## Global Existence and Convergence of Solutions to a Chemotactic Model with Logarithmic Sensitivity and Mixed Boundary Conditions

Wang Juan,, Yuan Zixia,

 基金资助: 电子科技大学中央高校基本科研业务费.  ZYGX2019J096

 Fund supported: the Fundamental Research Funds for the Central Universities, UESTC.  ZYGX2019J096

Abstract

This paper investigates the following chemotactic model with logarithmic sensitivity in a one-dimensional bounded domain: By using a Cole-Hopf type transformation, we transform the above singular repulsive chemotaxis model into a non-singular system of the form Then under some mixed boundary conditions, we prove the global existence and exponential convergence of solutions to the initial-boundary value problem of the above system with regular initial data.

Keywords： Chemotaxis ; Logarithmic sensitivity ; Mixed boundary conditions ; Global existence ; Exponential convergence

Wang Juan, Yuan Zixia. Global Existence and Convergence of Solutions to a Chemotactic Model with Logarithmic Sensitivity and Mixed Boundary Conditions. Acta Mathematica Scientia[J], 2020, 40(6): 1646-1669 doi:

## 1 引言

$$$\left\{ \begin{array}{lll} \partial_{t}u = D\triangle u-\nabla\cdot\big(\chi u \nabla \phi(v)\big), \quad & x\in \Omega, \quad t>0, \\ \partial_{t}v = \varepsilon \triangle v+g(u, v), \quad & x\in \Omega, \quad t>0, \\ \end{array} \right.$$$

$\bullet $$\varepsilon = 0 时, 且当 \overline{p}>0 时, 对于所有的 t>0 , 存在与 t 无关的常数 C>0 满足 且有 \|(p-\overline{p})(t)\|_{H^{2}}^2+ \|q(t)\|_{H^{2}}^2\leq \alpha e^{-\beta t}, 其中 \alpha , \beta 是与 t 无关的正常数. 研究动机.证明问题(1.3)-(1.4)解的全局存在性的主要困难在于: \bullet 在混合边界条件下, 对于方程 (1.3)_{1} 等号右边的二次非线性项, 通常的方法无效.例如, 在边界条件 p_x|_{x = 0, x = 1} = q|_{x = 0, x = 1} = 0 下, 对变换后的方程组 (1.3) 应用Lyapunov泛函, 得到守恒方程 其中 \eta(z) = z\ln(z)-z (参见文献[15, 32]), 然后借助修正的Sobolev不等式 控制相应的非线性项, 这里 [p]$$ p$在(0, 1)中的平均值.这一方法主要是基于$[p]$的守恒.然而在混合边界条件下, $[p]$不是守恒的, 从而在高阶估计中会出现边界项, 因此需要对这些边界项进行更细致的分析.

$\bullet$在标准的$L^{2}$能量框架下, 估计得不到闭合.例如, 因为得不到$\widetilde{p}_x$的耗散估计, 所以$\int_{0}^{1}\widetilde{p}\widetilde{p}_xq {\rm d}x$不能根据标准的$L^{2}$能量估计来处理.因此, 本文将构造一个新的能量框架:根据文献[14]中建立的$L^{r}\, (r = 3, 4)$估计得到$L^{2}$估计, 从而$L^{r}$估计的右边项刚好抵消$L^{2}$中不能被估计的项$\int_{0}^{1}\widetilde{p}\widetilde{p}_xq {\rm d}x$.与直接应用$L^{2}$估计相比, $L^{r}$估计的推导更为复杂.

$\bullet$由于缺少关于$\varepsilon$的一致估计, 在证明非扩散问题($\varepsilon = 0$)解的全局存在性的时候, 不能通过取极限$\varepsilon\rightarrow0$而由扩散问题($\varepsilon>0$)的存在性结论直接得到.

