Well-Posedness of the Keller–Segel System in Fourier–Besov–Morrey Spaces

Abstract

In this note, we investigate the Cauchy problem for Keller–Segel system with fractional diffusion for the initial data $(u_0,v_0)$ in the critical Fourier–Bessov–Morrey spaces ${\mathcal{F}\mathcal{N}}_{q,\mu ,r}^{2-2\alpha +d-\frac{d-\mu }{q}}({\mathbb{R}}^d)\times {\mathcal{F}\mathcal{N}}_{q,\mu ,r}^{2-\alpha +d-\frac{d-\mu }{q}}({\mathbb{R}}^d)$ with $1<\alpha \le 2$. The global well-posedness with a small initial data of the solution to Keller–Segel system of double-parabolic type is established.