## 三维 Keller-Segel-Stokes 系统的快速信号扩散极限的收敛速率

1电子科技大学数学科学学院 成都 611731

2西华大学理学院 成都 610039

## The Convergence Rate of the Fast Signal Diffusion Limit for a Three-Dimensional Keller-Segel-Stokes System

Yu Ting,1,*, Dong Ying,2

1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731

2School of Science, Xihua University, Chengdu 610039

 基金资助: 四川省自然科学基金(2022NSFSC1835)

 Fund supported: Natural Science Foundation of Sichuan Province(2022NSFSC1835)

Abstract

In this paper, We demonstrates that when the initial cell mass is small, the solution of the initial boundary value problem converges at an algebraic rate to the corresponding parabolic-elliptical Keller-Segel-Stokes system during the fast signal diffusion limit process by performing appropriate energy iterative estimation on the three-dimensional parabolic-parabolic Keller-Segel-Stokes system.

Keywords： Keller-Segel-Stokes; Fast signal diffusion limit; Decay estimate; Convergence rate

Yu Ting, Dong Ying. The Convergence Rate of the Fast Signal Diffusion Limit for a Three-Dimensional Keller-Segel-Stokes System[J]. Acta Mathematica Scientia, 2024, 44(4): 925-945

## 1 引言

$\left\{\begin{split}&\partial_{t}n=\Delta n-\nabla\cdot\big(nS(x,n,c)\nabla c\big),\ \ &x\in\Omega,t>0,\\&\partial_{t}c=\Delta c-c+n,\ \ &x\in\Omega,t>0,\\\end{split}\right.$

$\left\{\begin{split}&\partial_t{n}+u\cdot\nabla n=\Delta n-\nabla\cdot\Big(nS(x,n,c)\cdot\nabla c\Big)+f(x,n,c),&&x\in\Omega,\,t>0,\\&u\cdot\nabla c=\Delta{c}-c+n,\,\,&&x\in\Omega,\,t>0,\\&\partial_t{u}+\kappa(u\cdot\nabla u)+\nabla P=\Delta{u}+n\nabla\phi,\,\,&&x\in\Omega,\,t>0,\\&\nabla\cdot\ u=0,\,\,&&x\in\Omega,\,t>0,\\&\Big(\nabla n-nS(x,n,c)\cdot \nabla c\Big)\cdot \nu=\nabla c\cdot\nu=0,u=0,&&x\in\partial\Omega,\,t>0,\\&n(x,0)=n_{0}(x),u(x,0)=u_{0}(x),&&x\in\Omega\\\end{split}\right.$

## 2 主要结果

$\left\{\begin{split}&\partial_t{n_{\epsilon}}+u_{\epsilon}\cdot\nabla n_{\epsilon}=\Delta n_{\epsilon}-\nabla\cdot\Big(n_{\epsilon}S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_{\epsilon}\Big),&&x\in\Omega,\,t>0,\\&\epsilon\partial_t{c_{\epsilon}}+u_{\epsilon}\cdot\nabla c_{\epsilon}=\Delta{c_{\epsilon}}-c_{\epsilon}+n_{\epsilon},&&x\in\Omega,\,t>0,\\&\partial_t{u_{\epsilon}}+\nabla P_{\epsilon}=\Delta{u_{\epsilon}}+n_{\epsilon}\nabla\phi,\,\,&&x\in\Omega,\,t>0,\\&\nabla\cdot\ {u_{\epsilon}}=0,\,\,&&x\in\Omega,\,t>0,\\&\Big(\nabla n_{\epsilon}-n_{\epsilon}S(x,n_{\epsilon},c_{\epsilon})\cdot \nabla c_{\epsilon}\Big)\cdot \nu=\nabla c_{\epsilon}\cdot\nu=0,u_{\epsilon}=0,\,&&x\in\partial\Omega,\,t>0,\\&n_{\epsilon}(x,0)=n_{0}(x),c_{\epsilon}(x,0)=c_{0}(x),u_{\epsilon}(x,0)=u_{0}(x),&&x\in\Omega\\\end{split}\right.$