### 2 扩散问题(ε>0)解的全局存在性

$$$\left\{ \begin{array}{lll} p_t = p_{xx} + (pq)_x, \\ q_t = \varepsilon q_{xx} + \varepsilon (q^2)_x + p_x, \quad \varepsilon>0\\ \end{array} \right.$$$

$$$\left\{ \begin{array}{lll} (p, q)(x, 0) = (p_0, q_0)(x), \quad p_0(x)\geq0, \quad x\in(0, 1), \\ p|_{x = 0} = \overline{p}, \quad p_x|_{x = 1} = 0, \quad q|_{x = 0, x = 1} = 0, \\ \end{array} \right.$$$

$$$\|{\widetilde{p}}(t)\|_{L^{2}}^2+\|{q}(t)\|_{L^{2}}^2+\int_{0}^{t}\bigg(\int_{0}^{1}{\frac{(\widetilde{p}_x)^2}{\widetilde{p}+\overline{p}}}{\rm d}x +\|{\widetilde{p}_x}(t)\|_{L^{2}}^2+\varepsilon\|{q_x}\|_{L^{2}}^2\bigg ){\rm d}\tau\leq C,$$$

证明分为两个步骤.

$$$\frac{1}{2}\|{q(t)}\|_{L^{2}}^2+\int_{0}^{t}\bigg(\int_{0}^{1}{\frac{(p_x)^2}{p}}{\rm d}x +\varepsilon\|{q_x}\|_{L^{2}}^2\bigg){\rm d}\tau \leq C_{2},$$$

$$$\|(\widetilde{p}_x, q_x)(t)\|_{H^{1}}^2+\int_{0}^{t}\|\widetilde{p}_{xx}(\tau)\|_{H^{1}}^2 +\|q_{xx}(\tau)\|_{H^{1}}^2{\rm d}\tau\leq C,$$$

$\begin{eqnarray} \int_{0}^{1}|\widetilde{p}(x)|^2{\rm d}x & = &\int_{0}^{1}\bigg|\int_{0}^{x}\widetilde{p}'(y){\rm d}y\bigg|^2{\rm d}x \leq\int_{0}^{1}\int_{0}^{x}|\widetilde{p}'(y)|^2{\rm d}y{\rm d}x \\ & \leq&\int_{0}^{1}\int_{0}^{1}|\widetilde{p}'(y)|^2{\rm d}y{\rm d}x \leq\int_{0}^{1}|\widetilde{p}'(y)|^2{\rm d}x \leq\int_{0}^{1}|\widetilde{p}_x|^2{\rm d}x, \end{eqnarray}$

$\|\widetilde{p}\|_{L^{2}}^2\leq \|\widetilde{p}_x\|_{L^{2}}^2$.应用Hölder不等式, $(2.3)$式和$(2.43)$式, 可得

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2)+\|\widetilde{p}_{xx}\|_{L^{2}}^2+ \varepsilon\|q_{xx}\|_{L^{2}}^2\\ & = &-\int_{0}^{1}\big((\widetilde{p}q)_x+\overline{p}q_x\big)\widetilde{p}_{xx}{\rm d}x -\int_{0}^{1}\big(\widetilde{p}_x+\varepsilon(q^2)_x\big)q_{xx}{\rm d}x\\ &\leq&\frac{1}{2}\|\widetilde{p}_{xx}\|_{L^{2}}^2+\frac{\varepsilon}{2}\|q_{xx}\|_{L^{2}}^2+ C_{9}(\|\widetilde{p}\|_{L^{\infty}}^2\|q_x\|_{L^{2}}^2+ \|q\|_{L^{\infty}}^2\|\widetilde{p}_x\|_{L^{2}}^2 \\ &&+\|q\|_{L^{\infty}}^2\|q_x\|_{L^{2}}^2+\|\widetilde{p}_x\|_{L^{2}}^2+ \|q_x\|_{L^{2}}^2)\\ &\leq&\frac{1}{2}\|\widetilde{p}_{xx}\|_{L^{2}}^2+\frac{\varepsilon}{2}\|q_{xx}\|_{L^{2}}^2+ C_{10}(\|\widetilde{p}\|_{L^{2}}\|\widetilde{p}_x\|_{L^{2}}\|q_x\|_{L^{2}}^2+\|q\|_{L^{2}}\|q_x\|_{L^{2}}\|\widetilde{p}_x\|_{L^{2}}^2\\ &&+\|q\|_{L^{2}}\|q_x\|_{L^{2}}\|q_x\|_{L^{2}}^2+\|\widetilde{p}_x\|_{L^{2}}^2+ \|q_x\|_{L^{2}}^2+\|\widetilde{p}\|_{L^{2}}^2\|q_x\|_{L^{2}}^2)\\ &\leq&\frac{1}{2}\|\widetilde{p}_{xx}\|_{L^{2}}^2+\frac{\varepsilon}{2}\|q_{xx}\|_{L^{2}}^2+ C_{11}(\|\widetilde{p}_x\|_{L^{2}}^2 +\|q_x\|_{L^{2}}^2)(\|q_x\|_{L^{2}}^2+1), \end{eqnarray}$

