsilon}^{2} \le C_7\end{equation}

for all $t>0$ . Here we used the Young inequality in the last inequality of (3.6) due to $\frac{\theta-1}{\alpha\theta}\in(0,1)$ , which follows from $\theta\in\left(1,\frac1{1-\alpha}\right)$ . Then by setting

\begin{align*}y(t)&:=-\frac{1}{2\alpha}\|n_\epsilon^\alpha (\cdot,t)\|_{L^2(\Omega)}^2 + 2C_1\epsilon\|c_\epsilon (\cdot,t) \|_{L^2(\Omega)}^2,\\[3pt] g(t)&:=\frac{1-2\alpha}{4\alpha^2}\|\nabla n_\epsilon^\alpha (\cdot,t)\|_{L^2(\Omega)}^2 + C_1\|\nabla c_\epsilon (\cdot,t)\|_{L^2(\Omega)}^2,\end{align*}

and noticing that $4C_1\epsilon\int_{\Omega}c_{\epsilon}^2\le 4C_1\int_{\Omega}c_{\epsilon}^2 $ due to the fact $\epsilon\in(0,1)$ , we can conclude from (3.6) that

(3.7) \begin{equation}y'(t)+2y(t)+g(t)\leq C_{7}.\end{equation}

Since g(t) is nonnegative, we deduce from an ordinary differential inequality comparison argument that

(3.8) \begin{equation}y(t)\leq C_{8}:=\max\left\{\!-\frac{1}{2\alpha}\|n_0^\alpha\|_{L^2(\Omega)}^2+2C_1\|c_0\|_{L^2(\Omega)}^2, \frac{C_{7}}{2}\right\}\end{equation}

for all $t>0$ , which together with (3.7) yields that

\begin{equation*}\int_t^{t+1}g(s)ds\leq y(t)-y(t+1)-2\int_t^{t+1}y(s)ds+C_{7}\end{equation*}

for all $t\geq0$ . Due to $\alpha\in \left(0,\frac12\right)$ , we see from the Hölder inequality and the mass conservation (2.1) that

\begin{equation*}-y(t)\leq \frac{1}{2\alpha}\|n_\epsilon^\alpha (\cdot,t) \|_{L^2(\Omega)}^2 \le \frac{|\Omega|^{1-2\alpha}}{2\alpha}\|n_\epsilon (\cdot,t) \|_{L^1(\Omega)}^{2\alpha}=C_{9}:=\frac{|\Omega|^{1-2\alpha}}{2\alpha}\|n_0\|_{L^1(\Omega)}^{2\alpha},\end{equation*}

which together with (3.8) yields that

\begin{equation*}\frac{1-2\alpha}{4\alpha^2}\int_t^{t+1}\int_{\Omega}|\nabla n_\epsilon^{\alpha}|^2+C_1\int_t^{t+1}\int_{\Omega}|\nabla c_\epsilon|^2=\int_t^{t+1}g(s)ds\leq C_{10}:= C_{8}+3C_{9}+C_{7}\end{equation*}

for all $t\geq0$ , which entails (3.1).

Proof. For any fixed $\theta\in\left(1,\frac1{1-\alpha}\right)$ , we test equation (1.9) $_3$ by $u_\epsilon$ and employ the Hölder inequality, the Sobolev embedding $W^{1,2}(\Omega)\hookrightarrow L^{\frac{\theta}{\theta-1}}(\Omega)$ , the Poincaré inequality and the Young inequality to obtain that

\begin{align*}\frac{1}{2} \frac{d}{dt} \|u_\epsilon\|_{L^2(\Omega)}^2 + \|\nabla u_\epsilon\|_{L^2(\Omega)}^2 &= \int_{\Omega}n_{\epsilon}\nabla\phi\cdot u_\epsilon \\[3pt] &\leq\|\nabla\phi\|_{L^\infty(\Omega)}\|n_{\epsilon}\|_{L^\theta(\Omega)}\|u_\epsilon\|_{L^{\frac{\theta}{\theta-1}}(\Omega)}\\[3pt] &\leq C_1 \|n_{\epsilon}\|_{L^\theta(\Omega)}\|\nabla u_\epsilon\|_{L^{2}(\Omega)}\\[3pt] &\leq \frac12\|\nabla u_\epsilon\|_{L^{2}(\Omega)}^2+\frac{C_1^2}{2}\|n_{\epsilon}\|_{L^\theta(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t>0.\end{align*}

By means of the Young inequality and (3.5), we have

(3.10) \begin{align}\frac{d}{dt} \|u_\epsilon\|_{L^2(\Omega)}^2+ \|\nabla u_\epsilon\|_{L^2(\Omega)}^2&\leq {C_1^2}\|n_{\epsilon}\|_{L^\theta(\Omega)}^2={C_1^2}\|n_{\epsilon}^\alpha\|_{L^\frac{\theta}{\alpha}(\Omega)}^\frac{2}{\alpha}\nonumber\\[3pt] &\leq C_{2}\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}+C_{2}\leq \int_{\Omega}|\nabla n_{\epsilon}^\alpha|^2+C_{3} \qquad \mathrm{for\,\, all}\quad t>0\end{align}

due to $\frac{\theta-1}{\alpha\theta}\in(0,1)$ . Noticing that

\begin{equation*}\int_t^{t+1}\left(\int_{\Omega}|\nabla n_{\epsilon}^\alpha(\cdot,s)|^2+C_3\right)ds \le C_4\end{equation*}

due to Lemma 3.1, we then see from (3.10) and Lemma 2.3 that

\begin{equation*} \|u_\epsilon(\cdot,t)\|_{L^2(\Omega)}^2 \le C_5:= \|u_0\|_{L^2(\Omega)}^2+\frac{ C_4}{1-e^{-1}} \qquad \mathrm{for\,\, all}\quad t>0\end{equation*}

and thus that

\begin{equation*}\int_t^{t+1}\int_{\Omega}|\nabla u_\epsilon(\cdot,s)|^2ds\le \|u_0\|_{L^2(\Omega)}^2+ \int_t^{t+1}\left(\int_{\Omega}|\nabla n_{\epsilon}^\alpha(\cdot,s)|^2+C_3\right)ds \le C_6:= \|u_0\|_{L^2(\Omega)}^2+C_4\end{equation*}

for all $t\geq0$ . This completes the proof of Lemma 3.2.

Proof. Testing equation (1.9) $_2$ by $c_\epsilon^{p-1}$ , we obtain using the Hölder inequality that

\begin{equation*} \frac{\epsilon}{p}\frac{d}{dt}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2 + \frac{4(p-1)}{p^2}\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2+\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2= \int_{\Omega}n_{\epsilon}c_{\epsilon}^{p-1}\leq \|n_{\epsilon}\|_{L^r(\Omega)}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^{\frac{2(p-1)r}{p(r-1)}}(\Omega)}^{\frac{2(p-1)}{p}}\end{equation*}

for any $r>1$ and all $t>0$ . It then follows from the Gagliardo–Nirenberg inequality, the Young inequality and (2.2) that

\begin{align*}& \frac{\epsilon}{p}\frac{d}{dt}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2 + \frac{4(p-1)}{p^2}\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2+\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2\\[3pt]&\leq C_1\|n_{\epsilon}\|_{L^r(\Omega)}\left(\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^{\frac{2(pr-2r+1)}{pr}}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^{\frac{2}{p}}(\Omega)}^{\frac{2(r-1)}{pr}}+\|c_{\epsilon}^{\frac{p}{2}}\|_{L^{\frac{2}{p}}(\Omega)}^{\frac{2(p-1)}{p}}\right)\\[3pt] &= C_1\|n_{\epsilon}\|_{L^r(\Omega)}\left(\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^{\frac{2(pr-2r+1)}{pr}}\|c_0\|_{L^{1}(\Omega)}^{\frac{r-1}{r}}+\|c_0\|_{L^{1}(\Omega)}^{p-1}\right)\\[3pt] &\leq C_2\|n_{\epsilon}\|_{L^r(\Omega)}\left(\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^{\frac{2(pr-2r+1)}{pr}}+1\right)\\[3pt] &\leq \frac{4(p-1)}{p^2}\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2+ C_3\|n_{\epsilon}\|_{L^r(\Omega)}^{\frac{pr}{2r-1}}+C_3 \qquad \mathrm{for\,\, all}\quad t>0,\end{align*}

and thus that

(3.11) \begin{equation} \epsilon\frac{d}{dt}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2 +p\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2\leq p C_3\|n_{\epsilon}\|_{L^r(\Omega)}^{\frac{pr}{2r-1}}+pC_3 \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

Taking $r=\frac{p-2\alpha}{p-4\alpha}$ and applying the Gagliardo–Nirenberg inequality again, Lemma 3.1 and the mass conservation (2.1), we have

(3.12) \begin{align}\int_t^{t+1}\|n_{\epsilon}\|_{L^r(\Omega)}^{\frac{pr}{2r-1}}&=\int_t^{t+1}\|n_{\epsilon}^\alpha\|_{L^{\frac{r}{\alpha}}(\Omega)}^{\frac{2r}{r-1}}\nonumber\\&\le C_4 \int_t^{t+1}\left(\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{2}\|n_{\epsilon}^\alpha\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{r-1}}+\|n_{\epsilon}^\alpha\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2r}{r-1}}\right)\nonumber\\&= C_4 \int_t^{t+1}\left(\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{2}\|n_{\epsilon}\|_{L^{1}(\Omega)}^{\frac{2\alpha}{r-1}}+\|n_{\epsilon}\|_{L^{1}(\Omega)}^{\frac{2r\alpha}{r-1}}\right)\nonumber\\&\le C_5 \qquad \mathrm{for\,\, all}\quad t>0.\end{align}

Consequently, setting

\begin{equation*}y(t):=\epsilon\|c_{\epsilon}^{\frac{p}{2}}(\cdot,t)\|_{L^2(\Omega)}^2, \qquad g(t)=p\|c_{\epsilon}^{\frac{p}{2}}(\cdot,t)\|_{L^2(\Omega)}^2, \qquad h(t):=p C_3\|n_{\epsilon}(\cdot,t)\|_{L^r(\Omega)}^{\frac{pr}{2r-1}}+pC_3,\end{equation*}

we can use (3.11) and the fact $p\|c_{\epsilon}^{\frac{p}{2}}(\cdot,t)\|_{L^2(\Omega)}^2> py(t)$ for any $\epsilon\in(0,1)$ to deduce that

\begin{equation*}y'(t)+py(t)\leq y'(t)+g(t)\leq h(t)\end{equation*}

for all $t>0$ and thus from Lemma 2.3 and (3.12) that

\begin{equation*}y(t)\leq y(0)+\frac{C_6}{1-e^{-p}}=\epsilon\|c_0^{\frac{p}{2}}\|_{L^2(\Omega)}^2+\frac{C_6}{1-e^{-p}}<C_7:=\|c_0^{\frac{p}{2}}\|_{L^2(\Omega)}^2+\frac{C_6}{1-e^{-p}}\end{equation*}

for all $t>0$ with $C_6:=pC_3C_5+pC_3$ , which also implies that

\begin{equation*}\int_t^{t+1}g(s)ds\leq y(t)-y(t+1)+\int_t^{t+1}h(s)ds< C_7+C_6 \qquad \mathrm{for\,\, all}\quad t \ge 0.\end{equation*}

This completes the proof of Lemma 3.3.

Proof. We will deduce our desired result by investigating the combinational functional of the form

\begin{equation*}\int_{\Omega}n_{\epsilon} \mathrm{\,ln\,}n_{\epsilon}+K\epsilon\int_{\Omega}|\nabla c_\epsilon|^2+M\int_{\Omega}|\nabla u_\epsilon|^2\end{equation*}

with positive constants K and M to be determined.

For this purpose, since $\|A^{\frac12} u_\epsilon\|_{L^2(\Omega)}=\|\nabla u_\epsilon\|_{L^2(\Omega)}$ , we first apply the Helmholtz projection $\mathcal{P}$ to both sides of equation (1.9) $_3$ , multiply the result with $Au_\epsilon$ , integrate by parts over $\Omega$ and use the Hölder inequality, the $L^2$ boundedness of $\mathcal{P}$ , the Gagliardo–Nirenberg inequality and the Young inequality to obtain

\begin{align*} &\quad \frac{1}{2}\frac{d}{dt} \int_{\Omega} |\nabla u_\epsilon|^2 + \int_{\Omega} |Au_\epsilon|^2\\[3pt]& \le \|\mathcal{P}(u_\epsilon\cdot\nabla u_\epsilon)\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}+\|\mathcal{P}(n_\epsilon\nabla \phi)\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}\\[3pt]& \le C_1\| u_\epsilon\|_{L^4(\Omega)}\|\nabla u_\epsilon\|_{L^4(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}+C_1\|\nabla \phi\|_{L^\infty(\Omega)}\|n_\epsilon\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}\\[3pt] & \le C_2 \left(\| u_\epsilon\|_{L^2(\Omega)}^{\frac12} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\| u_\epsilon\|_{L^2(\Omega)}\right)\left(\|A u_\epsilon\|_{L^2(\Omega)}^{\frac12}\|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\|\nabla u_\epsilon\|_{L^2(\Omega)}\right) \\[3pt] &\qquad \cdot \|A u_\epsilon\|_{L^2(\Omega)} + C_2 \|n_\epsilon\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}\\[3pt] & \le \|A u_\epsilon\|_{L^2(\Omega)}^2+C_3 \|\nabla u_\epsilon\|_{L^2(\Omega)}^{4} +C_3\|\nabla u_\epsilon\|_{L^2(\Omega)}^{2} +C_3 \|n_\epsilon\|_{L^2(\Omega)}^2\end{align*}

for all $t>0$ . Here we used (3.9) in the last inequality. Then, we have

(3.13) \begin{equation}\frac{d}{dt}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2 \leq 2C_3 \|\nabla u_\epsilon\|_{L^2(\Omega)}^{4}+2C_3\|\nabla u_\epsilon\|_{L^2(\Omega)}^{2} +2C_3 \|n_\epsilon\|_{L^2(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

We next estimate the last term on the right-hand side of (3.13). Due to $n_{\epsilon}>0$ in $\bar{\Omega}\times(0,\infty)$ , we may test (1.9) $_1$ by $\mathrm{ln\,}n_{\epsilon}+1$ to see from the integration by parts over $\Omega$ and the Young inequality, as well as the upper estimate (1.11) for S, that

\begin{align*}\frac{d}{dt}\int_{\Omega} n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon}+\int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}&=\int_{\Omega}\nabla n_\epsilon\cdot\Big(S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_{\epsilon}\Big) \\[3pt] &\le \frac12 \int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}+\frac12 \int_{\Omega}n_{\epsilon}\Big|S(x,n_{\epsilon},c_{\epsilon})\Big|^2\cdot|\nabla c_{\epsilon} |^2\\[3pt] &\le \frac12 \int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}+\frac{C_S^2}{2}\int_{\Omega}n_{\epsilon} (1+n_{\epsilon})^{-2\alpha}|\nabla c_{\epsilon} |^2 \qquad \mathrm{for\,\, all}\quad t>0\end{align*}

and thus that

(3.14) \begin{equation}\frac{d}{dt}\int_{\Omega} n_{\epsilon}\mathrm{\,ln\,} n_{\epsilon}+\frac12\int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}\le \frac{C_S^2}{2}\int_{\Omega}n_{\epsilon}^{1-2\alpha}|\nabla c_{\epsilon} |^2 \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

