# 具有间接吸引信号产出的高维趋化增长系统解的有界性Boundedness in the Higher-Dimensional Chemotaxis-Growth System with Indirect Attractant Production

• 全文下载: PDF(589KB)    PP.650-656   DOI: 10.12677/AAM.2019.84072
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，其中是一个光滑有界区域，τ>0。利用能量方法和Moser迭代证明了在任意充分光滑的初值边界条件下，当µ足够大，该模型有唯一的全局有界经典解。

This paper deals with the following Chemotaxis-growth system of the Mountain Pain Beetle with in-direct attractant production: in a smoothly bounded domain ,   is positive. The energy method and Moser iteration are used to prove that under any sufficiently smooth initial boundary conditions, when µ is large enough, the model has a unique global-in-time classical solution.

1. 引言

$\left\{\begin{array}{cc}{u}_{t}=\Delta u-\nabla \cdot \left(u\nabla v\right)\text{+}\mu \left(u-{u}^{2}\right),& x\in \Omega ,t>0,\\ {v}_{t}=\Delta v+w-v,& x\in \Omega ,t>0,\\ \tau {w}_{t}+w=u,& x\in \Omega ,t>0,\\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,& x\in \partial \Omega ,t>0,\\ u\left(x,0\right)={u}_{0}\left(x\right),\text{\hspace{0.17em}}v\left(x,0\right)={v}_{0}\left(x\right)\text{\hspace{0.17em}},w\left(x,0\right)={w}_{0}\left(x\right),& x\in \Omega ,\end{array}$ (1.1)

$\left\{\begin{array}{cc}{u}_{t}=\Delta u-\nabla \cdot \left(u\nabla v\right),& x\in \Omega ,t>0,\\ {v}_{t}=\Delta v-v+u,& x\in \Omega ,t>0\\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,& x\in \partial \Omega ,t>0,\\ u\left(x,0\right)={u}_{0}\left(x\right),\text{\hspace{0.17em}}v\left(x,0\right)={v}_{0}\left(x\right),& x\in \Omega ,\end{array}$

$n=3$$\mu >0$ 时，Hu和Tao在文献 [7] 中证明了模型解的全局有界性。本文证明了 $n\ge 3$$\mu >{\delta }_{2}+C\left(p\right)\left(2{\delta }_{1}+\epsilon \right)$ 时，模型(1.1)解的全局有界性，其中 ${\delta }_{1}$${\delta }_{2}$$C\left(p\right)$ 满足(3.7)，(3.11)，(3.9)。当 $n\ge 2$ 时，Li和Tao [8] 研究了logistic源为 $u-{u}^{\alpha }$ 时的情况，系数满足 $\alpha >n/2$ 时模型解的全局有界性，维数 $n\ge 4$ 时，显然 $\alpha >2$ ，而本文 $\alpha =2$ 优于其结果。在Hu和Tao研究的基础上，Qiu，Mu和Wang [9] 将飞行甲壳虫的随机扩散项 $\Delta u$ 用非线性函数 $D\left(u\right)$ 来描述，考虑扩散系数 $D\left(u\right)$ 对模型的影响，当 $n\ge 3$ 时，假设扩散系数 $D\left(u\right)>{D}_{0}{u}^{\theta }$ 得到 $\theta >1-4/n$ 时模型解的全局有界性。而当 $n\ge 4$ 时，本文 $\theta =0$ 优于其结果。当 $n\ge 1$ 时，Zheng在文献 [10] 中假设扩散系数 $D\left(u\right)\ge {C}_{D}{\left(u+1\right)}^{m-1}$ 得到 $m>\left\{\begin{array}{cc}1-\frac{\mu }{\chi \left[1+{\lambda }_{0}{‖{v}_{0}‖}_{{L}^{\infty }\left(\Omega \right)}{2}^{3}\right]}& n\le 2,\\ 1& n\ge 3,\end{array}$ 时模型解的全局有界性。但是本文 $n\ge 3$是文献 [10] 的一个临界情况。