$\begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2+1)+\frac{1}{2}\|\widetilde{p}_{xx}\|_{L^{2}}^2+ \frac{\varepsilon}{2}\|q_{xx}\|_{L^{2}}^2\\ &\leq& C_{11}(\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2) (\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2+1). \end{eqnarray}$

$$$\|\widetilde{p}_x(t)\|_{L^{2}}^2+\|q_x(t)\|_{L^{2}}^2+\int_{0}^{t}(\|\widetilde{p}_{xx}(\tau)\|_{L^{2}}^2+ \|q_{xx}(\tau)\|_{L^{2}}^2){\rm d}\tau\leq C_{12}.$$$

$\begin{eqnarray} \|\widetilde{p}_{xx}\|_{L^{2}}^2 & = &\|\widetilde{p}_t-(\widetilde{p}q)_x-\overline{p}q_x\|_{L^{2}}^2\\ &\leq &C_{16}\big(|\widetilde{p}_t\|_{L^{2}}^2+\|\widetilde{p}_x\|_{L^{2}}^2\|q\|_{L^{\infty}}^{2}+ \|\widetilde{p}\|_{L^{\infty}}^{2}\|q_x\|_{L^{2}}^2+\overline{p}^2\|q_x\|_{L^{2}}^2\big)\\ &\leq& C_{17} \end{eqnarray}$

$$$\|q_{xx}\|_{L^{2}}^2 \leq C_{18}\big(\frac{1}{\varepsilon^2}\|q_t\|_{L^{2}}^2+ \frac{1}{\varepsilon^2}\|\widetilde{p}_x\|_{L^{2}}^2+ \|q\|_{L^{\infty}}^{2}\|q_x\|_{L^{2}}^2\big) \leq C_{19}.$$$

$$$\|\widetilde{p}_{xx}(t)\|_{L^{2}}^2+\|q_{xx}(t)\|_{L^{2}}^2+\int_{0}^{t}(\|\widetilde{p}_{xxx}(\tau)\|_{L^{2}}^2 +\|q_{xxx}(\tau)\|_{L^{2}}^2){\rm d}\tau\leq C_{20},$$$

$$$\|(\widetilde{p}, q)(t)\|_{H^{2}}^2\leq\alpha e^{-\beta t}.$$$

由(2.38)-(2.40)式和(2.46)-(2.51)式可得

$$$\frac{\rm d}{{\rm d}t}G_1(t)+\frac{3}{2}\overline{p}^2\|{\widetilde{p}_x}\|_{L^{2}}^2 +\frac{3}{2}\|\overline{p}{\widetilde{p}_x-\widetilde{p}\widetilde{p}_x}\|_{L^{2}}^2 +\|\widetilde{p}\widetilde{p}_x\|_{L^{2}}^2 +3\varepsilon\overline{p}^3\|q_x\|_{L^{2}}^2 \leq C_{19}\int_{0}^{1}\frac{(\widetilde{p}_x)^2}{\widetilde{p}+\overline{p}}{\rm d}x,$$$

$$$\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2)+\frac{1}{2}\|\widetilde{p}_{xx}\|_{L^{2}}^2+ \frac{\varepsilon}{2}\|q_{xx}\|_{L^{2}}^2 \leq C_{20}(\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2)$$$

$$$\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\widetilde{p}_t\|_{L^{2}}^2+\overline{p}\|q_t\|_{L^{2}}^2)+ \frac{1}{2}\|\widetilde{p}_{xt}\|_{L^{2}}^2+\frac{\varepsilon\overline{p}}{2}\|q_{xt}\|_{L^{2}}^2 \leq C_{21}(\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2).$$$