$S_{ij}(x,n_{\epsilon},c_{\epsilon})\in C^{2}(\overline \Omega\times[0,\infty)\times[0,\infty))$

$\vert S(x,n_{\epsilon},c_{\epsilon})\vert\leq\frac{C_{S}}{(1+n_{\epsilon})^{\alpha}},$

$\Vert n_{0}\Vert_{L^{1}(\Omega)}<\delta$

## 3 预备知识

$\Vert n_{\epsilon}(\cdot,t)\Vert_{L^{1}(\Omega)}=\Vert n_{0}\Vert_{L^{1}(\Omega)}$

$\Vert c_{\epsilon}(\cdot,t)\Vert_{L^{1}(\Omega)}\leq\max\Big\{\Vert n_{0}\Vert_{L^{1}(\Omega)}, \ \Vert c_{0}\Vert_{L^{1}(\Omega)}\Big\}.$

$\Vert n_{\epsilon}(\cdot,t)\Vert_{L^{s}(\Omega)}\leq C,\ \ \ t\in(0,\infty).$

$\Vert c_{\epsilon}(\cdot,t)\Vert_{L^{p}(\Omega)}\leq C(p,Q,c_{0}),\ \ \ t\in(0,\infty).$

$\Vert c_{\epsilon}(\cdot,t)\Vert_{L^{s}(\Omega)}\leq C,\ \ \ t\in(0,\infty).$

$\Vert\nabla c_{\epsilon}(\cdot,t)\Vert_{L^{2}(\Omega)}\leq C,\ \ \ t\in(0,\infty).$

$\Vert u_{\epsilon}(\cdot,t)\Vert_{L^{q}(\Omega)}\leq C,\ \ \ t\in(0,\infty).$

$\int_{0}^{t}\int_{\Omega}\partial_{t}{c_{\epsilon}c}\leq C(1+t),\ \ \ t\in(0,\infty).$

$\Vert\nabla c_{\epsilon}(\cdot,t)\Vert_{L^{m}(\Omega)}\leq C,\ \ \ t\in(0,\infty).$

$\begin{matrix}&\frac{\epsilon}{2m}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2m}-\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\nabla c_{\epsilon}\cdot\Delta\nabla c_{\epsilon}+\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2m} \\=&\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\nabla c_{\epsilon}\cdot \nabla n_{\epsilon}-\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\nabla c_{\epsilon}\cdot \nabla (u_{\epsilon}\cdot\nabla c_{\epsilon}).\end{matrix}$

$\begin{matrix}&\frac{\epsilon}{2m}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2m}+\frac{m-1}{2}\int_{\Omega}\vert\nabla c_{\epsilon}|^{2(m-2)}\vert\nabla\vert\nabla c_{\epsilon}\vert^{2}|^{2}+\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\vert D^{2}c_{\epsilon}\vert^{2}+\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2m} \\=&\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\nabla n_{\epsilon}\cdot \nabla c_{\epsilon}+(m-1)\int_{\Omega}(u_{\epsilon}\cdot\nabla c_{\epsilon})\vert\nabla c_{\epsilon}\vert^{2(m-2)}\nabla c_{\epsilon}\cdot\nabla\vert\nabla c_{\epsilon}\vert^{2} \\& +\int_{\Omega}(u_{\epsilon}\cdot\nabla c_{\epsilon})\vert\nabla c_{\epsilon}\vert^{2(m-1)}\Delta c_{\epsilon}+\frac{1}{2}\int_{\partial \Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\frac{\partial \vert\nabla c_{\epsilon}\vert^{2}}{\partial \nu}.\end{matrix}$