Noticing that

\begin{equation*}\int_{\Omega} n_{\epsilon}^2=\| \sqrt{n_\epsilon}\|_{L^4(\Omega)}^{4} \le C_4\|\nabla \sqrt{n_\epsilon}\|_{L^2(\Omega)}^{2}\| \sqrt{n_\epsilon}\|_{L^2(\Omega)}^{2} +C_4\| \sqrt{n_\epsilon}\|_{L^2(\Omega)}^{4}\le C_5 \int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}+C_5\end{equation*}

for all $t>0$ by the Gagliardo–Nirenberg inequality and the mass conservation (2.1), we deduce from the Young inequality that

\begin{equation*}\frac{C_S^2}{2}\int_{\Omega}n_{\epsilon} ^{1-2\alpha}|\nabla c_{\epsilon} |^2\leq \frac{1}{4C_5}\int_{\Omega} n_{\epsilon}^2 +C_6\int_{\Omega} |\nabla c_{\epsilon}|^{\frac{4}{1+2\alpha}} \qquad \mathrm{for\,\, all}\quad t>0,\end{equation*}

which together with (3.14) implies that

(3.15) \begin{equation}\frac{d}{dt}\int_{\Omega} n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon}+\frac{1}{4C_5}\int_{\Omega}n_{\epsilon}^2 \le\frac12+ C_6\int_{\Omega} |\nabla c_{\epsilon}|^{\frac{4}{1+2\alpha}} \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

For the right-hand side of (3.15), we see from the Gagliardo–Nirenberg inequality and the Young inequality again that

\begin{align*}\frac12+ C_6\int_{\Omega} |\nabla c_{\epsilon}|^{\frac{4}{1+2\alpha}}&=\frac12+ C_6\|\nabla c_\epsilon\|_{L^{\frac{4}{1+2\alpha}}(\Omega)}^{\frac{4}{1+2\alpha}}\\[3pt] & \le \frac12+C_{7}\|\Delta c_\epsilon\|_{L^{2}(\Omega)}^{\frac{2}{1+2\alpha}}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{1+2\alpha}}+C_{7}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{4}{1+2\alpha}}\\[3pt] & \le \frac{1}{16C_5}\int_{\Omega}|\Delta c_{\epsilon}|^2+ C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8} \qquad \mathrm{for\,\, all}\quad t>0\end{align*}

due to $\alpha\in\left(0,\frac12\right)$ , and thus that

(3.16) \begin{equation}\frac{d}{dt}\int_{\Omega} n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon}+\frac{1}{4C_5}\int_{\Omega}n_{\epsilon}^2 \le\frac{1}{16C_5}\int_{\Omega}|\Delta c_{\epsilon}|^2+ C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8} \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

To deal with the first integral on the right-hand side of (3.16), we test (1.9) $_2$ by $-\Delta c_{\epsilon}$ and integrate by parts over $\Omega$ to get

(3.17) \begin{align}\frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega} |\nabla c_{\epsilon}|^2+\int_{\Omega} |\Delta c_{\epsilon}|^2+\int_{\Omega} |\nabla c_{\epsilon}|^2=- \int_{\Omega}n_{\epsilon}\Delta c_{\epsilon}+ \int_{\Omega} (u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon} \qquad \mathrm{for\,\, all}\quad t>0.\end{align}

For the first integral on the right-hand side of (3.17), it is clear that

(3.18) \begin{align}- \int_{\Omega}n_{\epsilon}\Delta c_{\epsilon}\leq\frac14\int_{\Omega} |\Delta c_{\epsilon}|^2+\int_{\Omega} n_{\epsilon}^2 \qquad \mathrm{for\,\, all}\quad t>0.\end{align}

On the other hand, for the second integral on the right-hand side of (3.17), we apply the Hölder inequality to obtain

(3.19) \begin{align}\int_{\Omega} (u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon}\leq \|\Delta c_\epsilon\|_{L^2(\Omega)}\| {u_\epsilon}\|_{L^4(\Omega)}\|\nabla c_\epsilon\|_{L^4(\Omega)} \qquad \mathrm{for\,\, all}\quad t>0.\end{align}

Since

\begin{equation*} \int_{\Omega}|D^2 c_{\epsilon}|^2= \frac12 \int_{\partial\Omega} \nabla |\nabla c_{\epsilon} |^2 \cdot \nu - \int_{\Omega} \nabla \Delta c_{\epsilon} \cdot \nabla c_{\epsilon}= \frac12 \int_{\partial\Omega} \nabla |\nabla c_{\epsilon} |^2 \cdot \nu + \int_{\Omega} (\Delta c_{\epsilon})^2\end{equation*}

by the integration by parts and $\nabla c_{\epsilon}\cdot \nu=0$ , we can use the geometric property

(3.20) \begin{equation} \nabla |\nabla c_{\epsilon} |^2 \cdot \nu \le 2 C_{\Omega} |\nabla c_{\epsilon} |^2\end{equation}

with $C_{\Omega}$ an upper bound for the curvatures of $\partial\Omega$ (see Lemma 4.2 in [Reference Mizoguchi and Souplet22]), the trace theorem and the Gagliardo–Nirenberg inequality to see that

\begin{align*}\|D^2c_{\epsilon}\|_{L^2(\Omega)}^2& \le C_{\Omega}\|\nabla c_{\epsilon}\|_{L^{2}(\partial\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \\[3pt] & \le C_{\Omega}\|\nabla c_{\epsilon}\|_{W^{\frac34, 2}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \\[3pt] & \le C_9 \left( \|D^2c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac{11}6} \|c_{\epsilon}\|_{L^{1}(\Omega)}^{\frac16}+\|c_{\epsilon}\|_{L^{1}(\Omega)}^2 \right) + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \\[3pt] & \le \frac12 \|D^2c_{\epsilon}\|_{L^{2}(\Omega)}^2 + \frac{C_{10}}2 \left( \|c_{\epsilon}\|_{L^{1}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \right) \qquad \text{for all}\quad t\in(0,\infty)\end{align*}

and thus that

(3.21) \begin{equation}\|D^2c_{\epsilon}\|_{L^2(\Omega)}^2 \le C_{10} \Big( \|c_{\epsilon}\|_{L^{1}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \Big) \qquad \text{for all}\quad t\in(0,\infty).\end{equation}

It then follows from the Gagliardo–Nirenberg inequality that

\begin{align*}\|\nabla c_\epsilon\|_{L^4(\Omega)}^2&\leq C_{11}\|D^2 c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)}+C_{11}\|\nabla c_\epsilon\|_{L^2(\Omega)}^2\\&\le C_{11}C_{10}^{\frac12} \Big( \|c_{\epsilon}\|_{L^{1}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \Big) ^{\frac12}\|\nabla c_\epsilon\|_{L^2(\Omega)}+C_{11}\|\nabla c_\epsilon\|_{L^2(\Omega)}^2 \\&\le C_{12}\|\Delta c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)}+C_{12}\|\nabla c_\epsilon\|_{L^2(\Omega)}+C_{12}\|\nabla c_\epsilon\|_{L^2(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t>0,\end{align*}

which together with (3.19) by the Gagliardo–Nirenberg inequality, the Poincaré inequality and Lemma 3.2 entails that

(3.22) \begin{align} &\quad \int_{\Omega} (u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon} \nonumber\\[3pt]&\leq {C_{13}} \left(\|\Delta c_\epsilon\|_{L^2(\Omega)}^{\frac32}\|\nabla c_\epsilon\|_{L^2(\Omega)}^{\frac12}+\|\Delta c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)}^{\frac12} + \|\Delta c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)}\right)\| {u_\epsilon}\|_{L^4(\Omega)} \nonumber \\[3pt] & \le C_{14}\left(\|\Delta c_\epsilon\|_{L^2(\Omega)}^{\frac32}\|\nabla c_\epsilon\|_{L^2(\Omega)}^{\frac12}+\|\Delta c_\epsilon\|_{L^2(\Omega)} \|\nabla c_\epsilon\|_{L^2(\Omega)}^{\frac12} \right) \| \nabla{u_\epsilon}\|_{L^2(\Omega)}^{\frac12}\| {u_\epsilon}\|_{L^2(\Omega)}^{\frac12} \nonumber \\[3pt] & \qquad +C_{14} \|\Delta c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)} \| \nabla{u_\epsilon}\|_{L^2(\Omega)} \nonumber \\[3pt] & \le {\frac14}\|\Delta c_\epsilon\|_{L^2(\Omega)}^{2}+C_{15}\| \nabla{u_\epsilon}\|_{L^2(\Omega)}^{2}\| \nabla{c_\epsilon}\|_{L^2(\Omega)}^{2}+C_{15}\| \nabla{u_\epsilon}\|_{L^2(\Omega)}\| \nabla{c_\epsilon}\|_{L^2(\Omega)} \nonumber \\[3pt] & \le {\frac14}\|\Delta c_\epsilon\|_{L^2(\Omega)}^{2}+\frac12\|\nabla c_\epsilon\|_{L^2(\Omega)}^{2}+C_{16}\| \nabla{u_\epsilon}\|_{L^2(\Omega)}^{2}\| \nabla{c_\epsilon}\|_{L^2(\Omega)}^{2}+C_{16}\| \nabla{u_\epsilon}\|_{L^2(\Omega)}^{2}\end{align}

for all $t>0$ . Combining (3.18), (3.22) and (3.17), we can deduce that

\begin{align*}&\epsilon\frac{d}{dt}\int_{\Omega} |\nabla c_{\epsilon}|^2+\int_{\Omega} |\Delta c_{\epsilon}|^2+\int_{\Omega} |\nabla c_{\epsilon}|^2\leq 2\int_{\Omega} n_{\epsilon}^2\\[3pt] &\quad +2C_{16}\left(\int_{\Omega} |\nabla u_\epsilon|^2\right)\cdot\left(\int_{\Omega} |\nabla c_\epsilon|^2\right)+2C_{16}\int_{\Omega} |\nabla u_\epsilon|^2 \end{align*}

for all $t>0$ , which together with (3.16) and (3.13) yields that

\begin{align*}&\frac{d}{dt}\left( \int_{\Omega}n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon} +\frac{\epsilon}{16C_5}\int_{\Omega} |\nabla c_{\epsilon}|^2+\frac{1}{32C_3C_5}\int_{\Omega} |\nabla u_\epsilon|^2\right) + \frac{1}{16C_5}\int_{\Omega}n_{\epsilon}^2+\frac{1}{16C_5}\int_{\Omega} |\nabla c_{\epsilon}|^2\\[3pt] & \le \frac{C_{16}}{8C_5}\left(\int_{\Omega} |\nabla u_\epsilon|^2\right) \left(\int_{\Omega} |\nabla c_\epsilon|^2+1\right)+ \frac{1}{16C_5}\left(\int_{\Omega} |\nabla u_\epsilon|^2\right) \left(\int_{\Omega} |\nabla u_\epsilon|^2+1\right)\\ &\quad +C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}} + C_8 \\[3pt] & = \frac{1}{32C_3C_5}\int_{\Omega} |\nabla u_\epsilon|^2 \left(4C_3C_{16}\int_{\Omega} |\nabla c_{\epsilon}|^2 +2C_3\int_{\Omega} |\nabla u_\epsilon|^2+4C_3C_{16}+2C_3\right)\\ &\quad +C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8} \end{align*}

for all $t>0$ . Thus, by taking $K:=\frac{1}{16C_5}$ with $ M:=\frac{1}{32C_3C_5}$ and setting

\begin{equation*}y(t):=\int_{\Omega}n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon} +K\epsilon\int_{\Omega} |\nabla c_{\epsilon}|^2+M\int_{\Omega} |\nabla u_\epsilon|^2,\qquad g(t):=K\int_{\Omega}n_{\epsilon}^2+K\int_{\Omega} |\nabla c_{\epsilon}|^2\end{equation*}

and

\begin{equation*}h(t):=4C_3C_{16}\int_{\Omega} |\nabla c_{\epsilon}|^2 +2C_3\int_{\Omega} |\nabla u_\epsilon|^2+4C_3C_{16}+2C_3,\qquad m(t):= C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8},\end{equation*}

we have

(3.23) \begin{equation}y'(t)+g(t)\leq h(t)\left\{y(t)+\frac{|\Omega|}{e}\right\}+m(t) \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

Here we used the fact that

(3.24) \begin{equation}-\int_{\Omega}n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon}\leq\frac{|\Omega|}{e}, \qquad \mathrm{for\,\, all}\quad t>0\end{equation}

due to $\xi \mathrm{\,ln\,} \xi\geq-\frac{1}{e}$ for all $\xi>0$ .

According to the Gagliardo–Nirenberg inequality, the mass conservation (2.1) and Lemma 3.1, we can achieve that

\begin{align*}\int_{t-1}^{t}\|n_\epsilon(\cdot,s)\|_{L^{\frac{1}{1-\alpha}}(\Omega)}^{2}ds &=\int_{t-1}^{t}\|n_\epsilon^{\alpha}(\cdot,s)\|_{L^{\frac{1}{\alpha(1-\alpha)}}(\Omega)}^{\frac{2}{\alpha}}ds \\[3pt] &\le C_{17}\int_{t-1}^{t}\|\nabla n_\epsilon^{\alpha}(\cdot,s)\|_{L^{2}(\Omega)}^{2}\|n_\epsilon^{\alpha}(\cdot,s)\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2(1-\alpha)}{\alpha}}ds + C_{17}\int_{t-1}^{t}\|n_\epsilon^{\alpha}(\cdot,s)\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}ds \\[3pt] & \le C_{18}\int_{t-1}^{t}\|\nabla n_\epsilon^{\alpha}(\cdot,s)\|_{L^{2}(\Omega)}^{2}ds +C_{18} \\[3pt] & \le C_{19} \qquad \mathrm{for\,\, all}\quad t\geq1.\end{align*}

This together with Lemmas 3.1 and 3.2 implies that

\begin{equation*}\int_{t-1}^{t}\left(\|n_\epsilon(\cdot,s)\|_{L^{\frac{1}{1-\alpha}}(\Omega)}^{2}+\|\nabla c_\epsilon(\cdot,s)\|_{L^{2}(\Omega)}^{2}+\|\nabla u_\epsilon(\cdot,s)\|_{L^{2}(\Omega)}^{2}\right)ds \le C_{20} \qquad \mathrm{for\,\, all}\quad t\geq1,\end{equation*}

and thus that for each fixed $t>0$ , we can find $t_\star\equiv t_\star(t;\epsilon)\geq0$ such that $t_\star\in\big( (t-1)_+, t\big)$ and

\begin{align*}&\|n_\epsilon(\cdot,t_\star)\|_{L^{\frac{1}{1-\alpha}}(\Omega)}^{2}+\|\nabla c_\epsilon(\cdot,t_\star)\|_{L^{2}(\Omega)}^{2}+\|\nabla u_\epsilon(\cdot,t_\star)\|_{L^{2}(\Omega)}^{2}\\[3pt] & \le C_{21}:=\max \left\{C_{20},\|n_0\|_{L^{\frac{1}{1-\alpha}}(\Omega)}^{2}+\|\nabla c_0\|_{L^{2}(\Omega)}^{2}+\|\nabla u_0\|_{L^{2}(\Omega)}^{2} \right\}.\end{align*}

By means of the elementary inequality $\xi \mathrm{\,ln\,} \xi\leq\frac{1-\alpha}{\alpha e}\xi^{\frac{1}{1-\alpha}}$ for all $\xi>0$ , we infer that

\begin{equation*}\int_{\Omega}n_{\epsilon}(\cdot,t_\star) \mathrm{\,ln\,} n_{\epsilon}(\cdot,t_\star) \leq \frac{1-\alpha}{\alpha e}\int_{\Omega}n_{\epsilon}^{\frac{1}{1-\alpha}}(\cdot,t_\star)\leq C_{22}:=\frac{1-\alpha}{\alpha e}C_{21}^{\frac{1}{2(1-\alpha)}}\end{equation*}

and thus that

\begin{equation*}y(t_\star)\leq C_{23}:=C_{22}+ KC_{21}+MC_{21}.\end{equation*}

On the other hand, we can see from Lemmas 3.1, 3.2 and 3.3 that

\begin{equation*}\int_{t-1}^{t}h(s)ds =\int_{t-1}^{t}\left(4C_3C_{16}\int_{\Omega} |\nabla c_{\epsilon}(\cdot,s)|^2+2C_3\int_{\Omega} |\nabla u_\epsilon(\cdot,s)|^2+4C_3C_{16}+2C_3\right)ds\leq C_{24}\end{equation*}

and

\begin{equation*}\int_{t-1}^{t}m(s)ds=\int_{t-1}^{t}\left(C_{8}\| c_\epsilon(\cdot,s)\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8}\right) ds \leq C_{25}\end{equation*}

for all $t\geq1$ . Thus integrating (3.23) from $t_\star$ to t, we can deduce that

\begin{align*}y(t)&\leq \Big(y(t_\star)+\frac{|\Omega|}{e}\Big) e^{\int_{t_\star}^{t}h(s)ds}+\int_{t_\star}^{t}e^{\int_{s}^{t}h(\sigma)d\sigma} m(s)ds \\[3pt] & \le \Big(C_{23}+\frac{|\Omega|}{e}\Big) e^{C_{24}}+ \int_{t_\star}^{t}e^{C_{24}} m(s)ds \le C_{26} \qquad \mathrm{for\,\, all}\quad t>0.\end{align*}

Whereupon, this together with (3.24) yields our desired conclusion.