$\begin{array}{l}u\in {C}^{0}\left(\stackrel{¯}{\Omega }×\left[0,T{}_{\mathrm{max}}\right)\right)\cap {C}^{2,1}\left(\stackrel{¯}{\Omega }×\left(0,{T}_{\mathrm{max}}\right)\right),\\ v\in {C}^{0}\left(\stackrel{¯}{\Omega }×\left[0,T{}_{\mathrm{max}}\right)\right)\cap {C}^{2,1}\left(\stackrel{¯}{\Omega }×\left(0,{T}_{\mathrm{max}}\right)\right),\\ w\in {C}^{0,1}\left(\stackrel{¯}{\Omega }×\left[0,T{}_{\mathrm{max}}\right)\right),\end{array}$

$\Omega ×\left(0,\infty \right)$ 是模型(1.1)的经典解。特别的，存在一个常数 $C>0$ ，使得

$u\left(x,t\right)\le C$$v\left(x,t\right)\le C$$w\left(x,t\right)\le C,x\in \Omega$$t>0$

2. 先验估计

${‖\nabla v‖}_{{L}^{2p+2}\left(\Omega \right)}^{2p+2}\le {k}_{0}\left({‖{|\nabla v|}^{p-1}{D}^{2}v‖}_{{L}^{2}\left(\Omega \right)}^{2}+1\right)$

${\int }_{\Omega }u\left(\cdot ,t\right)\text{d}x\le C$${‖v\left(\cdot ,t\right)‖}_{{W}^{1,q}\left(\Omega \right)}\le C$${\int }_{\Omega }{u}^{2}\left(\cdot ,t\right)\text{d}x\le C$

3. 主要结论的证明

${‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}\le C$${‖\nabla v‖}_{{L}^{2p}\left(\Omega \right)}^{2p}\le C$${‖w‖}_{{L}^{p\text{+}1}\left(\Omega \right)}^{p\text{+}1}\le C$

 (3.1)

$\frac{1}{2}{\int }_{\partial \Omega }{|\nabla v|}^{2p-2}\frac{\partial {|\nabla v|}^{2}}{\partial n}\text{\hspace{0.17em}}\text{d}x\le \frac{1}{2}{C}_{\Omega }{‖{|\nabla v|}^{p}‖}_{{L}^{2}\left(\partial \Omega \right)}^{2}\le {C}_{1}{‖{|\nabla v|}^{p}‖}_{{W}^{r+\frac{1}{2},2}\left(\Omega \right)}^{2}$(3.2)

${‖{|\nabla v|}^{p}‖}_{{W}^{r+\frac{1}{2},2}\left(\Omega \right)}^{2}\le {C}_{2}{‖{|\nabla v|}^{p}‖}_{{L}^{2}\left(\Omega \right)}^{a}\text{+}{C}_{3}\le \frac{p-1}{{C}_{1}{p}^{2}}{\int }_{\Omega }{|\nabla {|\nabla v|}^{p}|}^{2}\text{d}x\text{+}{C}_{4}$

。 (3.3)

$\begin{array}{c}-{\int }_{\Omega }w\nabla v\cdot \nabla \left({|\nabla v|}^{2p-2}\right)\text{\hspace{0.17em}}\text{d}x=-\left(p-1\right){\int }_{\Omega }w{|\nabla v|}^{2\left(p-2\right)}\nabla v\cdot \nabla {|\nabla v|}^{2}\text{d}x\\ \le \frac{p-1}{4}{\int }_{\Omega }{|\nabla v|}^{2p-4}{|\nabla {|\nabla v|}^{2}|}^{2}dx+\left(p-1\right){\int }_{\Omega }{w}^{2}{|\nabla v|}^{2p-2}\text{d}x\\ \le \frac{p-1}{{p}^{2}}{\int }_{\Omega }{|\nabla {|\nabla v|}^{p}|}^{2}\text{d}x\text{+}\left(p-1\right){\int }_{\Omega }{w}^{2}{|\nabla v|}^{2p-2}\text{d}x.\end{array}$ (3.4)