$\begin{matrix}& \, \int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\nabla n_{\epsilon}\cdot \nabla c_{\epsilon} \\&=-\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)} n_{\epsilon}\Delta c_{\epsilon}-(m-1)\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-2)}n_{\epsilon}\nabla c_{\epsilon}\cdot\nabla\vert\nabla c_{\epsilon}\vert^{2} \\&\leq \sqrt{3}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)} n_{\epsilon}\vert D^{2} c_{\epsilon}\vert+(m-1)\int_{\Omega}n_{\epsilon}\vert\nabla c_{\epsilon}\vert^{2m-3}\vert\nabla\vert\nabla c_{\epsilon}\vert^{2}\vert \\&\leq \frac{1}{2}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\vert D^{2} c_{\epsilon}\vert^{2}+\frac{m-1}{4}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-2)}\vert\nabla\vert\nabla c_{\epsilon}\vert^{2}\vert^{2}+(m+\frac{1}{2})\int_{\Omega}n_{\epsilon}^{2}\vert\nabla c_{\epsilon}\vert^{2(m-1)}.\end{matrix}$

$\begin{matrix}&(m-1)\int_{\Omega}(u_{\epsilon}\cdot\nabla c_{\epsilon})\vert\nabla c_{\epsilon}\vert^{2(m-2)}\nabla c_{\epsilon}\cdot\nabla\vert\nabla c_{\epsilon}\vert^{2} \\\leq\ &\frac{m-1}{8}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-2)}\vert\nabla\vert\nabla c_{\epsilon}\vert^{2}\vert^{2}+2(m-1)\int_{\Omega}\vert u_{\epsilon}\vert^{2}\vert\nabla c_{\epsilon}\vert^{2m}\end{matrix}$

$\begin{matrix}\int_{\Omega}(u_{\epsilon}\cdot\nabla c_{\epsilon})\vert\nabla c_{\epsilon}\vert^{2(m-1)}\Delta c_{\epsilon}&\leq \sqrt{3}\int_{\Omega}\vert u_{\epsilon}\vert\vert\nabla c_{\epsilon}\vert^{2m-1} \vert D^{2} c_{\epsilon}\vert \\&\leq\frac{1}{2}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)} \vert D^{2} c_{\epsilon}\vert^{2}+\frac{3}{2}\int_{\Omega}\vert u_{\epsilon}\vert^{2}\vert\nabla c_{\epsilon}\vert^{2m}.\end{matrix}$

$\begin{matrix}\frac{1}{2}\int_{\partial \Omega}\vert\nabla c_{\epsilon}\vert^{2(m-1)}\frac{\partial \vert\nabla c_{\epsilon}\vert^{2}}{\partial \nu}&\leq C_{1}\int_{\partial \Omega}\vert\nabla c_{\epsilon}\vert^{2m}\leq C_{2}\Vert\vert\nabla c_{\epsilon}\vert^{m}\Vert_{W^{\frac{3}{4},2}(\Omega)}^{2} \\& \leq C_{3}\Vert\nabla\vert\nabla c_{\epsilon}\vert^{m}\Vert_{L^{2}(\Omega)}^{\frac{3(2m-1)}{3m-1}}\Vert\vert\nabla c_{\epsilon}\vert^{m}\Vert_{L^{\frac{2}{m}}(\Omega)}^{\frac{1}{3m-1}}+C_{3}\Vert\vert\nabla c_{\epsilon}\vert^{m}\Vert_{L^{\frac{2}{m}}(\Omega)}^{2} \\& \leq \frac{m-1}{4m^{2}}\int_{\Omega}\vert\nabla\vert\nabla c_{\epsilon}\vert^{m}\vert^{2}+C_{4}.\end{matrix}$

$\begin{matrix} &\epsilon\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2m}+\frac{m-1}{2m}\int_{\Omega}\vert\nabla\vert\nabla c_{\epsilon}\vert^{m}\vert^{2}+2m\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2m} \\\leq\ & C_{5}\int_{\Omega}\vert u_{\epsilon}\vert^{2}\vert\nabla c_{\epsilon}\vert^{2m}+C_{5}\int_{\Omega}n_{\epsilon}^{2}\vert\nabla c_{\epsilon}\vert^{2(m-1)}+C_{5}.\end{matrix}$