Proof. This is a direct consequence of Lemma 3.4, the Sobolev embedding $W^{1,2}(\Omega)\hookrightarrow L^{p}(\Omega)$ and the Poincaré inequality.

Proof. Firstly, testing equation (1.9) $_1$ by $n_{\epsilon}^{s-1}$ , integrating by parts over $\Omega$ and making use of the Young inequality, the Hölder inequality and the upper estimate (1.11) for S, we have

\begin{align*} \quad \frac{1}{s}\frac{d}{dt} \int_{\Omega} n_{\epsilon}^{s}+(s-1) \int_{\Omega} n_{\epsilon}^{s-2}|\nabla n_{\epsilon}|^{2} &= (s-1) \int_{\Omega} n_{\epsilon}^{s-1} \nabla n_{\epsilon}\cdot \big(S(x,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big) \nonumber\\[3pt] &\le \frac{(s-1)}{4}\int_{\Omega} n_{\epsilon}^{s-2}|\nabla n_{\epsilon}|^{2}+(s-1) C_{{\mathcal S}}^2 \int_{\Omega} n_{\epsilon}^{s-2\alpha}|\nabla c_{\epsilon}|^2\end{align*}

and thus that

(3.26) \begin{align}\frac{d}{dt} \int_{\Omega}n_{\epsilon}^{s}+\frac{3(s-1)}{s} \int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2& \le s(s-1)C_{S}^2\int_{\Omega}n_{\epsilon}^{s-2\alpha}|\nabla c_{\epsilon}|^2 \nonumber \\& \le s(s-1)C_{S}^2 \left(\int_{\Omega}n_{\epsilon}^{\widehat{r}(s-2\alpha)}\right)^{\frac{1}{\widehat{r}}} \left(\int_{\Omega}|\nabla c_{\epsilon}|^{\frac{2\widehat{r}}{\widehat{r}-1}}\right)^{\frac{\widehat{r}-1}{\widehat{r}}}\end{align}

for all $ t\in (0,\infty)$ . For the first integral on the right-hand side of (3.26), we can deduce that

(3.27) \begin{align}\left(\int_{\Omega}n_{\epsilon}^{\widehat{r}(s-2\alpha)}\right)^{\frac{1}{\widehat{r}}} =\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2\widehat{r}(s-2\alpha)}{s}}(\Omega)}^{\frac{2(s-2\alpha)}{s}}& \le C_1\left(\|\nabla n_{\epsilon}^{\frac{s}{2}}\|_{L^{2}(\Omega)}^{\frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}(s-2\alpha)}}\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2p}{s}}(\Omega)}^{\frac{p}{\widehat{r}(s-2\alpha)}} + \|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2p}{s}}(\Omega)}\right)^{\frac{2(s-2\alpha)}{s}} \nonumber \\& \le C_2 \left(\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 \right)^{\frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}s}} + C_2\end{align}

for all $t\in(0,\infty)$ by (3.25) and $s>2\alpha+\frac{p}{\widehat{r}}$ , while for the second one, we have

(3.28) \begin{align}&\quad \left(\int_{\Omega}|\nabla c_{\epsilon}|^{\frac{2\widehat{r}}{\widehat{r}-1}}\right)^{\frac{\widehat{r}-1}{\widehat{r}}}=\|\nabla c_{\epsilon}\|_{L^{\frac{2\widehat{r}}{\widehat{r}-1}}(\Omega)}^2 \nonumber\\ & \le C_3 \left( \|D^2c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} \|c_{\epsilon}\|_{L^{p}(\Omega)}^{\frac{(\widehat{r}-1)p}{\widehat{r}(p+2)}}+\|c_{\epsilon}\|_{L^{p}(\Omega)} \right)^2 \nonumber \\& \le C_4 \left( \left( \|c_{\epsilon}\|_{L^{1}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \right)^{\frac{p+2\widehat{r}}{2\widehat{r}(p+2)}} \|c_{\epsilon}\|_{L^{p}(\Omega)}^{\frac{(\widehat{r}-1)p}{\widehat{r}(p+2)}} + \|c_{\epsilon}\|_{L^{p}(\Omega)} \right)^2 \nonumber \\& \le C_5 \left(\int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} + C_6\end{align}

for all $t\in(0,\infty)$ by (3.21) and (3.25). Thus, by substituting (3.27) and (3.28) into (3.26), we can obtain

\begin{align*}& \frac{d}{dt}\int_{\Omega}n_{\epsilon}^{s}+\frac{3(s-1)}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 \\&\le C_7 \left(\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^{2}\right)^{\frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}s}} \left(\int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} + C_7 \left(\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2\right)^{\frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}s}}\\ &\quad + C_7 \left(\int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} + C_7\end{align*}

for all $t\in(0,\infty)$ . Noticing that

\begin{equation*}0< \frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}s}<1 \qquad \mathrm{and} \qquad 0< \frac{s(p+2\widehat{r})}{(p+2)(p+ 2\alpha \widehat{r})} <1\end{equation*}

due to $1< s< 2\alpha + \frac{p(p+ 2\alpha \widehat{r} +2 - 2 \alpha)}{p+2\widehat{r}}$ , we can use the Young inequality twice to obtain that

\begin{align*}& \frac{d}{dt}\int_{\Omega}n_{\epsilon}^{s} + \frac{3(s-1)}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 \\[3pt] & \le \frac{s-1}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 + C_{8} \left( \int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{s(p+2\widehat{r})}{(p+2)(2\alpha \widehat{r}+p)}} + C_{8} \left(\int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} + C_{8} \\[3pt] & \le \frac{s-1}{s} \int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 + \int_{\Omega}|\Delta c_{\epsilon}|^2 +C_{9}\end{align*}

for all $t\in(0,\infty)$ and thus that

(3.29) \begin{equation}\frac{d}{dt}\int_{\Omega}n_{\epsilon}^{s}+\frac{2(s-1)}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2\le \int_{\Omega}|\Delta c_{\epsilon}|^2 + C_{9}\end{equation}

for all $t\in(0,\infty)$ .

On the other hand, in order to absorb the integral on the right-hand side of (3.29), we multiply equation (1.9) $_2$ by $-\Delta c_{\epsilon}$ and integrate on $\Omega$ to obtain that

(3.30) \begin{equation} \frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^2 + \int_{\Omega}|\Delta c_{\epsilon}|^2 +\int_{\Omega}|\nabla c_{\epsilon}|^2 = -\int_{\Omega}n_{\epsilon} \Delta c_{\epsilon} +\int_{\Omega}(u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon}\end{equation}

for all $t\in(0,\infty)$ . Since the interpolation and the mass conservation (2.1) imply that

\begin{align*}\int_{\Omega}n_{\epsilon}^2=\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{4}{s}}(\Omega)}^{\frac{4}{s}}& \le C_{10}\|\nabla n_{\epsilon}^{\frac{s}{2}}\|_{L^2(\Omega)}^{\frac{2}{s}} \|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2}{s}}(\Omega)}^{\frac{2}{s}}+C_{10}\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2}{s}}(\Omega)}^{\frac{4}{s}} \\[3pt] & \le C_{11}\|\nabla n_{\epsilon}^{\frac{s}{2}}\|_{L^2(\Omega)}^{\frac{2}{s}}+C_{11} \le \frac{s-1}{2s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2+C_{12} \qquad \text{for all}\quad t\in(0,\infty),\end{align*}

we have

(3.31) \begin{equation} - \int_{\Omega} n_{\epsilon} \Delta c_{\epsilon}\le \frac{1}{4}\int_{\Omega}|\Delta c_{\epsilon}|^2 + \int_{\Omega}n_{\epsilon}^2\le \frac{1}{4}\int_{\Omega}|\Delta c_{\epsilon}|^2 + \frac{s-1}{2s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2+C_{12}\end{equation}

for all $t\in(0,\infty)$ . Similarly, it follows from

\begin{equation*} \|\nabla c_{\epsilon}\|_{L^4(\Omega)} \le C_{13} \|\Delta c_{\epsilon}\|_{L^2(\Omega)}^{\frac56} \|c_{\epsilon}\|_{L^1(\Omega)}^{\frac16} + C_{13}\|c_{\epsilon}\|_{L^1(\Omega)} \le C_{14} \|\Delta c_{\epsilon}\|_{L^2(\Omega)}^{\frac56} + C_{14} \end{equation*}

and the boundedness of $\|u_\epsilon(\cdot, t)\|_{L^4(\Omega)}$ obtained in Lemma 3.5 that

(3.32) \begin{align} \int_{\Omega}(u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon} & \le \|u_\epsilon\|_{L^4(\Omega)} \|\nabla c_{\epsilon}\|_{L^4(\Omega)} \|\Delta c_{\epsilon}\|_{L^2(\Omega)} \nonumber \\[3pt] & \le C_{14} \|u_\epsilon\|_{L^4(\Omega)} \left( \|\Delta c_{\epsilon}\|_{L^2(\Omega)}^{\frac{11}6} + \|\Delta c_{\epsilon}\|_{L^2(\Omega)} \right) \le \frac{1}{4}\int_{\Omega}|\Delta c_{\epsilon}|^2 + C_{15}\end{align}

for all $t\in(0,\infty)$ . Substituting (3.31) and (3.32) into (3.30), we can see

(3.33) \begin{equation} \epsilon\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^2+\int_{\Omega}|\Delta c_{\epsilon}|^2 +2\int_{\Omega}|\nabla c_{\epsilon}|^2\le \frac{s-1}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 + 2(C_{12}+ C_{15})\end{equation}

for all $t\in(0,\infty)$ .

Then combining (3.29) and (3.33), we obtain

\begin{equation*}\frac{d}{dt} \left\{ \int_{\Omega}n_{\epsilon}^{s} +\epsilon\int_{\Omega}|\nabla c_{\epsilon}|^2 \right\}+\frac{s-1}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 +2\int_{\Omega}|\nabla c_{\epsilon}|^2\leq C_{16} \qquad \text{for all}\quad t\in(0, \infty)\end{equation*}

with $C_{16}:= C_{9} + 2(C_{12}+ C_{15})$ . To establish the uniform bound for the functional $\int_{\Omega}n_{\epsilon}^{s} +\epsilon\int_{\Omega}|\nabla c_{\epsilon}|^2$ , we apply the Gagliardo–Nirenberg inequality, the Young inequality and the mass conservation (2.1) to gain

\begin{equation*}\int_{\Omega}n_{\epsilon}^{s}=\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{2}(\Omega)}^{2}\le C_{17}\left(\|\nabla n_{\epsilon}^{\frac{s}{2}}\|_{L^{2}(\Omega)}^{\frac{2(s-1)}{s}}\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2}{s}}(\Omega)}^{\frac2{s}}+ \|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2}{s}}(\Omega)}^{2} \right)\le\frac{s-1}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2+C_{18}\end{equation*}

for all $t\in(0,\infty)$ . Also noting $\epsilon\in(0,1)$ , we have

\begin{equation*}\frac{d}{dt}\left\{\int_{\Omega} n_{\epsilon}^{s} +\epsilon\int_{\Omega}|\nabla c_{\epsilon}|^2\right\}+ \left\{\int_{\Omega} n_{\epsilon}^{s} +\epsilon\int_{\Omega}|\nabla c_{\epsilon}|^2\right\}\leq C_{16}+C_{18}\end{equation*}

for all $t\in(0,\infty)$ . A direct calculation yields the desired result.

Proof. We can show the conclusion by an induction argument on k. Indeed, due to Lemmas 2.1 and 3.6, we see that the case $k=1$ is true. Then assuming that the conclusion is valid for some positive constant k, we can use Lemma 2.2 to obtain

\begin{equation*}\|c_\epsilon(\cdot,t)\|_{L^{s_k}(\Omega)}\le C_{1} \qquad \mathrm{for\,\, all} \quad t\in (0,\infty)\end{equation*}

with some $C_{1}>0$ , and thus have

\begin{equation*}\|n_{\epsilon}(\cdot,t)\|_{L^{s_{k+1}}(\Omega)} \le C_{2} \qquad \mathrm{for\,\, all}\quad t\in (0,\infty)\end{equation*}

with some positive constant $C_{2}$ by Lemma 3.6. This implies Corollary 3.1.

Proof. Let $\big\{ s_k\big\}_{k=0}^{\infty}$ be defined by Corollary 3.1. Then $ s_k> 2\alpha + \frac1{\widehat{r}} s_{k-1}$ , $(k=1, 2, 3, \cdots)$ , implies that

\begin{align*} & s_k > 2\alpha + \frac1{\widehat{r}} \left( 2\alpha + \frac1{\widehat{r}} s_{k-2}\right) > \cdots > 2\alpha \left(1 +\frac1{\widehat{r}} +\frac1{\widehat{r}^2} + \cdots + \frac1{\widehat{r}^{k-1}} \right)\\[3pt] &\quad + \frac1{\widehat{r}^k}s_0 = \frac{2\alpha\widehat{r}}{\widehat{r} -1 } + \frac1{\widehat{r}^k} \Big(s_0 - \frac{2\alpha\widehat{r}}{\widehat{r} -1} \Big)\end{align*}

for $k=1, 2, 3, \cdots$ . Noticing that $ \frac{2\alpha\widehat{r}}{\widehat{r} -1 }>4$ due to $\widehat{r}\in\big(1, \frac{2}{2-\alpha}\big)$ , we have $s_k\ge 4$ for k large enough. It then follows from Corollary 3.1 that $\|n_{\epsilon}(\cdot,t)\|_{L^{4}(\Omega)}$ is bounded in $(0, \infty)$ .

Proof. By repeating the proof of (3.30), (3.31) and (3.32) in Lemma 3.6 and using Corollary 3.2, we can deduce that

\begin{equation*} \frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^2 + \int_{\Omega}|\Delta c_{\epsilon}|^2 +\int_{\Omega}|\nabla c_{\epsilon}|^2 \le \frac{1}{2}\int_{\Omega}|\Delta c_{\epsilon}|^2 + \int_{\Omega}n_{\epsilon}^2 + C_1 \le \frac{1}{2}\int_{\Omega}|\Delta c_{\epsilon}|^2 + C_2\end{equation*}

for all $t\in(0,\infty)$ . Then a direct calculation can complete the proof of Lemma 3.7.

