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: $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}u+\Lambda ^{\alpha }u+\nabla \cdot (u\nabla \psi )=0 &{}\quad \text{ in \mathbb {R}^n\times (0,\infty ), -\Delta \psi =u &{}\quad \text{ in \mathbb {R}^n\times (0,\infty ), u(x,0)=u_0(x) &{}\quad \text{ in \mathbb {R}^n. \end{array}\right. } \end{aligned}$$ In the case $$1<\alpha \le 2$$ , we prove local well-posedness for any initial data and global well-posedness for small initial data in critical Besov spaces $$\dot{B}^{-\alpha +\frac{n}{p}}_{p,q}(\mathbb {R}^{n})$$ with $$1\le p<\infty $$ , $$1\le q\le \infty $$ , and analyticity of solutions for initial data $$u_{0}\in \dot{B}^{-\alpha +\frac{n}{p}}_{p,q}(\mathbb {R}^{n})$$ 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 $$\dot{B}^{-\alpha }_{\infty ,1}(\mathbb {R}^{n})$$ is also established. In the limit case $$\alpha =1$$ , we prove global well-posedness for small initial data in critical Besov spaces $$\dot{B}^{-1+\frac{n}{p}}_{p,1}(\mathbb {R}^{n})$$ with $$1\le p<\infty $$ and $$\dot{B}^{-1}_{\infty ,1}(\mathbb {R}^{n})$$ and show analyticity of solutions for small initial data in $$\dot{B}^{-1+\frac{n}{p}}_{p,1}(\mathbb {R}^{n})$$ with $$1<p<\infty $$ and $$\dot{B}^{-1}_{\infty ,1}(\mathbb {R}^{n})$$ , respectively.