$\begin{matrix} C_{5}\int_{\Omega} \vert u_{\epsilon}\vert ^{2} \vert \nabla c_{\epsilon} \vert ^{2m}&\leq C_{5}\Vert \vert u_{\epsilon}\vert ^{2}\Vert_{L^{2}(\Omega)}\Vert \vert \nabla c_{\epsilon}\vert^{2m}\Vert_{L^{2}(\Omega)} \\&=C_{5}\Vert u_{\epsilon}\Vert_{L^{4}(\Omega)}^{2}\Vert \vert \nabla c_{\epsilon}\vert^{m}\Vert_{L^{4}(\Omega)}^{2}\leq C_{6}\Vert \vert \nabla c_{\epsilon}\vert^{m}\Vert_{L^{4}(\Omega)}^{2} \\&\leq C_{7}\Big(\Vert\nabla\vert\nabla c_{\epsilon}\vert^{m}\Vert_{L^{2}(\Omega)}^{\frac{3(2m-1)}{3m-1}}\Vert \vert \nabla c_{\epsilon}\vert^{m}\Vert_{L^{\frac{2}{m}}(\Omega)}^{\frac{1}{3m-1}}+\Vert \vert \nabla c_{\epsilon}\vert^{m}\Vert_{L^{\frac{2}{m}}(\Omega)}^{2}\Big) \\&\leq\frac{m-1}{4m}\Vert\nabla\vert\nabla c_{\epsilon}\vert^{m}\Vert_{L^{2}(\Omega)}^{2}+ C_{8}.\end{matrix}$

$\begin{matrix}C_{5}\int_{\Omega}n_{\epsilon}^{2}\vert\nabla c_{\epsilon}\vert^{2(m-1)}&\leq C_{5}\Vert n_{\epsilon}\Vert_{L^{4}(\Omega)}^{2}\Vert\nabla c_{\epsilon}\Vert_{L^{4(m-1)}(\Omega)}^{2(m-1)} \\&\leq C_{9}\Vert\nabla c_{\epsilon}\Vert_{L^{4(m-1)}(\Omega)}^{2(m-1)}=C_{9}\Vert\vert\nabla c_{\epsilon}\vert^{m}\Vert_{L^{\frac{4(m-1)}{m}}(\Omega)}^{\frac{2(m-1)}{m}} \\&\leq C_{10}\Big(\Vert\nabla\vert\nabla c_{\epsilon}\vert^{m}\Vert_{L^{2}(\Omega)}^{\frac{3(2m-3)}{3m-1}}\Vert \vert \nabla c_{\epsilon}\vert^{m}\Vert_{L^{\frac{2}{m}}(\Omega)}^{\frac{m+2}{m(3m-1)}}+\Vert \vert \nabla c_{\epsilon}\vert^{m}\Vert_{L^{\frac{2}{m}}(\Omega)}^{\frac{2(m-1)}{m}}\Big) \\&\leq \frac{m-1}{4m}\Vert\nabla\vert\nabla c_{\epsilon}\vert^{m}\Vert_{L^{2}(\Omega)}^{2}+ C_{11}.\end{matrix}$

$\epsilon\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2m}+2m\int_{\Omega}\vert\nabla c_{\epsilon}\vert^{2m}\leq C_{12}.$

(1) 在 $C^{0}\big(\overline{\Omega}\times [0,\infty)\big)$ 中, $n_{\epsilon _{i}}\rightarrow n ;$

(2) 在 $L_{\rm loc}^{2}\big((0,\infty);W^{1,2}(\Omega)\big)$ 中, $n_{\epsilon_{i}}\rightharpoonup n ;$

(3) 在 $L_{\rm loc}^{\infty}\big((0,\infty);C^{0}(\overline{\Omega})\big)\bigcap L_{\rm loc}^{2}\big((0,\infty);W^{1,2}(\Omega)\big)$ 中, $c_{\epsilon_{i}}\rightarrow c ;$

(4) 在 $\bigcap _{m>3}L_{\rm loc}^{\infty}\big((0,\infty);W^{1,m}(\Omega)\big)\bigcap L_{\rm loc}^{\infty}\big(\Omega\times(0,\infty)\big)$ 中, $\nabla c_{\epsilon_{i}}\stackrel{*}{\rightharpoonup}\nabla c ;$