Proof. We apply $\nabla$ to equation (1.9) $_2$ and test the resulting equation by $|\nabla c_{\epsilon}|^{2(p-1)}\nabla c_{\epsilon}$ with $p>\frac32$ to get

\begin{align*}& \frac{\epsilon}{2p}\frac{d}{dt}\int_\Omega |\nabla c_{\epsilon}|^{2p} - \int_{\Omega}|\nabla c_{\epsilon}|^{2(p-1)}\nabla c_{\epsilon}\cdot\Delta\nabla c_{\epsilon} + \int_{\Omega}|\nabla c_{\epsilon}|^{2p} = \int_{\Omega}|\nabla c_{\epsilon}|^{2(p-1)}\nabla c_{\epsilon}\cdot\nabla n_{\epsilon}\\ &\quad - \int_\Omega |\nabla c_{\epsilon}|^{2(p-1)} \nabla c_{\epsilon}\cdot\nabla\big(u_\epsilon\cdot\nabla c_{\epsilon}\big),\end{align*}

which together with the pointwise identity $2\nabla c_{\epsilon}\cdot\nabla\Delta c_{\epsilon}=\Delta|\nabla c_{\epsilon}|^2-2|D^2 c_{\epsilon}|^2$ and the integration by parts yields that

(3.34) \begin{align} &\frac{\epsilon}{2p} \frac{d}{dt}\int_\Omega |\nabla c_{\epsilon}|^{2p} + \frac{p-1}2 \int_\Omega |\nabla c_{\epsilon}|^{2(p-2)} \left|\nabla|\nabla c_{\epsilon}|^2\right|^2 + \int_\Omega |\nabla c_{\epsilon}|^{2{(p-1)}}|D^2 c_{\epsilon}|^2 + \int_\Omega |\nabla c_{\epsilon}|^{2p} \nonumber \\[3pt] &= \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)} \nabla n_{\epsilon}\cdot\nabla c_{\epsilon} + (p-1) \int_\Omega(u_{\epsilon}\cdot\nabla c_{\epsilon})|\nabla c_{\epsilon}|^{2(p-2)} \nabla c_{\epsilon}\cdot\nabla|\nabla c_{\epsilon}|^2\nonumber\\ &\quad + \int_\Omega(u_{\epsilon}\cdot\nabla c_{\epsilon})|\nabla c_{\epsilon}|^{2{(p-1)}}\Delta c_{\epsilon} +\frac12 \int_{\partial\Omega}|\nabla c_{\epsilon}|^{2{(p-1)}}\frac{\partial|\nabla c_{\epsilon}|^2}{\partial\nu}\end{align}

for all $ t\in(0,\infty)$ . For the first term on the right-hand side of (3.34), it follows from the pointwise inequality $|\Delta c_{\epsilon}|^2\le 2|D^2 c_{\epsilon}|^2$ , the integration by parts and the Young inequality that

(3.35) \begin{align}& \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)} \nabla n_{\epsilon}\cdot\nabla c_{\epsilon} \nonumber \\[3pt] &=- \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)} n_{\epsilon}\Delta c_{\epsilon} - (p-1) \int_\Omega |\nabla c_{\epsilon}|^{2(p-2)} n_{\epsilon}\nabla c_{\epsilon}\cdot\nabla|\nabla c_{\epsilon}|^2 \nonumber \\[3pt] &\le \sqrt{2} \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)}n_{\epsilon}|D^2 c_{\epsilon}| + (p-1)\int_\Omega n_{\epsilon}|\nabla c_{\epsilon}|^{2p-3}\big|\nabla|\nabla c_\epsilon|^2\big| \nonumber \\[3pt] &\le \frac12 \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)}|D^2c_{\epsilon}|^2+\frac{p-1}4\int_\Omega |\nabla c_{\epsilon}|^{2(p-2)} \left|\nabla|\nabla c_{\epsilon}|^2\right|^2 + p\int_\Omega n_{\epsilon}^2|\nabla c_{\epsilon}|^{2(p-1)},\end{align}

while for the second and third terms, we have

(3.36) \begin{align}& (p-1) \int_\Omega(u_{\epsilon}\cdot\nabla c_{\epsilon})|\nabla c_{\epsilon}|^{2(p-2)} \nabla c_{\epsilon}\cdot\nabla|\nabla c_{\epsilon}|^2 \nonumber \\[3pt] & \le\frac{p-1}8 \int_\Omega |\nabla c_{\epsilon}|^{2(p-2)} \left|\nabla|\nabla c_{\epsilon}|^2\right|^2 + 2(p-1)\int_\Omega| u_{\epsilon}|^2|\nabla c_{\epsilon}|^{2p}\end{align}

and

(3.37) \begin{align} &\int_\Omega(u_{\epsilon}\cdot\nabla c_{\epsilon})|\nabla c_{\epsilon}|^{2(p-1)}\Delta c_{\epsilon}\le \sqrt{2} \int_\Omega|u_{\epsilon}||\nabla c_{\epsilon}|^{2p-1}|D^2c_{\epsilon}|\le \frac12\int_\Omega|\nabla c_{\epsilon}|^{2(p-1)}|D^2 c_{\epsilon}|^2 \nonumber\\[3pt] &\quad + \int_\Omega|u_{\epsilon}|^2|\nabla c_{\epsilon}|^{2p}.\end{align}

For the last term on the right-hand side of (3.34), we know from the geometry property (3.20), the trace theorem, the Gagliardo–Nirenberg inequality, the Young inequality and Lemma 3.7 that

(3.38) \begin{align} \frac12\int_{\partial\Omega}|\nabla c_{\epsilon}|^{2(p-1)}\frac{\partial|\nabla c_{\epsilon}|^2}{\partial\nu}& \le C_1\int_{\partial\Omega}|\nabla c_{\epsilon}|^{2p}=C_1\big\||\nabla c_{\epsilon}|^p \big\|^2_{L^{2}(\partial\Omega)} \nonumber \\[3pt] & \le C_2\big\||\nabla c_{\epsilon}|^p\big\|^2_{W^{\frac34,2}(\Omega)} \nonumber \\[3pt] & \le C_3 \big\|\nabla|\nabla c_{\epsilon}|^p \big\|^{\frac{4p-1}{2p}}_{L^{2}(\Omega)} \big\||\nabla c_{\epsilon}|^p \big\|^{\frac{1}{2p}}_{L^{\frac2{p}}(\Omega)} + C_3\big\||\nabla c_{\epsilon}|^p \big\|_{L^{\frac{2}p}(\Omega)}^2 \nonumber \\[3pt] & \le \frac{p-1}{4p^2} \int_\Omega\left|\nabla |\nabla c_{\epsilon}|^p \right|^2 + C_4\end{align}

for all $t\in(0,\infty)$ . Inserting (3.35)–(3.38) into (3.34), we deduce that

\begin{align*}&\epsilon\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^{2p} + \frac{p-1}{2p} \int_{\Omega}\big|\nabla |\nabla c_{\epsilon}|^p\big|^2 + 2p \int_\Omega |\nabla c_{\epsilon}|^{2p} \le C_5 \int_{\Omega}n_{\epsilon}^2|\nabla c_{\epsilon}|^{2(p-1)}\\ & \quad + C_5 \int_{\Omega} |u_{\epsilon}|^2 |\nabla c_{\epsilon}|^{2p} + C_5\end{align*}

for all $t\in(0,\infty)$ . Noticing that

\begin{align*}C_5 \int_{\Omega} n_{\epsilon}^2 |\nabla c_{\epsilon}|^{2 (p-1)} & \le C_5 \|n_\epsilon\|_{L^4(\Omega)}^2 \|\nabla c_{\epsilon} \|_{L^{4(p-1)}(\Omega)}^{2(p-1)} \\[3pt] & \le C_6\|\nabla c_{\epsilon} \|_{L^{4(p-1)}(\Omega)}^{2(p-1)} =C_6 \big\||\nabla c_{\epsilon}|^p \big\|_{L^{\frac{4(p-1)}{p}}(\Omega)}^{\frac{2(p-1)}{p}} \\[3pt] & \le C_7 \left( \big\|\nabla|\nabla c_{\epsilon}|^p\big\|^{\frac{2p-3}{p}}_{L^{2}(\Omega)} \big\||\nabla c_{\epsilon}|^p\big\|^{\frac1{p}}_{L^{\frac2{p}}(\Omega)} + \big\||\nabla c_{\epsilon}|^p\big\|^{\frac{2(p-1)}{p}}_{L^{\frac2{p}}(\Omega)} \right) \\[3pt] & \le \frac{p-1}{4p} \big\|\nabla|\nabla c_{\epsilon}|^p\big\|^2_{L^{2}(\Omega)} + C_{8}\end{align*}

by Corollary 3.2 and Lemma 3.7, and that

\begin{align*} C_5 \int_{\Omega}|u_\epsilon|^2 |\nabla c_{\epsilon}|^{2p} & \le C_5 \||u_\epsilon|^2\|_{L^2(\Omega)} \big\||\nabla c_{\epsilon}|^{2p} \big\|_{L^2(\Omega)} \\[3pt] & \le C_9 \big\||\nabla c_{\epsilon}|^{2p} \big\|_{L^2(\Omega)} \\[3pt] & \le C_{10} \left( \big\|\nabla|\nabla c_{\epsilon}|^p \big\|_{L^{2}(\Omega)}^{\frac{2p-1}{p}}\big\||\nabla c_{\epsilon}|^p \big\|_{L^{\frac2{p}}(\Omega)}^{\frac1{p}} + \big\||\nabla c_{\epsilon}|^{p}\big\|^2_{L^{\frac2{p}}(\Omega)} \right) \\[3pt] & \le \frac{p-1}{4p} \big\|\nabla|\nabla c_{\epsilon}|^p \big\|^2_{L^{2}(\Omega)} + C_{11}\end{align*}

by Lemmas 3.5 and 3.7 again, we have

\begin{equation*}\epsilon\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^{2p} + 2p \int_\Omega |\nabla c_{\epsilon}|^{2p} \le C_{12} \qquad \text{for all} \quad t\in(0,\infty).\end{equation*}

Thus, a direct calculation yields the boundedness of $\|\nabla c_{\epsilon}(\cdot,t)\|_{L^{2p}(\Omega)}$ for any $p>\frac32$ . Then in light of the Hölder inequality we can complete the proof of Lemma 3.8.

Proof. We know from Lemmas 3.8 and 3.5 that

\begin{equation*}\sup\limits_{\epsilon\in(0,1)}\|\nabla c_{\epsilon} \|_{L^{\infty}\left((0, \infty); L^4(\Omega)\right)} \le C \qquad \mathrm{and} \qquad\sup\limits_{\epsilon\in(0,1)}\| u_\epsilon \|_{L^{\infty}\left((0, \infty); L^4(\Omega)\right)} \le C.\end{equation*}

Therefore, in light of taking $d=2$ , $T=\infty$ , $\lambda=\infty$ , $q=4$ and $r=4$ in Theorem 1.1 of [Reference Wang, Winkler and Xiang30], we can complete the proof of Lemma 3.9. Here the convexity of $\Omega$ can actually be removed as pointed out by Remark (i) in [Reference Wang, Winkler and Xiang30] and the uniqueness of solution $(n,c,u.P)$ to the PE-fluid system (1.10) can be derived from a standard energy method and a bootstrap argument, and thus we omit the details.

Proof. Since

\begin{equation*} \int_{\Omega}\partial_tc_{\epsilon} c = \frac{d}{dt}\int_{\Omega}c_{\epsilon} c - \int_{\Omega}c_{\epsilon} \partial_t c \le \frac{d}{dt}\int_{\Omega}c_{\epsilon} c + \frac12 \left( \int_{\Omega} c_{\epsilon}^2 + \int_{\Omega} (\partial_t c) ^2\right) \qquad \text{for all} \quad t\in(0,\infty),\end{equation*}

we can see from the boundedness of $c_{\epsilon}$ and c, which follows from Lemma 3.8 and the Sobolev embedding, that

(4.2) \begin{align}\int_0^t \int_{\Omega}\partial_tc_{\epsilon} c&\le \int_{\Omega}c_{\epsilon}(\cdot, t) c(\cdot, t) - \int_{\Omega}c_0(\cdot) c(\cdot, 0) + \frac12 \left( \int_0^t \int_{\Omega} c_{\epsilon}^2 + \int_0^t \int_{\Omega} (\partial_t c) ^2\right) \nonumber \\[3pt] &\le \|c_{\epsilon}\|_{L^{\infty}(\Omega\times(0, \infty))} \|c\|_{L^{\infty}(\Omega\times(0, \infty))} |\Omega| + \frac12 \left( \|c_{\epsilon}\|_{L^{\infty}(\Omega\times(0, \infty))}^2 |\Omega| t + \int_0^t \int_{\Omega} (\partial_t c) ^2 \right) \nonumber \\[3pt] &\le C_1 (1+t) + \frac12 \int_0^t \int_{\Omega} (\partial_t c) ^2 \qquad \text{for all} \quad t\in(0,\infty).\end{align}

To establish the $L^2$ space-time estimate of $\partial_t c$ , we first derive the uniform estimates for the difference quotient

\begin{equation*}c_h(x,t):= \frac{c(x, t+h)-c(x,t)}{h} \qquad \text{for all} \quad t\in(\tau, \infty)\end{equation*}

with $\tau\in(0, \infty)$ and $h\in(-\tau, \infty)$ . It is easy to show that for each $t\in(\tau, \infty)$ , $c_h(\cdot,t) \in C^2(\overline{\Omega})$ is a classical solution of the homogeneous Neumann boundary-value problem for

\begin{equation*}-\Delta c_h(\cdot,t) + c_h(\cdot,t) = -u_h(\cdot, t)\cdot \nabla c(\cdot, t+h) - u(\cdot,t)\cdot\nabla c_h(\cdot,t) + n_h(\cdot, t)\end{equation*}

in $\Omega$ , where

\begin{equation*} u_h(\cdot, t) := \frac{u(x,t+h)-u(x,t)}{h}\qquad \mathrm{and} \qquad n_h(\cdot, t) := \frac{n(x,t+h)-n(x,t)}{h}. \end{equation*}

Then testing the above equation by $c_h(\cdot, t)$ , we obtain from integrating by parts over $\Omega$ , the solenoidality of u and the Hölder inequality that

\begin{align*}& \|\nabla c_h(\cdot, t)\|_{L^{2}(\Omega)}^2+\|c_h(\cdot, t)\|_{L^{2}(\Omega)}^2 \\[3pt] & =-\int_{\Omega}c_h(\cdot, t) u_h(\cdot, t)\cdot \nabla c(\cdot, t+h) - \frac12\int_{\Omega}u(\cdot,t)\cdot\nabla c_h^2(\cdot,t) + \int_{\Omega}c_h(\cdot, t) n_h(\cdot, t) \\[3pt] & =-\int_{\Omega}c_h(\cdot, t) u_h(\cdot, t)\cdot \nabla c(\cdot, t+h) + \int_{\Omega} c_h(\cdot, t) n_h(\cdot, t) \\[3pt] & \le \|c_h(\cdot, t)\|_{L^{4}(\Omega)}\|u_h(\cdot, t)\|_{L^{2}(\Omega)}\|\nabla c(\cdot, t+h)\|_{L^{4}(\Omega)} + \|c_h(\cdot, t)\|_{L^{2}(\Omega)}\|n_h(\cdot, t)\|_{L^{2}(\Omega)}\end{align*}

for all $t\in(\tau,\infty)$ , which together with the uniform boundedness of $\|\nabla c(\cdot, t+h)\|_{L^{4}(\Omega)}$ entails that

\begin{align*}& \|\nabla c_h(\cdot, t)\|_{L^{2}(\Omega)}^2+\|c_h(\cdot, t)\|_{L^{2}(\Omega)}^2 \\[3pt] & \le C_2 \|c_h(\cdot, t)\|_{L^{4}(\Omega)}\|u_h(\cdot, t)\|_{L^{2}(\Omega)} + \|c_h(\cdot, t)\|_{L^{2}(\Omega)}\|n_h(\cdot, t)\|_{L^{2}(\Omega)} \\[3pt] & \le C_{3}\Big(\|\nabla c_h(\cdot, t)\|_{L^{2}(\Omega)}^{\frac{1}2}\|c_h(\cdot, t)\|_{L^{2}(\Omega)}^{\frac{1}2}+\|c_h(\cdot, t)\|_{L^{2}(\Omega)}\Big)\|u_h(\cdot, t)\|_{L^{2}(\Omega)} +\|c_h(\cdot, t)\|_{L^{2}(\Omega)}\| n_h(\cdot, t)\|_{L^{2}(\Omega)} \nonumber\\[3pt] &\le \frac12 \Big( \|\nabla c_h(\cdot, t)\|_{L^{2}(\Omega)}^2+\|c_h(\cdot, t)\|_{L^{2}(\Omega)}^2\Big) + C_{4}\Big(\|u_h(\cdot, t)\|_{L^{2}(\Omega)}^2 + \|n_h(\cdot, t)\|_{L^{2}(\Omega)}^2 \Big)\end{align*}

for all $t\in(\tau,\infty)$ , and thus that

(4.3) \begin{align} &\int_\tau^t \int_{\Omega} |\nabla c_h| ^2 + \int_\tau^t \int_{\Omega} c_h^2 \le 2C_{4} \left( \int_\tau^t \int_{\Omega} n_h^2 + \int_\tau^t \int_{\Omega} |u_h|^2 \right) \nonumber\\[3pt]&\quad \le C_5 \left( \int_0^{t+1} \int_{\Omega} (\partial_tn) ^2 + \int_0^{t+1} \int_{\Omega} |\partial_tu|^2 \right),\end{align}

where in the last inequality we used the temporal version of Theorem 3(i) in Section 5.8.2, [Reference Evans7].

