Well-posedness and Gevrey Analyticity of the Generalized Keller-Segel System in Critical Besov Spaces

Abstract

In this paper, we study the Cauchy problem for the generalized Keller-Segel system with the cell diffusion being ruled by fractional diffusion: \begin{equation*} \begin{cases} \partial_{t}u+Λ^αu-\nabla\cdot(u\nabla ψ)=0\quad &\mbox{in}\ \ \mathbb{R}^n\times(0,\infty), -Δψ=u\quad &\mbox{in}\ \ \mathbb{R}^n\times(0,\infty), u(x,0)=u_0(x), \ \ &\mbox{in}\ \ \mathbb{R}^n. \end{cases} \end{equation*} In the case that $1<\alpha \le 2$, we prove local well-posedness for any initial data and global well-posedness for small initial data in critical Besov spaces ${\dot{B}}_{p,q}^{-\alpha +\frac{n}{p}}({\mathbb{R}}^n)$ with $1\le p<\infty$, $1\le q\le \infty$, and analyticity of solutions for initial data $u_0\in {\dot{B}}_{p,q}^{-\alpha +\frac{n}{p}}({\mathbb{R}}^n)$ with $1<p<\infty$, $1\le q\le \infty$. Moreover, the global existence and analyticity of solutions with small initial data in critical Besov spaces ${\dot{B}}_{\infty ,1}^{-\alpha }({\mathbb{R}}^n)$ is also established. In the limit case that $\alpha =1$, we prove global well-posedness for small initial data in critical Besov spaces ${\dot{B}}_{p,1}^{-1+\frac{n}{p}}({\mathbb{R}}^n)$ with $1\le p<\infty$ and ${\dot{B}}_{\infty ,1}^{-1}({\mathbb{R}}^n)$, and show analyticity of solutions for small initial data in ${\dot{B}}_{p,1}^{-1+\frac{n}{p}}({\mathbb{R}}^n)$ with $1<p<\infty$ and ${\dot{B}}_{\infty ,1}^{-1}({\mathbb{R}}^n)$, respectively.

24 pages

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• Optimal criteria for blowup of radial and $N$-symmetric solutions of chemotaxis systems

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