(5) 在 $C^{0}\big(\overline{\Omega}\times(0,\infty);\mathbb{R}^{3}\big)\bigcap C_{\rm loc}^{2,1}\big(\overline{\Omega}\times(0,\infty);\mathbb{R}^{3}\big)$ 中, $u_{\epsilon_{i}}\rightarrow u.$

## 4 衰减估计与收敛速率

$h(t)\leq {\rm e}^{-{\frac{t}{8C_{1}}}}h(0)\leq {\rm e}^{-{\frac{t}{8C_{1}}}}\big(\Vert n_{0}-\overline n_{0}\Vert_{L^{2}(\Omega)}^{2}+\frac{1}{8C_{1}}\Vert c_{0}-\overline n_{0}\Vert_{L^{2}(\Omega)}^{2}\big)\leq C_{10}{\rm e}^{-{\frac{t}{8C_{1}}}}.$

$g(t)\leq g(0){\rm e}^{-{\frac{t}{\epsilon}}}+\frac{C_{1}{\rm e}^{-{\frac{t}{\epsilon}}}}{\epsilon}\int_{0}^{t}{\rm e}^{({\frac{1}{\epsilon}}-\mu_{1})s}\mathrm{d}s\leq g(0){\rm e}^{-{\frac{t}{\epsilon}}}+2C_{1}{\rm e}^{-\mu_{1}t}\leq C_{2}{\rm e}^{-\min\{\mu_{1},1\}t}.$

$\Vert n_{\epsilon}(\cdot,t)-\overline n_{0}\Vert_{L^{\infty}(\Omega)}\leq C{\rm e}^{-\mu t},\ \ \ t\in(0,\infty)$

$\Vert u_{\epsilon}(\cdot,t)\Vert_{L^{\infty}}\leq C{\rm e}^{-\mu t},\ \ \ t\in(0,\infty).$

$({\rm e}^{t\Delta})_{t\geq 0}$ 为在 $\Omega$ 内的齐次 Neumann 热半群. 由

$n_{\epsilon}(\cdot,t)={\rm e}^{t\Delta}n_{0}-\int_{0}^{t}{\rm e}^{(t-s)\Delta}\Big(\nabla\cdot\big(n_{\epsilon}S(\cdot,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big)+u_{\epsilon}\cdot\nabla n_{\epsilon}\Big)(\cdot,s)\mathrm{d}s$

$\begin{matrix} & \ \Vert n_{\epsilon}(\cdot,t)-\overline n_{0}\Vert_{L^{\infty}(\Omega)} \\ &\leq \Vert {\rm e}^{t\Delta}( n_{0}-\overline n_{0})\Vert_{L^{\infty}(\Omega)}+\int_{0}^{t}\Vert {\rm e}^{(t-s)\Delta}\nabla\cdot\Big(n_{\epsilon}S(\cdot,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\Big)(\cdot,s)\Vert_{L^{\infty}(\Omega)}\mathrm{d}s \\ &\ \ \ \ +\int_{0}^{t}\Vert {\rm e}^{(t-s)\Delta}u_{\epsilon}(\cdot,s)\cdot\nabla n_{\epsilon}(\cdot,s)\Vert_{L^{\infty}(\Omega)}\mathrm{d}s \\ &=:Q_{1}+Q_{2}+Q_{3}. \end{matrix}$

$\Vert n-\overline n_{0}\Vert_{L^{\infty}(\Omega)}^{2}\leq C_{2}{\rm e}^{-\mu t}$

$\Vert \nabla(c-\overline n_{0})\Vert_{L^{p}(\Omega)}^{2}\leq C_{2}{\rm e}^{-\mu t}.$