For the first integral on the right-hand side of (4.3), we can use the maximal regularity of parabolic equations (see Theorem 2.3 in [Reference Giga and Sohr10]) and the trace theorem to obtain

(4.4) \begin{align} \|{\partial_t n}\|_{L^{2}((0,t+1); L^2(\Omega))}& \le C_6\Big(\|u\cdot\nabla n\|_{L^{2}((0,t+1); L^2(\Omega))} + \|\nabla\cdot (nS(x,n,c) \cdot \nabla c) \|_{L^{2}((0,t+1); L^2(\Omega))} \nonumber \\ & \qquad + \| nS(x,n,c) \cdot \nabla c \|_{L^{2}((0,t+1); W^{\frac12,2}(\partial\Omega))} + \|n_0\|_{W^{1,2}(\Omega)} \Big) \nonumber \\& \le C_7\Big(\!\|u\cdot\nabla n\|_{L^{2}((0,t+1); L^2(\Omega))} + \|nS(x,n,c) \cdot \nabla c\|_{L^{2}((0,t+1); W^{1,2}(\Omega))} {+} \|n_0\|_{W^{1,2}(\Omega)} \Big) \nonumber \\&\le C_8 \Big(\! \|{u}\|_{L^{\infty}(\Omega\times(0,\infty))} \|\nabla n\|_{L^2((0, t+1); L^{2}(\Omega))} {+} \|{\nabla n}\|_{L^2((0, t+1); L^{2}(\Omega))} \|{\nabla c}\|_{L^\infty(\Omega\times(0,\infty))} \nonumber\\& \qquad + \|n\|_{L^{\infty}(\Omega\times(0,\infty))} \|{\nabla c}\|_{L^2((0, t+1); W^{1,2}(\Omega))} + \|n_0\|_{W^{1,2}(\Omega)} \Big) \nonumber \\&\le C_{9}(1+t)^{\frac12} \qquad \text{for all}\quad t\in(0,\infty),\end{align}

where in the last inequality, we used the bounds for n, u and $\nabla c$ derived from the proofs of Lemmas 2.2, 2.3 and 5.3 in [Reference Wang, Winkler and Xiang30], respectively, and the growth estimates $\|\nabla n\|^2_{L^2((0, t+1);L^2(\Omega))}\leq C(1+t)$ and $\|D^2 c\|^2_{L^{2}((0, t+1);L^2(\Omega))}\leq C(1+t)$ obtained by following the proof of Lemma 3.4 in [Reference Wang, Winkler and Xiang30] and by a direct integral in equation (1.10) $_2$ , respectively. On the other hand, for the second integral on the right-hand side of (4.3), we can utilise the Sobolev maximal regularity of Stokes equation (see Theorem 2.1 in [Reference Giga and Sohr10]) and $\|n\|^2_{L^{2}((0, t+1);L^2(\Omega))}\leq C(1+t)$ to obtain that

\begin{align*}&\int_{0}^{t+1}\|\partial_t u(\cdot,s)\|_{L^{2}(\Omega)}^2ds+\int_{0}^{t+1}\| u(\cdot,s)\|_{H^{2}(\Omega)}^2ds\\&\qquad\leq C_{10} \int_{0}^{t+1}\Big(\big\|u\cdot\nabla u(\cdot,s)\big\|_{L^{2}(\Omega)}^2 + \big\|n\nabla\phi(\cdot,s)\big\|_{L^{2}(\Omega)}^2\Big)ds\\& \qquad\le C_{10}\|u\|_{L^{\infty}(\Omega\times(0, \infty))}^2\int_{0}^{t+1}\|\nabla u(\cdot,s)\|_{L^{2}(\Omega)}^2ds +C_{10}\|\nabla\phi\|_{L^{\infty}(\Omega)}^2\int_{0}^{t+1}\|n(\cdot,s)\|^2_{L^{2}(\Omega)}ds \\&\qquad \le C_{11}\int_{0}^{t+1}\Big(\| u(\cdot,s)\|_{H^{2}(\Omega)}\| u(\cdot,s)\|_{L^{2}(\Omega)}+\| u(\cdot,s)\|_{L^{2}(\Omega)}^2\Big)ds+C_{11}(1+t)\\& \qquad \le \frac12\int_{0}^{t+1}\| u(\cdot,s)\|_{H^{2}(\Omega)}^2ds +C_{12}\int_{0}^{t+1}\| u(\cdot,s)\|_{L^{2}(\Omega)}^2ds+C_{12}(1+t) \\&\qquad \le \frac12\int_{0}^{t+1}\| u(\cdot,s)\|_{H^{2}(\Omega)}^2ds + C_{13} (1+t) \qquad \mathrm{for\,\, all} \quad t\in (0,\infty)\end{align*}

and thus that

(4.5) \begin{equation} \int_0^{t+1}\|\partial_t u(\cdot,s)\|_{L^{2}(\Omega)}^2ds +\frac12 \int_0^{t+1}\| u(\cdot,s)\|_{H^{2}(\Omega)}^2ds \le C_{13}(1+t) \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation}

Consequently, inserting (4.4) and (4.5) into (4.3), we see that

\begin{equation*} \int_\tau^t \int_{\Omega} |\nabla c_h| ^2 + \int_\tau^t \int_{\Omega} c_h^2 \le C_{14}(1+t):=C_5\big( C_{9}^2(1+t) + C_{13} (1+t)\big) \qquad \mathrm{for\,\, all}\quad t\in(\tau, \infty) \end{equation*}

and that there exists $(h_i)_{i\in\mathbb N}$ satisfying $h_i\to 0$ and $c_{h_i} \to \partial_tc$ in $L^2(\Omega\times (\tau,t))$ as $i\to\infty$ with the same bound $\int_\tau^t \int_{\Omega} (\partial_tc)^2 \le C_{14}(1+t)$ . Since $C_{14}$ is independent of $\tau$ , we may take $\tau \searrow 0$ to obtain $\int_0^t \int_{\Omega} (\partial_tc)^2 \le C_{14}(1+t)$ , which together with (4.2) completes the proof of Lemma 4.1.

Proof. The proof is parallel to that of Lemma 5.2 in [Reference Li and Xiang19] and thus we omit the details here.

Proof. Applying the Helmholtz projection $\mathcal{P}$ to both sides of equation (4.1) $_3$ and testing the resulting equation with $A\widehat{u}$ , we obtain

\begin{align*}&\frac{1}{2}\frac{d}{dt}\|\nabla \widehat{u}\|_{L^2(\Omega)}^2 + \|A\widehat{u}\|_{L^2(\Omega)}^2\\[3pt]&\quad =-\int_{\Omega} \mathcal{P}\big(u_\epsilon\cdot\nabla \widehat{u} + \widehat{u}\cdot\nabla u - \widehat{n}\nabla \phi\big) \cdot A\widehat{u} \\[3pt] & \quad \leq C_1 \Big( \| u_\epsilon\|_{L^4(\Omega)}\|\nabla\widehat{u}\|_{L^4(\Omega)} + \|\nabla u\|_{L^2(\Omega)} \| \widehat{u}\|_{L^\infty(\Omega)} + \|\widehat{n}\|_{L^2(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)} \Big) \|A \widehat{u}\|_{L^2(\Omega)}\\[3pt] &\quad \le C_2\Big(\|A \widehat{u}\|_{L^2(\Omega)}^{\frac34}\| \widehat{u}\|_{L^2(\Omega)}^{\frac14} + \|A \widehat{u}\|_{L^2(\Omega)}^{\frac12}\| \widehat{u}\|_{L^2(\Omega)}^{\frac12}+\|\widehat{u}\|_{L^2(\Omega)} + \|\widehat{n}\|_{L^2(\Omega)} \Big)\|A \widehat{u}\|_{L^2(\Omega)}\\[3pt] &\quad \le \|A \widehat{u}\|_{L^2(\Omega)}^2+C_3 \|\widehat{u}\|_{L^2(\Omega)}^{2} +C_3 \|\widehat{n}\|_{L^2(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty),\end{align*}

where we used the uniform boundedness of $\| u_\epsilon\|_{L^4(\Omega)}$ and the boundedness of $\|\nabla u\|_{L^2(\Omega)}$ . It then follows from Lemma 4.2 and $\widehat{u}(\cdot,0)=0$ that

(4.6) \begin{equation}\|\nabla \widehat{u}(\cdot,t)\|_{L^2(\Omega)}^2 \leq 2C_3\int_0^{t}\Big( \|\widehat{n}(\cdot,s)\|_{L^2(\Omega)}^2 + \|\widehat{u}(\cdot,s)\|_{L^2(\Omega)}^{2}\Big)ds\le C_4 e^{C_4t}\epsilon\end{equation}

for all $t\in(0,\infty)$ . This completes the proof of Lemma 4.3.

Proof. Using the variation-of-constants representation of $\widehat{u}$ , we have

(4.7) \begin{align}A^{\theta}\widehat{u}(\cdot,t)&:= -\int^t_0{A}^{\theta} e^{-(t-s){A}}\mathcal{P}\big(u_\epsilon\cdot\nabla \widehat{u} + \widehat{u}\cdot\nabla u - \widehat{n}\nabla\phi \big)(\cdot,s) ds.\end{align}

Setting $ f(\cdot,s) := \mathcal{P}\big(u_\epsilon\cdot\nabla \widehat{u} + \widehat{u}\cdot\nabla u - \widehat{n}\nabla\phi \big)(\cdot,s)$ , we can use the Hölder inequality, the Gagliardo–Nirenberg inequality, the boundedness of $u_\epsilon$ , Lemmas 4.2 and 4.3 to obtain

\begin{align*}\|f(\cdot,s)\|_{L^2(\Omega)}&\leq \|u_\epsilon\|_{L^\infty(\Omega)}\|\nabla\widehat{u}\|_{L^2(\Omega)} + \|\widehat{u}\|_{L^4(\Omega)}\|\nabla{u}\|_{L^4(\Omega)} + \|\widehat{n}\|_{L^2(\Omega)}\|\nabla\phi\|_{L^\infty(\Omega)}\\[3pt]&\le C_1 \|\nabla\widehat{u}\|_{L^2(\Omega)} + C_1\|\nabla{u}\|_{L^4(\Omega)}\left(\|\nabla\widehat{u}\|_{L^2(\Omega)}^{\frac12}\|\widehat{u}\|_{L^2(\Omega)}^{\frac12}+\|\widehat{u}\|_{L^2(\Omega)}\right)+C_1\|\widehat{n}\|_{L^2(\Omega)}\\[3pt] &\le C_2e^{C_2s}\epsilon^{\frac12}+ C_2\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}e^{C_2s}\epsilon^{\frac12}\end{align*}

and thus have

(4.8) \begin{align}\|{A}^{\theta}\widehat{u}(\cdot,t)\|_{L^2(\Omega)}&\le C_3 \int^t_0(t-s)^{-\theta}e^{-\lambda(t-s)}\| f(\cdot,s)\|_{L^2(\Omega)}ds \nonumber \\[3pt] &\le C_4\epsilon^{\frac12}\int^t_0(t-s)^{-\theta}e^{-\lambda(t-s)+C_2s}ds \nonumber\\[3pt] &\qquad +C_4\epsilon^{\frac12}\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}(t-s)^{-\theta}e^{-\lambda(t-s)+C_2s}ds \nonumber \\[3pt] &\le C_5e^{C_5t}\epsilon^{\frac12} + C_4\epsilon^{\frac12}\left(\int^t_0\big((t-s)^{-\theta}e^{-\lambda(t-s)+C_2s}\big)^\frac43 ds\right)^{\frac34}\left(\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}^4 ds\right)^{\frac14} \nonumber \\[3pt] &\le C_5e^{C_5t}\epsilon^{\frac12} + C_6e^{C_6t}\epsilon^{\frac12}\left(\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}^4 ds\right)^{\frac14}\end{align}

for some $\lambda>0$ due to $\widehat{u}(\cdot, 0)=0$ and $\theta\in\big(\frac12,\frac34\big)$ . For the last integral on the right-hand side of (4.8), we apply the Gagliardo–Nirenberg inequality, (4.5) and the boundedness of $\|{u}\|_{L^\infty(\Omega\times(0,\infty))}$ to obtain

(4.9) \begin{align}\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}^4 ds&\le \int^t_0\Big(\|u(\cdot,s)\|_{H^2(\Omega)}^2\|u(\cdot,s)\|_{L^\infty(\Omega)}^2+\|u(\cdot,s)\|_{L^\infty(\Omega)}^4\Big) ds \nonumber\\[3pt]&\le \|{u}\|^2_{L^\infty(\Omega\times(0,\infty))}\int^t_0\|u(\cdot,s)\|_{H^2(\Omega)}^2ds+\|{u}\|^4_{L^\infty(\Omega\times(0,\infty))}|\Omega|t \nonumber\\[3pt]&\le C_7(1+t)\end{align}

for all $t\in(0,\infty)$ . This together with (4.8) and the fact $1+t\leq e^t$ yields that

\begin{equation*}\|{A}^{\theta}\widehat{u}(\cdot,t)\|_{L^2(\Omega)}\le C_8 e^{C_8t} \epsilon^{\frac12} \qquad \text{for all} \quad t\in(0, \infty).\end{equation*}

Meanwhile, the estimate of $\|\widehat{u}(\cdot,t)\|_{L^{\infty}(\Omega)}$ will follow from $D({A}^\theta)\hookrightarrow L^\infty(\Omega)$ due to $\theta\in\left(\frac12,\frac34\right)$ . This completes the proof of Lemma 4.4.

Proof. The proof is parallel to that of Lemma 5.4 in [Reference Li and Xiang19] and thus we omit the details here.