$(2.56)$式两边同时乘以$B_{1}$, 与(2.57)式和(2.58)式相加, 得到

$$$\frac{\rm d}{{\rm d}t}M(t)+N(t)\leq B_{1}C_{19}\int_{0}^{1}\frac{(\widetilde{p}_x)^2}{\widetilde{p}+\overline{p}}{\rm d}x,$$$

$\begin{eqnarray} M(t)& = &B_{1}G_1(t)+\frac{1}{2}(\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2+ \|\widetilde{p}_t\|_{L^{2}}^2+\overline{p}\|q_t\|_{L^{2}}^2), \\ N(t)& = &B_{1}\bigg(\frac{3}{2}\|\overline{p}{\widetilde{p}_x-\widetilde{p}\widetilde{p}_x}\|_{L^{2}}^2 +\|\widetilde{p}\widetilde{p}_x\|_{L^{2}}^2 +3\varepsilon\overline{p}^3\|q_x\|_{L^{2}}^2\bigg)+\frac{1}{2}\|\widetilde{p}_{xx}\|_{L^{2}}^2+\frac{\varepsilon}{2}\|q_{xx}\|_{L^{2}}^2\\ &&+\frac{1}{2}\|\widetilde{p}_{xt}\|_{L^{2}}^2+ \frac{\varepsilon\overline{p}}{2}\|q_{xt}\|_{L^{2}}^2-(C_{20}+C_{21}) (\|\widetilde{p}_x\|_{L^{2}}^2+\|q_x\|_{L^{2}}^2). \end{eqnarray}$

$(2.44)$式和Gagliardo-Nirenberg不等式可知, 存在一个与$t$无关的常数$C_{22}$, 使得

$$$\|\widetilde{p}(t)\|_{H^{2}}^2+\|q(t)\|_{H^{2}}^2 \leq \alpha e^{-\beta t}.$$$

### 2.2 $\overline{p} = 0$的情形

$\overline{p} = 0$时, 根据$p|_{x = 0} = 0$, $p_0>0$, 局部存在性结果和最大值原理可知$p>0$.$\widetilde{p} = p+R$, 应用扰动方法将问题(2.1)-(2.2)变换为如下初边值问题

$$$\left\{ \begin{array}{lll} \widetilde{p}_t-(\widetilde{p}q)_x+Rq_x = \widetilde{p}_{xx}, \\ q_t-\widetilde{p}_x = \varepsilon q_{xx} + \varepsilon (q^2)_x, \\ \widetilde{p}|_{x = 0} = R, \quad\widetilde{p}_x|_{x = 1} = 0, \\ q|_{x = 0, x = 1} = 0, \\ (p, q)(x, 0) = (p_0, q_0)(x), \end{array} \right.$$$

$$$\|(p, q)(t)\|_{H^{2}}^2+\int_{0}^{t}\|p(\tau)\|_{H^{3}}^2 +\|q(\tau)\|_{H^{3}}^2{\rm d}\tau\leq C,$$$

$$$\left\{ \begin{array}{lll} p_t-(pq)_x = p_{xx}, \\ q_t-p_x = \varepsilon q_{xx} + \varepsilon (q^2)_x, \\ p|_{x = 0} = 0, \quad p_x|_{x = 1} = 0, \\ q|_{x = 0, x = 1} = 0, \\ (p, q)(x, 0) = (p_0, q_0)(x). \end{array} \right.$$$