$\begin{matrix} & \ \int_{\Omega}\hat{n}^{r-2}(\hat{n}^{2}+\hat{c}^{2})\vert\nabla(c-\overline n_{0})\vert^{2} \\ &\leq \Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}+\Vert \hat {c}\Vert_{L^{r}(\Omega)}^{2}\Big)\Vert \hat {n}\Vert_{L^{m}(\Omega)}^{r-2}\Vert \nabla(c-\overline n_{0}) \Vert_{L^{\frac{2mr}{(r-2)(m-r)}}(\Omega)}^{2} \\ &\leq\Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}+\Vert \hat {c}\Vert_{L^{r}(\Omega)}^{2}\Big)\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{\frac{r(r-2)}{m}}\Vert n_{\epsilon}-n\Vert_{L^{\infty}(\Omega)}^{\frac{(r-2)(m-r)}{m}}\Vert \nabla(c-\overline n_{0}) \Vert_{L^{\frac{2mr}{(r-2)(m-r)}}(\Omega)}^{2} \\ &\leq C_{3}{\rm e}^{-\mu t}\Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}+\Vert \hat {c}\Vert_{L^{r}(\Omega)}^{2}\Big)\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{\frac{r(r-2)}{m}} \\ &\leq C_{4}{\rm e}^{-\mu t}\Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}+\Vert \hat {c}\Vert_{L^{r}(\Omega)}^{2}\Big)\Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{r-2}+1\Big) \\ &\leq C_{5}{\rm e}^{-\mu t}\Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{r}+\Vert \hat {c}\Vert_{L^{r}(\Omega)}^{r}\Big)+C_{5}{\rm e}^{-\mu t}\Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}+\Vert \hat {c}\Vert_{L^{r}(\Omega)}^{2}\Big). \end{matrix}$

$\begin{matrix} \int_{\Omega}\hat{n}^{r}\vert\nabla(c_{\epsilon}-\overline n_{0})\vert^{2}&=\int_{\Omega}\hat{n}^{r-2}\hat{n}^{2}\vert\nabla(c_{\epsilon}-\overline n_{0})\vert^{2} \\ &\leq \Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}\Vert \hat {n}\Vert_{L^{m}(\Omega)}^{r-2}\Vert \nabla(c_{\epsilon}-\overline n_{0}) \Vert_{L^{\frac{2mr}{(r-2)(m-r)}}(\Omega)}^{2} \\ &\leq\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{\frac{r(r-2)}{m}}\Vert n_{\epsilon}-n\Vert_{L^{\infty}(\Omega)}^{\frac{(r-2)(m-r)}{m}}\Vert \nabla(c_{\epsilon}-\overline n_{0}) \Vert_{L^{\frac{2mr}{(r-2)(m-r)}}(\Omega)}^{2} \\ &\leq C_{6}{\rm e}^{-\mu t}\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{\frac{r(r-2)}{m}} \\ &\leq C_{7}{\rm e}^{-\mu t}\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}\Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{r-2}+1\Big) \\ &\leq C_{8}{\rm e}^{-\mu t}\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{r}+C_{8}{\rm e}^{-\mu t}\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}. \end{matrix}$

$\begin{matrix} &\frac{\mathrm{d}}{\mathrm{d}t}\Vert \hat{n}^{\frac{r}{2}}\Vert_{L^{2}(\Omega)}^{2}+\frac{3(r-1)}{r}\Vert \nabla\hat{n}^{\frac{r}{2}}\Vert_{L^{2}(\Omega)}^{2} \\ \leq\ & C_{9}{\rm e}^{-\mu t}\Big(\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{r}+\Vert \hat {u}\Vert_{L^{r}(\Omega)}^{r}\Big)+C_{9}\overline n_{0}\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{r}+ C_{9}\Big({\rm e}^{-\mu t}+\overline n_{0}\Big)\Vert \hat {c}\Vert_{W^{1,r}(\Omega)}^{r} \\ &+C_{9}{\rm e}^{-\mu t}\Vert \hat {n}\Vert_{L^{r}(\Omega)}^{2}+C_{9}{\rm e}^{-\mu t}\Vert \hat {c}\Vert_{L^{r}(\Omega)}^{2}. \end{matrix}$

$M_{1}\leq C_{16}\epsilon^{\frac{4r}{3r^{2}-8r+12}}(1+t)^{\frac{r}{2}}\int_{0}^{t}{\rm e}^{-\frac{(t-s)}{2}}{\mathrm{d}s}\leq C_{16}\epsilon^{\frac{4r}{3r^{2}-8r+12}}(1+t)^{\frac{r}{2}}.$