Proof. Firstly, we multiply equation (1.9) $_1$ by $n_{\epsilon}-\overline{n}_0$ and integrate by parts over $\Omega$ to obtain

\begin{align*}\frac{1}{2}\frac{d}{dt}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + \|\nabla n_\epsilon\|_{L^2(\Omega)}^2&=\int_{\Omega}\nabla n_{\epsilon}\cdot \big(n_{\epsilon} S(x,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big)\\[3pt] &\quad \le \frac12\|\nabla n_\epsilon\|_{L^2(\Omega)}^2+\frac{C_S^2}{2}\int_{\Omega}n_{\epsilon}^2|\nabla c_{\epsilon}|^2\end{align*}

for all $t\in(0,\infty)$ , and thus have

\begin{align*}\frac{d}{dt}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + \|\nabla n_\epsilon\|_{L^2(\Omega)}^2&\le C_S^2\int_{\Omega}n_{\epsilon}^2|\nabla c_{\epsilon}|^2 \\[3pt] &\le 2C_S^2\int_{\Omega}(n_\epsilon-\overline{n}_0)^2|\nabla c_{\epsilon}|^2+ 2C_S^2\int_{\Omega}\overline{n}_0^2|\nabla c_{\epsilon}|^2 \\[3pt] &\le 2C_S^2 \|n_\epsilon-\overline{n}_0\|_{L^4(\Omega)}^2\|\nabla c_{\epsilon}\|_{L^4(\Omega)}^2+2C_S^2\overline{n}_0^2\|\nabla c_{\epsilon}\|_{L^2(\Omega)}^2\end{align*}

for all $t\in(0,\infty)$ . Since the Poincaré inequality and the interpolation entail that

\begin{equation*} \|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 = \|n_\epsilon-\overline{n_\epsilon}\|_{L^2(\Omega)}^2 \le C_1\|\nabla n_\epsilon\|_{L^2(\Omega)}^2 \end{equation*}

and

\begin{equation*}\|n_\epsilon-\overline{n}_0\|_{L^{\frac{10}9}(\Omega)}^2 \le 2\left(\|n_\epsilon\|_{L^1(\Omega)}^{\frac95}\|n_\epsilon\|_{L^\infty(\Omega)}^{\frac15} + \|\overline{n}_0\|_{L^{\frac{10}9}(\Omega)}^2\right) \le C_2 \left(\|n_0\|_{L^1(\Omega)}^{\frac95} + \|n_0\|_{L^1(\Omega)}^2\right),\end{equation*}

we deduce from the Gagliardo–Nirenberg inequality, the Young inequality, Lemma 3.8 and the interpolation that

\begin{align*}&2C_S^2 \|n_\epsilon-\overline{n}_0\|_{L^4(\Omega)}^2\|\nabla c_{\epsilon}\|_{L^4(\Omega)}^2 \\[3pt]&\le C_3\left(\|\nabla(n_\epsilon-\overline{n}_0)\|_{L^2(\Omega)}^{\frac54}\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^{\frac34}+\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^2\right)\left(\|\nabla c_{\epsilon}\|_{L^{14}(\Omega)}^{\frac7{6}}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac5{6}} \right) \\[3pt] & \le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2+ C_4\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^{2}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac{20}9}+C_4\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^{2}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac56} \\[3pt] &\le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2+ C_5\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^{2}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac56} \\[3pt] & \le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2+ C_6\|n_\epsilon-\overline{n}_0\|_{L^{2}(\Omega)}^{\frac76}\|n_\epsilon-\overline{n}_0\|_{L^{\frac{10}9}(\Omega)}^{\frac56}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac56} \\[3pt] & \le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2+ \frac{1}{4C_1}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + C_7 \|n_\epsilon-\overline{n}_0\|_{L^{\frac{10}9}(\Omega)}^2 \|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^2 \\[3pt] & \le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2 + \frac{1}{4C_1}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + C_8 \left(\|n_0\|_{L^1(\Omega)}^{\frac95} + \|n_0\|_{L^1(\Omega)}^2 \right) \|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^2\end{align*}

and thus that

(4.11) \begin{align}\frac{d}{dt}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 +\frac{1}{4 C_1}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2& \le C_9 \left(\|n_0\|_{L^1(\Omega)}^{\frac95} + \|n_0\|_{L^1(\Omega)}^2\right)\|\nabla c_{\epsilon}\|_{L^2(\Omega)}^2 \nonumber \\& \le C_9 \left(\delta^{\frac95} + \delta^2\right)\|\nabla c_{\epsilon}\|_{L^2(\Omega)}^2\le \frac{1}{4C_1}\|\nabla c_{\epsilon}\|_{L^2(\Omega)}^2\end{align}

for all $t\in(0,\infty)$ provided that $\delta$ is suitable small.

To absorb the right-hand side of (4.11), we test equation (1.9) $_2$ by $c_{\epsilon}-\overline{n}_0$ , integrate by parts over $\Omega$ and utilise the Hölder inequality and the Young inequality to obtain

\begin{equation*}\frac{\epsilon}{2}\frac{d}{dt}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + \|\nabla c_\epsilon\|_{L^2(\Omega)}^2+\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 \le \frac12\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2+ \frac12\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\end{equation*}

for all $t\in(0,\infty)$ , and then have

(4.12) \begin{equation}\epsilon\frac{d}{dt}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 +2 \|\nabla c_\epsilon\|_{L^2(\Omega)}^2+\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\le \|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\end{equation}

for all $t\in(0,\infty)$ . This together with (4.11) yields that

\begin{equation*}\frac{d}{dt}\Big(\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2+\frac{\epsilon}{ 8C_1}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\Big)+\frac{1}{ 8C_1}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 +\frac{1}{ 8C_1}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\le 0\end{equation*}

for all $t\in(0,\infty)$ . Without loss of generality, we can assume $0<\epsilon<\min\big\{1, 8C_1\big\}$ and set

\begin{equation*}y(t):=\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2+\frac{\epsilon}{ 8C_1}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\end{equation*}

to obtain

(4.13) \begin{equation}y'(t)+\frac{1}{ 8C_1}y(t)\le 0 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation}

Integrating (4.13) from 0 to t, we conclude that

\begin{equation*}y(t)\le e^{-\frac{t}{ 8C_1}}y(0)\le \Big(\|n_0-\overline{n}_0\|_{L^2(\Omega)}^2+\frac{1}{ 8C_1}\|c_0-\overline{n}_0\|_{L^2(\Omega)}^2\Big)e^{-\frac{t}{8C_1}}\le C_{10} e^{-\frac{t}{ 8C_1}}\end{equation*}

for all $t\in(0,\infty)$ , which entails (4.10).

Proof. By repeating the proof of (4.12), we can see from Lemma 4.6 that

\begin{equation*}\epsilon\frac{d}{dt}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 +2 \|\nabla c_\epsilon\|_{L^2(\Omega)}^2+\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\le \|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\le C_1e^{-\mu_1t}\end{equation*}

for all $t\in(0,\infty)$ and some positive constants $C_1$ and $\mu_1$ . Then by setting

\begin{equation*}y(t):=\|c_\epsilon(\cdot,t)-\overline{n}_0\|_{L^2(\Omega)}^2\end{equation*}

and letting $0<\epsilon<\min\big\{1, \frac1{2\mu_1}\big\}$ , we have

\begin{equation*}\epsilon y'(t) + y(t) \le C_1e^{-\mu_1t}\end{equation*}

and thus

\begin{equation*}y(t)\le y(0)e^{-\frac{t}{\epsilon}} + \frac{C_1e^{-\frac{t}{\epsilon}}}{\epsilon} \int_0^t e^{(\frac{1}{\epsilon}-\mu_1)s}ds\le y(0)e^{-\frac{t}{\epsilon}} + 2C_1e^{-\mu_1t}\le C_2 e^{-\min\{\mu_1, 1\}t}\end{equation*}

for all $t\in(0,\infty)$ . This completes the proof of Lemma 4.7.

Proof. For any fixed $q>p>1$ , we use the Gagliardo–Nirenberg inequality, the $W^{1,q}$ boundedness of $c_{\epsilon}$ and Lemma 4.7 to get

\begin{align*}\|c_{\epsilon}-\overline{n}_0\|_{W^{1,\,p}({\Omega})}& \le C_1\|c_{\epsilon}-\overline{n}_0\|^{\theta_1}_{W^{1,q}({\Omega})}\|c_{\epsilon}-\overline{n}_0\|^{1-\theta_1}_{L^{2}({\Omega})}+C_1\|c_{\epsilon}-\overline{n}_0\|_{L^{2}({\Omega})} \\[2pt]& \le C_1\big( \|c_{\epsilon}\|_{W^{1,q}({\Omega})} + \|\overline{n}_0\|_{W^{1,q}({\Omega})}\big)^{\theta_1} \|c_{\epsilon}-\overline{n}_0\|^{1-\theta_1}_{L^{2}({\Omega})}+C_1\|c_{\epsilon}-\overline{n}_0\|_{L^{2}({\Omega})} \\[2pt] & \le C_2 \|c_{\epsilon}-\overline{n}_0\|^{1-\theta_1}_{L^{2}({\Omega})} + C_1\|c_{\epsilon}-\overline{n}_0\|_{L^{2}({\Omega})} \\[2pt] & \le C_3e^{-\mu t}\end{align*}

for some $\mu>0$ and all $t\in(0,\infty)$ , where $ \theta_1 = \frac{q(p-1)}{(q-1)p}\in(0,1)$ .

Proof. Let $\big(e^{t\Delta}\big)_{t\ge0}$ be the homogeneous Neumann heat semigroup in $\Omega$ . Since

\begin{equation*}n_{\epsilon}(\cdot,t)=e^{t\triangle}n_0-\int_0^t e^{(t-s)\triangle}\Big(\nabla\cdot\big(n_{\epsilon} S(\cdot,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big)+u_\epsilon\cdot\nabla n_{\epsilon}\Big)(\cdot,s)ds\end{equation*}

for all $t\in(0,\infty)$ , we can use the fact $e^{t\triangle}\overline{n}_0=\overline{n}_0$ to obtain

(4.16) \begin{align}\|n_{\epsilon}(\cdot,t)-\overline{n}_0\|_{L^{\infty}({\Omega})} &\le \|e^{t\triangle}(n_0-\overline{n}_0)\|_{L^{\infty}({\Omega})}+\int_0^t \|e^{(t-s)\triangle}\nabla\cdot\big(n_{\epsilon} S(\cdot,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big)(\cdot,s)\|_{L^{\infty}({\Omega})}ds \nonumber\\[3pt] &\qquad +\int_0^t \|e^{(t-s)\triangle}u_\epsilon(\cdot,s)\cdot\nabla n_{\epsilon}(\cdot,s)\|_{L^{\infty}({\Omega})}ds \nonumber\\[3pt] &:= J_1 + J_2 + J_3, \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{align}

To estimate $J_1$ , $J_2$ and $J_3$ , we first deduce from the asymptotics of the Neumann heat semigroup (see Lemma 1.3 in [Reference Winkler35]) that

\begin{equation*}J_1\le k_1 e^{-\lambda_1 t} \|n_0-\overline{n}_0\|_{L^{\infty}({\Omega})}\le C_1 e^{-\lambda_1 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

due to $\int_{\Omega}(n_0-\overline{n}_0)dx=0$ ,

\begin{equation*}J_2\le k_4C_S\int_0^t\Big(1+(t-s)^{-\frac12-\frac13}\Big)e^{-\lambda_1(t-s)}\|n_{\epsilon}(\cdot,s)\cdot\nabla c_{\epsilon}(\cdot,s)\|_{L^{3}({\Omega})}ds \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

and

\begin{align*}J_3& = \int_0^t \|e^{(t-s)\triangle}\nabla\cdot \big((n_{\epsilon}-\overline{n}_0)u_\epsilon\big)(\cdot,s)\|_{L^{\infty}({\Omega})}ds \\& \le k_4\int_0^t\Big(1+(t-s)^{-\frac12-\frac1{q}}\Big)e^{-\lambda_1(t-s)}\|(n_{\epsilon}(\cdot,s)-\overline{n}_0)u_\epsilon(\cdot,s)\|_{L^{q}({\Omega})}ds \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{align*}

for any fixed $2<q<3$ with some positive constants $k_1, k_4$ and $\lambda_1$ . Then since Corollaries 3.2 and 4.1 entail that

\begin{align*}\|n_{\epsilon}(\cdot,s)\cdot\nabla c_{\epsilon}(\cdot,s)\|_{L^{3}({\Omega})}&\le \|n_{\epsilon}(\cdot,s)\|_{L^{4}({\Omega})}\|\nabla c_{\epsilon}(\cdot,s)\|_{L^{12}({\Omega})} \\[3pt] & =\|n_{\epsilon}(\cdot,s)\|_{L^{4}({\Omega})}\|\nabla \big(c_{\epsilon}(\cdot,s)-\overline{n}_0\big)\|_{L^{12}({\Omega})}\le C_2 e^{-\mu_1 s}\end{align*}

for some $\mu_1>0$ , we can further estimate $J_2$ as

\begin{equation*}J_2\le C_3\int_0^t\Big(1+(t-s)^{-\frac56}\Big)e^{-\lambda_1(t-s)} e^{-\mu_1 s}ds\le C_4 e^{-\min\{\lambda_1, \mu_1\} t}\end{equation*}

for all $t\in(0,\infty)$ . Next, we can take a similar procedure to further estimate $J_3$ . Indeed, it follows from the interpolation, Lemmas 4.6, 3.5 and Corollary 3.2 that

\begin{align*}& \|(n_{\epsilon}(\cdot,s)-\overline{n}_0)u_\epsilon(\cdot,s)\|_{L^{q}({\Omega})} \\[3pt] &\le \|(n_{\epsilon}(\cdot,s)-\overline{n}_0)u_\epsilon(\cdot,s)\|^{\frac{3-q}{q}}_{L^{\frac32}({\Omega})}\|(n_{\epsilon}(\cdot,s)-\overline{n}_0)u_\epsilon(\cdot,s)\|^{\frac{2q-3}{q}}_{L^{3}({\Omega})}\\[3pt] &\le C_5\Big(\|n_{\epsilon}(\cdot,s)-\overline{n}_0\|_{L^{2}({\Omega})} \|u_\epsilon(\cdot,s)\|_{L^{6}({\Omega})} \Big)^{\frac{3-q}{q}} \Big(\|n_{\epsilon}(\cdot,s)-\overline{n}_0\|_{L^{4}({\Omega})} \|u_\epsilon(\cdot,s)\|_{L^{12}({\Omega})} \Big)^{\frac{2q-3}{q}} \\[3pt] & \le C_6 e^{-\mu_2 s}\end{align*}

for some $\mu_2>0$ , which entails that

\begin{equation*}J_3\le C_7\int_0^t\Big(1+(t-s)^{-\frac12-\frac1{q}}\Big)e^{-\lambda_1(t-s)} e^{-\mu_2 s}\le C_8 e^{-\min\{\lambda_1, \mu_2\} t}\end{equation*}

for all $t\in(0,\infty)$ . Inserting $J_1, J_2$ and $J_3$ into (4.16), we can achieve (4.14).

In order to get (4.15), it is sufficient to show that

\begin{equation*}\|A^\beta u_\epsilon(\cdot,t)\|_{L^{2}({\Omega})}\le Ce^{-\mu t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

with $\beta\in(\frac12,1)$ and $\mu>0$ due to the embedding $D(A^\beta)\hookrightarrow L^\infty(\Omega)$ . For this purpose, we first test equation (1.9) $_3$ by $u_\epsilon$ , integrate by parts over $\Omega$ and employ the solenoidality of $u_\epsilon$ , the Hölder inequality, Lemma 4.6 and Lemma 3.5 to obtain that

\begin{align*}\frac{1}{2} \frac{d}{dt} \|u_\epsilon\|_{L^2(\Omega)}^2 + \|\nabla u_\epsilon\|_{L^2(\Omega)}^2&= \int_{\Omega}n_{\epsilon}\nabla\phi\cdot u_\epsilon \\[3pt] &=\int_{\Omega}(n_{\epsilon}-\overline{n}_0)\nabla\phi\cdot u_\epsilon \\[3pt] &\le \|\nabla \phi\|_{L^\infty(\Omega)}\|n_{\epsilon}-\overline{n}_0\|_{L^2(\Omega)}\|u_\epsilon\|_{L^2(\Omega)}\le C_9 e^{-\mu_3 t}\end{align*}

for all $t\in(0,\infty)$ with some $\mu_3>0$ . Due to $u_\epsilon=0$ on $\partial\Omega$ , it then follows from the Poincaré inequality that

\begin{equation*}\frac{d}{dt} \|u_\epsilon(\cdot,t)\|_{L^2(\Omega)}^2+ \frac{1}{C_{10}}\| u_\epsilon(\cdot,t)\|_{L^2(\Omega)}^2 \le 2C_9 e^{-\mu_3 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Thus, without loss of generality, we may assume $\mu_3<\frac{1}{C_{10}}$ to deduce that

(4.17) \begin{align}\|u_\epsilon(\cdot,t)\|_{L^2(\Omega)}^2 \le e^{-\frac{t}{C_{10}}}\|u_0\|_{L^2(\Omega)}^2+C_{11} e^{-\mu_3 t}\le C_{12}e^{-\mu_3 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{align}