$\begin{eqnarray} && \frac{1}{2}\frac{\rm d}{{\rm d}t}(\|p\|_{L^{2}}^2+\|q\|_{L^{2}}^2)+ \|p_x\|_{L^{2}}^2+\varepsilon\|q_x\|_{L^{2}}^2 = \int_{0}^{1}p_x q {\rm d}x-\int_{0}^{1}p q p_x {\rm d}x\\ &\leq&\frac{1}{4}\|p_x\|_{L^{2}}^2+\|q\|_{L^{2}}^2+\frac{1}{4}\|p_x\|_{L^{2}}^2+ \|p\|_{L^{2}}^2\|q\|_{L^{\infty}}^2\\ &\leq&\frac{1}{2}\|p_x\|_{L^{2}}^2+C_{25}\|q_x\|_{L^{2}}^2+C_{26}\|p\|_{L^{2}}^2\|q_x\|_{L^{2}}^2, \end{eqnarray}$

$$$\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|p\|_{L^{2}}^2+\|q\|_{L^{2}}^2)+ \frac{1}{2}\|p_x\|_{L^{2}}^2+\varepsilon\|q_x\|_{L^{2}}^2\leq C_{25}\|q_x\|_{L^{2}}^2+C_{26}\|p\|_{L^{2}}^2\|q_x\|_{L^{2}}^2.$$$

$$$\|p(t)\|_{L^{2}}^2+\int_{0}^{t}\|p_x(\tau)\|_{L^{2}}^2{\rm d}\tau\leq C_{27},$$$

$$$\quad \quad \|p_{xx}\|_{L^{2}}^2\leq C_{39}\big(\|p_t\|_{L^{2}}^2+\|p\|_{L^{\infty}}^2\|q_x\|_{L^{2}}^2+ \|q\|_{L^{\infty}}^2\|p_x\|_{L^{2}}^2\big) \leq C_{40}$$$

$$$\|q_{xx}\|_{L^{2}}^2\leq C_{41}(\|q_t\|_{L^{2}}^2+\|p_x\|_{L^{2}}^2+ \|q\|_{L^{\infty}}^2\|q_x\|_{L^{2}}^2)\leq C_{42},$$$

$$$\|p_{xx}(t)\|_{L^{2}}^2+\|q_{xx}(t)\|_{L^{2}}^2+\int_{0}^{t}(\|p_{xxx}(\tau)\|_{L^{2}}^2+ \|q_{xxx}(\tau)\|_{L^{2}}^2){\rm d}\tau\leq C_{43}.$$$

设$R$为待确定的正参数.在方程$(2.63)_1$两边同时乘以$\widetilde{p}$, 在方程$(2.63)_2$两边同时乘以$-Rq$, 关于$x $$(0, 1) 上积分, 两式相减可得 $$\frac{\rm d}{{\rm d}t}\Big(\frac{1}{2}\|{\widetilde{p}}\|_{L^{2}}^2+\frac{R}{2}\|{q}\|_{L^{2}}^2\Big)+ \|{\widetilde{p}_x}\|_{L^{2}}^2+\varepsilon R\|{q_x}\|_{L^{2}}^2 = -\int_{0}^{1}\widetilde{p} q\widetilde{p}_x {\rm d}x.$$ 方程 (2.63)_1 两边同时乘以 \widetilde{p}^{2} , 关于 x$$ (0, 1)$上积分, 利用分部积分可得

$$$\frac{\rm d}{{\rm d}t}\bigg(\frac{1}{3}\int_{0}^{1}\widetilde{p}^{3}{\rm d}x\bigg) +2\int_{0}^{1}\widetilde{p}(\widetilde{p}_x)^{2}{\rm d}x = -2\int_{0}^{1}\widetilde{p}^{2}q\widetilde{p}_x {\rm d}x +2R\int_{0}^{1}\widetilde{p}q\widetilde{p}_x {\rm d}x.$$$

