ABSTRACT

Solutions $(u,v)$ to the chemotaxis system $\begin{array}{rl}& u_t=\mathrm{\nabla }\cdot \left(\right(u+1{)}^{m-1}\mathrm{\nabla }u-u(u+1{)}^{q-1}\mathrm{\nabla }v),\\ & \tau v_t=\mathrm{\Delta }v-v+u\end{array}$in a ball $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, wherein $m,q\in \mathbb{R}$ and $\tau \in \{0,1\}$ are given parameters with m−q>−1, cannot blow up in finite time provided u is uniformly-in-time bounded in $L^p\left(\mathrm{\Omega }\right)$ for some $p>p_0:=\frac{n}{2}(1-(m-q\left)\right)$. For radially symmetric solutions, we show that, if u is only bounded in $L^{p_0}\left(\mathrm{\Omega }\right)$ and the technical condition $m>\frac{n-2p_0}{n}$ is fulfilled, then, for any $\alpha >\frac{n}{p_0}$, there is C>0 with $u(x,t)\le C|x{|}^{-\alpha }\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x\in \mathrm{\Omega }\mathrm{a}\mathrm{n}\mathrm{d}t\in (0,T_{max}),$$T_{max}\in (0,\mathrm{\infty }]$ denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any $\alpha <\frac{n}{p_0}$, then u would already be bounded in $L^p\left(\mathrm{\Omega }\right)$ for some $p>p_0$.

摘要

解决方案$(你,v)$趋化系统$\begin{array}{rl}& {你}_{吨}=\mathrm{\nabla }\cdot \left(\right(你+1{)}^{米-1}\mathrm{\nabla }你-你(你+1{)}^{q-1}\mathrm{\nabla }v),\\ & \tau v_{吨}=\mathrm{\Delta }v-v+你\end{array}$在一个球中$\mathrm{\Omega }\subset {\mathbb{R}}^n$,$n\ge 2$, 其中$米,q\in \mathbb{R}$和$\tau \in \{0,1\}$给定参数m − q >−1，如果u在时间上一致，则不会在有限时间内爆炸${\mathrm{大号}}^p\left(\mathrm{\Omega }\right)$对于一些$p>p_0:=\frac{n}{2}(1-(米-q\left)\right)$. 对于径向对称解决方案，我们表明，如果u仅在${\mathrm{大号}}^{p_0}\left(\mathrm{\Omega }\right)$和技术条件$米>\frac{n-2p_0}{n}$满足，那么，对于任何$\alpha >\frac{n}{p_0}$, 有C >0$你(X,吨)\le C|X{|}^{-\alpha }\mathrm{F}\mathrm{○}\mathrm{r}\mathrm{一个}\mathrm{l}\mathrm{l}X\in \mathrm{\Omega }\mathrm{一个}\mathrm{n}\mathrm{d}吨\in (0,{吨}_{最大限度}),$${吨}_{最大限度}\in (0,\mathrm{\infty }]$表示最大存在时间。从某种意义上说，这基本上是最优的，如果这个估计值适用于任何$\alpha <\frac{n}{p_0}$，那么你已经被限制在${\mathrm{大号}}^p\left(\mathrm{\Omega }\right)$对于一些$p>p_0$.