$-{\int }_{\Omega }w{|\nabla v|}^{2p-2}\Delta v\text{d}x\le \sqrt{n}{\int }_{\Omega }w{|\nabla v|}^{2p-2}|{D}^{2}v|\text{ }\text{d}x\le \frac{1}{4}{\int }_{\Omega }{|\nabla v|}^{2p-2}{|{D}^{2}v|}^{2}\text{d}x\text{+}n{\int }_{\Omega }{w}^{2}{|\nabla v|}^{2p-2}\text{d}x$(3.5)

$\begin{array}{c}{\int }_{\Omega }{w}^{2}{|\nabla v|}^{2p-2}\text{d}x\le \frac{1}{4\left(n+p-1\right){k}_{0}}{\int }_{\Omega }{|\nabla v|}^{2p+2}\text{d}x+\frac{2}{p+1}{\left(\frac{4\left(n+p-1\right)\left(p-1\right){k}_{0}}{p+1}\right)}^{\frac{p-1}{2}}{\int }_{\Omega }{w}^{p+1}\text{d}x\\ \text{ }\le \frac{1}{4\left(n+p-1\right)}{\int }_{\Omega }{|\nabla v|}^{2p-2}{|{D}^{2}v|}^{2}\text{d}x+{\delta }_{1}{\int }_{\Omega }{w}^{p+1}\text{d}x+C，\end{array}$ (3.6)

$\frac{1}{2p}\frac{\text{d}}{\text{d}t}{‖\nabla v‖}_{{L}^{2p}\left(\Omega \right)}^{2p}+{\int }_{\Omega }{|\nabla v|}^{2p}\text{d}x+\frac{1}{2}{\int }_{\Omega }{|\nabla v|}^{2p-2}{|{D}^{2}v|}^{2}\text{d}x\le {\delta }_{1}{\int }_{\Omega }{w}^{p+1}\text{d}x+C,$ (3.7)

$\frac{\tau }{p+1}\frac{\text{d}}{\text{d}t}{‖w‖}_{{L}^{p+1}\left(\Omega \right)}^{p+1}+{\int }_{\Omega }{w}^{p+1}\text{d}x={\int }_{\Omega }u{w}^{p}\text{d}x\le \frac{1}{2}{\int }_{\Omega }{w}^{p+1}\text{d}x+C\left(p\right){\int }_{\Omega }{u}^{p+1}\text{d}x$(3.8)

$\frac{\tau }{p+1}\frac{\text{d}}{\text{d}t}{‖w‖}_{{L}^{p+1}\left(\Omega \right)}^{p+1}+\frac{1}{2}{\int }_{\Omega }{w}^{p+1}\text{d}x\le C\left(p\right){\int }_{\Omega }{u}^{p+1}\text{d}x$(3.9)

$\frac{1}{p}\frac{\text{d}}{\text{d}t}{‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}+\left(p-1\right){\int }_{\Omega }{u}^{p-2}{|\nabla u|}^{2}\text{d}x=-{\int }_{\Omega }\nabla \cdot \left(u\nabla v\right){u}^{p-1}\text{d}x+\mu {\int }_{\Omega }{u}^{p}\text{d}x-\mu {\int }_{\Omega }{u}^{p+1}\text{d}x.$ (3.10)

$\frac{1}{p}\frac{\text{d}}{\text{d}t}{‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}\text{+}\frac{1}{p}{‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}\le \frac{1}{2}{\int }_{\Omega }{|\nabla v|}^{2p-2}{|{D}^{2}v|}^{2}\text{d}x\text{+}\left(\mu +\frac{1}{p}\right){\int }_{\Omega }{u}^{p}\text{d}x-\left(\mu -{\delta }_{2}\right){\int }_{\Omega }{u}^{p+1}\text{d}x+C.$ (3.11)

