| 摘要: |
| 本文研究了一类双重抛物型凯勒 - 塞格尔方程组， $u_t=\Delta u-\nabla(f(u)\nabla v),~\tau v_t=\Delta v-\gamma v+u$ on $\mathbb{R}^n(n\ge 2)$, 在全空间 $\mathbb{R}^n(n\ge 2)$上的柯西问题。其中，函数 $f(u)$为非负函数，且满足$f(u)\le k{{u}^{\alpha }}$（常数 $k>0$ and $\alpha>\frac{2}{n+1}$）。当扩散系数参数$\tau$满足$\tau(u_0) > 0$（其中 $\tau_0(u_0) > 0$）时，我们证明了：对于所有属于贝索夫空间 $\dot{\mathop{B}}\,_{p,\infty }^{-(\frac{2}{\alpha}-\frac{n}{p})}(\mathbb{R}^{n})(p\ge1)$的初始值$u_0$，该方程组均存在整体解；同时，我们还推导出对应的衰减估计式： $\sup\limits_{t>0}t^{\frac1\alpha -\frac{n}{2p}}{\left\| u(t) \right\|_p}<\infty$,特别地，当 $1<\alpha<2$ 且初始值 $u_0\in M^{\frac{n}{2}}\cap \dot{B}^{-(\frac{2}{\alpha}-1)}_{n,\infty}$时，该方程组的解关于时间是连续的。 |
| 关键词: 趋化性 抛物 - 抛物型 扩散速率参数 整体存在性 |
| DOI： |
| 分类号:primary 35Q92; 35B40 |
| 基金项目: |
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| The global existence in Cauchy problem for chemotaxis model with diffusion parameter |
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yangchunxiao, wangkexin
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| Abstract: |
| This paper investigates the Cauchy problem for a doubly parabolic Keller-Segel system $u_t=\Delta u-\nabla(f(u)\nabla v),~\tau v_t=\Delta v-\gamma v+u$ on $\mathbb{R}^n(n\ge 2)$, with a nonnegative function $f(u)\le k{{u}^{\alpha }}$ for $k>0$ and $\alpha>\frac{2}{n+1}$. We establish global existence of solutions for all initial data $u_0$ belonging to Besov space $\dot{\mathop{B}}\,_{p,\infty }^{-(\frac{2}{\alpha}-\frac{n}{p})}(\mathbb{R}^{n})(p\ge1)$ when the diffusion rate parameter $\tau \ge \tau_0(u_0)$ for some $\tau_0(u_0) > 0$, and also derive the corresponding decay estimates $\sup\limits_{t>0}t^{\frac1\alpha -\frac{n}{2p}}{\left\| u(t) \right\|_p}<\infty$. Specially, for $1<\alpha<2$, the solution is continuous in time provided that $u_0\in M^{\frac{n}{2}}\cap \dot{B}^{-(\frac{2}{\alpha}-1)}_{n,\infty}$ |
| Key words: Chemotaxis Parabolic-parabolic Diffusion rate parameter Global existence |