$\begin{matrix} M_{2}&\leq C_{17}\int_{0}^{t}{\rm e}^{-\frac{(t-s)}{2}}\Vert \hat {c}(\cdot,s)\Vert_{W^{1,2}(\Omega)}^{\frac{2(p-r)}{p-2}}\Vert \hat {c}(\cdot,s)\Vert_{W^{1,p}(\Omega)}^{\frac{p(r-2)}{p-2}}{\mathrm{d}s} \\ &\leq C_{18}\int_{0}^{t}{\rm e}^{-\frac{(t-s)}{2}}\Vert \hat {c}(\cdot,s)\Vert_{W^{1,2}(\Omega)}^{\frac{2(p-r)}{p-2}}{\mathrm{d}s} \\ &\leq C_{19}\Big(\int_{0}^{t} \Vert\hat{c}(\cdot,s)\Vert_{W^{1,2}(\Omega)}^{2}{\mathrm{d}s}\Big)^{\frac{p-r}{p-2}}\Big(\int_{0}^{t}{\rm e}^{-\frac{p-2}{r-2}\frac{(t-s)}{2}}{\mathrm{d}s}\Big)^{\frac{r-2}{p-2}} \\ &\leq C_{20}(1+t)^{\frac{p-r}{p-2}}\epsilon^{\frac{p-r}{p-2}}. \end{matrix}$

$\begin{matrix} M_{3}&\leq C_{21}\int_{0}^{t}{\rm e}^{-\frac{(t-s)}{2}}\Vert \hat {c}(\cdot,s)\Vert_{L^{2}(\Omega)}^{\frac{4}{r}}\Vert \hat {c}(\cdot,s)\Vert_{L^{\infty}(\Omega)}^{\frac{2(r-2)}{r}}{\mathrm{d}s} \\ &\leq C_{22}\Big(\int_{0}^{t}\Vert \hat {c}(\cdot,s)\Vert_{L^{2}(\Omega)}^{2}{\mathrm{d}s}\Big)^{\frac{2}{r}}\Big(\int_{0}^{t}{\rm e}^{-\frac{r}{r-2}\frac{(t-s)}{2}}{\mathrm{d}s}\Big)^{\frac{r-2}{r}} \\ &\leq C_{23}(1+t)^{\frac{2}{r}}\epsilon^{\frac{2}{r}}. \end{matrix}$

\begin{align*} h(t)&\leq C_{24}(1+t)^{\frac{r}{2}}\epsilon^{\frac{4r}{3r^{2}-8r+12}}+C_{24}(1+t)^{\frac{p-r}{p-2}}\epsilon^{\frac{p-r}{p-2}}+C_{24}(1+t)^{\frac{2}{r}}\epsilon^{\frac{2}{r}}\\ &\leq C_{25}(1+t)^{\frac{r}{2}}\epsilon^{\frac{2}{r}},\ 2<r\leq 6 \end{align*}

\begin{align*}h(t)&\leq C_{24}(1+t)^{\frac{r}{2}}\epsilon^{\frac{4r}{3r^{2}-8r+12}}+C_{24}(1+t)^{\frac{p-r}{p-2}}\epsilon^{\frac{p-r}{p-2}}+C_{24}(1+t)^{\frac{2}{r}}\epsilon^{\frac{2}{r}}\\ &\leq C_{25}(1+t)^{\frac{r}{2}}\epsilon^{\frac{4r}{3r^{2}-8r+12}}, \ r>6. \end{align*}

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This paper deals with convergence of a solution for the parabolic-parabolic Keller-Segel system {(u(lambda))(t) = Delta u(lambda) - chi del. (u(lambda) del v(lambda)) in Omega x (0, infinity), lambda(v(lambda))(t) = Delta v(lambda) - v(lambda) + u(lambda) in Omega x (0, infinity), where Omega is a bounded domain in R-n (n >= 2) with smooth boundary, chi, lambda > 0 are constants, to that for the parabolic-elliptic-Keller-Segel system (u(t) = Delta u - chi del. (u del v) in Omega x (0, infinity), 0 = Delta v - v + u in Omega x (0, infinity) as lambda SE arrow 0. (C) 2018 Elsevier Inc. All rights reserved.

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