Following the proof of (3.13), we have

\begin{align*}& \frac12\frac{d}{dt}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2+\|A u_\epsilon\|_{L^2(\Omega)}^2 \\[3pt] &=-\int_\Omega \mathcal{P}(u_\epsilon\cdot\nabla u_\epsilon)\cdot Au_\epsilon+\int_\Omega \mathcal{P}(n_\epsilon\nabla \phi)\cdot Au_\epsilon\\[3pt] &=-\int_\Omega \mathcal{P}(u_\epsilon\cdot\nabla u_\epsilon)\cdot Au_\epsilon+\int_\Omega \mathcal{P}\big((n_\epsilon-\overline{n}_0)\nabla \phi\big)\cdot Au_\epsilon\\[3pt] & \le C_{13} \Big(\| u_\epsilon\|_{L^2(\Omega)}^{\frac12} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\| u_\epsilon\|_{L^2(\Omega)}\Big)\Big(\|A u_\epsilon\|_{L^2(\Omega)}^{\frac12}\|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\|\nabla u_\epsilon\|_{L^2(\Omega)}\Big)\|A u_\epsilon\|_{L^2(\Omega)} \\[3pt] &\qquad + C_{13} \|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}\\[3pt] & \le \frac12\|A u_\epsilon\|_{L^2(\Omega)}^2+C_{14} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{2}\left( \left(\| u_\epsilon\|_{L^2(\Omega)}^{\frac12} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\| u_\epsilon\|_{L^2(\Omega)}\right)^4 \right.\\[3pt] & \qquad \left.+\left(\| u_\epsilon\|_{L^2(\Omega)}^{\frac12} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\| u_\epsilon\|_{L^2(\Omega)}\right)^2 \right)+C_{14}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\end{align*}

for all $t\in(0,\infty)$ , which together with (4.17), Lemmas 4.6 and 3.4 yields that

\begin{equation*}\frac{d}{dt}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2+\|A u_\epsilon\|_{L^2(\Omega)}^2 \le C_{15}e^{-\mu_4 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

with some $\mu_4>0$ . Since

\begin{equation*}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2\le C_{16}\|A u_\epsilon\|_{L^2(\Omega)}\| u_\epsilon\|_{L^2(\Omega)}+C_{16}\| u_\epsilon\|_{L^2(\Omega)}^2\le \|A u_\epsilon\|_{L^2(\Omega)}^2 + C_{17}\| u_\epsilon\|_{L^2(\Omega)}^2\end{equation*}

for all $t\in(0,\infty)$ , we have

\begin{equation*}\frac{d}{dt}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2 + \|\nabla u_\epsilon\|_{L^2(\Omega)}^2 \le C_{15}e^{-\mu_4 t} + C_{17}\| u_\epsilon\|_{L^2(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

This together with (4.17) warrants that

(4.18) \begin{align}\|\nabla u_\epsilon\|_{L^2(\Omega)} \le C_{18}e^{-\mu_5 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{align}

with some $\mu_5>0$ . Noticing that $\mathcal{P}\big( \overline{n}_0 \nabla\phi \big)=0$ , we can use the variation-of-constants representation

\begin{equation*} u_\epsilon(\cdot, t) = e^{-tA}u_0 + \int^t_0 e^{-(t-s){A}}\mathcal{P}\big(u_\epsilon\cdot\nabla u_\epsilon - (n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s) ds\end{equation*}

of $u_\epsilon$ to get

\begin{align*}\|A^\beta u_\epsilon\|_{L^2(\Omega)} &\le \|A^\beta e^{-tA}u_0\|_{L^2(\Omega)}+\int^t_0 \|{A}^{\beta} e^{-(t-s){A}}\mathcal{P}\big(u_\epsilon\cdot\nabla u_\epsilon - (n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s) \|_{L^2(\Omega)}ds \\&\le K_1t^{-\beta}e^{-\mu_6 t}\|u_0\|_{L^2(\Omega)}\\[3pt] &\quad +K_1\int^t_0 (t-s)^{-\beta}e^{-\mu_6 (t-s)}\|\mathcal{P}\big(u_\epsilon\cdot\nabla u_\epsilon - (n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s)\|_{L^2(\Omega)}ds\end{align*}

for all $t\in(0,\infty)$ with some $K_1>0$ and $\mu_6>0$ by Lemma 2.3(i) in [Reference Yu, Wang and Zheng40]. Since Lemma 3.9, (4.18) and Lemma 4.6 entail that

\begin{align*}& \|\mathcal{P}\big(u_\epsilon\cdot\nabla u_\epsilon - (n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s)\|_{L^2(\Omega)} \\[3pt]& \le \|(u_\epsilon\cdot\nabla u_\epsilon) (\cdot,s)\|_{L^2(\Omega)} + \|(n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s)\|_{L^2(\Omega)} \\[3pt] & \le C_{19} \| u_\epsilon\|_{L^\infty(\Omega)}\|\nabla u_\epsilon\|_{L^2(\Omega)}+C_{19}\| n_{\epsilon}-\overline{n}_0\|_{L^2(\Omega)}\|\nabla\phi\|_{L^\infty(\Omega)} \\[3pt] & \le C_{20}e^{-\mu_7 s} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{align*}

with some $\mu_7>0$ , we have

(4.19) \begin{align}\|A^\beta u_\epsilon\|_{L^2(\Omega)}&\le K_1t^{-\beta}e^{-\mu_6 t}\|u_0\|_{L^2(\Omega)} + K_1C_{20}\int^t_0 (t-s)^{-\beta}e^{-\mu_6 (t-s)}e^{-\mu_7 s}ds \nonumber \\[3pt] &\le C_{21} e^{-\mu_6 t} + C_{22} e^{-\mu_7 t} \nonumber \\[3pt] & \le C_{23} e^{-\mu_8 t} \qquad \mathrm{for\,\, all}\quad t\in(1,\infty)\end{align}

with some $\mu_8>0$ . On the other hand, it is clear that

\begin{equation*}\|A^\beta u_\epsilon\|_{L^2(\Omega)}\le \|A^\beta u_0\|_{L^2(\Omega)} + C_{22} e^{-\mu_7 t} \le C_{24} e^{-\mu_8 t} \qquad \mathrm{for\,\, all}\quad t\in(0,1],\end{equation*}

which together with (4.19) implies (4.15). This completes the proof of Lemma 4.8.

Proof. We will deduce our desired result by analysing an entropy-like evolution estimate involving $\widehat{n}$ , $\widehat{u}$ and $c_{\epsilon}$ of the form

\begin{equation*}K\|\widehat{n} (\cdot,t)\|_{L^2(\Omega)}^2 + \epsilon\|c_{\epsilon}(\cdot,t)\|_{L^{2}(\Omega)}^{2}+\|\widehat{u} (\cdot,t)\|_{L^2(\Omega)}^2\end{equation*}

with some K to be determined. For this purpose, testing equation (4.1) $_1$ by $\widehat{n}$ and integrating by parts over $\Omega$ , we obtain

\begin{align*} & \frac{1}{2}\frac{d}{dt} \|\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 \nonumber \\[3pt] &= \int_{\Omega} \Big( (n-\overline{n}_0)\widehat{u} + \widehat{n}S(x,n_{\epsilon},c_{\epsilon}) \nabla(c_{\epsilon}-\overline{n}_0) + (n-\overline{n}_0)S(x,n_{\epsilon},c_{\epsilon}) \nabla\widehat{c} + \overline{n}_0S(x,n_{\epsilon},c_{\epsilon}) \nabla\widehat{c} \Big) \cdot\nabla\widehat{n} \nonumber\\[3pt] \ &\qquad + \int_{\Omega} n\big(S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c)\big) \cdot \nabla (c-\overline{n}_0)\cdot\nabla\widehat{n} \nonumber\\[3pt] \end{align*}

\begin{align*}\\[-40pt] &\le { \frac14}\|\nabla\widehat{n}\|_{L^{2}(\Omega)}^2 + C_1 \|n-\overline{n}_0\|_{L^{\infty}(\Omega)}^2\|\widehat{u}\|_{L^{2}(\Omega)}^2 + C_1 \|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^2\|\widehat{n}\|_{L^{4}(\Omega)}^2 \nonumber \\[3pt] & \quad + C_1 \|n-\overline{n}_0\|_{L^{\infty}(\Omega)}^2\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2 + C_1 \overline{n}_0^2\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2\\[3pt] &\quad + C_1 \int_{\Omega}n^2 \big|S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c) \big|^2 |\nabla (c-\overline{n}_0)|^2.\end{align*}

Noticing that the Gagliardo–Nirenberg inequality entails that

\begin{align*} &C_1 \|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^2\|\widehat{n}\|_{L^{4}(\Omega)}^2\\[3pt] &\le C_2\|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^2\left(\|\nabla\widehat{n}\|_{L^{2}(\Omega)}^{\frac12}\|\widehat{n}\|_{L^{2}(\Omega)}^{\frac12}+\|\widehat{n}\|_{L^{2}(\Omega)}\right)^2\\[3pt] &\le \frac18\|\nabla\widehat{n}\|_{L^{2}(\Omega)}^{2}+C_3\left(\|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^4+\|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^2\right)\|\widehat{n}\|_{L^{2}(\Omega)}^2\end{align*}

and

\begin{align*} & C_1 \int_{\Omega}n^2 \big|S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c) \big|^2 |\nabla (c-\overline{n}_0)|^2 \\&\le 2 C_1 \int_{\Omega}n^2 \left( \big|S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c_{\epsilon})\big|^2 + \big|S(x,n,c_{\epsilon})-S(x,n,c)\big|^2 \right) |\nabla (c-\overline{n}_0)|^2 \nonumber \\&\le 2 C_1 \int_{\Omega}n^2 \left( \big| \nabla S(x,\xi,c_{\epsilon})\big|^2 |\widehat{n}|^2 + \big| \nabla S(x,n, \eta) \big|^2 |\widehat{c} |^2 \right) |\nabla (c-\overline{n}_0)|^2 \nonumber \\& \le C_4 \|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^2 \left( \|\widehat{n}\|_{L^{4}(\Omega)}^2+ \|\widehat{c}\|_{L^4(\Omega)}^2\right)\\& \le \frac18\|\nabla\widehat{n}\|_{L^{2}(\Omega)}^{2} + \gamma\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2 {+} C_5\left(\!\|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^4+\|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^2\right)\left(\!\|\widehat{n}\|_{L^{2}(\Omega)}^{2}+\|\widehat{c}\|_{L^{2}(\Omega)}^{2}\right)\!,\end{align*}

where $\xi$ lies between $n_{\epsilon}$ and n, and $\eta$ lies between $c_{\epsilon}$ and c, respectively, while $\gamma>0$ is to be determined, we can use Corollary 4.1 and Lemma 4.8 to deduce that

(4.20) \begin{align} & \frac{d}{dt} \|\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 \nonumber \\& \le 2\gamma\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2 + C_6\big(e^{-\mu_1t} + \bar{n}_0^2\big)\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2 + C_6 e^{-\mu_1t}\|\widehat{c}\|_{L^{2}(\Omega)}^2 + C_6 e^{-\mu_1t}\left(\|\widehat{n}\|_{L^{2}(\Omega)}^2+\|\widehat{u}\|_{L^{2}(\Omega)}^2\right)\end{align}

for all $t\in(0,\infty)$ with some $\mu_1>0$ .

We next test equation (4.1) $_2$ by $\widehat{c}$ and use the integration by parts over $\Omega$ to obtain

\begin{align*}& \frac{\epsilon}{2}\frac{d}{dt} \|c_{\epsilon}\|_{L^2(\Omega)}^2 + \|\nabla \widehat{c}\|_{L^2(\Omega)}^2+ \|\widehat{c}\|_{L^2(\Omega)}^2 \\& = \epsilon\int_{\Omega}\partial_tc_{\epsilon} c +\int_{\Omega} \big(\widehat{n} - \widehat{u}\cdot\nabla (c-\overline{n}_0) \big) \widehat{c} \nonumber\\ & \le \epsilon\int_{\Omega}\partial_tc_{\epsilon} c + \frac12\|\widehat{c}\|_{L^{2}(\Omega)}^2 + \|\widehat{n}\|_{L^{2}(\Omega)}^2 + \|\nabla (c-\overline{n}_0)\|_{L^{4}(\Omega)}^2\|\widehat{u}\|_{L^{4}(\Omega)}^2 \\ &\le \epsilon\int_{\Omega}\partial_tc_{\epsilon} c + \frac12\|\widehat{c}\|_{L^{2}(\Omega)}^2 + \|\widehat{n}\|_{L^{2}(\Omega)}^2 + \frac14\|\nabla\widehat{u}\|_{L^{2}(\Omega)}^2 + C_7\Big(\|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^4\\[2pt] &\quad +\|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^2\Big) \|\widehat{u}\|_{L^{2}(\Omega)}^2.\end{align*}

Due to $\int_{\Omega}\widehat{n}=0$ , the Poincaré inequality and Corollary 4.1 imply that

(4.21) \begin{align}& \epsilon\frac{d}{dt} \|c_{\epsilon}\|_{L^2(\Omega)}^2 + 2\|\nabla\widehat{c}\|_{L^2(\Omega)}^2 + \|\widehat{c}\|_{L^2(\Omega)}^2 \nonumber \\[3pt]& \le 2\epsilon\int_{\Omega}\partial_tc_{\epsilon} c + \frac12\|\nabla\widehat{u}\|_{L^{2}(\Omega)}^2 + C_8 \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 + C_8 e^{-\mu_2t}\|\widehat{u}\|_{L^2(\Omega)}^2\end{align}

for all $t\in(0,\infty)$ with some $\mu_2>0$ .

Finally, multiplying equation (4.1) $_3$ by $\widehat{u}$ and using the Poincaré inequality again since $\widehat{u}=0$ on $\partial\Omega$ and $\int_{\Omega}\widehat{n}=0$ , we see that

\begin{align*} \frac{1}{2}\frac{d}{dt} \|\widehat{u}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{u}\|_{L^2(\Omega)}^2 =\int_{\Omega} \widehat{n} \nabla\phi \cdot \widehat{u}& \le \|\nabla\phi\|_{L^{\infty}(\Omega)}\|\widehat{u}\|_{L^2(\Omega)}\|\widehat{n}\|_{L^2(\Omega)} \nonumber\\ & \le C_9 \|\nabla \widehat{u}\|_{L^2(\Omega)} \|\nabla\widehat{n}\|_{L^2(\Omega)} \le \frac1{2}\|\nabla\widehat{u}\|_{L^2(\Omega)}^2 + C_{10} \|\nabla\widehat{n}\|_{L^2(\Omega)}^2\end{align*}

and thus that

(4.22) \begin{equation} \frac{d}{dt} \|\widehat{u}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{u}\|_{L^2(\Omega)}^2\le C_{11} \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 \qquad \text{for all} \quad t\in(0,\infty).\end{equation}

We now take $T_1$ large enough such that

\begin{equation*}C_6 e^{-\mu_1t}<\frac{1}{ 6(C_8 + C_{11})} \qquad \text{for all} \quad t\in(T_1, \infty)\end{equation*}

and $\delta$ suitable small such that

\begin{equation*} C_6\overline{n}^2_0<\frac{1}{ 6(C_8 + C_{11})}. \end{equation*}

Then by setting

\begin{equation*} \gamma=\frac{1}{12(C_8 + C_{11})} \qquad \text{and} \qquad K= 2 (C_8+ C_{11}),\end{equation*}

we deduce from (4.20)–(4.22) that

\begin{align*} & \frac{d}{dt}\Big(K\|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\Big) + \frac{K}2 \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{c}\|_{L^2(\Omega)}^2 + \frac12 \|\nabla\widehat{u}\|_{L^2(\Omega)}^2 + \frac23 \|\widehat{c}\|_{L^2(\Omega)}^2 \nonumber\\ &\le 2\epsilon \int_{\Omega}\partial_tc_{\epsilon} c + C_{12} e^{-\mu t} \Big(\|\widehat{n}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\Big) \qquad \quad \text{for all} \quad t\in(T_1, \infty)\end{align*}

with $\mu=\min\big\{\mu_1, \mu_2\big\}$ , which together with the Poincaré inequality yields that

\begin{align*} & \frac{d}{dt}\Big(K \|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\Big) + C_{13} \Big( K \|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2 \Big) \nonumber\\ &\qquad + \frac14\Big( K\|\nabla\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{c}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{u}\|_{L^2(\Omega)}^2 + \|\widehat{c}\|_{L^2(\Omega)}^2 \Big) \nonumber\\ &\le C_{14} e^{-\mu t}\Big(K\|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\Big) + C_{13} \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+ 2\epsilon \int_{\Omega}\partial_tc_{\epsilon} c.\end{align*}