### 3 非扩散问题($\varepsilon=0$)解的全局存在性

$$$\left\{ \begin{array}{lll} \widetilde{p}_t-(\widetilde{p}q)_x-\overline{p}q_x = \widetilde{p}_{xx}, \\ q_t-\widetilde{p}_x = 0, \\ (\widetilde{p}, q)(x, 0) = (p_0-\overline{p}, q_0)(x), \\ \widetilde{p}|_{x = 0} = 0, \, \, \widetilde{p}_x|_{x = 1} = 0, \\ \end{array} \right.$$$

$$$q_{xt} = \widetilde{p}_t-\overline{p}q_x-(\widetilde{p}q)_x.$$$

$$$\|q_x(t)\|_{L^{2}}^2+\int_{0}^{t}\|q_x(\tau)\|_{L^{2}}^2{\rm d}\tau \leq C_{52},$$$

$$$\|\widetilde{p}_t(t)\|_{L^{2}}^2+\|q_t(t)\|_{L^{2}}^2+ \int_{0}^{t}\|\widetilde{p}_{xt}(\tau)\|_{L^{2}}^2 {\rm d}\tau \leq C_{58}.$$$

$$$\|\widetilde{p}_{xx}(t)\|_{L^{2}}^2+\|q_{t}(t)\|_{L^{2}}^2+ \int_{0}^{t}\|\widetilde{p}_{xt}(\tau)\|_{L^{2}}^2 {\rm d}\tau \leq C_{59}.$$$

$$$\frac{1}{2}\frac{\rm d}{{\rm d}t}\|q_{xx}\|_{L^{2}}^2+\overline{p}\|q_{xx}\|_{L^{2}}^2 \leq\frac{\overline{p}}{2}\|q_{xx}\|_{L^{2}}^2+C_{61}(\|\widetilde{p}_{xt}\|_{L^{2}}^2+ \|q_x\|_{L^{2}}^2)+C_{62}\|\widetilde{p}_x\|_{L^{2}}^2\|q_{xx}\|_{L^{2}}^2.$$$

$$$\|q_{xx}(t)\|_{L^{2}}^2+ \int_{0}^{t}\|q_{xx}(\tau)\|_{L^{2}}^2 {\rm d}\tau \leq C_{63},$$$

### 3.2 $\overline{p} = 0$的情形

$$$\left\{ \begin{array}{lll} \widetilde{p}_t-(\widetilde{p}q)_x+q_x = \widetilde{p}_{xx}, \\ q_t-\widetilde{p}_x = 0, \\ \widetilde{p}|_{x = 0} = 1, \quad \widetilde{p}_x|_{x = 1} = 0, \\ (\widetilde{p}, q)(x, 0) = (p_0+1, q_0)(x).\\ \end{array} \right.$$$

$$$\|(\widetilde{p}, q)(t)\|_{H^{2}}^2+\int_{0}^{t}\|\widetilde{p}(\tau)\|_{H^{3}}^2 \leq C,$$$

$$$\|\widetilde{p}_t(t)\|_{L^{2}}^2+\|q_t(t)\|_{L^{2}}^2+\int_{0}^{t}\|\widetilde{p}_{xt}(\tau)\|_{L^{2}}^2{\rm d}\tau \leq C_{76},$$$

$$$\|\widetilde{p}_{xx}(t)\|_{L^{2}}^2+\|q_{t}(t)\|_{L^{2}}^2+\int_{0}^{t}\|\widetilde{p}_{xt}(\tau)\|_{L^{2}}^2{\rm d}\tau \leq C_{77}.$$$

$$$q_{xt} = \widetilde{p}_t+q_x-(\widetilde{p}q)_x.$$$

$$$\|q_{xx}(t)\|_{L^{2}}^2\leq C_{81}.$$$

$(3.33)$式和$(3.37)$式知

$$$\|\widetilde{p}_{xx}(t)\|_{L^{2}}^2+\|q_{xx}(t)\|_{L^{2}}^2+ \int_{0}^{t}\|\widetilde{p}_{xxx}(\tau)\|_{L^{2}}^2\leq C_{82}.$$$

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