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left[\frac{1}{2p}{‖\nabla v‖}_{{L}^{2p}\left(\Omega \right)}^{2p}+\frac{\tau \left(2{\delta }_{1}+\epsilon \right)}{p+1}{‖w‖}_{{L}^{p+1}\left(\Omega \right)}^{p+1}+\frac{1}{p}{‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}\right]+{\int }_{\Omega }{|\nabla v|}^{2p}\text{d}x+\frac{\epsilon }{2}{\int }_{\Omega }{w}^{p+1}\text{d}x+\frac{1}{p}{‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}\\ \le \left(\mu +\frac{1}{p}\right){\int }_{\Omega }{u}^{p}\text{d}x-\left[\mu -{\delta }_{2}-C\left(p\right)\left(2{\delta }_{1}+\epsilon \right)\right]{\int }_{\Omega }{u}^{p+1}\text{d}x+C,\end{array}$

${‖\nabla v‖}_{{L}^{2p}\left(\Omega \right)}^{2p}\le C$${‖w‖}_{{L}^{p+1}\left(\Omega \right)}^{p+1}\le C$${‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}\le C$

${‖\nabla v\left(\cdot ,t\right)‖}_{{L}^{\infty }\left(\Omega \right)}\le C$

${‖u\left(\cdot ,t\right)‖}_{{L}^{\infty }\left(\Omega \right)}\le C$${‖w\left(\cdot ,t\right)‖}_{{L}^{\infty }\left(\Omega \right)}\le C$

$\begin{array}{c}\frac{1}{p}\frac{\text{d}}{\text{d}t}{‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}+\left(p-1\right){\int }_{\Omega }{u}^{p-2}{|\nabla u|}^{2}\text{d}x\le \left(p-1\right){\int }_{\Omega }{u}^{p-1}|\nabla u||\nabla v|\text{d}x+\mu {\int }_{\Omega }{u}^{p-1}\left(u-{u}^{2}\right)\text{\hspace{0.17em}}\text{d}x\\ \le C{\int }_{\Omega }{u}^{p-1}|\nabla u|\text{\hspace{0.17em}}\text{d}x+\mu {\int }_{\Omega }{u}^{p-1}\left(u-{u}^{2}\right)\text{\hspace{0.17em}}\text{d}x\le \frac{p-1}{4}{\int }_{\Omega }{u}^{p-2}{|\nabla u|}^{2}\text{d}x\\ +C{\int }_{\Omega }{u}^{p}\text{d}x+\mu {\int }_{\Omega }{u}^{p-1}\left(u-{u}^{2}\right)\text{\hspace{0.17em}}\text{d}x\\ \le \frac{p-1}{4}{\int }_{\Omega }{u}^{p-2}{|\nabla u|}^{2}\text{d}x+C{\int }_{\Omega }{u}^{p}\text{d}x-\mu {\int }_{\Omega }{u}^{p+1}\text{d}x.\end{array}$

$\frac{1}{p}\frac{\text{d}}{\text{d}t}{‖u‖}_{{L}^{p}\left(\Omega \right)}^{p}+\frac{1}{p}{\int }_{\Omega }{u}^{p}\text{d}x+\frac{p-1}{{p}^{2}}{\int }_{\Omega }{|\nabla {u}^{\frac{p}{2}}|}^{2}\text{d}x\le C$

${‖u\left(\cdot ,t\right)‖}_{{L}^{p}\left(\Omega \right)}\le C$

${‖u\left(\cdot ,t\right)‖}_{{L}^{\infty }\left(\Omega \right)}\le C$

${‖w\left(\cdot ,t\right)‖}_{{L}^{\infty }\left(\Omega \right)}\le C$

NOTES

*通讯作者。

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