For simplicity, we set

\begin{equation*}y(t):= K \|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\end{equation*}

and

\begin{equation*}g(t):= \frac14\Big( K\|\nabla\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{c}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{u}\|_{L^2(\Omega)}^2 + \|\widehat{c}\|_{L^2(\Omega)}^2 \Big)\end{equation*}

as well as

\begin{align*}h(t):= C_{13} \|c_{\epsilon}\|_{L^2(\Omega)}^2+ 2 \int_{\Omega}\partial_tc_{\epsilon} c,\end{align*}

we have

\begin{equation*}y'(t) + C_{13} y(t) + g(t)\le C_{14} e^{-\mu t}y(t) + \epsilon h(t) \qquad \quad \text{for all} \quad t\in(T_1, \infty).\end{equation*}

If we further take $T_2$ large enough such that $2 C_{14} e^{-\mu t} < C_{13}$ for all $t>T_2$ , we can conclude that

\begin{equation*}y'(t) + \frac{ C_{13}}{2}y(t) + g(t) \le \epsilon h(t) \qquad \mathrm{for\,\, all}\quad t\in(T_0,\infty)\end{equation*}

with $T_0:=\max\big\{T_1, T_2\big\}$ . It then follows from Lemmas 4.1, 4.2 and the uniform boundedness of $c_{\epsilon}$ that

\begin{align*}y(t)&\le y(T_0)+ \epsilon \int^t_{T_0} h(s)ds = y(T_0) + \epsilon\left( C_{13} \int^t_{T_0}\|c_{\epsilon}\|_{L^2(\Omega)}^2ds+2 \int^t_{T_0}\int_{\Omega}\partial_tc_{\epsilon} c ds\right) \nonumber\\[3pt] &\le \left((K+1) C_{15} e^{C_{15} T_0}\epsilon+\epsilon\|c_{\epsilon}(\cdot, T_0)\|_{L^2(\Omega)}^2\right) + \epsilon C_{15} (1+t):= C_{16} (1+t)\epsilon\end{align*}

for all $t\in(T_0,\infty)$ , which also implies that

\begin{equation*}\int^t_{T_0} g(s)ds\le y(T_0) + \epsilon \int^t_{T_0}h(s)ds \le C_{16} (1+t)\epsilon \qquad \mathrm{for\,\, all}\quad t\in(T_0,\infty).\end{equation*}

This together with the Poincaré inequality yields the desired results on $(T_0,\infty)$ , while on $(0, T_0)$ , the conclusions follow from Lemma 4.2 directly. This completes the proof of Lemma 4.9.

Proof. We first multiply equation (4.1) $_1$ by $\widehat{n}^{\,p-1}$ with $p>2$ and integrate by parts over $\Omega$ to obtain

\begin{align*} & \frac{1}{p}\frac{d}{dt} \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac{4(p-1)}{p^2}\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 \nonumber \\[3pt] & = (p-1) \int_{\Omega} \Big( (n-\overline{n}_0)\widehat{n}^{\,p-2}\widehat{u} + \widehat{n}^{\,p-1}S(x,n_{\epsilon},c_{\epsilon}) \nabla(c_{\epsilon}-\overline{n}_0) + (n-\overline{n}_0) \widehat{n}^{\,p-2} S(x,n_{\epsilon},c_{\epsilon}) \nabla\widehat{c} \nonumber \\[3pt] & \qquad + \overline{n}_0 \widehat{n}^{\,p-2} S(x,n_{\epsilon},c_{\epsilon}) \nabla\widehat{c} \Big) \cdot\nabla\widehat{n} +(p-1) \int_{\Omega} n \widehat{n}^{\,p-2} \big(S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c)\big) \nabla (c-\overline{n}_0)\cdot\nabla\widehat{n} \nonumber \\[3pt] & \le \frac{p-1}4 \int_{\Omega} \widehat{n}^{\,p-2}|\nabla\widehat{n}|^2 + C_1 \|n-\overline{n}_0\|_{L^\infty(\Omega)}^2\int_{\Omega}\widehat{n}^{\,p-2} |\widehat{u}|^2 + C_1 \int_{\Omega}\widehat{n}^p |\nabla(c_{\epsilon}-\overline{n}_0)|^2 \nonumber \\[3pt] & \qquad + C_1 \|n-\overline{n}_0\|_{L^\infty(\Omega)}^2\int_{\Omega}\widehat{n}^{\,p-2} |\nabla\widehat{c}|^2 + C_1 \overline{n}_0\int_{\Omega}\widehat{n}^{\,p-2}|\nabla\widehat{c}|^2 \\[3pt] & \qquad + C_1\|n\|_{L^{\infty}(\Omega)}^2 \int_{\Omega} \widehat{n}^{\,p-2} \big(\widehat{n}^{\,2} + \widehat{c}^{\,2}\big) |\nabla(c-\overline{n}_0)|^2. \end{align*}

Noting that Lemmas 4.8, 3.9 and Corollary 4.1 entail that

\begin{equation*}\|n-\overline{n}_0\|_{L^\infty(\Omega)}^2 \le C_2 e^{-\mu t} \quad \mathrm{and}\quad \|\nabla(c-\overline{n}_0)\|_{L^q(\Omega)}^2 \le C_2 e^{-\mu t}\end{equation*}

for all $t\in(0,\infty)$ and any $q>1$ with some $\mu>0$ , we can use the Hölder inequality, the boundedness of $n_{\epsilon}$ and n, Corollary 4.1 and the Young inequality to obtain that for any fixed $q_0>p$ ,

\begin{align*}\\[-40pt] & \int_{\Omega} \widehat{n}^{\,p-2} \left(\widehat{n}^{\,2} + \widehat{c}^{\,2}\right) |\nabla(c-\overline{n}_0)|^2 \\[3pt]&\le \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right) \|\widehat{n}\|_{L^{q_0}(\Omega)}^{p-2} \|\nabla(c-\overline{n}_0)\|_{L^{\frac{2pq_0}{(p-2)(q_0-p)}}(\Omega)}^2\\[3pt] &\le \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right) \|\widehat{n}\|_{L^{p}(\Omega)}^{\frac{p(p-2)}{q_0}}\|n_{\epsilon}-n\|_{L^{\infty}(\Omega)}^{\frac{(p-2)(q_0-p)}{q_0}}\|\nabla(c-\overline{n}_0)\|_{L^{\frac{2pq_0}{(p-2)(q_0-p)}}(\Omega)}^2\\[3pt] &\le C_3 e^{-\mu t} \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right) \|\widehat{n}\|_{L^{p}(\Omega)}^{\frac{p(p-2)}{q_0}}\\[3pt] &\le C_4 e^{-\mu t} \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right) \left(\|\widehat{n}\|_{L^{p}(\Omega)}^{p-2}+1\right)\\[3pt] &\le C_5 e^{-\mu t}\left(\|\widehat{n}\|_{L^{p}(\Omega)}^p + \|\widehat{c}\|_{L^{p}(\Omega)}^p\right) + C_5e^{-\mu t} \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right).\end{align*}

Similarly, we have

\begin{equation*}\int_{\Omega}\widehat{n}^p |\nabla(c_{\epsilon}-\overline{n}_0)|^2\le C_6e^{-\mu t}\|\widehat{n}\|_{L^{p}(\Omega)}^p + C_6e^{-\mu t}\|\widehat{n}\|_{L^{p}(\Omega)}^2\end{equation*}

and thus deduce from the Young inequality that

\begin{align*} & \frac{d}{dt} \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac{3(p-1)}{p} \|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 \nonumber \\[3pt] & \le C_7 e^{-\mu t} \Big(\|\widehat{n}\|_{L^p(\Omega)}^p + \|\widehat{u}\|_{L^p(\Omega)}^p\Big) + C_7 \overline{n}_0 \|\widehat{n}\|_{L^{p}(\Omega)}^{p} + C_7 \big( e^{-\mu t} + \overline{n}_0\big) \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^{p}\nonumber \\[3pt] & \qquad + C_7e^{-\mu t}\|\widehat{n}\|_{L^{p}(\Omega)}^2+C_7e^{-\mu t}\|\widehat{c}\|_{L^{p}(\Omega)}^2. \end{align*}

Without loss of generality, we assume $ C_7>1$ and take suitable small $\delta$ such that $\overline{n}_0<1$ . Then since

\begin{align*}\|\widehat{n}\|_{L^p(\Omega)}^p = \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2& \le C_8\left(\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^{\frac{2(p-2)}{p}} \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{4}{p}} + \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^2\right) \\[3pt] &\le \frac{p-1}{ C_7 p}\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + C_9 \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^2 \end{align*}

and

\begin{equation*} \begin{split}\|\widehat{n}\|_{L^p(\Omega)}^2 =\|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^{\frac4{p}}& \le C_{10}\left(\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^{\frac{p-2}{p}} \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{2}{p}} + \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}\right)^{\frac4{p}}\\[3pt] &\le \frac{p-1}{2C_7p}\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + C_{11}\|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{8}{p^2-2p+4}}+C_{11}\|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{4}{p}} \end{split} \end{equation*}

we can use the fact

\begin{equation*} \|\widehat{u}\|_{L^p(\Omega)}^p \le C_{12}\Big(\|\widehat{u}\|_{L^2(\Omega)}^2\|\nabla\widehat{u}\|_{L^2(\Omega)}^{p-2}+\|\widehat{u}\|_{L^2(\Omega)}^p \Big), \end{equation*}

Lemma 4.9 and Corollary 4.2 to obtain

\begin{align*} & \frac{d}{dt} \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac12 \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 \nonumber \\ & \le \frac{d}{dt} \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac{p-1}{2p}\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac{C_9}{2} \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^2 \nonumber \\ & \le \left( C_7C_9 e^{-\mu t} + C_7 C_9 + \frac{C_9}{2}\right) \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4p}(\Omega)}^2 + C_7 e^{-\mu t} \|\widehat{u}\|_{L^p(\Omega)}^p + C_7 \big( e^{-\mu t} + 1\big) \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^{p} \nonumber \\ & \quad + C_7C_{11}e^{-\mu t}\|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{8}{p^2-2p+4}}+C_7C_{11}e^{-\mu t}\|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{4}{p}}+C_7e^{-\mu t}\|\widehat{c}\|_{L^p(\Omega)}^2 \nonumber \\ & \le C_{13} (1+t)^{\frac{p}2}\epsilon^{\frac{p}2} + C_{13}(1+t)^{\frac{2p}{p^2-2p+4}}\epsilon^{\frac{2p}{p^2-2p+4}}+C_{13}(1+t)\epsilon \nonumber \\ & \qquad +C_7 \big( e^{-\mu t} + 1\big) \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^p+ C_7 e^{-\mu t}\|\widehat{c}\|_{L^p(\Omega)}^2 \nonumber \\ &\le 3C_{13}(1+t)^{\frac{p}2}\epsilon^{\frac{2p}{p^2-2p+4}} + C_7 \big( e^{-\mu t} + 1\big) \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^p + C_7 e^{-\mu t}\|\widehat{c}\|_{L^p(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty) \end{align*}

due to $p>2$ and $\epsilon\in(0,1)$ . By setting

\begin{equation*}y(t):=\|\widehat{n}(\cdot,t)\|_{L^p(\Omega)}^p =\|\widehat{n}^{\,\frac{p}2}(\cdot,t)\|_{L^2(\Omega)}^2,\end{equation*}

we have

\begin{equation*}y'(t) + \frac12y(t)\le C_{14} (1+t)^{\frac{p}2}\epsilon^{\frac{2p}{p^2-2p+4}} + C_{14} \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^p + C_{14}\|\widehat{c}\|_{L^p(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Thus, a direct calculation implies that

(4.23) \begin{equation}y(t) \le C_{14} \epsilon^{\frac{2p}{p^2-2p+4}} \int_0^t (1+s)^{\frac{p}2}e^{ -\frac{(t-s)}{2}} ds + C_{14} \int_0^t e^{ -\frac{(t-s)}{2}} \left(\|\widehat{c}(\cdot, s)\|_{W^{1,\, p}(\Omega)}^{p} + \|\widehat{c}(\cdot, s)\|_{L^p(\Omega)}^2 \right)ds \end{equation}

for all $t\in(0,\infty)$ due to $y(0)=0$ . For the first integral on the right-hand side of (4.23), we have

\begin{equation*} \int_0^t (1+s)^{\frac{p}2}e^{-\frac{(t-s)}{2}} ds\le (1+t)^{\frac{p}2}\int_0^t e^{-\frac{(t-s)}{2}} ds \le (1+t)^{\frac{p}2},\end{equation*}

while for the second one, we can make use of the interpolation inequality, Lemmas 3.8, 4.9 and the Hölder inequality to deduce that

\begin{align*}\int_0^t e^{ -\frac{(t-s)}{2}} \|\widehat{c}(\cdot, s)\|_{W^{1,\, p}(\Omega)}^{p} ds& \le C_{15} \int_0^t e^{ -\frac{(t-s)}{2}} \|\widehat{c}(\cdot, s)\|_{W^{1, 2}(\Omega)}^{\frac{2(q-p)}{q-2}} \|\widehat{c}(\cdot, s)\|_{W^{1, q}(\Omega)}^{\frac{q(p-2)}{q-2}} ds \nonumber \\&\le C_{16} \int_0^t e^{ -\frac{(t-s)}{2}} \|\widehat{c}(\cdot, s)\|_{W^{1, 2}(\Omega)}^{\frac{2(q-p)}{q-2}} ds \nonumber \\&\le C_{17} \left(\int_0^t \|\widehat{c}(\cdot, s)\|_{W^{1, 2}(\Omega)}^{2} ds\right)^{\frac{q-p}{q-2}} \left(\int_0^t e^{-\frac{q-2}{p-2}\frac{(t-s)}{2}} ds\right)^{\frac{p-2}{q-2}} \nonumber \\&\le C_{18} (1+t)^{\frac{q-p}{q-2}}\epsilon^{\frac{q-p}{q-2}} \qquad \quad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{align*}

with $q>p+2>4$ . Similarly, for the third integral, we also have

\begin{align*}\int_0^t e^{-\frac{(t-s)}{2}}\|\widehat{c}(\cdot, s)\|_{L^{ p}(\Omega)}^{2} ds& \le C_{19} \int_0^t e^{-\frac{(t-s)}{2}}\|\widehat{c}(\cdot, s)\|_{L^{ 2}(\Omega)}^{\frac{4}{p}} \|\widehat{c}(\cdot, s)\|_{L^{\infty}(\Omega)}^{\frac{2(p-2)}{p}} ds \nonumber \\&\le C_{20} \left(\int_0^t \|\widehat{c}(\cdot, s)\|_{L^{ 2}(\Omega)}^{2} ds\right)^{\frac{2}{p}} \left(\int_0^t e^{-\frac{p}{p-2}\frac{(t-s)}{2}} ds\right)^{\frac{p-2}{p}} \nonumber \\&\le C_{21} (1+t)^{\frac{2}{p}}\epsilon^{\frac{2}{p}} \qquad \quad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{align*}

These three estimates together with (4.23) yield that

\begin{equation*}y(t)\le C_{22} (1+t)^{\frac{p}2}\epsilon^{\frac{2p}{p^2-2p+4}} + C_{22} (1+t)^{\frac{q-p}{q-2}} \epsilon^{\frac{q-p}{q-2}} + C_{22} (1+t)^{\frac{2}{p}}\epsilon^{\frac{2}{p}}\le C_{23} (1+t)^{\frac{p}{2}} \epsilon^{\frac{2}{p}}\end{equation*}

for all $ t\in(0,\infty)$ . This completes the proof of Lemma 4.11.